An effective field theory calculation of the parity violating...

4 March 1999

Ž .Physics Letters B 449 1999 1–5

An effective field theory calculation™of the parity violating asymmetry in nqp™dqg

David B. Kaplan a,1, Martin J. Savage b,2, Roxanne P. Springer b,3,4, Mark B. Wise c,5

a Institute for Nuclear Theory, UniÕersity of Washington, Seattle, WA 98195, USAb Department of Physics, UniÕersity of Washington, Seattle, WA 98195, USA

c California Institute of Technology, Pasadena, CA 91125, USA

Received 27 August 1998; revised 14 December 1998Editor: J.-P. Blaizot

Abstract

Weak interactions are expected to induce a parity violating pion–nucleon coupling, hŽ1. . This coupling should bep NN™measurable in a proposed experiment to study the parity violating asymmetry A in the process nqp™dqg . Weg

compute the leading dependence of A on the coupling hŽ1. using recently developed effective field theory techniques andg p NN

find an asymmetry of A sq0.17 hŽ1. at leading order. This asymmetry has the opposite sign to that given byg p NN

Desplanques and Missimer. q 1999 Published by Elsevier Science B.V. All rights reserved.

Recently an improved measurement of the parityviolating asymmetry, A , in the angular distributiong

of 2.2 MeV gamma rays from the radiative capture™of polarized cold neutrons nqp™dqg , was pro-

w xposed 1 . With u the angle between the neutronsg

spin and the photon momentum, the asymmetry Ag

is defined by

1 d Gs1qA cosu . 1Ž .g sg

G dcosusg

The current experimental limit on this asymmetryŽ . y8 w xparameter is A sy 1.5"4.8 =10 2 , whileg

1 E-mail: [email protected] E-mail: [email protected] On leave from the Department of Physics, Duke University,

Durham NC 27708.4 E-mail: [email protected] E-mail: [email protected]

the proposed experiment expects to measure A withg

a precision of "5=10y9. Interest in A is moti-g

w xvated by a recent measurement 3 of the cesiumanapole moment that appears to give weak couplingparameters that are inconsistent with other low en-

w xergy parity violating measurements 4,5 . In order toobtain the theoretically cleanest determination of theweak parameters in the nucleon-meson Lagrangedensity it is clear that measurements in few nucleonsystems are desirable, thereby eliminating densityeffects that could arise in nuclei and are difficult tocalculate. Reviews of the the subject can be found in

w xRefs. 6–10 .A is sensitive to the weak parity violating pion–g

nucleon coupling, and its measurement may providea clean determination of this important parameter. Inthis letter we present a calculation of A usingg

recently developed effective field theory techniquesw x11,12 . This method allows us to calculate processes

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00032-5

( )D.B. Kaplan et al.rPhysics Letters B 449 1999 1–52

in the two nucleon sector in a systematic fashion,and if carried out to higher orders, is expected to beable to reach the same level of precision as conven-tional nuclear physics techniques, but without modeldependence.

The effective field theory method was used tocalculate the electromagnetic form factors of the

w xdeuteron in Ref. 12 and the basic tools for perform-ing the calculations in this letter were developedthere. The only new interaction needed is the parityviolation pion–nucleon coupling which appears inthe interaction Lagrange density,

hŽ1. e 3 i jp NN † i jLL sy N t p N . 2Ž .pnc '2

1 ŽN is the doublet of spin nucleon fields, i.e.,2

. iN sp and N sn , t , is1,2,3 are the three Pauli1 2

matrices in isospin space and p j, js1,2,3 are theŽ .three real pion fields. The D Is1 weak pion–

nucleon coupling hŽ1. is simply related to thep NN

notation of Desplanques, Donoghue and HolsteinŽ . w x Ž1. 6DDH 7 , h s f , but is of opposite sign top NN p

w xthat used in Ref. 13 . The strong pion–nucleoninteraction Lagrange density we have used is

g A †LL s N sP=p N , 3Ž .pc fpi i 'where pst p r 2 , g sq1.25 is the axial cou-A

pling constant and f s132 MeV is the pion decayp

constant.Predictions based on effective field theory are

made in a chiral and momentum expansion. For coldneutron capture nqp™dqg with the neutron andproton essentially at rest the relevant momentum Qis determined by the binding energy of the deuteron,Bs2.224 MeV and is M B s45.70 MeV, where( N

M is the nucleon mass. The power counting treatsNŽQrL and m rL where L is the scaleQC D p QC D QC D

characteristic of short range nucleon–nucleon inter-.actions as small and takes Q;m . This is similarp

6 w x Ž . w xThere are typographical errors in Refs. 6,10 . In Eq. 3 of 6'Ž . w xand Eq. 7 of 10 the replacement f r2™ f r 2 should bep p

made. In this letter the symbol f is reserved for the pion decayp

w xconstant. The signs of coupling constants used in Ref. 23 arew xconsistent with those used in Ref. 7 only if the strong coupling is

negative, in which case redefining all meson fields M™y Mw xwill give rise to the signs used in Ref. 7 .

in spirit to applications of chiral perturbation theoryin the single nucleon sector and for pion self interac-tions. However, the power counting is unlike thatused in conventional chiral perturbation theory be-cause of the large scattering lengths that occur in the1S and 3S NN scattering channels. These large0 1

scattering lengths render the leading order interac-tions nonperturbative and cause the two-body opera-tors to develop large anomalous dimensions. The

w xexpansion is described in detail in Refs. 11,12,14 .At lowest order in the Q power counting LL , Eq.pncŽ .2 , is the only parity violating interaction that occursw x13,14 . Other terms, such as the parity violatingtwo-body operators are not relevant until higher or-

w xder in the Q expansion 14 .At leading order in the Q expansion we find the

matrix element for cold neutron capture nqp™dqg can be written as

) )TMMse X N t s sPq e d Pe gŽ . Ž .2 2

) )

ysPe g qPe d NŽ . Ž .i) k )i jk j T 2 3 2q ie Ye e d q e g N t t s NŽ . Ž . Ž .i) k )i jk T 2 2 jq ie We e d e g N t s s N .Ž . Ž . Ž .

4Ž .Ž . < <In Eq. 4 , es e is the magnitude of the electron

Ž .charge, N is the doublet of nucleon spinors, e g isŽ .the polarization vector for the photon, e d is the

polarization vector for the deuteron and qis theoutgoing photon momentum. The terms X and Y areparity conserving while the term W is parity violat-ing. Note that for the parity conserving term Y theneutron and proton are in a 1S state while for the0

parity conserving term X and the parity violatingterm W they are in a 3S state. Interference between1

the parity conserving and parity violating amplitudesis possible if the neutron is polarized. At leadingorder, X and Y are calculated from the sum ofFeynman diagrams in Fig. 1 and from wavefunctionrenormalization associated with the deuteron interpo-

w xlating field 12 , giving

p 2k1Xs0 , Ysy 1yg a , 5Ž . Ž .03( Mg N

1 Ž .where k s k yk s2.35294 is the isovector1 p n2

nucleon magnetic moment in units of nuclear magne-tons, a sy23.714"0.013 fm is the NN 1S scat-0 0

( )D.B. Kaplan et al.rPhysics Letters B 449 1999 1–5 3

Fig. 1. Graphs contributing to the parity conserving amplitude fornq p™ dqg at leading order in the effective field theory expan-sion. The solid lines denote nucleons and the wavy lines denotephotons. The light solid circles correspond to the nucleon mag-netic moment coupling to the electromagnetic field. The crossedcircle represents an insertion of the deuteron interpolating fieldwhich is taken to have sup3S quantum numbers.1

tering length, and gs M B . This expression for Y( N

yields the nqp™dqg capture cross section25 2 28pag k a 11 0

ss 1y , 6Ž .5 ž /g aÕM 0N

where a is the fine structure constant and Õ is theŽmagnitude of the neutron velocity in the proton rest

.frame . This leading order result agrees with thew xresults of Bethe and Longmire 15,16 when terms in

their expression involving the effective range areneglected, and it is about 10% less than the experi-mental value for the cross section. The agreementbetween the the expanded Bethe-Longmire and lead-ing order effective field theory results is to be ex-pected, as the only experimental input to either is thevalue of g . In the power counting appropriate to theeffective field theory approach the effective rangeenters at next order in the Q counting. However,other effects also occur at this order. For example, atwo body operator involving the magnetic field.Including just the effective range, as in the full Be-the-Longmire result, does not represent a system-atic improvement of the theoretical expression in

Ž . 7Eq. 6 .

7 The strong interaction amplitude has been computed beyondw xleading order in effective field theory by Park et al. in Ref. 17 ,

w xand more recently by Savage et al. in Ref. 18 . Such calculationsaccount for the 10% discrepancy found at lowest order.

In terms of the amplitudes X, Y and W the parityviolating asymmetry is,

)

2 M Re YqX WŽ .NA sy , 7Ž .g 2 2 2g < < < <2 X q Y

where g 2rM is the photon energy. At leading orderN

in the Q expansion W follows from the sum ofdiagrams in Fig. 2. We find that

'pg mpŽ1.Wsg hA p NN 23p f m qgŽ .p p

2 2m 2g mp py ln q1 q , 8Ž .3 2ž /m2g g m qgŽ .p p

Ž . Ž1.where g is defined in Eq. 3 , h is defined inA p NNŽ .Eq. 2 , and m ,140 MeV is the pion mass. Thep

first term in the square brackets comes from the oneloop diagrams while the second and third termscome from the two loop diagrams with the neutronand proton rescattering through a contact term.Therefore, at leading order in the effective field

Fig. 2. Graphs contributing to the parity violating amplitude fornq p™ dqg at leading order in the effective field theory expan-sion. The solid lines denote nucleons, the dashed lines denotepions and the wavy lines denote photons. The solid squares denote

Ž Ž . .an insertion of LL Eq. 2 while the solid circles correspondp nc

to the minimal electromagnetic coupling. The crossed circle repre-sents an insertion of the deuteron interpolating field which istaken to have sup3S quantum numbers.1

( )D.B. Kaplan et al.rPhysics Letters B 449 1999 1–54

theory Q expansion the numerical value of the asym-metry A is,g

A sq0.17 hŽ1. , 9Ž .g p NN

where we estimate the theoretical errors of our ap-proximations to be at the 30% level. A naive dimen-

w xsional analysis estimate of the weak coupling 13< Ž1. < y7yields h ;5=10 , arising largely from thep NN

w xstrange quark operators 19 , consistent with the bestŽ Ž .guess of DDH a recent calculation in the SU 3

Ž1. y7 w x.Skyrme model yields h ;q1.3=10 20 .p NN< < y7Hence an asymmetry A ;0.8=10 could rea-g

sonably be expected. This is consistent with thepresent experimental bound and would be easily

w xaccessible to the experiment proposed in Ref. 1 .Ž .The calculated asymmetry in Eq. 9 is somewhat

larger in magnitude than previous calculations thatŽ1. w xhave found A ;y0.11 h 21–25 . It is also ofg p NN

the opposite sign to the currently accepted theoreticalw xprediction 24 , and we suggest that there is an error

in that calculation. Note that, as stated in a footnotew xin Ref. 24 , the result of Desplanques and Missimer

also disagrees with the sign computed previously inw xRefs. 21,25 . This sign disagreement is of serious

concern and warrants an independent recalculation ofŽ .Eq. 9 .

The Feynman diagrams that contribute to W con-tain a contribution from exchange currents where thephoton couples to the exchanged pion. This contribu-tion by itself is ultraviolet divergent, yet the sum ofdiagrams is finite. This means that the value of theexchange current contribution alone is dependent on

Ž .the ultraviolet regulator albeit, only logarithmicallyand subtraction the scheme adopted. In potentialmodels, the short distance behavior of the potentialregulates the ultraviolet behavior. Many differentmodels for the short distance behavior of the poten-tial give the same low energy physics and the size ofthe exchange current contribution to W depends onhow the short distance physics is modeled. For theparity conserving amplitude Y the exchange contri-bution does not occur until next-to-leading order inthe Q expansion. A similar situation occurs there andagain the value of the exchange current contributionalone is not a meaningful quantity since in theeffective field theory approach that contribution isultraviolet divergent and its value depends on theregulator and subtraction scheme used.

In the Q expansion the leading contribution to WŽ .' 'is ;1r Q . At next-to-leading order i.e., ; Q

the parity violating S-wave to P-wave two bodyoperators contribute. Since their coefficients are notknown it is not possible at this time to improve thecalculation of W by going beyond leading order inthe Q expansion. However, the same coefficientsappear in other parity violating observables so that asystematic analysis of higher order effects may bepossible.

To conclude, we have computed the parity violat-ing symmetry A in the radiative capture of polar-g

™ized cold neutrons nqp™dqg at leading order ineffective field theory. The number we find is A sg

q0.17 hŽ1. . The estimated theoretical errors arep NN

Q30%, based on the comparison with data of previ-w xous calculations of scattering phase shifts 11 , elec-w xtromagnetic form factors of the deuteron 12 , and

w xdeuteron-Compton scattering 27 . Unless the cou-Ž1. Žpling h is anomalously small as suggested byp NN

18 w x.the circular polarization experiments in F 26 , theasymmetry A could provide a relatively clean deter-g

mination of hŽ1. , due to the absence of many-bodyp NN

effects in the deuteron. If hŽ1. is much smaller thanp NN

naive estimates suggest then there will be additionalcontributions from parity violating two body opera-tors that would need to be included.

Acknowledgements

We would like to thank Wick Haxton, BarryHolstein and Bruce McKellar for useful discussions.This work is supported in part by the U.S. Dept. ofEnergy under Grants No. DOE-ER-40561, DE-FG03-97ER4014 and DE-FG02-96ER40945.

References

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S.C. Bennet, D. Cho, B.P. Masterson, J.L. Roberts, C.E.Ž .Tanner, C.E. Wieman, Science 275 1997 1759.

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w x12 D.B. Kaplan, M.J. Savage, M.B. Wise, nucl-thr9804032,submitted to Phys. Rev. C.

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Ž .1991 403.w x20 U.G. Meissner, H. Weigel, nucl-thr9807038.w x Ž .21 G.S. Danilov, Phys. Lett. B 35 1971 579; Sov. Phys. JETP

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nucl-thr9809023.

4 March 1999


A new measurementof the strength of the superallowed Fermi branch

in the beta decay of 10C with GAMMASPHERE 1

B.K. Fujikawa a, S.J. Asztalos a,b, R.M. Clark a, M.-A. Deleplanque-Stephens a,P. Fallon a, S.J. Freedman a,b,c, J.P. Greene c, I.-Y. Lee a, L.J. Lising a,b,

A.O. Macchiavelli a, R.W. MacLeod a, J.C. Reich a,b, M.A. Rowe a,b, S.-Q. Shang a,F.S. Stephens a, E.G. Wasserman a

a Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAb UniÕersity of California, Berkeley, CA 94720, USA

c Argonne National Laboratory, Argonne, IL 60439, USA

Received 3 August 1998; revised 21 December 1998Editor: J.P. Schiffer

Abstract

We report a new measurement of the strength of the superallowed 0q™0q transition in the b-decay of 10C. Theexperiment was done at the LBNL 88-inch cyclotron using forty-seven GAMMASPHERE germanium detectors. Thetechnique used in this measurement was similar to that of an earlier experiment, but the systematic corrections weresignificantly different. The measured branching ratio: 1.4665"0.0038=10y2 is used to compute the superallowed Fermif t, which gives the weak vector coupling constant and the u to d element of the Cabibbo–Kobayashi–Maskawa quarkmixing matrix. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Beta decay

The most precise value of the u to d element ofŽ .the Cabibbo-Kobayashi–Maskawa CKM quark

mixing matrix is obtained from measurements ofsuperallowed 0q™0q Fermi b-decays in nuclearsystems. Specifically, these decay rates determinethe nucleon weak vector coupling constant G giv-V

2 2 < < 2Ž .ing V : G sG V 1qD where G is theud V F ud R F

Fermi coupling constant obtained from the muon

1 This work was supported in part by the US DOE undercontract numbers DE-AC03-76SF00098 and W-31-109-ENG-38.

Ž .lifetime and D is a nucleus independent ‘‘inner’’R

radiative correction. The conserved vector currentŽ .CVC hypothesis implies that superallowed f t-val-ues within isospin-1 multiplets are related to G by:V

Kf 1qd 1yd t'FF ts , 1Ž . Ž . Ž .R C 22 < <G MV V

< < 2where M s2 is the vector matrix element, f isV

the familiar Fermi statistical rate function, d is theRŽ .nucleus dependent ‘‘outer’’ radiative correction, dC

< < 2is the charge dependent correction to M s2 dueV

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01603-7

( )B.K. Fujikawa et al.rPhysics Letters B 449 1999 6–11 7

to isospin symmetry breaking, and K is the usualb-decay constant. The corrected FF t include nuclearand radiative effects. Precise determination of GV

requires precision measurements of the partial 0q™

0q half-life, the b endpoint energy, and reliablew xtheoretical calculations of d and d . Ref. 1 sum-R C

marized the status of measurements and calculationsfor 10 C, 14 O, 26Alm, 34 Cl, 38 K m, 42 Sc, 46 V, 50 Mn,and 54 Co. The constancy of FF t for these nineprecisely measured superallowed decays support the

< <CVC hypothesis. This review suggests V sud

0.9740"0.0005. Together with the two other ele-ments in the first row of the CKM matrix taken from

w xRef. 2 , this tests the unitarity of the CKM matrix.< < 2 < < 2 < < 2 < < 2The result, V q V q V ' V s0.9972"ud u s ub

0.0013, is more than two standard deviations fromthe unitarity constraint.

A violation of CKM unitarity would require theStandard Model to be extended. A more mundaneexplanation is unaccounted systematic uncertaintiesin the difficult theoretical calculations needed toextract V . The calculation of the isospin symmetryud

breaking correction d is regarded as the most prob-C

lematic. Fig. 1 shows the most precisely measuredFF t as a function of daughter nucleus charge Z.

Ž .Fig. 1. The FF t solid circles of the nine precisely measuredŽ10 14 26 m 34 38 m 42 46superallowed decays C, O, Al , Cl, K , Sc, V,

50 54 .Mn, and Co plotted as a function of the daughter nucleuscharge Z. The solid line is the weighted average. The dashed lineis the result of a linear fit and the open square is the extrapolationof this fit to zero charge. The closed circle at Zs5 includes thepresent measurement and the open circle is the 10 C FF t using thesuperallowed branching ratio from this measurement alone.

Possible unaccounted Z-dependent corrections moti-vated extrapolations to zero charge using second and

w xthird order polynomials fits to d -corrected 3 orCw x w xuncorrected 4 f t values. The authors of Ref. 3

argue against a Z-dependence, but the statisticalstrength of their conclusion is weak. The agreement

w xof the extrapolated values 4 with unitarity suggeststhat incomplete isospin corrections might explain thediscrepancy. The f t for the superallowed Fermi b

decay of 10 C is of particular interest: 10 C has thelowest nuclear charge of a superallowed Fermi de-cay. Moreover, all existing calculations agree that dC

for 10 C is small. The potential of 10 C motivated thepresent work in which we measured the 0q to 0q

branching ratio. Our final result has slightly largererror and differs by about a standard deviation fromthe best previous measurement. The systematic cor-rections in the present work are well under controland differ in significant ways from previous experi-ments.

The necessary experimental inputs are the total10 Ž q .half-life, the branching fraction for C 0 ,g.s. ™

10 Ž q . qB 0 ,1.74 MeV qe qn , and the superallowedŽend-point energy. The half-life 19.290"0.012 s

w x. Ž5 and the recently revised endpoint energy 885.86w x."0.12 keV 6,7 are known to high precision; the

limiting experimental input is the 0q™0q branchingratio. Fig. 2 shows the 10 B and 10 C levels importantfor this measurement. The b decay of 10 C goes to

10 Ž q . 10 Ž q .the B 0 ,1.740 MeV or the B 1 ,0.718 MeV10 Ž q .state. The allowed decay to the B 1 ,2.154 MeV

Žlevel is known to be small experimentally -8=y6 .10 as expected from the meager available energy.

The forbidden b decay to the 10 B ground state issuppressed by about 10y10. The decay to the10 Ž q .B 0 ,1.740 MeV state is followed with g-rays at1022 keV and 718 keV. The direct ground state

10 Ž q .decay of the B 0 ,1.740 MeV level is magneticy12 w xoctupole, with an estimated branch below 10 8 .

10 Ž q .The decay to the B 1 ,0.718 MeV state is fol-lowed by a single 718 keV g-ray. Therefore the0q™0q branching ratio is the same as the g-rayintensity ratio:

I 1022 keV Y 1022 keV e 718 keVŽ . Ž . Ž .gbs s

I 718 keV Y 718 keV e 1022 keVŽ . Ž . Ž .g

2Ž .

( )B.K. Fujikawa et al.rPhysics Letters B 449 1999 6–118

Fig. 2. The relevant energy levels of 10 B and 10 C.

Ž . 10where Y g is the g-ray yields from C b-decayŽ .and e g full energy g-ray detection efficiencies.

This experiment was performed with the GAM-w xMASPHERE 9 detector at the Lawrence Berkeley

National Laboratory 88-Inch Cyclotron. Three mea-surements are required: a measurement of the g-rayyield ratio following b-decay, the full energy g-raydetection efficiency ratio, and the 2=511 keV pileupbackground to the 1022 keV g-ray peak. For theb-decay measurement, the 10C source is produced

10 Ž .10 2with the B p,n C reaction using a 325 mgrcmthick target of 99.5% enriched 10 B on a 600 mgrcm2

thick carbon backing and a 250 nA 8 MeV protonbeam. The b delayed g-rays from 10 C decay weredetected by forty-seven GAMMASPHERE germa-nium detectors. The usual BGO Compton suppres-sors were turned off in order to avoid possiblesystematic effects from ‘‘false vetoes’’ by an unre-lated g-ray. A 35 second beam-onrbeam-off cyclewith a 1 second delay was used. We use a technique

w xemployed by a previous experiment 3 for measur-ing the g-ray efficiency ratio. The efficiency is mea-sured in situ with the g-rays of interest by tagging g

10 Ž qcascades prepared by exciting the B 1 ,2.154.MeV state. A reduced intensity 10 nA proton beam

10 Ž X.10 )is used to populate this state with B p,p B . The10 Ž q . 10 Ž q .B 1 ,2.154 MeV ™ B 0 ,1.740 MeV transition

10 Ž qis tagged with the 414 keV g-ray. The B 0 ,1.740

. 10MeV state then decays to the B ground state byemitting one 1022 keV g-ray and one 718 keV g-ray.The distribution of these g-rays are isotropic becausethe cascade begins with the 0q state.

Fig. 3a shows the b-delayed g-ray energy spec-trum. We use the following procedure to determinethe g-ray yields. The region around the g-ray peak isfit to a function imitating the peak and a smoothunderlying background. The peak is modeled by aGaussian having small exponential tails and thebackground is taken as a quadratic polynomial with aresolution smoothed step function. The step functionaccounts for the discontinuity in the backgroundcaused by scattering of g-rays in inactive material infront of the detector. Peaks for background radiationare included. The fit is performed by minimizing ax-square statistic for Poisson distributed histograms

Ž . Ž .Fig. 3. a The b-delayed g-ray energy spectrum. b The promptŽ .g-ray energy spectrum. c The prompt g-ray energy spectrum

gated by the 414 keV g-ray. The peaks at 718 keV and 1022 keVin the delayed spectrum are from the b-decay of 10 C. Theremaining peaks are due to position annihilation, room back-ground, neutron activation, and background proton reactions, pri-

10 Ž .7marily: B p,a Be. The peaks at 718 keV and 1022 keV in thegated spectrum are used to measure the relative efficiency. The

Ž .residual 414 keV g-ray peak in c disappears after corrections aremade for accidentals and Compton background.


w x w x10 with MINUIT 11 . The fitting procedure is usedonly for determining the background; the yields arecomputed by subtracting the fitted background fromthe data.

Fig. 3b shows the g-ray spectrum from the10 Ž X.10 )B p,p B reaction. The gating process is asfollows. A fit to the 414 keV peak is performedusing the method described. The result is used todetermine the energy window, defined to be "1s

centered about the peak. The 414 keV peak is on asmooth background, which includes Compton scat-tering of higher energy g-rays. The effect of thisCompton background is estimated by taking elevenadditional energy gates below and twelve above the414 keV peak. The Compton background is deter-mined for each gate and a quadratic polynomialinterpolation is used to estimate the Compton back-ground under the 414 keV g-ray peak. The back-ground under the 414 keV peak also includes a smalldouble escape peak from the 1436 keV g-ray which

10 Ž q . 10 Ž qis emitted in the B 1 ,2.154 MeV ™ B 1 ,0.718.MeV transition. Since the 1436 keV g-ray is always

emitted with a 718 keV g-ray and never with a 1022keV g-ray, a small correction is applied to the effi-ciency ratio. This correction is determined from thenumber of counts in the single escape peak and the

w xratio of double to single escapes from an EGS4 12Monte Carlo simulation. The accidental g yg co-414

incidences were corrected for by subtracting countsobtained in non-coincident time gates. The accidentalgates were normalized to the coincidence gate bytaking advantage of the fact that it is impossible fortwo 718 keV g-rays to be in true coincidence. Thenormalization factor is chosen such that the g y718

g coincidences disappears in the subtracted spec-718

trum.Since the 1022 keV g-ray is emitted during the

slow down of the recoiling 10 B, a small correction ismade to account for the kinematical change in solidangle and the Doppler energy shift. The overallcorrection is reduced because of the symmetry ofGAMMASPHERE. The size of the correction wascalculated with Monte Carlo integration using the

10 Ž X.10 )differential B p,p B cross sections from Ref.w x13 , the lifetimes and cascade branching ratios from

w x w xRef. 14 , and the stopping powers from Ref. 15 .The number of background 2=511 keV pileup

counts in the 1022 keV g-ray peak is measured using

19 Ne as a source of positrons. The 19 Ne source is19 Ž .19prepared in situ with the F p,n Ne reaction by

bombarding a 325 mgrcm2 thick PbF target on a600 mgrcm2 thick carbon foil backing with a 100nA 8 MeV proton beam. Like the 10 C decay mea-surement, a 35 second bombardment and countingcycle was used. The 19 Ne decay is similar to 10 Cwith a 17.239"0.014 s half-life and a similar b

Ž . 19endpoint energy 1705.38"0.80 keV . The Ne isa source of 511 keV annihilation g-rays with no true1022 keV g-ray, and the entire peak at 1022 keV isdue to pileup. In order to normalize the 10 Ne data tothe 10 C data, we use the following technique. TheGAMMASPHERE data stream contains a 1 MHzclock and the absolute time of each trigger is knownto 1 ms. Using this information, we determine the

Ž .number N 2=511 of 511 keV g-rays within a 1 mstime bin that follow a 511 keV g-ray with an arbi-trary delay for each detector. Like the pileup of twoannihilation g-rays this is a purely random process.Neglecting, for the moment, small dead time correc-

Ž . Ž .tions which are later corrected for , N 2=511 sŽ .Ž .R PT R Pt where R is the rate of 511511 511 bin 511

keV g-rays in a single detector, T is the countingtime, and t s1 ms is the bin width. Similarly, thebin

number of 2=511 keV pileup counts in the energyŽ . Ž .Žspectrum is given by: Y 2=511 s R PT R P511 511

.t where t is the pileup rejection time in thepu pu

GAMMASPHERE amplifiers. The rate independentratio

Y 2=511 tŽ . pus 3Ž .

N 2=511 tŽ . bin

is used to compute the pileup correction.With the exception of the 2=511 keV pileup,

Ž .random pileup does not affect the ratio in Eq. 2 .However, the summing of g-rays from a singlecascade is a possible systematic effect. Specifically,both the 718 keV and 1022 keV g-rays can depositenergy into the same detector, in effect removingcounts from the full energy peaks. The effect cancels

w xto first order in the efficiency ratio and an EGS4 12Monte Carlo simulation indicates that this effect is

Ž .y0.032 3 %. However, due to the smallness of the10 C 0q™0q branch, this effect is significant in thedecay measurement. Only about 1.5% of the 718keV g-rays are emitted with a 1022 keV g-ray, but all

( )B.K. Fujikawa et al.rPhysics Letters B 449 1999 6–1110

Table 1Summary of the experimental corrections made in the measure-ment of the superallowed branching ratio

Correction Size Affects

Ž .Accidental Coincidences y 1.94"0.02 % EfficiencyŽ .Compton Background y 0.049"0.008 % EfficiencyŽ .Double Escape Peak y 0.020"0.004 % EfficiencyŽ .Kinematic Shift y 0.019"0.051 % EfficiencyŽ .2=511 keV Pileup y 1.25"0.19 % b-decayŽ .Summing y 0.032"0.003 % EfficiencyŽ .q 0.44"0.05 % b-decay

120 Ž .Sb background y 0.23"0.11 % b-decay

1022 keV g-rays come with a 718 keV g-ray. Sys-tematically, more 1022 keV g-rays will be removedfrom the full energy peak. The summing correctionfor a single detector is estimated by measuring coin-cidences between different detectors in the GAM-MASPHERE array. Neglecting, for the moment,threshold corrections, small variations in detectorssizes, and the small gyg angular correlation, thesumming correction is equal to

Y 1022PxŽ .1q

Y 1022Ž .f s 4Ž .SUM 1qY 718Px r718Ž .

Ž . Ž .where Y g is the total g-ray yield and Y gPx isthe g-ray yield when there is a coincident event in asecond detector. The correction for detector size andg y g angular correlation are straight forward.

Ž .The detector size is simply scaled by Y 718 keV .10 Ž q .Since the transition B 0 ,1.740 MeV ™

10 Ž q .B 1 ,0.718 MeV is pure M1 and the transition10 Ž q 10 Ž q .B 1 ,0.718 MeV™ B 3 ,g.s. is primarily E2,

Ž .the gyg angular correlation is equal to: P cosu sŽ . w x1 y 0.0714 P P cosu 16 . The correction for2

threshold is more problematic. GAMMASPHEREŽ .uses constant fraction discrimination CFD whose

thresholds are not easily described. To avoid thisproblem, we enforce a software threshold at 417

w xkeV, well above the CFD threshold. An EGS4 12Monte Carlo simulation of GAMMASPHERE is usedto correct for the fraction of events below 417 keV.

The room background was measured for 17.5hours after the run. No g-rays were found in theregion of 718 keV. However, a background g-ray,with an energy of 1022.6"0.4 keV, was observed.

Based upon constraints on half-life, intensity, andassociated g-rays, the only possible source is the b

decay of 120Sb. This was probably produced through120 Ž .120proton activation, Sn p,n Sb, of the aluminium

alloy foil lining the inside of the GAMMASPHEREscattering chamber. In addition to a 1023 keV g-ray,the b decay of 120Sb emits a 197.3 keV g-ray withnearly equal intensity. We use this 197.3 keV g-rayto scale the background spectrum to the 10 C decaydata in order to subtract the 120Sb contamination.

The summary of all corrections are shown inTable 1. The strength of the 10 C superallowed 0q™

0q branch is determined to bey2bs 1.4665"0.0038 stat "0.0006 syst =10Ž . Ž .5Ž .

where the systematic error dominated by the uncer-tainty in the EGS4 threshold correction in the sum-ming correction. A comparison with previous experi-ments is shown in Table 2. This result is about one

w xstandard deviation from the results of Ref. 3 . Al-though the techniques used in this measurement is

w xsimilar that of Ref. 3 , the are significant differ-ences. This measurement was made without the useof Compton suppressors, which resulted in a slightlylarger correction for Compton background, butavoided all possible systematic effects from ‘‘falsevetoes’’. The systematic effects from ‘‘false vetoes’’and summing was the largest correction applied to

w xthe measurement of Ref. 3 . In addition, the correc-tion for 2=511 keV pileup background in the pre-sent experiment was measured in situ using 19 Ne asa source of positrons. The previous experiment madean estimation of the size of the 2=511 keV pileupcorrection from the 511 keV rate and the averagepileup rejection time. Our result for b along withprevious measurements of the b endpoint energy

Table 2Comparison of 10 C superallowed 0q™0q branching ratios

Branching ratio Ref.y2Ž . w x1.465"0.014 =10 17y2Ž . w x1.473"0.007 =10 18y2Ž . w x1.465"0.009 =10 8

y2Ž . w x1.4625"0.0025 =10 3y2Ž .1.4665"0.0038 =10 This worky2Ž .1.4645"0.0019 =10 World average


and the total lifetime gives an FF t-value 3068.9"8.510 < <s for C, which by itself would yield V s0.9745ud

"0.0014, using the usual radiative corrections andŽ .the isospin breaking corrections d s0.16 3 % fromC

w xRef. 1 . Thus, under these conditions, the unitaritytest would be satisfied for the mass-10 data alone,< < 2V s0.9983"0.0029, but the error is large. Thepresent experiment seems to favor a Z dependence

w xcorrection of Ref. 4 but the statistics are not suffi-cient for a definite conclusion.

References

w x1 E. Hagberg et al., in: T. Minamisono, Y. Nojiri, T. Sato, K.Ž .Matsuta Eds. , Non-Nucleonic Degrees of Freedom De-

tected in Nucleus, World Scientific, 1996.w x Ž .2 Particle Data Group, Phys. Rev. D 54 1996 1.

w x Ž .3 G. Savard et al., Phys. Rev. Lett. 74 1995 1521.w x Ž .4 D.H. Wilkinson, Nucl. Inst. Meth. A 335 1993 201.w x Ž .5 P.H. Barker, G.D. Leonard, Phys. Rev. C 41 1990 246.w x6 S.C. Baker, M.J. Brown, P.H. Barker, Phys. Rev. C 40

Ž .1989 940.w x7 P.H. Barker, private communication.w x Ž .8 M.A. Kroupa et al., Nucl. Inst. Meth. A 310 1991 649.w x Ž .9 I.-Y. Lee, Nucl. Phys. A 520 1990 641c.

w x Ž .10 S. Baker, R.D. Cousins, Nucl. Inst. Meth. 221 1984 437.w x Ž .11 F. James, M. Roos, Comp. Phys. Comm. 10 1975 343.w x12 W.R. Nelson, H. Hirayama, D.W.O. Rogers, SLAC Report-

265, 1985, unpublished.w x Ž .13 B.A. Watson et al., Phys. Rev. 187 1969 1351.w x Ž .14 F. Ajzenberg-Selove, Nucl. Phys. A 490 1988 1.w x15 J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and

Range of Ions in Solids, Pergamon Press, 1985.w x Ž .16 H. Frauenfelder, R.M. Steffen, in: K. Siegbahn Ed. , a-, b ,

and g-Ray Spectroscopy, North-Holland, 1968.w x17 D.C. Robinson, J.M. Freeman, T.T. Thwaites, Nucl. Phys. A

Ž .181 1972 645.w x Ž .18 Y. Nagai et al., Phys. Rev. C 43 1991 R9.

4 March 1999


Self-gravitating cosmic rings

Gerard Clement 1´ ´Laboratoire de GraÕitation et Cosmologie RelatiÕistes, UniÕersite Pierre et Marie Curie, CNRSrUPRESA 7065,´

Tour 22-12, Boıte 142, 4 place Jussieu, 75252 Paris cedex 05, Franceˆ

Received 1 December 1998Editor: L. Alvarez-Gaume

Abstract

The classical Einstein-Maxwell field equations admit static horizonless wormhole solutions with only a circular cosmicstring singularity. We show how to extend these static solutions to exact rotating asymptotically flat solutions. For a suitablerange of parameter values, these solutions describe charged or neutral rotating closed cosmic strings, with a perimeter of theorder of their Schwarzschild radius. q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 04.20 Jb; 04.40 Nr; 98.80 Cq

The Einstein-Maxwell field equations couplinggravity to electromagnetism admit outside sources avariety of stationary axially symmetric solutions,among which the Kerr-Newman family of solutionsw x1 depending on three parameters M, Q and Jwhich, from the consideration of the asymptoticbehaviour of these field configurations, may be iden-tified as their total mass, charge, and angular mo-mentum. A fourth physical characteristic, the totalmagnetic moment m, is related to these three param-eters by the same ‘‘anomalous’’ relation

Mmg'2 s2 1Ž .

JQ

as in the case of an elementary particle such as theelectron. It would be tempting to interpret theseclassical configurations as the fields generated by an

1 E-mail: [email protected]

isolated elementary particle, were it not for the nu-merical values of these parameters. The Kerr-New-man metrics correspond to regular black-hole space-times if

M 2 GQ2 qa2 , 2Ž .where 2 a'JrM. However in the case of elemen-

< < 2 Ž Ž .1r2tary particles J ;m where m s "crG isP P. < <the Planck mass and Q ;m , so thatP

< < < < < < 22a r Q ; Q rM;m rm ;10P e

Ž .where m is the electron mass , and thereforee

M 2 -Q2 qa2 . 3Ž .So the field configurations generated by elementaryparticles are not of the black hole type, but exhibitnaked singularities, contrary to the cosmic censor-

2 We use gravitational units Gs1, cs1.


( )G. ClementrPhysics Letters B 449 1999 12–16´ 13

w xship paradigm 2 . For this reason, it is generallybelieved that there is no viable classical model for

Želementary particles except possibly for neutral.spinless particles in the framework of Einstein’s

general relativity.In the charged spinless case Js0, the relation

QrM;m rm tells us that electromagnetism is pre-P e

ponderant, and that the naked point singularity in thespherically symmetric metric originates from that ofthe Coulomb central field. This point singularity isultimately responsible for the divergences whichplague both classical and quantum electrodynamics.A way to regularize these divergences is to replacethe zero-dimensional point particles of traditionalfield theory by the one-dimensional fundamental ob-jects of string theory.

String-like objects also occur as classical solu-tions to field theories with spontaneously brokenglobal or local symmetries. Such symmetry breakingtransitions are believed to have occurred during theexpansion of the universe, leading to the formation

w xof large, approximately straight cosmic strings 3 .The long-range behavior of the metric generated by a

w xstraight cosmic string 4 is given by an exact sta-tionary solution of the vacuum Einstein equationswith a line source carrying equal mass per unitlength and tension. In the case of closed cosmicstrings, or rings, this tension will cause the string tocontract, precluding the existence of stationary solu-tions, unless the tension is balanced by other forces.

w xA possibility is that of vortons 6 , rotating loops ofw xsuperconducting current-carrying string 5 stabilized

by the centrifugal force. Another possibility has beenw xadvocated by Bronnikov and co-workers 7 , that of

ring wormhole solutions to multi-dimensional fieldmodels. In the case of these static solutions, thegauge field energy-momentum curves space nega-tively to produce a wormhole, at the neck of whichsits a closed cosmic string, which cannot contractbecause its circumference is already minimized.

In this Letter, we first rederive such static ringwormhole solutions to the Einstein-Maxwell fieldequations. Then, using a recently proposed spin-gen-

w xerating method 8 , we construct from these staticsolutions new rotating Einstein-Maxwell ring solu-tions with only a cosmic ring singularity. Thesesolutions depend on four parameters, the values ofwhich can be chosen such that the elementary parti-

Ž . Ž .cle constraints 1 slightly generalized to gf2 andŽ .3 are satisfied. However, it then turns out that for

< < 2the ‘‘elementary’’ orders of magnitude J ;Q ;

m2 , the mass of these objects cannot be small, but isP

also of the order of the Planck mass. We show that,Ž < < 2 .in the case of large quantum numbers J 4m , aP

subclass of these solutions describes macroscopiccharged or neutral rotating cosmic rings, also satisfy-

Ž .ing the elementary particle constraint 3 , but withg/2.

Under the assumption of a timelike Killing vectorfield E , the spacetime metric and electromagnetict

field may be parametrized by

22 i y1 i jds s f dtyv dx y f h dx dx ,Ž .i i j

F sE Õ , F i j s f hy1r2e i jk E u , 4Ž .i0 i k

where the various fields depend only on the threespace coordinates x i. The stationary Einstein-Maxwell equations are equivalent to the three-di-

w xmensional Ernst equations 9

2f = EEs=EEP =EEq2c =c ,Ž .2f = cs=cP =EEq2c =c , 5Ž .Ž .

12f R h sRe EE , q2c EE , c ,Ž .i j i EE , j Ž i j.2 Ž .

y2 EEc , c , ,Ž i j.

where the scalar products and Laplacian are com-puted with the metric h , the complex Ernst poten-i j

tials EE and c are defined by

EEs fyccq ix , csÕq iu , 6Ž .

and x is the twist potential

E xsyf 2 hy1r2 h e jk lE v q2 uE ÕyÕE u .Ž .i i j k l i i

7Ž .

Ž .These equations are invariant under an SU 2,1 groupw xof transformations 10 . The class of electrostatic

Ž .solutions EE and c real depending on a single realpotential can be reduced, by a group transformation,

Ž .to EEsEE constant, which solves the first Eq. 5 .0Ž .The form of the solution of the second Eq. 5

( )G. ClementrPhysics Letters B 449 1999 12–16´14

depends on the sign of EE . Representative solutions0

are

EE sy1 , c scoth s , f s1rsinh2sŽ .0 0 0

EE s0 , c s1rs , f s1rs 2 8Ž .0 0 0

EE sq1 , c scot s , f s1rsin2sŽ .0 0 0

Ž . Ž 2 .where the potential s x is harmonic = ss0 ;other electrostatic solutions depending on a single

Ž .potential may be obtained from these by SU 2,1transformations. We note that the electric and gravi-

Ž .tational potentials 8 are singular for ss0 if EE s0Ž .y1 or 0, and for ssnp n integer if EE s q1.0

As we wish to obtain axisymmetric ring-like solu-Ž .tions, we choose oblate spheroidal coordinates x, y ,

Ž .related to the familiar Weyl coordinates r, z by

1r2 1r22 2rsn 1qx 1yy ,Ž . Ž .zsn xy . 9Ž .In these coordinates, the three-dimensional metricds 2 'h dx idx j

i j

2 2dx dy2 2 2 k 2 2ds sn e x qy qŽ . 2 2ž /1qx 1yy

2 2 2q 1qx 1yy dw 10Ž . Ž .Ž .

Ž .depends on the single function k x, y . Now, follow-w xing Bronnikov et al. 7 , we assume the harmonic

potential s to depend only on the variable x, whichyields

EE a 22 01qx

2 ksss qaarctan x , e s , 11Ž .0 2 2ž /x qy

where s and a are integration constants. We note0

that the reflexion xlyx is an isometry for theŽ .three-dimensional metric 10 , which has two points

at infinity xs"`. The full four-dimensional metricŽ . Ž4 is quasi-regular i.e. regular except on the ring

.xsys0, see below for xgR if

< < < <s ) a pr2 for EE sy1 or 00 0

< < < < < <nq a r2 p-s - nq1y a r2 p a -1Ž .Ž . Ž .0

for EE sq1 12Ž .0

for some integer n. If these conditions are fulfilled,this metric describes a wormhole spacetime with twoasymptotically flat regions connected through the

Ž .disk xs0 zs0,r-n . There is no horizon. TheŽpoint singularity of the spherically symmetric Reis-

.sner-Nordstrom solution is here spread over the ring¨Ž .xsys0 zs0,rsn , near which the behavior of

the spatial metric

21yEE a2 2 2 2 2 2 20ds ,n x qy dx qdy qdwŽ . Ž .13Ž .

Ž 2is that of a cosmic string with deficit angle p EE a0. Žy1 , which is negative in all cases of interest it can

< <be positive only for EE sq1, a )1, correspond-0.ing to a singular solution . This ring singularity

disappears in the limit of a vanishing deficit angleŽ < < .EE sq1, a ™1 , where the solution reduces to a0

Reissner-Nordstrom solution with naked point singu-¨larity. The asymptotic behaviours of the gravitationaland electric potentials at the two points at infinity arethose of particles with masses and charges

c "`Ž .0M s.an , Q s"an ; 14Ž ." "

f "`( Ž .0

Ž . 2 2the three cases 8 lead respectively to Q -M" "

for EE sy1, Q2 sM 2 for EE s0, and Q2 )M 20 " " 0 " "

for EE sq1. The vanishing of the sum of the0

outgoing electric fluxes at xs"` shows that thering xsys0 is uncharged.

A case of special interest is EE sq1, s spr2,0 0

corresponding to a symmetrical wormhole metricw x Ž .13 . The mass of this particle M san sin apr2"

does not depend on the point at infinity considered,and is positive, even though the deficit angle isnegative. For the physical characteristics of this par-ticle to be those of a spinless electron, we should

< < 2take a ;m rm , and n;m rm , of the order ofe P P e

the classical electron radius.Now we proceed to generate asymptotically flat

rotating solutions from these static axisymmetric so-lutions, using the simple procedure recently proposed

( )G. ClementrPhysics Letters B 449 1999 12–16´ 15

w x Žin Ref. 8 . This procedure S generalized in Ref.w x.11 involves three successive transformations:

Ž . Ži The electrostatic solution real potentials EE,2 k .c , e is transformed to another electrostatic solu-

ˆ2 kˆ ˆŽ . Ž .tion EE, c , e by the SU 2,1 involution P :

y1qEEq2c 1qEEˆ ˆEEs , cs ,

1yEEq2c 1yEEq2c

ˆ2 k 2 ke se . 15Ž .In the case of asymptotically flat fields with largedistance monopole behavior, if the gauge is chosen

Ž . Ž Ž ..2so that f ` s 1qc ` , then the asymptotic be-haviors of the resulting electric and gravitationalpotentials are those of the Bertotti-Robinson solution

ˆ ˆ 2w x12 , cAr, fAr .ˆ2 kˆ ˆŽ . Ž .ii The static solution EE, c , e is transformed

to a uniformly rotating frame by the global coordi-nate transformation dw s dw

X q V dtX , dt s dtX,ˆX ˆ Xleading to gauge-transformed complex fields EE , c ,

ˆX2 ke . While such a transformation on an asymptoti-cally Minkowskian metric leads to a non-asymptoti-cally Minkowskian metric, it does not modify theleading asymptotic behavior of the Bertotti-Robinsonmetric, so that the transformed fields are againasymptotically Bertotti-Robinson.

ˆXX X 2 kˆ ˆŽ . Ž .iii The solution EE , c , e is transformedŽ X X 2 kX.back by the involution P to a solution EE , c , e

which, by construction, is asymptotically flat, butnow has asymptotically dipole magnetic and gravi-

w xmagnetic fields. As shown in Ref. 8 , the combinedtransformation S transforms the Reissner-Nordstrom¨family of solutions into the Kerr-Newman family.

The static ring wormhole solutions of the preced-ing section have two distinct asymptotically flat re-gions x™"`. To apply the general spin-generatingprocedure S to such wormhole spacetimes, we musttherefore select a particular region at infinity. e. g.x™q`, and gauge transform the static solutionŽ Ž ..EE ,c x to0 0

EE x sc2 EE y2cdc x yd2 ,Ž . Ž .0 0

c x scc x qd 16Ž . Ž . Ž .0

2 Ž .with the parameters c s 1rf q` , d s0Ž . Ž . Ž .ycc q` such that c q` s0, f q` s1. A0

perturbative approach to the generation of slowlyw xrotating ring wormhole solutions 13 shows that the

asymmetry between the two points at infinity thusintroduced is a necessary feature of spinning ringwormholes.

To recover the spacetime metric from the spin-Ž X X 2 kX . Ž .ning potentials EE , c , e , we compute from 7

X Ž .the partial derivative E v x, y , which is a rationaly w

function of y, and integrate it with the boundaryX Ž . Žcondition v x,"1 s0 regularity on the axis rsw

. Ž .0 . The resulting spacetime metric is of the form 4 ,Ž .10 with the metric functions

X y2 2 2 ˆ 2< <f s D 1yV r rf f ,Ž .Xy1 2 kX

< < 2 y1 2 kf e s D f e ,

vX sVn 2 1yy2Ž .w

=

2 2 2 2ˆ< <D 1qx f jŽ .0y 22 2 2 2 2ˆ 1qxb c f yV rŽ .

bqh hya , 17Ž .ž /c

ˆ 2 2Ž . Ž . Ž . Ž .where bsdq1, f x s f x rb c x , D x s02 2 2 2 2Ž Ž .. Ž .D x, y x with 1yy s f rV n 1qx , and0 0

D x , y s1qV 2n 2 bc a 2 b2r2 c2 qj 1yy2Ž . Ž .Ž .y iVn bhc y ,

2 ˆ 2 2 2j x s 1qx r2 fya b r2c ,Ž . Ž .h x sxqa 2by1 rcyarcc . 18Ž . Ž . Ž .

Ž .We can show that zeroes of D x, y , correspond-w xing to strong Kerr-like ring singularities 11 of the

stationary solution, are absent if

a bc)0 . 19Ž .This quasi-regularity condition which, in the casesEE sy1 or 0, is equivalent to assuming the static0

mass M to be positive, can always be satisfied byqchoosing the appropriate sign for the constant c inŽ . Ž .16 . The metric 17 is of course still singular on therotating cosmic ring xsys0, with the same deficit

Ž 2 .angle p EE a y1 as in the static case. The gravita-0

tional potential f X vanishes on the stationary limitŽ .surfaces f x s"Vr, where the full metric is regu-

Ž .lar. However the spinning solution 17 is horizon-w xless, just as the corresponding static solution 11 .

This spinning solution is by construction asymptoti-

( )G. ClementrPhysics Letters B 449 1999 12–16´16

cally flat for x™q`, but not for x™y`, wherethe asymptotic metric behaves as

2X 2 y2 y2 2 2ds ,yl r dtq Vr4 r q4 z dwŽ . Ž .Ž .y16Vy2 l 6r 4 dr 2 qdz 2 q l4r 4dw 2 20Ž .Ž .2 2 ˆŽ Ž ..with l sV r4 f y` . The study of geodesic mo-

Ž .tion in the metric 17 shows that all test particlescoming from x™q` are reflected back by aninfinite potential barrier, so that there is no loss ofinformation to x™y`.

From the asymptotic behaviours of the spinningmetric and electromagnetic field, we obtain the cor-responding mass, angular momentum, charge, andmagnetic dipole moment,

Ms narc by1qt , Jsnb Mqd ,Ž . Ž . Ž .Qs narc 1yt , msnb Qyd , 21Ž . Ž . Ž . Ž .with bsVna 2 b2rc2, t sb 2c2r2 a 2 b, dsn cŽ 2 .1yEE a r3a b.0

Can the values of these parameters correspond tothose of elementary particles? Combining the abovevalues we obtain

M 2 yQ2 ya2 sn 2 b 2 yEE a 2 ya2 . 22Ž .Ž .0

Ž .The quasi-regularity condition 19 implies d)0, so2 2 2 Ž .that a )n b , hence the inequality 3 is satisfied

for EE G0. The gyromagnetic ratio0

M QydŽ .gs2 23Ž .

Q MqdŽ .can be very close to 2 for very small values of d .One would then expect that the values of the inde-pendent free parameters n , a , b , and t can be

Ž .adjusted so that the four physical parameters 21take their elementary particle values. However, ow-

Ž .ing to the regularity constraint 12 , which stronglyrestricts the range of allowed parameter values, itturns out that if for instance the charge and angular

Ž .momentum are of order unity in Planck units andg,2, then the mass of these spinning ring ‘‘par-

ticles’’ should be at least of the order of the Planckmass.

Ž < < 2 .In the case of large quantum numbers J 4m ,PŽ .our classical solutions 17 describe macroscopic

closed cosmic strings with negative deficit angle, butpositive total mass. These cosmic strings satisfy the

Ž .elementary particle constraint 3 if EE G0. The0

exotic line source xsys0 is spacelike only if itlies outside the stationary limit surface, which furtherrestricts the parameter values. A specially interestingcase is ts1, corresponding to a neutral spinning

Ž .cosmic string Qs0 . In this case we can show that< < Žthe ring source is spacelike for b -1r2 leading

. < <to drM)4r3 ; for EE s0, the range of b can be0< <narrowed down to 0.29- b -0.40. These neutral

strings have a proper perimeter of the order of or< <smaller than their Schwarzschild radius Ms b n , a

rotation velocity which is close to 1, and a magnetic< < 2moment m Gn r3, corresponding to a current in-

tensity of the order of the Planck intensity. Thestability of these exact closed string solutions re-mains to be investigated.

References

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w x Ž .2 R. Penrose, Nuovo Cimento 1 1969 252.w x Ž .3 T.W.B. Kibble, J. Phys. A 9 1976 1387.w x Ž .4 A. Vilenkin, Phys. Rev. D 23 1981 852.w x Ž .5 E. Witten, Nucl. Phys. B 249 1985 557.w x Ž .6 R.L. Davis, E.P.S. Shellard, Phys. Lett. B 209 1988 485; B.

Ž .Carter, Phys. Lett. B 238 1990 166.w x Ž .7 K.A. Bronnikov, V.N. Melnikov, Grav. &Cosm. 1 1995

155; K.A. Bronnikov, J.C. Fabris, Class. Quant. Grav. 14Ž .1997 831.

w x Ž .8 G. Clement, Phys. Rev. D 57 1998 4885.´w x Ž . Ž .9 F.J. Ernst, Phys. Rev. 167 1968 1175; 168 1968 1415.

w x Ž . Ž .10 G. Neugebauer, D. Kramer, Ann. Phys. Leipzig 24 196962.

w x11 G. Clement, gr-qcr9810069.´w x Ž .12 B. Bertotti, Phys. Rev. 116 1959 1331; I. Robinson, Bull.

Ž .Acad. Polon. Sci., Ser. Math. Astr. Phys. 7 1959 351.w x13 G. Clement, M. Haouchine, in preparation.´

4 March 1999


On intersection of domain walls in a supersymmetric model

S.V. Troitsky a, M.B. Voloshin b,c

a Institute for Nuclear Research of the Russian Academy of Sciences, 60th October AnniÕersary Prospect 7a, Moscow 117312, Russiab Theoretical Physics Institute, UniÕersity of Minnesota, Minneapolis, MN 55455, USA

c Institute of Theoretical and Experimental Physics, Moscow 117259, Russia

Received 30 December 1998Editor: H. Georgi

Abstract

We consider a classical field configuration, corresponding to intersection of two domain walls in a supersymmetricmodel, where the field profile for two parallel walls at a finite separation is known explicitly. An approximation to thesolution for intersecting walls is constructed for a small angle at the intersection. We find a finite effective length of theintersection region and also an energy, associated with the intersection. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

The problem of domain walls has a long historyw xboth in its general field aspect 1 and in a possible

w xrelevance to cosmology 2 . Recently the interest toproperties of domain walls has been given a newboost in supersymmetric models, which naturallypossess several degenerate vacua, thus allowing for a

w xmultitude 3 of domain wall type configurationsinterpolating between those vacua. Furthermore it

w xhas been realised 4,5 that at least some of thesewalls have rather distinct properties under supersym-metry and in fact their field profiles satisfy firstorder differential equations in analogy with the Bo-

Ž . w xgomol’nyi–Prasad–Sommerfield BPS equations 6 .For this reason such configurations are called‘‘BPS-saturated walls’’, or ‘‘BPS walls’’. It was also

w xnoticed 7 that, quite typically, there may exist acontinuous set of the BPS walls, all degenerate inenergy, interpolating between the same pair of vacua.

w xA further analysis 8 of such set in a specific model

revealed that the solutions in the set can be inter-preted as two elementary BPS walls parallel to eachother at a finite separation, the distance between thewalls being the continuous parameter labeling thesolutions. Since all the configurations in the set havethe same energy, equal to the sum of the energies ofthe elementary walls, one encounters here a remark-able situation where there is no ‘potential’ interac-

Žtion between the elementary walls. This property isprotected by supersymmetry and holds in all orders

.of perturbation theory.The existence of multiple vacua in supersymmet-

ric models naturally invites a consideration of morecomplicated field configurations, than just two vacuaseparated by a domain wall, namely those withco-existing multiple domains of different degenerate

w xvacua 3 . This leads to the problem of intersectingdomain walls. It should be noted, that by far notevery conceivable configuration of domains is stable,e.g. any intersection of domain walls in a one-fieldtheory is unstable, and the conditions for stability of


( )S.V. Troitsky, M.B. VoloshinrPhysics Letters B 449 1999 17–2318

the intersections of the walls in multiple-field theo-w xries 3 generally allow only a well defined set of

intersection angles, depending on relation betweenthe energy densities of the intersecting walls. As isdiscussed further in this paper, for the elementary

w xBPS walls, considered in Ref. 8 , i.e. non-interactingŽin the parallel configuration zero intersection angle

.b , at least a finite range of values for b , includingbs0, is allowed by the stability conditions for aquadruple intersection, shown in Fig. 1. When viewedas an intersection of world surfaces of the walls inthe space-time, rather than as a static spatial configu-ration, the intersection describes scattering of mov-ing walls, and in the case of a collision of spatially

Ž Ž .parallel walls or kinks in a 1q1 dimensional.model the angle b translates into bsÕrc with Õ

being the relative velocity of the walls, for small b ,i.e. for non-relativistic collisions. The purpose of thepresent paper is a more detailed study of the fieldprofile for the intersection configuration of suchtype. We consider here the same supersymmetric

w xmodel as in Ref. 8 and use the explicit solution forparallel walls, found there, to construct an approxi-mation to the classical field profile for intersectingwalls, which is valid to the first order in b for smallb. This approximation gives the energy of the staticfield configuration up to b 2 inclusive, while for thecase of collision of the walls the corresponding resultis obviously the action for the collision ‘‘trajectory’’.

Our main findings are as follows. There exists awell defined gap D x between the vertices of pair-wise ‘meet’ of the walls, so that a more detailedpicture of the intersection looks as shown in actualdetail in Fig. 3 further in the paper. In the collisionkinematics this gap is in time and corresponds to afinite time delay in the scattering process. This clas-sical quantity clearly can also be translated into aphase shift in a quantum mechanical description of

Fig. 1. Quadruple vertex for four different vacuum domains.

the scattering. The gap depends on the intersectionangle, and is given by

1 l0D xs , 1Ž .

b m

where m is a mass parameter for the masses ofquanta in the model, and l is a dimensionless0

constant, depending on the ratio of the couplingconstants in the model. We also find that there is a

Žfinite angle-dependent energy in the static configu-.ration associated with the intersection:

l0´sb m , 2Ž .

m

where m is the energy density of each of the elemen-Ž .tary walls. Naturally, in a 3q1 dimensional theory

´ is in fact the energy per unit length of the intersec-Ž .tion, while in a 2q1 dimensional case it is an

Ž .energy localized at the intersection. In a 1q1Ždimensional model only the collision kinematics for

.two elementary kinks is possible, thus ´ has theŽ .meaning of a finite action phase shift associated

Ž .with the collision. We believe that the Eqs. 1 andŽ .2 are quite general at small b , and the only modeldependence is encoded in the dimensionless quantityl .0

The rest of the paper is organized as follows. InSection 2 we present the supersymmetric model un-der consideration and describe the solutions for theBPS walls. In Section 3 we consider intersectingdomain walls within an Ansatz allowing us to con-struct the field profiles in the limit of small intersec-tion angle, and we find the characteristics of theintersection. In Section 4 we discuss applicability ofthe findings of this paper when some of the restric-tions, assumed here, are relaxed.

2. Domain walls in a SUSY model

The specific SUSY model under consideration isthat of two chiral superfields F and X with thesuperpotential

m21 3 2W F , X s Fy lF ya F X . 3Ž . Ž .3

l

Here m is a mass parameter and l and a arecoupling constants. The phases of the fields and of

( )S.V. Troitsky, M.B. VoloshinrPhysics Letters B 449 1999 17–23 19

the W are assumed to be adjusted in such a way thatall the parameters are real and positive, and thelowest components of the superfields F and X aredenoted correspondingly as f and x throughout thispaper. The model has the Z =Z symmetry under2 2

independent flip of the sign of either of the fields:F™yF , X™yX. The vacuum states in thismodel are found as stationary points of the superpo-

Ž . Ž .tential function 3 : E W f , x rEf s 0 andŽ .E W f, x rExs0, and are located at fs"mrl,

Ž .xs0 labeled here as the vacua 1 and 2 , and at' Ž .fs0, xs"mr l a the vacua 3 and 4 . The

locations and the labeling of the vacuum states areshown in Fig. 2. Throughout this paper the fermionicsuperpartners of the bosons are irrelevant and alsoonly real components of the fields appear in theconsidered configurations. Therefore for what fol-lows it is appropriate to write the expression for thepart of the Lagrangian describing the real parts of thescalar fields:

22m2 2 2 2Ls Ef q Ex y yl f ya xŽ . Ž . ž /l

y4a 2 f 2 x 2 . 4Ž .w xAs discussed in detail in Ref. 8 , in this model

exists a continuous set of BPS walls interpolatingbetween the vacua 1 and 2, all having the energy

8 3 2density m s m rl . These configurations with12 3

Ž .positive negative x can be interpreted as parallelelementary BPS walls, i.e. connecting the vacua 1

Fig. 2. The diagram showing locations of the four vacua and theirŽ .labeling in the SUSY model with the superpotential of Eq. 3 .

Ž . Ž .and 3 1 and 4 and the vacua 3 and 2 4 and 2located at a finite distance from each other. Theenergy m of each of the elementary walls is msm r2. The only non-BPS domain wall in this model12

is the one connecting the vacua 3 and 4, and its'energy is given by lra m . The first-order equa-12

tions for the BPS walls are conveniently written interms of dimensionless field variables f and h,defined as

m mfs f , xs h .'l l a

For a domain wall perpendicular to the z axis thew xBPS equations read as 8

d f dh 22 2s1y f yh , sy f h . 5Ž .

d z d z r

Here the notation is used rslra and the massparameter m is set to one.

Ž . w xAlthough Eqs. 5 are solved 8 in quadraturesfor arbitrary r, it is only for rs4 that the non-triv-ial solution can be written in terms of elementaryfunctions. It is this explicitly solvable case of rs4that we consider for the most part in this paper forthe sake of presenting closed expressions wherever

Ž .possible. The set of solutions to Eqs. 5 in this casew xreads as 8

a e2 z y1Ž .f z s ,Ž . z 2 zaq2 e qa e

2 e z2h z s , 6Ž . Ž .z 2 zaq2 e qa e

where a is a continuous parameter, 0FaF` label-ing the solutions in the set, and the overall transla-tional freedom is fixed here by centering the config-

Ž .uration at zs0 in the sense that f zs0 s0. Theinterpretation of these configurations becomes trans-parent, if one introduces the notation:

cosh ssay1 . 7Ž .Ž .Then the expressions 6 can be written as

zys zqs1f z s tanh q tanh ,Ž . 2 ž /2 2

zys zqs12h z s 1y tanh tanh . 8Ž . Ž .2 ž /2 2


Ž .In this form it is clear that at large s i.e. at small athe functions f and h differ from their values in oneof the vacua only near zs"s, i.e. the configurationsplits into the elementary walls, corresponding to the

ŽŽ . Ž .transition between the vacua 1™3 f ,h s y1,0Ž . Ž ..™ f ,h s 0,1 at zsys and the transition be-

Žtween the vacua 3™2 at zss. For definiteness weŽ . .refer to the solution with positive h z . Remark-

ably, the function f is simply a sum of the profilesŽ . Ž . Žfor the elementary walls: f z s f zqs q f z13 32

.ys with

z1f z s tanh y1 ,Ž .13 2 ž /2

z1f z s tanh q1 . 9Ž . Ž .32 2 ž /2

Ž .while the function h z is not a linear superpositionof the profiles of h for elementary walls:

z12h z s 1q tanh ,Ž .13 2 ž /2

z12h z s 1y tanh , 10Ž . Ž .32 2 ž /2

Ž .but rather for large s the profile of h z is exponen-Ž .tially in s close to a linear superposition of the

elementary wall profiles separated by the distanceŽ . Ž .2 s. Thus at large s small a the solution 6 de-

scribes two far separated elementary walls.The caveat of parameterizing the solution in terms

of s for arbitrary positive a is that, according to Eq.Ž .6 , at as1 the parameter s bifurcates into thecomplex plane and becomes purely imaginary, reach-

Ž Ž .ing ss"i pr2 at as0. The functions f z andŽ . . 1h z obviously are still real at 0FaF1.

3. Intersecting domain walls

In order to describe a static intersection of twoelementary walls at a small angle b , we make the

1 It can be noted that the profile of the fields at as1 is givenw xby the special solution to the BPS equations first found in Ref. 7

at any r, such that r )2.

natural Ansatz that the parameter a in the solutionŽ .6 is a ‘‘slow’’ function of the coordinate x, i.e.

Ž Ž . . Ž Ž . .f™ f a x , z and h™h a x , z . We then substi-tute this Ansatz in the expression for the energy ofthe fields and find

2da2Esm d x 2qF a , 11Ž . Ž .H ž /d x

with

12F a sŽ . 22a y1Ž .

=

2 2'1 3 a arctan a y15y q .22 2ž /'a 2 a y1

12Ž .

The interpretation of this expression for the energybecomes quite simple at small a, if one writes it in

Ž . Ž Ž ..terms of the x dependent parameter s cf. Eq. 7 ,assuming that a-1:

1Esm d x 2qH 2sinh s

=

23 s d s52cosh sy q .2ž / ž /2 sinh s cosh s d x

13Ž .Ž .2At large s the weight function for d srd x in the

latter expression rapidly reaches one, and for thetrajectory ssb xr2, corresponding to two wallsbeing at large distance and inclined towards eachother at the relative slope 2 b , the energy per unitlength of x coincides in order b 2 with that of two

2(independent walls: 2 m 1q br2 .Ž .In order to find the relation between a and x at

arbitrary separation between the walls within ourAnsatz, one needs to solve the variational problem

Ž .for the energy integral, given by Eq. 11 . The

2 At small b we make no distinction between the angle and theslope. When discussing large b , or higher order terms, we implythat b is the relative slope, so that the full opening angle between

Ž .the walls is 2 arctan br2 .


solution is quite straightforward: introduce a functionŽ .s x , such that

ds dasyF a 14Ž . Ž .

d x d x

ŽThe minus sign here ensures that s is growing.when a decreases. Then the solution of the varia-

tional problem for s is a linear function of x:

ssb xr2 . 15Ž .The overall shift in x and s is chosen so that ss0and xs0 when as` i.e. at the center of the

Ž .intersection. The slope of the s x is determined byŽ .noticing that at a™0 the function F a behaves as

Ž . y1 Ž .F a sa qO a , thus at small a the slope of s

Ž .coincides with that of s x , defined above as br2 atlarge s.

At this point we address the question about theaccuracy of our Ansatz at small b , and consider thefull second-order differential equations for the fields

Ž .f and x following from the Lagrangian in Eq. 4 .Ž .Since at fixed a the profile given by Eqs. 6 satis-

fies also the second-order equations in the z vari-able, the mismatch in the full two-dimensional equa-tions is given by the second derivatives in x only:

2 Ž Ž . . 2 2 Ž Ž . . 2E f a x , z rE x and E h a x , z rE x . One canreadily see however that these quantities are of order

2 Ž Ž . .b . Indeed, e.g. for the term with f a x , z oneŽ . Ž .finds using Eqs. 14 and 15

E 2 f a x , zŽ .Ž .2E x

b 2 E 2 f a, z F X a E f a, zŽ . Ž . Ž .s y .2 2ž /F a E a4 F a E a Ž .Ž .

Thus our Ansatz correctly approximates the actualsolution in the first order in b. Once it is establishedthat the correction f Ž2. to the solution within thei

Ž 2 . ŽAnsatz is O b with f generically denoting thei.fields f and x , it is clear that the corrections to the

energy can start only in the order b 4. Indeed, thevariation of the energy, minimized within the Ansatz,dErdf is of order of the mismatch in the fieldi

Ž 2 .equations, i.e. O b . Thus the error in the foundŽ . Ž2. Ž 4.energy dErdf f is O b .i i

A remark concerning further details of the dis-cussed solution is due in relation with the singularityin a at xs0. Formally, the described solution is

only specified so far at xG0. At negative x one apriori can choose one of two options: symmetricallyreflect the profile at positive x, or also flip the signof the field h at negative x. The first option would

Žcorrespond to the domains of the same vacuum i.e..the vacuum 3 or the vacuum 4 at small z on both

sides of the intersection, while the second optiondescribes the change from the vacuum 3 to thevacuum 4 at the intersection. The first configuration

Žhowever is unstable the translational zero mode.develops a nodal line , and only the second type

configuration should be chosen.One can notice that the profile of the fields within

our Ansatz depends in fact on the scaling variablessb xr2. This obviously implies that a ‘longitudi-nal’ interval of d x between some fixed characteristicvalues of the fields in the configuration scales asby1. The behavior of the parameter a in the expres-

Ž .sion 6 for the field profile as a function of s isdetermined by the equation

daF a sy1 . 16Ž . Ž .

ds

This equation can be readily solved numerically, andthe resulting picture of the intersection is presentedin Figs. 3 and 4.

Fig. 3. Contour plot of the energy density at the intersection of thedomain walls as described within our Ansatz in terms of thescaling variable b xr2. Darker shading corresponds to largerenergy density.


Fig. 4. Three dimensional plot of the energy density at theintersection.

It is seen from the figures that the four crests ofthe energy density profile do not intersect at onepoint, rather there is a finite gap between the pair-wise intersections of the energy crests for the ‘ini-tial’ and the ‘final’ walls. In order to quantify this

Ž . Ž .gap, we find from Eqs. 16 and 7 the relationbetween s and s as

`

ss F a da 17Ž . Ž .H1rcosh s

and then find numerically that at large s this relationgives sssqd with ds1.7527 . . . . Since at larges the distance between the elementary walls is natu-rally identified as 2 s, one concludes that when theintersection of two walls is extrapolated from largedistances it comes short of the actual center of thesolution by the distance 2drb. The gap l between0

the apparent pairwise collision vertices is then twiceŽ .this distance. In this way we obtain the Eq. 1 with

l s4 ds7.0108 . . . in the particular model with0

rs4.In order to find the proper expression for the

energy associated with the intersection, one can con-sider the problem of intersecting walls as a boundaryvalue problem in a large box with the size L in the x

Ž .direction so that the boundaries are at xs"Lr2 ,and at the boundaries the positions of the walls in thez direction are specified as ss"lr2, where s is the

Ž .parameter in the expression 8 for the profile.Ž Ž .Clearly the sign of h z at xsqLr2 should bechosen opposite to that at xsyLr2 in order to

.have an intersection configuration. We also assumethat the separation l between the walls at the bound-

aries satisfies l<L in order to correspond to b<1.For free walls, with no energy associated with theintersection, the energy of such configuration wouldbe determined by the total length of the walls in thebox:

2l2 2'E s2 m L q l f2 m Lqm L . 18Ž .0 ž /L

For the solution within our Ansatz, however, theenergy is given by

b 2

Es2 m Lqm L , 19Ž .4

Ž .where br2 is the slope in the solution s x sb xr2. At xsLr2 one can use the asymptoticrelation sssqd , and find from the boundary con-dition for s the relation for b :

b l 2 ds q .

2 L L

Ž .Upon substituting this relation in Eq. 19 one finds aŽ .finite difference between the expressions 19 and

Ž .18 for the energy: ´sEyE , given by our result0Ž .in Eq. 2 .

If instead of the spatial coordinate x one uses thetime coordinate t, the discussed configuration de-

Žscribes a collision of two parallel walls or two kinksŽ . .in 1q1 dimensional theory . In this case the pa-

rameter of the intersection b is identified as theŽ .relative velocity Õ of the walls, and Eq. 1 describes

the time delay in the process of scattering of theŽ .walls, while the quantity ´ in Eq. 2 gives the

additional action with respect to the free motion ofthe walls, and thus is equal to the phase of thetransmission amplitude in a quantum mechanical de-scription of the scattering.

4. Discussion

The description of intersection of elementary do-main walls is found here for small intersection an-gles and for a particular value of the parameter r,rs4, in a specific SUSY model. At present we canonly speculate how our results are modified beyondthese restrictions. Although the same Ansatz, asused here, can be applied in the limit of small b atany value of r, one inevitably has to resort to


numerical analysis because of lack of an explicitsimple expressions for the profiles of the fields,except for the trivial case rs1. In particular, theexistence of the gap at the intersection appears to berelated to the existence of the special bifurcation

w xsolution 7 for parallel walls. Indeed a non-zerodifference d between s and s, arising from the

Ž .integral in Eq. 17 , comes mostly from the integra-tion over the values of a past the bifurcation point,i.e. from as1 to as`. The bifurcation solution onthe other hand exists in the considered model onlyfor rG2, and at rs2 the solution corresponds toŽ .x z s0, so that the bifurcation point is located at

xs0 for such r. Our preliminary numerical study ofthe dependence of the size of the gap on r indicatesthat indeed the gap appears to arise starting withrs2 and becomes larger with increasing r, whereasat r-2 we have found no apparent gap. Thisconjecture about such behavior is somewhat sup-ported by the fact that at rs1 the elementary wallsare not interacting and simply ‘‘go through’’ at theintersection with no gap whatsoever.

As to the dependence on the intersection parame-ter b , our Ansatz becomes inapplicable when b isnot small, and the behavior of the correspondingconfigurations is not known. Here we can only men-

w xtion that the stability conditions 3 for the wallintersections allow a static quadruple intersectionwith any b if r)1. Indeed in this case the energy

Ž . 'of the non-BPS wall 3™4 , m s2 r m is larger34

than the sum of the energies of the elementary walls,and thus the quadruple intersection cannot split intotriple ones, since the stability conditions for tripleintersections can not be satisfied. On the contrary, atr-1 such splitting is possible with the critical value

2Ž .( 'of b defined as b r 2 1q b r2 s r .Ž .c c

BPS solitons are known to present effective low-energy degrees of freedom in some supersymmetric

Ž .field theories. These effective ‘‘dual’’ theories ex-

ploit the fact that elementary solitons do not interactwith each other at zero energies, both classically andquantum mechanically. This is just the case in themodel considered, where the energy of parallel ele-mentary domain walls does not depend on distancebetween them, and this degeneracy is not lifted byquantum corrections. We demonstrated here by anexplicit calculation that some highly nontrivial inter-action between BPS domain walls arises at nonzeromomenta. If this observation holds for general case,to obtain dynamical information from dual models ofthat class appears to be an extremely complicatedtask.

Acknowledgements

Ž .One of us SVT acknowledges warm hospitalityof the Theoretical Physics Institute at the Universityof Minnesota, where this work was done. The workof SVT is supported in part by the RFFI grant96-02-17449a and in part by the U.S. Civilian Re-search and Development Foundation for Independent

Ž .States of FSU CRDF Award No. RP1-187. Thework of MBV is supported in part by DOE under thegrant number DE-FG02-94ER40823.

References

w x Ž .1 T.D. Lee, G.C. Wick, Phys. Rev. D 9 1974 2291.w x2 Ya.B. Zeldovich, I.Yu. Kobzarev, L.B. Okun, Zh. Eksp. Teor.

Ž . Ž Ž . .Fiz. 67 1974 3 Sov. Phys. JETP 40 1974 1 .w x Ž .3 M.B. Voloshin, Phys. Rev. D 57 1998 1266.w x Ž .4 G. Dvali, M. Shifman, Nucl. Phys. B 504 1997 127.w x Ž .5 G. Dvali, M. Shifman, Phys. Lett. B 396 1997 64; B 407

Ž . Ž .1997 452 E .w x Ž .6 E. Bogomol’nyi, Sov. J. Nucl. Phys. 24 1976 449; M.K.

Ž .Prasad, C.H. Sommerfield, Phys. Rev. Lett. 35 1975 760.w x Ž .7 M. Shifman, Phys. Rev. D 57 1998 1258.w x Ž .8 M.A. Shifman, M.B. Voloshin, Phys. Rev. D 57 1998 2590.

4 March 1999


Global structure of evaporating black holes

Maulik K. Parikh a,1, Frank Wilczek b,2

a Joseph Henry Laboratories, Princeton UniÕersity, Princeton, NJ 08544, USAb School of Natural Sciences, Institute for AdÕanced Study, Princeton, NJ 08540, USA

Received 6 January 1999Editor: M. Cvetic

Abstract

By extending the charged Vaidya metric to cover all of spacetime, we obtain a Penrose diagram for the formation andevaporation of a charged black hole. In this construction, the singularity is time-like. The entire spacetime can be predictedfrom initial conditions if boundary conditions at the singularity are known. q 1999 Published by Elsevier Science B.V. Allrights reserved.

PACS: 04.70.Dy; 04.20.Gz; 04.60.-m; 04.40.Nr

1. Introduction

It is challenging to envision a plausible globalstructure for a spacetime containing a decaying blackhole. If information is not lost in the process of blackhole decay, then the final state must be uniquelydetermined by the initial state, and vice versa. Thus apost-evaporation space-like hypersurface must liewithin the future domain of dependence of a pre-evaporation Cauchy surface. One would like to havemodels with this property that support approximateŽ .apparent horizons.

In addition, within the framework of general rela-tivity, one expects that singularities will form inside

w xblack holes 1 . If the singularities are time-like, one

1 E-mail: [email protected] E-mail: [email protected]

can imagine that they will go over into the world-linesof additional degrees of freedom occurring in aquantum theory of gravity. Ignorance of the nature ofthese degrees of freedom is reflected in the need to

Žapply boundary conditions at such singularities. Onthe other hand, boundary conditions at future space-like singularities represent constraints on the initialconditions; it is not obvious how a more completedynamical theory could replace them with something

.more natural.In this paper, we use the charged Vaidya metric to

obtain a candidate macroscopic Penrose diagram forthe formation and subsequent evaporation of acharged black hole, thereby illustrating how pre-dictability might be retained. We do this by firstextending the charged Vaidya metric past its coordi-nate singularities, and then joining together patchesof spacetime that describe different stages of theevolution.


( )M.K. Parikh, F. WilczekrPhysics Letters B 449 1999 24–29 25

2. Extending the charged Vaidya metric

w xThe Vaidya metric 2 and its charged generaliza-w xtion 3,4 describe the spacetime geometry of unpo-

larized radiation, represented by a null fluid, emerg-ing from a spherically symmetric source. In mostapplications, the physical relevance of the Vaidyametric is limited to the spacetime outside a star, witha different metric describing the star’s internal struc-

w xture. But black hole radiance 5 suggests use of theVaidya metric to model back-reaction effects for

w xevaporating black holes 6,7 all the way upto thesingularity.

The line element of the charged Vaidya solutionis

2 M u Q2 uŽ . Ž .2 2ds sy 1y q du y2 du dr2ž /r r

qr 2 du 2 qsin2u df 2 . 1Ž .Ž .

Ž .The mass function M u is the mass measured atŽ .future null infinity the Bondi mass and is in general

a decreasing function of the outgoing null coordi-Ž .nate, u. Similarly, the function Q u describes the

charge, measured again at future null infinity. WhenŽ . Ž .M u and Q u are constant, the metric reduces to

the stationary Reissner-Nordstrom metric. The corre-¨sponding stress tensor describes a purely electricCoulomb field,

Q uŽ .F sq , 2Ž .r u 2r

and a null fluid with current

1 E Q22k sk= u , k sq yMq . 3Ž .a a 2 ž /E u 2 r4p r

In particular,

2 21 2 M u Q u Q uŽ . Ž . Ž .T s 1y quu 2 2 2ž /r8p r r r

21 E Q u E M uŽ . Ž .q y2 . 4Ž .

r E u E u

Like the Reissner-Nordstrom metric, the charged¨Vaidya metric is beset by coordinate singularities. Itis not known how to remove these spurious singular-

Žities for arbitrary mass and charge functions forw x.example, see 9 . We shall simply choose functions

for which the relevant integrations can be done andcontinuation past the spurious singularities can becarried out, expecting that the qualitative structurewe find is robust.

Specifically, we choose the mass to be a decreas-ing linear function of u, and the charge to be propor-tional to the mass:

M u 'auqb'u , Q u 'hu , 5Ž . Ž . Ž .˜ ˜

< < < <where a-0 and h F1, with h s1 at extremality.We always have uG0. With these choices, we can˜

Ž .find an ingoing advanced time null coordinate, Õ,with which the line element can be written in a‘‘double-null’’ form:

g u ,rŽ .˜2 2 2 2 2ds sy du dÕqr du qsin u df . 6Ž .Ž .˜

a

Thus

2 21 2u h u du˜ ˜ ˜dÕs 1y q q2 dr .2ž /g u ,r r arŽ .˜

7Ž .

Ž .The term in brackets is of the form X u,r duq˜ ˜Ž . Ž . Ž .Y u,r dr. Since X u,r and Y u,r are both homo-˜ ˜ ˜

geneous functions, Euler’s relation provides the inte-Ž . Ž . Ž .grating factor: g u,r sX u,r uqY u,r r. Hence˜ ˜ ˜ ˜

E Õ r 2

s 8Ž .uE r 3 2 2 2r q r y2urqh u˜ ˜Ž .

2 a

12 2 2r y2urqh u˜ ˜Ž .E Õ 2 as . 9Ž .

uE u 3 2 2 2r q r y2urqh u˜ ˜Ž .2 a

From the sign of the constant term of the cubic,we know that there is at least one positive zero.

( )M.K. Parikh, F. WilczekrPhysics Letters B 449 1999 24–2926

Then, calling the largest positive zero rX, we mayŽ X.Ž 2 .factorize the cubic as ryr r qb rqg . Hence

h 2 u3 u2˜ ˜Xgsy )0 , gyb r sy )0 ,X2 ar 2 a

uX

byr s -0 . 10Ž .2 a

Consequently, the cubic can have either three posi-tive roots, with possibly a double root but not a triple

Ž .root, or one positive and two complex conjugateroots. We consider these in turn.

2.1. Three positiÕe roots

When there are three distinct positive roots, theŽ .solution to Eq. 8 is

ÕsAln ryrX qBln ryr qC ln ryr ,Ž . Ž . Ž .2 1

11Ž .

where rX)r )r )0, and2 1

qrX 2

As )0 ,X Xr yr r yrŽ . Ž .2 1

yr 22

Bs -0 ,Xr yr r yrŽ . Ž .2 2 1

qr 21

Cs )0 . 12Ž .Xr yr r yrŽ . Ž .1 2 1

We can push through the rX singularity by defining anew coordinate,

BrA CrAXÕr AV Õ 'e s ryr ryr ryr ,Ž . Ž . Ž . Ž .2 2 1

13Ž .

which is regular for r)r . To extend the coordi-2

nates beyond r we define2

ArBV Õ 'k q yVŽ . Ž .1 2 2

ArB CrBXsk q r yr r yr ryr ,Ž . Ž . Ž .2 2 1

14Ž .

where k is some constant chosen to match V and2 2X Ž .V at some r )r)r .V r is now regular for1 2 1

r )r)r . Finally, we define yet another coordi-2 1

nate,

BrCV Õ 'k q y V ykŽ . Ž .Ž .1 1 2

ArC BrCXsk q r yr r yr ryr ,Ž . Ž . Ž .1 2 1

15Ž .

which is now free of coordinate singularities forr-r . A similar procedure can be applied if the2

cubic has a double root.

2.2. One positiÕe root

When there is only one positive root, Õ is singularonly at rsrX:

X 1 2ÕsAln ryr q Bln r qb rqgŽ . Ž .2

2CyBb 2 rqbq arctan . 16Ž .

2 2ž /( (4gyb 4gyb

We can eliminate this coordinate singularity by in-troducing a new coordinate

Br2 AXÕr A 2V Õ 'e s ryr r qb rqgŽ . Ž . Ž .

=2CyBb

exp q2(A 4gyb

=2 rqb

arctan , 17Ž .2ž /(4gyb

which is well-behaved everywhere. The metric nowreads

A du2 2 2ds syg u ,r dVqr dV . 18Ž . Ž .˜

V u ,r aŽ .˜

In all cases, to determine the causal structure of thecurvature singularity we express dV in terms of duwith r held constant. Now we note that, since u is˜the only dimensionful parameter, all derived dimen-sionful constants such as rX must be proportional to


powers of u. For example, when there is only posi-˜Ž .tive zero, Eq. 17 yields

XV yr B b rq2gdV sdu q˜r X 2u ryr 2 A r qb rqg˜

2CyBb 1q 22(A 4gyb 2 rqb

1q2ž /(4gyb

y2 r= . 19Ž .

2(4gyb

Thus, as r™0, and using the fact that AqBs1,we have

Q2 uŽ .2 2ds ™y du , 20Ž .2r

so that the curvature singularity is time-like.

3. Patches of spacetime

Our working hypothesis is that the Vaidya space-time, since it incorporates radiation from the shrink-ing black hole, offers a more realistic backgroundthan the static Reissner spacetime, where all back-re-action is ignored. In this spirit, we can model theblack hole’s evolution by joining patches of the

Ž .collapse and post-evaporation Minkowski phasesonto the Vaidya geometry.

To ensure that adjacent patches of spacetimematch along their common boundaries, we can calcu-

Ž .late the stress-tensor at their light-like junction. Theabsence of a stress-tensor intrinsic to the boundaryindicates a smooth match when there is no explicitsource there. Surface stress tensors are ordinarilycomputed by applying junction conditions relatingdiscontinuities in the extrinsic curvature; the appro-priate conditions for light-like shells were obtained

w xin Ref. 8 . However, we can avoid computing mostof the extrinsic curvature tensors by using the Vaidya

metric to describe the geometry on both sides of agiven boundary, because the Reissner-Nordstrom and¨Minkowski spacetimes are both special cases of theVaidya solution.

Initially then, we have a collapsing charged spher-ically symmetric light-like shell. Inside the shell,region I, the metric must be that of flat Minkowskispace; outside, region II, it must be the Reissner-Nordstrom metric, at least initially. In fact, we can¨describe both regions together by a time-reversedcharged Vaidya metric,

2 M Õ Q2 ÕŽ . Ž .2 2ds sy 1y q dÕ q2 dÕ dr2ž /r r

qr 2dV 2 , 21Ž .where the mass and charge functions are step func-tions of the ingoing null coordinate:

M Õ sM Q ÕyÕ , Q Õ shM Õ . 22Ž . Ž . Ž . Ž . Ž .0 0

s Ž .The surface stress tensor, t , follows from Eq. 4 .Õ Õ

Thus

1 Q20st s M y . 23Ž .Õ Õ 02 ž /2 r4p r

The shell, being light-like, is constrained to move at45 degrees on a conformal diagram until it hascollapsed completely. Inside the shell, the spacetimeis guaranteed by Birkhoff’s theorem to remain flatuntil the shell hits rs0, at which point a singularityforms.

Meanwhile, outside the shell, we must have theReissner-Nordstrom metric. This is appropriate for¨all r)r . Once the shell nears r , however, oneq qexpects that quantum effects start to play a role. For

Ž < < .non-extremal h -1 shells, the Killing vectorchanges character – time-like to space-like – as theapparent horizon is traversed, outside the shell. Thispermits a virtual pair, created by a vacuum fluctua-tion just outside or just inside the apparent horizon,to materialize by having one member of the pairtunnel across the apparent horizon. Thus, Hawkingradiation begins, and charge and energy will streamout from the black hole.

We shall model this patch of spacetime, regionIII, by the Vaidya metric. This must be attached tothe Reissner metric, region II, infinitesimally outsidersr . A smooth match requires that there be noqsurface stress tensor intrinsic to the boundary of the

( )M.K. Parikh, F. WilczekrPhysics Letters B 449 1999 24–2928

two regions. The Reissner metric can be smoothlymatched to the radiating solution along the us0

Ž .boundary if bsM in Eq. 5 .0Ž . Ž .Now, using Eqs. 7 and 17 , one can write the

Vaidya metric as

g 2 u ,r AŽ .˜2 2ds sy dV22 M u Q uŽ . Ž .

21y q V2ž /r r

g u ,r AŽ .˜q2 dV dr . 24Ž .22 M u Q uŽ . Ž .

1y q V2ž /r r

Ž .We shall assume for convenience that g r has onlyone positive real root, which we call rX. Then, since

Ž X. Ž .V and g both contain a factor ryr , Eq. 17 , theabove line element and the coordinates are both

Ž .well-defined for r)r u . In particular, rs` is˜qpart of the Vaidya spacetime patch. Moreover, theonly solution with ds2 sdrs0 also has dVs0, sothat there are no light-like marginally trapped sur-faces analogous to the Reissner r . In other words,"

the Vaidya metric extends to future null infinity,IIq, and hence there is neither an event horizon, nora second time-like singularity on the right of theconformal diagram.

The singularity on the left exists until the radia-tion stops, at which point one has to join the Vaidyasolution to Minkowski space. This is easy: bothspacetimes are at once encompassed by a Vaidyasolution with mass and charge functions

M u s auqb Q u yu , Q u shM u .Ž . Ž . Ž . Ž . Ž .0

25Ž .

As before, the stress tensor intrinsic to the boundaryŽ .at u can be read off Eq. 4 :0

21 auqbŽ .st s auqb y , 26Ž . Ž .uu 2 2 r4p r

which is zero if u sybra, i.e., if us0. This says˜0

simply that the black hole must have evaporatedcompletely before one can return to flat space.

Collecting all the constraints from the precedingparagraphs, we can put together a possible conformal

Ždiagram, as in Fig. 1. We say ‘‘possible’’ because asimilar analysis for an uncharged hole leads to a

Fig. 1. Penrose diagram for the formation and evaporation of acharged black hole.

space-like singularity; thus our analysis demonstratesthe possibility, but not the inevitability, of the be-

.haviour displayed in Fig. 1. Fig. 1 is a Penrosediagram showing the global structure of a spacetimein which a charged imploding null shock wave col-lapses catastrophically to a point and subsequentlyevaporates completely. Here regions I and IV are flatMinkowski space, region II is the stationary Reiss-ner-Nordstrom spacetime, and region III is our ex-¨tended charged Vaidya solution. The zigzag line onthe left represents the singularity, and the straightline separating region I from regions II and III is theshell. The curve connecting the start of the Hawking


Ž .radiation to the end of the singularity is r u ,˜qwhich can be thought of as a surface of pair creation.The part of region III interior to this line mightperhaps be better approximated by an ingoing nega-tive energy Vaidya metric.

From this cut-and-paste picture we see that, givensome initial data set, only regions I and II and part ofregion III can be determined entirely; an outgoingray starting at the bottom of the singularity marks theCauchy horizon for these regions. Note also thatthere is no true horizon; the singularity is naked.However, because the singularity is time-like, Fig. 1has the attractive feature that predictability for theentire spacetime is restored if conditions at the singu-larity are known. It is tempting to speculate that,with higher resolution, the time-like singularity mightbe resolvable into some dynamical Planck-scale ob-ject such as a D-brane.

Acknowledgements

F.W. is supported in part by DOE grant DE-FG02-90ER-40542.

References

w x Ž .1 R. Penrose, Phys. Rev. Lett. 14 1965 57.w x Ž .2 P.C. Vaidya, Proc. Indian Acad. Sci. A 33 1951 264.w x Ž .3 J. Plebanski, J. Stachel, J. Math. Phys. 9 1967 269.w x Ž .4 W.B. Bonnor, P.C. Vaidya, Gen. Rel. and Grav. 1 1970 127.w x Ž .5 S.W. Hawking, Commun. Math. Phys. 43 1975 199.w x Ž .6 W.A. Hiscock, Phys. Rev. D 23 1981 2823.w x Ž .7 R. Balbinot, in: R. Ruffini Ed. , Proceedings of the Fourth

Marcel Grossmann Meeting on General Relativity, North-Hol-land, Amsterdam, 1988, p. 713.

w x Ž .8 C. Barrabes, W. Israel, Phys. Rev. D 43 1991 1129.`w x9 F. Fayos, M.M. Martin-Prats, J.M.M. Senovilla, Class. Quant.

Ž .Grav. 12 1995 2565.

4 March 1999


Can conformal transformationschange the fate of 2D black holes? 1

J. Cruz a,2, A. Fabbri b,3, J. Navarro-Salas a,4

a Departamento de Fısica Teorica and IFIC, Centro Mixto UniÕersidad de Valencia-CSIC. Facultad de Fısica, UniÕersidad de Valencia,´ ´ ´Burjassot-46100, Valencia, Spain

b Department of Physics, Stanford UniÕersity, Stanford, CA 94305-4060, USA


Abstract

By using a classical Liouville-type model of two dimensional dilaton gravity we show that the one-loop theory impliesthat the fate of a black hole depends on the conformal frame. There is one frame for which the evaporation process neverstops and another one leading to a complete disappearance of the black hole. This can be seen as a consequence of the factthat thermodynamic variables are not conformally invariant. In the second case the evaporation always produces the samestatic and regular end-point geometry, irrespective of the initial state. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 04.60qnKeywords: Black holes; Back-reaction; Solvable models; End-point geometry

The understanding of the dynamical evolution of black holes is an important ingredient in the formulation ofw xa consistent theory of quantum gravity. Since the work of Callan-Giddings-Harvey-Strominger 1 , the study of

two-dimensional models for black hole formation and evaporation has increased a lot and it has been very usefulto analyze quantum aspects of black hole physics. In particular, the existence of exactly solvable one-loop

w xmodels 2,3 has allowed to study back reaction effects in an analytical setting. One of the central properties ofthe CGHS model is that the Hawking temperature is independent of the mass and many aspects of the quantumevolution of the black holes are indeed associated with this fact. Therefore it is interesting to consider other

1 Work partially supported by the Comision Interministerial de Ciencia y Tecnologıa and DGICYT.´ ´2 E-mail: [email protected] Supported by an INFN fellowship. E-mail: [email protected] E-mail: [email protected]


( )J. Cruz et al.rPhysics Letters B 449 1999 30–38 31

w xmodels giving rise to black hole solutions with a more realistic Hawking temperature. In Ref. 4 it was analyzeda model with a classical gravitational action conformally related to the Polyakov-Liouville action

N1 1 22 2 bf'Ss d x yg Rfq4l e y =f . 1Ž . Ž .ÝH i2p 2 is1

w xThis model, closely related to the one introduced by Mann 5 , has solutions which resemble the string 2D blackw xholes 1 , but the Hawking temperature depends on the mass. Moreover it also permits to construct an associated

w xsolvable semiclassical model and it was also pointed out in Ref. 4 that the one-loop solution predicts that theHawking evaporation never stops.

Ž .The black hole solutions of the model 1 are, in an appropriate Kruskal-type gauge, of the form

ydxqdxy2ds s , 2Ž .2l b

q yqCx xC

1bfe s , 3Ž .2l b

q yqCx xC

< <where the constant C is proportional to the black hole mass. Although we cannot recover Minkowski spacetimefor any value of C, there is a simple way to get a flat geometry. By performing a conformal rescaling of themetric

gmng ™ , 4Ž .mn J fŽ .

where

1bfJ f s e y1 , 5Ž . Ž . Ž .

b

the new geometry, in conformal gauge ds2 sye2 rdxqdxy, is

ydxqdxy2ds s , 6Ž .21 l C

q yy y x xb C b

and it is clear that for Csl2b we obtain a Minkowskian ground state, which was not present in the original5 Ž . 2model. Moreover expanding the solutions 6 around Csl b we have, to leading order

ydxqdxy2ds s , 7Ž .y2 y2 2 2 q yb l Cybl yl x xŽ .

l2 Cwhich implies that, for C;bl , the solutions are similar to the CGHS black hole solutions with Ms y .b

2b l

The above discussion serves to motivate the analysis of the conformally rescaled model through the transforma-Ž . Ž .tion 4-5 . Due to the existence of a classical Minkowski ground state one would expect that the black holes 6

Ž w x w x.may now decay completely as, for instance, in Ref. 2 and 3 . It has already been stressed that the

5 This is a particular case of a general procedure to construct a theory with a flat ground state starting from an arbitrary 2D dilaton gravityw xtheory 6 .

( )J. Cruz et al.rPhysics Letters B 449 1999 30–3832

w xthermodynamical variables can depend upon the conformal frame 7 and in this paper we shall explicitly showthat the evaporation process can be indeed very different.

Ž .In terms of the rescaled metric the action 1 transforms intobf 2 N1 be 4l 12 22 bf bf'Ss d x yg Rfq =f q e e y1 y =f . 8Ž . Ž . Ž . Ž .ÝH ibf2p b 2e y1 ıs1

It is also interesting to note that for bs0 we recover the CGHS model after the trivial redefinition f™ey2 f.Let us now consider the formation of a black hole by collapse of an infalling shock wave at xqsxq. For0

xq-xq the metric is given by0

dxqdxy2ds s , 9Ž .2 q yl x x

with Minkowskian coordinates s " defined by l x "s"e" ls "

. After the incoming shock-wave, xq)xq , the0

metric is described as

ydxqdxy2ds s , 10Ž .21 l C

q q y yy y x qD x qDŽ . Ž .b C b

with

l2bq qD sx y1 , 11Ž .0 ž /C

1 1 1yD s y , 12Ž .q 2ž /x Cl b0

C"'bC" " " " sand the new asymptotically flat coordinates s are defined as x qD s"e . A simpleŽ .˜ (b

calculation leads to the following expression for the stress tensor of the shock-wave

CDyf q qT s d x yx , 13Ž .Ž .qq 0

b

and, in terms of the asymptotically flat coordinates s ", we have

C lf q qq qT s y d s ys . 14Ž .Ž .s s 02ž /blb

lCThis means that the energy of the wave is Ms y , which turns out to be, by energy conservation, equal tob

2lb

Ž .the ADM mass of the black hole solution 6 . This expression is in accordance with the mass formula forw x mgeneric 2D models 8,9 , with an appropriate normalization of the Killing vector k at spatial infinity

l2 CŽ . Ž . Ž .k sy , and differs in the constant shift y from the mass formula for the solutions 2 , 3 . In fact, usingb

2l b

w x Žthe arguments of 10 one can show that the mass is conformally invariant up to a constant shift related to the6.choice of the ground state . We have to note that a different normalization of the Killing vector gives rise to a

different expression for the mass which is not compatible with energy conservation.

6 w xSuch a constant shift was not considered in Ref. 10 because it was supposed there that the ground state of the rescaled theory is alwaysobtained by conformally transforming the vacuum of the original one. This is not what happens here.


The flux of radiation measured by inertial observers at future null infinity sq™` can be worked out as˜follows

Nf y y² :y y � 4T sy s ,s , 15Ž .˜s s˜ ˜ 24

� y y4where s ,s is the Schwartzian derivative. One obtains˜y2

Cys(NC C bf y² :y yT s 1y 1qD e . 16Ž .s s (˜ ˜ 48b b� 0

Ž y .At late times s ™` , the flux of radiation approaches a constant thermal value with an associated Hawking˜temperature given, in terms of the mass, by

1T s lqbM . 17Ž . Ž .H 2p

We must stress that in obtaining this expression we have used the normalization of the Killing vector alreadylused to compute the mass and that for small masses M< one recovers the constant temperature of the< <b

CGHS black hole.We have also to point out that due to the shift Dq relating the asymptotically flat coordinates sq and sq˜

C qsq ˜'bbls q Ž .before and after the collapse e qlD s l e there exists also an additional semiclassical(Ž C /² f : q

q qincoming flux T . This flux is positive and vanishes at s ™` and one therefore could expect that it does˜s s˜ ˜² f :q qnot affect the evaporation process. Despite the presence of such a non-vanishing T we will be able, later,s s˜ ˜

to construct evaporating solutions where this flux does not appear.Ž . Ž .Let us now consider the semiclassical one-loop theory in both models 1 and 8 . One can construct a

Ž .solvable semiclassical theory, maintaining the classical free field equation E E 2 rybf s0, by adding aq yparticular local counterterm to the standard non-local Polyakov effective action

N1 1 22 2 bf'Ss d x yg Rfq4l e y =fŽ .ÝH i2p 2 is1

N 22 y1'y d x yg RI Rqb 2 Rfyb =f . 18Ž . Ž .Ž .H96p

w xWe have introduced a slightly different counterterm to that of 4 because now we can also preserve theŽ Ž ..remaining unconstrained classical equation of motion see 19 . For the theory defined by the classical action

Ž . Ž . Ž . Ž .8 a solvable one-loop theory can be obtained by transforming 18 with the rescaling 4 – 5 . The neww x Ž .semiclassical theory recovers the BPP model 3 in the limit bs0. In Kruskal coordinates 2 rsbf the

Ž .equations of motion derived from 18 leads to the Liouville equation for the field 2 r

E E 2 rsyl2be4r , 19Ž .q y

and the constraint equationsN

2 r 2 y2 r fe E e sb T y t , 20Ž ." "" "ž /12Ž ".where t x are the boundary contributions coming from the non-local Polyakov term. The simplest solution"

can be obtained when T f s0s t"" "

ydxqdxy2ds s , 21Ž .2l b

q yqCx xC


2 Ž .and, for C-0 and l b-0, the metric 21 represents a black hole in thermal equilibrium. As it was illustratedw xin Ref. 4 for the CGHS case we can study the evaporation process of the black hole by considering the

Ž . q qdynamical evolution of the solution 21 when the incoming flux is turned off at x sx . This can be exactly0

achieved in the present theory by assuming the following boundary conditions

1q q

qt s Q x yx , 22Ž .Ž .x 02q q4 x qDŽ .

t ys0 , 23Ž .x

where

21

nq12q1 Cx ln0q qD syx q , 24Ž .0 2l Cqlb M� 0

and

Nb2n s1y . 25Ž .

12

The evaporating solution for xq)xq is0

ydxqdxy2ds s , 26Ž .1yn nq1

2 22 q q 2 q q y ybl l x qlD Cqlb M l x qlD l x qlDŽ . Ž . Ž .Ž .q1 1

22 22n Cqlb MŽ . l n

where

1yn21qn2 ql b l x CŽ .0 1qnyD s 1y , 27Ž .n2 q 2ž /C x Cqlb M0 1qnn

and there is additionally a shock wave along xqsxq0

1yn 2 n

1qn 1qn2l nq1 Cqlb M 1Ž .f q qT s y d x yx . 28Ž .Ž .qq 0q12b b x0� 0

2qCx n0

ŽThis shock wave can be eliminated with an appropriate choice of the parameter M M turns out to be the energy.of the shock wave in the classical limit . The main property of the evaporating solution is that the apparent

Ž .horizon E fs0q

l4b ny1Ž .y yl x sylD q , 29Ž .2 n2 q qCqlb M nq1 l x qlDŽ . Ž .Ž .


and the singularity curve

2 n4 2 q q y yl bq Cqlb M l x qlD l x qlD s0 , 30Ž . Ž . Ž .Ž .never meet each other, implying that the Hawking evaporation never ceases.

The point now is to see what happens in the evaporation process of the conformally rescaled model withl2b-C-0, which is the range of variation of C valid for both models. The rescaled dynamical solution, afterswitching off the incoming flux at xq)xq is0

y11ynnq122 q q 21 l l x qlD Cqlb MŽ . 22 q q y y q yds sy y y l x qlD l x qlD dx dx .Ž . Ž .1 1b

2 22 2n Cqlb M n l bŽ .31Ž .

Ž .The curvature singularity of 31

ny112 n22 2 q q 4 2 q q y y2n l Cqlb M l x qlD yl by Cqlb M l x qlD l x qlD s0 , 32Ž . Ž . Ž . Ž .Ž . Ž .

Ž . q qis hidden behind the apparent horizon 29 immediately after x sx , but they intersect each other at the point0

21

ny12 21 2n l b

q qx syD q , 33Ž .int 2l nq1 Cqlb MŽ . Ž .� 02 n

11yn

23 2l b ny1 2n l bŽ .y yx syD q . 34Ž .int 2 22 Cqlb M nq1Ž .Ž .Cqlb M nq1 � 0Ž .Ž .

Remarkably, the evaporating solution can be matched at the end-point xysxy with a static stable and regularint

solution.To show this let us first analyze the static radiationless solutions of the model. With the following boundary

conditions

1"t s , 35Ž .x 2"4 xŽ .

one can check that the solutions

ydxqdxy2ds s , 36Ž .1yn nq1

21 l C2 22 q y 2 q yy yl x x q yl x xŽ . Ž .2b nC nl b

Ž .where C is an integration constant and n is given by 25 , are stable with respect to the asymptoticallyRindlerian coordinates s " defined by

C"" " sl x s"e . 37Ž .lb


The requirement of stability with respect to these coordinates seems natural because in terms of them theŽ . Žsolutions 36 are independent of the time coordinate similar considerations using asymptotically Rindler

ˆw x.coordinates in solvable models of 2D gravity can also be found in Ref. 11 . For C-C, wherenq1 n222Csyl yb , 38Ž . Ž .nq1 ny1

2 2nq1 ny1Ž . Ž .ˆ ˆthe spacetime geometry has a naked singularity, for CsC the solution is completely regular and for C)C

there are null singularities at xqxys0. Alternatively one can also choose these solutions to be stable withŽ .respect to the asymptotically Minkowskian frame although in these coordinates the solutions 36 are no longer

Ž w x. 7 Ž .static the same prescription was adopted in Ref. 12 . In this case the boundary functions 35 must besubstituted by

2nyn2 q3"t s , 39Ž .x 2"16 xŽ .

and n is now given by

Nb2y

8ns . 40Ž .Nb2y

24

Ž . Ž .Analogously, in order to eliminate the incoming flux in the evaporating solution 26 the boundary function 22would be substituted by

2nyn2 q3q q

qt s Q x yx . 41Ž .Ž .x 02q q16 x qDŽ .Ž .As we have already mentioned the evaporating solution 31 can be just matched with the static and regular

ˆsolution with CsC1yn1yn21 l 222 q q y y˜ds sy y l x qlD yl x ylDŽ . Ž .ˆb nC

y1nq1nq1C 22q q y y q y˜q l x qlD yl x ylD dx dx , 42Ž . Ž .Ž .2nl b

where

˜y y yD snD q ny1 x , 43Ž . Ž .int

and there is emission of a‘‘thunderpop’’ along the null line xysxyint

1ynf y yT s d x yx . 44Ž .Ž .yy inty y2nb x qDŽ .int

We can obtain the same result analizing the evaporation process of a black hole formed by gravitational

7 We must point out that in the classical limit both descriptions coincide.


ˆ q qŽ . Ž .collapse. The static radiationless solutions 36 CFC can be matched at the shock wave line x sx with an0

evaporating solution

ydxqdxy2ds s ,1yn nq1

n2 22 q q y 2 q q y y1 l l x qlD yl x Cqlb M l x qlD yl x ylDŽ . Ž . Ž . Ž .Ž . Ž .y q 2 ny12b nl bn Cqlb MŽ .

2yyl xŽ .45Ž .

with2

C 1qnq qD sx y1 , 46Ž .0 2ž /Cqlb M

23l b C 1qnyD s 1y , 47Ž .n 2q 2 ž /Cqlb Ml x CŽ .0

and the energy momentum tensor takes the expression2

2nq1 Cqlb M nq1f q qT s y1 d x yx . 48Ž .Ž .qq 0q ž /2b x C0

Ž .The singularity curve of 45ny1

22 2 q q y 4nl Cqlb M l x qlD yl x yl bŽ . Ž .Ž .Ž .2 n n2 q q y yq Cqlb M l x qlD yl x ylD s0 , 49Ž . Ž . Ž .Ž . Ž .

is hidden behind the apparent horizon

l4b 1ynŽ .ny yyl x slD q , 50Ž . Ž .2 n2 q qCqlb M 1qn l x qlDŽ . Ž .Ž .ˆŽ .provided the mass M is above a critical mass M vanishing for CsC , and the black hole shrinks until thecr

intersection point1

2 nn2 411qn 2l b l 1ynŽ .ny1yq q y y nx syD ql D y , 51Ž . Ž .nint 22 2ž /Cqlb M 1qnŽ .Ž . Cqlb M 1qnŽ .Ž .

12

ny 2lD l bŽ . ny1yx syint 2ž /l Cqlb M 1qnŽ .Ž .

=

1y2 n n

2 42l b l 1ynŽ .ny1y . 52Ž .22 2ž /Cqlb M 1qnŽ .Ž . Cqlb M 1qnŽ .Ž .


y y Ž .The evaporating solution can also be matched across x sx with the regular static solution 42 where nowint

1yny y yD sD yl x , 53Ž .Ž .int

and with emission of a ‘‘thunderpop’’ at xysxyint

221yn Cqlb M 1ynf y yT s y1 d x yx . 54Ž .Ž .yy inty ž /ˆ2b x Cint

We want to remark that the different result for the process of black hole evaporation in the initialŽ . Ž .Liouville-type model 1 and in the rescaled one 8 can be seen as a consequence of the different relation

Ž .between thermodynamic variables in each of these models. In the model 1 the temperature is proportional tothe black hole mass and therefore goes to zero when the black hole mass becomes very small preventing the

Ž . Ž .complete evaporation. On the contrary in the model 8 due to the shift in the temperature 17 it approaches alconstant non-vanishing value when the black hole mass goes to zero and consequently the black hole2p

evaporates completely. In this second model the evaporation always ends with the same remnant geometry,Žirrespective of the initial state or the type of evaporation process thermal bath removal or gravitational

. w xcollapse . We find that this end-point geometry is everywhere regular, as in other models 2,3 for which theevaporation process can be followed analytically, suggesting an underlying general behaviour. Moreover the

Žexact solvability of these models can be used to analyze other physical aspects in an analytical setting critical.behaviour, thermality, etc . We will consider these questions and details of this work in a future publication.

Acknowledgements

J. C. acknowledges the Generalitat Valenciana for a FPI fellowship. J. C. and J. N-S. want to thank P.Navarro, J. M. Izquierdo and A. Mikovic for useful discussions. A. F. wishes to thank N. Kaloper for interestingremarks.

References

w x Ž .1 C.G. Callan, S.B. Giddings, J.A. Harvey, A. Strominger, Phys. Rev. D 45 1992 1005.w x Ž . Ž .2 J.G. Russo, L. Susskind, L. Thorlacius, Phys. Rev. D 46 1993 3444; Phys. Rev. D 47 1993 533.w x Ž . Ž .3 S. Bose, L. Parker, Y. Peleg, Phys. Rev. D 52 1995 3512; Phys. Rev. Lett. 76 1996 861.w x Ž .4 J. Cruz, J. Navarro-Salas, Phys. Lett. B 387 1996 51.w x Ž .5 R.B. Mann, Nucl. Phys. B 418 1994 231.w x Ž .6 M.O. Katanaev, W. Kummer, H. Liebl, Nucl. Phys. B 486 1997 353.w x Ž .7 K.C.K. Chan, gr-qcr9701029; K.C.K. Chan, J.D.E. Creighton, R.B. Mann, Phys. Rev. D 54 1996 3892; H. Liebl, D.V. Vassilevich,

Ž .S. Alexandrov, Class. Quant. Grav. 14 1997 889.w x Ž .8 R.B. Mann, Phys. Rev. D 47 1993 4438.w x Ž .9 J. Gegenberg, G. Kunstatter, D. Louis-Martinez, Phys. Rev. D 51 1995 1781.

w x Ž .10 M. Cadoni, Phys. Lett. B 395 1997 10.w x Ž .11 A. Fabbri, J.G. Russo, Phys. Rev. D 53 1996 6995.w x Ž .12 R. Balbinot, A. Fabbri, Class. Quant. Grav. 14 1997 463.

4 March 1999


Two-boundaries AdSrCFT correspondence in dilatonic gravity

Shin’ichi Nojiri a,1, Sergei D. Odintsov b,2

a Department of Mathematics and Physics, National Defence Academy, Hashirimizu, Yokosuka 239, Japanb Tomsk Pedagogical UniÕersity, 634041 Tomsk, Russia

Received 29 November 1998Editor: P.V. Landshoff

Abstract

Ž . Ž .We discuss dilatonic gravity bulk theory from the point of view of generalized AdSrCFT correspondence.Self-consistent dilatonic background is considered. It may be understood as two boundaries space where AdS boundary

Ž .appears as infinite boundary and new singular boundary occurs at short distances. The two-point correlation function andconformal dimension for minimal and dilaton coupled scalar are found. Even for minimal scalar, the conformal dimension isfound to be different on above two boundaries. Hence, new CFT appears in AdSrCFT correspondence at short distances.AdSrCFT correspondence may be understood as interpolating bulk theory between two different CFTs. q 1999 ElsevierScience B.V. All rights reserved.

w x Ž .AdSrCFT correspondence 1–3 gives an interesting framework to relate classical bulk theory with theconformal field theory living on the infinite boundary. In the original version of AdSrCFT correspondence, it

Ž . w xhas only one boundary AdS . However, there are some indications 4 that singularities which appear and inw xbulk and in boundary theories could mean the opening of a new space. For example, it happens 4 that singular

branes become regular in a dual, conformally rescaled, frame. That may indicate that there naturally appearsŽ . w xsecond boundary vacuum in AdSrCFT correspondence 4,5 . In other words, brane probably should not reside

w xonly at the infinite boundary of AdS 5 and it is better to imagine the brane is everywhere.In the present letter, we discuss AdSrCFT correspondence for dilatonic theories in the space with two

Žboundaries. We concentrate on the behaviour of correlation function of scalar field in different cases minimalŽ w x ..scalars, dilaton coupled scalars see 7 for introduction on new short distance boundary. It is shown that CFT

on this boundary is different from the one on infinite AdS boundary. The shift of conformal dimension due tomass and curvature coupling is also calculated. Finally, we give some remarks about another representation ofdilatonic gravity under discussion as higher derivative gravity.



( )S. Nojiri, S.D. OdintsoÕrPhysics Letters B 449 1999 39–4740

We start from the following action of dilatonic gravity in dq1 dimensions:

dq1 mn'Ssy d x yg RyLya g E fE f . 1Ž .Ž .H m n

In the following, we assume l2 'yL and a to be positive. From the variation of the metric g mn , we obtain 3

1 L 1rs0sR y g Rq g qa E fE fy g g E fE f 2Ž .mn mn mn m n mn r sž /2 2 2

and from that of dilaton f

mn'0sE yg g E f . 3Ž .Ž .m n

We assume that solutions for g and f depend only on one of the coordinate, say y'x dmn

g sg y , fsf y 4Ž . Ž . Ž .mn mn

and g has the following formmn

d dy12 m n 2 i jds s g dx dx s f y dy qg y h dx dx 5Ž . Ž . Ž .Ý Ýmn i j

m ,ns0 i , js0

Ž . Ž . Ž .Here h is the metric in the flat Lorentzian background. Then the equations of motion 2 and 3 take thei j

following forms:X 2 2d dy1 g l aŽ . 2X0sy q fq f 6Ž . Ž .ž /8 g 2 2

XX X X X 2 2dy1 g dy1 f g dy1 dy4 g l aŽ . Ž . 2X0sy q y q fy f 7Ž . Ž .ž /2 g 4 fg 8 g 2 2X

dgX0s f . 8Ž .(ž /f

X Ž . Ž . Ž . Ž . Ž .Here expresses the derivative with respect to y. Eq. 6 corresponds to m,n s d,d in 2 and Eq. 7 toŽ . Ž . Ž . Ž . Ž . Ž .m,n s i, j . The case of m,n s 0,i or i,0 is identically satisfied. Integrating 8 , we find

fX

f sc . 9Ž .( dg

Ž . Ž .Substituting 9 into 6 , we can solve it algebraically with respect to f :2Xd dy1 gŽ . Ž .

fs . 10Ž .2ac2 24 g l q dž /g

Ž . Ž . Ž .We find that Eq. 7 is automatically satisfied when we substitute 9 and 10 . Therefore g can be an arbitraryŽ .function of y but this corresponds to the degree of the freedom of the reparametrization of y in the metric 5 .

3 1mn l l h l h l h hnThe conventions of curvatures are given by R s g R ; R s y G q G y G G q G G ; G s gmn mn m l,k mk ,l m l kh mk lh m l 2

= g q g y g .Ž .mn ,l ln ,m ml ,n

( )S. Nojiri, S.D. OdintsoÕrPhysics Letters B 449 1999 39–47 41

We can fix it by choosing

gsy . 11Ž .Ž . Ž .Then we find from 9 and 10 ,

d dy1Ž .fs 12Ž .2ac

2 24 y l q dž /y

22 2° ¶d dy1 1 dy1 2ac 2acŽ . Ž . ~ •fsc dy sf q ln q1" q1 y1 . 13Ž .(H )0 2 d 2 d2 ž /¢ ß2 da l y l yacdq2 24 y l q) dž /y

Ž . dd dy1Ž .From Eq. 13 , we find the dilaton field behaves as f™. ln y when y™0. Since the string(a

coupling constant ef should be small in order that the supergravity picture is consistent, the dilaton field cannotŽ .grow up positively. Therefore the y sign should be chosen in the " sign in Eq. 13 and we find

1 d dy1Ž .f™ ln y . 14Ž .(

2 a

Ž . Ž . Ž .The metric given by 5 , 11 and 12 becomes that of the usual anti-de Sitter space in the limit where a or cvanishes.

Ž . Ž .The backreaction from the non-trivial background of dilaton 13 to the metric as in 12 changes thestructure of the spacetime near the region where ys0. When y is small, the metric behaves as

dy1d dy1Ž .2 dy2 2 i jds ; y dy qy h dx dx . 15Ž .Ý i j24ac i , js0

The metric tells that the distance l between the point with finite ysy and that of ys0 is finite:0

1 1dy2 dy d dy1 dy1Ž . Ž .ys0 0 2 22 2ls dss dy y s y . 16Ž .H H 02 2ž / ž /4ac 4dacysy 00

This should be compared with the case of usual anti-de Sitter space where the distance is infinite. In the limitŽ .where a or c vanishes, the distance l in 16 becomes infinite as expected. It ahould be also noted that there is

a curvature singularity at ys0 since the scalar curvature is given by

dq1 LŽ .2 yd 2 ydRsy yac y ;ac y . 17Ž .

dy1

The infinite boundary discussed in AdSrCFT correspondence lies at ys`. When y™`, f and f in theŽ . Ž .solution 12 , 13 behave as

d dy1Ž .ydf™ 1qOO y ,Ž .Ž .2 24l y

dyf™f 1qOO y . 18Ž .20 ž /ž /


Therefore the geometry of spacetime asymptotically approaches to that of AdS, which tells the correlationfunctions of matter fields on the boundary corresponding to ys` do not change with those on the boundary ofAdS.

After Wick rotating the spacetime signature by changing x 0 ™ ix 0, as an example, we consider free masslessscalar whose action is given by

1x dq1 mn'S s d x g g E xE x 19Ž .H m n2

and consider the correlation function in the neighborhood of the boundary ys` by solving the followingequation

dy2d 2 dy1yq1

2 2 2 22'g IG y , X ;A E y E G y , X q E G y , X s0 ,Ž . Ž . Ž .Ý0 y y iž / A0 is0

2 dy1l 2mn 2 i i' 'g I'E g g E , A '2 , X ' x yx . 20Ž .Ž .Ž . Ý(m n 0 1 2d dy1Ž . is0

The solution of above equation near the infinite boundary is given by

ydd 4

2 y y1 2 ydG y , X sG y y qX qOO y . 21Ž .Ž . Ž .20 2½ 5ž /A0

Ž . w xHere G is a constant. Eq. 21 gives a correlation function at ys` as in Ref. 2 . In the limit of y™`, we0

obtain

d yd2 y 2G ys0, X sG y X . 22Ž . Ž .Ž . 20

This correlation function is nothing but that of the operators with the conformal dimension d in some kind ofconformal field theory.

Since there is a singularity at ys0, we need to check if the fields at the boundary ys` have a uniqueŽ . 'extension in the bulk spacetime y-`. As an example, we consider free massless scalar in 19 . Since g is

given by

dy2d dy1 yŽ .'g s , 23Ž .2ac24 l q) dž /y

the square-integrability requires

dyx;o y when y™`4ž /

dyx;o y when y™0 . 24Ž .2ž /


If the Laplace equation I xs0 has a square integrable solution whose boundary value at ys` vanishes, theuniqueness is broken since we can add the solution to any given solution. The solution can be written using the

� i 4Fourier transformation with respect to the coordinates x , is1, PPP ,d as follows:

dy11i d ix y , x s d kf y exp i k x . 25Ž . Ž .Ž . ÝH k id ž /

is022pŽ .

Then the Laplace equation is rewritten as

dy1

2d 2 2(2 ac d dy1 k yq1 Ž .22'0s g I x™ E y l q E f y y f y . 26Ž . Ž . Ž .(y y k kd 2ž / 2y(d dy1Ž . ac

2l q( dy

) Ž . Ž Ž ..Multiplying f y the complex conjugate of f y and integrating with respect to y, we obtaink k

d° ¶dy1q1

22 2 2` (2 y ac d dy1 k yŽ .2 22~ •0s dy l q E f y q f yŽ . Ž .H ( y k kd 22y(d dy10 Ž . ac

2l q¢ ß( dy

2ac)y2 yf y E f y . 27Ž . Ž . Ž .( k y kd dy1Ž .

y™0

Ž . Ž Ž . . Ž . Ž .Here we assume Eq. 24 and that x vanishes at the boundary ys` f ` s0 . Eq. 27 tells that if f yk k

does not vanish at ys0 4, there can be non-trivial square-integrable solutions whose boundary value at ys`

Ž i.vanishes. This situation is very different from that in the usual AdS and the boundary value x ys`, x cannotŽ . Ž . Ž .uniquely determine the value of x in the bulk y-`. Eq. 27 tells, however, f y sE f y s0 everywherek y k

Ž Ž . .if x vanishes at ys0 f 0 s0 and there is no any non-trivial square-integrable solution. Note that thekŽ . Ž i.square-integrability requires x vanish at ys` due to 24 . This implies that the value x ys0, x at ys0

Ž .can uniquely determine the value of x in the bulk y)0. Since there is a curvature singularity at ys0 17 ,ys0 can be regarded as a boundary, which is similar to the case of Schwarzschild spacetime although the

5 Ž .singularity discussed here is naked . From the metric in 5 , the new boundary at ys0 has also the topologyof d-dimensional Minkowski space in the Minkowski signature. In the usual AdSrCFT correspondence, the

Ž .boundary at ys` can be regarded as a brane in superstring or M-theory. Since the solution 13 tells that thereŽis a source of dilaton at ys0, the object at ys0 could be considered as brane with a dilatonic hair. For

w x.classification of non-singular branes see 6 . If it is so, it is not so unnatural to expect that some kind ofŽconformal field theory is realized on the boundary at ys0 like it happened in the example of 3d AdS gravity

w x.in Ref. 5 .

4 Ž . Ž . Ž .More exactly, if f y ; yln y when y;0, the boundary term in 27 becomes finite. The more singular behaviour of f y can be' kŽ . Ž .consistent with the conditon of the square integrability in 24 . In such a case, the boundary term, and therefore the bulk integration, in 27

diverges.5 Ž . Ž . d 2 2 .If a -0, there is a horizon since f y in 12 diverges at y sy ac rl . In this case, the singularity is not naked.


Ž .We now consider the correlation function of free massless scalar whose action is given by 19 . We onlyconsider the neighborhood of the boundary y;0 and solve the following equation for correlation function

dy2 dy1y2 2 2 2'g IG y , X ;AE yE G y , X q E G y , X s0 ,Ž . Ž . Ž .Ž . Ýy y iA is0

2 dy1ac 2mn 2 i i' 'g I'E g g E , A'2 X ' x yx . 28Ž .Ž .Ž . Ý(m n 1 2d dy1Ž . is0

Ž .The solution of 28 near the boundary y;` is given byd

y24

2 dy1 2G y , X sG y qX . 29Ž .Ž . 0 2 2ž /dy1 AŽ .Ž . w xHere G is a constant. Eq. 29 would give a correlation function on the boundary 2 . In fact, in the limit of0

y™0, we obtaind

y2 2 2G ys0, X sG X . 30Ž . Ž .Ž . 0

Ž . Ž .The correlation function 30 which is different from one on the infinite boundary is nothing but that of thedoperators with the conformal dimension in some kind of conformal field theory. Especially when ds2, the2

Ž 1 2 .correlation function is that of the product of left-moving and right-moving free fermions OO x , x 'Ž . ) Ž ) .Ž 1 2 .c z c z zsx q ix .As a more general case, we consider the correlation function of massless dilaton coupled scalar whose action

is given by

1 ax dq1 mn'S s d x g exp 2b fyf g E xE x . 31Ž . Ž .H2 0 m n(2 d dy1Ž .

Ž .Here b is a parameter which is now introduced. From 14 , we find near the boundary

abexp 2b fyf ;y . 32Ž . Ž .0( d dy1Ž .

Then in order to find the correlation function in the neighborhood of the boundary y;0, we should solve theŽ .following equation instead of 28 :

bqdy1 dy1yf f 2 bq1 f 2 2 f 2'g I G y , X ;AE y E G y , X q E G y , X s0 ,Ž . Ž . Ž .Ž . Ýy y iA is0

af 2 b fyf mnŽ .0(' 'g I 'E e g g E . 33Ž .Ž .d dy1m nž /

Ž .The solution of 33 near the boundary y;0 is given byd b

y y2 dy14

f 2 dy1 2G y , X sG y qX . 34Ž .Ž . 0 2 2ž /dy1 AŽ .In the limit of y™0, we obtain

d by y

f 2 2 2 dy1G ys0, X sG X . 35Ž . Ž .Ž . 0


d bŽ .The correlation function 35 is that of the operators with the conformal dimension q . It is interesting2dy1

Ž .that the conformal dimension is shifted by the parameter b which comes from the dilaton coupling in 31 . Inw xthe usual AdSrCFT correspondence, such a shift comes from the mass of the scalar field 2 .

It would be also interesting to investigate the effect from the mass term. We add the following dilatonŽ .dependent mass term to the action 31 :

2m am dq1 2'S sy d x g exp 2g fyf x . 36Ž . Ž .H 0(2 d dy1Ž .

Ž .Then the equation coresponding to 28 is given bya

f 2 2g fyf m 2Ž .0('g I ym e G y , XŽ .Ž .d dy1ž /bqdy1 dy1 2y m

bq1 m 2 2 m 2 gqdy3 m 2;AE y E G y , X q E G y , X y y G y , X s0 . 37Ž .Ž . Ž . Ž .Ž . Ýy y iA Ais0

mŽ 2 .In order that G y, X corresponds to the correlation function of the conformal field theory on the boundary,mŽ 2 .G y, X should have the following form in the limit of y™0:

y r

mG ; . 38Ž .n2XŽ .Ž . Ž . Ž .Substituting 38 into 37 , we find that g in 36 cannot be arbitrary but g should be given by

gsbydq2 . 39Ž .Ž . Ž .When the Eq. 39 is satisfied, solution of 36 near the boundary y;0 is given by

2 2 2'b q4 m rAd2 2 2'yb" b q4 m rA y .2 dy14

2m 2 dy1 2G y , X sG y y qX . 40Ž .Ž . 0 2 2ž /dy1 AŽ .In the limit of y™0, we obtain

2 2 2' 2 2 2yb" b q4 m rA 'b q4 m rAdy .2m 2 2 2 dy1G ys0, X sG y X . 41Ž . Ž .Ž . 0

2 2 2b q4 m r A'dŽ .The correlation function 41 is that of the operators with the conformal dimension " .2dy1

w xWe can also consider the coupling of the matter scalar field x with the scalar curvature R 7 :2m a

R dq1 2'S sy d x g R exp 2d fyf x . 42Ž . Ž .H 0(2 d dy1Ž .Ž . Ž . Ž .Near the boundary y;0 the behavior of R is given by 17 . Comparing 17 with 32 , we can identify near the

boundary

da2R;ac exp y2 fyf . 43Ž . Ž .0( dy1Ž .

Therefore by the following replacement,

g™dyd , m2 ™m2ac2 44Ž .Ž . Ž . Ž .we can use the results in 39 , 40 and 41 .


Ž .Finally we show that the gravity theory in 1 can be rewritten as a higher derivative gravity theory withoutdilaton. When we rescale the metric by

g ™e rg , 45Ž .mn mn

Ž .the action 1 is rewritten after the partial integration as follows

dy1dy1 dr Ž .

dq1 r mn mn2'Ssy d x yg e RyLe q g E rE rya g E fE f . 46Ž .H m n m nž /4

Choosing r as

ars2 f 47Ž .( d dy1Ž .

we obtain

dq1 dy1 2'Ssy d x yg F RyLF . 48� 4Ž . Ž .HHere

aF'exp y f . 49Ž .(ž /d dy1Ž .

By using the equation of motion with respect to F , we can solve F with respect to the scalar curvature R:

dy12F s R . 50Ž .

dq1 LŽ .dq 1Ž . Ž .Substituting 50 into 48 , we obtain the higher derivative gravity theory which contains -power of the2

scalar curvature R:

dq1dy12 dy1 2dq1 'Ssy y d x yg yR . 51Ž . Ž .Hž /dq1 dq1 LŽ .

In the usual AdSrCFT correspondence, we believe that the conformal symmetry on the boundary manifoldŽ . Ž . Ž . Ž .comes from the SO d,2 symmetry in the anti-de Sitter space but the spacetime given by 5 , 11 and 12 has

Ž . Ž . Ž . Ž .no the SO d,2 symmetry. The obtained correlation functions 30 , 35 and 41 , however, seem to be those inŽ .some kind of conformal field theory. We should note that the action 51 is invariant under the global scale

transformation with a constant parameter c

g ™ecg , 52Ž .mn mn

which might be the origin of the conformal symmetry on the boundary.Ž .In summary, we discussed CFT which appears on short distance ys0 boundary of dilatonic spacetime

under consideration in generalized AdSrCFT correspondence. Dilatonic gravity may be considered as bulktheory interpolating between two different CFTs living at the boundaries ys0 and ys`. It could be really

Ž .interesting to study AdSrCFT correspondence for the spacetimes with second boundary of above sort inw x Ž .Ns4 conformal supergravity 9–11 more exactly gauged supergravity where at ys` boundary NNs4

w xsuper Yang-Mills theory lies 8 as dilaton naturally appears there.


Acknowledgements

We would like to acknowledge helpful discussions with T. Hollowood, P. van Nieuwenhuizen and E.Mottola. The work by SDO has been partially supported by T-8, LANL and RFBR project n96-02-16017.

References

w x Ž .1 J.M. Maldacena, Adv. Theor. Math. Phys. 2 1998 231. hep-thr9711200.w x Ž .2 E. Witten, Adv. Theor. Math. Phys. 2 1998 505. hep-thr9802150.w x Ž .3 S. Gubser, I. Klebanov, A. Polyakov, Phys. Lett. B 428 1998 105. hep-thr9802109.w x4 H.J. Boonstra, K. Skenderis, P.K. Townsend, hep-thr9807137.w x5 K. Behrndt, hep-thr9809015.w x Ž .6 G.W. Gibbons, G.T. Horowitz, P.K. Townsend, Class. Quant. Grav. 12 1995 287.w x Ž .7 S. Nojiri, S.D. Odintsov, Phys. Rev. D 57 1998 2363.w x8 H. Liu, A.A. Tseytlin, hep-thr9804083.w x Ž .9 M. Kaku, P.K. Townsend, P. van Nieuwenhuisen, Phys. Rev. D 17 1978 3179.

w x Ž .10 E. Bergshoeff, M. de Roo, B. de Wit, Nucl. Phys. B 182 1981 233.w x Ž .11 E.S. Fradkin, A.A. Tseytlin, Phys. Repts. 119 1985 233.

4 March 1999


Extended superconformal algebras on AdS3

Katsushi ItoYukawa Institute for Theoretical Physics, Kyoto UniÕersity, Kyoto 606-8502, Japan

Received 6 November 1998; revised 19 December 1998Editor: M. Cvetic

Abstract

We study a supersymmetric extension of the Virasoro algebra on the boundary of the anti-de Sitter space-time AdS .3Ž < .Ž1. Ž < .Ž1.Using the free field realization of the currents, we show that the world-sheet affine Lie superalgebras osp 1 2 , sl 1 2

Ž < .Ž1.and sl 2 2 provide the boundary NNs1,2 and 4 extended superconformal algebras, respectively. q 1999 Elsevier ScienceB.V. All rights reserved.

The duality between the type IIB string theory on AdS =S3 =M 4, where M 4 sK 3 or T 4, and two-dimen-34 w xsional NNs4 superconformal field theory on a symmetric product of M 1–3 , is one of interesting examples

w xof the AdSrCFT correspondence 4 . Conformal symmetry on the boundary of AdS , firstly introduced by3w xBrown and Henneaux 5 , is realized as the chiral algebra of the boundary conformal field theory associated with

Ž . w x w xthe SL 2, R Chern-Simons theory 6 . On the other hand, Giveon et al. 3 constructed the boundary Virasoroalgebra from the string theory on AdS . The generators of the algebra are identified as the global charges3

Ž . w xassociated with the world-sheet sl 2, R current algebra, which is expressed in terms of the free fields 7 . TheyŽ . 3 4also constructed a part of the NNs4 superconformal algebra from the superstrings on AdS =S =T . Their3

boundary algebra, however, is not manifestly supersymmetric since the supercurrents are introduced by thew xbosonization of the world-sheet fermions and the algebra is defined up to the picture changing operator 8 .

In the present paper, we study the boundary extended superconformal algebra realized in a manifestlyŽ .supersymmetric way. By replacing the world-sheet affine Lie algebra sl 2, R to an affine Lie superalgebra

Ž .which includes the subalgebra sl 2, R and employing the free field realization of the currents, we will obtainthe extended superconformal algebra which acts on the boundary AdS superspace. In particular, we will show3

Ž < .Ž1. Ž < .Ž1. Ž < .Ž1.that the world-sheet affine Lie superalgebras osp 1 2 , sl 1 2 and sl 2 2 provide the boundary NNs1,2and 4 extended superconformal algebras, respectively.

We begin with reviewing the boundary Virasoro algebra associated with AdS . The anti-de Sitter space AdS3 32 2 2 2 2 2,2 Ž .is the hypersurface yU yV qX qY syl embedded in the flat space R . In the coordinates r,t ,f

defined by Us lcosh rsint , Vs lcosh rcost , Xs lsinh rcosf, Ys lcosh rsinf, the metric is given by

ds22 2 2 2 2sycosh r dt qsinh r df qdr . 1Ž .2l


( )K. ItorPhysics Letters B 449 1999 48–55 49

Ž . Ž . Ž . w xIn the AdS space, there is SL 2, R =SL 2, R symmetry. The generators of SL 2, R read 13 L R L

1 i" iuL s iE , L s ie coth2 rE y E . E , 2Ž .0 u "1 u Õ rž /sinh2 r 2

Ž .where ustqf, Õstyf. SL 2, R is generated by L , which are obtained by ulÕ. Define theR 0,"1˜Ž .coordinates f,g ,g by

fs lcosh ry it ,iu iÕgs tanh re , gs tanh re . 3Ž .

Ž .The metric 1 becomes˜2 2 2 2f˜ds s l df qe dg dg . 4Ž .Ž .

Ž .The SL 2, R generators areL

1 ˜L s EfygE , L syE ,0 g y1 g2

˜2 2 y2f˜L sgEfyg E q l e E . 5Ž .1 g g

˜ Ž .Thus in the limit f™`, the derivative with respect to g in L , decouples and the SL 2, R generators become1

holomorphic with respect to g . 1 These currents are regarded as the zero-mode part of the Wakimoto realizationw x Ž .7 of the affine Lie algebra sl 2, R at level k:

'H z syi 2 a Ewq2gb ,Ž . q

2'J z sb , J z s i 2 a Ewgyg bykEg , 6Ž . Ž . Ž .y q q

' Ž Ž . Ž .. Ž .where a s ykq2 . b z ,g z is the bosonic ghost system with conformal weights 1,0 and has theqŽ . Ž . Ž . Ž . Ž .operator product expansion OPE b z g w s1r zyw q PPP .w z is a free boson with the OPE

Ž . Ž . Ž .w z w w sylog zyw q PPP . Then

dz dz1L sy H z , L s" J z , 7Ž . Ž . Ž .E E0 "1 "2 2p i 2p i

Ž . w xgenerate sl 2, R . For integer n, it is shown in Ref. 3 that the generators

dzL s LL z , 8Ž . Ž .En n2p i

where

1yn2 n ny1 n nq1Ž . Ž .n nq1 ny1LL z sy Hg y J g q J g , 9Ž . Ž .n y q2 2 2

satisfy the Virasoro algebrac

3w xL , L s myn L q m ym d , 10Ž . Ž . Ž .m n mqn mqn ,012Ž .with central charge cs6kp. Here the contour integral in 8 is taken around zs0 and p is defined by

dz Egp' . 11Ž .E

2p i g

Ž .We now consider an NN supersymmetric extension of this Virasoro algebra 10 , which is realized in theŽ .superspace g ,j , PPP ,j . Here j , PPP ,j denote the Grassmann odd coordinates. The supercharges are1 NN 1 NN

1 Here we consider the Euclidean version of the theory, which is obtained by replacing t by y it .E

( )K. ItorPhysics Letters B 449 1999 48–5550

realized as the conserved charges of the anti-commuting currents with spin one on the world-sheet, whose zeroŽ < . Ž < . Ž < . Ž < .modes should obey the Lie superalgebras osp 1 2 , osp 2 2 ssl 1 2 and sl 2 2 for NNs1,2,4 supersymme-

tries respectively. Hence we expect that conformal field theory whose chiral algebra is the affine LieŽ .superalgebra with a subalgebra sl 2, R , provides a natural supersymmetric generalization of the boundary

conformal symmetry. In the following we will construct some boundary superconformal algebras explicitly. TheŽ .Ž . iŽ .Ž .affine Lie superalgebra g is generated by the bosonic currents J z agD , H z is1, PPP ,r anda 0

Ž .Ž . iŽ . Ž . Ž .fermionic currents j z bgD . Here H z are the Cartan currents and D D the set of even odd roots.b 1 0 1w xThe free field representation 9 of an affine Lie superalgebra with rank r is obtained by r free bosons

iŽ .Ž . Ž Ž . Ž .. Ž .f z is1, PPP ,r , pairs of bosonic ghosts b z ,g z with conformal weights 1,0 for positive even rootsa a

Ž Ž . Ž .. Ž .a and pairs of fermionic ghosts h z ,j z with conformal weights 1,0 for positive odd roots a . The OPEsa aiŽ . jŽ . Ž . Ž . Ž . Ž . Ž . Ž .X X Xfor these fields are f z f w syd log zyw q PPP , b z g w sd r zyw q PPP , h z j wi j a a a ,a a a

Ž .Xsd r zyw q PPP .a ,aŽ < .Firstly we consider the NNs1 superconformal symmetry. The Lie superalgebra osp 1 2 contains odd roots

2 Ž < .Ž1."a and even roots "2a with a s1. The free field realization of affine Lie superalgebra osp 1 2 at1 1 1

level yk is

j z s y2kq2 Ejq2 ia jEwqghy2jgb ,Ž . Ž .a q1

yk2J z s Egy ykq2 jEjq ia gEwyg bygjh ,Ž . Ž .2 a q1 2

j z shy2jb , J z sb ,Ž . Ž .ya y2 a1 1

a PH z syia Ewq2gbqjh , 12Ž . Ž .1 q

'Ž . Ž . Ž . Ž . Ž .where h,j s h ,j , b ,g s b ,g and a s ykq3 . This algebra contains the sl 2, R currenta a 2 a 2 a q1 1 1 1

Ž . Ž . Ž . Ž . Ž . Ž .algebra which is generated by H z sa PH z , J z sJ z . Then the generators LL z defined by 91 " " 2 a n1

Ž .satisfy the Virasoro algebra 10 . Concerning the supercurrent, let us introduce the conserved charges

1 dzG s j z . 13Ž . Ž .E"1r2 " a1' 2p i2

Ž . Ž < .From the OPE for the currents, it is shown that L ns0,"1 and G satisfy the Lie superalgebra osp 1 2 .n "1r2Ž .Higher modes of the supercurrent are obtained by extracting the simple pole term in the OPE between LL zn

Ž . Ž .and j z , which is denoted by GG z . The result isya ny1r21

n ny1ny1 n ny1 ny2 n'GG z s j g y j g y 2 n ny1 Hg jqJ g jyJ g j . 14Ž . Ž . Ž .Ž .ny1r2 a ya q y1 1' '2 2

We define the generators G byny1r2

dzG s GG z . 15Ž . Ž .Eny1r2 ny1r22p i

It is shown that the generators L and G obey the NNs1 superconformal algebra in the NS sector:n r

c3w xL , L s myn L q m ym d ,Ž . Ž .m n mqn mqn ,012

nw xL ,G s yr G ,n r nqrž /2

c1 12� 4G ,G s2 L q r y d , n ,mgZ, r ,sgZq 16Ž .Ž .Ž .r s rqs rqs ,04 23


Ž .with the central charge cs6kp. For example, the third eq. in 16 follows from OPEs

4 J w y4 J wŽ . Ž .2 a y2 a1 1j z j w s q PPP , j z j w s q PPP ,Ž . Ž . Ž . Ž .a a ya ya1 1 1 1zyw zyw

2k y2a PH wŽ .1j z j w s q q PPP ,Ž . Ž .a ya 21 1 zywzywŽ .

y2gj w g wŽ . Ž .j z g w s q PPP , j z j w s q PPP ,Ž . Ž . Ž . Ž .a a1 1zyw zyw

y2j w 1Ž .j z g w s q PPP , j z j w s q PPP , 17Ž . Ž . Ž . Ž . Ž .ya ya1 1zyw zyw

Ž ny1 ny2 n. ny2 Ž .and Hg qJ g yJ g jsykr2g Egj . Note that the third term in 14 is necessary to obtain theq ycorrect superconformal algebra, although in the bosonic case the combination yHgyJ qJ g 2 becomes theq ytotal derivative and may be regarded as the longitudinal part. Let us take the zero mode part of the supercurrentand the Virasoro generator. Regarding b and h as the derivatives ErEg and ErEj respectively, we find

1 E En nG ; g y2g j ,ny1r2 ž /' Ej Eg2

E nq1 Enq1 nL ;yg y g j . 18Ž .n Eg 2 Ej

Thus, in this classical limit, the NNs1 superconformal algebra is realized on the superspace with coordinatesŽ .g ,j .

Now we generalize this result to the case of NNs2,4 extended superconformal algebras by consideringŽ < .Ž1. Ž < .Ž1. Ž < .affine Lie superalgebras sl 1 2 , sl 2 2 , respectively. The Lie superalgebra sl N M has even roots

Ž .Ž . Ž .Ž . Ž .Ž" e y e 1 F i - j F N , " d y d 1 F a - b F M and odd roots " e y d i s 1, PPP , N, a si j a b i a. Ž . Ž . Ž .1, PPP , M . Here e d are the orthonormal basis with positive negative metric e Pe sd d Pd syd .i a i j i j a b ab

Ž . ŽWe take the simple roots as a se ye , is1, PPP , Ny1 , a se yd , a sd yd asi i iq1 N N 1 Nqa a aq1. Ž < .Ž1.1, PPP , My1 . The OPEs for the currents of the affine Lie superalgebra sl N M at level k are given by

J z J wŽ . Ž .e ye e yei j k l

d 1yd J w yd 1yd J w qd d e ye PH wkd d Ž . Ž . Ž . Ž . Ž . Ž .j ,k i , l e ye i , l k , i e ye j ,k i l i jj ,k i , l i l k is q q PPP ,2 zywzywŽ .

J z J wŽ . Ž .d yd d yda b c d

d 1yd J w yd 1yd J w yd d d yd PH wykd d Ž . Ž . Ž . Ž . Ž . Ž .d ,a c ,b d yd b ,c a ,d d yd b ,c a ,d a bb ,c a ,d c b a ds q2 zywzywŽ .q PPP ,

j z j wŽ . Ž .e yd d yei a b j

d 1yd J w qd 1yd J w yd d d ye PH wŽ . Ž . Ž . Ž . Ž .Ž .ykd d a ,b i , j e ye i , j a ,b d yd i , j a ,b a ii , j a ,b i j b as y2 zywzywŽ .q PPP ,


d j wŽ .j ,k e ydi aJ z j w s q PPP ,Ž . Ž .e ye e ydi j k a zyw

yd j wŽ .i ,k d yda jJ z j w s q PPP ,Ž . Ž .e ye d ydi j a k zyw

d j wŽ .a ,c e ydk bJ z j w s q PPP ,Ž . Ž .d yd e yda b k c zyw

yd j wŽ .b ,c d yea kJ z j w s q PPP ,Ž . Ž .d yd d yea b c k zyw

kd i ji jH z H w s q PPP ,Ž . Ž . 2zywŽ .

a iJ w a i j wŽ . Ž .a ai iH z J w s q PPP , H z j w s q PPP . 19Ž . Ž . Ž . Ž . Ž .a azyw zyw

Ž . Ž . Ž . Ž . Ž .For Ms2, H z sa PH z , J z sJ z become the currents of the sl 2, R subalgebra. Then itNq1 " " a Nq 1

Ž . Ž . Ž .Ž1.Ž Ž .Ž1.will be shown that 9 for gsg obeys the Virasoro algebra 10 . The other even subalgebra u N sl 2aNq 1

.for Ns2 turns out to be an affine Lie algebra symmetry on the boundary.Ž Ž < .. Ž < .Ž1.Let us consider the NNs2 sl 1 2 case. The free field realization of the affine Lie superalgebra sl 1 2 at

level k is given by

j z sh , J z sb yj h , j z sh ,Ž . Ž . Ž .ya a ya a a a qa ya ya a qa1 1 2 2 1 1 2 1 2 1 2

j z skEj q ia a PEwj y j qj g b yj j h ,Ž . Ž .a a q 1 a a qa a a a a a qa a qa1 1 1 1 2 1 2 2 1 1 2 1 2

J z sy kq1 Eg y ia a PEwg yg 2 b yj h ,Ž . Ž .a a q 2 a a a a qa a2 2 2 2 2 1 2 1

j z skEj q kq1 Eg j q ia a qa PEwj q ia a PEwg jŽ . Ž . Ž .Ž .a qa a qa a a q 1 2 a qa q 2 a a1 2 1 2 2 1 1 2 2 1

qj j h qj g b qj g 2 b ,a a qa a a qa a a a a a1 1 2 1 1 2 2 2 1 2 2

ii i i iH z syia Ew qa j h qa g b q a qa j h , 20Ž . Ž . Ž .q 1 a a 2 a a 1 2 a qa a qa1 1 2 2 1 2 1 2

'where a s ky1 . Let us introduce the operatorq

1yn2 n ny1 n nq1Ž . Ž .n nq1 ny1LL z s a PHg y J g q J g , 21Ž . Ž .n 2 ya a2 22 2 2

which corresponds to the Virasoro generator. Here gsg . As in the case of NNs1, from the OPEs betweena2

Ž . Ž .LL z and j w , one may construct the operators corresponding to the supercurrents:n " a1

GG z sy ny1 j g n qnj g ny1 ,Ž . Ž .ny1r2 ya ya ya1 2 1

n ny1GG z sy ny1 j g ynj gŽ . Ž .ny1r2 a a qa1 1 2

yn ny1 ya PHg ny1j qJ g nj yJ g ny2j . 22Ž . Ž .Ž .2 a qa ya a qa a a qa1 2 2 1 2 2 1 2

Ž . Ž .By extracting the simple pole term in the OPE between GG z and GG w , one may read off the operatorny1r2 1r2Ž . Ž .TT z , which corresponds to the U 1 current. The result isn

n ˜ ny1TT z sy a q2a PHg ynj h g , 23Ž . Ž . Ž .˜n 1 2 a qa a1 2 1


˜where j sj qg j , h sh yg h . After introducing these operators, we can show that the˜a qa a qa a a a a a a qa1 2 1 2 2 1 1 1 2 1 2

generators

dz dzL s LL z , T s TT z , ngZŽ . Ž . Ž .E En n n n2p i 2p i

dz dz1G s GG z , G s GG z , rgZq 24Ž . Ž . Ž .Ž .E Er r r r 22p i 2p i

satisfy the NNs2 superconformal algebra with central charge cs6kp:c

3w xL , L s myn L q m ym d ,Ž . Ž .m n mqn mqn ,012m m

w xL ,G s yr G , L ,G s yr G ,m r mqr m r mqrž / ž /2 2

w xL ,T synT ,m n mqn

w xT ,G sG , T ,G sG ,m r mqr m r mqr

cw xT ,T s md ,m n mqn ,03

� 4G ,G s G ,G s0,� 4r s r s

c1 12G ,G sL q rys T q r y d , 25Ž . Ž .� 4 Ž .r s rqs rqs rqs ,02 46

where n,mgZ and r,sgZq1r2.Ž Ž < ..Similar construction can be done for NNs4 sl 2 2 case. In terms of free fields, the currents of the affine

Ž < .Ž1.Lie superalgebra sl 2 2 at level k are given by

J z sb , j z sh yg h ,Ž . Ž .ya a ya a a a qa1 1 2 2 1 1 2

J z sb yj h yj h , j z sh ,Ž . Ž .ya a a a qa a qa a qa qa ya ya a qa3 3 2 2 3 1 2 1 2 3 1 2 1 2

j z sh yg h , j z sh ,Ž . Ž .ya ya a qa a a qa qa ya ya ya a qa qa2 3 2 3 3 1 2 3 1 2 3 1 2 3

J z skEg q ia a PEwg yg 2 b qg j h yg j h qg j hŽ .a a q 1 a a a a a a a a qa a qa a a qa a qa1 1 1 1 1 1 2 2 1 1 2 1 2 1 2 3 2 3

yg j h yj h yj h ,a a qa qa a qa qa a qa a a qa qa a qa1 1 2 3 1 2 3 1 2 2 1 2 3 2 3

j z s kq1 Ej q ia a PEwj qj b yj g b yj j h qj b ,Ž . Ž .a a q 2 a a qa a a a a a a qa a qa a qa a2 2 2 1 2 1 2 3 3 2 2 3 2 3 2 3 3

J z sy kq2 Eg y ia a PEwyg 2 b qj h qj h ,Ž . Ž .a a q 3 a a a qa a a qa qa a qa3 3 3 3 2 3 2 1 2 3 2 3

ii i i i iH z syia Ew qa g b qa g b qa j h q a qa j hŽ . Ž .q 1 a a 3 a a 2 a a 1 2 a qa a qa1 1 3 3 2 2 1 2 1 2

i iq a qa j h q a qa qa j h , 26Ž . Ž . Ž .2 3 a qa a qa 1 2 3 a qa qa a qa qa2 3 2 3 1 2 3 1 2 3

'where a s k . The currents for positive non-simple roots are obtained from those for the simple roots.qIntroduce the operators

1yn2 n ny1 n nq1Ž . Ž .n nq1 ny1LL z s a PHg y J g q J g ,Ž .n 3 ya a3 32 2 2

GG1 z sy ny1 j g n qnj g ny1 ,Ž . Ž .ny1r2 ya ya ya ya ya1 2 1 2 3

GG 2 z sy ny1 j g n qnj g ny1 ,Ž . Ž .ny1r2 ya ya ya2 2 3


1 n ny1GG z sy ny1 j g ynj gŽ . Ž .ny1r2 a qa a qa qa1 2 1 2 3

ny1˜ n˜ ny2˜yn ny1 a PHg j qJ g j yJ g j ,Ž . 3 a qa qa ya a qa qa a a qa qaž /1 2 3 3 1 2 3 3 1 2 3

2 n ny1GG z sy ny1 j g ynj gŽ . Ž .ny1r2 a a qa2 2 3

ny1˜ n˜ ny2˜yn ny1 a PHg j qJ g j yJ g j ,Ž . 3 a qa ya a qa a a qaž /2 3 3 2 3 3 2 3

q n ˜ ny1TT z sJ g qnj h g ,Ž . ñ a a qa qa a1 1 2 3 2

y n ˜ ny1TT z sJ g qnj h g ,Ž . ñ ya a qa a qa1 2 3 1 2

n n10 n ny1 ny1˜ ˜TT z s a PHg q j h g y j h g , 27Ž . Ž .˜ ñ 1 a qa qa a qa a qa a2 1 2 3 1 2 2 3 22 2

where gsg anda3

j sj qg j ,a qa a qa a a2 3 2 3 3 3

j sj qg j qg g j qg j ,a qa qa a qa qa a a qa a a a a a qa1 2 3 1 2 3 1 2 3 1 3 2 3 1 2

h sh yg h ,ã qa a qa a a qa qa1 2 1 2 3 1 2 3

h sh yg h qg g h yg h . 28Ž .ã a a a qa a a a qa qa a a qa2 2 1 1 2 1 3 1 2 3 3 2 3

Then the generators

dzL s LL z ,Ž .En n2p i

dz dza a a aG s GG z , G s GG z ,Ž . Ž .E Er r r r2p i 2p i

dz dz1" 1 2 " 0 0T sT " iT s TT z , T s TT z , ngZ,rgZq 29Ž . Ž . Ž .Ž .E En n n n n n 22p i 2p i

can be shown to satisfy NNs4 superconformal algebra with central charge cs6kp:c

3w xL , L s myn L q m ym d ,Ž . Ž .m n mqn mqn ,012i iL ,T synT ,m n mqn

m ma a a aL ,G s yr G , L ,G s yr G ,m r mqr m r mqrž / ž /2 2

ci j i jk k i jT ,T s i´ T q md d ,m n mqn mqn ,012

a a1 1i a i b i a ) i bT ,G sy s G , T ,G s s G ,Ž . Ž .b bm r mqr m r mqr2 2

a b a bG ,G s G ,G s0,� 4 � 4r s r s

ca b ab i i 2 abG ,G sd L y rys s T q 4r y1 d d , 30Ž . Ž . Ž . Ž .� 4 abr s rqs rqs rqs ,06

where i, j,ks1,2,3, a,bs1,2, n,mgZ and r,sgZq1r2.


In the present work, we have constructed boundary extended superconformal symmetry from the world-sheetŽ . w xaffine Lie superalgebra with a subalgebra sl 2, R . The Wakimoto type free field realization 9 is shown to be

useful for the construction of manifestly supersymmetric extended superconformal algebras. We have writedown only the NS sector of superconformal algebras. The R-sector of the algebra is obtained by the spectral

w xflow 10 in the case of NNs2,4 extended superconformal algebras. For NNs1 case, it is necessary to find theworld-sheet current which represents the supercurrent in the R-sector.

The present construction would be generalized to other affine Lie superalgebras associated with AdS Lie3w xsupergroups 11 . The corresponding conformal algebras would be linearly extended superconformal algebras.

On the other hand, the Chern-Simons theories based on general AdS Lie supergroups lead to non-linearly3w xextended superconformal algebras constructed in Ref. 12 . Hence it is an interesting problem to examine the

boundary conformal field theory for general AdS supergroups. Finally, in order to relate the present34 w xconstruction to the type IIB string theory on AdS =S =T 13 or K 3, we need to clarify the hidden affine3 3

Lie superalgebra symmetry in the type IIB theory. These subjects will be discussed elsewhere.

Acknowledgements

This work is supported in part by the Grant-in-Aid from the Ministry of Education, Science and Culture,Ž .Priority Area: ‘‘Supersymmetry and Unified Theory of Elementary Particles’’ a707 .

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hep-thr9802150.w x Ž .5 J.D. Brown, M. Henneaux, Comm. Math. Phys. 104 1986 207.w x Ž .6 O. Coussaert, M. Henneaux, P. van Driel, Class. Quant. Grav. 12 1995 2961; M. Banados, K. Bautier, O. Coussaert, M. Henneaux,˜

Ž .M. Ortiz, Phys. Rev. D 58 1998 085020, hep-thr9805162; M. Banados, M.E. Ortiz, hep-thr980689; K. Behrndt, I. Brunner, I.˜Gaida, hep-thr9806195.

w x Ž . Ž .7 M. Wakimoto, Commun. Math. Phys. 92 1984 455; D. Bernard, G. Felder, Commun. Math. Phys. 127 1990 145.w x Ž .8 D. Friedan, E. Martinec, S. Shenker, Nucl. Phys. B 271 1986 93.w x Ž . Ž . Ž .9 M. Bershadsky, H. Ooguri, Phys. Lett. B 229 1989 374; K. Ito, Int. J. Mod. Phys. A 1992 4885; Phys. Lett. B 259 1991 73; J.

Ž .Rasmussen, Nucl. Phys. B 510 1998 688, hep-thr9706091.w x Ž .10 A. Schwimmer, N. Seiberg, Phys. Lett. B 184 1987 191.w x Ž .11 M. Gunaydin, G. Sierra, P.K. Townsend, Nucl. Phys. B 274 1986 429.¨w x Ž . Ž . Ž .12 V. Knizhnik, Theor. Math. Phys. 66 1986 68; M. Bershadsky, Phys. Lett. B 174 1986 285; K. Schoutens, Nucl. Phys. B 314 1989

Ž . Ž .519; P. Bowcock, Nucl. Phys. B 381 1992 415; K. Ito, J.O. Madsen, J.L. Petersen, Nucl. Phys. B 398 1993 425; Phys. Lett. B 318Ž . Ž .1993 315; E.S. Fradkin, V. Ya Linetsky, Phys. Lett. B 282 1992 352; A. Sevrin, K. Thielmans, W. Troost, Nucl. Phys. B 407Ž .1993 459.

w x13 I. Pesando, hep-thr9809145; J. Rahmfeld, A. Rajaraman, hep-thr9809164.

4 March 1999


Invariant wave equations, propagatorsand generalized coherent states over de Sitter space

S.A. Pol’shin 1

Department of Physics, KharkoÕ State UniÕersity, SÕobody Sq., 4, KharkoÕ 310077, Ukraine

Received 16 December 1998Editor: P.V. Landshoff

Abstract

The invariant wave equations in the de Sitter space for spin 0 and 1r2 massive particles are constructed. The explicitsolutions of these equations are obtained in the form of so-called ‘‘plane waves’’ without use of the separation of variablesmethod. The connection with the generalized coherent states for the de Sitter group is established. Invariant spin 0 and 1r2

Ž .two-point functions are constructed which are the anti commutators of the corresponding secondly quantized fields. q 1999Elsevier Science B.V. All rights reserved.

Keywords: Invariant wave equations; Plane waves; Generalized coherent states; de Sitter space; Conformal transformations; Two-pointfunctions

The construction of the quantum field theory inŽ .the de Sitter space dS is the important part of the

construction of the consistent quantum field theoryw xin curved space-time 1 . In particular, a great atten-

tion was paid to the construction of the invariantpropagators for the various spin particles in the dS

w x w xspace 2 . In the recent paper 3 the new method ofthe construction of the two-point functions for thespin zero particles over dS space was proposed. Thismethod is based on the employing of the ‘‘planewaves’’ which are solutions of the corresponding

ŽdS-invariant wave equation Klein-Gordon equation.in this case . The advantage of this approach is the

closed connection of the obtained two-point func-


Ž .tions with the anti commutator of the correspondingsecondly quantized fields.

In the present paper this approach is generalizedto the massive spin 1r2 particles on the dS space,and the new interpretation is given for the case ofspin zero particles using the generalized coherent

Ž .states CS for the de Sitter group. We start from theirreducible representations of the dS group and con-struct the dS-invariant equations for the massive spin0 and 1r2 particles performing the restriction to thespaces of the corresponding irreducible representa-tions. The results of preceding works on this subjectŽ w x .see 4 and references therein are revised and essen-tially corrected. Solutions of the dS-invariant Diracequation in the form of ‘‘plane waves’’ are obtained,which generalize the ‘‘plane waves’’ for spinlessparticles. It is shown that the ‘‘plane waves’’ forspinless particles within the coordinate-independent


( )S.A. Pol’shinrPhysics Letters B 449 1999 56–59 57

multiplier coincide with the CS for the coset spacewhich is the dS space; earlier the CS for the Anti-deSitter group were constructed, which correspond to

Ž . Ž . Ž . w xthe coset space SO 3,2 rSO 3 mSO 2 5 . It isshown that the two-point functions for the spin zero

w xparticles studied in Refs. 3,6 are the scalar productof two CS. The four-parametric invariance of the‘‘plane waves’’ for the spin 1r2 particles is found;using this property the invariant spinor two-pointfunctions are constructed.

Indices m,n , . . . run values 0 up to 3; indicesi,k, . . . run values 1,2,3. The four-vector indices maybe omitted; then the dot P denotes the scalar product

Žusing the Galilean metric tensor h sdiag q1,mn

.y1,y1,y1 . The 3-vectors are picked out by thebold type.

The de Sitter space is the four-dimensional hyper-sphere in the fictitious five-dimensional space withthe pseudoeuclidean metric of signatureŽ .qyyyy . The equation determining the hyper-

Ž 5.2 2sphere is xPxy x syR , where R is the ra-dius of hypersphere. The internal metric reads

x m xn x m xn

mn mng sh y , g sh q ,mn mn 2 2 2R x R

Ž 2 .1r2where xs 1qxPxrR . The symmetry groupŽ .of the de Sitter space is SO 4,1 ; its irreducible

infinite-dimensional representations are well-knownw x7 . They are denoted as p and are determined bym , s

the two numbers m and ss0,1r2,1, . . . , where s isspin; m corresponds to mass. If s is integer thenm2 )0; if s is half-integer then m2 )1r4R2; ifss0 then m2 )y2rR2. We introduce the magni-

1 1r22tude ms m y ; it is real if ss1r2.2Ž .4R

The Klein-Gordon equation describing confor-mally coupled field of a mass m

Icq m2 q2 Ry2 cs0 ,Ž .y1r2 1r2 mn

Is yg E yg g E , 1Ž . Ž . Ž .Ž .m n

corresponds to the representation p for spinlessm ,0

particles.The Dirac equation

2 imiG E cy my cs0 ,m ž /R

1m m m nG sxg q g ,g x . 2Ž .n2 R

corresponds to the representation p for spinm ,1r2w x1r2 particles 12 . One can show that the above

equation may be reduced to the general-covariantDirac equation with the vierbein

x m xn

Ž m .n mn n Ž m .r nre sh q , e e sgŽ m .2R xq1Ž .using the transformation

1qgP´ x m

mVs , ´ s . 3Ž .1r2 R xq1Ž .1y´P´Ž .Ž .Squaring Eq. 2 gives that the Klein-Gordon equa-

tion

im 22

Iqm y q cs02ž /R R

for the spin 1r2 particles is satisfied too. It is easy toŽ .show that Eq. 1 is satisfied by the functions

Ž0."Ž .w x;s , wherek 0

skPx3Ž0."w x ;s s x" , s syimRyŽ .k 0 2ž /R

4Ž .

and k m is a constant unit four-vector: kPks1.w x Ž .These functions are well-known 3 . Eqs. 2 are

satisfied by the functions

s y1r20kPxŽ1r2." "w x ;l s x" u k ;l , 5Ž . Ž . Ž .k ž /R

"Ž .where u k;l , ls1,2 are the basic Dirac spinorsobeys the equality

gPk u" k ;l s"u" k ;l .Ž . Ž . Ž .With R™` these solutions turns into the usualplane waves in the Minkowski space and mPk hasm

the meaning of the wave vector. These solutions aremuch simpler than those ones obtained by separationof variables in the general-covariant Dirac equationw x8 .

Functions w Ž0.y are the generalized CS for the dek

Sitter group; they are analogous to ones for the spaceŽ . Ž . w xSO n,1 rSO n constructed in Ref. 9 . However,

such a way doesn’t permit the equivalent descriptionof particles and antiparticles. To overcome this trou-ble we shall use the realization of the dS group asthe group of conformal transformations of three-di-

( )S.A. Pol’shinrPhysics Letters B 449 1999 56–5958

mensional Euclidean space 2; we denote vectors ofthese space as w. To construct the system of CS, weshall use the representation of the dS group on thefunctions depending on w:

s

y1T g f w s a g f w ,Ž . Ž . Ž . Ž .Ž .s w g

wherey1r3i

y1E wga g s ,Ž .w kE w

w¨w is the conformal action of the dS group ongŽ .the three-dimensional Euclidean space; ggSO 4,1

is an arbitrary element of the dS group. Then one canw xshow that the generalized CS are 11

< : Ž0."x ;s ," sF x ;sŽ .ws2 Ž0."s 1yw w x ;s , 6Ž . Ž . Ž .k w

where

1qw 2 2wmk s ," .w 2 2ž /1yw 1yw

Their transformation propertiessŽ0." Ž0."

XF x ;s s a g F x ;s 7Ž . Ž . Ž . Ž .Ž .w g w w

Ž .follow from the construction of the functions 6 asthe coherent states, where x¨x is an action of theg

dS group on the dS space and wX sw y1 . FormulagŽ .7 may be proved straightforwardly if one representthe action of the dS group on the dS space as thetranslations and rotations of two different Lemaitrecoordinate systems.

The scalar product of two CS is the dS-invarianttwo-point function over the dS space:

2 1² < :x ;s ," x ;s ,"0 0

1 23 Ž0." Ž0."s d wF x ;s F x ;sŽ . Ž .H w 0 w 03R

d3k1 21 Ž .0 q Ž0.qs" w x ;s w x ;sŽ . Ž .ŽH k 0 k 08 00 kk )0

1 2Ž0.y Ž0.yyw x ;s w x ;s . 8Ž .Ž . Ž . .k 0 k 0

Ž .The second integral in 8 coincides with the two-point function on the dS space considered in Refs.

2 There exist two different such a realizations; first one leads usto the CS for particles, and the second one leads to the CS forantiparticles. They transforms to each other under the replacementw ™ywrw2.

w x Ž .3,6 . In general, integrals in 8 diverge; one canmake them meaningful by the passage to the com-

w x w xplex dS space 6 or to the generalized functions 3 .Let us construct the spinors

1r2" 2 "u w;l s 1yw u k ;l .Ž . Ž . Ž .˜ w

Using the explicit form of basic Dirac spinors"Ž . Ž w x.u k;l see, for instance, 10 one can show that if

g belongs to the group of translations and dilatationsof the 3-vector w then

1r2y1 " " "y1U g u w;l s a g u w ;l ,Ž . Ž . Ž .˜ ˜ Ž .Ž .w g

9Ž .Ž .where g¨U g is the 4-spinor representation of the

dS group. Let us construct now the following spinorfunctions:

F Ž1r2." x ;l sF Ž0." x ;s y1r2 u" w;l .Ž . Ž . Ž .˜w w 0

Ž . Ž .Using 7 and 9 one obtains that under the transfor-mations from the group of translations and dilata-

Ž1r2."Ž .tions of the 3-vector w functions F x;lw

transforms, within the constant matrix transforma-Ž0."Ž .tion, just as the functions F x;s :w 0

Uy1 g F Ž1r2." x ;lŽ . Ž .w g

s0" Ž1r2."Xs a g F x ;l , 10Ž . Ž . Ž .Ž .w w

where wX sw y1 . Let us define the spin 1r2g

matrix-valued differential form as1 2"V x , x ;wŽ .

s8dw1 ndw 2 ndw 3

=1 2Ž1r2." Ž1r2."F x ;l mF x ;l , 11Ž . Ž . Ž .Ý w w

l

where the line above denotes the Dirac conjugation:† 0wsw g . It is easy to show that1 2"V x , x ;wŽ .

y10 1 2 3s k dk ndk ndkŽ .w w w w

=1 2Ž1r2." Ž1r2."w x ;l mw x ;l 12Ž . Ž . Ž .Ý k kw w

l

Then these differential forms are Lorentz-invariant.Ž .It follows from 9 that if g belongs to the group of

translations and dilatations of the 3-vector w then1 2 1 2" "

y1V x , x ;w sU g V x , x ;w U g .Ž . Ž .ž / ž /g g g

13Ž .One can prove that any transformation from the dSgroup may be decomposed into the transformation

( )S.A. Pol’shinrPhysics Letters B 449 1999 56–59 59

from the group of translations and dilatations of the3-vector w, and some Lorentz transformation. Thisyields that spin 1r2 differential forms are com-

Ž .pletely dS-invariant in the sense of 13 . The sec-ondly quantized spin 1r2 field may be constructed

Ž1r2."Ž .from the functions w x;l using the fermionick

creation-destruction operators which obey the canon-ical commutation relations. Then it is easy to show

1that the anticommutator of the field in the point x2and the Dirac-conjugated field in the point x may be

1 2"Ž .3expressed using the integral H V x, x;w . TheR

problem of the convergence of this propagator re-quires an additional studies.

References

w x1 N.D. Birrell, P.C.W. Davies, Quantum fields in curved space.Cambridge, 1982.

w x Ž .2 Cl. Gabriel, Ph. Spindel, J. Math. Phys 38 1997 622; B.Ž .Allen, Th. Jacobson, Commun. Math. Phys. 103 1986 669.

w x Ž .3 J. Bros, U. Moschella, Rev. Math. Phys. 8 1996 327;gr-qcr9511019.

w x4 E. Capelas de Oliveira, E.A. Notte Cuello, Gravitation: theŽ .Spacetime Structure, P.S. Letelier, W.A. Rodrigues Eds. ,

World Scientific, 1994, p. 381; E.A. Notte Cuello, E. CapelasŽ .de Oliveira, Int. J. Theor. Phys. 36 1997 1231.

w x Ž . Ž .5 A. El Gradechi, S. De Bievre, Ann. Phys USA 235 1994`1; hep-thr9210133.

w x6 M.B. Mensky, Method of induced representations, Space-timeŽand the conception of particles, Nauka, Moskow, 1976 in

.Russian .w x Ž .7 J. Dixmier, Bull. Soc. Math. de France 89 1960 9; S.

Ž .Strom, Arkiv for fysik 30 1965 455.¨ ¨w x Ž .8 V.M. Villalba, Modern Phys. Lett. A 8 1993 2351; gr-

qcr9306019.w x9 A.M. Perelomov, Generalized Coherent States and Their

Applications B, Springer, 1986.w x10 L.H. Ryder, Quantum Field Theory, Cambridge Univ. Press,

Cambridge, 1985.w x11 S.A. Pol’shin, hep-thr9811163, Generalized coherent states

and invariant propagators for massive spin 0 and 1r2 fieldsover de Sitter space.

w x12 S.A. Pol’shin, Group Theoretical Examination of the Rela-tivistic Wave Equations on Curved Spaces II, De Sitter andAnti-de Sitter Spaces, preprint gr-qcr9803092.

4 March 1999


Comment on multigraviton scattering in the matrix model

Robert Echols 1, Joshua P. Gray 2

Santa Cruz Institute for Particle Physics, UniÕersity of California, Santa Cruz, CA 95064, USA

Received 19 November 1998Editor: M. Cvetic

Abstract

We show by explicit calculation that the matrix model effective action does not contain the term Õ2 Õ2 Õ2 rR7r7, in the12 23 13

limit R4r, contradicting a result reported recently. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 11.25.-wKeywords: Matrix theory

1. Introduction

w x w x w xThe conjectures of 1 and 2 along with the arguments provided by 3,4 , and numerous other pieces ofŽ w x .evidence for reviews, see 5 and references therein give one reason to believe that finite N matrix theory

Ž .describes the discrete light-cone quantization DLCQ of M-theory with DLCQ supergravity as its low energylimit. Although there are still many open questions in the matrix formulation of M-theory, we would like to

w xfocus on whether the three graviton scattering calculation of 6 shows a discrepancy between the matrix modeland supergravity. 3

w xIt has recently been reported in Ref. 7 that supergravity and the matrix model do not disagree onw xmulti-graviton scattering. In this note we will show that the term computed in Ref. 7 does indeed have a

supersymmetric cancellation and that the matrix model effective action does not contain a term of the formÕ2 Õ2 Õ2 rR7r7.12 23 13

It is worthwhile to review the problems which arise when one tries to compare three graviton scattering in thematrix model picture with supergravity, setting the stage for our notation which will be used in this note.

w xBriefly, the authors of 6 considered the case of three gravitons; two separated a distance r from each other and

1 E-mail: [email protected] E-mail: [email protected] See note added for the resolution to this discrepancy.


( )R. Echols, J.P. GrayrPhysics Letters B 449 1999 60–67 61

another a distance R from the other two in the limit R4r. A term in the supergravity S-matrix for threegraviton scattering in the small momentum transfer limit was shown to be

k Pk k Pk k PkŽ . Ž . Ž .1 2 1 3 2 31Ž .2 2q q1 2

where k are the ith graviton momenta and q are the two relevant momenta transfer. In the language of matrixi 1,2

theory, this corresponds to taking the Fourier transform of the two-loop effective potential

Õ2 Õ2 Õ212 13 23

2Ž .7 7R rŽ .where Õ s Õ yÕ , etc. refer to the relative velocities of the D0-branes. The two scales R and r arise from12 1 2

Ž .integrating out the massive degrees of freedom introduced by giving the diagonal generators of SU 3 vacuumexpectation values:

² a: a3 a8X srd d qRd d 3Ž .i i1 i2˙ ˙a 8 8 3 3Ž .where X are the 9 SU 3 -valued fields describing the bosonic coordinates. Since X sX T qX T , one can˙i i i i

2 ˙ 3 ˙ 8work out Õ , etc. in terms of X and X12 i i

22 3˙Õ ; X 4Ž .Ž .23 i

2 22 3 8 8 3˙ ˙ ˙ ˙Õ ; X q 3 X y6 X X 5Ž .Ž . Ž .13 i i i i

2 22 3 8 8 3˙ ˙ ˙ ˙Õ ; X q 3 X q6 X X 6Ž .Ž . Ž .12 i i i i

Multiplying these three together yields the expected result for matrix theory22 4 6 4 2 22 2 2 3 8 3 3 8 3 8 3˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙Õ Õ Õ ; X X q X q X X y X X X . 7Ž .Ž . Ž . Ž . Ž . Ž . Ž .ž /12 13 23 i i i i i i i i

w xIn Ref. 6 it was argued that matrix theory was incapable of reproducing the term,4 28 3˙ ˙X XŽ . Ž .i i

8Ž .7 7R rw xwith the correct powers of R and r at two-loops. In Ref. 7 , it was argued that this term can arise at two-loops

from vertices with three massive bosons in the form of the setting-sun diagram, as well as from other two-loopinteractions. After describing the background field method used in this note, we go on to show that the one-loop

w xeffective operator needed to arrive at the conclusion of 7 does indeed cancel among bosons and fermions. By˙8exploiting the fact that X only couples to fields of scale R, we integrate out these most massive modes to findi

that the first term containing coupling between the heavy and light states without supersymmetric cancellations˙ 8 4 a 2 9Ž . Ž . w x Ž .has the form X X rR as described in Ref. 6 . Then integrating over the light SU 2 modes of scalei i

˙ 8Ž .r as1,2 , we demonstrate that the term in the matrix model effective action with four powers of X and thei˙ 8 4 ˙ 3 2 9 5Ž . Ž .least suppression in R is X X rR r .i i

2. Contributions to the low energy effective action

The matrix model Lagrangian is obtained from the dimensional reduction of NNs1 supersymmetricw xYang-Mills theory in Ds9q1 down to Ds0q1 dimensions 1 . For our purposes it will be useful to

initially keep the action in its ten dimensional form expressed as

1 i10 a mn a a m aSs d x y F F q C G D C 9Ž .H mn mž /4 g 2

( )R. Echols, J.P. GrayrPhysics Letters B 449 1999 60–6762

where the field strength is given by

F a sE Aa yE Aa q f abcAb Ac , 10Ž .mn m n n m m n

and the 32=32 dimensional Dirac matrices G satisfy the usual algebra G ,G s2 g with metric� 4m n mn

Ž . ag sdiag q1,y1, . . . , y1 . The 32 component Majorana-Weyl adjoint spinor C has only 16 real physicalmn

components off mass shell. We should mention that the center of mass motion of the D0 particles has beenŽ .removed and we will be considering the SU 3 theory with the gauge index as1–8.

w xTo calculate the one loop contributions to the effective action, we will use the background field method 8and break the gauge field up into a classical background field and a fluctuating quantum field,

Aa ™X a qAX a , 11Ž .m m m

and choose our gauge fixing condition, D mAX a s0, to be covariant with respect to the background field,m

D sE y it aX a. By only keeping terms quadratic in the quantum fields, one obtains the gauge-fixed Lagrangianm m m

in the Feynman-‘t Hooft gauge:

LLsLL qLL X qLL qLL . 12Ž .B A c c

The first piece of the Lagrangian just contains the background gauge field,

1a mn aLL sy F F 13Ž .B mn4 g

whereas the other pieces are quadratic in their respective quantum fields and contain the background gauge fieldin the background covariant derivative squared, D2, as well as in the background field strength F b :rs

1 mnac acX Xa 2 mn b rs b c

XLL sy A y D g q F JJ t A 14Ž . Ž . Ž .Ž .½ 5A m rs n2 g

ac ac1 a 2 b rs b cLL s C y D q F S t C 15( Ž . Ž . Ž .Ž .c rs2

aba 2 bLL sc y D c 16Ž . Ž .c

where

JJ rs s i d rd s yd sd r 17Ž . Ž .Ž .ab a b a b

imn m nw xS s G ,G . 18Ž .

4

The one loop effective action is obtained by evaluating the functional integral for the quantum fields,

Xi G w X x 10Xe s DDA DDC DDC DDc DDc exp i d x LL qLL qLL qLL , 19Ž .Ž .H H B A c cž /

giving

1 i mn10 a mn a 2 mn b rs bw xG X s d x y F F q ln Det y D g q F JJ tŽ . Ž .H mn rsž /4 g 2

i2 b rs b 2y ln Det y D q F S t y i ln Det y D . 20Ž . Ž . Ž .Ž .rs8

1For the fermion functional integration the extra factor arises from the fermion field having 16 real components4

instead of 32 complex ones.


To compute the determinants for the different fields, it is useful to expand D2,

yD2 syE 2 qD qD 21Ž .1 2

where

D s it a E X ma qX aE m 22Ž .Ž .1 m m

D sX at aX mb t b . 23Ž .2 m

At this point it is convenient to dimensionally reduce to 1-D while choosing X a s0, so D s0. By letting0 1a a 1 a 2 a Ž . Ž . Ž .X ™rd d qRd d qX we can break SU 3 ™U 1 =U 1 givingm 3 m 8 m m

D syr 2 t 3 t 3 y2 rX at at 3 yR2 t 8 t 8 y2 RX at at 8 yX at aX ib t b 24Ž .2 1 2 i

with the Latin index going 1–9 and fields X a depending only on time. It is important to note that in 1-D, r andi

R are dynamical variables and we are holding them fixed in the spirit of doing a Born-Oppenheimerapproximation. The magnetic moment interaction for the bosons

mnB b rs b

D s F JJ t 25Ž .Ž .J rs

dimensionally reduced becomesmnB b 0 i b

D s2 E X JJ t 26Ž .Ž .J 0 i

since we will be working in a flat direction. Similarly for the fermions one has

Dc s2 E X bS0 i t b . 27Ž .Ž .J 0 i

Ž .The general form of a determinant in 20 can be written

Tr ln yE 2 qD qD . 28Ž .Ž .0 2 J

Ž .Because we are interested in the limit R4r and will be letting only the most massive modes scale R run in1 33 3 2 2 8 8 2 2Ž .the loop gauge index as4–7 then t t r s r and t t R s R . It is convenient to rescale, r™2 r, R4 4

2™ R and define'3

1D s 29Ž .F 2 2 2yE yR yr0

in addition to

D sy4rX at at 3 30Ž .r 1

4a a 8

D sy RX t t 31Ž .R 2'3

DX syX at aX ib t b 32Ž .2 i

then the trace becomesX2 2 2Tr ln yE yR yr qTr ln 1qD D qD qD qD . 33Ž . Ž .Ž .0 F 2 r R J

The first piece involving yE 2 yR2 yr 2 is a constant and the second contains the one loop quantum corrections0

to the effective action which we will evaluate below by expanding the logarithm for various numbers of externalbackground fields. We will find that the first non-zero terms contain four derivatives even if one just integratesover the most massive modes, R.


2.1. Terms with no deriÕatiÕes

We will display in this section a supersymmetric cancellation between bosons and fermions for all operators2 2 Ž .which can be constructed from yD . Even before considering the expansion of yD in 21 , it is

straightforward to see that all terms in the one loop effective action with no derivatives cancel. This is becausethe determinants of the bosons and fermions differ only by derivative terms, and there are an equal number ofbosonic and fermionic factors in the determinant. Given that a non-derivative operator is particularly important

w xin the analysis of 7 , we show explicitly in this section how non-derivative operators are cancelled.The operator in question has the form

r 2Ž1. b bd LL s x x 34Ž .eff 1 13R

Ž . Ž .where the gauge index, bs1-2, for the small mass SU 2 subgroup scale r . Such a term arises fromŽ .expanding the logarithm in 33 and is given by

1 w xy Tr D D D D 35Ž .F r F r2

or in frequency space

dw dw 1 113 a 3 b 2 a bw xy8Tr t t t t r x w x yw 36Ž . Ž . Ž .H H1 1 1 1 2 2 2 22p 2p w yRŽ . wyw yRŽ .1

2 Ž .where we have dropped r in D for the leading 1rR behavior. Integrating 36 in the limit w ™0 and thenF 1Ž . 2Fourier transforming gives 34 . Now the important point to notice is that D arises from yD which occurs inr

Ž .each determinant for the gauge, fermion and ghost fields 20 . However, they each give a different contributionw 3 a 3 b x ab Ž . Ž .to Tr t t t t ;d d j , where d j is the number of components for the various fields

X cc Ad j s32, d j s10, d j s1. 37Ž . Ž . Ž . Ž .2 Ž .Now it becomes clear that all terms coming from yD in each of the three determinants appearing in 20 will

cancel. To be explicit one gets2i i r

b b10 y 32 y i 1 x x s0. 38Ž . Ž . Ž . Ž .1 132 8 R

A similar result holds for any number of external fields without derivatives involving D , D , DX .r R 2

a 2 ˙a 2( ) ( )2.2. Cancellation of F or X0 i i

In this section, we show that terms with two derivatives cancel as well. This result is familiar in higherdimensions, where it is well known that the kinetic terms of the fields are not renormalized.

Based on the arguments given above the only possible non-vanishing term with two external fields containstwo derivatives and is given by

1 w xy Tr D D D D . 39Ž .F J F J2

Ž . w 0 i 0 j xThe supersymmetric cancellation of 39 between bosons and fermions requires the determination of Tr S Si j w 0 i 0 j x i j Ž .s8 g for fermions and Tr JJ JJ s2 g for bosons. Putting the term into 20 gives

i i dw dw 1 11a b 2 a ibw x2 y 8 y i 0 Tr t t w X w X yw s0Ž . Ž . Ž . Ž . Ž .H H1 i 1 1 2 2 2 22 8 2p 2p w yRŽ . wyw yRŽ .1

40Ž .


which shows that the 2-point contribution to the effective action at one-loop is zero. We can also generalize thisresult to show that all possible non-derivative insertions on a loop with two derivatives will not give acontribution to the effective action.

2.3. V 4 rR7

Since all terms with two derivatives, no derivatives, or a mixture cancel by the arguments given above, theonly possible non-vanishing term with four external fields is the four derivative term given by

41y Tr D D 41Ž . Ž .F J4

or in frequency space8 8 8 8dw dw dw dw y w qw qw X y w qw qw w X w w X w w X wŽ . Ž . Ž . Ž . Ž .2 3 4 2 3 4 i 2 3 4 2 j 2 3 k 3 4 l 4H 4 2 2 22 2 2 2 2w x2p wqw yR wqw qw yR wqw qw qw yR w yRŽ . Ž . Ž . Ž .2 2 3 2 3 4

42Ž .with the prefactor

4 41 4 8 0 iy 2 Tr t Tr JJ 43Ž . Ž . Ž .4

for the gauge boson case. An identical result holds for fermions if one replaces the Lorentz generator trace with

w 0 i 0 j 0 k 0 l x i j k l i k jl i l jkTr S S S S s2 g g yg g qg g 44Ž .Ž .whereas for the gauge bosons one finds

w 0 i 0 j 0 k 0 l x i j k l i l jkTr JJ JJ JJ JJ s g g qg g . 45Ž .Ž .Ž .Now using 20 and the low energy approximation w ,w ,w ,w ™0, we get1 2 3 4

227i dw 128y F . 46Ž .Ž . H0 i 42 24 2p w yRŽ .The integral can be performed in the complex plane using the usual qie prescription for handling the poles.

˙ 8 4 8 4 4Ž . Ž .Defining X s F 'V , one is left with the result that the first non-vanishing contribution to thei 0 i

effective potential has four derivatives,

V 4Ž1.d LL ; , 47Ž .eff 7R

even when the gauge group experiences multiple levels of breaking.

2.4. V 4 x 2 rR9 and V 4 Õ 2 rR9r 5

Looking at possible insertions with two background fields on a massive loop with four derivatives givesterms of the form,

4 XTr D D D 48Ž . Ž .F J 2

4 25y Tr D D D D 49Ž . Ž . Ž .F J F R2

4 25y Tr D D D D 50Ž . Ž . Ž .F J F r2

4y5Tr D D D D D D . 51Ž . Ž . Ž . Ž .F J F R F r


Ž . Ž . 4 2 9 Ž .The operators in 48 and 49 lead to terms of the form V x rR with x being a light field scale r inw x Ž . Ž .agreement with 6 , whereas the operators in 50 and 51 give terms with more powers of R in the

denominator. At this point in our analysis, one might worry that we have thrown out the vertices coupling threeŽ .quantum fields two of mass R and one of mass r with one background field which was found to be importantw xin the result of 7 . However, by considering the x’s as background plus quantum fields, the effective operator

V 4 xX 2rR9 contains the sum of all non-vanishing vertices with up to four derivatives constructible from such a4 X 2 9 Ž . Xvertex. We can now use V x rR in the path integral 19 and integrate over the light modes x to generate

V 4 Õ2

, 52Ž .9 5R r2 ˙ 3 2Ž . Ž .where Õ ' X . Clearly 52 has the wrong dependence on R and r to reproduce the term of interest in thei

supergravity scattering amplitude.

3. Comment on the Eikonal approximation

Ž w x .When analyzing D0-brane scattering most authors see e.g. 7,9 and references therein have chosen to usean explicit background given by xsÕtqb where Õ is a relative velocity of the D0-branes and b an impactparameter. Such an approach allows one to construct the exact propagator as a power series in b, Õ, and t. Byorganizing the calculation along the lines suggested by our analysis above, we can exhibit the cancellation of allV 4 Õ2rR7r7 contributions to the effective action. The point, again, is to take advantage of the large R limit. Inthe functional integral, one first does the integration over the fields with mass of order R. As explained inSection 2.1 terms involving only D2 cancel, allowing one to write a simplified expression for the effectiveaction which only depends on the difference of the derivative terms between bosons and fermions

i iB cw xG X s Tr ln 1qD D y Tr ln 1qD D , 53Ž .F J F J2 8

where D 'yDy2 is the propagator for the heavy fields and is a function of the background and the lightFŽ .fields. Again, terms with two derivatives of the background or light fields cancel as in 40 . Terms with four

derivatives and factors of r 2 expanded up from the heavy propagator yield precisely the structure V 4 x 2rR9. Soagain, there are no terms of the form V 4 Õ2rR7r7 in the effective action.

This of course does not mean that there are not individual diagrams with the behavior V 4 Õ2rR7r7. However,w xwe see explicitly from this analysis that there are cancellations between bosons and fermions. In Ref. 7 , a

particular diagram with this behavior was exhibited. But we see that this contribution is cancelled by diagramsinvolving fermions.

4. Discussion

We have shown by explicit calculation in an arbitrary background that the operators one first encounters inthe matrix model effective action after integrating out just the most massive modes contain four derivatives and

4 2 9 w x 4 2 9 5are of the form V x rR as discussed in Ref. 6 . When we use such an operator to construct V Õ rR r attwo loops by integrating over the modes of scale r, one finds the wrong scaling with R and r to correspondwith the term V 4 Õ2rR7r7 in the supergravity S-matrix. In our analysis, we found no velocity independent terms

w x w x 4indicating that fermionic contributions were missed in the work of 7 . In fact, in Ref. 12 the missingfermionic piece was identified.

4 While this work was being completed we received word of this result.


What does one conclude about the correspondence between the matrix model and supergravity for threegraviton scattering? It is possible that a proper treatment of various subtleties of DLCQ supergravity will show

w xcomplete agreement with the finite N matrix model 12 . It is likely that at large N, the supergravity predictionw x 6 Ž .is recovered. The recent work of 10 showing that there is a non-renormalization theorem for Õ in SU 2

indicates that the matrix model-DLCQ supergravity correspondence is working for the Õ6 terms, but usingw x 6 Ž .reasoning similar to 11 it is not hard to show that some Õ terms are renormalized in SU N for NG4. For

now, we will have to wait and see how the issue of three graviton scattering is resolved.

w xNote added: Concurrent with the appearance of this note on hep-th, the work of 13 showed conclusivelyŽ w x.that supergravity and the matrix model do agree for 3-graviton scattering including the effect of recoil 14 .w xThe results reported in this note coincide with these findings since 13 have no terms of the form

2 2 2 7 7 w xÕ Õ Õ rR r in the effective action of supergravity or the matrix model. The source of the error in Ref. 612 13 23w xoccurs in extracting the S-matrix from the matrix model effective action 15 .

Acknowledgements

We are indebted to Michael Dine for extremely useful conversations and encouragements, Washingtonw xTaylor for sending us an early preprint of his work with Mark Van Raamsdonk 12 , and Gautum Mandal for

w x w xclarifying some of the aspects of 7 and 12 with us. This work is supported in part by the U.S. Department ofEnergy. R.E. is grateful for GAANN fellowship support.

References

w x Ž .1 T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys. Rev. D 55 1997 5112. hep-thr9610043.w x Ž .2 L. Susskind, hep-thr9704080, Another Conjecture about M atrix Theory.w x Ž .3 N. Seiberg, Phys. Rev. Lett. 79 1997 3577. hep-thr9710009.w x n4 A. Sen, hep-thr9709220, D0 Branes on T and Matrix Theory.w x5 T. Banks, Matrix Theory, hep-thr9710231; D. Bigatti, L. Susskind, Review of Matrix Theory, hep-thr9712072; W. Taylor, Lectures

Ž .on D-Branes, Gauge Theory and M atrices , hep-thr9801182; A. Bilal, Matrix Theory: A Pedagogical Introduction, hep-thr9710136.w x6 M. Dine, A. Rajaraman, hep-thr9710174, Multigraviton Scattering in the Matrix Model.w x7 M. Fabbrichesi, G. Ferretti, R. Iengo, hep-thr9806018, Supergravity and matrix theory do not disagree on multi-graviton scattering.w x8 M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.w x Ž .9 K. Becker, M. Becker, Nucl. Phys. B 506 1997 48. hep-thr9705091.

w x10 S. Paban, S. Sethi, M. Stern, hep-thr9806028, Supersymmetry and Higher Derivative Terms in the Effective Action of Yang-MillsTheories.

w x11 M. Dine, R. Echols, J. Gray, hep-thr9805007, Renormalization of Higher Derivative Operators in the Matrix Model.w x12 W. Taylor, M. Van Raamsdonk, hep-thr9806066, Three-graviton scattering in Matrix theory revisited.w x13 Y. Okawa, T. Yoneya, hep-thr9806108, Multibody Interreactions of D-Particles in Supergravity and Matrix Theory.w x14 Y. Okawa, T. Yoneya, hep-thr9808188, Equations of Motion and Galilei Invariance in D-Particle Dynamics.w x15 M. Dine, R. Echols, J.P. Gray, hep-thr9810021, Tree Level Supergravity and the Matrix Model.

4 March 1999


Non-supersymmetric stable vacua of M-theory

Micha Berkooz a,1, Soo-Jong Rey b,2

a School of Natural Sciences, Institute for AdÕanced Study, Princeton, NJ 08540, USAb Physics Department, Seoul National UniÕersity, Seoul 151-742, South Korea

Received 8 December 1998Editor: M. Cvetic

Abstract

We discuss the stability of non-supersymmetric compactifications of M-theory and string theory of the form AdS=X,and their dual non-supersymmetric interacting conformal field theories. We argue that some of the difficulties in controlling

Ž1rN-corrections disappear in the cases that the large N dual conformal field theory has no invariant marginal operators and.in some cases with no exactly marginal operators only . We provide several examples of such compactifications of M-theory

down to AdS . q 1999 Published by Elsevier Science B.V. All rights reserved.4

1. Introduction

A powerful tool in the study of large N limits ofconformal field theories is their description in terms

w xof certain string theory or M-theory vacua 1 . Thesevacua are of the form AdS =X , where X is ap q q

compact Einstein manifold and pqq adds up to 10or 11. A precise prescription of how to derive con-formal dimensions and correlation functions from the

w xsupergravity side was given in Refs. 2,3 and hasbeen extensively developed since.

Of particular interest is the description of vacuathat break supersymmetry completely. The interest isdrawn from both field theoretic and M-theory pointsof view. From the field theoretic point of view, onewould like to have some control over non-supersym-metric interacting conformal field theories as a toolfor better understanding non-supersymmetric field


theories in general. The existence of such conformalw xfixed points is well established 4 but only in the

perturbative regime. There is no concrete proposalfor a strongly coupled non-supersymmetric confor-mal field theory in d)2.

The issue is even more interesting from the pointof view of M-theory. The supergravity backgroundthat is dual to the large N limit of such conformalfield theory will be a vacuum with no supersymme-try which is nevertheless stable. The understandingof stable non-supersymmetric vacua would be animportant step forward in order for string theories tomake contact with reality.

A procedure for obtaining non-supersymmetricsupergravity vacua that may be dual to ds4 non-supersymmetric conformal field theories has been

w xdiscussed in Ref. 5 in the context of Type IIB stringŽtheory orbifolds related issues have been studied

w x .subsequently in Ref. 6 . As discussed in that work,there might be a non-vanishing dilaton tadpole, which

Žwould destabilize the vacuum unless the dilatonw xpotential remains flat, as was suggested recently 7


( )M. Berkooz, S.-J. ReyrPhysics Letters B 449 1999 68–75 69

.for some cases . This implies that the true quantumvacuum may be quite far away in the space of vacua.One indeed expects that this would be a genericproblem of orbifolds at weak string coupling. Thephenomenon is the well-known Dine-Seiberg prob-

w xlem re-emerging in this context 8 .In this paper, we will discuss a construction in

M-theory of vacua which break supersymmetry com-pletely, but nevertheless overcome this specific prob-lem: there is no candidate for a field which mightdevelop a significant tadpole when taking into ac-

1 Žcount possible corrections although we do notN

.know how to actually compute them in M-theory .The usefulness of M-theory for freezing moduli was

w xalready observed in Ref. 9 . We will focus primarilyon compactification down to AdS .4

In the following section, we will discuss an argu-ment why a certain class of vacua of M-theory maybe close to a stable one in a sense that will be madeprecise there. In Section 3 we discuss a concrete setof non-supersymmetric vacua of M-theory of theform AdS times an Einstein manifold M which4 7

fall into this class.

2. A conformal field theory argument for stability

Suppose we are given an M-theory vacuum whichis dual, as N™`, to a conformal field theory. Onecan then argue that if the conformal field theorycontains no invariant marginal operators then theeffects of the tadpoles is to modify the true confor-mal fixed point as well as the true supergravityvacuum, whether supersymmetric or not, by effectsthat are only of order 1rN.

The main assumption that goes into the argumentis what we will call calculability. We will assumethat given a supergravity vacuum at large N limitthere exists a well-defined calculational proceduresuch that this vacuum corresponds to the classicaltheory. This is a very natural assumption from thepoint of view of supergravity side but when trans-lated to the conformal field theory side it becomesvery powerful.

Since we do not know how to quantize suchvacua of M-theory or string theories with Ramond-

1Ramond flux, we cannot compute the corrections.N

Nevertheless, the assumption of calculability can

carry us some steps forward. The implication of thisassumption is that gauge symmetries of the classicalsupergravity background can be broken only sponta-

1neously, if at all, by the corrections. In that case,N

any supergravity field that is charged under thegauge symmetries would not have a linear tadpoleŽ ) ) .of the form LL s PPP ql fqlf q PPPSUGRA

since such a tadpole will necessarily break gaugesymmetries explicitly. Since we cannot generate alinear tadpole for the charged fields, we can at most

Ž .change their mass term by 1rN corrections . This inturn will change their dimension around the confor-mal fixed point but will not destabilize it. Higherorder corrections to these fields will not even changethe dimension. Our only concern, therefore, are lin-ear tadpoles for fields that are neutral under all theglobal symmetries of the dual conformal field theory,Ži.e., the associated particles are not charged underthe gauge symmetries of the classical supergravity

.background . If the field has non-zero mass, viz., itŽ 2 .satisfies the equation of motion D qm fs0A dS

with m/0, then a tadpole of the form LL sSUGRACLL q fq PPP in the Lagrangian will only shift the0 N

expectation value of f by an amount proportional to1 but otherwise the solution will still retain the AdSN

symmetries. This will not be true for a massless fieldCwhere now the equation of motion is Dfq s0.N

One concludes that if there are no massless invariant1scalar fields, then any -corrections to the equationsN

of motion does not destabilize the solutionA parallel argument can be made in the terms of

the field theory. The large N limit of a givenquantum field theory is defined in terms of rescaledcouplings such that in terms of these couplings the

Ž .b-function is finite i.e. N-independent . We willdenote the vector of such rescaled couplings of thetheory by g . The statement of finite b function ist

that the RG equation is

g sF g 2.1Ž . Ž .˙t 0 t

without any explicit dependence on N. Taking into1account correction we can expand the b functionN

by

1 1g sF g q F g q F g q PPP .Ž . Ž . Ž .˙t 0 t 1 t 2 t2N N

2.2Ž .

A familiar example is that of the ‘t Hooft effective

( )M. Berkooz, S.-J. ReyrPhysics Letters B 449 1999 68–7570

coupling in the large N limit of QCD. The relevantterm in the b-function at finite N is

2 4g sb Ng q PPP ,Ž .YM 0 YM

which implies that the b-function for the ‘t Hooftcoupling g 2 sg 2N is independent of N.eff

If the classical supergravity vacuum is such thatthe dual theory at large N is conformal, viz. thespacetime is of the type AdS =M , then thep 11yp

functional form of F is constrained to be0

F g sMPg ,Ž .0 t t

where M is a matrix which encodes the dimensionsof the operators around the fixed point at large N.We also relabeled the couplings such that the largeN fixed point is at g s0.t

Concentrating on the invariant operators, if there1are no marginal operators, the effect of the correc-N

tions to these operators will be small and control-1lable. The reason is that even when we take the N

corrections into account there still is a fixed pointŽ .close by. Expanding F g around the large N fixed1 t

point, the renormalization group equation, to leading1order in -correction and g , istN

1g sMPg q F 0 q PPP 2.3Ž . Ž .˙ t t 1Nand is solved by

1) y1g sy M PF 0 q PPP . 2.4Ž . Ž .t 1N

This is possible because under the assumption thatthere are no marginal operators the matrix M isinvertible. Shifting the value of the fixed point interms of the coupling is exactly analogous to givingsmall expectation value to non-zero mass fields.

Under more limited circumstances, the same lineof argument can be applied to the case where theconformal field theory at the large N limit hasmarginal operators but not truly marginal ones 3. Forexample, if the large N limit renormalization groupflow near g s0 is of the formt

12g sa g q bq PPP ,˙t t N

where a and b are numbers, then there exists a1 b b) 2fixed point at g sy provided -0.t N a a

3 We are indebted to N. Seiberg for discussion of this point.

We see that this kind of instability in non-super-symmetric theories occurs only when there aremarginal operators and, in some cases, when thereare marginal but no exactly marginal operators. Inthe case where in large N there are truly marginaloperators then the effect of F on the submanifold 4

1

of truly marginal deformations is pronounced. Sincethere is no flow on this submanifold at leading orderin 1rN, the sub-leading terms control the flow andthere may simply not be a fixed point or it may befar away from our starting point. This will be thecase with any typical vacua that is derived in stringperturbation theory, where the dilaton will be associ-ated with an invariant marginal operator. We willtherefore discuss M-theory and specific vacua thereofand show that they do not have any marginal pertur-bations.

One might similarly obtain four-dimensional con-formal field theory from Type IIB on AdS =M 5,5

but not in the perturbative regime. The generic prob-lem, as we have explained above, is the tadpole forthe dilaton field. However, there cannot be such atadpole for the dilaton if it is fixed at the self-dual

Ž .point under SL 2,Z . In this case, the discrete gaugesymmetry prevents generation of such a tadpole.

3. ds3 non-supersymmetric conformal field the-ory and AdS =X4 7

The most extensive and varied list of M-theorycompactification of the form AdS =X 11yp is avail-p

able for the case ps4. A large class of vacua isŽknown and the stability in the sense that all fields

satisfy the Breitenlohner-Freedman unitarity boundw x .10 of many of them has been analyzed. In particu-lar, it has been shown that there are many non-super-symmetric yet stable vacua. To achieve the goal ofthis paper, we are interested in examining the ques-tion whether these vacua have massless fields onAdS space that are invariant under all the gauge4

symmetries of the supergravity theory. If a specificvacuum does not contain any massless, gauge-singletscalar field, one would then conclude that the vac-uum passes the tadpole-hurdle alluded in the previ-

4 In the space of coupling constants.


1ous section even after the -corrections are takenN

into account. Such a vacuum may be dual to anon-supersymmetric conformal field theory.

A subset of the non-supersymmetric vacua of theform AdS =X 7 is obtained by ‘‘skew-whiffing’’4

w xsupersymmetric compactification vacua 11,12 . Theadvantage of this method is that it guarantees thestability of the resulting non-supersymmetric vacuaw x13 , at leading N. This is evidently so as the massspectrum of the bosonic fields that might potentiallycause an instability remains unchanged from that ofthe initial supersymmetric vacua and hence is stable.The disadvantage from our perspectives is that, inthe vacua with more supersymmetries, one oftenencounters exactly marginal operators. According toour previous argument such operators, if exist andare invariant, would cause an instability to the non-

Žsupersymmetric vacua those obtained by ‘‘skew.whiffing’’ once 1rN-corrections are made. On the

other hand, exactly marginal operators become scarcefor compactifications with less supersymmetries. Wewill henceforth focus on non-supersymmetric vacuaobtained from ‘‘skew-whiffing’’ of compactifica-tions with smaller supersymmetries.

3.1. Basics of ‘‘skew-whiffed’’ non-supersymmetriccompactification Õacua

Let us briefly review known facts about the massspectrum of ‘‘skew-whiffed’’ compactification vacua.The ‘‘skew-whiffing’’ procedure is to reverse theorientation of the compactification manifold X 7.This is done by replacing e m by ye m. The resultinga a

background generated in this way is clearly a solu-tion of the supergravity equations of motion. How-ever, the amount of the residual supersymmetry pre-served by the ‘‘skew-whiffed’’ vacuum changes ingeneral. Indeed, apart from the most symmetricchoice such as X 7 sS7, no supersymmetry is left

7 Žpreserved after the orientation reversal of X in so7 .far as the manifold X is smooth .

Starting from the original supersymmetric vac-uum, the mass spectrum of the ‘‘skew-whiffed’’vacuum is obtained by interchanging the negativeand the positive parts of the spectrum of the appro-priate Laplacian operators on the compact manifold

7 w xX 13,11 . Since the spectrum of these operatorsdetermines the masses of supergravity fields on

AdS , the masses of some of the fields may change.4

For example, the spectrum of the spinors changes,corresponding to the fact that the new vacuum typi-cally leaves out no residual supersymmetry at all.

Being non-supersymmetric, one might suspect thatthe ‘‘skew-whiffed’’ vacua generically contain somefields violating the unitarity bound and developtachyonic instability. The supergravity fields that canpotentially violate the unitarity bound are actuallythose with quantum numbers J P s0q on AdS .4

Ž .They arise from traceless deformations of the met-ric field on the compact manifold X. Fortunately, themass spectrum of these fields does not change fromthat of the starting supersymmetric vacuum and hencethe non-supersymmetric vacuum will remain stable

w xas well 13 , at least for large N.

3.2. Generic problems

The argument given above holds only in the strict1N™` limit. Clearly, the -corrections will be dif-N

ferent for the supersymmetric vacua and for their‘‘skew-whiffed’’ ones. In the above discussion, wehave utilized the underlying supersymmetry to arguethat both the starting vacuum and the ‘‘skew-whiffed’’ one are stable at large N. However, thevery same supersymmetry may also be a source ofpotentially dangerous operators. One needs to beaware of that the following problems might arise:1. fields saturating unitarity bound may be present:

Since the J P s0q part of mass spectrum is thesame as in the starting supersymmetric compacti-fication, there might be fields that saturate the

Žunitarity bound and corresponding operator withdimensions-3r2 in the dual conformal field the-

.ory . The reason for this is that, in highly super-symmetric field theories, such operators occur

Žquite commonly for example, in three-dimen-sional NN s 8 gauge theory, the operator

�i j k4 .trX X X has the scaling dimension 3r2 . Toavoid this problem, we will consider compactifi-cations with less supersymmetries, for whichgenerically we do not expect to find such opera-tors. We then ‘‘skew-whiff’’ these compactifica-tions and begin with the resulting non-supersym-metric vacua.( )2. exactly marginal operators may be present:Since many of the dimensions in the ‘‘skew-


whiffed’’ compactifications are inherited from thesupersymmetric ones, there might be masslessparticles in the spectrum, inherited from exactlymarginal operators in the original supersymmetricones. As before, in supersymmetric gauge theo-

Ž .ries, it is quite common that exactly marginalscalar operators are present. Our main concern iswhether a given compactification vacuum has anymassless fields invariant under all supergravitygauge symmetries. While certainly a much morerestricted class, it is again advantageous to focuson ‘‘skew-whiffing’’ of compactifications withless supersymmetries as they would not haveexactly marginal operators in general.Before we proceed to specific examples, let us

identify which modes of the scalar field might bepotentially dangerous in the sense that they willcorrespond to massless gauge singlet fields on theAdS . Since our foregoing discussion will be based4

w xon results from 11 , we first note that the definitionw xof mass in Ref. 11 is shifted from that we will be

w xusing momentarily. In Ref. 11 , the mass spectrumŽ 2 2 . Žis defined according to Dy8m qM Ss0 Eq.

Ž . w x.3.2.22 in Ref. 11 , where D is the scalar Lapla-cian on the AdS space, m2 is a parameter associ-4

ated with the compactification, and M 2 is the mass.˜In what follows, we will define the mass M via

˜ 2Ž .DqM Ss0, viz.,

˜ 2 2 2M sM y8m .

We are interested in identifying possible masslessmodes. The modes that are potentially dangerous todevelop tadpoles are those of J P s0q. Those ofJ P s0y cannot develop a tadpole as such a tadpolewould break parity, viz., the symmetry x1 ™yx1, C™yC, where x1 is one of the spatial coordinates inAdS and C is the three-form potential of M-theory.4

Under our assumption of calculability, we expectthis to be a symmetry of the dual conformal fieldtheory. This symmetry then forbids generation of thetadpoles for the parity-odd fields.

Hence, potentially dangerous modes that mightlead to massless states are

1r2qŽ1. 2 2 20 :D q36m y12m D q9mŽ .0 0

0qŽ2 . :D y12m2 .L

The first will give rise to a dangerous mode for2 Ž .D s72m D s0 is omitted from the spectrum0 0

and the second will give a dangerous mode forD s12m2.L

3.3. Examples

We will now present three examples of compacti-fication of the form AdS =X 7, which are free from4

massless, gauge-singlet scalar fields. The full spec-trum is known in the literature only for the first case.For the other two, the full spectrum is not known. Assuch, it is not clear whether there will be a field thatsaturates the unitarity bound. However since thesewill the ‘‘skew-whiffed’’ vacuum of a lower super-symmetry compactification, we do not expect such ascalar field in such compactifications to be present.

3.3.1. Example 1: AdS =S7 rZ4 2

There are two types of AdS =S7rZ compactifi-4 2Žcations even before taking into account different

w x.torsion classes 14,15 related by ‘‘skew-whiffing’’.M-theory on S7 is dual to three-dimensional NNs8

Ž . w xSU N super-Yang Mills theory 16 . The supersym-metry charges would transform in one of the two

Ž .spinor representations of SO 8 , say, 8 . Since wes

take a quotient by Z which acts as y1 on the 82 Õ

spinor, we can lift it to act as q1 on the 8 spinor,s

in which case we still have NNs8 supersymmetry.Alternatively, if it is lifted to act as y1 on the 8 s

spinor, then supersymmetry is broken completely.Even though we have not casted it this way, this isbasically equivalent to the ‘‘skew-whiffing’’ process.

It is easy to see that this compactification has noscalar field that can violate the unitarity bound. The

w xonly scalar field that lies at the unitarity bound 16for S7 is trX �iX jX k4, but it is projected out by Z2

quotient. One can also verify that there is no scalarfield that can generate dangerous tadpoles: the onlymarginal operator with the J P s0q quantum num-ber is in the symmetric traceless 6-tensor representa-

Ž .tion of SO 8 .

3.3.2. Example 2: the squashed 7-sphereIt is known that there is one supersymmetric and

w xone non-supersymmetric squashed 7-spheres 12 .The supersymmetric one has four-dimensional NNs1supersymmetry. The spectrum of Laplacians relevant


for the J P s0q states is the same in both cases andŽ . Ž . Ž .is given by Eqs. 8.4.2 , 8.4.9 and 8.4.10 in Ref.

w x11 :20 2

D s m C0 G9

20 92D s m C q orŽ .L G9 5

2 1r220 8 12m C q " C q ,Ž .G G9 5 20ž /'5

where C sC q3C . Since we interested inG SOŽ5. SUŽ2.gauge-singlet fields, C s0 and the masses of theG

relevant J P s0q obtained in this way turn outalways non-zero.

( )3.3.3. Example 3: N k,lŽ . w xThe manifold N k,l is defined 17 as a coset

w Ž . Ž .x w Ž . Ž .xSU 3 =U 1 r U 1 =U 1 . The integers k and lŽ .parameterize different embeddings of the two U 1 ’s

w Ž . Ž .xinto the maximal torus of SU 3 =U 1 . For morew xdetails, the reader is referred to 17 . The supersym-

Ž .metric compactifications on N k,l have NNs1 onAdS , and we will be interested in the ‘‘skew-4

whiffed’’ counterpart of them.Ž . Ž .To examine whether there can be SU 3 =U 1

invariant massless field that might destabilize thevacuum, we again check possible massless fields inthe spectrum of J P s0qŽ1 . and J P s0qŽ2 .. For asymmetric space, however, J P s0qŽ1 . does not leadto any dangerous modes. The reason is that thesemodes originate from the supergravity fields that are

7 Žscalars on X hence eigenstates of the scalar Lapla-.cian , and the only invariant mode is the constant

mode for which D s0. Such a constant mode,0

however, is not physical.The analysis of the 0qŽ2 . mode is also straightfor-

ward. We are again interested only in modes whichŽ . Ž .are invariant under SU 3 =U 1 . The analysis of

w xthese modes is discussed in full detail in Ref. 17 .The fact that there is no invariant massless scalarfield is essentially encoded in the computations ofw x17 , where the authors find only a unique solutionfor every value of k and l. Nevertheless, for com-pleteness we briefly outline here the computation ofthe determinant of the mass matrix for these fluctua-tions. The result would be non-zero, indicating thatthere are no massless invariant fields.

To analyze the allowed metrics that preserve theŽ . Ž .GsSU 3 =U 1 symmetry, we first decompose

the adjoint representation of this group into irre-

Ž Ž . Ž ..ducible representations under the ad U 1 =U 1 .After we discard the subspace of G which generatesH, the remaining linear space is tangent to X 7 at theH-orbit that passes through the identity of G. Themetric of this space splits into a sum of metrics on

Ž . Ž .each of the U 1 =U 1 invariant subspaces. Thefreedom that we now have is to multiply each suchcomponent by an arbitrary number.

Ž . w Ž . Ž .x w Ž . Ž .xFor N k,l s SU 3 =U 1 r U 1 =U 1 , thistangent space splits into four irreducible representa-tions, three of them are of real-dimension 2 and onewhich is of real-dimension 1. These will be parame-

˙ ˙Ž . Ž . Ž .terized by indices a,b,.. , A, B,.. , A, B,.. and Z,respectively. Denoting by g 0 the G-invariant metricŽ Ž ..on ad g the invariant metric on M is of the7

form:

g sa g 0 , g sb g 0 , g sg g 0 ,ab ab ZZ ZZ A B A B

g sd g 0 . 3.1Ž .˙ ˙ ˙ ˙A B A B

The solution to Einstein’s equations is given in Ref.w x17 , where it is written in terms of the variables

a 2 a 2 ag 3 pqqas , bs , usq ,2 2 'bdd g 2

ad 3 pqqÕsy .'bg 2

The solution is unique.Once we have found the solution of Einstein

equations, the allowed G-invariant fluctuations arefluctuations of a , b , g and d subject to the con-straint that the total volume is invariant

Da Db Dg Dd2 q q2 q2 s0.

a b g d

Under small fluctuations of the three independentscale parameters a , g and d , we find that

2 2g d3 1a 2 2dR s abq y1ya ybŽ .b 2 42a

Da2y aÕqbuŽ .a

Dg21 12 2q 1qb ya y aÕqbuŽ . Ž .4 2g

Dd21 12 2q 1yb qa y aÕqbuŽ . Ž .4 2d

3.2Ž .


2 2g d Da2 1 1Z 2 2dR s aÕqbu q u q ÕŽ .Z 2 22 aa

2 2g d Dg21 1 2 2q aÕqbu q Õ quŽ .2 22 ga

2 2g d Dd21 12 2q aÕqbu qÕ q uŽ .2 22 da

3.3Ž .2 2g d Da

1 1A 2 2 2dR s b q1ya y uŽ .B 4 22 aa

Dg3 1 2 2 2q aq yb y1ya yuŽ .2 4

g

Dd1 12 2 2q b y1qa y u 3.4Ž . Ž .4 2

d

2 2g d Da˙ 1 1A 2 2 2dR s a q1yb y ÕŽ .B 4 22 aa

Dg1 12 2 2q a y1qb y ÕŽ .4 2

g

Dd3 1 2 2 2q bq ya y1yb yÕ ,Ž .2 4

d

3.5Ž .where only three out of the four variational equationsare actually independent.

Choosing the three independent equations appro-priately, we can cast these equations into a form

DVim mdR sMm i Vi

˙where msa, A, A and V sa ,g ,d . From these ex-i

pressions, we can deduce the equations of motion forthe fluctuations Da ,Dg and Dd . The result is thatthe mass matrix is the matrix M m, up to multiplica-i

tions by a non-degenerate matrix. Since we areinterested in zero modes, this will not matter sincewe can evaluate the determinant of M, using the

w xvalues of a ,g ,d of the solution in Ref. 17 . Theresult is that there are no zero mass fields, i.e., thereare no massless invariant fields that can acquiredangerous tadpoles.

Finally, a remark is in order. One might alsoprompt to explore stable higher-dimensional non-su-persymmetric compactifications, especially, of theform AdS =X 4 and the corresponding dual confor-7

mal field theories. In the previous version of thisletter, we have indicated AdS =S2 =S2 might be a7

possible non-supersymmetric compactification thatenables us to address the stability along the lines ofSection 2. It turned out that this vacuum is actually

1not a good starting point for -expansion: the com-N

pactification is unstable under fluctuations for whichthe volume of the first S2 shrinks and that of thesecond S2 expands while keeping the total volumeof S2 =S2 fixed. Hence, barring the possibility oforbifold construction, it then appears that there is nostable non-supersymmetric compactification to AdS .7

Acknowledgements

We would like to thank O. Aharony, J. Distler, S.de Alwis, M.J. Duff, O. Ganor, S. Gubser, A. Ka-pustin, I. Klebanov, N. Seiberg and F. Zamora forilluminating discussions. The work of MB is sup-ported by NSF grant NSF PHY-9513835. The workof SJR is supported by KOSEF InterdisciplinaryResearch Grant and SRC-Program, KRF Interna-tional Collaboration Grant, Ministry of EducationBSRI 98-2418 Grant, SNU Research Fund, and TheKorea Foundation for Advanced Studies Faculty Fel-lowship.

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w x2 E. Witten, hep-thr9802150, Anti de Sitter Space and Holog-raphy.

w x3 S.S. Gubser, I. Klebanov, A. Polyakov, hep-thr9802190,Gauge Theories Correlators from Non-critical String Theory.

w x Ž .4 T. Banks, Z. Zaks, Nucl. Phys. B 196 1982 189.w x Ž .5 S. Kachru, E. Silverstein, Phys. Rev. Lett. 80 1998 4855.w x6 A. Lawrence, N. Nekrasov, C. Vafa, On Conformal Field

Theories in Four Dimensions, hep-thr9803015; M. Bershad-Ž .sky, Z. Kakushadze, C. Vafa, Nucl. Phys. B 523 1998 59.

w x7 S. Kachru, J. Kumar, E. Silverstein, hep-thr9807076, Vac-uum Energy Cancellation in a Non-supersymmetric String.

w x Ž .8 M. Dine, N. Seiberg, Phys. Lett. B 162 1985 299.w x9 M. Dine, E. Silverstein, hep-thr9712166, New M-Theory

Backgrounds with Frozen Moduli.w x Ž .10 P. Breitenlohner, D.Z. Freedman, Ann. Phys. 144 1982

Ž .173; 144 1982 249; L. Mezincescu, P.K. Townsend, Ann.Ž .Phys. 160 1985 406.

w x11 M.J. Duff, B.E.W. Nilsson, C.N. Pope, Phys. Rep. 130Ž .1986 1.


w x12 M.J. Duff, B.E.W. Nilsson, C.N. Pope, Phys. Rev. Lett. 50Ž . Ž . Ž .1983 2043: 51 1983 846 E .

w x13 M.J. Duff, B.E.W. Nilsson, C.N. Pope, Phys. Lett. B 139Ž .1984 154.

w x Ž .14 E. Witten, J. High Energy Phys. 98r07 1998 6.w x15 S. Sethi, A Relation Between Ns8 Gauge Theories in Three

Dimensions, hep-thr9809162; M. Berkooz, A. Kapustin,New IR Dualities in Supersymmetric Gauge Theories inThree Dimensions, hep-thr9810257; C. Ahn, H. Kim, B.-H.

Ž .Lee, H.-S. Yang, Ns8 SCFT and M-theory on AdS 4 =

RP 7, hep-thr9811010.

w x Ž . Ž16 O. Aharony, Y. Oz, Z. Yin, M-Theory in AdS P =S 11y.P and Superconformal Field Theories, hep-thr9803051; J.

Gomis, Anti-de-Sitter Geometry and Strongly Coupled StringTheory, hep-thr9803119; E. Halyo, Supergravity on

Ž .AdS 5r4 = Hope Fibration and Conformal Field Theories,hep-thr9803193; R. Entin, J. Gomis, Spectrum of ChiralOperators in Strongly Coupled Gauge Theories, hep-thr9804060.

w x Ž .17 L. Castellani, R. D’Auria, P. Fre, Nucl. Phys. B 239 1984Ž .610; L. Castellani, L.J. Romans, Nucl. Phys. B 238 1984

683.

4 March 1999


Screening in supersymmetric gauge theories in two dimensions

A. Armoni a,1, Y. Frishman b,2, J. Sonnenschein a,3,4

a School of Physics and Astronomy, BeÕerly and Raymond Sackler Faculty of Exact Sciences, Tel AÕiÕ UniÕersity,Ramat AÕiÕ, 69978, Israel

b Department of Particle Physics, Weizmann Institute of Science, 76100 RehoÕot, Israel


Abstract

We show that the string tension in NNs1 two-dimensional super Yang-Mills theory vanishes independently of therepresentation of the quark anti-quark external source. We argue that this result persists in SQCD and in two-dimensional2

gauge theories with extended supersymmetry or in chiral invariant models with at least one massless dynamical fermion.We also compute the string tension for the massive Schwinger model, as a demonstration of the method of the

calculation. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

w xIn a paper by Gross et al. 1 it was conjecturedthat two dimensional NNs1 Super Yang-Mills ex-hibits a screening nature for any representation of theexternal sources. In particular the adjoint vector mul-tiplet can screen completely an external source whichtransforms in the fundamental representation.

SYM is an interesting theory, since it is probably2

the simplest non-trivial supersymmetric model in1q1 dimensions. It’s dynamics is similar to theadjoint fermions model, but SUSY simplifies its

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Work supported in part by the Israel Science Foundation, the

US-Israel Binational Science Foundation and the Einstein Centerfor Theoretical Physics at the Weizmann Institute.

behavior. The model was analyzed lately in Ref.w x2,3 .

In this letter we present a short proof that thetheory is indeed screening. We show that that the

Žstring tension vanishes in this model though we.don’t exclude less than linear confining potential .

An evidence to this phenomenon was given in Ref.w x w x4 . The proof follows an idea presented in Ref. 5which was generalized to the non-Abelian case in

w xRef. 6 . We use a local chiral rotation to eliminatethe external source from the action. The chiral rota-tion affects terms which are not chiral invariant. Inthe case of SYM it is the interaction term of the2

gluino and the pseudo-scalar. The string tension isthen computed using the change in the Hamiltoniandensity.

As a demonstration of the technique, we computethe string tension in the massive Schwinger model. Adifferent derivation in the fermionic basis was given

w xin Ref. 7 .


( )A. Armoni et al.rPhysics Letters B 449 1999 76–80 77

The expression for the string tension in the mas-sive Schwinger model was calculated by using

w xbosonization 5

s smm 1ycos 2p q rq qO m2 ,Ž .Ž .Ž .Q ED e x t d yn

1Ž .

Ž .where m is the electron mass, m s gexp g r3r2Ž2p g is the gauge coupling and g is the Euler

.number and q ,q are the external and dynami-e x t d yn

cal charges respectively.w xThe expression in massive QCD is 62

ke x t 2s smm 1ycos 4pl qO mŽ .ÝQC D R iž /ž /kd yni

2Ž .

where m ;g, l are the isospin eigenvalues of theR i

dynamical representation, k and k are the affinee x t d yn

current algebra levels of the external and dynamicalrepresentations, respectively. This expression is validonly for the fundamental and the adjoint representa-tions. Other representations were also discussed in

w xRef. 6 .Ž .Note that when ms0 the string tensions 1 and

Ž .2 vanish. In this case the theories are screening dueto chiral invariance of the actions.

Ž 2 . Ž . Ž .The O m term in 1 and 2 indicates that theseexpressions are only the leading terms in mass per-turbation theory and are valid when m<g. Thenext to leading order correction, in the Abelian case,

w xwas derived in Ref. 7 .The derivation of the string tension in the massive

Schwinger model is as follows. Consider the parti-tion function of two dimensional massive QED

12 2Zs DA DC DC exp i d x y FH Hm mn2žž 4 g

mqC iEuCymCCyq A Cg C , 3Ž .d yn m / /where q is the charge of the dynamical fermions.d yn

Let us add an external electron-positron source withe x t Ž Ž .charge q at "L, namely j sq d xqL ye x t 0 e x t

Ž .. e x t mŽ .d xyL , so that the change of LL is yj A x .m

Note that by choosing je x t which is conserved,m

E m je x t s0, the action including the coupling to them

external current is gauge invariant.

Now, to eliminate this charge we perform a local,space-dependent left-handed rotation

1Ž .i a Ž x . 1yg 5C™e C 4Ž .2

1Ž .yi a Ž x . 1qg 5C™C e , 5Ž .2

where g 5 sg 0g 1. The rotation introduce a change inthe action, due to the chiral anomaly

a x qŽ . d yn2dSs d x F , 6Ž .H4p

1 mnwhere F is the dual of the electric field Fs e F .mn2

The new action of the original fields is

12 2Ss d x y F qC iEuCH mn2ž 4 g

1myCE a x g 1yg CŽ . Ž .m 52

yi a Ž x .g m5ymC e Cyq A Cg Cd yn m

yq d xqL yd xyL AŽ . Ž .Ž .e x t 0

a x qŽ . d ynq F 7Ž ./4p

The external source and the anomaly term aresimilar, both being linear in the gauge potential. Thisis the reason that the u-vacuum and electron-positronpair at the boundaries are the same in two-dimen-

w xsions 5 . In the following we assume us0, asotherwise we absorb it in a . Choosing the A s01

gauge and integrating by parts the anomaly termlooks like an external sourceqd yn

A E a x 8Ž . Ž .0 12p

This term can cancel the external source by thechoice

qe x ta x s2p u xqL yu xyL . 9Ž . Ž . Ž . Ž .Ž .

qd yn

Let us take the limit L™`. The form of the action,in the region B of yL-x-L is

12 2S s d x y F qC iEuCHB mn2ž 4 gB

qe x tyi 2p g m5ymC e Cyq A Cg C 10Ž .q d yn md y n /

( )A. Armoni et al.rPhysics Letters B 449 1999 76–8078

Thus the total impact of the external electron-positronpair is a chiral rotation of the mass term. This termcan be written as

q qe x t e x tyi2p g 5C e Cscos 2p CCqd y n ž /qd yn

qe x ty isin 2p Cg C 11Ž .5ž /qd yn

The string tension is the vacuum expectation valueŽ .v.e.v. of the Hamiltonian density in the presence ofthe external source relative to the v.e.v. of theHamiltonian density without the external source, inthe L™` limit.

ss-HH)y-HH ) 12Ž .0

The change in the vacuum energy is due to the massterm. The change in the kinetic term which appears

Ž . w xin 7 does not contribute to the vacuum energy 6 .Thus

qe x tssmcos 2p -CC)ž /qd yn

qe x tymsin 2p -C ig C)ym-CC)5ž /qd yn

13Ž .

Thus, the values of the condensates -CC) and-Cg C) are needed. The easiest way to compute5

w xthese condensates is Bosonization 5 , but it can alsow xbe computed directly in the fermionic language 8

exp gŽ .-CC)syg 14Ž .3r22p

-Cg C)s0, 15Ž .5

Ž . Ž .The condensates 14 and 15 were computed inthe massless Schwinger model. However, the correc-tions to these expressions will affect the string ten-

Ž .sion only by terms higher in mrg. Eq. 15 is due toŽ .parity invariance with our choice us0 . The result-

Ž .ing string tension is Eq. 1 .Ž .Though Eq. 1 gives only the leading term in a

w xmrg expansion and might be corrected 7 , whenq is an integer multiple of q the string tensione x t d yn

is exactly zero, since in this case the rotated actionŽ . Ž .10 is the same as 3 .

2. Super Yang-Mills

The same technique can be used to prove screen-w xing in SYM . In this case the action is 92

12 2Ss d x tr y F q ilDulH mn2ž 4 g

1 2q D f y2 igflg l , 16Ž .Ž .m 5 /2

Žwhere A is the gluon field, l the gluino a Majo-m

.rana fermion and f a pseudo-scalar, are the compo-nents of the vector supermultiplet and transform as

Ž .the adjoint representation of SU N . Also D sEc m m

w xy i A ,. .m

Ž .The action 16 is invariant under SUSY' 'd A syigeg g 2 l dfsye 2 lm 5 m

1 imn mdl s ee F q g eD fmn m' '2 2 g 2

We now introduce an external current. The exter-nal source breaks explicitly supersymmetry. How-ever, this breaking does not affect our derivation. Weassume a semi-classical quark anti-quark pair whichpoints in some direction in the algebra. Without lossof generality this direction can be chosen as the ‘3’

Ž .direction ‘isospin’ . The additional part in theLagrangian is y tr j e x tA m where j a e x t sm 0w Ž .x a3Ž Ž . Ž .. w Ž .xC R d d xqL yd xyL and C R ise x t e x t

a c-number which depends on the representation ofŽ w x.the external source see Ref. 6 . The interaction

term can be eliminated by a left-handed rotation inŽthe ‘3’ direction, of the gluino field we are using a

spherical basis, and so we can perform appropriate.complex transformation also for real fermions

13Ž .i a Ž x . 1yg T5˜l™lse l 17Ž .2

13Ž .yi a Ž x . 1qg T˜ 5l™lsle 18Ž .2

T 3 is in the 3 direction of the adjoint representation

T 3 sdiag m ,m , . . . , m 2Ž .1 2 N y1c

( )A. Armoni et al.rPhysics Letters B 449 1999 76–80 79

The chiral rotation introduces an anomaly termŽ . 3tr a x T r4p F, which is used to cancel the exter-

nal charges. Note that the chiral rotation introducesadditional terms. However, these terms involve morederivatives and therefore do not affect the stringtension. This situation is very similar to the Abelian

w xcase and QCD 6 .2Ž . Ž . Ž Ž .The choice a x s2p C R rN u xqL ye x t c

Ž ..u xyL leads to an action which is similar to theŽ .original 16 , but has a chiral rotated term. The

information of the external source is now trans-formed into a rotation angle.

The terms which are relevant to the computationof the string tension are those which appear in theinteraction Lagrangian. In this case, it is the gluinopseudo-scalar term

˜ ˜tr 2 iflg l™ tr 2 iflg l 19Ž .5 5

Let us see how this change influences the Hamil-tonian vacuum energy. In the original theory, with-out the external source, -H )s0 since the theory0

2 Žis supersymmetric and H ;Q where Q is the0.supercharge . In particular it means that there is no

- tr flg l) condensate. Here we assume that5

SUSY is not broken dynamically. The numericalw xanalysis of Ref. 3 indicates that this is indeed the

case.Let us compute the Hamiltonian density of the

rotated theory. In the regime yL-x-L

˜ ˜-HH)s2 ig- tr flg l) 20Ž .5

By using the fact that T 3 is diagonal, and thevacuum state is color symmetric, we get

1˜ ˜- tr flg l)s cos amŽ .Ý5 a2N y1c a

1- tr flg l)yi sin am - tr fll)Ž .Ý5 a2N y1c a

21Ž .

Ž .where as lim a x . The first term on the rightL™`

hand side vanishes since as argued before- tr flg l)s0, and the second term vanishes5

since the isospin eigenvalues, m , come in pairs ofa

opposite signs.

Thus -HH)s0 and the string tension is zero.Note that though we used the classical expression

for the external current and the effective Hamiltonianmay include other terms, these terms cannot changethe value of the string tension. It is so because thistheory contains only one dimension-full parameter,the gauge coupling g, and therefore the string ten-sion is some number times g 2. We showed that thisnumber is zero and higher terms in g which mayappear in the effective Hamiltonian cannot affect thestring tension.

The meaning of the last result is that a quarkanti-quark pair located at xs"` does not generatea linear potential. In the non supersymmetric case, itis a consequence of infinitely many adjoint fermionswhich are produced from the vacuum, as there is nomass gap, that are attracted to the external source,form a soliton in the fundamental representation andresult in screening it. We do not have a constructionof the soliton, but such must occur as a result of the

w xsituation implied by the equivalence in Ref. 10 . Webelieve that similar mechanism occurs in the super-symmetric case too. A complementary argumentw x11,4 is that due to loop effects, the intermediategauge boson acquires a mass M 2 ;g 2N , whichc

leads to a Yukawa potential between the externalquark anti-quark pair.

The above result can be generalized to theorieswith extended supersymmetry and additional mas-sive or massless matter content.

We argue that any supersymmetric gauge theoryin two dimensions is screening. Technically, thereason is that the gluino is coupled to other fields in

Žsuch a way that -HH)s0 guaranteed if SUSY is.not broken dynamically and therefore there are no

non-trivial chiral condensates. However, since thestring tension is proportional to chiral condensates,SUSY leads to zero string tension. Physically, itfollows from the fact that the gluino is an adjointmassless fermion. Since it does not acquire mass,external sources are screened, as in the non-super-symmetric massless model.

In fact, the essential requirement for a screeningnature of the type argued above, is to have amongthe charged particles at least one massless particlewhose masslessness is protected by an unbrokensymmetry. The symmetry can be gauge symmetrycombined with supersymmetry or chiral symmetry.

( )A. Armoni et al.rPhysics Letters B 449 1999 76–8080

References

w x1 D.J. Gross, I.R. Klebanov, A.V. Matytsin, A.V. Smilga,Ž .Nucl. Phys. B 461 1996 109.

w x Ž .2 Y. Matsumura, N. Sakai, T. Sakai, Phys. Rev. D 52 19952446.

w x3 F. Antonuccio, O. Lunin, S. Pinsky, hep-thr9803170, Non-Ž .perturbative Spectrum of Two-Dimensional 1, 1 Super

Yang-Mills at Finite and Large N, OHSTPY-hep-thr-98-005.w x Ž .4 A. Armoni, J. Sonnenschein, Nucl. Phys. B 502 1997 516.

w x Ž .5 S. Coleman, R. Jackiw, L. Susskind, Ann. Phys. 93 1975267.

w x6 A. Armoni, Y. Frishman, J. Sonnenschein, Phys. Rev. Let.Ž .80 1998 430.

w x Ž .7 C. Adam, Phys. Let. B 394 1997 161.w x Ž .8 C. Jayewardena, Helv. Phys. Acta 61 1988 633; I. Sachs,

Ž .A. Wipf, Helv. Phys. Acta 65 1992 652.w x Ž .9 S. Ferrara, Lett. Nuovo Cim. 13 1975 629.

w x Ž .10 D. Kutasov, A. Schwimmer, Nucl. Phys. B 442 1995 447.w x Ž .11 Y. Frishman, J. Sonnenschein, Nucl. Phys. B 496 1997

285.

4 March 1999


Symmetries and structure of skewed and double distributions

A.V. Radyushkin 1, 2

Physics Department, Old Dominion UniÕersity, Norfolk, VA 23529, USAJefferson Lab, Newport News, VA 23606, USA

Received 6 November 1998; revised 11 December 1998Editor: H. Georgi

Abstract

² X < Ž . < :Extending the concept of parton densities onto nonforward matrix elements p OO 0, z p , one can use doubleŽ . Ž . Ž . Ž . Ž . Ž . Ž .distributions DDs f x,a ;t ,F x, y;t or skewed off&nonforward parton distributions SPDs H x,j ;t ,FF X,t . The˜ z

Ž . Ž .use of DDs is crucial for understanding interplay between X x and z j dependences of SPDs and securing the property˜that Nth moments of SPDs are Nth degree polynomials in the skewedness parameters z or j . Proposing simple ansatze for¨DDs, we derive model expressions for SPDs satisfying all known constraints. We argue that for small skewedness, one can

w x Žobtain SPDs from the usual parton densities by averaging the latter with an appropriate weight over the region Xyz , X orw x.xyj , xqj . q 1999 Elsevier Science B.V. All rights reserved.˜ ˜

1. Introduction

² < ŽNonforward matrix elements p y r OO 0,. < : 2z p of light-cone operators which appear inz s0

applications of perturbative QCD to deeply virtualŽ .Compton scattering DVCS and hard exclusive elec-

w xtroproduction processes 1–5 can be parametrizedby two basic types of nonperturbative functions. The

Ž . Ž . w xdouble distributions DDs F x, y;t 2,3 specify theSudakov light-cone ‘‘plus’’ fractions xpq and yrq

of the initial hadron momentum p and the momen-tum transfer r carried by the initial parton. Treatingthe proportionality coefficient z as an independentparameter one can introduce description in terms of

Ž .the nonforward parton distributions NFPDs

1 Also Laboratory of Theoretical Physics, JINR, Dubna, Rus-sian Federation.


Ž .FF X;t with Xsxqyz being the total fraction ofz

the initial hadron momentum taken by the initialparton. The shape of NFPDs explicitly depends onthe parameter z characterizing the skewedness of thenonforward matrix element. This parametrization by

Ž . w xFF X;t is similar to that proposed by X. Ji 1 whoz

Ž .introduced off-forward parton distributions OFPDsŽ .H x,j ;t in which the parton momenta and the˜

skewedness parameter j'rqrPq are measured inŽunits of the average hadron momentum Ps pq

X.p r2. There are one-to-one relations between OF-w xPDs and NFPDs 3 , so it is convenient to treat them

as particular forms of skewed parton distributionsŽ .SPDs .

In our approach, DDs are the primary objectsproducing SPDs after an appropriate integration. Weshow that using the support and symmetry propertiesof DDs one can easily establish important features ofSPDs such as nonanalyticity at border points Xs


( )A.V. RadyushkinrPhysics Letters B 449 1999 81–8882

Ž . N Nz ,0 xs"j , polynomiality of their X and x˜ ˜moments in skewedness parameters z and j , etc.We also discuss simple models for DDs which resultin realistic models for SPDs.

2. Double distributions and their symmetries

2 ²For z s0, the nonforward matrix elements py< Ž . < :r OO 0, z p depend on the relative coordinate z

Ž . Ž .through pz and rz . In the forward case, whenrs0, one obtains the usual quark helicity-averageddensities by Fourier transforming the relevant matrix

Ž .element with respect to pzX

2² < < :p ,s c 0 zE 0, z ; A c z p ,sŽ . Ž . Ž .ˆ z s0a a

1X Ž .yi x p zsu p ,s zu p ,s e f xŽ . Ž . Ž .ˆ ŽH a0

Ž .i x p zye f x dx , 1Ž . Ž ..a

X XŽ . Ž . Ž .where E 0, z; A is the gauge link, u p ,s ,u p,sare the Dirac spinors and we use the notation g z a

a

'z. In the nonforward case, we can use the doubleˆŽ .Fourier representation with respect to both pz and

Ž .rz :X X

2² < < :p ,s c 0 zE 0, z ; A c z p ,sŽ . Ž . Ž .ˆ z s0a a

² X X < < : 2' p ,s OO 0, z p ,sŽ . z s0a

X Xsu p ,s zu p ,sŽ . Ž .ˆ

=1 1 yi xŽ p z .y i yŽ r z . ˜dy e F x , y ;tŽ .H H a

0 y1

= ˜u 0FxqyF1 dx q ‘‘K ’’-term , 2Ž . Ž .˜where the ‘‘K ’’-term stands for the hadron helicity-

w xflip part 1,3 . For any Feynman diagram, the spec-tral constraints y1FxF1, 0FyF1, 0FxqyF1

w xwere proved in the a-representation 3 using thew xapproach of Ref. 6 . The support area for the double

˜ Ž .distribution F x, y;t is shown on Fig. 1a. Compar-aŽ .ing Eq. 1 with the rs0 limit of the DD definition

Ž .2 gives ‘‘reduction formulas’’ relating DD˜ Ž .F x, y;ts0 to quark and antiquark densitiesa

1yx ˜ <F x , y ;ts0 dys f x ,Ž . Ž .H x ) 0a a0

1 ˜ <F x , y ;ts0 dysyf yx . 3Ž . Ž . Ž .H x - 0a ayx

Ž .Fig. 1. a Support region and symmetry line ys xr2 for y-DDs˜ ˜Ž . Ž . Ž .F x, y;t ; b support region for a-DDs f x,a .

Hence, the positive-x and negative-x components of˜ Ž .the double distribution F x, y;t can be treated asa

nonforward generalizations of quark and antiquarkdensities, respectively. If we define the ‘‘untilded’’

˜Ž . Ž . Ž .DDs by F x, y;t s F x, y;t ; F x, y;t sa a x ) 0 a˜ Ž .yF yx,1yy;t , then x is always positive anda x - 0

the reduction formulas

1yx<F x , y ;ts0 dys f x 4Ž . Ž . Ž .H x / 0a ,a a ,a

0

have the same form in both cases. The new antiquarkdistributions also ‘‘live’’ on the triangle 0Fx, y, xqyF1. Taking zszy, we can interpret functions

Ž .F x, y;t as probability amplitudes for an outgoinga,a

parton to carry fractions xpq and yrq of the ‘‘plus’’components of the external momenta r and p. Note,

Ž .that extraction of two separate components F x, y;ta˜Ž . Ž .and F x, y;t from the quark DD F x, y;t as itsa a

x)0 and x-0 parts is unambiguous. In principle,we cannot exclude the third possibility that the func-

˜Ž .tions F x, y;t have singular terms at xs0 propor-

( )A.V. RadyushkinrPhysics Letters B 449 1999 81–88 83

Ž . Ž .tional to d x or its derivative s . Such terms haveno projection onto the usual parton densities. We

Ž .will denote them by F x, y;t . They may be inter-M

preted as coming from the t-channel meson-ex-Ž .change type contributions see Fig. 2b . In this case,

the partons just share the plus component of themomentum transfer r: information about the magni-tude of the initial hadron momentum is lost if theexchanged particle can be described by a pole propa-

Ž 2 .gator ;1r tym . Hence, the meson-exchangeM

contributions to DDs may look like

wq yŽ .MqF x , y ;t ;d x orŽ . Ž .M 2m y tM

wy yŽ .MXyF x , y ;t ;d x , etc. , 5Ž . Ž . Ž .M 2m y tM

"Ž .where w y are the functions related to the distri-M

bution amplitudes of the relevant mesons M ". Thetwo examples above correspond to x-even and x-odd

˜Ž .parts of the double distribution F x, y;t . The singu-lar terms can also be produced by diagrams contain-

Ž . w xing a quartic pion vertex Fig. 2c 7 .A more symmetric description is obtained if non-

forward matrix elements are treated as functions of˜Ž . Ž . Ž . wPz and rz . The relevant a-DDs f x,a ;t thisa

terminology distinguishes them from y-DDsŽ .xF x, y;t are defined by

X² < < :p c yzr2 zc zr2 pŽ . Ž .ˆa a

1 < <1y xX yi xŽP z .y i a Ž r z .r2su p zu p dx eŽ . Ž .ˆ H H< <y1 y1q x

= ˜ ˜f x ,a ;t da q‘‘k ’’-terms . 6Ž . Ž .a

˜ Ž .The support area for f x,a ;t is shown in Fig. 1b.aŽ . Ž .Again, the usual densities f x and f x are ob-a a

˜ Ž .tained by integrating f x,a ; ts0 over verticala

˜ Ž .lines xsconst. Hence, we can split f x,a ; t intoa

three components

f x ,a ; t s f x ,a ; t u x)0Ž . Ž . Ž .a a

y f yx ,ya ; t u x-0Ž . Ž .a

q f x ,a ; t , 7Ž . Ž .M

Ž .where f x,a ; t is the xs0 singular term. Due toM

hermiticity and time-reversal invariance properties ofnonforward matrix elements, the a-DDs are even

˜ ˜Ž . Ž .functions of a : f x,a ;t s f x,ya ;t . For oura aŽ .original y-DDs F x, y;t , this corresponds to sym-a,a

metry with respect to the interchange yl1yxyyw xestablished in Ref. 8 . In particular, the functions

"Ž . "Ž .w y for singular contributions F x, y;t areM M"Ž . "Ž .symmetric w y sw 1yy both for x-even andM M

x-odd parts. The matrix element of the a-quark partSŽ . Ž .w Ž . �OO yzr2, zr2 ' ir2 OO yzr2, zr2 y z ™a a4xyz of the flavor-singlet quark operator can be

parametrized either by y-DDs or by a-DDs

² X X < S < : 2p ,s OO yzr2, zr2 p ,sŽ . z s0aX Xsu p ,s zu p ,sŽ . Ž .ˆ

=1 < <1y x yi xŽP z .y i a Ž r z .r2dx eH H

< <y1 y1q x

= S ˜ Sf x ,a ; t daq ‘‘k ’’-termŽ .a aX Xsu p ,s zu p ,sŽ . Ž .ˆ

=11 1yx Ž . Ž .yi x p z yi yy1r2 r zŽ .dx eŽH H20 0

Ž . Ž .i x p z qi yy1r2 r zŽ .ye .=F S x , y ;t dyq ‘‘K S ’’-term. 8Ž . Ž .a a

Both x)0 and x-0 parts in this case are describedSŽ .by the same untilded function F x, y;t sa x / 0

SŽ . Ž . Ž .F x, y;t qF x, y;t . The a-DDs f x,a ; t area a aŽ .even functions of a and, from Eq. 8 , odd functions

V Ž .of x. The valence quark functions f x,a ; t re-a

Ž . Ž . Ž .Fig. 2. a Parton picture in terms of y-DDs; b,c F -type contributions; d parton picture in terms of a-DDs.M


Fig. 3. Integration lines for integrals relating SPDs and DDs.

1 w Ž . � 4xlated to OO yzr2, zr2 q z™yz operators area2

even functions of both a and x.

3. Parton interpretation and models for doubledistributions

Ž .The structure of the integral 4 has a simpleŽ .graphic illustration see Fig. 3 : integrating DDs over

Ž .a line orthogonal to the x axis, we get f X . Hence,Ž . Ž Ž ..the profile of F x, y or f x,a in x-direction is

Ž . Ždictated by the shape of f x . The profile in y or.a direction gives the distribution of the momentum

Ž .transfer r. Thus, we can write a-DDs as f x,a sŽ . Ž . Ž .h x,a f x , where h x,a is an even function of a

whose integral over a is normalized to 1. TheŽ .a-profile of h x,a should be similar to that of a

Ž . < <symmetric distribution amplitude DA . Since a Fx, it makes sense to introduce the variable bsarxwith x-independent limits: y1FbF1. The sim-plest model is to assume that the profile in the

Ž .b-direction is a universal function g b for all x.Ž . Ž . ŽPossible simple choices for g b may be d b no

3 2. Ž . Žspread in b-direction , 1yb asymptotic limit415 2 2. Ž . Žof nonsinglet quark DAs , 1yb that of gluon16

.DAs , etc. In variables x,a , this gives2 2x yaŽ .

3Ž0. Ž1.h x ,a sd a , h x ,a s ,Ž . Ž . Ž . 4 31yxŽ .22 2x yaŽ .

15Ž2.h x ,a s . 9Ž . Ž .16 51yxŽ .

Ž .Furthermore, models for ‘‘tilded’’ a-DDs f x,ashould be even in x for the gluon and nonsingletquark distributions and odd in x in the singlet quarkcase. A model involving nonzero t values

2 2x ya tŽ .f x ,a ;t sh x ,a f x expŽ . Ž . Ž .i i 2ž /4 xxl

10Ž .Ž . Ž .with h x,a sd a and experimental valence den-V Ž . w xsities f x was used in Ref. 9 .u,d

4. Relations between double and skewed distribu-tions

The characteristic feature of DDs is the absenceŽ .of the z-dependence in the y-DDs F x, y and j-de-

Ž .pendence in f x,a . The alternative way is to intro-w x Ž .duce SPDs 1,3 by using z or j and total momen-

Ž .tum fractions X'xqyz or x'xqja as inde-˜pendent variables:

1 1yxiFF X s dx d xqz yyX F x , y dy .Ž . Ž . Ž .H Hz i0 0

11Ž .The relation between NFPDs and y-DDs can be

illustrated on the ‘‘DD-life triangle’’ 0Fx, y, xqyŽ . Ž .F1 see Fig. 3a . To get FF X , one should inte-z

Ž .grate F x, y over y along a straight line xsXyz y.The upper limit of the y-integration is determined byintersection of this line either with the line xqys1Ž . Ž .this happens if X)z or with the y-axis if X-z .


One can also write the relation between OFPDs˜ ˜Ž . w x Ž .H x,j ;t 1 and a-DDs f x,a ;t˜

1 < <1y xH x ,j ;t s dx d xqjayxŽ . Ž .˜ ˜H H

< <y1 y1q x

= f x ,a ;t da . 12Ž . Ž .˜Ž . Ž .Eq. 12 allows to construct H x,j ;t both for posi-˜

Ž .tive and negative values of j . Since a-DDs f x,a ;t˜Ž .are even functions of a , the OFPDs H x,j ;t are˜

˜ ˜Ž . Ž . Ževen functions of j : H x,j ;t sH x,yj ;t see˜ ˜w x.also 10 . For definiteness, j will be assumed to be

positive. The integration line xsxyja consists of˜two parts corresponding to positive and negative

˜ Ž . Ž .values of x. Substituting f x,a by f x,a ora aŽ . Ž Ž ..f x,a , respectively see Eq. 7 , we geta

H x ,j ;t sH x ,j ;t u yjFxF1Ž . Ž . Ž .˜ ˜ ã a

yH yx ,j ;t u y1FxFjŽ . Ž .˜ ã

qH x ,j ;t u yjFxFj ,Ž . Ž .˜ ˜M

13Ž .

Ž .where H x,j ;t comes from integration of the˜MŽ .singular term f xyja ,a over xrjye-a-˜ ˜M

xrjqe and˜

H x ,j ;t su jFxF1Ž . Ž .˜ ã ,a

=

1yx

1yjf xyja ,a daŽ .˜H a ,a1yx

y1qj

qu yjFxFjŽ .˜

=xrjye˜

f xyja ,a da .Ž .˜H a ,a1yxy

1qj

14Ž .˜ Ž .The OFPD H x,j ;t is in a one-to-one corre-ã

˜ aŽ .spondence with the ‘‘tilded’’ NFPD FF X intro-z

˜ aw x Ž .duced in our paper 3 . The support of FF X isz

y1qzFXF1 and it is related to the untildedŽ .components 11 by

˜ a aFF X sFF X u 0FXF1Ž . Ž . Ž .z z

ayFF zyX u y1qzFXFzŽ . Ž .z

qFF M X u 0FXFz . 15Ž . Ž . Ž .z

In the middle region 0FXFz , the componentsa,a aŽ . Ž .FF X appear only through the difference FF Xz z

aŽ . w xyFF zyX . In a recent paper 11 , Golec-Biernatz

˜ aŽ .and Martin proposed to split our function FF Xz

into overlapping 0FXF1 and y1qzFXFz

ˆ aŽ .parts to introduce ‘‘off-diagonal’’ ‘‘quark’’ FF Xza a˜ ˆŽ . Ž . Žsu 0FXF1 FF X and ‘‘antiquark’’ FF zyz z

˜ a. Ž . Ž .X syu 0FXF1 FF zyX distributions bothz

˜ aŽ .of which include the same middle part of FF X .z

˜ aŽ .They also argued that the decomposition of FF Xza aŽ . Ž .in the middle region into FF X and FF zyXz z

w xparts made in Ref. 3 amounts to ‘‘doubling thequark degrees of freedom’’. Of course, if there wereonly one value of z in the nature, one would never

˜ aŽ .get an idea about how much of FF X should beza a MŽ . Ž . Ž .attributed to FF X ,FF X or FF X . The crucialz z z

missing element is the interplay between z and X˜ aŽ .dependences. Our decomposition of FF X is basedz

aŽ .on splitting the underlying y-DDs F x, y intox)0, x-0 and xs0 parts. Using DDs, one canget NFPDs for all possible z ’s and X ’s, and that iswhy the DDs produce an unambiguous decomposi-tion: DDs ‘‘know’’ not only what is the shape of˜ aŽ .FF X for a particular z , but also how this shapez

would change if one would take another z . Thesimplest illustration of interplay between X and z

M Ž . Ždependences is provided by NFPDs FF X su 0z

. Ž . < <FXrzF1 w Xrz r z corresponding to singularM Ž .parts of DDs. Knowing FF X at some zsz , wez 0

can obtain its shape for any other z by rescaling.Vice Õersa, writing a formal inversion of the basic

Ž .relation 11

` `

w xF x , y s dX D Xyxyz y FF X dz ,Ž . Ž .H H zy` y`

16Ž .

Ž . Ž .where the mathematical distribution D z is de-fined by

`1im z< <w xD z s m e dm , 17Ž .H2

y`2pŽ .M Ž . Ž . Ž . < <taking FF X su 0FXrzF1 w Xrz r z andz

using the following property of the D-function

`

w xD ayz b dzsd a d b , 18Ž . Ž . Ž .Hy`

Ž . Ž . Ž . Ž .we obtain from Eq. 16 that F x, y sd x w y .M


Thus, SPDs contain information originating fromtwo different sources: meson-exchange type contri-

M Ž .butions FF X due to singular xs0 parts of DDsza aŽ . Ž .and the functions FF X , FF X obtained by scan-z z

a aŽ . Ž .ning the x/0 parts of DDs F x, y , F x, y . Thesupport of exchange contributions is restricted to0FXFz , so that they become invisible in the

M Ž .forward limit z™0. Up to rescaling, a FF Xz

function has the same shape for all z . On the otherhand, interplay between X and z dependences of

a aŽ . Ž .FF X , FF X is quite nontrivial and their supportz z

in general covers the whole 0FXF1 region for allz including the forward limit zs0 in which they

a aŽ . Ž .convert into the usual densities f x , f x . Thelatter are known from inclusive measurements, which

a aŽ . Ž .restricts the models for FF X , FF X . Note thatz za aŽ . Ž .the functions F x, y and F x, y are independent

a aŽ . Ž .as are their z-sensitive scans FF X and FF X .z z

Alternatively, one can use as independent functionsa aŽ . Ž . Žtheir sum F x, y qF x, y part of quark singlet

a a. Ž . Ž .functions and the difference F x, y yF x, yŽ .valence functions . Extending the DDs onto they1FxF1 segment does not require extra dynami-cal information: one should take into account only

SŽ .the symmetry properties: the singlet term f x,aaV Ž .must be odd in x while the valence term f x,aa

must be even in x. As a result, the singlet contribu-˜ S,aŽ .tion FF X is an odd function of Xyzr2 whilez

˜ V ,aŽ .the valence one FF X is an even function ofz

Xyzr2.SPDs possess a property which forces the use of

Ž . NDDs. According to Eq. 11 , the X moment ofŽ .FF X must be a polynomial in z of a degree notz

˜ NŽ .larger than N. For OFPDs H x,j ;t , their x mo-˜ ˜ments are Nth order polynomials of j . As explained

w xby X. Ji 10 , this restriction follows from a simplefact that the Lorentz indices of the nonforward ma-trix elements of a local operator O m1 . . . mN can becarried either by P m i or by r m i:

l Nq² < < :Pyrr2 f 0 E f 0 Pqrr2Ž . Ž .Ž .N

Nyk kN q qs P r AŽ . Ž .Ý N kž /kks0

NN Nq ks P j A , 19Ž . Ž .Ý N kž /k

ks0

N Ž . Ž .where 'N!r Nyk !k!. Our derivation 12 ofž /kOFPDs from a-DDs automatically satisfies the con-

Ž .dition 19 :

1 NH x ,j ;t x dxŽ .˜ ˜ ˜Hy1

N1Nks j dxÝ Hž /k y1ks0

=< <1y x Nyk kf x ,a x a da . 20Ž . Ž .H

< <y1q x

Ž .Hence, A ’s in 19 are given by double momentsN k

of a-DDs. Since DDs dictate a nontrivial interplaybetween N and k dependences of A ’s, the use ofN k

DDs is an unavoidable step in building parametriza-tions of SPDs.

The use of DDs also reveals some importantproperties of skewed distributions. Due to cusp at theupper corner of the DD-life triangle, the length of theintegration line nonanalytically depends on X forXsz . Hence, unless the DD vanishes in a finiteregion around the upper corner, the X-dependence ofthe relevant NFPD must be nonanalytic at the borderpoint Xsz . Furthermore, the length of the integra-tion line vanishes when X™0. As a result, the

a,aŽ .components FF X vanish at Xs0 if the relevantza,aŽ .double distribution F x, y is not too singular for

aŽ .small x. The combined contribution of FF X andza a˜Ž . Ž .FF zyX into the total function FF X in thisz z

case is continuous even at the nonanalyticity pointsw xXs0 and Xsz . As emphasized in Ref. 3 , be-

Ž .cause of the 1rX and 1r Xyz factors containedin hard amplitudes, this property is crucial for pQCDfactorization in DVCS and other hard electroproduc-tion processes. Note, that there is also the exchange

M Ž . Ž . Ž .contribution FF X . If it comes from a d x w yz

Ž .term with w y vanishing at the end-points ys0,1,M Ž .the FF X part of NFPD vanishes at Xs0 andz

˜ aŽ .Xsz . The total function FF X is then continuousz

˜Ž Ž .at these nonanalyticity points OFPDs H x,j ;t in˜.this case are continuous at xs"j . In the quark

singlet case, the DDs should be odd in x, hence theXŽ . Ž .singular term involves d x w y . One can get a

XŽ .continuous SPD in this case only if w y vanishesat the end points. Such a restriction might be toostrong to be satisfied in all cases. In particular, anessentially discontinuous behavior of singlet quark


OFPDs for xs"j was obtained in a nonperturba-˜Ž . w xtive chiral soliton model 12 .

5. Models for skewed distributions

The properties discussed above can be illustratedby SPDs constructed using model DDs specified in

Ž0.Ž .Section 3. In particular, the narrow model F x, yŽ . Ž .sd yyxr2 f x gives NFPDs

u XGzr2 Xyzr2Ž .Ž0.FF X s f , 21Ž . Ž .z ž /1yzr2 1yzr2

Ž .resulting from the forward distribution f X 'Ž .FF X by shift and rescaling. Due to an emptyzs0

upper corner, this model DD gives NFPD with noexplicit nonanalyticity at Xsz . The OFPD equiva-

Ž . Ž0.Žlent of Eq. 21 is the simplest model H x,j ;ts˜. Ž .0 s f x in which OFPDs have no j-dependence.

This result can be obtained directly by using theŽ0.Ž . Ž . Ž .model f x,a sd a f x for the a-DDs. An-

w xother example is the model 13,14 in which NFPDsŽ . Ž .do not depend on z , i.e., FF X s f X . Using thez

Ž . Ž .inversion formula 16 and Eq. 18 , we obtainŽ . Ž . Ž .F x, y sd y f x which violates the mandatory

yl1yxyy symmetry. Unlike the j-independentansatz for OFPDs, the z-independent ansatz for NF-PDs should not be used.

For the ‘‘ valence quark’’-oriented ansatzŽ1. Ž1. yaŽ . Ž . Žf x,a , the following choice f x sA x 1y.3x is both close to phenomenological quark distribu-

Ž .tions and allows to calculate the integral 12 explic-< <itly. For x Gj , we have˜

˜ Ž1V .H x ,jŽ .˜ < <x Gj˜

2ya 2ya˜sA 2ya j 1yx x qxŽ . Ž .˜ Ž .½ 1 2

2 2ya 2yaq j yx x yx u xŽ .˜ ˜Ž . Ž .1 2

q x™yx , 22Ž . Ž .˜ ˜ 5˜ Ž . Ž . Žwhere A s 3 A G 1 y a r2 G 4 y a , x s x q˜1

. Ž . Ž . Ž .j r 1qj and x s xyj r 1yj . For the˜2

middle yjFxFj region,˜Ž1V . 2ya˜ ˜H x ,j sA x 2ya j 1yxŽ . Ž . Ž .�˜ ˜< <x Fj˜ 1

2q j yx q x™yx .Ž . 4Ž . ˜ ˜23Ž .

As expected, these expressions are explicitly nonana-lytic for xs"j .

SŽ .The singlet quark distributions f x,a should beodd functions of x. Still, we can use the f Ž1. type

Ž1S . Ž1.Ž . Ž < < . Ž .model, but take f x,a s f x ,a sign x .x / 0˜ SŽ . Ž .Note, that the integral 12 producing H x,j in˜

< <the x Fj region would diverge for a™xrj if˜ ˜aG1, which is the usual case at sufficiently large

2 SŽ .Q . However, since f x,a is odd wrt x™yxand even wrt a™ya , the singularity at asxrj˜can be integrated using the principal value prescrip-tion which gives the x™yx antisymmetric version

Ž . Ž .of Eqs. 22 and 23 . As far as a-2, the resultingfunctions are finite for all x and continuous at˜xs"j . The use of the principal value prescription˜is equivalent to imposing a subtraction procedure for

Ž .the divergent second integral in Eq. 14 defininga,aŽ .H x,j .˜For small x, we can neglect the x-dependence of

Ž .the profile function h x,a and take the modelŽ . Ž . Ž . Ž .f x,a s f x r a with r a being a symmetric

weight function on y1FaF1 whose integral overa equals 1. In the region where both x and j are˜

Ž .small, we can approximate Eq. 12 by

1˜ ˜H x ;j s f xyja r a daq . . . , 24Ž . Ž . Ž . Ž .˜ ˜Hy1

Ž .i.e., the OFPD H x;j is obtained in this case by˜˜Ž . Ž .averaging the usual forward parton density f x

Ž .extended onto y1FxF1 over the region xyjF˜Ž .xFxqj with the weight r a . For NFPDs, the˜

Ž Ž . . Ž .integrand is f Xyz 1qa r2 r a , i.e., the aver-age is taken over the region XyzFxFX.

The imaginary part of hard exclusive meson elec-troproduction amplitude is determined by the skeweddistributions at the border point. For this reason, the

Ž . w Ž .xmagnitude of FF z or H j ,j , and its relation toz

Ž .the forward densities f x has a practical interest.Ž . Ž .Assuming the infinitely narrow weight r a sd a ,

Ž . Ž . Ž .we have FF X s f Xyzr2 q . . . and H x,j sz

Ž . Ž . Ž .f x . Hence, both FF z and H j ,j are given byz

Ž .f x r2 because zsx and jsx r2q . . . . SinceB j B j B jŽ .the argument of f x is twice smaller than in deep

inelastic scattering, this results in an enhancementŽ . yafactor. In particular, if f x ;x for small x, the

Ž . Ž . aratio FF z rf z is 2 . The use of a wider weightz

Ž .function r a produces further enhancement. For


3 2 yaŽ . Ž . Ž .example, taking r a s 1ya and f x ;x4

Ž . Ž . wŽ .Ž .xwe get FF z rf z s 1r 1 y ar2 1 y ar3z

which is larger than 2 a for 0-a-2. Due to evolu-tion, the effective parameter a is an increasing func-tion of Q2. As a result, the above ratio increaseswith Q2.

Ž .Possible profiles of f x,a in the a-direction areŽ w x.restricted by inequalities see 14,10,15,16 relating

skewed and forward distributions. For quark OFPDsw x Ž w x.15 see also 16 ,

1 xqj xyj˜ ˜qH x ,j F f fŽ .˜ ( 2 ž / ž /1qj 1yj1yj

1s f x f x . 25( Ž . Ž . Ž .1 22(1yj

Ž0.Ž . Ž . Ž .If one uses the narrow model f x,a s f x d a

w Ž0.Ž . Ž .xcorresponding to H x,j s f x , the inequality˜ ˜Ž . Ž . yaŽ .b25 is satisfied for any function f x of x 1yx

Ž1.Ž .type with aG0, b)0. For the model f x,a s2 2Ž . w Ž . xA x ya corresponding to Eq. 22 with as0

Ž .which has a wider profile, the inequality 25 isŽ4.Ž .exactly saturated. If one takes the model f x,a

2 2Ž . Ž .sx f x d x ya with an extremely wide profile,1Ž4.Ž . � Ž . Ž . Ž . Žthe result H x,j s f x r 1qj q f x r 1˜ 1 22

.4 Ž .yj violates 25 .

6. Summary

In this paper, we treated double distributions asthe basic objects for parametrizing nonforward ma-trix elements. An alternative description in terms ofskewed distributions was obtained by an appropriateintegration of relevant DDs. The use of DDs helps toestablish important features of SPDs such as theirnonanalyticity at the border points Xsz and xs˜"j . DDs are crucial for securing the property thatthe moments of SPDs should be polynomial in theskewedness parameter. For these reasons, the use of

DDs is unavoidable in constructing consistent mod-els of SPDs.

Acknowledgements

I acknowledge stimulating discussions and com-munication with I.I. Balitsky, A.V. Belitsky, S.J.Brodsky, J.C. Collins, L.L. Frankfurt, K. Golec-Biernat, X. Ji, L. Mankiewicz, A.D. Martin, I.V.Musatov, G. Piller, M.V. Polyakov, M.G. Ryskin, A.Schafer, A. Shuvaev, M.A. Strikman, O.V. Teryaev¨and C. Weiss. This work was supported by the USDepartment of Energy under contract DE-AC05-84ER40150.

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4 March 1999


Vacuum stability bounds in the two-Higgs doublet model

Shuquan Nie 1, Marc Sher 2

Nuclear and Particle Theory Group, Physics Department, College of William and Mary, Williamsburg, VA 23187, USA

Received 29 November 1998; revised 29 December 1998Editor: H. Georgi

Abstract

In the standard model, the requirements of vacuum stability and the validity of perturbation theory up to the unificationscale force the mass of the Higgs boson to be approximately between 130 GeV and 180 GeV. We re-examine these

Ž .requirements in the non-supersymmetric two-Higgs doublet model, in the light of the large top quark mass, and constrainthe masses of the Higgs bosons in this model. It is found that the mass of the charged Higgs boson must be lighter than 150GeV. This bound is below the lower bound in the popular model-II two-Higgs doublet model, and thus we conclude that thismodel cannot be valid up to the unification scale. The bounds on the neutral Higgs scalars are also discussed. q 1999Published by Elsevier Science B.V. All rights reserved.

The mass of the Higgs boson in the standardmodel is, at first sight, completely arbitrary. It de-pends on the scalar self-coupling, l, which is a freeparameter. However, rather stringent bounds on the

w x Ž .mass can be obtained 1,2 by requiring that a l

Žremain perturbative up to a large scale generallytaken to be the unification scale of approximately

16 .10 GeV–the precise value doesn’t much matter ,Ž .and b that the vacuum of the standard model

remain stable up to that large scale. The first condi-tion gives an upper bound to the Higgs mass ofapproximately 180 GeV. The second condition is


virtually identical to requiring that l remain positiveup to the large scale, and that gives a lower bound tothe Higgs mass of approximately 130 GeV. Thus,these two conditions strongly constrain the mass ofthe Higgs boson to be between 130 and 180 GeV.

It is easy to see where these constraints arise. Therenormalization group equation for the scalar self-coupling is of the form dlrd tsal2 ybg 4 , whereY

g is the top quark Yukawa coupling. If l is large,Y

the first term dominates, and l blows up; if it issmall, the second term dominates, and l becomesnegative, leading to a vacuum instability. Only for l

near the fixed point of this equation does l remainpositive and finite from the electroweak to unifica-tion scale.

The most straightforward extension of the stan-Ž .dard model is the two-Higgs doublet model 2HDM .


( )S. Nie, M. SherrPhysics Letters B 449 1999 89–9290

Ž w xIt includes two complex scalar doublets see Ref. 1.for a review ,

xq1

F s ,1 'ž /f q ix r 2Ž .1 1

xq2

F s 1Ž .2 'ž /f q ix r 2Ž .2 2

Of the eight real fields, three must become thelongitudinal components of the W " and Z bosonsafter the spontaneous symmetry breaking. Linearcombinations of x " and x " are absorbed into the1 2

longitudinal parts of the W " bosons and a linearcombination of x and x give mass to the Z boson.1 2

Five physical Higgs scalars will remain: a chargedscalar x " and three neutral scalars f , f and the1 2

other linear combination of x and x , called x 0.1 2

Given how stringently the Higgs mass in thestandard model is constrained, one might ask howstringently the masses of the scalars in the two-doublet model are constrained. There are many more

Žself-couplings which could potentially diverge by.the unification scale and more directions in field

space where an instability could arise. In this paper,we examine these constraints in the two-doubletmodel. This is not new—the constraints have been

w xexamined before 1,3 , but the top quark mass wasŽunknown at the time and only values up to about

.130 GeV for the top quark mass were considered .A potential danger with additional Higgs doublets

is the possibility of flavor-changing neutral currentsŽ .FCNC . It is well known that FCNC are highlysuppressed relative to the charged current processes,so it would be desirable to ‘‘naturally’’ suppressthem in these models. If all quarks with the samequantum numbers couple to the same scalar multi-plet, then FCNC will be absent. This led Glashow

w xand Weinberg 4 to propose a discrete symmetrywhich force all the quarks of a given charge tocouple to only one doublet. There are two suchpossible discrete symmetries in the 2HDM,

I f ™yf II f ™yf ,di ™ydiŽ . Ž .2 2 2 2 R R

2Ž .In model I, all quarks couple to the same doublet,and no quarks couple to the other doublet. In modelII, the Qs2r3 quarks couple to one doublet and theQsy1r3 quarks couple to the other doublet.

The most general potential subject to one of thediscrete symmetries, for two doublets of hyperchargeq1, is

22 q 2 q qVsm F F qm F F ql F FŽ .1 1 1 2 2 2 1 1 1

2q q qql F F ql F F F FŽ . Ž . Ž .2 2 2 3 1 1 2 2

2 22 1q q q< <ql F F q l F F q F FŽ . Ž .4 1 2 5 1 2 2 12

3Ž .the vacuum expectation values of F and F can be1 2

written as

0 0' 'F s1r 2 , F s1r 2 4Ž .1 2Õ Õž / ž /1 2

2 2 2' Ž .with Õ qÕ s 2 G s 247 GeV . The masses of1 2 F

the physical scalars are given by12 2 2

"m sy l ql Õ qÕ ,Ž . Ž .x 4 5 1 22

m 02 syl Õ2 qÕ2 , 5Ž .Ž .x 5 1 2

212 2(m s AqBq AyB q4C ,Ž .f 2 ž /212 2(m s AqBy AyB q4C , 6Ž . Ž .h 2 ž /

2 2 Žwhere As2l Õ , Bs2l Õ and Cs l ql q1 1 2 2 3 4.l Õ Õ .5 1 2

It is required that all scalar boson masses-squaredmust be positive.This implies that

i l -0Ž . 5

ii l ql -0Ž . 4 5

iii l )0Ž . 71 Ž .iv l )0Ž . 2

v 2 l l )l ql qlŽ . ( 1 2 3 4 5

In the standard model the positivity of the scalarboson mass-squared implies that l)0. To ensurevacuum stability for all scales up to M , one mustX

Ž 2 . 2 2 2have l q )0 for all q from M to M . Simi-Z X

larly, to ensure vacuum stability in the 2HDM up toM 2 , one must require that all of the five constraintsX

be valid up to M 2. If the condition l or l )0 isX 1 2

violated, the potential will be unstable in the f or1Ž .f direction. if the condition v is violated, the2

potential will be unstable in some direction in thef yf plane. If the condition l ql -0 is vio-1 2 4 5

lated, a new minimum which breaks charge will beformed. What if the condition l -0 is violated?5

( )S. Nie, M. SherrPhysics Letters B 449 1999 89–92 91

Ž .The minimum in Eq. 4 will only be the minimum ifl -0; if l )0, then the minimum will be of the5 5

Ž .form of Eq. 4 with Õ ™ iÕ . This is obvious from2 2

the form of the potential, since F ™ iF will only2 2

change the sign of l . However, the phase of F is5 2

constrained by the Yukawa interactions; having acomplex vacuum expectation value leads to CP vio-lation in the Higgs sector. Since such models tend tohave too large a value of e

Xre and the neutronelectric dipole moment, we will assume that theHiggs sector conserves CP, and thus must havel -0 at the weak scale. If the sign of l changes at5 5

a high scale, then a new minimum which violates CPforms at that scale, and thus we require that condi-

Ž .tion i be valid at all scales. Therefore, it is requiredthat all of the constraints should be valid up to M .X

At the same time it is physically reasonable toŽ .demand that all l’s be finite or perturbative up to

M .X

Starting with l ,l ,l ,l ,l and tanb at the1 2 3 4 5

electroweak scale, the renormalization group equa-tions are integrated numerically to check whetherone of the five constraints is violated or whether anyof the couplings become nonperturbative beforereaching M . The results give an allowed region inX

the six-dimensional parameter space. To explore thisregion, we choose the six parameters to be the fourphysical scalar masses, tanb and l . For a point in3

this parameter-space to be acceptable, all of theabove constraints, as well as perturbativity, must be

Fig. 1. The allowed region in the neutral scalar mass plane, withm "s m 0 s100 GeV and tanb s2.x x

Fig. 2. The allowed region in the neutral scalar mass plane forŽ . 0various values of the charged Higgs mass in GeV , with the x

mass chosen to be 100 GeV and tanb s2.

satisfied at all scales up to M . In practice, theX

physical scalar masses and tanb can be easily mea-Žsured the vector-vector-scalar coupling depends di-

.rectly on tanb , whereas one would need to measurethe triple-Higgs vertex in order to determine l . As a3

result, we consider the other five parameters asstarting points, and see if any initial values of l3

give acceptable values. In this way, we determine ifa given point in the five-dimensional space of thescalar masses and tanb is acceptable.

Fig. 3. The allowed region in the neutral scalar mass plane forvarious values of tanb , with the x 0 and x " masses chosen to be100 GeV.

( )S. Nie, M. SherrPhysics Letters B 449 1999 89–9292

Fig. 4. The allowed region in the neutral scalar mass plane forŽ .various values of the pseudoscalar Higgs mass in GeV , with the

x " mass chosen to be 100 GeV and tanb s2.

It is, of course, difficult to plot a region infive-dimensional space. However, the basic featurescan easily be seen with a few examples. Let us firstconsider the case in which m "sm 0 s100 GeV.x x

We choose tanbs2, and plot the allowed region inthe neutral scalar mass plane. The region is shown inFig. 1. For m between 40 and 88 GeV, one seesh

that the value of m must lie below 180 GeV andf

above a value which ranges from 130 GeV to 100GeV; this bound is very similar to the result in thestandard model. However, there are no solutions inwhich m is greater than 88 GeV or below 40 GeV.h

Below the region, l becomes negative, above the1

region l becomes non-perturbative, to the left and1Ž .to the right of the region, the constraint v is

violated.One can now vary some of the three input param-

eters to see how this region changes. As the chargedHiggs mass increases, the region shrinks dramati-cally, disappearing when it reaches 140 GeV, asshown in Fig. 2. As tanb increases, the region shiftsto smaller values of m , as shown in Fig. 3. This ish

not surprising since m becomes small as tanb™`.h

As tanb decreases, the size of the allowed regionshrinks, since the top quark Yukawa coupling is

getting larger, leading to an instability. Finally, vary-ing m 0 gives the result in Fig. 4. As in the chargedx

Higgs case, the allowed region disappears when thepseudoscalar mass exceeds 140 GeV.

The most important result is seen from Fig. 2,where this is a stringent upper bound on the chargedHiggs mass. By optimizing tanb and m 0 , we findx

that the maximum allowed value for the chargedHiggs mass is 150 GeV.

What are the experimental constraints? As dis-w xcussed in Ref. 5 , there are very few constraints on

the neutral scalar masses. If one takes the x 0 massto be 100 GeV, the only constraints come from theBjorken process, eqey™Z ) ™Zh, and the rate canbe significantly reduced by judicious choice of themixing angle. So no bounds on m are relevant.h

w x "There is, however, a strong bound 5,6 on mx

coming from B™X g . In Model II, this processs

forces the charged Higgs mass to be greater than 165GeV. This in inconsistent with our upper bound.Model I, however, has no such constraint, and thecharged Higgs mass could be as light as 45 GeV.

We conclude that the popular two-Higgs doubletmodel, Model II, can not be valid up to the unifica-tion scale. Model I is not excluded, however we dofind that the charged Higgs mass must be lighterthan 150 GeV, the lightest neutral scalar must belighter than 110 GeV and the pseudoscalar must belighter than 140 GeV for the model to be valid up tothe unification scale.

References

w x Ž .1 M. Sher, Physics Reports 179 1989 273.w x Ž .2 M. Sher, Phys. Lett. B 317 93 317; J.A. Casas, J.R. Es-

Ž . Ž .pinosa, M. Quiros, Phys. Lett. B 324 95 324; B 382 96Ž .382; J.R. Espinosa, M. Quiros, Phys. Lett. B 353 95 353.

w x Ž .3 M. Sher, Perspectives in Higgs Physics I, G. Kane Ed. ,World Scientific, 1994.

w x Ž .4 S.L. Glashow, S. Weinberg, Phys. Rev. D 15 1977 15.w x5 F.M. Brozumati, C. Grueb, hep-phr9810240, in: Proc. XXIX

Intern. Conf. on High Enegy Physics, Vancouver, BC, Canada,July 1998.

w x6 T.M. Aliev, E.O. Iltan, hep-phr9803272; F.M. Borzumati, A.Djouadi, hep-phr9806301.

4 March 1999


Generalized Gordon identities, Hara theoremand weak radiative hyperon decays

Elena N. Bukina a, Vladimir M. Dubovik a, Valery S. Zamiralov b

a Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russiab D.V. Skobeltsyn Institute of Nuclear Physics, Moscow State UniÕersity, Moscow, Russia

Received 20 November 1998Editor: R. Gatto

Abstract

Ž .It is shown that an alternative form of the parity-nonconserving PNC transition electromagnetic current resolves partly apuzzle with the Hara theorem. Its new formulation has allowed PNC weak radiative hyperon transitions of charged hyperonsSq´pqg and Jy´Syqg revealing hitherto the unseen transition toroid dipole moment. q 1999 Published by ElsevierScience B.V. All rights reserved.

1. Introduction

The weak radiative decays seem to have first beenw xanalyzed theoretically in Ref. 1 . Unitary symmetry

arrived, a theorem was proved by Hara that decayasymmetry of the charged hyperons vanished in the

Ž . w xexact SU 3 2 . Since the experimental discoveryf

of a large negative asymmetry in the radiative decayq w x w x Ž .S ´pqg 3 , confirmed later 4 see Table 1 ,

the explanation of the net contradiction betweenexperimental results and the Hara theorem prediction‘‘has constituted a constant challenge to theorists’’w x5 . The Hara theorem was formulated at the hadronlevel. But quark models while more or less succeed-ing in describing experimental data on branching

Ž w x.ratios and asymmetry parameters see e.g. 5 didnot nevertheless reproduce the Hara claim withoutmaking vanish all asymmetry parameters in the

Ž .SU 3 symmetry limit. The origin of this discrep-f

ancy is not clear up to now although many authors

w xhave investigated this problem thoroughly 6–9 , andis a real puzzle as similar calculations say that ofbaryon magnetic moments are known to be rather

Ž .consistent at the quark and hadron level. Also SU 3 f

symmetry breaking effects can hardly be so largew xdue to the well-known Ademollo-Gatto theorem 10

to be able to explain for this puzzle.We shall try to show that the discrepancy between

the Hara theorem predictions and the quark modelresult may be overcome by an alternative multipoleparametrization of the parity-nonconserving transi-tion electromagnetic current which includes not onlythe dipole transition moment but also a contribution

w xof the toroid dipole moment 11,12 . The toroiddipole moment naturally arises in the parity-violatingŽ .PV part of the transition radiative matrix elementand leads to reformulation of the Hara theorem. Weshall also show that the Vasanti result for the single-

w xquark radiative transition 13 is reproduced in ourscheme while going from the hadron to quark level


( )E.N. Bukina et al.rPhysics Letters B 449 1999 93–9694

Table 1Hyperon radiative transitions, experiment

Transition BR AsymmetryqS ™ pg 1.23"0.06 y0.76"0.080S ™ ng y y0L ™ ng 1.63"0.14 y0J ™ Lg 1.06"0.16 q0.44"0.440 0J ™ S g 3.56"0.43 q0.20"0.32

y yJ ™ S g 0.128"0.023 q1.0"1.3

allowing at the same time any sign of the asymmetryparameter.

2. PV electromagnetic transition current andtoroid dipole moment

Let us consider the PV electromagnetic transitioncurrent of two particles with spin and parity 1r2q.Its possible form is not unique as the most generalexpression can be written in terms of 5 Lorentzstructures g g , P g , k g , s kng and ie -m 5 m 5 m 5 mn 5 mnrl

Ž . Žg P k , where P s p q p , k s p yn r l m 1 2 m m 1. Ž .w xp , s s ir2 g ,g . Due to electromagnetic2 m mn m n

current conservation and generalized Gordon identi-Ž w x.ties see, e.g., 11

u ie P k g g y iDms k½2 mnls n l s 5 mn n

2 ˆq k g ykk g u s0 2.1Ž .5ž /l m m 5 1

2 2 ˆu yik s k qDm k g ykk½ ž /2 l mn n l m m

2q k P y k P k g u s0 2.2Ž . Ž .5l m n n m 5 1

where Dmsm ym and u , u are Dirac spinors1 2 1 2

of the baryons with masses m , this transition1,2

current can be reduced to say one of the followingw xforms 11,12

J Ž A. kŽ .m n

ehs u23 2 2(2p 1yk rMŽ . l

=1

2 P V 2k g ykk G kŽ .Ž .l m m 1 l2M

1P V 2q s k G k g u , 2.3Ž .Ž .mn m 2 l 5 1M

J Ž A. kŽ .m n

ehs u23 2 2(2p 1yk rMŽ . l

=1

Ž .ds k G 0Ž .mn m½ M

2k P y k P kŽ .l m n n m Ž . Ž .d 2 dq G k yG 0Ž .Ž .l2 2M k yDmŽ .l

iŽ .T 2q e P k g g G k g u ,Ž .mnls n l s 5 l 5 12 5M

2.4Ž .2 2(where hs 1yDm rM , Msm qm .1 2

Ž Ž .The corresponding parity-conserving PC cur-Ž .rent can be obtained from Eq. 2.3 just by multiply-

ing every structure by g and changing the super-5. P C, P Vscript PV to PC. In terms of G the decay1,2

w xasymmetry is written as 1

2 Re G P V ) 0 G P C 0Ž . Ž .Ž .2 2as . 2.5Ž .2 2P V P C< < < <G 0 q G 0Ž . Ž .2 2

Ž .But the formfactors introduced by Eq. 2.3 donot correspond to the well-defined multipole expan-

w xsion of currents 11–16 . That is why we would likeŽ .to base our discussion on the Eq. 2.4 which, as has

w xbeen shown explicitly in Ref. 14 , does correspondto a definite multipole expansion in a properly cho-sen reference system, where k 2 sDm2 yk 2. Thism

reference system is given by the equality of kineticŽ .energies e.k.e. of both the baryons involved and

enables us to write the nonrelativistic reduction inw xthe form 14

Žd . 2G k u is k g u AŽ .m 2 mn n 5 1 m

Žd . 2 q ˙w x™G k f sf Eq iDmA ™dEydA ,Ž .e .k .e . 2 1

2.6Ž .ŽT . 2G k u ie g P k u AŽ .m 2 mnrl n r l 1 m

ŽT . 2 w x™G k k= k=s AŽ .e .k .e .

™GŽT . k 2 fqs f ==B. 2.7Ž . Ž .e .k .e . 2 1

Here d is the transition dipole moment, E and Bare the electric and magnetic fields, respectively,f are Pauli spinors of the baryons involved.1,2

( )E.N. Bukina et al.rPhysics Letters B 449 1999 93–96 95

One can see that indeed the parametrization givenŽ .by the Eq. 2.4 is a multipole one where the dipole

transition and toroid dipole transition moments aregiven, respectively, by

ds erM GŽd . 0 s erM GŽd .Dm2 , 2.8Ž . Ž . Ž . Ž . Ž .e .k .e

Ts erM 2 GŽT . 0 s erM 2 GŽT .Dm2 .Ž . Ž .Ž . Ž . e .k .e .

2.9Ž .Žd .Ž 2 .The derivatives of formfactors G k andl

ŽT .Ž 2 .G k define the corresponding transition aver-l

aged radii. Since

k 2 yDm2lP V 2 Žd . ŽT . 2G k sG 0 q G k 2.10Ž . Ž .Ž . Ž .2 l lMDm

we obtain that

erM GV 0 sdyDmT . 2.11Ž . Ž . Ž .2

Note that Dm here has pure kinematical origin,that is with Dms0 the decay discussed would notgo. This formula partly resolves a puzzle with the

Ž .Hara theorem. Indeed, in the SU 3 limit:f

Ø The dipole transition moments of charged hy-peron decays should vanish and presumably stay

w xsmall due to the Ademollo-Gatto theorem 10Ž .even in the presence of SU 3 breaking terms;f

Ø The toroid transition dipole moments defined byŽ .the Eq. 2.9 need not to be zero for these decays

as their contributions decouple automatically inthe limit Dms0.

So the toroid transition dipole moment of Sq´pqg may be at the origin of the large asymmetry

w xobserved 4 .

3. The extension of the Hara theorem

In order to state our result in another way, wewrite the PV part of the radiative transition matrixelement with the Lorentz structure O T sm

Ž .ie P k g in the framework of the SU 3 sym-mnl r n l r fw xmetry approach following strictly 2 as

MsJ ŽT .e qh.c.m m

T 2 T 1 3 T 1s a B O B qB O B½ ž 3 m 1 2 m 1

1 T 2 1 T 3qB O B qB O B /1 m 3 1 m 2

T 3 T 1 2 T 1qc B O B qB O Bž 1 m 2 1 m 3

1 T 3 1 T 2qB O B qB O B e 3.1Ž .5/2 m 1 3 m 1 m

Table 2Hyperon radiative transitions, theory

w x w x Ž .Transition PNC amplitude in Ref. 7 in Ref. 2 from Eq. 3.1

1q TS ™ pg y b 0 c30 d T1 1 1S ™ ng b a a

3 2 2 2' ' '0 d T3 1 1L ™ ng b a a

6 6 6' ' '0 d T1 1 1J ™ Lg b y a a

6 6 6' ' '0 0 d T5 1 1J ™ S g y b y a a

3 2 2 2' ' '5y y TJ ™ S g b 0 c3

b Ž . 3 2where B is the SU 3 baryon octet, B sp, B sa f 1 1

Sq, etc., and aT and cT are up to a factor the toroiddipole moments of the neutral and charged hyperonradiative transitions, respectively.

Here positive signs in front of every baryon bilin-ear combination arise due to Hermitian properties ofthe relevant Lorentz structure.Now all 6 PV radiativetransitions are open in contrast to the Hara result

MsJ Žd .e qh.c.m m

d 2 d 1 3 d 1 1 d 2 1 d 3sa B O B qB O B yB O B yB O B3 m 1 2 m 1 1 m 3 1 m 2ž /3.2Ž .

based on another Lorentz structure form is k gmn n 5w x2 which in turn comes to hadrondynamics fromQED. We display in the Table 2 the results of Eq.Ž .3.1 together with the result of a traditional single-quark radiative transition which we have taken fromw x T7 . The parameter c ;T in the 3rd column of theTable 2 opens a possibility to account for largenonzero asymmetry in the charged hyperon radiative

Ž .decays even in the SU 3 symmetry limit for thef

corresponding coupling constants.

4. New derivation of the Vasanti formula

Radiative hyperon decays were analyzed in Ref.w x13 also at the quark level upon taking into accountchiral invariance considerations. We shall try to red-

w xerive the main result of 13 , namely, that the PVsingle-quark radiative transition s™dqg is pro-

Ž .portional to m ym , using the Lorentz structures d

( )E.N. Bukina et al.rPhysics Letters B 449 1999 93–9696

OT. At the quark level we write for the s´dqg

transition matrix element

Msdg AqBg ie P k g e g s 4.1Ž . Ž .5 5 mnlr n l r m 5

Ž .and upon using Eq. 2.1 , where now all quarkquantities are assumed, arrive at

Msd A m qm qB m ym g is k e s,Ž . Ž .s d s d 5 mn n m

4.2Ž .w xi.e., in fact, the main Vasanti result 13 is repro-

Ž .duced. The factors m "m arrive due to the gen-s d

eralized Gordon identities. The relative signs of Aand B are not fixed here so it is possible to obtain anegative value of the asymmetry parameter. With thechiral invariance induced one gets exactly the Vas-

w x Ž .anti formula 13 since AsB. Note that Eq. 4.2Ž . w xwith AsB was obtained in Ref. 13 upon assum-

Ž . Ž .ing i chiral invariance, ii validity of the originalHara theorem. We have proved in fact that theintroduction of the toroid structure at the quark levelis in some way equivalent to the chiral invariance

w xapproach of 13 and to the diagram approach resultw xof 17 . This result dictates the insertion of the factor

Ž . Žm ym into the parameter c see the 2nd columns d

of the Table 2, and single-quark transition terms inw x w x.Ref. 7,8 , and other works cited in Ref. 5 to assure

the correct behaviour of the corresponding quark PVtransition amplitudes. And Õice Õersa the results ofw x13,17 together with the generalized Gordon identi-ties have just shown that at the quark level it is atoroid dipole moment which is generated with itscharacteristic Lorentz structure OT s ie P k g .mnlr n l r

5. Summary and conclusion

To resolve a contradiction between the experi-ments giving large negative asymmetry in Sq´pqg , the Hara theorem, predicting zero asymmetry for

q y y Ž .S ´pqg and J ´S qg in the exact SU 3 f

symmetry, and quark models which cannot repro-duce the Hara theorem results without making vanish

Ž .all asymmetry parameters in the SU 3 symmetryf

limit, we have considered a parity-violating part ofthe transition electromagnetic current in the alterna-tive form allowing a well-defined multipole expan-sion. Part of it which is connected with the Lorentzstructure ie P k g enables us to reformulate themnlr n l r

Hara theorem thus opening a possibility of nonzeroasymmetry parameters for all 6 weak radiative hy-peron decays and revealing hitherto unseen transitiontoroid dipole moments. Our result is consistent withthe traditional results of the single-quark transition

Ž .models see column 2 of the Table 2 if the relevantparameter has an intrinsic kinematical factor Dm.We also have reproduced the Vasanti formula at thequark level.

References

w x Ž .1 R.E. Behrends, Phys. Rev. 111 1958 1691.w x Ž .2 Y. Hara, Phys. Rev. Lett. 12 1964 378.w x Ž .3 L.K. Gershwin et al., Phys. Rev. 188 1969 2077.w x Ž .4 Particle Data Group, Phys. Rev. D 54 1996 1-I.w x Ž .5 J. Lach, P. Zenczykowski, Int. J. Mod. Phys. A 10 1995

3817.w x Ž .6 A.N. Kamal, Riazuddin, Phys. Rev. D 28 1983 2317.w x Ž .7 R.C. Verma, A. Sharma, Phys. Rev. D 38 1988 1443.w x Ž .8 A.N. Kamal, R.C. Verma, Phys. Rev. D 26 1982 190.w x Ž .9 P. Zenczykowski, Phys. Rev. D 44 1991 1485.

w x Ž .10 M. Ademollo, R. Gatto, Phys. Rev. Lett. 13 1964 264.w x Ž .11 V.M. Dubovik, A.A. Cheshkov, Sov. J. Part. Nucl. 5 1974

318.w x Ž .12 V.M. Dubovik, V.V. Tugushev, Phys. Rep. 187 1990 145.w x Ž .13 N. Vasanti, Phys. Rev. D 13 1976 1889.w x14 E.N. Bukina, V.M. Dubovik, V.E. Kuznetzov, preprint JINR,

P2-97-412, Dubna, Russia, 1997.w x15 G. Barton, Introduction to Dispersion Techniques in Field

Theory, Benjamin, 1965.w x16 R.G. Sachs, Nuclear Theory, Cambridge, 1953; Phys. Rev.

Ž .Lett. 13 1964 286.w x Ž .17 N.G. Deshpande, G. Eilam, Phys. Rev. D 26 1982 2463.

4 March 1999


Coherent exclusive exponentiation CEEX:the case of the resonant eqey collision 1

S. Jadach a,b, B.F.L. Ward a,c,d, Z. Was a,b

a CERN, Theory DiÕision, CH-1211 GeneÕa 23, Switzerlandb Institute of Nuclear Physics, ul. Kawiory 26a, Krakow, Poland´

c Department of Physics and Astronomy, The UniÕersity of Tennessee, KnoxÕille, TN 37996-1200, USAd SLAC, Stanford UniÕersity, Stanford, CA 94309, USA

Received 13 October 1998; revised 14 December 1998Editor: R. Gatto

Abstract

Ž .We present the first-order coherent exclusive exponentiation CEEX scheme, with the full control over spin polarizationfor all fermions. In particular it is applicable to difficult case of narrow resonances. The resulting spin amplitudes and thedifferential distributions are given in a form ready for their implementation in the Monte Carlo event generator. Theinitial-final state interferences are under control. The way is open to the use of the exact amplitudes for two and more hardphotons, using Weyl-spinor techniques, without giving up the advantages of the exclusive exponentiation, of the Yennie-Frautschi-Suura type. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction: What is the problem?

q yThe problem addressed in this work is: How to describe, consistently in the process e e ™ ff , the coherentŽ . Ž .emission of initial state radiation ISR and final state radiation FSR of soft and hard photons, providing for

Ž .cancellations of infrared IR divergences from real and virtual photon emission to infinite perturbative orderŽ .exponentiation , at the level of completely exclusiÕe multiphoton differential distributions, i.e. in the form

Ž .suitable for implementation in Monte Carlo MC event generators? In addition we are looking for the solutionthat is friendliest to narrow s-channel resonances.

Ž . w xThis work is firmly rooted in the work of Yennie, Frautschi and Suura YFS on QED exponentiation 1 andw xits further developments in Refs. 2–4 . The present work definitely goes beyond the scope of these previous

papers – the main difference is the consequent use of spin amplitudes in the exponentiation. Our work is closew xin spirit, although not in technical details, to seminal papers of Greco et al 5,6 . on QED exponentiation for

1 Work supported in part by the US DoE contract DE-FG05-91ER40627 and DE-AC03-76SF00515, Polish Government grants KBN2P03B08414, KBN 2P03B14715, Maria Skłodowska-Curie Fund II PAArDOE-97-316, and Polish-French Collaboration within IN2P3.


( )S. Jadach et al.rPhysics Letters B 449 1999 97–10898

w xnarrow resonances. However, it should be stressed that, contrary to Refs. 5,6 , all our differential multiphotonŽ .distributions are completely exclusive important for MC implementation and we do include hard photons

w xcompletely and systematically. In this context, the work of Ref. 7 should also be mentioned. It implementsQED interferences among eq and ey fermion lines, the analog of the ISR–FSR interferences, for the first timein the exclusive exponentiation. It does not, however, use spin amplitudes for exponentiation as consequently asdoes the present work; it is also rather strongly limited to exact first order exponentiation in the YFS framework.

w xIt is sort of half-way between the present work and the older ones of Refs. 2,3 . At the technical level, the2 Ž .methods used here for the construction of the spin amplitudes are essentially those of Kleiss and Stirling KS

w x w x8,9 , with the important supplement of Ref. 10 , providing for total control of complex phases andror fermionw xspin quantization frames. The MC implementation of the present work will soon be available 11 and it will

w xreplace two MC programs: KORALB 12 , where fermion spin polarizations are implemented exactly, but therew xis no exponentiation, and KORALZ 13 , where exponentiation is included, but the treatment of spin effects is

simplified 3.The present work is essential for any present experiments in eqey colliders and future eqey and mqmy

colliders, where the most important new features for data analysis will be inclusion of ISR–FSR interferencesŽ .and in the next step the exact matrix element for emission of 2 and 3 hard photons, in the presence of many

additional soft ones.

2. Basic KSrrrrrGPS spinors and photon polarizations

Ž .The arbitrary massless spinor u p of momentum p and chirality l is defined according to KS methodsl

w x w x Ž .8,9 . In the following we follow closely the notation of Ref. 10 in particular we also use zsz . In thex

Ž .above framework every spinor is transformed out of the two constant basic spinors u z , of opposite chiralityl

ls", as follows

12u p s pu u z , u z shu u z , h sy1, hz s0. 1Ž . Ž . Ž . Ž . Ž . Ž .l yl q y'2 pPz

Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .The usual relations hold: zuu z s0, v u z su z , u z u z szuv , puu p s0, v u p su p ,l l l l l l l l l l l1Ž . Ž . Ž . Žu p u p spuv , where v s 1qlg . Spinors for the massive particle with four-momentum p withl l l l 52

2 2 .p sm and spin projection lr2 are defined in terms of massless spinorsm m

u p ,l su p q u z , Õ p ,l su p y u z , 2Ž . Ž . Ž . Ž . Ž . Ž . Ž .l z yl yl z l' '2 pz 2 pz

2 Ž . Ž 2 .where p 'p'pyz m r 2z p is the light-cone projection p s0 of the p obtained with the help of theˆz z

constant auxiliary vector z .w xThe above definition is supplemented in Ref. 10 with the precise prescription on spin quantization axes,Ž .translation from spin amplitudes to density matrices also in vector notation and the methodology of connecting

Ž .production and decay for unstable fermions. We collectively call these rules global positioning of spin GPS .Thanks to these we are able to easily introduce polarizations for beams and implement polarization effects for

Ž .final fermion decays of t leptons, t-quarks , for the first time also in the presence of emission of many ISR andFSR photons!

2 We have evaluated several techniques based on Weyl-spinor techniques and we concluded that the technique of KS is best suited for ourŽ .needs exponentiation .

3 The ISR–FSR interference is also neglected in KORALZ, in the main mode with the exponentiation switched on.

( )S. Jadach et al.rPhysics Letters B 449 1999 97–108 99

Ž . Ž . Ž .The GPS rules determining spin quantization frame for u p," and Õ p," of Eq. 2 are summarized asŽ . Ž .follows: a In the rest frame of the fermion, take the z-axis along yz . b Place the x-axis in the plane defined

Ž .by the z-axis from the previous point and the vector h , in the same half-plane as h . c With the y-axis,complete the right-handed system of coordinates. The rest frame defined in this way we call the GPS frame of

w xthe particular fermion. See Ref. 10 for more details. In the following we shall assume that polarization vectorsof beams and of outgoing fermions are defined in their corresponding GPS frames.

The inner product of the two massless spinors is defined as follows)

s p , p 'u p u p , s p , p 'u p u p sy s p , p . 3Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .q 1 2 q 1 y 2 y 1 2 y 1 q 2 q 1 2

The above inner product can be evaluated using the Kleiss-Stirling expressiony1r2 y1r2 m n r ss p ,q s2 2 pz 2 qz pz qh y ph qz y ie z h p q 4Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .q mnrs

Ž . Ž .in any reference frame. In particular, in the laboratory frame we typically use zs 1,1,0,0 and hs 0,0,1,0 ,which leads to the following ‘‘massless’’ inner product

2 3 0 1 0 1 2 3 0 1 0 1( (s p ,q sy q q iq p yp r q yq q p q ip q yq r p yp . 5Ž . Ž .Ž . Ž . Ž . Ž . Ž . Ž .q

Ž .Eq. 2 immediately provides us also with the inner product for massive spinors

u p ,l u p ,l sS p ,m ,l , p ,m ,l ,Ž . Ž . Ž .1 1 2 2 1 1 1 2 2 2

u p ,l Õ p ,l sS p ,m ,l , p ,ym ,yl ,Ž . Ž . Ž .1 1 2 2 1 1 1 2 2 2

Õ p ,l u p ,l sS p ,ym ,yl , p ,m ,l ,Ž . Ž . Ž .1 1 2 2 1 1 1 2 2 2

Õ p ,l Õ p ,l sS p ,ym ,yl , p ,ym ,yl , 6Ž . Ž . Ž . Ž .1 1 2 2 1 1 1 2 2 2

where

2z p 2z p2 1S p ,m ,l , p ,m ,l sd s p , p qd m qm . 7Ž . Ž . Ž .1 1 1 2 2 2 l ,yl l 1z 2z l ,l 1 2( (1 2 1 1 2 ž /2z p 2z p1 2

In our spinor algebra we shall exploit the completeness relations2pu qms u p ,l u p ,l , pu yms Õ p ,l Õ p ,l , ku s u k ,l u k ,l , k s0. 8Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ý Ý Ý

l l l

ŽFor a circularly polarized photon with four-momentum k and helicity ss"1 we adopt the KS choice seew x. 4also Ref. 14 of polarization vector

m mu k g u b u k g u zŽ . Ž . Ž . Ž .) )s s s sm me b s , e z s , 9Ž . Ž . Ž .Ž . Ž .s s' '2 u k u b 2 u k u zŽ . Ž . Ž . Ž .ys s ys s

2 Ž . Ž w x.where b is an arbitrary light-like four-vector b s0. The second choice with u z not exploited in Ref. 8s

often leads to simplifications in the resulting photon emission amplitudes. Using the Chisholm identity 5

mu k g u b g s2u b u k q2u k u b , 10Ž . Ž . Ž . Ž . Ž . Ž . Ž .s m s s s ys ys

mu k g u z g s2u z u k y2u k u z , 11Ž . Ž . Ž . Ž . Ž . Ž . Ž .s m s s s ys ys

4 w xContrary to other papers on Weyl spinor techniques 8,15 we keep here the explicitly complex conjugation in e . This conjugation iscancelled by another conjugation following from Feynman rules, but only for outgoing photons, not for beam photon, as in the Compton

w xprocess, see Ref. 16 .5 mŽ . Ž .For b sz the identity is slightly different because of the additional minus sign in the ‘‘line-reversal’’ rule, i.e. u k g u z ss s

m m mŽ . Ž . Ž . Ž . Ž . Ž .yu z g u k , in contrast to the usual u k g u b squ b g u k .ys ys s s ys ys


Ž .we get two useful expressions, equivalent to Eq. 9 :

'2)

eu k ,b s u b u k qu k u bŽ . Ž . Ž . Ž . Ž .Ž .s s s ys ysu k u bŽ . Ž .ys s

'2)

eu k ,z s u z u k yu k u z . 12Ž . Ž . Ž . Ž . Ž . Ž .Ž .s s s ys ys'2z k

In the evaluation of photon emission spin amplitudes we shall use the following important building block –the elements of the ‘‘transition matrices’’ U and V defined as follows

w k p p s1 2u p ,l eu k ,b u p ,l sU sU k , p ,m , p ,m ,Ž . Ž . Ž . Ž .Ž .1 1 s 2 2 s l l l ,l 1 1 2 21 2 1 2

w k p p s1 2Õ p ,l eu k ,z Õ p ,l sV sV k , p ,m , p ,m . 13Ž . Ž . Ž . Ž . Ž .Ž .1 1 s 2 2 s l l l ,l 1 1 2 21 2 1 2

Ž . 6In the case of u z the above transition matrices are rather simple :s

2z p $2s k ,p , 0Ž .q 1( 2z k

q 'U k , p ,m , p ,m s 2 , 14Ž . Ž .1 1 2 2 2z p 2z p 2z p $1 2 1m ym , s k ,pŽ .2 1 q 2(( (2z p 2z p 2z k2 1

)y qU k , p ,m , p ,m s yU k , p ,m , p ,m , 15Ž . Ž . Ž .l ,l 1 1 2 2 l ,l 2 2 1 11 2 2 1

V s k , p ,m , p ,m sUs k , p ,ym , p ,ym . 16Ž . Ž . Ž .l ,l 1 1 2 2 yl ,yl 1 1 2 21 2 1 2

Ž .The more general case with u b looks a little bit more complicated:s

2zb 2z k 2zb 2zbs p ,k s b , p q m m , m s k , p q m s p ,kŽ . Ž .Ž . Ž .ˆ ˆ ˆ ˆq 1 y 2 1 2 1 q 2 2 q 1( ( (2z p 2z p 2z p 2z p1 2 1 2'2

qU k , p ,m , p ,m s ,Ž .1 1 2 2 s k ,bŽ . 2z k 2z k 2zb 2z kym s b , p q m s p ,b , s p ,b s k , p q m mŽ . Ž .Ž . Ž .ˆ ˆ ˆ ˆ1 y 2 2 y 1 y 1 q 2 1 2( ( (2z p 2z p 2z p 2z p1 2 1 2

17Ž .Ž . Ž .with the same relations 15 and 16 . In the above the following numbering of elements in matrices U and V is

adopted

qq qyŽ . Ž .l ,l s . 18� 4Ž . Ž .1 2 yq yyŽ . Ž .

When analysing the soft real photon limit we shall exploit the following important diagonality property 7

k p p k p pU sV sb k , p d , 19Ž . Ž .Ž . Ž .s l l s l l s l l1 2 1 2 1 2

u k pu u z 2z pŽ . Ž .s s' 'b k , p s 2 s 2 s k , p , 20Ž . Ž . Ž .ˆs s(u k u z 2z kŽ . Ž .ys s

6 w xOur U and V matrices are not the same as the M-matrices of Ref. 9 , but rather products of several of those.7 Ž . Ž Ž ..)Let us also keep in mind the relation b k, p sy b k, p , which can save time in the numerical calculations.ys s


Ž .which also holds in the general case of u b , wheres

2'2 m (b k , p s s b , p s p ,k q 2bz 2z k . 21Ž . Ž . Ž . Ž . Ž . Ž .ˆ ˆs ys sž /s k ,b 2z pŽ . ˆys

3. Born spin amplitudes

y qŽ . Ž . Ž . Ž .Let us calculate lowest order spin amplitudes for e p e p ™ f p f p . For the moment we require1 2 3 4Ž .f/e. Using our basic massive spinors of Eq. 2 with definite GPS helicities and Feynman rules, we define

m e , B f , BÕ p ,l g G u p ,l u p ,l g G Õ p ,lŽ . Ž . Ž . Ž .2 2 1 1 3 3 m 4 4p p p p p 21 2 3 4B X sB X s ie ,Ž . Ž . Ýl l l l l 2 2 21 2 3 4 X yM q iG X rMB B BBsg ,Z

Ge , B s v g e , B , G f , B s v g f , B , 22Ž .Ý Ýl l l l

ls" ls"

f , B Ž .where g are the usual chiral lsq1,y1sR, L coupling constants of the vector boson Bsg ,Z tol

fermion f in units of the elementary charge e.Ž .Spinor products are reorganized with the help of the Chisholm identity 10 , which applies assuming that

Ž .electron spinors are massless, and the inner product of Eq. 7 :X Xe , B f , B e , B f , Bd g g T T qg g U Ul ,yl l yl l l l l l l l l l l1 2 1 1 3 1 2 4 1 1 3 2 1 4p 2B X s2 ie , 23Ž . Ž .Ýl 2 2 2X yM q iG X rMB B BBsg ,Z

where

T su p ,l u p ,l sS p ,m ,l , p ,0,l ,Ž . Ž . Ž .l l 3 3 1 1 3 3 3 1 13 1

XT sÕ p ,l Õ p ,l sS p ,0,yl , p ,ym ,yl ,Ž . Ž . Ž .l l 2 2 4 4 2 2 4 4 42 4

XU su p ,l Õ p ,yl sS p ,m ,l , p ,0,l ,Ž . Ž . Ž .l l 3 3 2 2 3 3 3 2 23 2

U su p ,yl Õ p ,l sS p ,0,yl , p ,ym ,yl . 24Ž . Ž . Ž . Ž .l l 1 1 4 4 1 1 4 4 41 4

We understand that the total s-channel four-momentum X is always the four-vector that enters the s-channelvector boson propagators. Let us stress that the above Born spin amplitudes will be used for p , which do noti

necessarily obey the four-momentum conservation p qp sp qp . This is necessary because, in the1 2 3 4

presence of the bremsstrahlung photons, the relation Xsp qp sp qp does not usually hold. Furthermore,1 2 3 4

any of the p may, and occasionally will, be replaced by the momentum k of one of photons. In this case, thei

spinor into which k enters as an argument is understood to be massless.

4. First order, one virtual photon

Ž 1.The OO a contribution with one virtual and zero real photon reads

Ž1. p p 2 2 pMM X sB X 1qQ F s,m qQ F s,m qMM X , 25Ž . Ž . Ž . Ž . Ž . Ž .0 l l e 1 g f 1 g box l

where F is the standard electric form-factor regularized with photon mass. We omit, for the moment, the1

magnetic form-factor F ; this is justified for light final fermions. It will be restored in the future. In F we keep2 1

the exact final fermion mass.


In the present work we use spin amplitudes for g-g and g-Z boxes in the small mass approximation2 2 w xm rs™0,m rs™0, following Refs. 17,18 ,e f

g e , Bg f , B T T X qg e , Bg f , B U X U al yl l l l l l l l l l l1 1 3 1 2 4 1 1 3 2 1 4p 2MM X s2 ie d d Q QŽ . ÝBox l l ,yl l ,yl e f2 2 2 1 2 3 4 pX yM q iG X rMB B BBsg ,Z

= 2 2d f M ,m ,s,t ,u yd f M ,m ,s,u ,t , 26Ž .ž / ž /l ,l BDP B g l ,yl BDP B g1 3 1 3

2 2 2 2 Ž . w xwhere M sM y iM G , M sm , and the function f is defined in eq. 11 of Ref. 18 . TheZ Z Z Z g g BDP

Mandelstam variables s,t and u are defined as usual. Since in the rest of our calculation we do not use2 Žm rs™0, we therefore intend to replace the above box spin amplitudes with the finite-mass results. NB: Forf

8 w x .the g-g box the spin amplitudes with the exact final fermion mass were given in Ref. 12 .

5. First order 1-photon, ISR alone

In order to introduce the notation gradually, let us first consider the 1-photon emission matrix elementseparately for ISR. The first order, 1-photon, ISR matrix element from the Feynman rule reads

eQeISR p p k w1 2MM s Õ p ,l M pu qmyku eu k u p ,lŽ . Ž . Ž .Ž .Ž .1 l l s 2 2 1 1 s 1 11 2 2kp1

eQe wq Õ p ,l eu k ypu qmqku M u p ,l , 27Ž . Ž . Ž . Ž .Ž .2 2 s 2 1 1 12kp2

Ž .where M is the annihilation scattering spinor matrix including final state spinors . The above expression we19 Ž .split into soft IR parts proportional to pu "m and non-IR parts proportional to ku. Employing the

Ž .completeness relations of Eq. 13 to those parts we obtain:

eQ eQe eISR p p k p p k p p k p p p p1 2 1 2 1 1 2 2 1 2MM s B U y V BŽ . Ž .Ý ÝŽ .1 l l s 1 rl s rl s l r 1 l r1 2 2 1 2 12kp 2kp1 2r r

eQ eQe ek p k k p k p k p k2 1 2 1y B U q V B , 28Ž .Ž . Ž .Ý Ý1 rl s rl s l r 1 l r2 1 2 12kp 2kp1 2r r

p p1 2 Ž . Ž .where B sÕ p ,l M u p ,l . The summation in the first two terms gets eliminated due to the1 l l 2 2 1 1 11 2

Ž .diagonality property of U and V, see Eq. 19 , and leads to

ISR p p k Ž1. p p Ž1. p p k1 2 1 2 1 2MM ss k B qr k ,Ž . Ž .Ž .1 l l s s 1 l l l l s1 2 1 2 1 2

eQ eQe eŽ1. p p k k p k k p k p k p k1 2 2 1 2 1r k sy B U q V B ,Ž . Ž . Ž .Ý Ýl l s 1 rl s rl s l r 1 l r1 2 2 1 2 12kp 2kp1 2r r

22 2b k , p b k , p e Q p pŽ . Ž .s 1 s 2 e 1 22Ž1. Ž1.< <s k seQ yeQ , s k sy y . 29Ž . Ž . Ž .s e e s ž /2kp 2kp 2 kp kp1 2 1 2

The soft part is now clearly separated and the remaining non-IR part, necessary for the CEEX, is obtained. Thecase of final state one real photon emission can be analysed in a similar way.

8 It seems, however, that the g-Z box for the heavy fermion is missing in the literature.9 w xThis kind of separation was already exploited in Refs. 19 . We thank E. Richter-Was for attracting our attention to this method.


6. First order 1-photon ISRHFSR

The first order, ISRqFSR, 1-photon matrix element, with explicit split into IR and non-IR parts, readsŽ1. p k Ž1. p Ž0. p Ž1. p k Ž0. p kMM ss k B Pyk qs k B P qr Pyk qr P , 30Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .1 ls s l s l ls ls

p p p p p1 2 3 4where we use the compact notation ' , and the lowest order Born spin amplitudes B arel l l l l1 2 3 4

Ž .defined in Eq. 23 . The other ingredients are the initial state non-IR part:

yeQ eQe eŽ1. p k k p k k p p p k k p p k p p1 2 3 4 2 1 3 4r X s U B X q V B XŽ . Ž . Ž .Ž . Ž .Ý Ýls s l r rl l l s rl l rl l1 2 3 4 2 1 3 42kp 2kp1 2r r

31Ž .and the final state non-IR part

eQ eQf fŽ0. p k k k p p p k p k p k p p p k3 1 2 4 4 1 2 3r X sy U B X q V B X .Ž . Ž . Ž .Ž . Ž .Ý Ýls s rl l l rl s l r l l l r3 1 2 4 4 1 2 32kp 2kp3 4r r

32Ž .The FSR s-factor

22 2b k , p b k , p e Q p pŽ . Ž .s 3 s 4 f 3 42Ž0. Ž0.< <s k syeQ qeQ , s k sy y 33Ž . Ž . Ž .s f f s ž /2kp 2kp 2 kp kp3 4 3 4

we define analogously to the ISR case.

7. Coherent exclusive exponentiation, zero and first order

Ž 0.Spin amplitudes in the zero-th order coherent exclusive exponentiation, OO a , we define as followsCEEX

X 2`Ž0. p k k k a B Ž p ,..., p . p ` ` `1 2 n 4 1 4 1 2 nMMMMM . . . se B X s k s k . . . s k , 34Ž . Ž . Ž . Ž . Ž .ÝŽ .n ls s s l ` s 1 s 2 s n21 2 n 1 2 np qpŽ .� 4` 3 4

where the s-channel four-momentum in the resonance propagator is X sp qp yÝn ` k . The partition `` 1 2 is1 i iŽ .is defined as a vector ` ,` , . . . ,` where ` s1 for ISR and ` s0 for FSR photon, see the analogous1 2 n i i

w xconstruction in Refs. 5,6 . For a given partition X is therefore the total incoming four-momentum minus`� 4 nfour-momenta of ISR photons. The coherent sum is taken over set ` of all 2 partitions – this set is explicitly

the following

� 4` s 0,0,0, . . . ,0 , 1,0,0, . . . ,0 , 0,1,0, . . . ,0 , 1,1,0, . . . ,0 , . . . 1,1,1, . . . ,1 . 35� 4Ž . Ž . Ž . Ž . Ž . Ž .Ž . w xIn Eq. 34 we profit from the Yennie-Frautschi-Suura 1 fundamental proof of factorization of all virtual IR

10 Ž .corrections in the form-factor exp aB , where4

B p , . . . , p sQ2 B p , p qQ2B p , p qQ Q B p , p qQ Q B p , pŽ . Ž . Ž . Ž . Ž .4 1 4 e 2 1 2 f 2 3 4 e f 2 1 3 e f 2 2 4

yQ Q B p , p yQ Q B p , p ,Ž . Ž .e f 2 1 4 e f 2 2 3

24d k i 2 pqk 2 qykB p ,q ' q . 36Ž . Ž .H2 2 2 3 2 2ž /k ym q ie k q2kpq ie k y2kqq ie2pŽ .g

10 In the LL approximation it is, of course, the doubly-logarithmic Sudakov form-factor.


In the above we assume that IR singularities are regularized with a finite photon mass m which enters into allg

Ž .B ’s and implicitly into s-factors and the real photon phase space integrals, in the following discussion .22 Ž .2The auxiliary factor FsX r p qp is, from the formal point of view, not really necessary. Note that the3 4

F-factor does not affect the soft limit; it really matters if at least one very hard FSR photon is present. However,Ž 1.the F-factor is very useful, because it is present in the photon emission matrix element, both in OO a and also

Ž .in all orders in the leading logarithmic LL approximation. It has also been present for a long time now in thew x‘‘crude distribution’’ in the YFS-type Monte Carlo generators, see for instance Ref. 3 . It is therefore natural to

Ž 0.include it already in the OO a exponentiation. Otherwise, this F-factor will be included order by order.However, in such a case, the convergence of perturbative expansion will be deteriorated. As we shall see below,the introduction of the F-factor will slightly complicate the first order exponentiation.

Ž 1.The complete set of spin amplitudes for emission of n photons we define in OO a as follows:CEEX

nŽ p , . . . , p . Ž .Ž1. p k k k a B ` p 1 p1 41 2 n 4 iMMMMM . . . se s k B X 1qd qRRRRR XŽ . Ž . Ž .Ž .Ý ŁŽ .n ls s s s i l ` Virt Box l `1 2 n i žis1� 4`

nŽ` . p kj jq RRRRR X ,Ž .Ý 1 ls `j /

js1

21 p qp qv kŽ .3 4 jŽ .Žv . p k v p k pRRRRR X ' r X q y1 B X , vs"1. 37Ž . Ž . Ž . Ž .1 ls ls lv 2s k ž /Ž . p qpŽ .s 3 4

The IR-finite d Ž1. and RRRRR are determined unambiguously by identifying for ns0 the above equation withVirt BoxŽ . Ž 1.Eq. 25 , up to terms of OO a . We obtain

d Ž1. s sQ2 F s,m qQ2F s,m yQ2aB s,m yQ2aB s,m . 38Ž . Ž . Ž . Ž . Ž . Ž .Virt e 1 g f 1 g e 2 g f 2 g

The RRRRR is obtained from MM by means of the substitution 11Box Box

2 2f M ,m ,s,t ,u ™ f M ,m ,s,t ,u y f m ,t ,u , 39Ž .Ž .ž / ž /BDP B g BDP B g IR g

where

2 2 t m2 1 tgf m ,t ,u s B m ,t y B m ,u s ln ln q ln . 40Ž . Ž .Ž . Ž .IR g 2 g 2 g ž / ž /ž /'p p u 2 utu

Žv . Ž . Ž .Similarly the IR-finite RRRRR is determined uniquely by identifying, for ns1, Eq. 37 with Eq. 30 . In1Ž .2 Ž .2particular the factor Fy1s p qp qv k r p qp y1 is a consequence of the introduction of the3 4 j 3 4

Ž . Ž .F-factor in Eq. 34 . If it was not included, then the 1-photon part in Eq. 37 would not reduce to the amplitudeŽ . Ž .of Eq. 30 . Thanks to the presence of Fy1, for ns1, we recover in Eq. 37 the correct first order amplitudeŽ .of Eq. 29 .

For very narrow resonances the photon emission in the decay process is separated from the photon emissionin the production process by very large time-space distance. The ISR)FSR interference is therefore strongly

0 'suppressed, typically by GrM factors. Since our real photons are present down to arbitrarily low k se s r2min

<G , the effects due to the resonance complex phase in the emission of the real photons are taken into accountnumerically and exactly. For Õirtual photons we have to sum up analytically certain subset of the ISR)FSR

11 In the above procedure of subtracting IR divergences, there is no reference to cut on photon energy, only reference to the photon mass,similar to the YFS exponentiation on squared spin-summed amplitudes.


w xinterferences to infinite order following Greco et al. 5,6 . In practice the rule is: multiply each part of the spinŽ Ž ..amplitude proportional to Z-propagator by the additional factor exp d s,t,u where:G

a t M 2 y iM G ysZ Z Zd s,t ,u sy2Q Q ln ln 41Ž . Ž .G e f 2ž / ž /p u MZ

Ž 1. Ž .In OO a the above exponential factor induces the additional subtraction in the g-Z box: MMMMM s,t,u ™boxŽ . Ž .MMMMM s,t,u yd s,t,u . Strictly speaking the above improvement is not really necessary, because we wouldbox G

have obtained it order-by-order, through higher order virtual non-IR correction. In practice, however, it ismandatory. If we had not made it, then the ISR)FSR interference contribution to A at Z peak fromFBŽ 1.OO a would be dramatically wrong, i.e. 0.5% instead of 0.05%!CEEX

8. Differential cross sections and the YFS form-factor

Ž r .The master formula for the unpolarized OO a total cross section is given by the standard quantum-CEEXŽmechanical expression of the type ‘‘matrix element squared modulus times phase space’’ contrary to typical

.‘‘parton shower’’ approach` 1 1 2Ž .Ž r . r p k k k1 2 ns s dt p qp ; p , p , k , . . . ,k MMMMM . . . , 42Ž . Ž .Ý ÝH Ž .n 1 2 3 4 1 n n ls s s1 2 nn! 4ns0 l ,s , . . . ,s s"1 n

Ž .where the Lorentz invariant phase space LIPS integration element isn n 3d pj4 Ž4.dt P ; p , p ,... p ' 2p d Py p . 43Ž . Ž . Ž .Ý ŁH Hn 1 2 n j 30ž / 2 p 2pjs1 Ž .js1 j

The above total cross section is perfectly IR-finite, as can be checked with a little bit of effort by analyticalpartial differentiation 12 with respect the photon mass

EŽ r .s s0. 44Ž .

E mg

Ž .Furthermore, the integral of Eq. 42 is perfectly implementable in the Monte Carlo form, using a method veryw xsimilar to those in Ref. 3 . Traditionally, however, the lower boundary on the real soft photons is defined using

0 'the energy cut condition k )´ s r2 in the laboratory frame. The practical advantage of such a cut is the lowerphoton multiplicity in the MC simulation, and consequently a faster computer program 13. If the above energycut on the photon energy is adopted, then the real soft-photon integral between the lower LIPS boundary defined

Žby m and that defined by ´ can be evaluated by hand and summed up rigorously the only approximation isg

˜. Ž Ž ..m rm ™0 into an additional overall factor exp 2aB p , . . . , p , whereg e 4 1 4

˜ 2 ˜ 2 ˜ ˜ ˜B p , . . . , p sQ B p , p qQ B p , p qQ Q B p , p qQ Q B p , pŽ . Ž . Ž . Ž . Ž .4 1 4 e 2 1 2 f 2 3 4 e f 2 1 3 e f 2 2 4

˜ ˜yQ Q B p , p yQ Q B p , p ,Ž . Ž .e f 2 1 4 e f 2 2 3

3 2d k y1 p qŽ .B p ,q ' y . 45Ž . Ž .H2 0 2 ž /0 kp kqk 8pk -´ s r2'

12 w x w xThis method of validating IR-finiteness was noticed by G. Burgers 20 . The classical method of Ref. 1 relies on the techniques of theMelin transform.

13 0 'The disadvantage of the cut k )´ s r2 is that in the MC it has to be implemented in different reference frames for ISR and for FSRw x– this costs the additional delicate procedure of bringing these two boundaries together, see Ref. 11 andror discussion in the analogous

w xt-channel case in Ref. 4 .


Ž r . ya B4 Ž r . Ž .Let us introduce M se MMMMM without virtual IR singularities and, altogether, the above reorganizationn n

yields the new expression for the unpolarized total cross-section` 1 1 2Ž p , . . . , p . Ž .Ž r . Y r p k k k1 4 1 2 ns s dt p qp ; p , p , k , . . . ,k e M . . . 46Ž . Ž .Ý ÝH Ž .n 1 2 3 4 1 n n ls s s1 2 nn! 4ns0 l ,s s"i

˜Ž . Ž . Ž .where Y p , . . . , p s2aB p , . . . , p q2a R B p , . . . , p is the conventional YFS form-factor defined1 4 4 1 4 4 1 4

analytically in terms of logs and Spence functions – we do not show it here explicitly due to lack of space, seew xRefs. 7,21,22,11 . In the YFS form-factors we keep the final fermion mass exact. The fully exclusive

Ž . w xdifferential cross section of Eq. 46 is already implemented in the Monte Carlo event generator KKKK 11 .Ž 2 .The extension of the above exponentiation procedure to OO a and beyond requires more work, butCEEX

does not pose any conceptual problem. It will be implemented in the future version of the KKKK Monte Carlo.

9. Fermion spin polarization and photon spin randomization

The great advantage of working with spin amplitudes is the easiness of introduction of full spin polarizationsfor all particles. The general case of the total cross section with polarized beams and decays of unstable final

w xfermion being sensitive to spin polarization 23–25,10 reads` 1 Ž p , . . . , p .Ž r . Y 1 4s s dt p qp ; p , p , k , . . . ,k eŽ .Ý H n 1 2 3 4 1 nn!ns0

=3

wŽ .a b a b Ž r . p k k k r p k k k c d c d1 2 n 1 2 n ˆ ˆ´ ´ s s M . . . M . . . s s h h ,ˆ ˆÝ Ý Ý Ž . Ž .1 2 l l l l n ls s s n ls s s l l l l 3 41 1 2 2 1 2 n 1 2 n 3 3 4 4

s a ,b ,c ,ds0 l ,li i i

47Ž .where, for ks1,2,3, s k are Pauli matrices and s 0 sd is the unit matrix. The components ´ a,´ b,a,bsˆ ˆl,m l,m 1 2

1,2,3 are the components of the conventional spin polarization vectors of ey and eq respectively, defined in theŽ w x . 0so-called GPS fermion rest frames see Ref. 10 for the exact definition of these frames . We define ´ s1 in aî

ˆŽ .non-standard way i.e. p P´ sm . The polarimeter vectors h are similarly defined in the appropriate GPSî i e iˆ ˆŽ .rest frames of the final unstable fermions p Ph sm . Note that, in general, h may depend in a non-triviali i f i

w xway on momenta of all decay products, see Refs. 25,24 for details. We did not introduce polarimeter vectorsfor bremsstrahlung photons, i.e. we take advantage of the fact that the high energy experiment is completelyblind to photon spin polarizations.

Ž .Let us finally touch briefly upon one very serious problem and its solution. In Eq. 47 the single spinŽ1. nŽ . Ž n .amplitude M already contains 2 nq1 terms due to 2 ISR–FSR partitions . The grand sum over spins inn

Ž . n 4 4 nq16 2 nq16Eq. 47 counts 2 4 4 s2 terms! Altogether we expect up to N;n2 operations in the CPU timeŽ . y q y qexpensive complex 16 bytes arithmetics. Typically in e e ™m m the average photon multiplicity with

k 0 )1MeV is about 3, corresponding to N;107 terms. In a sample of 104 MC events there will be a couple ofevents with ns10 and Ns1012 terms, clearly something that would ‘‘choke’’ completely any modern, fastworkstation. There are several simple tricks that help to soften the problem; for instance, objects such as

a aÝ ´ s and the s-factors are evaluated only once and stored for multiple use. This is however not sufficient.â i ll

What really helps to substantially speed up the numerical calculation in the Monte Carlo program is thefollowing trick of photon spin randomization. Instead of evaluating the sum over photon spins s , is1, . . . , ni

Ž . Ž .in Eq. 47 , we generate randomly one spin sequence of s , . . . , s per MC event and the MC weight is1 n

calculated only for this particular spin sequence! In this way we save one hefty 2 n factor in the calculationtime 14. Mathematically this method is correct, i.e. the resulting cross section and all MC distribution will be the

14 The other 2 n factor due to coherent summation over partitions cannot be eliminated, unless we give up on narrow resonances.


Ž . Žsame as if we had used in the MC weight the original Eq. 47 see a formal proof of the above statement inw x.Sect. 4 of Ref. 26 . Let us stress again that it is possible to apply this photon spin randomization trick because

Ž .a the typical high energy experiment is blind to photon spin polarization, so that we did not need to introduceŽ . Ž .in Eq. 47 the polarimeter vectors for photons, and b For our choice of photon spin polarizations the cross

section is rather weakly sensitive to them, so the method does not lead to significant loss in the MC efficiency.

10. Conclusions

We presented the first order coherent exclusive exponentiation CEEX scheme, with the full control over spinpolarization for all fermions. This new method of exponentiation is very general and has many immediate andlonger term advantages. The immediate profit will be the inclusion of the ISR–FSR interferences andavailability of the exact distributions for multiple hard photons without giving up on exclusive, YFS-style,exponentiation. In particular it is applicable to difficult case of the narrow resonances. The resulting spin

Žamplitudes and the differential distributions are readily implemented in the MC event generator. Numerical.results will be presented elsewhere.

Acknowledgements

Ž .We thank the CERN Theory Division and all four LEP collaborations for support. One of us Z.W.acknowledges specially the support of the ETH L3 group during the final work on the paper preparation. Usefuldiscussions with E. Richter-Was are warmly acknowledged. We also thank W. Płaczek for correcting themanuscript.

References

w x Ž . Ž .1 D.R. Yennie, S. Frautschi, H. Suura, Ann. Phys. NY 13 1961 379.w x Ž .2 S. Jadach, B.F.L. Ward, Phys. Rev. D 38 1988 2897.w x Ž .3 S. Jadach, B.F.L. Ward, Comput. Phys. Commun. 56 1990 351.w x Ž .4 S. Jadach, E. Richter-Was, B.F.L. Ward, Z. Was, Comput. Phys. Commun. 70 1992 305.w x Ž .5 M. Greco, G. Pancheri-Srivastava, Y. Srivastava, Nucl. Phys. B 101 1975 234.w x Ž . w Ž . x6 M. Greco, G. Pancheri-Srivastava, Y. Srivastava, Nucl. Phys. B 171 1980 118 Erratum: B 197 1982 543 .w x Ž .7 S. Jadach, W. Płaczek, B.F.L. Ward, Phys. Lett. B 390 1997 298, also hep-phr9608412; The Monte Carlo program BHWIDE is

available from http:rrhephp01.phys.utk.edurpubrBHWIDE.w x Ž .8 R. Kleiss, W.J. Stirling, Nucl. Phys. B 262 1985 235.w x Ž .9 R. Kleiss, W.J. Stirling, Phys. Lett. B 179 1986 159.

w x10 S. Jadach, B.F.L. Ward, Z. Was, Global Positioning of Spin GPS Scheme for Half Spin Massive Spinors, July 1989, preprintCERN-TH-98-235, submitted to Phys. Lett. B.

w x11 S. Jadach, B.F.L. Ward, Z. Was, KKKK Monte Carlo for fermion pair production at LEP2 and Linear Colliders, 1998, to be published inComput. Phys. Commun., in preparation.

w x Ž .12 S. Jadach, Z. Was, Comput. Phys. Commun. 36 1985 191.w x Ž .13 S. Jadach, B.F.L. Ward, Z. Was, Comput. Phys. Commun. 79 1994 503.w x Ž .14 Zhan Xu, Da-Hua Zhang, L. Chang, Nucl. Phys. B 291 1987 392.w x Ž . Ž . Ž . Ž .15 CALKUL Collaboration, Phys. Lett. B 103 1981 124; B 105 1981 215; B 114 1982 203; Nucl. Phys. B 206 1982 53, 61; B 239

Ž .1984 382.w x Ž .16 A. Gongora, R.G. Stuart, Z. Phys. C 42 1989 617.´w x Ž .17 Z. Was, Acta Phys. Polon. B 18 1987 1099.w x Ž .18 R.W. Brown, R. Decker, E.A. Paschos, Phys. Rev. Lett. 52 1984 1192.w x Ž .19 E. Richter-Was, Z. Phys. C 61 1994 323.w x20 G. Burgers, private communication, 1990, unpublished.


w x Ž .21 S. Jadach, W. Płaczek, M. Skrzypek, B.F.L. Ward, Phys. Rev. D 54 1996 5434.w x Ž .22 S. Jadach, W. Płaczek, M. Skrzypek, B.F.L. Ward, Z. Was, Phys. Lett. B 417 1998 326.w x Ž . w Ž . x23 S. Jadach, Z. Was, Acta Phys. Polon. B 15 1984 1151 Erratum: B 16 1985 483 .w x Ž .24 S. Jadach, Z. Was, R. Decker, J.H. Kuhn, Comput. Phys. Commun. 76 1993 361.¨w x Ž .25 S. Jadach, Acta Phys. Polon. B 16 1985 1007.w x26 S. Jadach, Guide to practical Monte Carlo methods, 1998, available from http:rrhome.cern.chr ; jadach, unpublished.

4 March 1999


Search for DCC in relativistic heavy-ion collisionthrough event shape analysis

B.K. Nandi a, G.C. Mishra a, B. Mohanty a, D.P. Mahapatra a, T.K. Nayak b

a Institute of Physics, Bhubaneswar, Indiab Variable Energy Cyclotron Centre, Calcutta, India

Received 8 June 1998; revised 13 January 1999Editor: L. Montanet

Abstract

Event shape analysis has been used to look for DCC signals in simulated ultra-relativistic heavy-ion collision data at SPSenergy. A simple redistribution of particles, with two detectors to detect charged particles and photons, is seen to result inthe same flow direction with the flow angle difference peaking at zero. However, events where the neutral pion fraction hasbeen modified according to the DCC probability distribution, show the flow angles in two detectors to be almost 908 apart.The results presented here show that the technique is complementary to the one based on the discrete wavelet transformation.Together the techniques are seen to provide a very powerful tool for DCC search in ultra relativistic heavy ion collision.q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

In heavy-ion collisions at ultra-relativistic ener-gies there is a rapid expansion of the collision debrisin the longitudinal direction leading to a super cool-ing of the interior interaction region. This is expectedto lead to the formation of unconventionally orientedvacuum structures as allowed by the chiral symmetryw x1–3 . These are called the Disoriented Chiral Con-

Ž .densates DCC . DCC formation results in largefluctuation in the neutral pion fraction. The probabil-ity distribution of the neutral pion fraction is charac-terized by,

1P f s 1Ž . Ž .'2 f

where

N 0 N r2p gfs ; 2Ž .

0 q yN qN qN N r2qNp p p g ch

N and N are multiplicities for photons and chargedg ch

particles, respectively. This assumes all charged par-ticles are pions and all photons come from p 0

decay.Detection of this interesting phenomenon of DCC

formation is expected to provide valuable informa-tion on the vacuum structure of strong interactionand chiral phase transition. Therefore, it has becomea very interesting aspect of heavy ion collision stud-ies and a number of experiments have been plannedat RHIC and LHC energies.

In an experiment involving the search of localisedDCC one essentially tries to detect a state with a


( )B.K. Nandi et al.rPhysics Letters B 449 1999 109–113110

large and localised fluctuation in the ratio of thenumber of photons to charged particles. In principlethere should be regions in the detected phase spacewhere the number f should be very much differentfrom 1r3. A typical event structure would be verysimilar to the anti-centauro event as reported by the

w xJACEE collaboration 4 . In view of this, a typicalexperiment would consist of two detectors to detectthe charged particles and the photons respectively.These two detectors should have complete overlap inh and f with as much h coverage as possible. Fromthe detected hit patterns of charged and neutral parti-cles one tries to see whether there is any localfluctuation of f indicating the signature of localisedDCC.

As far as DCC search goes many analysis meth-w xods have been proposed 1,5–9 . However, the most

attractive method has been the one based on theŽ .Discrete wavelet transforms DWT proposed by

w xHuang et al. 6 . This has the beauty of analysing aspectrum at different length scales with the ability offinally picking up the right scale at which there is afluctuation. Because of this advantage, although therehas been no claim regarding the observation of DCC,the method was successfully applied to filter outinteresting events with large photon to charge parti-

w xcle fluctuation in WA98 experiment 7,8 .In the present paper, using simulated data, at-

tempts have been made to show the power of yetanother analysis technique which has been used suc-

w xcessfully for flow analysis in heavy-ion data 10,11 .The method is based on the simple fact that localisedDCC formation is expected to lead to an event shapeanisotropy which is expected to be out of phase forone detector corresponding to the other. In otherwords it means, whenever there is large number ofcharge particles recorded in a DCC region in onedetector, there should be a depletion in the numberof neutral particles in the same zone of the otherdetector. Therefore, from an event shape analysisusing both detectors one can, in principle, look forDCC signals. On the other hand, in the DWT analy-sis, one tries to look at the neutral pion fraction f atdifferent resolutions constructing what are known as

Ž .the Father function coefficients FFC and MotherŽ .function coefficients MFC . Without DCC, the FFC

distribution for simulated generic events has beenshown to be a Gaussian. However with DCC, the

distribution goes to a non-Gaussian shape with sev-w xeral events appearing in the wings 7 The events

which lie beyond the generic Gaussian can be pickedup as DCC-like events. One can also construct thepower spectrum for the FFCs and look for theirvariation as a function of the length scale. For genericevents with only a statistical fluctuation in the num-bers of charged and neutral pions the FFC powerspectrum shows a flat curve without a structure atany scale. However, with DCC-like fluctuation, gen-erated over a given domain of phase space, thepower spectrum is expected to show an enhancementat scales below the specified domain size.

In the present case we have carried out simulationof DCC-like and pure flow-type events and haveapplied the technique of event shape analysis whichdistinguishes very clearly both class of events. Incase of simulated DCC-like events, particularly whentheir fraction in a large number of events is compara-tively small, the present method, together with thatbased on DWT, has been found to be successful infinding out DCC-type signature.

2. Modeling DCC and event anisotropy

For DCC production, a procedure which is similarw xto the one employed in Ref. 5 has been followed.w xVENUS 4.12 event generator 12 has been used for

the simulation of DCC type events at SPS energyŽ .Pb on Pb . In this, the charge of the pions is

Ž q y 0 0.interchanged pairwise p p lp p , in a se-lected hyf zone according to the DCC probability

Ž .distribution as given in Eq. 1 event by event.Finally the p 0s are allowed to decay.

For the present study, DCC events have beensimulated in a range 3.0FhF4.0 with a domainsize having Dfs908. For the analysis 20,000 eventswere generated. A similar amount of VENUS eventswere also generated for comparison. However, tosimulate what happens in a true experimental situa-tion it is also necessary to include detector relatedeffects. For photon and charge particle detection thedetection efficiencies were taken to be about 70"5%and 95"2% respectively. It is also known thatcharged particles sometimes lead to photon-like sig-nals and such a contamination in an experimental

w xsituation can be as high as 25% 13 . Following this,

( )B.K. Nandi et al.rPhysics Letters B 449 1999 109–113 111

it was decided to include a 25% charged particlecontamination in the photon signal. Finally, we havealso prepared several sets of data with different DCC

Ž .fractions 10–100% mixing generic VENUS andDCC type events in an appropriate manner.

To introduce event anisotropy in every event, asimple toy model has been employed. Here thedistributions for both charged and neutral particlesare generated from VENUS, distorted according to aprocedure as given below. First of all a flow direc-tion is selected at random, distributed uniformlybetween 08 and 3608. About each flow direction,corresponding to a given event, a Gaussian particledistribution, with a s of 108 is generated by pickingVENUS generated particles at random. Here thecharge conservation in every localised region is en-sured since all three types of particles are selectedwith equal probability.

Constructed as above, both localised DCC andŽ .simple event anisotropy indicating flow are mod-

eled in different events and the results were analysedusing the standard flow analysis as employed else-

w xwhere 11 . In the present study the method ofŽ .Fourier analysis with ns2 elliptic flow has been

employed.

3. Method of analysis

The particle distribution in a given detector can beŽ .written as a set points h ,f ,is1, N showing thei i

hit pattern in the detector. In the second order ellipticflow, for each event, one tries to construct the sums

Xs cos 2f 3Ž . Ž .Ý i

Ys sin 2f 4Ž . Ž .Ý i

The flow angle F is determined from these twosums using the expression

1 y1Fs tan YrX 5Ž . Ž .2

In the absence of any detector imperfections andother geometrical effects, the distribution of F , takenover a large number of events is expected to be flatwithout any peaks or bumps spread over 08 to 1808.This is because the flow direction varies randomlyfrom event to event. However, when the events arerealigned, with respect to the flow angle in each

Ževent, one can see the characteristic peaks at 0 and.1808 in the azimuthal distribution of particles.

In case of two detectors with the same phaseŽ .space hyf coverage, one detecting photons and

other detecting charged particles the situation is veryinteresting. If there is genuine flow in a particularevent both detectors would show the effect in termsof F angles getting aligned in the same direction.

Ž .Therefore the distribution of C sF yF , the1 2

difference between the flow angles, as obtained forthe two detectors is expected to be peaking at zero.However, in case of DCC being prominent in aparticular region, there will be more photons de-tected in one detector. The other detector is expectedto show less charged particles in the same region ofphase space. Therefore an event shape analysis isexpected to show two flow angles for both detectors

Žwhich will be out of phase with ns2 the angular.difference is expected to peak at 908 . This is the

most important result based on which the presentanalysis is carried out.

4. Results and discussions

Ž .Results shown in Fig. 1 a–c correspond to thecase with only DCC in which one can notice

Ž .Fig. 1. The distribution of C for simulated DCC a–c and flowŽ . Ž .d–f . a Anti-correlation between event planes determined for

Ž . Ž .charged particle and photon detectors. b c Correlations be-tween event planes obtained considering two sub events for thephoton and charged particle detectors respectively. The dotted

Ž .lines in a–c show the distributions for generic VENUS events.Ž .In case of flowy events d shows correlation between event

Ž . Ž . Ž .planes obtained from the two detectors. e and f are same as bŽ .and c .

( )B.K. Nandi et al.rPhysics Letters B 449 1999 109–113112

Ž Ž ..Fig. 1 a a clear peak for the angle between theevent planes for the two detectors, C , at 908. Fig.Ž . Ž .1 b and c show the angle between the event planes

as obtained for two sub-events in each of the detec-tors separately It is important to notice that bothdetectors show ‘‘flow’’, but one with respect to theother clearly shows an anti-flow type behavior. Inthe same figure we have also presented the datacorresponding to pure VENUS events for compari-son. One can notice there is neither any signature offlow nor DCC-like fluctuation.

Ž .Results shown in Fig. 1 d–f show the same plotsfor flow type events which have no DCC-like fluctu-ation. Here, the individual detectors are seen to showthe same effect as one with respect to the other.

In Fig. 2 we have presented event shape analysisresults for 20000 events having different fractions ofDCC-type fluctuation ranging from 10% to 100%.One can notice that the expected peak around 908

gets weaker as DCC fraction decreases. This isprimarily because of an overwhelming majority ofgeneric events that contribute uniformly to the C

distribution over the entire angular range. So in orderto look for any DCC-like signature one has to sup-press the contribution of these events filtering out theinteresting DCC-like ones.

Fig. 2. The distribution of C for simulated DCC and VENUSŽ . Ž .with varying DCC fractions a 100% of the events DCC, b 75%

Ž . Ž .of the events DCC, c 50% of the events DCC, d 25% of theŽ . Ž .events DCC, e 15% of the events DCC f 10% of the events

DCC.

Fig. 3. FFC distribution for simulated generic VENUS and pureDCC-like events. The solid and the dotted lines correspond to theDCC-like and VENUS events respectively.

In Fig. 3 we have shown the FFC distribution forpure generic and pure DCC-like events at scalejs1. We find that the FFC distribution for DCC-likeevents is broader in comparison to that for genericevents. In fact, one can show that there is pile up ofa large number of events within the width of thegeneric distribution with decrease in the fraction ofDCC-like events. But it is those events that liebeyond the width of the generic distribution, aboutwhich some definite conclusion regarding theirDCC-like nature can be made. In such a case, whena fraction of the events contain the DCC type signal,it is very difficult to notice their signature using the

w xpower spectrum analysis 7 which employs the eventaveraging procedure. This is because the interesting

Fig. 4. The distribution of C as obtained from the two detectorsexcluding events lying within 1.5 sigma of the FFC distribution ofVENUS. The dotted lines shows the distribution of C as abovebut for pure VENUS events. Errors though not shown are of the

'order of N at every point.

( )B.K. Nandi et al.rPhysics Letters B 449 1999 109–113 113

events lying beyond the width of the generic distri-bution, contributing significantly to the power spec-trum, are overwhelmingly outnumbered by those ly-ing within the width.

A simple method to filter out the contribution of agreat majority of non-DCC type events is to apply acut on the width of the FFC distribution and considerthe events lying above the cut. With this in view, inthe present case, we have applied a cut of "1.5s inthe FFC distribution. The C distribution of thefiltered events for the case of 25% of DCC-likeevents is shown in Fig. 4. One can clearly notice thepeaking at Cs908 indicating a DCC-like signature.The pure VENUS events with the same cut in theirFFC distribution are also shown for comparison.They are seen to contribute uniformly over the entireangular range.

This clearly shows that the DWT and the eventshape analysis applied together can be a very power-ful method for the search of DCC. However, oneneeds to judiciously use an appropriate cut on theFFC distribution to filter out a great many uninterest-ing effects. Before concluding, a word must be men-tioned regarding the errors. One can notice, thespectral distribution shows a histogram where the

'error in the entry at every angle goes as N . Byfiltering out a large number of events, which aremostly distributed uniformly over the entire angularrange, one essentially reduces the background al-though the statistical errors increase slightly.

5. Conclusion

In the present paper it has been demonstrated thatthe technique of event shape analysis can be veryeffectively employed to look for the signature ofDCC formation in relativistic heavy ion collisiondata. Here, in the absence of any rigorous theoreticalmodel to simulate DCC formation, an isospin fluctu-

ation has been introduced locally on the VENUSgenerated events to generate a charged to neutralpion asymmetry. At least for the case with a single,large, DCC domain, it seems to be a very effectivetechnique for looking at DCC signature. However, incases where only a certain fraction of the events areexpected to be of DCC type, together with thetechnique of DWT which provides a first hand filter,the present technique provides a very powerful probefor DCC signal.

References

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Ž .1980 151; Y. Takahasi et al., JACEE Collaboration, in: L.Ž .Jones Ed. , Proc. 7th Int’l. Symp. on Very High Energy

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4 March 1999


Evidence for a J PC s1qq Is1 meson at 1640 MeV

C.A. Baker a, C.J. Batty a, P. Blum c, D.V. Bugg d, R.P. Haddock e, C. Hodd d,¨C. Holtzhaußen c, L. Montanet b, D. Odoom d, C.N. Pinder a, I. Scott d, B.S. Zou d

a Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UKb CERN, CH-1211 GeneÕa 4, Switzerland

c UniÕersitat Karlsruhe, D-76021 Karlsruhe, Germany¨d Queen Mary and Westfield College, London E1 4NS, UK

e UniÕersity of California, Los Angeles, CA 90024, USA

Received 6 October 1998Editor: K. Winter

Abstract

0 P C qqFrom an analysis of pp™4p at a beam momentum of 1940 MeVrc, we find evidence for a J s1 resonance atŽ . Ž .1640 "12 stat "30 syst MeV with Gs300"22"40 MeV; it is observed decaying both to sp with Ls1 and to

Ž .f 1270 p with Ls1. q 1999 Published by Elsevier Science B.V. All rights reserved.2

PACS: 13.75.Cs; 14.20.GK; 14.40Keywords: Mesons; Resonances; Annihilation

0We present a study of pp™4p at 1940 MeVrc.Data were taken at LEAR by the Crystal BarrelCollaboration. A systematic search is in progress forresonances in the mass range 1600–2400 MeVrc2

and the present study reveals a 1qq Is1 candidatein the 1600–1700 mass range, where several other

Ž . Ž .resonances are well known: the r 1690 , p 1670 ,3 2Ž . w x Ž . w x Ž . w x Ž .r 1700 1 , a 1660 2 , p 1740 3 and p 18001 2

w x Ž .4 . A radial excitation of a 1260 is to be expected1

here.A full description of the detector has been given

w xearlier 5 . It is equipped for efficient detection ofboth photons and charged particles over nearly 4p

solid angle. For present purposes, it has been usedwith a trigger demanding neutral final states. The p

beam is defined by a coincidence including solidstate counters close to the entrance of a 4.4 cm liquidhydrogen target. Non-interacting p are rejected by aveto counter downstream of the target; this counteralso eliminates most of pp elastic scattering. Multi-wire chambers and a jet drift chamber surroundingthe target over 98% of 4p are used as further vetos.Photons are detected in a barrel of 1380 CsI crystalscovering 98% of the solid angle; after the Lorentzboost, this corresponds to 95% in the centre of mass.Photons are detected with high efficiency down to 10MeV with a resolution in energy E of 2.5%rE1r4,where E is in GeV. The angular resolution is 20mrad in both polar and azimuthal angles. A trigger

w xmodule 6 sums the total energy measured in the CsI


( )C.A. Baker et al.rPhysics Letters B 449 1999 114–121 115

crystals and quickly rejects events which give lessenergy than a threshold set ;150 MeV below thepp total energy.

Event reconstruction follows standard proceduresw xreported in earlier analyses of neutral final states 7 .

The data sample consists of 4.1 million triggers. Forthe present study, exactly 8 photons are required,

Ž .satisfying energy-momentum balance 8C with aconfidence level )10% and with unambiguous pair-

0 Žing of photons to make 4p . Ambiguous pairing. 0occurs for 3.3% of events . The 4p channel is quite

prolific and backgrounds from other channels arelow. A Monte Carlo study using GEANT shows auniform acceptance except for obvious loss of pho-tons into 128 entrance and exit holes for the beam.Angular distributions are symmetric backwards andforwards in the centre of mass, as they should be bycharge conjugation invariance.

The Monte Carlo study shows that the largestbackground is from hp 0p 0p 0 and is -1%; thischannel has a branching fraction a factor 2.7 lowerthan 4p 0 at 1940 MeVrc. Another possible back-

Ž 0 0 0ground arises from 9g events vp p p or0 0.vhp p where one photon is lost; however, a

search in the data for v™p 0g reveals no signal atthe 0.7% level. Events of the form hp 0, h™3p 0

have been rejected with a cut on 3p 0 mass below600 MeVrc2. This rejects 1294 events from 27881.

0After kinematic fitting to pp™4p , the datasample consists of 26,587 events. The maximum

w xlikelihood analysis described below 8 uses 84,967Monte Carlo events simulating the acceptance andsatisfying identical selection criteria to data. Afterkinematic fitting, we believe that uncertainties due to

Žpurely experimental effects energy and angular reso-.lution of the photons are simulated adequately by

the Monte Carlo and that systematic uncertainties inacceptance have negligible effect in the amplitudeanalysis compared with uncertainties in the ingredi-ents to be included in the fits.

Ž . Ž . 0 0Fig. 1 a and b show the 2p and 3p spectra2 Ž . Ž .of ssM and c and d the scatter plots of s v.12

s and s , where the digits label p 0. The only34 123Ž .conspicuous signals are due to f 1270 and2

Ž . 0p 1670 . In the 3p mass spectrum, there is also a2

slight shoulder at ;2100 MeV. Histograms on Fig.Ž . Ž .1 a and b are from the maximum likelihood fit.

Ž . Ž .Dashed curves show phase space. Fig. 1 e and f

are scatter plots from the fit, for comparison withŽ . Ž .data in Fig. 1 c and d .

The main channels contributing are:

pp™ss 1Ž .™ f 1270 s 2Ž . Ž .2

™ f 1270 f 1270 3Ž . Ž . Ž .2 2

™p 1670 p , p ™ f 1270 p 4Ž . Ž . Ž .2 2 2 Ls0

w x™p 1670 p , p ™ sp 5Ž . Ž .Ls22 2

™a 1640 p , a ™ f 1270 p 6Ž . Ž . Ž .1 1 2 Ls1

w x™a 1640 p , a ™ sp . 7Ž . Ž .Ls11 1

Here L is the orbital angular momentum in the decayof the resonance; s is a shorthand for the pp

S-wave amplitude, for which we use a slightly up-dated version of the parametrisation of Zou and

w xBugg 9 ; the original parametrisation up to 1400MeV has now been extended to 1800 MeV using a

w xdirect fit to CERN-Munich data 10 . Channel 3 isabove the available centre of mass energy, 2409MeV, but still contributes strongly. We use Particle

Ž . w xData Group PDG 1 masses and widths forŽ . Ž .f 1270 and p 1670 , but have tried varying these2 2

parameters over the range of errors quoted by thePDG. These variations have no appreciable effect onconclusions.

The amplitude analysis will follow the methods ofw xRef. 7 and we now give details. There are too many

partial waves to fit both production and decay. Wefit the decays in full, since they give the primaryinformation for spin-parity analysis of resonances.No attempt is made to fit the production process. Let

Ž . Ž .us take as examples channels 4 and 5 with parti-cles 1,2,3 making the p and particles 12 the f into2 2

Ž .which it decays. Suppose the p of channel 4 is2

produced with component of spin m along the beamdirection. The amplitude for its production is written

G exp id Y m a ,bŽ . Ž .m 2f s . 8Ž .m 2 2M ys y iMG m ys y imgŽ . Ž .123 12

Here G is a coupling constant and d a phase form

production of this channel; M, G refer to the massand width of the p and m,g are the mass and width2

of the f . Also Y are spherical harmonics in terms of2

polar angle a illustrated in Fig. 2 and azimuthalangle b about the beam direction. They are the

( )C.A. Baker et al.rPhysics Letters B 449 1999 114–121116

2 Ž . 0 Ž . 0Fig. 1. Mass spectra of a 2p , b 3p combinations; crosses show data and full histograms the fit. The vertical axis shows the numberŽ 0 0 . Ž . Ž .of combinations in each case 6 for 2p , 4 for 3p . The dashed histograms show phase space. Scatter plots of data: c s v. s and d12 34

Ž . Ž . Ž . Ž . Ž . Ž .s horizontally v. s vertically ; e and f are scatter plots for the fit, to be compared with c and d respectively.12 123

decay angles of f ™pp with respect to the beam2

direction after two Wick rotations. The details of thew xWick rotation are given in Ref. 3 . In outline, the

steps are as follows. Particle momenta are first trans-


Fig. 2. Decay angles used in the analysis.

formed to the centre of mass frame. The WickŽ .rotation then consists of three steps: i a rotation

through polar angle t and azimuth f to the directionŽ .of production of the X'p ; ii a Lorentz boost to2

Ž .its rest frame; iii a rotation back again throughangles yf and yt in the rest frame of the p . The2

effect of the second rotation is to cancel the quantummechanical rotation matrices required for the firstrotation. Amplitudes are invariant under the boost tothe rest frame of the resonance. A second Wickrotation is then made using angles g ,e at which thef appears after the first Wick rotation, The surviv-2

ing angular dependence is then simply throughmŽ . Ž . Ž . mŽ .Y a ,b of Eq. 8 . For channel 5 , Y a ,b is2 2

mŽ .replaced by Y g ,e describing the decay with Ls2

2 to sp .Amplitudes for one channel are summed coher-

ently over all combinations of p 0. For channelsŽ . Ž .4 – 7 , there are 12 such combinations; for f s2

there are 6 combinations and for ss there are 3.Cross sections are summed incoherently over mvalues. Interferences between channels are treatedapproximately, retaining only dominant contribu-tions. We illustrate the procedure with one example:

Ž . Ž .interference between channels 1 and 2 , ss andŽ .f 1270 s . Let their amplitudes for ms0 be A and2

B. Each of these amplitudes arises in principle frommany initial pp partial waves, with varying magni-tudes and phases. The interference term is taken to

Ž ) .be 2c Re A B , where c is a fitted parameterA B A B

lying between q1 and y1, and allowing for partialcoherence between A and B. In practice, if two ormore partial waves contribute with different depen-dence on production angles, the interference termwill tend to average over the production process to asmall value. Furthermore, the interferences con-tribute only over the small part of the scatter plots of

Ž . Ž .Fig. 1 c and d where A and B are both large. Forthe latter reason, interference terms mostly have asmall numerical effect on log likelihood. We exam-ine all interferences individually and keep only thosewhere log likelihood improves by more than 2 stan-dard deviations, summed over m values. Only asmall number survive, and those which do are almost

Ž < < .fully coherent c G0.75 , suggesting that one pppartial waves then dominates. We adopt the approxi-mation that the phase d of each channel is indepen-dent of m, so as to treat correctly the limit of just

< <one pp partial wave. Wherever c -1, the fittedphase is an average over m values. This approximatetreatment of interferences is a matter of practical

Ž .necessity. For a 1640 , the main objective of the1

present paper, only one interference survives, be-Ž .tween its sp decay and the f 1270 s channel. It2

turns out that the overlap on the scatter plots be-tween these two processes is larger than for otherprocesses and this particular interference does im-prove log likelihood by quite a significant amount,namely 24.4.

We define log likelihood in the standard way sothat a change of 0.5 corresponds to one standard

Ž .deviation for one degree of freedom. For p 1670 ,2

the ratio of ms"1 and ms0 decay amplitudes isconstrained to be identical for both decay modes;

Ž .likewise for a 1640 .1

Table 1 shows changes in log likelihood wheneach channel is dropped from the final fit and theremainder are re-optimised. It also shows branchingratios for each channel, keeping interferences be-tween combinations within the channel, but droppinginterferences between different channels. Branchingratios do not add up to 100% because of the interfer-ences between channels. It is obvious from the first 5

Ž . Ž .entries of Table 1 that all of reactions 1 – 5 makesubstantial contributions and are required.

After these technicalities, we return to the physics.Ž .The shoulder at Ms2.1 GeV in Fig. 1 b attracted

Ž .our attention. This is close to the mass of a 20504


Table 1Changes in log likelihood when each channel is dropped from thefit; also branching fractions including interferences within eachchannel, but excluding interferences between different channels.In entries 4 onward, the spectator pion is not listed in order tokeep notation uncluttered

Ž .Channel D ln L BranchingŽ .fraction %

ss 437 28.2Ž .s f 1270 698 43.82

Ž . Ž .f 1270 f 1270 630 22.42 2

Ž . w Ž . xp 1670 ™ f 1270 p 310 20.82 2 Ls0Ž . w xp 1670 ™ sp 271 6.22 Ls2Ž . w Ž . xp 1670 ™ f 1270 p 12.6 0.72 2 Ls2

Ž .All p 1670 508 27.72

Ž . w xa 1640 ™ sp 79.0 10.21 Ls1Ž . w Ž . xa 1640 ™ f 1270 p 20.5 2.81 2 Ls1

Ž .Both a 1640 152.0 13.01

Ž . w xa 1260 ™ sp 28.5 5.01 Ls1Ž .p 1690 72.7 5.9

and it seemed possible that this resonance and otherswith J P s3q or 2q might be responsible. To checkthis, we added to the amplitude analysis furtherresonances one by one with all J P from 0y to 4q,decaying with all possible L values to sp orŽ .f 1270 p . Each was assigned a trial width of 2502

MeV and its mass was scanned in 40 MeV stepsfrom 1600 to 2300 MeV. None showed any optimumnear 2.1 GeV.

The explanation which emerges for this shoulderis straightforward. It is due to a triple interference

0 Ž .amongst 3p involving three simultaneous f 12702

combinations. For three such resonances amongstparticles 12, 23 and 13,

s ss qs qs y3m2 s4.78 GeV 2 . 9Ž .123 12 23 13 p

When one allows further for the available phasespace, this triple interference gives quite a gooddescription of the shoulder, though some small sys-tematic discrepancies remain above 2 GeV.

The scans did, however, reveal a surprisinglystrong 1q signal at 1640 MeV. Its width optimises at

Ž .300"22 MeV, as shown in Fig. 3 a , with a system-atic error which we estimate as "40 MeV from awide variety of fits. There is almost no correlationbetween fitted mass and width.

Ž .b shows the variation of log likelihood with themass of this resonance for decays separately to sp

Ž . w Ž . x Ž .dashed curve and f 1270 p dotted . The2 Ls1

full curve shows the fit including both decay chan-wnels. Blatt-Weisskopf centrifugal barriers have been

Ž . qFig. 3. Variation of log likelihood with a the width of the 1Ž .state, b its mass. The dashed and dotted curves show results

w x w Ž . xincluding sp or f 1270 p decays respectively; theLs 1 2 Ls1

full curves are with both decays included.


included for both decays with a radius of interactionxof 0.8 fm, but their effects are small .

The obvious question is whether this peak may bedue to cross-talk with other signals, notably

Ž .p 1670 . To study this possibility, we have made2

four tests, all negative.Ž .The first is to test cross-talk between p 16702

Ž .and a 1640 . We have used the Monte Carlo events1

to generate several data samples weighted with theŽ . Ž .cross section we fit just to channels 1 – 5 . This

simulates the basic fit without any a . These data1

samples have then be fitted including in addition the1q channels. In all cases, the fitted 1q component is-1% and there is no peak at 1640 MeV in the mass

Ž .scan. So cross-talk with p 1670 definitely fails to2

account for the 1q signal.A second check has been made on the basis of the

angular dependences expected for 1q decays. Let ustake as an example the sp decay. The ms0 ampli-tude is proportional to cosg and the ms1 amplitude

Ž .is proportional to sing exp ie . We have replaced theŽ .angles g ,e with a variety of other wrong angles,

e.g. t , a , b and corresponding angles for the f s2

and f f channels. In all cases, the peak in log2 2

likelihood drops dramatically to a height typically25, compared with the much larger peak of about100 for the correct angles. Our conclusion is that thedata do have the angular dependence characteristicof 1q, although this cannot be readily displayedbecause of the 12 combinations.

Fig. 4 shows corresponding scans for other quan-q y Ž .tum numbers. For 2 and 1 exotic , there are only

very weak effects. For 3q there is a distorted peakŽ .close to 1640 MeV upper dashed curve , but with

much poorer log likelihood than for 1q. A 3q reso-nance is to be expected in the Ls3 set of qqexcitations around 2040 MeV, but not at 1640 MeV.Our third check is therefore that the observed 3q

peak may be explained as cross-talk with 1q. Thedetection efficiency of the detector drops close to thebeam direction because of entrance and exit holes for

mŽ .the beam and cabling. The Y g ,e dependence for3

3q has similarities with Y m for 1q, and the loss of1

detection efficiency near cosgs"1 allows 3q tomasquerade as 1q to some extent. A simulation hasbeen run using the Monte Carlo events weighted

Ž . Ž . Ž .with the fit to reactions 1 – 5 plus a 1640 . If1Ž . Ž .these events are then fitted by reactions 1 – 5 plus

Fig. 4. Variation of log likelihood with mass for alternativeq y w xquantum numbers: 2 ™ f p with Ls1 and 3, 1 ™ f p ,2 2 Ls2

q w x w x y3 ™ sp and f p , and 0 ™sp . The full curveLs 3 2 Ls1q Ž .shows the variation for 1 when a 1260 is also included in the1

fit.

a 3q resonance instead of 1q, the 3q curve of Fig. 4is reproduced well and so is the magnitude of the

Žfitted signal 6.4% from data compared to 5.9% from.the Monte Carlo . The fourth point is that if the fit is

allowed to include both 1q and 3q signals, it is theq Ž1 signal which dominates strongly 12.7% com-

. qpared with the value 13.0% of Table 1 and the 3signal drops to only 2.1%. The improvement in loglikelihood from adding the 3q component is only7.2. Our conclusion is that the 3q peak of Fig. 4 isan artefact and that the 1q signal is genuine.

Fig. 4 also shows an optimum for 0y at 1690MeV. It is a possibility that this is due to a 0y peaknear this mass observed by the VES group at 1740

w xMeV 3 . Simulation shows that ‘leakage’ fromŽ . yp 1670 can account for half of the 0 signal of2

Fig. 4. For quantum numbers 0y there is no angulardependence, and there is the further possibility ofconfusion with ss . Therefore we are not confident


that the 0y peak corresponds to real physics. With itincluded in the fit, the 1q signal optimises at 1635MeV; with it excluded, the optimum is at 1650 MeV.We therefore assign a compromise mass of 1640MeV to the 1q signal, with a statistical error of "12MeV and a systematic error of "30 MeV.

We have extended the scan for 1q™sp of Fig.Ž .3 b down to 1150 MeV, in order to search forŽ . w xa 1260 . According to the PDG 1 , the decay of1Ž .a 1260 to sp is extremely weak, with a branching1

ratio 0.003"0.003. We do observe a small enhance-Ž . Ž .ment near the a 1260 . When a 1260 is included1 1

in the amplitude analysis with PDG mass and widthŽ .our final fit , there is very little effect on the

Ž .evidence for a 1640 , as shown by the full curve on1Ž .Fig. 4. We have searched also for p 1300 but found

no significant evidence for its presence.In summary, the evidence we have presented forŽ . Ž .a 1640 depends on i the mass scan shown in Fig.1Ž . Ž .3 b and Fig. 4, ii the correct angular dependence

Ž .observed in two decay channels, iii simulations ofpossible ‘leakage’ effects. We have considered thepossibility that the 1q signal is due to faulty detectorperformance, but can see no reason why such a faultshould peak in the 3p 0 mass range narrowly around

Ž .1650 MeV. The observed a 1640 signal is statisti-1

cally a 12 standard deviations effect and is roughlyŽ .half the strength of p 1670 . We acknowledge that2

it requires confirmation elsewhere. It is unfortunatethat the fits to the mass spectra and scatter plots of

Ž .Fig. 1 show almost no change when a 1640 is1

included in the fit. It is therefore the angular depen-dence which is providing the evidence for its pres-ence.

There has been earlier evidence from Pernegr etw x P C qqal. 11 for a J s1 resonance at 1650 MeV

with G,400 MeV, decaying to rp with Ls2.w xDaum et al. 12 also reported tentative evidence for

a 1q resonance near 1700 MeV in this decay chan-nel. The VES group finds evidence for a strong 1qq

peak with Is1 at 1650 MeV in the rp D-wave,and less well defined signals in the rp S-wave and

w xthe sp P-wave 13 . A similar signal was observedŽ . w x w xearlier in the f 1285 p channel 14 . Lee et al. 151

have reported evidence for an a at about 1700 MeV1Ž .in the f 1285 p channel. The Delphi collaboration1

reports tentative evidence for something at the highw xmass end of the 3p mass spectrum in t™3pn 16 .t

w xMore recently 17 , the E852 group has reportedevidence for an Is1 J P C s1qq resonance in the

Ž .mass range 1.67 to 1.75 GeV, decaying to f 1285 p1Ž . Ž .and possibly a 980 r 770 , with a width near 4000

MeV.Ž .A radial excitation of a 1260 is to be expected1

around 1650 MeV, and decay modes to sp andŽ .f 1270 p are likely. To evaluate relative branching2

ratios, we follow the procedures developed in earlier0 w xstudies of annihilation to 5p 18 . It is necessary to

evaluate the decays of the a resonance as they1

would appear in isolation, without interferences withthe spectator pion. From the coupling constants fittedto the data, we find, after integrating over the avail-able phase space, a branching ratio:

BR a 1640 ™ f pŽ .Ž .1 2s 24"7 %. 10Ž . Ž .

BR a 1640 ™spŽ .Ž .1

This is slightly lower than the 27% one woulddeduce directly from Table 1, which includes inter-ferences with the spectator pion. The ms"1branching fraction is a factor 3 stronger than forms0.

Ž .For p 1670 we find a ratio of decay amplitudes2Ž .to f 1270 p with Ls2 and Ls0:2

Dsy0.18"0.06, 11Ž .

S

in agreement with the magnitude found by Daum etw xal. 12 : 0.22"0.1. Our data require destructive

interference between these two decays, but Daum etal. do not quote a phase angle.

We find a ratio of branching ratios for decays ofŽ .p 1670 to sp with Ls2 and to f p with Ls0:2 2

w xBR p 1670 ™ spŽ .Ž .Ls22s0.24"0.10. 12Ž .w xBR p 1670 ™ f pŽ .Ž .2 2 Ls0

This is in good agreement with the value of Daum etal. of 0.2"0.1.

Acknowledgements

We thank the Crystal Barrel Collaboration forallowing use of the data. We wish to thank thetechnical staff of the LEAR machine group and of allthe participating institutions for their invaluable con-


tributions to the success of the experiment. Weacknowledge financial support from the British Parti-cle Physics and Astronomy Research CouncilŽ .PPARC , the German Bundesministerium fur Bil-¨dung, Wissenschaft, Forschung und Technologie, andthe U.S. Department of Energy.

References

w x Ž .1 Particle Data Group, Phys. Rev. D 54 1996 1.w x2 A. Abele et al., to be submitted to Zeit. f. Physik.w x3 D. Amelin, Hadron’97 Conference Proceedings, to be pub-

lished.w x Ž .4 D. Amelin et al., Phys. Lett. B 356 1995 595.

w x Ž .5 E. Aker et al., Nucl. Instr. and Methods A 321 1992 69.w x Ž .6 C.A. Baker et al., Meth. Phys. Res. A 394 1997 180.w x Ž .7 J. Adomeit et al., Zeit. Phys. C 71 1996 227.w x8 D. Odoom, Ph.D. thesis, University of London, 1998.w x Ž .9 B.S. Zou, D.V. Bugg, Phys. Rev. D 48 1993 3948.

w x10 D.V. Bugg, A.V. Sarantsev, B.S. Zou, Nucl. Phys. B 471Ž .1996 59.

w x Ž .11 J. Pernegr et al., Nucl. Phys. B 134 1978 436.w x Ž .12 C. Daum et al., Nucl. Phys. B 182 1981 269.w x Ž .13 D.V. Amelin et al., Phys. Lett. B 356 1995 595.w x14 Yu.P. Gouz et al., Report at the XXVI International Confer-

ence on High Energy Physics, Dallas, USA, August 5–12,1992.

w x Ž .15 J.H. Lee et al., Phys. Lett. B 323 1994 227.w x Ž .16 P. Abreu et al., Phys. Lett. B 426 1998 411.w x Ž .17 A. Abele et al., Phys. Lett. B 380 1996 453.w x18 D. Ryabchikov, Hadron Spectroscopy, AIP Conf. Proc. 432,

Amer. Inst. Phys., 1998, p 527.

4 March 1999


Observation of the decay f™vp 0

M.N. Achasov, S.E. Baru, A.V. Berdyugin, A.V. Bozhenok, D.A. Bukin, S.V. Burdin,T.V. Dimova, S.I. Dolinsky, V.P. Druzhinin 1, M.S. Dubrovin, I.A. Gaponenko,V.B. Golubev, V.N. Ivanchenko, A.A. Korol, M.S. Korostelev, S.V. Koshuba,

E.V. Pakhtusova, A.A. Polunin, E.E. Pyata, A.A. Salnikov, V.V. Shary,S.I. Serednyakov, Yu.M. Shatunov, V.A. Sidorov, A.N. Skrinsky, Z.K. Silagadze,

Yu.S. VelikzhaninBudker Institute of Nuclear Physics and NoÕosibirsk State UniÕersity, 630 090 NoÕosibirsk, Russia

Received 28 November 1998; revised 6 January 1999Editor: L. Montanet

Abstract

The reaction eqey™vp 0 ™pqpyp 0p 0 has been studied with SND detector at VEPP-2M eqey collider in thevicinity of the f meson resonance. The observed interference pattern in the energy dependence of the cross section is

0 Ž 0. Ž q1.9 . y5consistent with existence of the decay f™vp with a branching ratio of B f™vp s 4.8 "0.8 =10 . The realy1.7

and imaginary parts of the decay amplitude were measured. The f™vp 0 decay was observed for the first time. q 1999Published by Elsevier Science B.V. All rights reserved.

PACS: 13.65.q i; 14.40.Cs

1. Introduction

Recently in experiments with SND and CMD-2detectors at VEPP-2M eqey collider the study off-meson rare decays with branching ratios of 10y4 –

y5 w x10 became available 1–3 . One of such decays isan OZI and G-parity violating f™vp 0 decay. Thepredicted branching fraction of this decay is of the

y5 w xorder of 5=10 4,5 . It varies within wide limitsdepending on the nature of r, v and f-mesonmixing and existence of direct f™vp 0 transition.Since the nonresonant cross section of the process


eqey™vp 0 in the vicinity of the f-meson peak isrelatively large, the decay f™vp 0 reveals itself asan interference pattern in the cross section energydependence. This allows to extract from the databoth real and imaginary parts of the decay amplitude.

The f™vp 0 decay has not been observed yet.w xIn our preliminary study 1,2 an upper limit for the

decay probability was placed at a level of the theo-retical prediction of 5=10y5.

2. Experiment

The experiment was carried out with SND detec-tor at VEPP-2M in 1996–1997. SND is a general

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00043-X

( )M.N. AchasoÕ et al.rPhysics Letters B 449 1999 122–127 123

w xpurpose non-magnetic detector 6 , based on 3.6 tthree layer spherical electromagnetic calorimeter with

Ž .1620 NaI Tl crystals covering 90% of 4p solidangle. The energy resolution for electromagnetic

4(showers is s rEs4.2%r E GeV , the angularŽ .E

resolution is about 1.58. The angles of charged parti-cles are measured by two cylindrical drift chamberscovering 95% of 4p solid angle. The angular accu-racy of charged track measurements is about 0.48 and2.08 in azimuth and polar directions respectively.

Seven successive data taking runs were performedat 14 energy points in the energy range 2 Es980–1060 MeV. The total integrated luminosity D Ls3.7pby1 corresponds to 6.3=106 produced f mesons.The integrated luminosity was measured using eqey

™eqey and eqey™gg reactions with the accu-w xracy of 3% 7

3. Event selection

For a search for the f™vp 0 decay we studiedthe cross section of the process

eqey™vp 0 ™pqpyp 0p 0 1Ž .

in the vicinity of the f-meson. Events with twocharged particles and four or more photons wereselected for analysis. To suppress beam backgroundthe production point of charged particles was re-quired to be within 0.5 cm from the detector centerin the azimuth plane and "7.5 cm along the beam

Ždirection the longitudinal size of the interaction.region s is about 2 cm .z

Since for the process under study a probability tofind spurious photons in the events is rather largeŽ .about 15% , nearly all main f-meson decays are asource of background in our search:

eqey™f™KqKy ; 2Ž .

eqey™f™K K , K ™pqpy ; 3Ž .S L S

eqey™f™pqpyp 0 ; 4Ž .

eqey™f™hg ,h™pqpyp 0 . 5Ž .

The main source of the nonresonant background isthe process

eqey™pqpyp 0p 0 6Ž .Ž .which has the same final state as the process 1 , but

without the vp 0 intermediate state. The interferenceŽ . Ž .between the processes 1 and 6 is significant only

in the small part of the decay phase space, wherepqpyp 0 invariant mass is within the v-mesonwidth. Estimated contribution of the interference termin the energy region under study does not exceed 2%

Ž .of the cross section of the process 1 and wasneglected in the further analysis. It was treated in-stead as an additional source of systematic error in

Ž .the nonresonant cross section of the process 1 .To suppress the background from the processes

Ž . Ž .2 and 3 , the following cuts were imposed:Ø average dErdx losses in the drift chamber

Ž . Ž .dErdx-4 dErdx , where dErdx aremin min

average dErdx losses of a minimum ionizingparticle;

Ø the spatial angle between charged particles is lessthan 1408.

The first condition suppresses events of the processŽ . Ž .2 with slow bf0.25 charged kaons having largedErdx losses in the SND drift chamber. The second

Ž .condition suppresses events from the process 3with a minimum angle between pions from the KS

decay close to 150 degrees.For each selected event independent kinematic fits

were performed under three following assumptionsabout the reaction mechanism:1. The event originates from the process eqey™

pqpyp 0. The value of the likelihood functionx is calculated.3p

2. The event originates from the process eqey™

pqpyp 0g . The photon recoil mass M is calcu-rec

lated.3. The event is due to the process eqey™

pqpyp 0p 0. The value of the likelihood functionx is calculated together with M is the recoil4p 3p

mass of the p 0-meson closest to the v-mesonmass.

In the events where the number of photons exceedsthat required by a certain hypothesis, extra photonsare considered as spurious ones and rejected. To dothat all possible subsets with a correct number ofphotons were tested and the one, corresponding to a

( )M.N. AchasoÕ et al.rPhysics Letters B 449 1999 122–127124

Fig. 1. Distribution over x for experimental events and simula-3pq y 0 q y 0 Žtion of the process e e ™ vp ™p p 2p shaded his-

.togram . The cut is indicated by the arrow.

maximum likelihood of a certain hypothesis, wasselected.

The distribution of experimental and simulatedŽ .events of the process 1 over the parameter x , is3p

shown in Fig. 1. One can see a considerable contri-Ž .bution from the process 4 , producing a peak at low

x . Fig. 2 shows the experimental M spectrum,3p recŽ .where a clear h-meson peak from the reaction 5 is

Fig. 2. Distribution over the parameter M for experimentalrec

events and simulation of the process eq ey™ vp 0 ™pqpy 2p 0

Ž .shaded histogram . The cut is indicated by the arrow.

seen. To suppress the background from the processesŽ . Ž .4 and 5 the following cuts were applied:

x )25, M )620 MeV.3p rec

4. Data analysis

Fig. 3 shows the distribution of experimental andŽ .simulated events of the process 1 over x . Con-4p

siderable difference between the tails of measuredand simulated spectra indicates the presence of abackground surviving the cuts. One can see that theexperimental peak in Fig. 3 is broader than a simu-lated one. It means that the simulation of the x4p

distribution is not precise. In Fig. 4 the distributionsover the p 0 recoil mass M for experimental3p

events with x -20, simulated events of the pro-4p

Ž . Ž .cess 6 the rpp mechanism was assumed , andŽ .simulation of the process 1 with the 11% admixture

Ž .of the process 6 are presented. The last distributionis in good agreement with the experimental one. Forfurther analysis two additional cuts were applied:

< <x -40, M y782 -100.4p 3p

The total of 4500 events which survived the cutswere divided into four classes:

< <1. x -20, M y782 -35;4p 3p

< <2. x -20, M y782 )35;4p 3p

Fig. 3. Distribution over x for experimental events and simula-4p

Ž . Ž .tion of the process 1 shaded histogram .


Fig. 4. Distribution over parameter M for experimental events3p

Ž . Ž . Žpoints with errors , simulation of the process 6 shaded his-. Ž .togram , and simulation of the process 1 with an 11% admixture

Ž . Ž .of the process 6 histogram .

< <3. x )20, M y782 -35;4p 3p

< <4. x )20, M y782 )35;4p 3p

with the relative populations of 0.69:0.09:0.16:0.06respectively.

The visible cross section for each class s wasvis i

fitted according to the following formulae:

s E sa s E qb s E qs E ,Ž . Ž . Ž . Ž .vis i i vp i 4p f i

s E s´ B v™3p s qA EymŽ . Ž . Ž .Ž .vp 0 f

=

2m Gf f

1yZ 1qd . 7Ž . Ž .Df

Here s , s are the visible cross sections of thevp 4p

Ž . Ž .processes 1 and 6 , a , b are the probabilities fori iŽ . Ž .events of the processes 1 or 6 to be found in the

Ž .i-th class Ýa s1 and Ýb s1 at Esm , s isi i f f i

the visible cross section of the resonant backgroundin the i-th class, s is the nonresonant cross section0

of the process eqey™vp 0 at Esm , A is itsf

slope, ´ is the detection efficiency for the processŽ .1 at Esm , Z is the complex interference ampli-f

2 2 Ž .tude, m ,G , D sm yE y iEG E are the mass,f f f f f

width and f-meson propagator function respectively,Ž .B v™3p s0.888 is the branching ratio of the

w xv™3p decay 9 , d is a radiative correction calcu-w xlated according to 8 .

The fitting was performed for all four classessimultaneously. The class 1 with a small resonant

background was the most important for the determi-nation of s , A, Z. Classes 2–4 were used to0

determine the background from f decays: s , s ,f2 f 3

s . In the fit it was assumed that for the resonantf4

background the distribution over M is independent3p

of x , and therefore the background for the class 14p

can be obtained from the expression: s ssf1 f 2Ž .s rs . The validity of this relation was checkedf3 f 4

with the statistical accuracy of 15% for simulatedevents and specially selected experimental events of

Ž . Ž .the processes 2 – 5 . The cross section of the reso-nant background s in the resonance maximumf1

Ž .was found to be s s 31"17 pb, that is aboutf1

4% of the visible cross section s in the class 1.vis 1Ž .The contribution of the process 6 was deter-

mined using the value of the nonresonant part of thecross section in the class 2 and the ratios a ra s2 1

0.058 and b rb s1.05, obtained from simulation.2 1Ž .The cross section s E was fitted by a sum of a4p

linear function and an interference term similar toŽ . Ž .s E in the expression 7 . For the OZI-rule andvp

G-parity double suppressed f™pqpyp 0p 0 de-cay, the main mechanism is a transition of the f

meson into pqpyp 0p 0 via a virtual photon. Thecalculated real part of the interference amplitude Zdue to this decay mechanism is equal to 0.127. In thefit the real and imaginary parts of the interference

Ž .amplitude of the process 6 were set to 0.127 and0.0 respectively, while their errors were set to 0.07and 0.15 in agreement with the accuracy of theoreti-cal estimations and experimental data on anotherdouble suppressed decay f™pqpy. The nonreso-

Ž .nant cross section 6 was found to be equal toŽ . Ž .11"4 % of the cross section of the process 1 . Inthe class 1 the contribution of this process is smaller:4.7%.

Since the simulated distribution over the x may4p

differ from the experimental one, the coefficients a1

and a were determined from the fit. Other coeffi-3

cients a and b were derived from the relationsi i

described above, with the additional assumption thatŽ .the distributions over x in the processes 1 and4p

Ž .6 are the same. The coefficients a , b vary slowlyi i

with energy, and their energy dependence approxi-mated by linear functions was obtained by simula-tion.

Ž .The detection efficiency ´ for the process 1 wasalso obtained from simulation. To estimate a system-

( )M.N. AchasoÕ et al.rPhysics Letters B 449 1999 122–127126

atic error in ´ , we processed data with softer cutsand studied the class of events with one chargedparticle and four photons. It was found, that thedetection efficiency obtained by Monte Carlo simula-tion must be corrected by y11%. The correction ismainly due to the inaccuracy in the simulation ofx distribution and loss of charged particles during4p

track reconstruction. With this correction, the detec-tion efficiency is equal to 17.6"1.8% with a sys-tematic error of 10%.

The total number of fit parameters describingenergy dependence of the cross sections in all fourclasses of selected events is 11. For each class thecross section was measured in 14 energy points. Atx 2rn s43r45 the following values of main fitD

parameters were obtained:

s s 8.2"0.2"0.9 nb,Ž .0

As 0.088"0.009"0.011 nbrMeV,Ž .Re Z s0.104"0.028"0.006,Ž .Im Z sy0.118"0.030"0.009, 8Ž . Ž .where the first error is a statistical one obtainedduring the fitting, and the second is systematic to bediscussed below.

The visible cross section for the class 1 and thefitted curve are shown in Fig. 5. Despite a 4%resonant background, the interference pattern is

Fig. 5. Energy dependence of the visible cross section for theŽ .process 1 and optimal curve describing data.

clearly seen. Another representation of the interfer-< < cence amplitude is Zs Z e with

< <Z s0.158"0.030"0.009,

cs y49"10"4 8. 9Ž . Ž .

The branching ratio for the decay f™vp 0 canbe obtained from the following relation:

< < 2s Z00 q1.9 y5B f™vp s s 4.8 "0.8 =10 ,Ž . Ž .y1 .7sf

Ž q y. 2where s s12p B f™e e rm s4220 nb is thef f

w xcross section in the f-meson peak 9 .The following sources of systematic errors were

considered: inaccuracy of the detection efficiencyŽ .estimation, interference between the processes 1

Ž . qand 6 , contribution from the nonresonant e ey™3pg reaction, inaccuracy of Monte Carlo simula-tion of the M distribution, and possible deviations3p

from linear energy dependence of the nonresonanteqe y™3pg cross section. The total systematicerror in the nonresonant cross section s was esti-0

mated to be 11%. It is determined mainly by the10% uncertainty in the detection efficiency.

Of the factors listed above, the main systematicerror in the measured interference amplitude Z isintroduced by a possible nonlinearity of the energy

Ž .dependence of the cross section of the process 1 . ItŽ . Ž .is equal to 4% for Re Z and 7% for Im Z . OtherŽ .factors contribute to Re Z only at a level of 2%.

This nonlinearity also determines the error in theslope A.

To estimate systematic errors caused by a possibledetector instability during lengthy data taking, wedivided data into three subsets and processed themseparately. It was found that all three subsets areconsistent with the fit obtained from the summarydata. The fit parameters obtained from each subset

Ž .agree well with each other and with 8 . To checkthe stability of the obtained results, we loosenedcuts: for the parameter x the cuts 50 and 100 were4p

<used instead of 20 and 40, for the parameter M y3p

<782 – 50 and 100 instead of 35 and 100. As a result,the detection efficiency increased up to 20% and theresonant background in the maximum became s sf1Ž .0.20"0.05 nb, that is 20% of the visible cross


Ž .section 1 . However the obtained interference ampli-Ž . Ž .tude was Re Z s0.108"0.026, Im Z sy0.121

Ž ."0.029 which does not contradict 8 , thus confirm-ing the validity of the background subtraction proce-

Ž .dure. As a final result we take 8 , because it wasobtained with a lower resonant background. Thedifference between the results was considered as anestimate of the systematic error of the resonant back-ground subtraction. Thus, the total systematic errorof the interference amplitude Z is equal to 6% for

Ž . Ž .Re Z and 8% for Im Z .

5. Conclusion

The nonresonant cross section of the processeqey™vp 0 obtained in this work is in agreement

w xwith our old result 10 on the neutral v-mesonŽ q y 0 0 0 . Ždecay mode: s e e ™vp ™p p g s 8.7"

.1.0"0.7 nb. The measured nonresonant cross sec-tion s is almost twice higher than the expected0

Ž . 0value, calculated with only r 770 ™vp transitiontaken into account. The agreement may be signifi-cantly improved if the known radial excitations ofthe r meson are taken into account.

Ž .The measured interference amplitude Z 8 isw xclose to the lower limit of theoretical predictions 5 ,

although contributions from radial excitations of r

w xnot considered in Ref. 5 , may affect its value.Another important remark is a small value of themeasured real part of the interference amplitude

Ž .Re Z , which could be hardly explained by thew xstandard fyv mixing model 5 . For example, the

y5 w xvalue of 8.2=10 predicted in Ref. 4 for the

branching ratio of the decay f™vp 0, is 1.5 timeshigher than one measured in this work.

Ž .The interference amplitude 9 measured in thiswork is five standard deviations above zero. Thus,we claim the existence of the decay f™vp 0 withthe branching ratio of

B f™vp 0 s 4.8q1 .9 "0.8 =10y5 .Ž . Ž .y1 .7

Acknowledgements

The authors are grateful to S.I.Eidelman for use-ful discussions and valuable comments. This work issupported in part by The Russian Fund for BasicResearches, grants No. 97-02-18561 and 96-15-96327.

References

w x1 S.I. Serednyakov, in: Proc. of HADRON97, Upton, NY,August 24–30, 1997, pp. 26–35.

w x2 M.N. Achasov et al., Preprint Budker INP 97-78, Novosi-birsk, 1997, e-print hep-exr9710017.

w x Ž .3 R.R. Akhmetshin et al., Phys. Lett. B 415 1997 445.w x Ž .4 V.A. Karnakov, Yad. Fiz. 42 1985 1001.w x5 N.N. Achasov, A.A. Kozhevnikov, Int. J. Mod. Phys. A 7

Ž .1992 4825.w x6 V.M. Aulchenko et al., in: Proc. Workshop on Physics and

Detectors for DAFNE, Frascati, April, 1991, p. 605.w x7 M.N. Achasov et al., Preprint Budker INP 96-47, Novosi-

birsk, 1996.w x Ž .8 E.A. Kuraev, V.S. Fadin, Sov. J. Nucl. Phys. 41 1985 466.w x Ž .9 C. Caso et al. Particle Data Group , Europ. Phys. Jour. C 3

Ž .1998 1.w x Ž .10 S.I. Dolinsky et al., Phys. Reports 202 1991 99.

4 March 1999


Jrc , cX and Drell-Yan production

in S-U interactions at 200 GeV per nucleon

NA38 Collaboration

M.C. Abreu a,1, J. Astruc b, C. Baglin c, A. Baldit d, M. Bedjidian e, P. Bordalo a,2,A. Bohrani f, A. Bussiere c, P. Busson f, J. Castor d, T. Chambon d, C. Charlot f,`

B. Chaurand f, I. Chevrot d, D. Contardo e, E. Descroix e,3, A. Devaux d,O. Drapier e, B. Espagnon d, J. Fargeix d, R. Ferreira a, F. Fleuret f, P. Force d,J. Gago a,2, C. Gerschel b, P. Gorodetzky g,4, J.Y. Grossiord e, A. Guichard e,J.P. Guillaud c, R. Haroutunian e, D. Jouan b, L. Kluberg f, R. Kossakowski c,

G. Landaud d, C. Lourenco a,h, L. Luquin d, R. Mandry e, S. Mourgues d,F. Ohlsson-Malek e,5, S. Papillon b, J.R. Pizzi e, C. Racca g, S. Ramos a,2,A. Romana f, B. Ronceux c, P. Saturnini d, S. Silva a, P. Sonderegger h,2,

X. Tarrago b, J. Varela a,6

a LIP, AÕ. Elias Garcia 14, P-1000 Lisbon, Portugalb IPN, IN2P3-CNRS and UniÕersite de Paris-Sud, F-91406 Orsay Cedex, France´

c LAPP, IN2P3-CNRS, F-74941 Annecy-le-Vieux Cedex, Franced LPC Clermont-Ferrand, IN2P3-CNRS and UniÕersite Blaise Pascal, F-63177 Aubiere Cedex, France´ `

e IPN Lyon, IN2P3-CNRS and UniÕersite Claude Bernard, F-69622 Villeurbanne Cedex, France´f LPNHE, Ecole Polytechnique, IN2P3-CNRS, F-91128 Palaiseau Cedex, France

g IRes, IN2P3-CNRS and UniÕersite Louis Pasteur, F-67037 Strasbourg Cedex, France´h CERN, CH-1211 GeneÕa 23, Switzerland

Received 15 December 1998Editor: L. Montanet

1 Also at UCEH, Universidade de Algarve, Faro, Portugal.2 Also at IST, Universidade Tecnica de Lisboa, Lisbon, Portugal.´3 Now at Universite Jean Monnet, Saint-Etienne, France.´4 Now at PCC College de France, Paris, France.`5 Now at ISN, Grenoble, France.6 Now at CERN, Geneva, Switzerland.

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00057-X

( )M.C. Abreu et al.rPhysics Letters B 449 1999 128–136 129

Abstract

A detailed study of Jrc , cX and Drell-Yan production in S-U collisions has been performed by experiment NA38 at the

CERN SPS. This paper presents production cross sections and their centrality dependence, based on the largest sample ofS-U events collected by the experiment. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

Statistical QCD predicts that, at sufficiently highenergy density, matter should undergo a transitionfrom the hadronic phase to a medium of partonicdegrees of freedom, where quarks and gluons are nolonger confined to specific hadrons. The formationof such a medium should induce considerable changesin the normal pattern of heavy quarkonia production.In particular, the production of Jrc mesons shouldbe suppressed, either due to the colour screening of

Ž .the cc potential c melting by ‘Debye screening’ ordue to the breaking of the cc bound state by scatter-ing with deconfined energetic gluons. Recent de-

w xscriptions of these scenarios can be found in Ref. 1 .Jrc production in high energy nucleus-nucleus

interactions has been studied by experiment NA38,at the CERN SPS, since 1986. The first resultsobtained with 200 GeV per nucleon oxygen and

w xsulphur beams revealed 2,3 that Jrc production isindeed suppressed in ion collisions, both relative tothe p-A case and as a function of the transverseenergy, E , released in the ion collisions. However,T

it was later found that the suppression pattern ob-served in S-U collisions was compatible with a

w xnatural extrapolation of the p-A trend 4 , whereQGP formation is certainly not expected. To fullyclarify this issue, NA38 collected a significantlylarger sample of sulphur-uranium events. The corre-sponding results, reported in this paper, provide avery important element in the study of charmoniumproduction and suppression, together with the results

w x w xobtained in p-A 5 and in Pb-Pb 6 interactions.

2. Apparatus and data reduction

The sulphur data reported here were collected atthe CERN-SPS in 1990, 1991 and 1992. The main

features of the detector are recalled hereafter. Specialemphasis is put on some differences with respect tothe setup used in 1987, described in detail in Ref.w x2 .

The main component of the NA38 detector was amuon spectrometer composed of a toroidal magnetbetween 2 sets of multiwire proportional chambersand 4 trigger hodoscopes. In the geometrical config-uration used to collect the S-U data, muon pairs were

Ž .accepted in the dimuon lab rapidity range 2.8–4.0.The muon spectrometer was protected from the highmultiplicity environment of the target region by a520 cm long hadron absorber. It was made of 40 cm

Ž .of aluminum oxyde Al O followed by 480 cm of2 3Ž .carbon graphite disks in ‘setup 1’, while in ‘setup

2’ the last 80 cm of carbon were replaced by iron.Besides absorbing all the hadrons produced in the

Žinteraction including a large fraction of pions and.kaons before their decay into muons , the heavier

absorber imposes a higher threshold on the energythat the muons must have to trigger the experiment.In setup 2, therefore, the low mass muon pairs arestrongly reduced at the trigger level, allowing toincrease the intensity of the beam without affectingthe acquisition dead-time. This effectively increases

Ž .the statistics in the Jrc mass region see Table 1 ,degrading very little the mass resolution at the Jrc

peak, from ss142 MeV to 147 MeV, with anoperating current of 4000 A in the spectrometermagnet.

In both setups the non-interacting beam wasstopped by a 400 cm long tungsten and uranium plug

Table 1Data samples collected with the two setups

Beam intensity Interactions Recorded SelectedŽ .ions per burst per burst events events

6 6 6 6Setup 1 ;9=10 ;1.4=10 9.5=10 2.1=107 7 7 6Setup 2 ;8=10 ;1.5=10 1.5=10 2.5=10

( )M.C. Abreu et al.rPhysics Letters B 449 1999 128–136130

located 165 cm downstream from the centre of thetarget.

Ž .The active target system in setup 1 setup 2Ž .consisted of 10 12 uranium sub-targets, each ofŽthem 1 mm thick i.e. 1.7% of a S-U interaction

.length . The targets were surrounded by 32 ringscintillators, used to identify the sub-target where theinteraction occurred and to tag, for offline rejection,events where a spectator fragment has interacted in asubsequent sub-target. The small transverse dimen-

Ž 2 .sions of the targets 1=2 mm and their largeŽ .relative spacing 24 mm were chosen to minimize

Žthe interaction of secondary particles and the show-.ering of photons and electrons in the downstream

sub-targets.The first sub-target was larger than the others

Ž 2 .10=10 mm to intercept 100% of the incidentbeam, allowing to determine the targetting efficiencyof the whole system. The sulphur beam had a spotsize of s ,300 mm and s ,600 mm. To helpx y

centering the beam on the sub-targets, the targetsystem was aligned between two small quartz detec-tors, each 3 mm thick and subdivided into fourquadrants.

A 15 radiation length electromagnetic calorimeterprovided an estimate of the centrality of the collisionfrom the measurement of the neutral transverse en-ergy. It was made of 1 mm diameter scintillatingplastic fibers embedded in a 14 cm thick lead con-verter in a 1:2 volume ratio. It measured the trans-

Ž .verse energy E in the lab pseudo-rapidity rangeT

1.7–4.1. The neutral transverse energy was obtainedfrom the measured value after correcting for thecontribution from charged hadrons. This chargedcomponent amounted to ;40% according to aMonte-Carlo simulation which used a ratio 2:1 of

w xhadronic to electromagnetic energy flux 7 .Ž .A beam hodoscope BH was used to count and

identify the beam particles. It was made of twoconsecutive planes of plastic scintillators, one di-vided in 16 and the other in 14 slabs. It was located33 m upstream of the target, where the beam spot

Ž .was large enough s ,3 mm, s ,8 mm to inter-x y

cept most of the slabs and induce acceptable individ-ual counting rates. The average beam intensity andthe number of interactions in the 4.5 s spill is shownin Table 1 for the two data sets. At high luminosities,the probability of having two interactions in the 20

ns wide reading gate of the detectors is not negligi-ble. If not rejected, these double interactions wouldspoil the transverse energy measurement. Events aretagged as ‘beam pile-up’ when more than one inci-

w xdent ion crosses the BH in the 20 ns gate 2 , butsince the target thickness was ;20% l , there isI

only a small fraction of this event sample where twointeractions actually occurred. The ‘interaction pile-up’ sub-sample is identified through the shape analy-sis of the electromagnetic calorimeter signals.

The final analysis event sample is selected accord-w xing to the criteria described in Refs. 2,3,8 which,

briefly summarized, require two reconstructed trackswith a common origin in the target region, no rein-teraction of a spectator fragment and no pile-up.Table 1 reports the number of events before and afterapplying the selection cuts.

ŽBesides prompt dimuons vector meson decays.and qq annihilation , the measured opposite-sign

muon pair event sample also includes muon pairsoriginating from meson decays. The correlated muonpairs from the simultaneous semileptonic decays ofD and D mesons must be purely estimated byMonte-Carlo simulation. On the contrary, the uncor-related muon pairs resulting from decays of pionsand kaons also contribute to the sample of like-signmuon pairs. This source of background can, there-fore, be estimated from the collected data using therelation

qq yy'N s2=R = N =N ,bg bg

qqŽ yy. Žwhere N N is the number of positive nega-.tive muon pairs. Since this relation is only valid if

the muon acceptance in the apparatus is independentof its charge, a fiducial cut is applied which rejectsevents where at least one muon, if oppositely charged,would not have been accepted. R reflects eventualbg

charge correlations among the numbers of positiveand negative muons. For high multiplicity events,like those produced in S-U collisions, R is ex-bg

pected to be unity. The prompt dimuon signal isobtained from N sNqyyN , where Nqy issignal bg

the number of collected events with opposite-signmuon pairs.

Fig. 1 shows that the number of backgroundevents, N , is a very small fraction of the number ofbg

collected opposite-sign events, Nqy, in the mass


Fig. 1. Background fraction in the opposite-sign muon pair sam-ple, as a function of the mass.

window of relevance for this paper: less than 1% inthe Jrc peak and less than 3% in the c

X massregion.

3. Data analysis

The aim of this study is to obtain the yield of Jrc

and cX mesons through their dimuon decay and to

compare with the corresponding yield of Drell-Yanpairs. The analysis is restricted to the kinematicaldomain where the acceptance of the spectrometer isabove 10% of its maximum value, i.e. a dimuon labrapidity between 3 and 4, and a dimuon polar decayangle in the Collins-Soper reference frame limited to< Ž . <cos u -0.5. Fig. 2 shows the muon pair invari-CS

ant mass spectrum, as collected with setup 2, afterbackground subtraction.

The numbers of Jrc , cX and Drell-Yan events

are extracted by fitting the signal dimuon mass dis-tribution to a superposition of line shapes represent-ing these three contributions. The line shapes areobtained by a Monte-Carlo procedure that generatesevents and simulates the acceptance and smearingeffects, as induced by the detector. These simulatedevents are treated in the same way as the real eventsin what concerns reconstruction and selection crite-ria.

The charmonia resonances are generated withŽ .gaussian rapidity distributions of width 0.6 and

Ž .with transverse momenta given by m PK m rTT 1 T

Ž .with Ts236 MeV . The quoted parameter valueswere found through an iteration procedure that re-sulted in a good agreement between the resultingrapidity and p shapes and the measured Jrc distri-T

w xbutions 9 . The charmonia events are generatedŽ .according to a uniform cos u distribution.CS

Ž .The correlated mass and rapidity distributions ofthe Drell-Yan events are generated with the LOexpression, using the GRV LO 92 set of parton

w xdistribution functions 10 . The polar decay angle2Ž .follows the usual 1qcos u distribution whileCS

the transverse momentum is generated with the samefunction as the Jrc , but with a variable dimuonmass.

It is important to note that the results presented inthis paper are completely insensitive to the exactshape of the p distributions used to generate theT

events. Indeed, the acceptance of the muon spec-trometer is rather flat as a function of p , down toT

p s0 for dimuon masses above ;2 GeVrc2, mak-T

ing the integrated acceptances independent of thegeneration function.

ŽThe intermediate mass continuum between the f

.and the Jrc peaks produced in p-A and S-U inter-actions has been analysed as a superposition of muonpairs originating from the Drell-Yan mechanism andfrom semi-leptonic decays of D and D mesonsw x9,11 . From those studies we know that the mass

Fig. 2. Signal dimuon mass spectrum. The analysis reported in thispaper is limited to the window above 2.9 GeVrc2.


spectra can be very well described by the superposi-tion of these expected sources, if the normalisationof the DD contribution is fitted from the data.

In order to keep the charmonia analysis as insensi-tive as possible to the physical understanding of thelower mass continuum, the dimuon mass distribu-tions are fitted in the window above 2.9 GeVrc2.The normalisation of the DD curve is previouslyfixed from a fit to the mass window 1.5–2.0 GeVrc2

and, as can be appreciated from Fig. 2, has a negligi-ble effect in the analysis of the high mass region.

The analytical curves presented in Fig. 2 areempirical parametrisations of the mass distributionsof the four simulated physical processes, as theyappear after the acceptance and smearing effects

w xinduced by the spectrometer 12 .In view of the extremely high statistics collected

in the Jrc peak, the Monte-Carlo simulation shapesmust be slightly adjusted to precisely reproduce themeasured Jrc shape, to prevent any bias on theresults of the fit in terms of yields. This is done byleaving as free parameters in the fit the mean, m, and

w xwidth, s , of the Jrc ‘‘gaussian’’ 12 . These twoparameters are also used to adjust the analyticalfunction that describes the c

X simulated shape, im-posing m

X s m q 0.589 GeVrc2 and sX s

X Ž X .s s rs , where s s is the width of thesim sim sim simŽ X .simulated Jrc c peak.

A maximum likelihood fit to the dimuon massdistribution provides the values of m and s , thenumber of Jrc events, N , and the ratios N XrNc c c

and N rN , where N X and N are the numbersc DY c DY

of cX and Drell-Yan events, respectively. The result

of the best fit is included in Fig. 2.

4. Absolute cross sections and ratios

The absolute cross section of Jrc production inS-U collisions was determined from an event samplecollected with setup 1, for which the luminosity has

w xbeen precisely measured 9 .The incident flux of Sulphur ions is determined

by the beam hodoscope. A small fraction of thebeam flux, 3.15%, interacts in the quartz beam detec-tor, BI, located just upstream from the target. Theseinteractions are identified by the active target systemand rejected from the analysis event sample. The

calculation of the integrated luminosity takes intoaccount this absorption factor.

The effective length of the target is calculatedŽfrom the length of the first sub-target which inter-

.cepts all the beam profile using the ratio betweenthe number of events produced in all the sub-targetsand in the first sub-target. This ratio is computedusing only the events with a muon pair in the massrange 2.7–3.5 GeVrc2, where the background con-tribution can be neglected.

The efficiency of the algorithm that recognises thesub-target where the interaction took place dependson the number of charged particles released in thecollisions and, therefore, on E . Very peripheral S-UT

collisions, with E below 13 GeV, have a sub-targetT

recognition efficiency lower than 15% and wererejected. We have estimated the fraction of eventssurviving this cut, 92"1%, using the E distribu-T

tion before applying the sub-target recognition algo-rithm. We must correct for this ‘acceptance’ to ob-

Ž .tain the total all E cross sections. For the selectedT

event sample, the average efficiency to identify theinteraction sub-target is 84"3%.

The fraction of events that survive the pile-upselection cut is 92"1%, while for 87"1% of theevents no reinteraction is detected in a downstreamsub-target.

To calculate the Jrc absolute cross section, wealso need to take into account the track reconstruc-

Ž . Žtion efficiency 95"4% , the trigger efficiency 94."4% and the acceptance of setup 1 for Jrc pro-

duction, 19.8%. The total systematic error affectingthe absolute cross sections amounts to ;8%.

From a sub-sample of events collected with thesetup 1, in the kinematical domain 3.0-y-4.0 and< Ž . <cos u -0.5, we obtain Bs s7.78"0.04"0.62CS c

Žmb for the Jrc cross section multiplied by the.branching ratio into muons .

This measurement is in good agreement with thew xvalue previously published 13 by NA38, based on

the data collected in 1987, once extrapolated to theŽ .whole cos u domain and to x )0.CS F

From the high statistics event sample collectedŽ .with setup 2 see Table 2 , we have measured very

precisely the ratios Bs XrBs s0.76"0.04% andc c

Bs rs s25.1"0.8, integrating the Drell-Yanc DY

cross section between 2.9 and 4.5 GeVrc2. It isimportant to note that systematical errors arising


Table 2Acceptances and number of events of setup 2, after quality and

X Žkinematical selection cuts, for Jrc , c and Drell-Yan in the2 .2.9–4.5 GeVrc mass window

Ž .A % Events

Jrc 16.0 113452X

c 18.4 936DY 17.1 4808

from normalisation uncertainties do not affect theseratios.

From these ratios and the Jrc absolute crosssection, we have computed the c

X and the Drell-Yancross sections: Bs X s59.1"6.2"4.7 nb and sc DY

Žs310"10"25 nb in the mass window 2.9–4.52 .GeVrc .

5. Centrality dependence of charmonia produc-tion

In the context of using charmonia suppression asa signature of QGP formation in nucleus-nucleuscollisions, it is very important to understand how theproduction yields change from peripheral to centralcollisions. This study is considerably facilitated by

Ž .using as a reference the production of high massDrell-Yan dileptons, a well known process, linearlyproportional to the number of nucleon-nucleon colli-sions and not expected to be affected by QGP forma-tion.

We use the neutral transverse energy measured bythe electromagnetic calorimeter, E , to study char-T

monia production in S-U interactions as a function ofthe centrality of the collision. The transverse energydistribution for Drell-Yan pairs with an invariantmass above 4 GeVrc2 is shown in Fig. 3. Thespectrum is corrected for the target identificationinefficiency and is normalized to the integrated crosssection value.

The total sample of muon pair events is dividedinto 5 approximately equipopulated sub-samples, ac-cording to the transverse energy of the event. Table 3includes the E ranges of each bin. Events aboveT

ŽE s88 GeV, in the tail of the E distribution seeT T.Fig. 3 , have not been used in this centrality depen-

dence study.

Fig. 3. Transverse energy distribution for events with M)42 Ž . Ž .GeVrc , before open squares and after filled circles sub-target

inefficiency correction.

The mass distributions obtained for each sub-sam-ple of events are fitted as previously explained, usingas free parameters the total number of Jrc eventsand the ratios Bs rs and Bs XrBs . The m andc DY c c

s parameters are fixed to the values obtained fromthe total sample of events. Table 3 shows the fittedvalues.

Fig. 4 shows the E dependence of the ratiosT

Bs rs and Bs Xrs . Both ratios decrease sig-c DY c DY

nificantly as E increases, i.e. as the interactionT

becomes more and more central.The Bs XrBs ratio has been shown to be con-c c

w xstant from pp to p-U 5 , with an average value of1.64"0.03%. The band on the right side of Fig. 4has been derived from the Bs rs pattern assum-c DY

ing that the same value would still hold in S-Ureactions. The c

Xrc ratio measured in S-U colli-sions is more than a factor of 2 lower than the p-Avalue, and the difference is seen to be more pro-

Table 3Centrality dependence of N and of the ratios c rDY and c

Xrcc

XE N Bs rs Bs rBsT c c DY c c

Ž . Ž .GeV %

13–88 113288"543 25.2"0.8 0.76"0.0813–34 18265"154 29.7"2.4 1.15"0.1834–50 20969"167 26.9"1.8 0.91"0.1750–64 21240"168 24.7"1.6 0.77"0.1564–77 23430"177 23.7"1.4 0.56"0.1477–88 17684"154 22.6"1.5 0.39"0.15


Ž . X Ž .Fig. 4. Ratio of Jrc left and c right to Drell-Yan cross sections as a function of E . The Drell-Yan cross section is given in the massT

range 2.9–4.5 GeVrc2.

nounced in the most central interactions. While inp-A collisions the nuclear medium has exactly thesame influence on the two charmonia states, as if

Ž .they were the same preresonant state, in the case ofX Ž .S-U collisions the Jrc and c fully formed reso-

nances clearly suffer different interactions with thesurrounding matter.

It is important to compare these results on thecentrality dependence of charmonia production withmodel calculations and with results from other exper-

iments. However, it is not easy to directly comparethe E distributions measured by NA38 with theoret-T

ical calculations of E or with other centrality re-T

lated variables, measured by other experiments. Thisemphasizes the importance of converting the mea-sured transverse energy in a variable more directlyrelated to the geometry of the nucleus-nucleus colli-sions, as the impact parameter, b.

The relation between the neutral transverse energyand the impact parameter of the reaction has been

Ž . Ž .Fig. 5. Correlation between E and the impact parameter of the reaction left and the length of matter crossed by the preresonantTŽ .charmonia state right .


Table 4Average values of E , impact parameter and L, for each E bin.T T

The error bar given with L is the r.m.s. value corresponding to thewidth of the E binT

² : Ž . Ž . Ž .E GeV b fm L fmT

25.4 7.5"0.7 4.8"0.642.2 6.0"0.7 6.0"0.457.2 4.8"0.8 6.7"0.470.6 3.6"1.0 7.3"0.482.1 2.6"1.0 7.6"0.3

determined with a geometrical model that properlydescribes the S-U E spectra. This simulation isT

based on an improved version of the model de-w xscribed in Ref. 8 . The transverse energy is com-

puted as being proportional to the number of woundedw xnucleons 14,15 , calculated for each impact parame-

ter from the overlap of the nuclear density distribu-tions. We have used the Woods-Saxon nuclear densi-

w xties with the numerical values tabulated in Ref. 16 .The calculation incorporates gaussian fluctuations inthe correlation between the number of wounded nu-cleons and the released transverse energy. The result-ing correlation between E and the impact parameterT

can be seen on the left side of Fig. 5, while theaverage values of b, and of E , are collected inT

Table 4, for each of the five centrality bins used inthis paper.

w xIt has been empirically observed 4 that the mea-sured Jrc yields follow a simple exponential scal-ing as a function of the length of nuclear matter, L,

crossed by the charmonia state. This purely geomet-rical variable, fully determined from the impact pa-rameter of the collision, has been determined by ageometrical simulation similar to the one describedabove. The calculated correlation between E and LT

is presented on the right side of Fig. 5, while theaverage values of L in our five E bins are includedT

in Table 4.The dependence of charmonia production on the

centrality of the S-U interactions, as parametrised bythe variable L, is shown in Fig. 6. The scaling of the

w xJrc yields on L suggests 4 that the suppression ofcharmonia production relative to the Drell-Yan yieldis due to final state absorption of the cc states whilemoving through the nuclear matter. Within this inter-pretation, the pattern of charmonium suppressionshould be, to first order, an exponentially decreasingfunction of L.

Fitting the Jrc data to the expressionŽ .exp yr s L , and using an average nuclear den-abs

sity of rs0.17 nucleonsrfm3, we obtain an absorp-tion cross section of s c s5.68"1.92 mb. Onlyabs

statistical errors were taken into account since thisvalue is fully determined by the relative variation ofthe Jrc yield from peripheral to central S-U colli-sions. A more refined treatment of the collision

Ž w x.geometry see Ref. 15 leads to an absorptioncross-section ;13% higher than the value obtainedin the simplified procedure described here.

If we apply the same procedure to the cX data we

find a four times bigger absorption cross section,

Fig. 6. Dependence of the Bs rs and Bs Xrs ratios on the variable L. The line on the left figure is the best fit to the expressionc DY c DYŽ .exp yr s L . The band in the right figure corresponds to the same line, scaled by 1.64"0.03%.abs


quantifying the stronger suppression of cX produc-

tion in S-U collisions.

6. Conclusions

Charmonia and Drell-Yan production cross sec-tions have been measured in S-U interactions, at's s20 GeV. In the kinematical domain defined by

< Ž . <3.0-y-4.0 and cos u -0.5 we obtain Bs sCS c

7.78"0.04"0.62 mb, Bs XrBs s0.76"0.04%c c

and Bs rs s25.1"0.8, integrating the Drell-c DY

Yan cross section between 2.9 and 4.5 GeVrc2. TheJrc production cross-section per nucleon-nucleoncollision decreases with increasing centrality. A sig-nificantly steeper decreasing pattern is observed forthe c

X resonance.

References

w x Ž .1 H. Satz, in: A. Zichichi Ed. , International School of Subnu-clear Physics, Erice, Italy, 1997, BI-TP-97-47, hep-phr9711289.

w x2 NA38 Collaboration, C. Baglin et al., Phys. Lett. B 220Ž .1989 471.


w x Ž .4 C. Gerschel, J. Huefner, Z. Phys. C 56 1992 71.w x5 M.C. Abreu et al., NA38 Collaboration, accepted by Phys.

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Clermont-Ferrand, 1993; L. Luquin, Ph.D. Thesis, UniversiteBlaise Pascal, Clermont-Ferrand, 1995.


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w x Ž .10 M. Gluck, E. Reya, A. Vogt, Zeit. Phys. C 53 1992 127;¨Ž .Phys. Lett. B 306 1993 391.

w x11 M.C. Abreu et al., NA38 Collaboration, Nucl. Phys. A 566Ž .1994 77c; A. Borhani, Ph.D. thesis, Ecole Polytechnique,Palaiseau, 1996.

w x12 F. Fleuret, Ph.D. thesis, Ecole Polytechnique, Palaiseau,1997.


w x Ž .14 A. Bialas et al., Nucl. Phys. B 111 1976 461.w x Ž .15 D. Kharzeev et al., Z. Phys. C 74 1997 307.w x16 C.W. de Jager et al., Atomic Data and Nuclear Data Tables

Ž .14 1974 485.

4 March 1999


The atmospheric neutrino flavor ratiofrom a 3.9 fiducial kiloton-year exposure of Soudan 2

W.W.M. Allison c, G.J. Alner d, D.S. Ayres a, G. Barr c,1, W.L. Barrett f, C. Bode b,P.M. Border b, C.B. Brooks c, J.H. Cobb c, R.J. Cotton d, H. Courant b,D.M. Demuth b, T.H. Fields a,2, H.R. Gallagher c, C. Garcia-Garcia d,3,

M.C. Goodman a, R. Gran b, T. Joffe–Minor a, T. Kafka e, S.M.S. Kasahara b,W. Leeson a, P.J. Litchfield d, N.P. Longley b,4, W.A. Mann e, M.L. Marshak b,

R.H. Milburn e, W.H. Miller b, L. Mualem b, A. Napier e, W.P. Oliver e,G.F. Pearce d, E.A. Peterson b, D.A. Petyt d, L.E. Price a, K. Ruddick b,

M. Sanchez e, J. Schneps e, M.H. Schub b,5, R. Seidlein a,6, A. Stassinakis c,J.L. Thron a, V. Vassiliev b, G. Villaume b, S. Wakely b, D. Wall e, N. West c,

U.M. Wielgosz c

a Argonne National Laboratory, Argonne, IL 60439, USAb UniÕersity of Minnesota, Minneapolis, MN 55455, USA

c Department of Physics, UniÕersity of Oxford, Oxford OX1 3RH, UKd Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK

e Tufts UniÕersity, Medford, MA 02155, USAf Western Washington UniÕersity, Bellingham, WA 98225, USA

Received 4 January 1999Editor: L. Montanet

Abstract

We report a measurement of the atmospheric neutrino flavor ratio, R, using a sample of quasi-elastic neutrino interactionsŽ .occurring in an iron medium. The flavor ratio tracksrshowers of atmospheric neutrinos in a 3.9 fiducial kiloton-year

Ž . Ž .exposure of Soudan 2 is 0.64"0.11 stat. "0.06 syst. of that expected. Important aspects of our main analysis have beenchecked by carrying out two independent, alternative analyses; one is based upon automated scanning, the other uses amultivariate approach for background subtraction. Similar results are found by all three approaches. q 1999 Elsevier ScienceB.V. All rights reserved.

1 Now at CERN, Geneva, Switzerland.2 Now at Fermilab, Batavia, IL 60510, USA.3 Now at IFIC, E-46100 Burjassot, Valencia, Spain.4 Now at Physics Department, Colorado College, Colorado Springs, CO 80903, USA.5 Now at Cypress Semiconductor, Minneapolis, MN, USA.6 Now at Lucent Technologies, Naperville, IL 60566, USA.


( )W.W.M. Allison et al.rPhysics Letters B 449 1999 137–144138

1. Introduction

The flavor ratio in sub-GeV atmospheric neutrinointeractions as measured in underground detectorshas sensitivity to a breakdown of the Standard Modelin the neutrino sector. The flavor double ratio, de-fined as

n qn r n qnŽ .Ž .m m e e DataR s , 1Ž .tn qn r n qnŽ .Ž .m m e e MC

factors out the dependence on the absolute flux andin principle provides a measurement with small sys-tematic errors. In practice the pure n and n ratesm e

cannot be measured directly and the experimentsmeasure

tracks r showersŽ . Ž . DataRs 2Ž .

tracks r showersŽ . Ž . MC

for the iron calorimeters or

m ring r e ringŽ . Ž . DataRs 3Ž .

m ring r e ringŽ . Ž . MC

for the water Cherenkov detectors. The measuredvalues of R depend on the acceptance and misidenti-fication in each individual experiment and are thusnot expected to be equal to each other or to R .tHowever a measurement different from 1.0 in anyexperiment is evidence of an anomaly.

w xSix experiments have reported results on R 1–6 .These measurements suggest a value of R signifi-cantly lower than unity. The highest statistics on thismeasurement come from the water Cherenkov exper-iments, Kamiokande, IMB, and SuperKamiokande.Three iron calorimeter experiments, NUSEX, Frejus,and Soudan 2, have reported results. Our previous

w x Ž .q0.05Ž .result 6 , R s 0.72 " 0.19 stat. syst. wasy0.07

based on an exposure of 1.52 kton-years. The confir-mation of the low atmospheric flavor ratio with goodstatistical significance in a calorimeter would pro-vide additional evidence that there is no significantunexpected source of systematic error in water detec-tors. In this paper we report a value of Rs0.64"

Ž . Ž .0.11 stat. "0.06 syst. obtained in a 3.9 fiducialkiloton-year exposure of Soudan 2.

There are three stages involved in our analysis.First a sample of contained events is identified.These are then classified for neutrino flavor. Finallya background subtraction is made and a value of Rcalculated. Each of these stages, particularly theflavor classification, could introduce bias into theflavor ratio measurement. We have therefore checkedthe procedure by using different analyses. Theseanalyses give consistent results and confirm the va-lidity of our principal result.

Sections 3-5 describe our principal result, theflavor ratio as measured in a sample of quasi-elasticinteractions from a 3.9 fiducial kiloton-year expo-sure. The main analysis relies on physicist scanningfor the verification of containment and for the flavordetermination. By performing an analysis in which

Žcomputer programs largely replace the scanning the.Automated Analysis , we have verified that the main

procedure does not introduce biases due to subjectiv-ity in the scanning. We have also checked our methodof background subtraction and R calculation by anadditional analysis in which an alternative method

Žfor background estimation is used the Multivariate.Analysis . These analyses are described in Section 6.

2. The Soudan 2 detector

The Soudan 2 detector is a 963 ton fine-grainedgas tracking calorimeter located in the Soudan Un-derground Mine State Park in Soudan, Minnesota.The detector currently operates with 90% live timeand has been taking data since 1989. It consists of224 iron modules, each 1 meter x 1 meter x 2.7meters in size, and occupies a volume 8 meters widex 5.5 meters high x 16.1 meters long. Each modulehas a mass of 4.3 tons. Ionization deposited in theplastic tubes of a module drifts in an electric field tothe faces of the module where it is detected byvertical anode wires and horizontal cathode strips.The third coordinate of the charge deposition isdetermined from the drift time in the module. Thecalorimeter modules operate in proportional mode;the measured pulse height is proportional to theionization deposited in the tube. Pulse height mea-surements are used for particle identification. Moredetails of the module construction and performance

w xcan be found in Refs. 7,8 .

( )W.W.M. Allison et al.rPhysics Letters B 449 1999 137–144 139

The detector is surrounded by a 1700 m2 activeshield mounted on the cavern walls. The shield isdesigned to identify particles entering or exiting thecavern. It tags events associated with cosmic raymuons passing close to the detector. It has a mea-sured efficiency of 95% for cosmic muons crossing ashield element. The complete shield covers about

w x97% of the total solid angle. Ref. 9 contains moreinformation about the shield.

The most recent 2.4 kton-yrs of data were ob-tained with a number of improvements to the detec-tor modules and shield. Additional layers added tothe shield provided a third layer of shield coveragefor much of the floor and ceiling sections.

3. Data analysis

The data described in this paper come from a 3.9fiducial kiloton-year exposure taken between April1989 and January 1998. This corresponds to a totalexposure of 4.8 kiloton-years. During this time some100 million triggers were recorded.

3.1. Contained eÕent selection

In the initial stage of our data analysis a sample ofcontained events is selected. A contained event isone in which all tracks and the main body of anyshowers are located within the fiducial volume, de-fined by a 20 cm depth cut on all sides of thedetector. All events are processed by software filters.Events that satisfy the filter criteria are then scannedby physicists to finalize the containment selection.Data events are interspersed with Monte Carlo eventsso that on an event by event basis the scanner doesnot know if he or she is scanning Monte Carlo ordata. The scanning is performed in two stages withthree independent scans carried out at each stage.The contained event selection is fully described in

w xRefs. 6,10 .The Monte Carlo sample used in this analysis is

5.45 times the size of the expected neutrino sample.The Monte Carlo simulates neutrino interactions inthe detector; background processes are not simulated.

The Monte Carlo detector simulation reproduces theactual performance of the Soudan 2 detector to ahigh degree of accuracy. The real detector geometryis simulated, as are local variations in the detectorperformance, particularly pulse height and drifting.Background noise in the detector is included byoverlaying Monte Carlo events onto randomly initi-ated triggers generated throughout the exposure ofthis data set. For the first 2.2 kton-yrs exposure theMC and data events were combined prior to thescanning while for the latter 1.7 kton-yrs exposureMonte Carlo events were inserted into the data streamduring data acquisition at the Soudan mine. Data andMC events are analyzed identically at each stage ofthe data reduction.

3.2. FlaÕor classification

During scanning, events are classified into one ofthree categories: single track, single shower, andmultiprong. The single track category is further sub-divided into mu-like tracks and protons as describedin the following paragraph. The track and showercategories include primarily n and n quasi-elasticm e

scattering respectively; they are largely equivalent tothe ‘single ring’ category in the water Cherenkovexperiments. In addition to the lepton, events inthese categories may contain recoil nucleons at thevertex andror small showers from muon decay attrack endpoints. Events with two or more particlesŽ .other then recoil nucleons emerging from the pri-mary vertex, or single track events which are chargedpions having visible scatters, are classified as multi-prong.

Proton tracks can be identified because they arestraight and highly ionizing. All tracks are fitted to astraight line trajectory and the track residual andaverage pulse height are calculated. Tracks with lowfit residuals and high average pulse height are classi-fied as protons. There is some overlap betweenprotons and short, low energy muons where most ofthe observed track has b<1. The separation algo-rithm is tuned to minimize the incorrect tagging ofmuons as protons. Muon tracks are incorrectly classi-fied 4% of the time and 80% of protons are correctlyidentified. Fig. 1 shows the track residual versus


Fig. 1. Proton Identification. Fit residual versus average pulseheight for Monte Carlo events. The left plot shows the results

Ž .from 2-d fits to the track in the event usually a muon . The rightŽ .plot shows the results from the fit to the recoil usually a proton

in track q recoil events. The band at low fit residual in each plotis due to protons and the band at low average pulse height tomuons.

Ž .average pulse height for MC tracks mostly muonsŽ .and recoils mostly protons .

4. The flavor ratio

4.1. Shield classifications

Contained events are a mixture of neutrino inter-actions and background processes. Neutral particleswhich are produced by the interaction of cosmic raymuons in the rock surrounding the detector cavernare the principal source of background. These parti-

Ž .cles neutrons and photons can produce containedevents if they travel into the fiducial volume of thedetector before interacting. Such events are usuallyaccompanied by large numbers of charged particleswhich strike the active shield located at the cavernwalls. The presence of shield activity accompanyinga contained event therefore provides a tag for back-ground events.

The shield information allows us to identify twoseparate event samples in our data. An event withzero shield hits is referred to as ‘gold’; such an event

is a neutrino candidate. Events with two or moreshield hits are referred to as ‘rock’ events; theycomprise a shield-tagged background sample. Table1 gives the number of ‘gold’, ‘rock’, and MonteCarlo events in each of the scanned categories. Dataevents with one shield hit are a mixture of neutrinoevents with a random shield hit, stopping muons thatpass the containment tests, and multiple shield hitevents where shield hits are missing due to shieldinefficiency. Consequently the single shield hit eventsinclude both neutrino signal and backgrounds, andwe have excluded them from our analysis. The lossof neutrino events due to random shield hits issimulated by rejecting Monte Carlo events with shieldhits in the randomly triggered background event.

4.2. Background corrections

It is possible that some muon interactions in therock produce contained events unaccompanied byshield hits, due either to shield inefficiency or be-cause the interaction did not produce any chargedparticles which entered the shield. The number ofsuch interactions is determined by examining thedistributions of event depths in the detector, wherethe event depth is defined as the minimum distancebetween the event vertex and the detector exterior,excluding the detector floor. These are shown in Fig.2.

We fit the depth distributions to determine theamount of background present in the gold sample.

Ž .An extended maximum likelihood EML fit, thatcorrectly handles bins with small numbers of events,is performed which describes the data distributionsas a sum of background and neutrino distributions.The shapes of the neutrino and background depthdistributions are obtained from the MC simulationand the rock samples respectively. From the rock

Table 1Ž .Raw numbers of gold, rock shield-tagged background and Monte

Carlo events in each of the 4 categories

Event type Track Shower Multiprong Proton

Gold 95 151 125 49Rock 278 472 232 277MC 749 729 711 82


Fig. 2. Depth distributions. Gold data are crosses. The rockŽ .distributions shaded histograms are normalized to the amount of

background present in the gold sample as determined by the depthŽ .fit. The MC distributions open histograms are normalized to the

number of neutrino events present in the gold data as determinedby the depth fit. The dashed histogram shows the best fit to thedata.

sample, we have determined that the trackrshowerratio for background is 0.59"0.04. We have previ-ously shown that the trackrshower ratio of the back-ground does not vary as a function of shield hit

w xmultiplicity 6 . We therefore expect background pre-Ž .sent in the gold zero shield hit sample to occur in

this same trackrshower ratio and we include thisexpectation as a constraint in the fit. Since the‘flavor ratio’ in the background is very similar tothat measured for neutrino events, background sub-traction does not produce a large change in ourmeasured ratio.

Early data had some contamination of the showersample from electrical breakdown inside the mod-ules. A cut requiring at least 9 hits on all showerswas used to remove this contamination. A minimumof 6 hits were required on all tracks.

4.3. Calculation of the flaÕor ratio

The results of the depth fits are that 76.9"10.8of the gold tracks and 116.3"12.8 of the goldshowers are due to neutrino interactions. We usethese numbers to calculate the background correctedatmospheric neutrino flavor ratio. Table 2 shows the

Ž .flavor ratios with and without ‘raw’ the back-ground subtraction.

The systematic error due to the background sub-traction has two components.1. Many of the single tracks in the background

sample are protons, which come from neutronsentering the fiducial volume of the detector andelastically scattering. Hence the proton classifica-tion, which removes single protons from the tracksample and places them in a separate category,serves as a background correction even before thedepth fits are performed. An alternative approachto the one we have taken is to leave the singleprotons in the track sample and determine theamount of background solely from the depth fits.The resulting value for the flavor ratio differsfrom our main value by d Rsq0.023; the fulldifference is taken as a component of the system-atic error.

2. Our method assumes that any background presentin the gold sample behaves identically to theshield-tagged background of the rock sample. Wehave investigated how the results change if thisassumption is not valid. For instance, if the zero-shield hit shower background has a different depthdistribution than the rock shower sample then the

Ždepth fit which assumes the rock distribution for.the background will incorrectly estimate the

amount of background present. We have consid-ered a number of such effects and have deter-mined the resulting uncertainty on R to be d Rs q0.041.y0.030

The total error on R due to the background subtrac-tion can be obtained by adding these two contribu-

Table 2Data used in the calculation of the corrected flavor ratio. TheMonte Carlo numbers in parentheses are normalized to the detec-tor exposure. The error on the flavor ratio is statistical only

Number of gold tracks 95Number of gold showers 151

Ž .Number of mc tracks 749 137.4Ž .Number of mc showers 729 133.8

Corrected number of n tracks 76.9"10.8Corrected number of n showers 116.3"12.8

Raw value of R 0.61"0.09

Corrected value of R 0.64"0.11


tions in quadrature: d Rs q0.047. Systematic errorsy0.038

due to the uncertainty in the expected flavor ratio,Monte Carlo, and scanning procedure are calculated

w xto be d Rs"0.040 6 . Here we have taken thesystematic uncertainty in the expected flavor ratio tobe d RrRs3%. Adding this in quadrature to theerror from the background subtraction results in atotal systematic error of d Rs q0.062. Our primaryy0.056

measurement of the flavor ratio is therefore R sŽ . Ž .0.64"0.11 stat. "0.06 syst. .

5. Alternative analysis methods

We have used two other methods to determine theflavor content of the atmospheric neutrino flux.

5.1. MultiÕariate discriminant analysis

This method uses the same event sample as themain analysis but uses additional event variables toprovide discrimination between background and neu-trino interactions. There are several quantifiable dif-ferences between background events and neutrinoevents. In comparison with neutrino interactions,

Ž .background events are on average closer to thedetector exterior, of lower energy, traveling preferen-tially downward, and are more likely to be traveling

Ž .into the detector rather than outward . The corre-sponding variables are the event depth, energy, zenith

Žangle, and ‘inwardness’ defined as the cosine of theangle between the event direction and the inward-pointing normal vector to the nearest face of the

.detector . These event variables are combined, usingthe method of multivariate discriminant analysisw x11,12 , to form a single variable.

Fig. 3 shows the distributions of the discriminantvariable for MC neutrino, rock, and gold events. Theimproved discrimination between the expected neu-trino and rock distributions is clearly visible in thefirst and last bins. A fit of the discriminant variabledistributions for the gold events to a sum of thedistributions for rock and Monte Carlo events thengives the amount of background for track and showerevents separately and thus a value of R. Thetrackrshower ratio in the background sample is againused to provide a constraint on the zero shield hitbackground determined from the fit, as in the main

Fig. 3. Distributions of the multivariate discriminant variablecombining event depth, energy, zenith angle, and inwardness.Track distributions are at top, showers at bottom. Gold data are

Ž .crosses. The rock distributions shaded histograms are normalizedto the amount of background present in the gold sample asdetermined by the fit to the multivariate distributions. The MC

Ž .distributions open histograms are normalized to the number ofneutrino events present in the gold data as determined by the fit tothe multivariate distributions.

analysis. The result of the fit, using the four vari-Ž .ables described above is Rs0.61"0.10 stat. only .

Unlike the event depth, the distributions of othervariables in this analysis would be influenced bypossible new physics. This analysis therefore shouldonly be regarded as a test of the null hypothesis, i.e.no new physics, for which the value expected isRs1.0.

5.2. Automated CEV selection and flaÕor determina-tion

A third analysis that uses software for both eventselection and flavor determination has been devel-

w xoped and is described in detail elsewhere 13 . It hasbeen applied to a subsample of the data correspond-ing to a 2.7 fiducial kiloton-year exposure. Since thismethod has been applied to a smaller data set and thecontained event selection is done differently, theevent sample used in this analysis is not identical tothat used in the main analysis. The important virtuesof this method are that it almost entirely eliminatesthe role of subjective decisions and permits the useof much larger Monte Carlo samples. The shield is


used to separate data events into neutrino candidatesŽ . ŽGOLD events and a background sample ROCK

.events , using the same principles as in the mainanalysis. The ‘GOLD’ and ‘ROCK’ samples here arenot the same as the gold and rock samples describedpreviously since the contained event samples aredifferent and shield information is used differently.Events are required to have more than 10 hits. Thedistinction between track and shower events is madeon the basis of L, a variable derived from a spheri-cal harmonic analysis of inter-hit correlations in eachevent. Fig. 4 shows distributions of L for the dataand all true n and n interactions as given by them e

Monte Carlo. All events are included and no attemptis made to distinguish neutral current interactions,inelastics and quasi-elastics. From the Monte Carlowe expect neutral current interactions to account for12% of the events in the final data sample.

Events are separated into ‘n -like’ and ‘n -like’m e

sub-samples by application of an energy dependentŽcut on L. Of n interactions neutral and chargedm

.current , 78.3% are correctly tagged as n -like whilem

80.7% of n interactions are correctly tagged ase

n -like.e

A depth fit is performed to the GOLD data interms of a combination of ROCK and Monte Carlo

Ž . ŽFig. 4. Distributions of L for n dashed histogram and n solidm e.histogram Monte Carlo events which survive event selection cuts.

ŽAll events produced by the given neutrino flavor charged and.neutral current are included. Background corrected GOLD data

are shown as crosses. The deficit of n -like events relative tom

n -like events in the data is clearly evident.e

Table 3Effects of the flavor cut on the Monte Carlo, rock, and datadistributions in the automated analysis

n -like n -likem e

Monte Carlo 17353 15915Rock 328 410Contained events 109 171

due to n-interactions 74.9"13.5 111.5"15.7due to rock 34.1"12.1 59.5"14.4

w xdistributions 13 . The results of this procedure aresummarized in Table 3. The ratio of ratios is Rs

Ž . Ž .0.62"0.14 stat. "0.05 syst. . This figure is calcu-lated using all events: quasi-elastic, inelastic andneutral current.

6. Conclusion

The flavor ratio of atmospheric neutrinosŽ .datarMC has been measured from a 3.9 fiducialkiloton-year exposure of the Soudan 2 detector to be

Ž . Ž .0.64"0.11 stat. "0.06 syst. . This result is ob-tained after applying a background correction to asample of 246 quasi-elastic neutrino candidates. Theprobability of a statistical fluctuation to Rs0.64 orbelow is less than 4=10y3. Two other independentanalyses have been performed. These check the con-tained event selection, flavor determination, andbackground correction procedures of our main analy-sis. Both alternative analyses confirm the validity ofthe main analysis. The good agreement of these threerather different methods gives confidence that theeffect is not an artifact of a particular analysis. Thismeasurement is in good agreement with the previ-ously published result from this experiment as wellas the results from the water Cherenkov experiments.

Acknowledgements

This work was undertaken with the support of theU.S. Department of Energy, the State and Universityof Minnesota, and the U.K. Particle Physics andAstronomy Research Council. We would also like tothank: the Minnesota Department of Natural Re-sources for allowing us to use the facilities of theSoudan Underground Mine State Park; the staff of


the Park, particularly Park Managers P. Wannarkaand J. Essig, for their day to day support; and MessrsB. Anderson, J. Beaty, G. Benson, D. Carlson, J.Eininger and J. Meier of the Soudan Mine Crew fortheir work in the installation and running of theexperiment.

References

w x1 Kamiokande Collaboration: K.S. Hirata et al., Phys. Lett. BŽ . Ž .205 1988 416; B 280 1992 146. Kamiokande Collabora-

Ž .tion: Y. Fukuda et al., Phys. Lett. B 335 1994 237.w x2 IMB Collaboration: D. Casper et al., Phys. Rev. Lett. 66

Ž .1991 2561. IMB Collaboration: R. Becker-Szendy et al.,Ž .Phys. Rev. D 46 1992 3720.

w x3 Super Kamiokande Collaboration: Y. Fukuda et al., Phys.Ž .Rev. Lett. 81 1998 1562.

w x4 NUSEX Collaboration: M. Aglietta et al., Europhys. Lett. 8Ž .1989 611.

w x Ž .5 Frejus Collaboration: K. Daum et al., Z. Phys. C 66 1995417.

w x6 Soudan 2 Collaboration: W.W.M. Allison et al., Phys. Lett.Ž .B 391 1997 491.

w x7 Soudan 2 Collaboration: W.W. M Allison et al., Nucl. In-Ž .strum. Methods A 376 1996 36.

w x8 Soudan 2 Collaboration: W.W. M Allison et al., Nucl. In-Ž .strum. Methods A 381 1996 385.

w x Ž .9 W.P. Oliver et al., Nucl. Instrum. Methods A 276 1989371.

w x10 H. Gallagher, Neutrino Oscillation Searches with the Soudan2 Detector, Ph.D. thesis, University of Minnesota, 1996.

w x11 R. Granadesikan, Methods for Statistical Data Analysis ofMultivariate Observations, 2nd ed., Wiley, 1997.

w x12 J. Schneps, Multivariate analysis of atmospheric neutrinodata: Part 1 - The ratio of ratios, Soudan 2 Internal MemoPDK-690, November 1997, unpublished.

w x13 A. Stassinakis, A Study of the Atmospheric Neutrino ContentUsing the Soudan 2 Detector, D. Phil. thesis, Oxford Univer-sity, 1997; W.W. M Allison et al., Atmospheric NeutrinoFlavour Detection in the Soudan 2 Detector, a New Experi-mental Analysis, in preparation.

4 March 1999


0Study of the process pp™hhp from 1350 to 1940 MeVrc

A.V. Anisovich d, C.A. Baker a, C.J. Batty a, D.V. Bugg b, R.P. Haddock c,C. Hodd b, V.A. Nikonov d, C.N. Pinder a, A.V. Sarantsev d, V.V. Sarantsev d,

I. Scott b, B.S. Zou b

a Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UKb Queen Mary and Westfield College, London E1 4NS, UK

c UniÕersity of California, Los Angeles, CA 90024, USAd PNPI, Gatchina 188350, St. Petersburg district, Russia


Abstract

Data on the final state hhp 0 are presented for beam momenta of 1350, 1525, 1642, 1800 and 1940 MeVrc. There isevidence for the presence of a broad high mass 2q contribution in hh; the amplitude analysis fits it as a resonance with massMs1980"50 MeV and width Gs500"100 MeV. It has the curious feature that it is produced dominantly with orbitalangular momentum Ls1 in the final state and with component of spin "1 along the beam direction. It contributes typically

Ž .;15y20% of the integrated cross section. In addition, the presence of a 1660 is confirmed at all beam momenta. Also2Ž .f 2100 ™hh is confirmed. q 1999 Published by Elsevier Science B.V. All rights reserved.0


We present evidence for resonances in hp and0hh channels observed in the reaction pp™hhp at

five p momenta from 1350 to 1940 MeVrc. Wefind a new broad hh resonance with J P s2q, massM s1980" 50 MeV and width G s 500"100MeV. We also confirm the hh resonance at 2100-

w x2120 MeV observed in earlier data 1,2 and deter-mine its spin to be 0q. We confirm the presence ofthe Is1hp J P C s2qq resonance at 1660"15MeV, discovered in an earlier analysis of Crystal

w xBarrel data at 1940 MeVrc 3,4 .

The data were taken by the Crystal Barrel Collab-oration at LEAR using a trigger demanding neutralfinal states. An average of 9=106 triggers weretaken at each momentum. The detector has been

w xdescribed fully in an earlier publication 5 . A 4.4 cmlong liquid hydrogen target is situated at the centreof the cylindrical detector, which covers most of 4p .A veto counter downstream of the target eliminatesnon-interacting p and also pp elastic scattering atsmall angles. Immediately surrounding the target is asilicon vertex detector, used here simply to veto


( )A.V. AnisoÕich et al.rPhysics Letters B 449 1999 145–153146

events containing charged particles. Next is a jet driftchamber, again used here as a veto. Photons aredetected in a barrel of 1380 CsI crystals, 16 radiationlengths long covering 98% of 4p and pointing to-wards the target; because of the Lorentz boost, this is95% in the centre of mass. The angular resolution inthe laboratory reference system is ;"20 mrad inboth polar and azimuthal angles; the energy resolu-tion D ErE is 2.5%rE1r4 with E in GeV, and thedetector is highly efficient down to photon energiesbelow 20 MeV. A beam intensity of typically 2=105

y1s was used. Incident p were defined by a coinci-dence between a small multiwire chamber and a Sicounter of diameter 5 mm, close to the entrance ofthe target. Cross sections are determined with rela-tive errors F3% from 1350 to 1800 MeVrc fromthe beam counts, target length and density, and thereconstruction efficiencies estimated by a MonteCarlo simulation. The error may be "10% at 1940MeVrc, since these data were taken in a number ofearlier runs under different beam conditions. Theerror on absolute normalisation arising from the un-certainty in target length is "2.5%.

Event reconstruction follows the techniques usedw x 0earlier 6 . To select the hhp channel, we demand

exactly 6 photons satisfying a 7C kinematic fit withconfidence level )10%; events fitting 3p 0 orhp 0p 0 with confidence level )0.1% are rejected,and also those few events fitting 3h, h

Xp 0p 0 or

hXhp 0 with confidence level )1%. A Monte Carlo

simulation using GEANT shows that the reconstruc-tion efficiency is almost constant at 21–22% forhhp 0 at all momenta. The worst backgrounds arise

0 0 Ž 0 . 0 0 0from vp p events v™p g or hp p p whereone photon is lost in the former case and two in thelatter. These channels have branching fractions largerthan hhp 0 by factors of 16 and 9 respectively.Residual backgrounds are 3.4% and 2.8% respec-tively at 1800 MeVrc. That from hp 0p 0 is 0.5%.Other small backgrounds and 1% wrong pairings ofphotons to hhp 0 produce a total background of7.8"0.6% at 1800 MeVrc, dropping at lower beammomenta. Numbers of events and backgrounds aresummarised in Table 1. The distribution of back-ground events on the Dalitz plot has been evaluatedand appears to be random. It has therefore beenapproximated by a phase space distribution. It hasbeen included in the amplitude analysis, but has

Table 1Numbers of events and background levels

Ž .Beam momentum Events Background %Ž .MeVrc

1940 5832 8.01800 5672 7.81642 4929 6.71525 4425 5.41350 4640 4.2

almost no effect; setting it to zero, log likelihoodchanges by only 1 or 2 at each momentum, a negligi-ble amount.

Ž .Dalitz plots and projections on to M hp andŽ . Ž .M hh are shown in Figs. 1 and 2. The a 980 and0Ž .a 1320 are clearly visible as horizontal and vertical2

Ž .bands in hp ; f 1500 ™hh produces a strong di-0

agonal band. In Fig. 2 at 1940 MeVrc, a diagonalŽ .band due to f 2100 ™hh interferes with the cross-J

Ž .ing a 980 bands; its centre is indicated by the0

diagonal line. It is less conspicuous at lower beammomenta in Fig. 1. The Dalitz plots for data exhibitstatistical fluctuations typically 20% per bin, whilethe fit is smooth; this accounts for some apparentdiscrepancies.

The amplitude analysis fits the following chan-nels:

pp™a 980 h 1Ž . Ž .0

™a 1320 h 2Ž . Ž .2

™a 1660 h 3Ž . Ž .2

™ f 975 p 4Ž . Ž .0

™ f 1270 p 5Ž . Ž .2

™ f 1500 p 6Ž . Ž .0

™ f 2100 p or f 2050 p 7Ž . Ž . Ž .J 4

™ f 1980 p 8Ž . Ž .2

™ f 1770 p . 9Ž . Ž .0

Ž .The a 980 is fitted with a Flatte formula using´0w xparameters determined previously 7 . Channels 2, 4,

5 and 6 are fitted with Breit-Wigner amplitudes withconstant width using masses and widths taken from

w x Ž .the Particle Data Tables 8 . Parameters of a 166020w xare taken from Ref. 4 . Other data on pp™3p at

the same beam momenta, not shown here, display a

( )A.V. AnisoÕich et al.rPhysics Letters B 449 1999 145–153 147

Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Fig. 1. Dalitz plots for a – d data at 1800, 1642, 1525 and 1350 MeVrc, e – h fits, i – l projection on to M hp , m – p projectionŽ .on to M hh . Data are shown with error bars; histograms show fits.

Ž .strong f 1270 signal. From its known branching2

ratios to hh and p 0p 0, we calculate that it con-tributes 5–7% of the hhp 0 channel, and fits to these

Ž .data agree closely with this estimate. The f 975 ™0

hh makes only minor contributions at the right-handedge of each Dalitz plot and has no repercussions onphysics conclusions. It is introduced for consistency

Ž . 0with the f 975 signal observed in our 3p data0Ž .not shown here and with GAMS data on pp™hh

w x9 , which determine its coupling constant to hh. WeŽ .have also tried adding a 1450 and an a in the0 0

mass range 1650–1800 MeV. There is no significant

Ž .evidence for either. Contributions from f 1370 can0

be calculated from our data on 3p 0 and are -1%.w xThe decays of resonances are fitted as in Ref. 6

using the Wick rotation and the angles of Fig. 3. Inbrief, centre of mass momenta are rotated throughpolar angle t and azimuthal angle f to the directionin which resonance X is produced and are thenboosted to the rest frame of X. Then they are rotatedback through angles yf ,yt . Quantum mechanicalrotation matrices needed for the first rotation arecancelled by the second, and amplitudes are invariant

w xunder the Lorentz boost 10 . Let us take as an


Ž . Ž . Ž . Ž .Fig. 2. a The Dalitz plot for a beam momentum of 1940 MeVrc; the diagonal line marks the location of f 2100 ; b the fit; c0Ž . Ž . Ž .projection on to M hh ; d projection on to M hp .

3 Ž .example the amplitude for P pp™ f 2100 p . For2 2

initial helicity m, this takes the form

2 my mX mX

XŽ . Ž . Ž .Gexp id C B Y t ,f Y a ,bm ,m 1 1 2f s .Ým 2

X M y sy iMGm sy2

10Ž .

Here mX is the component of spin of the resonancealong the beam direction, G is a coupling constant,C X are Clebsch-Gordan coefficients, Y are spheri-m ,m

cal harmonics, and d is a phase for this channel; B1

is the Blatt-Weisskopf centrifugal barrier factor forproduction with Ls1.

We fit production and decay in full for channelsŽ .1, 3, 7, 8 and 9. For f 2100 , the centre of massJ

momentum is F256 MeVrc, so angular momentaLs0 and 1 in the final state are sufficient, and the

Ž . Ž .same is true for a 1660 . The a 980 h channel2 0Ž .requires up to Ls3; for f 1500 and the very weak0

Ž . Ž .f 1770 there is no evidence for L)0. For a 13200 2Ž .and f 1270 , approximations to the production pro-2

cess must be made. For Ls3, a total of 13 partialwaves would be required for each and the statisticsare inadequate. For these resonances, we averageover the production process by omitting the t depen-dence and the Clebsch-Gordan coefficient. The de-cay dependence on a , b and mX are retained in full.Interferences with other channels then need to behandled approximately. Consider as an exampleŽ . Ž .a 1320 interfering with f 1500 . Let their ampli-2 0


Fig. 3. Angles used in the Wick rotation. The dashed line indicatesthe direction of a decay particle from resonance X in the restframe of X after the Wick rotation.

tudes be A and B. Each may be produced frommany initial pp states. The interference term is

Ž ) .therefore fitted as 2cRe A B , where c is a coher-ence factor fitted in the range y1 to q1 and allow-ing for partial coherence between A and B. The

Ž .interference of the crossing a 1320 bands is treated2

likewise. Only those interferences improving loglikelihood by more than 2 standard deviations areretained. For the purposes of the present paper,

Ž . Ž .interferences between a 1320 , f 975 and2 0Ž .f 1270 play only a minor role and have little effect2

on physics conclusions. The important interferenceŽ . Ž .of a 980 with f 2100 is treated exactly.0 J

We now return to the data and results of fits. ForŽ . Ž .f 2100 and the broad f 1980 , we are concernedJ 2

with the bottom left corner of the Dalitz plots. There,we find almost no dependence of data on productionangle t for M G1450 MeV. This allows us tohh

describe the decay using either angle a or aX of Fig.

3. It is well known that cosaX varies linearly with s

along a resonance band in the Dalitz plot. Interfer-ences with other intersecting bands appear as pertur-bations to the cosa

X dependence. To remove them,we display results against cosa , where this depen-dence is averaged over t and hence disappears.

Figs. 4 show the dependence of cross sections on< < Ž . Ž .cosa for a series of hh mass bands. In a – g , dataare at a beam momentum of 1525 MeVrc, so that

Ž .the a 1320 band is almost completely excluded;2

results at other momenta are very similar. In Fig.Ž . Ž .4 a , the f 1500 produces an almost flat angular0

wdistribution, as expected for spin 0. Fits includingXŽ .f 1525 give a negligible contribution, and we also2

know from data at 1940 MeVrc for the KqKyp 0

w x xfinal state 11 that its contribution is very small . InŽ . Ž .Figs. 4 b – g , with increasing mass, an angular

dependence gradually develops requiring Js2 or 4.The signal is too broad in mass to be fitted byŽ . Ž .f 2050 . In the last slice, Fig. 4 h shows results for4

M G2050 MeV at the highest beam momentum ofhh

Ž .1940 MeVrc. The f 2100 makes a large contribu-0

< < ŽFig. 4. Decay angular distributions ds rd cosa in arbitrary. Ž . Ž .units for several ranges of M ; a – g are at a beam momentumhh

Ž . Ž .of 1525 MeVrc for mass intervals: a 1450–1550 MeV, bŽ . Ž . Ž .1550-1640 MeV, c 1640–1690 MeV, d 1690–1750 MeV, eŽ . Ž . Ž .1750–1820 MeV, f 1820–1900 MeV, g 1900–2000 MeV; h

is the distribution for M s2050–2200 MeV at a beam momen-hh

tum of 1940 MeVrc. Curves show the fit.


tion there and some flattening of the angular distribu-tion is noticeable near cosas0.

A combined fit to all five beam momenta requiresa broad 2q state in hh with mass 1980"50 MeVand Gs500"100 MeV, where errors cover sys-tematics. Fig. 5 shows the log likelihood distributionas a function of mass and width. An alternative fit

Ž .was tried with a broad f instead of f 1980 . At0 2

1525, 1642 and 1800 MeVrc log likelihood im-Ž . Ž .proved very little ;10 when f 1980 was intro-0

Ž .duced. On the contrary, f 1980 improves log likeli-2Žhood by typically 62–84 at four momenta see Table

.3 below . We regard that as highly significant: statis-tically more than 10 standard deviations. The fits to

< <the cosa distribution are considerably poorer withŽ .f 1980 omitted. The scatter of the points about the2

fit in Fig. 4 are somewhat above statistics, but the fitŽ .rejects any significant f 2050 contribution and we4

have been unable to identify further structure in the

Ž . Ž .Fig. 5. Variation of log likelihood with a M and b G forŽ .f 1980 .2

Table 2Ž . Ž .Branching fractions % for five beam momenta MeVrc

Process 1350 1525 1642 1800 19401Ž .f 1980 p D 2.9 1.3 1.7 2.1 0.42 23Ž .f 1980 p P 2.9 4.0 5.6 2.0 1.92 1

XqŽ .f 1980 p 2 m s0 0.1 0.2 0.0 0.2 0.02XqŽ .f 1980 p 2 m s1 7.4 10.3 7.8 4.7 5.92

3Ž .f 1980 p F 6.8 5.2 8.2 6.9 6.82 3Ž .a 980 h 17.2 14.8 14.6 14.5 16.00Ž .a 1320 h 30.3 25.6 24.2 26.2 29.22Ž .a 1660 h 5.4 7.5 8.9 10.7 13.42Ž .f 975 p 3.3 1.9 4.3 2.7 5.40Ž .f 1270 p 7.5 5.9 7.0 7.1 5.42Ž .f 1500 p 9.7 11.9 7.3 10.1 6.10Ž .f 2100 p 0.8 3.8 5.6 8.9 8.60

q Ž .2 component. We also know the f 2050 contribu-4

tion to be negligible from high statistics data on0 w xpp™3p at the same beam momenta 12 ; those

data place a tight limit on any contribution from0 0 0Ž .pp™ f 2050 p , f ™p p .4 4

The data of Fig. 4 immediately reveal a veryŽ .curious property of the f 1980 . It is dominated2

strongly by the amplitude with final Ls1 and mX s"1. This amplitude is proportional to sinacosa . Itis obvious that the mX s"2 amplitude must besmall, since it varies as sin2a and would peak in Fig.4 at cosas0. Likewise, the mX s0 amplitude, pro-portional to 1.5cos2ay0.5, has a zero at cos2as1r3; its contribution is obviously small.

In more detail, the data have been fitted in twoalternative ways, which give very similar conclu-sions. In the first method, they are fitted as describedabove with three decay amplitudes having mX s0,"1 and "2. The mX s"1 amplitude is totallydominant over mX s0 by at least a factor 10; mX s"2 is negligible. In the second approach, we fitindividual partial waves having Ls0 and 1 in thefinal state, coming from initial states 1D , 3P , 3P ,2 1 23F and 3F . Branching fractions are summarised in2 3

wTable 2. They do not add up to exactly 100%xbecause of interferences . The Ls0 component from

1D has a strikingly small cross section, because the2< 0Ž . < 2data are incompatible with the Y a distribution2

it would require. What Table 2 does not reveal is thatcontributions to the mX s0 and "2 amplitudes inter-fere destructively so that the total intensities fromthese mX values are very small.


Before commenting on this result, we turn toŽ .f 2100 . Even if it is fitted with Js2 or Js4, itsJ

contribution is consistent with a flat angular distribu-tion superposed on the broad 2q signal beneath it.We find no significant evidence at any of the fivebeam momenta for departure from isotropy for eitherproduction or decay. This suggests Js0. A fit withJs0, Ls0 alone is excellent. A fit with Js2 or 4is very poor unless final states with Ls1 dominatestrongly over Ls0; a combination of 3P , 3P , 3F1 2 2

and 3F is capable of simulating a flat angular3

distribution for both production and decay. We nowargue that that the dependence of the cross sectionon beam momentum favours Ls0.

Near threshold, the production cross section forLs0 should vary as krk 2 , where k is the centre ofi n

mass momentum with which the resonance is pro-duced and k is the centre of mass momentum ofi n

Ž .the initial p. Fig. 6 a shows the production cross

Ž . Ž . Ž . Ž .Fig. 6. Cross sections of a f 2100 b a 1660 . Full linesJ 2

show the k dependence expected for Ls0 and the dashed line inŽ .a the Ls1 cross section predicted with a radius of 0.8 fm forthe Blatt-Weisskopf centrifugal barrier.

Ž .section for f 2100 as a function of k. The full lineJ

shows the expected variation for Ls0, normalisedto the point at 1800 MeVrc. It is in fair agreementwith the data, although the cross section seems tosaturate at large k, either because of a form factor orbecause of competition with other S-wave reactions.The dashed curve shows the expected dependence

3Ž .2 2for Ls1: k B rk . It disagrees strongly with1 i nŽ .the trend of the data. We draw from Fig. 6 a the

inference that Ls0 is dominant. Then Js2 and 4are excluded by ;80 in log likelihood. If Js4 isincluded with Ls1, it gives a worse fit than Js2

Žby typically 30 in log likelihood statistically )7.standard deviations ; but if its mass and width are

Ž .fixed at PDG values for f 2050 , the fit is worse by4Ž .a further 32. So any contribution from f 2050 is4

certainly small.The combined evidence from k dependence and

Ž .angular dependence require that f 2100 has Js0.Jw xThis is consistent with an analysis 2 of JrC™

Ž .g 4p and also with the second of two solutionsw x y qfound by GAMS 13 for data on p p ™hh. In

both cases, there is a strong peak at 2100 MeV. Ourw xown data on pp™hh 14 , also show a strong

Ž . w xf 2100 signal. The E760 group 1 finds a very0

prominent peak at this mass in hh, though no spindetermination is made. In view of their high statis-tics, we fit to their mass Ms2104 MeV and widthGs216 MeV, which are close to our best fit: Ms2115"15"15 MeV, Gs210"10"20 MeV.

Finally we require at all beam momenta a contri-Ž .bution from a 1660 . Its production cross section,2

Ž .shown by the full line in Fig. 6 b is similar to the kŽ .dependence full line expected for Ls0 in the final

state. The mX s0 amplitude dominates strongly; mX

s"1 is almost negligible and mX s"2 is com-pletely negligible. This suggests dominant produc-tion from the initial state 1D with Ls0. The two2Ž .a 1660 bands both have angular distributions close2

to 1.5cos2aX y0.5 and have a strong constructive

interference close to the right-hand edge of the Dalitzplot; this is the fortunate feature which identifiesthem. We cannot improve on the earlier determina-

w xtion 3,4 of mass and width of this resonance fromthe data at 1940 MeVrc. For lower beam momenta,when the mass is scanned, log likelihood gets rapidlyworse for M-1630 MeV, so this is a lower limit.For masses above 1660 MeV, log likelihood flattens


Table 3Ž .Changes DS in log likelihood at the five beam momenta MeVrc

when these contributions are omitted and others are re-optimised;log likelihood is defined so that DSs0.5 corresponds to a onestandard deviation change per degree of freedom

Channel 1350 1525 1642 1800 19401Ž .f 1980 p D 12.5 11.0 11.9 27.3 4.12 23Ž .f 1980 p P 9.3 17.5 20.7 11.8 14.52 1

XqŽ .f 1980 p 2 ,m s1 23.4 45.8 32.2 23.7 28.023Ž .f 1980 p F 21.1 19.5 28.1 25.4 24.22 3

Ž . Ž .f 1980 p Ls1 45.8 53.9 54.7 47.2 37.92Ž .all f 1980 p 62.0 72.4 84.7 81.1 43.22

Ž .a 980 h 82.6 91.0 69.1 93.6 87.30Ž .a 1320 h 105.1 116.2 201.1 228.5 255.52Ž .a 1660 h 35.5 51.1 44.4 72.8 42.72Ž .f 975 p 11.0 14.0 36.8 30.8 74.50Ž .f 1270 p 25.1 30.6 54.2 44.2 37.72Ž .f 1500 p 29.1 51.1 56.2 69.6 62.90Ž .f 2100 p 4.3 21.4 56.2 77.8 92.70

out because of the limited available phase space inthe hp channel.

Many systematic checks have been made on thefits. Angular distributions have been examined forevery resonance band and are all fitted well. Table 3illustrates the changes in log likelihood when eachchannel is dropped from the fit and others are re-op-

Ž . Ž . Ž .timised. Dropping f 2100 or f 1980 or a 16600 2 2

produces large changes in log likelihood at severalmomenta. At 1350 MeVrc, we find also a definite

Ž . w xsmall contribution from f 1770 15 . This and pos-0Ž .sible contributions from f 1710 are the subject ofJ

an accompanying paper on beam momenta 600–1200w xMeVrc 16 .

Now we comment briefly on possible interpreta-tions of the results. The large contributions to the

0 Ž . Ž .hhp channel from f 1500 and f 2100 are very0 0w xstriking, both here and in E760 data 1 . Both reso-Ž . w xnances are also present in JrC™g 4p 2 . The

Ž .f 1500 is likely to have a strong glueball compo-0

nent. We would like to raise the possibility thatŽ .f 2100 also has a significant glueball component.0

The latest QCD calculations predict a 2qq glue-w xball with mass ;2250 MeV 17,18 . On the other

hand, mixing between the glueball and nearby qqw xstates is very probable. Bugg and Zou 19 have

qproposed that the 2 glueball mixes with qq statesto form a broad state around 2 GeV. They argue that

Ž .its mixing with f 1710 can explain the large uJ

signal in JrC radiative decays; and its overlap withss states above 2 GeV can explain the anomalous

w xff states observed by Etkin et al. 20 from 2020 toŽ .2340 MeV. The broad f 1980 which we find could2

then be a candidate for the mixed glueball-qq stateproposed by Bugg and Zou. The strong production ofŽ . Xf 1980 with Ls1, m s"1 is intriguing, but we2

do not have a firm explanation. A possible explana-tion is dominant production from the 2q initial statehaving z-component of spin ms0; Clebsch-Gordancoefficients are such that this produces purely mX s"1 in the final state.

Ž .A broad f 1930 with Gs460 MeV has also2

been reported by the Omega group at low p inTw xcentral production 21 , an obvious place to see a

glue-rich state. The BES group also made a prelimi-Ž .nary report at Hadron’97 of a broad f 2010 with2Ž . w xG;350 MeV in new data on JrC™g 4p 22 .

Acknowledgements

We thank the Crystal Barrel Collaboration forallowing use of the data. We wish to thank thetechnical staff of the LEAR machine group and of allthe participating institutions for their invaluable con-tributions to preparing and running the experimentand analysing data. We acknowledge financial sup-port from the British Particle Physics and Astronomy

Ž .Research Council PPARC , and the U.S. Depart-ment of Energy. The St. Petersburg group wishes toacknowledge financial support from PPARC and IN-TAS grant RFBR 95-0267.

References

w x Ž .1 T.A. Armstrong et al., Phys. Lett. B 307 1993 394.w x Ž .2 D.V. Bugg et al., Phys. Lett. B 353 1995 378.w x3 J. Luedemann, Ph.D. thesis, University of Bochum, 1995.w x4 A. Abele et al., submitted to Zeit. f. Phys.w x Ž .5 E. Aker et al., Nucl. Instr. A 321 1992 69.w x Ž .6 J. Adomeit et al., Zeit. Phys. C 71 1996 227.w x7 D.V. Bugg, V.V. Anisovich, A. Sarantsev, B.S. Zou, Phys.

Ž .Rev. D 50 1994 4412.w x Ž .8 Particle Data Group, C. Caso et al., Euro. Phys. J. 1998 1.w x Ž .9 F. Binon et al., Nu. Cim. A 78 1983 313.

w x Ž .10 C. Bourelly, E. Leader, J. Soffer, Phys. Rep. 59 1980 95.


w x11 M. Doser, Hadron’97 Proceedings, to be published.w x12 A.V. Anisovich et al., submitted to Phys. Lett. B.w x Ž .13 D. Alde et al., Nucl. Phys. B 269 1986 485.w x14 A.V. Anisovich et al., submitted to Phys. Lett. B.w x Ž .15 S.J. Lindenbaum, R.S. Longacre, Phys. Lett. B 274 1992

Ž .492; B.V. Bolonkin et al., Nucl. Phys. B 309 1988 426; D.Ž .Alde et al., Phys. Lett. B 284 1992 457; A.V. Anisovich,

Ž .V.V. Anisovich, A.V. Sarantsev, Phys. Lett. B 395 1997Ž .123; Zeit. Phys. A 359 1997 173.

w x Ž .16 A.V. Anisovich et al., Phys. Lett. B 449 1999 , accompany-ing paper.

w x Ž .17 G.S. Bali et al., Phys. Lett. B 309 1993 378.w x18 J. Sexton, A. Vaccarino, D. Weingarten, Phys. Rev. Lett. 75

Ž .1995 4563.w x Ž .19 D.V. Bugg, B.S. Zou, Phys. Lett. B 396 1997 295.w x Ž .20 A. Etkin et al., Phys. Lett. B 201 1988 568.w x Ž .21 D. Barberis et al., Phys. Lett. B 397 1997 339.w x22 X. Shen, Hadron’97 Proceedings, to be published.

4 March 1999


0ž /Observation of f 1770 ™hh in pp™hhp reactions0

from 600 to 1200 MeVrc

A.V. Anisovich d, C.A. Baker a, C.J. Batty a, D.V. Bugg b, R.P. Haddock c,C. Hodd b, V.A. Nikonov d, A.V. Sarantsev d, V.V. Sarantsev d, I. Scott b,

B.S. Zou b

a Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UKb Queen Mary and Westfield College, London E1 4NS, UK

c UniÕersity of California, Los Angeles, CA 90024, USAd PNPI, Gatchina 188350, St. Petersburg district, Russia


Abstract

0We present data on pp™hhp at beam momenta of 600, 900, 1050, and 1200 MeVrc. At the higher three momenta, a0Ž . Ž .signal is clearly visible due to pp™ f 1770 p , f 1770 ™hh. It has mass 1770"12 MeV and width 220"40 MeV,0 0

where errors cover systematic uncertainties as well as statistics. q 1999 Published by Elsevier Science B.V. All rightsreserved.


0We investigate pp™hhp at p momenta from600 to 1200 MeVrc. Data were taken at LEAR bythe Crystal Barrel Collaboration. Data from 1350 to1940 MeVrc are reported in an accompanying paperw x Ž .1 . They provide evidence for f 2100 ™hh,0Ž . Ž .a 1660 ™hp and a broad f 1980 ™hh. The2 2

first resonance lies above the mass range of presentdata, but is an important element in 0q spectroscopy.

Ž .The a 1660 is strong at higher momenta but, be-2

cause its threshold is at ;2200 MeV, it makes onlya minor contribution to present data up to 1200

Ž .MeVrc. The f 1980 plays an important role, since2

it is broad and extends into the present mass range.We shall provide evidence for an Is0 J P C s0qq

resonance at 1770 MeV, decaying to hh. This reso-

nance is important in trying to unravel the mysteriesŽ . w xconcerning the f 1710 2 , hence in establishingJ

the systematics of 0q mesons.At lower masses, there has been earlier evidence

from the Crystal Barrel Collaboration for an Is0resonance whose mass is variously estimated from

w x1300 to 1380 MeV 3–6 . There is also an Is1w xresonance at 1450 MeV 7 and finally the well

Ž .known f 1500 . At higher mass, the WA102 collab-0Ž .oration has published evidence for an f 2020 reso-0

w xnance decaying to 4p 8 .The experimental conditions and data processing

are identical to those for the higher momenta re-w xported in Ref. 1 , and we refer to that paper for

experimental details. The largest backgrounds are



from hp 0p 0p 0 and vp 0p 0, two photons being lostin the former case and one in the latter. Numbers ofevents and estimated backgrounds are shown in Table1. Using cross sections observed in these other chan-nels, we find that the backgrounds are close to aphase space distribution. They are included in fittingthe data, but omitting them has negligible effect onthe analysis, where log likelihood changes by -4.

Fig. 1 shows Dalitz plots at the four beam mo-menta and mass projections on to hh and hp .Histograms superimposed on the projections are theresult of the maximum likelihood fit described be-

Ž .low. There are conspicuous peaks due to f 1500 ,0Ž . Ž .a 980 and a 1320 . The feature of interest is a0 2

shoulder at ;1770 MeV in hh, clearly visible inŽ . Ž . Ž .Figs. 1 f , g and h at 900, 1050 and 1200 MeVrc.

It is barely visible at the higher momenta reported inw xRef. 1 . At 1350 MeVrc, there is also a shoulder at

;1770 MeV, and the amplitude analysis gives asmall optimum there, but the significance is found tobe much less than at lower momenta.

We shall report here a full partial wave analysisof production and decays of resonances. Amplitudeswe find to be significant are listed in Table 2. Inmaking this choice, we are guided by past experiencethat it requires roughly 225 MeVrc of momentumper unit of orbital angular momentum L in theproduction process. Present data are consistent withthis guideline. The last column of Table 2 showsmomenta in the decay process at the highest beam

Ž .momentum of 1200 MeVrc. For the a 980 h chan-0

nel, we require up to Ls3. There is no evidence forthe presence of Ls4; this agrees with the expecta-

ytion that the first 4 qq state will lie at a mass of2300 MeV or above, i.e. a beam momentum of at

Ž .least 1650 MeVrc. The a 980 is fitted with a0

Flatte form using parameters derived earlier from´0 0 w x Ž .data on pp™hp p at rest 9 . The a 980 signal0

is slightly stronger in the data than the fit, particu-

Table 1Numbers of events and estimated background levels

Momentum Events BackgroundŽ . Ž .MeVrc %

600 2922 2.8900 9023 3.9

1050 6607 3.31200 9594 3.7

Ž .larly in Fig. 1 i ; the fit can be improved by reducingŽ . Ž .the width of the a 980 by ;20%. The f 1500 p0 0

channel is described successfully with waves up toŽ .Ls2. The strong a 1320 h channel requires up to2

Ls3, but again there is no evidence for Ls4; the2q initial state with helicity 0 makes quite a signifi-

Ž .cant Ls3 contribution: 4–8% at 900 MeVrc andabove.

Figs. 2 and 3 show examples of angular distribu-tions in bands centred on resonances. The fits, shownby histograms, are equally good at other momenta.Fig. 2 shows the production angular distribution interms of the centre of mass angle t at which theresonance is produced. Fig. 3 shows decay angular

Ž .distributions of a resonance, e.g. a 1320 ™hp , in2

terms of its decay angle a with respect to the beamdirection.

0 Ž .Our data on the 3p final state not shown hereŽ . 0contain strong f 1270 p signals and lead us to2

Ž .expect weak f 1270 ™hh contributions at the level2

of 3–5% in present data; these are barely visible inthe projections. We find that this weak channel canbe approximated with Ls0 production only; thisapproximation has no impact on the interesting hh

mass range around 1770 MeV. We have tried includ-Ž .ing f 1370 in the fit, but no significant contribu-0

tion is required.We find that most partial waves follow within

errors a smooth dependence on centre of mass mo-mentum, but with fluctuations which are typically

Ž .25% for the larger waves ;8% branching ratioand 50% for smaller waves contributing branchingratios of only 2%. Contributions below 1% are

0 0dropped. We have data on pp™hp p withw xroughly a factor 12–16 higher statistics 10 . There,

many resonances are visible, but phase variationswith momentum are mostly small, i.e. resonances arephase coherent. In present data, phase variations areagain small, although there are fluctuations of typi-

Ž .cally " 10y25 8 from one momentum to another.The fit is made by the maximum likelihood method.Our definition is such that a change of log likelihoodof 0.5 corresponds to one standard deviation for achange in one parameter. The acceptance of thedetector is simulated by Monte Carlo events, pro-cessed through exactly the same selection procedureas data; statistics of accepted Monte Carlo events areroughly 4 times those of data events.


Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Fig. 1. a – d Dalitz plots at 600, 900, 1050 and 1200 MeVrc, e – h Projections on to M hh , i – l Projections on to M hp .

The fitting algorithm is sufficiently fast that theamplitude analysis produces a fit in roughly 60seconds of computing. This is sufficiently fast that

Žwe have been able to examine a large number ).1500 of trials with varying ingredients in order to

study the systematics of the fit, its stability and thesensitivity to every channel. Although some ambigui-

Žties are present at individual momenta particularlybetween Ls0 and Ls2 and between Ls1 and

.Ls3 , we have located only one solution having a

smooth variation of magnitude, and particularlyphase, with momentum. The Dalitz plots of Fig. 1are reproduced so closely by the fit that no differ-ence from data is visible within statistics.

The hh signal at 1770 MeV is visible in hh

Ž . Ž . Ž .projections of Figs. 1 f , g and h at 900, 1050 and1200 MeVrc. Fits without any resonance in this

Ž . Ž .mass range are shown in Fig. 4 a – c . There is anobvious surplus of events between 1700 and 1800MeV. We have tried fitting it with Js0 or 2.


Table 2Partial waves included in the analysis; the third column shows thecentre of mass momentum on resonance for the highest beammomentum of 1200 MeVrc

Channel Partial waves CM momentumŽ .MeVrc

1 3 1 3Ž .a 980 h S , P , D , F 7340 0 1 2 31 3 1Ž .f 1500 p S , P , D 5390 0 1 21 1Ž . Ž . Ž .a 1320 h D Ls0 and 2 , S Ls2 4982 2 03 q 3Ž . Ž .P , 2 ms0 and 1 , F Ls11 3

q Ž .2 Ls3,ms01 3Ž .f 1770 p S , P 3230 0 11Ž .f 1270 p D 6902 21 3 q 3Ž . Ž .f 1980 p D , P , 2 ms0 and 1 , F 1002 2 1 3

Variations of log likelihood with its mass are shownŽ . Ž .in Figs. 5 a – c . There is a strong, well defined

Ž .optimum for Js0 full curve . Optimum masses are1758"10 MeV at a beam momentum of 900MeVrc, 1770"8 at 1050 MeVrc and 1775"8MeV at 1200 MeVrc. Branching fractions are re-spectively 5.5%, 11.7% and 10.1% at these threemomenta. At 1350 MeVrc, there is a shallow mini-

Fig. 2. A comparison of the fit at 1200 MeVrc with angularŽ .distributions unnormalised against cost , where t is the centre of

Ž . Ž .mass production angle; a a 65 MeV wide band around a 980 ,0Ž . Ž . Ž .b a 118 MeV wide band around a 1320 , c an hh band from2

Ž . Ž .1640 to 1840, d a 120 MeV wide band around f 1500 . Points0

with error bars are data and the histogram shows the fit.

Fig. 3. As Fig. 2 for decay angular distributions with respect to thebeam in the rest frame of the resonance.

mum at the same mass and width, but the fittedŽ .f 1770 contribution is only 1.5%. Data at 6000

MeVrc are too close to threshold to show an opti-Ž . Ž .mum in the mass or width of f 1770 . Column a0

of Table 3 lists changes in log likelihood whenŽ .f 1770 is omitted from the fit.0

ŽSpin 0 gives a better optimum than Js2 shown.by dashed curves in Fig. 5 . Differences of logŽ .likelihood are given in column b of Table 3. How-

ever, spin 2 still shows some optimum at a similarmass to spin 0. We find this is due to feed-throughfrom mis-identified spin 0 signal. The detector hasholes around the beam with an opening angle of 128.This incomplete coverage allows some cross-talkbetween spin 0 and spin 2. In order to examine this,we have adopted two simulation procedures. In thefirst, we have used Monte Carlo events to generatedata samples with the same partial wave amplitudes

Ž .as are fitted to f 1770 plus other channels. Then0Ž .we have fitted these samples replacing f 1770 by a0

2q resonance. This simulation reproduces the magni-Ž . Ž .tudes of the dashed curves of Figs. 5 a and b at

900 and 1050 MeVrc quite well.The fits with spin 2 use five partial waves 1D ,2

3 q Ž . 3P , 2 ms0 and 1 and F , while fits with spin 01 3


Ž . Ž . Ž .Fig. 4. Fits without any f 1770 at beam momenta of a 900, bJŽ .1050 and c 1200 MeVrc.

use two partial waves 1S and 3P . Statistically, we0 1

would expect the fits with spin 2 to be better thanthose with spin 0 by 3 in log likelihood, because ofthe 6 extra parameters. In practice, we find this to bean under-estimate, since certain combinations of am-plitudes producing a Js2 resonance with Ls0and 1 can simulate closely the flat angular distribu-tion of the dominant 1S amplitude producing Js0.0

To examine this, our second simulation is to fit fakeJs2 resonances of width 220 MeV to the simulateddata sets. What we observe in 10 such trials is thatthe fake 2q signal improves log likelihood by an

Ž .Fig. 5. Variations of log likelihood with the mass of f 1770 , full0Ž . Ž .line Js0, dashed line Js2, a at 900 MeVrc, b at 1050

Ž . Ž .MeVrc, c at 1200 MeVrc; d the variation of log likelihoodŽ .with width at 1050 MeVrc full curve and 1200 MeVrc

Ž .dashed .

average of 17"3. This sets a scale of what may befitted incorrectly by spin 2; any improvement smallerthan this cannot be believed. Our conclusion is that afake 2q signal produced with Ls0 or 1 from 5initial partial waves can be expected to fit the databetter than a fake Js0, produced from 2 initialpartial waves, by approximately 12 in log likelihood.If we use this result, the conclusion is that Js0 ispreferred over Js2 by ;46 in log likelihood at900 MeVrc, by ;46 at 1050 MeVrc and by ;41at 1200 MeVrc. These are statistically each over 9standard deviations. If, pessimistically, we ignore thedifference in the number of fitted partial waves and

Table 3Ž . Ž .Changes to log likelihood of the basic fit, a dropping f 1770 ,0

Ž . Ž . Ž . Ž . Ž . Ž .b replacing f 1770 by f 1770 , c adding f 1712 , or d0 2 2Ž .adding f 17120

Ž . Ž . Ž . Ž .Momentum a b c dŽ .MeVrc

900 y81.4 y33.8 16.0 3.11050 y101.6 y34.2 19.3 4.81200 y83.0 y28.9 17.5 3.51350 y7.7 y5.4 19.2 3.4


use just raw differences in log likelihood betweenspin 0 and 2, the discrimination is still over 8standard deviations at each momentum.

Ž .Fig. 5 d shows log likelihood versus the width ofŽ .the f 1770 at 1050 and 1200 MeVrc, where the0

data are most definitive. At other beam momenta, theoptimum is somewhat less well defined and flattensout at large widths. From our studies with a varietyof ingredients, we estimate that the systematic errorson the mass and width of the resonance are respec-tively "12 and "40 MeV.

Ž .There has been previous evidence for f 1770 . A0

small peak has been observed at 1750 MeV forw xJs0 in two sets of data on pp™K K 11,12 .S S

Ž .Secondly, a partial wave analysis of JrC™g 4p

w xhas fitted 4p peaks at 1780 MeV with Js0 13 .w xThe GAMS collaboration has reported 14 an

Ž .X 1740 decaying to hh at 1744"15 MeV, but theyfind a width -90 MeV. Their angular distribution isflat, but spin 0 is not claimed because of uncertaintyabout the production mechanism. E760 MeV data

0w x15 on pp™hhp show peaks at 1500, 1748"10and 2100 MeV; they do not report a spin-parityanalysis. In view of the fact that we observe strong

Ž . Ž . 0production of f 1500 and f 2100 in our hhp0 0

data, it seems plausible that their peak at ;1750MeV may be the same resonance as we report here.Their width of 264"25 MeV is acceptably close toours. The BES group observes an f at 1781 MeV in0

Ž q y. w xJrC™g K K 16 .qTwo 0 qq resonances are to be expected in this

Žmass range. One is the radial excitation of f 13000.y1380 . The second is the missing ss state from a

q Ž . Ž .0 nonet made up of f 1300y1380 , a 1450 and0 0) Ž .K 1430 . The resonance we observe could be ei-

ther of these or a linear combination. The L3 collab-w xoration 17 has reported a narrow signal at 1793"18

MeV in K K , but without a spin-parity determina-S Sw xtion. A more recent analysis 18 gives Ms1770"

20 MeV, Gs235"47 MeV, very close to valuesreported here.

There is a long-standing debate over the spin ofŽ .f 1710 , reported by the Particle Data Group asJ

having a mass of 1712"8 MeV and a width of133"14 MeV. The mass we observe, 1770 MeV,seems too far removed from 1712 MeV to be thesame resonance. It seems likely that there is oneresonance at 1710 MeV with Js0 or 2 and a

second at 1770 MeV. This is what the BES collabo-w xration found 16 . There is then the possibility that

Ž .f 1710 is also present weakly in our data. At 900J

MeVrc, there is a small surplus of events in the hh

Ž .projection of Fig. 1 f from 1600 to 1700 MeV. At1200 MeVrc there is again some indication of extraevents in four bins close to 1700 MeV. Our fits

Ž .include the 500 MeV wide f 1980 demanded by2w xthe data of Ref. 1 at higher beam momenta. In Ref.

w x q1 , the 2 signal is visible over the whole massrange from 1550 to 2000 MeV. The cosa depen-

Ž .dence of Fig. 3 a at 1200 MeVrc is again clearevidence for the presence of non-zero spin, but itdoes not appear to be narrowly concentrated in mass.

Ž .Again it is to be associated with f 1980 . At lower2

beam momenta, where the phase space is limited,Ž .there is the possibility of confusion between f 19802

Ž .and f 1710 . At all beam momenta, we have tried2Ž . Ž .adding to the present fit f 1712 or f 1712 with0 2

the PDG width of 133 MeV. Improvements to the fitŽ . Ž .are listed in columns c and d of Table 3 respec-

tively. They are not significant, compared with ourtest with fake signals; they make no visible improve-ment to mass projections. If one tries to fit the extra

Ž . Ž .events of Fig. 1 h with f 1710 , the optimum massJ

is 1690 MeV but the width required is F40 MeVand optimises at 25 MeV. This width is so much less

Ž .than the PDG value for f 1710 that we have noJŽ .confidence in the presence of an additional f 1710 .J

Ž .An upper limit 90% confidence level on its branch-ing ratio at all momenta is 4%.

Ž .In summary, there is strong )8s evidence ateach of three momenta for an f with Ms1770"0

12 MeV and Gs220"40 MeV. There is no signif-icant evidence for a further resonance at 1710 MeV.

Acknowledgements

We thank the Crystal Barrel Collaboration forallowing use of the data. We wish to thank thetechnical staff of the LEAR machine group and of allthe participating institutions for their invaluable con-tributions to the success of the experiment. Weacknowledge financial support from the British Parti-cle Physics and Astronomy Research CouncilŽ .PPARC , and the U.S. Department of Energy. TheSt. Petersburg group wishes to acknowledge finan-


cial support from PPARC and INTAS grant RFBR95-0267.

References

w x1 A.V. Anisovich et al., to be submitted to Physics Letters.w x Ž .2 Particle Data Group, Euro. Phys. Journ. 3 1998 1.w x Ž .3 V.V. Anisovich et al., Phys. Lett. B 323 1994 233.w x Ž .4 C. Amsler et al., Phys. Lett. B 355 1995 425.w x Ž .5 A. Abele et al., Nucl. Phys. A 609 1996 562.w x Ž .6 C. Amsler et al., Phys. Lett. B 322 1994 431.w x Ž .7 C. Amsler et al., Phys. Lett. B 333 1994 277.

w x Ž .8 D. Barberis et al., Phys. Lett. B 413 1997 225.w x9 D.V. Bugg, V.V. Anisovich, A. Sarantsev, B.S. Zou, Phys.

Ž .Rev. D 50 1994 4412.w x10 A.V. Anisovich et al., to be submitted to Phys. Lett. B.w x Ž .11 B.V. Bolonkin et al., Nucl. Phys. B 309 1988 426.w x Ž .12 A. Etkin et al., Phys. Rev. D 25 1982 1786, 2446.w x Ž .13 D.V. Bugg et al., Phys. Lett. B 353 1995 378.w x Ž .14 D. Alde et al., Phys. Rev. B 284 1992 457.w x Ž .15 T.A. Armstrong et al., Phys. Lett. B 307 1993 394.w x Ž .16 J.Z. Bai et al., Phys. Rev. Lett. 77 1996 3959.w x Ž .17 M. Acciarri et al., Phys. Lett. B 363 1995 118.w x18 S. Braccini, XXIX Int. Conf. on High Energy Physics,

Vancouver, 1998.

11 March 1999


Equation of state of hadronic matter with dibaryonsin an effective quark model

Ricardo M. Aguirre a,1, Martin Schvellinger a,b,c,2

a ( )Physics Department, UniÕersity of La Plata, C.C. 67, 1900 La Plata, Argentinab The NuclearrHigh Energy Physics Research Center, Hampton UniÕersity, Hampton, VA 23668, USA

c Thomas Jefferson National Accelerator Facility, 12000 Jefferson AÕenue, Newport News, VA 23606, USA

Received 8 July 1998; revised 4 November 1998Editor: W. Haxton

Abstract

The equation of state of symmetric nuclear matter with the inclusion of non-strange dibaryons is studied. We pay specialattention to the existence of a dibaryon condensate at zero temperature. These calculations have been performed in anextended quark-meson coupling model with density-dependent parameters, which takes into account the finite size ofnucleons and dibaryons. A first-order phase-transition to pure dibaryon matter has been found. The corresponding criticaldensity is strongly dependent on the value of the dibaryon mass. The density behavior of the nucleon and dibaryon effectivemasses and confining volumes have been discussed. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 21.65.q f; 12.39.Ba; 24.85.qp; 14.20PtKeywords: Nuclear matter; Bag model; Dibaryons; Equation of state; Bose condensate

Nuclear matter properties have been modeled byseveral kind of local relativistic effective lagrangianswhich basically use point-like representations of nu-cleons and mesons as the relevant degrees of free-dom. The differences are essentially coming from thenucleon-scalar meson interaction. It should be desir-able to describe nuclear matter from a fundamentaltheory like QCD, but it is well known that in the lowenergy limit QCD becomes non-perturvative. Thissituation has motivated the development of several

1 E-mail: [email protected] On leave from University of La Plata. E-mail:

[email protected] and [email protected]

effective models of quark interaction, among themw xthere is the MIT bag model 1 . This model predicts

the existence of some multibaryons and strange ex-w x w xotics 2 . The early work of Jaffe 3 has concentrated

w xthe attention of theoretical studies 2,4 and thefurther development of experimental research look-ing for signals of strange and non-strange dibaryons.

w xIn Ref. 3 it has been shown that in the scheme ofthe MIT bag model the gluon-exchange force shouldbe responsible of the existence of a stable six-quarksbound state. This particle, the so-called H-particle, isa flavor singlet Jps0q dihyperon with a mass of2150 MeV and strangeness y2. The problem hasalso been treated in the non-perturbative frameworkof the Skyrme model where dibaryons have been


( )R.M. Aguirre, M. SchÕellingerrPhysics Letters B 449 1999 161–167162

w xconsidered as axially symmetric skyrmions 5 .Non-strange as well as strange dibaryons and anothermultibaryons have been studied as multiskyrmionsw x6 . The experimental activity concerning to thissearch has been increased in the last years, recentexperiments have been developed at TRIUMF and

w xCELCIUS 7 . Non-strange dibaryons which have asmall width have been described as promising candi-

w xdates for experimental searches 8 .On the other hand, models based on the quantum

field theory of hadrons including non-strangedibaryons as effective degrees of freedom have been

w xextensively studied 9 , obtaining very interestingeffects on the binding energy per particle as well as

Ž .on the equation of state EOS of the system. Inthese hadronic effective lagrangians nucleons anddibaryons are treated as point-like particles repre-sented by two independent effective fields, both ofthem interact by the exchange of scalar and vectorneutral mesons using two different sets of couplingconstants.

The purpose of this work is to include finite sizeeffects on the EOS, taking into account the quarkstructure of the particles. As we will show later,several features of our results are in agreement with

w xthose obtained in Ref. 9 . Therefore, this fact seemsto support that the dibaryon condensate is essentiallya model-independent issue. We have selected the

Ž . w xso-called quark-meson coupling model QMC 10 ,which in some way satisfies the above mentionedrequirements. In this scheme we can perform a si-multaneous description of nucleons and dibaryonsusing only current quarks and mesons as effectivedegrees of freedom. The model was early developed

w xby Guichon 10 and it has been extensively appliedw xto calculate nuclear matter 11 as well as finite

w xnuclei 12 properties with successful results. It hasalso been used to evaluate nucleon structure func-

w xtions 13 . The naturalness of the QMC model hasw xbeen studied using the dimensional analysis 14 .

Recently the density dependence of the parametersof the MIT bag model have been phenomenologi-

w xcally investigated 15 , as well as by imposing aselfconsistent relationship with the quantum field

w xtheory of hadrons 16,17 .In this work we have developed an extension of

w xthe QMC model described in Ref. 16 , by includingdibaryons represented as spherical MIT bags confin-

ing six quarks. The extended QMC lagrangian den-Ž . Ž .sity with quark fields q x coupled to scalar s xa

Ž .and vector v x neutral mesons, is written as fol-m

lows

LL x sLL x qLL x qLL 0 x , 1Ž . Ž . Ž . Ž . Ž .QMC N D Mesons

LL x s LL 0 x yB QŽ . Ž .Ž .N N 1 V N

31y q x q x D , 2Ž . Ž . Ž .Ý a a SN2

as1

LL x s LL 0 x yB QŽ . Ž .Ž .D D 2 V D

61y q x q x D . 3Ž . Ž . Ž .Ý a a SD2

as1

Here Q and Q are the radial non-overlappingV N V D

step functions which schematically confine the quarksinside spherical bags for nucleons and dibaryons,respectively. B and B are the so-called MIT bag1 2

constants associated with these particles. Within thestandard QMC treatment B is a constant, however itcan be considered as function of the baryonic den-sity, r . The terms proportional to the surface deltaB

functions D and D ensure a zero flux of quarkSN SD

current through the bag surface. In this lagrangianwe have defined the following terms

30 mLL x s q x ig E ym qg s xŽ . Ž . Ž .ŽÝN a m a s

as1

yg g mv x q x , 4Ž . Ž . Ž ..v m a

60 mLL x s q x ig E ym qg s xŽ . Ž . Ž .ŽÝD a m a s

as1

yg g mv x q x , 5Ž . Ž . Ž ..v m a

10 m 2 2LL x s E s x E s x ym s xŽ . Ž . Ž . Ž .Mesons m s2

1 mny F x F xŽ . Ž .mn4

1 2 mq m v x v x . 6Ž . Ž . Ž .v m2

Here g and g are the quark-meson couplings v

Ž . Ž .constants associated with s x and v x , respec-m

tively. In order to get the minimal set of free parame-ters we have assumed that quarks do not distinguishbaryon or dibaryon bags, i.e. the coupling constantsare the same in both of the equations for baryons anddibaryons. In that follows we deal with u and d

( )R.M. Aguirre, M. SchÕellingerrPhysics Letters B 449 1999 161–167 163

massless quarks, furthermore we introduce the indexns1, 2 to label quantities related to nucleons anddibaryons, respectively.

The normalized quark wave function for the fun-damental state in a spherical bag of radius R isn

given by

q r ,t sNN eyi en tr RnŽ .n n

xj y rrRŽ . a0 n n= , 7Ž .'ž /ib sPrj y rrRŽ .ˆ 4pn 1 n n

where r is the distance from the center of the n bag,x is the quark spinor and the normalization con-q

stant is

ynNN s .n 3 2 )2 R j y V V y1 qR m r2( Ž . Ž .n 0 n n n n a

8Ž .

The parameter associated with the quark mass is)m sm yg s , and the energy eigenvalue is writ-a a s

ten as

e sV qg nv R , 9Ž .n n v n

22 )(where V s y q R m , and s ,v are theŽ .n n n a

mean values of meson fields calculated in theMean Field Approximation. The y variable is fixedn

by the boundary condition at the bag surfaceŽ . Ž . w xj y s b j y as in Ref. 1 and b s0 n n 1 n n

) )V yR m r V qR m . The mass associ-(Ž . Ž .n n a n n a

ated with the bag is given by

3nV yzn 0n 4 3M s q p B R . 10Ž .n n n3Rn

The B are the bag constants already introduced inn

Ž . Ž .Eqs. 2 and 3 , while z takes into account the0n

zero point energy and the center of mass correctionof the bag.

The usual procedure in QMC is to fix the nucleonbag parameters at zero baryon density to reproducethe experimental nucleon mass M s M s 9391 N

MeV, simultaneously it is required that the equilib-Ž .rium condition dM s rdRs0 must be fulfilled. A1

similar method should be applied to dibaryons, how-ever the experimental value of in-vacuum dibaryonmass M has not been definitely confirmed yet, andD

at the present, only theoretical estimates are avail-

able. Therefore, we leave M as a parameter of theD

model. Since the B are related to vacuum propertiesn

we assume that B sB sB, at all densities. Al-1 2

though the parameters z could also have a density0n

w xdependence, in a previous work 16 it was foundthat z remains approximately constant at the baryon01

densities below four times the nuclear matter satura-tion density. Consequently we have taken z as0n

constants fixed at zero density for each kind of bag.Under these assumptions one can immediately get a

Ž .1r3relation for masses and radii: R s M rM R .2 2 1 1

In order to obtain the density dependence of Bwe have stablished an explicit relationship betweennuclear matter observables evaluated in the extendedQMC model and in pure hadronic models, as it has

w xbeen described in detail in Ref. 16 . In the presentwork we have selected the Zimanyi-Moszkowski

w x Žmodel 18 to describe the hadronic sector as in Ref.w x.16 . Considering that the effective nucleon masspredicted by the extended QMC model and thehadronic model must be the same, together with thefact that the outward quark-momentum on the bagsurface must be compensated by the hadronic mo-mentum going inside, one gets the following equa-tions

M sM ) , 11Ž .1

P sP , 12Ž .bag 0

which are valids at each value of baryon density.M ) and P are the effective nucleon mass and0

pressure of nuclear matter in the hadronic model,Ž . 2while P sy dM rdR r4p R is the pressure inbag 1 1 1

the nucleon bag. Instead of the standard equilibriumcondition for the bags E MrE Rs0 we have pro-

Ž .posed Eq. 12 which takes into account many-bodyŽ .effects. To derive Eq. 12 we have noted that

mn mn mn ˜E T sE T qE T Q , 13Ž .m QMC m bag m mesons V

1mn n nE T sy yP qB D n y E qqD , 14Ž . Ž .Ž .m bag D S S2

E T mn s Iqm2 sE nsŽ .m mesons s

y E F ml qm2 v l E nv 15Ž .Ž .m v l

Ž . <where P sy1r2nPE qq is the pressure ex-surfaceD˜erted by the quarks on the bag surface and Q is theV

˜complementary step function. The presence of Q V


comes from a cancellation of the explicit mesoncontribution inside the bag, however the presence ofmesons remains in the density dependence of thearguments of quark wave functions. For the QMC

Ž .model the second term of Eq. 13 is strictly zero,which is due to the fact that in the QMC lagrangianthere is a static bag surrounded by free mesons. Asthe baryon density increases this picture is not ade-quate, since at high densities the mesons are far fromthe free field approximation. On the other hand, inthe limit of point-like hadrons we have T mn sT mn

hadrons f m

qT mn , where T mn comes from the free meson sector0 f m

and T mn is the contribution from baryons coupled to0

mesons. Explicit evaluation shows that E T mn sm f m

E T mn , therefore by energy-momentum conserva-m mesonsŽ .tion one can replace the second term of Eq. 13 by

yn E T mn. By integrating on the external bag vol-n m 0Ž .ume we obtain the relation proposed in the Eq. 12 .

Using these equations and m s550 MeV, m ss v

783 MeV, g s4.576, g s2.222 and R0 s0.8 fms v 1Ž .we have obtained the density behavior of BsB rB

shown in Fig. 1.Once we have dynamically derived the parameter

B as a function of the baryon density, we canperform the calculations for nuclear matter withdibaryons. Firstly, we evaluate the energy per baryonof a system composed by symmetric nuclear matterwith density r , and dibaryons with density r .N D

Since each dibaryon carries baryon number two, thebaryon density must be r sr q2 r . Using theB N D

Ž .quark wave functions of Eq. 7 one can constructŽ . Žthe antisymmetrized symmetrized nucleon di-

Ž .Fig. 1. B dashed line and the nucleon bag radius, R, for theŽ .values Qs0, 0.25, 0.33, 0.47 and 0.5 solid lines as functions of

the baryon density, r . The arrow indicates the increasing QB

values.

.baryon physical states and calculate the expectationvalue of the energy density HH derived from the

Ž .lagrangian of Eq. 1 . For uniform matter the energyper baryon is given by

esHHrrB

4 2 2 2 21 M m s m v1 s vs F h q q qM r ,Ž . 2 D2r 2 2pB

16Ž .

where hsk rM , k is the Fermi momentum forF 1 F

the nucleons and it is related to the nucleon density3 2 2 2Ž . Ž(by r s2k r3p , while F h sh h q1 2h qN F

2. Ž .(1 y log hq h q1 . Chemical potentials for nu-2 2(cleons and dibaryons are given by m s k qMN F 1

q3g r rm and m sM q6 g r rm , respec-v B v D 2 v B v

tively. Although the dibaryons are bosons, the corre-sponding particle number is conserved because theycarry baryon charge and since in our lagrangian wehave not included any decay mechanism. The pres-sure of hadronic matter at zero temperature can bewritten as

P sm r qm r yer . 17Ž .had N N D D B

The mean values of the meson fields have beenevaluated by minimizing the energy, i.e. EerEss0and EerEvs0, obtaining

21 M dF dM1 1ssy 2 4M F h yp hŽ . Ž .1 F2 2½ dh dsm ps

dM2qr , 18Ž .D 5ds

2together with vs3g r rm .v B v

In order to obtain numerical results we havearbitrarily selected the value of the in-vacuumdibaryon mass M s1970 MeV. We have used theD

meson masses, free nucleon bag radius and the previ-ously described procedure to calculate several in-vacuum quantities such as the parameters z s3.27,01

z s6.27, as well as the dibaryon bag radius RŽ0.s02 2

0.954 fmy1, the usual boundary condition eigen-value y Ž0.s2.042 and the bag parameter BŽ0.sn

0.5546 fmy4. When the constants z , z have been01 02

fixed and the density dependent parameter B hasbeen obtained, we must find the effective bag radii


Ž . Ž .R r and R r . To do that we have imposed the1 B 2 B

equilibrium condition for bags immersed in hadronicŽ .medium in similar way as Eq. 12 , but now using

Ž .Eq. 17 in the right hand side. After a little algebrait gives

4p R4 P sy3M R q4 3V yzŽ .n had n n n 0n

2 a V y1 qVŽ .n n ny3a , 19Ž .n 2V V y1 qaŽ .n n n

) Ž .where we have defined a sm R r . It must ben a n B

noticed that this condition becomes the standardequilibrium requirement of QMC model for vanish-ing r .B

Solving the boundary condition at the bag surfaceŽ . Ž .simultaneously with Eqs. 18 and 19 , for a given

baryon density and several values of the dibaryonabundance Qsr rr we have evaluated the bagD B

radii, masses, energy density and pressure as func-tions of r and Q. The coupling constants have beenB

fixed in order to reproduce at Qs0 the saturationdensity r s0.15 fmy3 and the binding energy per0

particle e s16 MeV, obtaining g s4.6876 and0 s

g s2.2967. In Fig. 1 the results for the nucleon bagv

radius R are shown. From this we can see that at1

low densities the radius is an increasing function ofr and it becomes a smoothly decreasing function atB

the saturation nuclear density. Furthermore, at densi-ties below 1.4r , R is higher than in-vacuum ra-0 1

dius, notwithstanding for Qs0 the relative incre-ment of R remains under the value 2%, as it is1

expected from y-scaling arguments in the analysis ofw xquasi-elastic electron scattering 19 . Therefore, the

use of this radius and the nucleon effective mass M1

is equivalent to have a bigger radius and the freenucleon mass, as it has been proposed in the refer-

w xence of Sick 19 . Density variation of the radius isless pronounced as Q increases. A similar behaviorfor the dibaryon bag radius has been found.

To investigate the existence of a phase-transitionfrom symmetric nuclear matter to a pure dibaryonicstate we have used the Gibbs criterion. If there is adynamical mechanism combining two nucleons togive one dibaryon, at the phase-transition point theconditions 2m sm and P sP must be ful-N D nuc dib

filled. Although we have not included such a reac-tion channel in our model, we are concerned with thestable initial and final phases rather than with any

particular mechanism of dibaryon formation. HereP and P denote the pressure in pure nuclear andnuc dib

pure dibaryon matter, respectively. In Fig. 2 thequantities 2m for Qs0 and m for Qs0.5 areN D

represented as functions of the corresponding pres-sures. The intersection point of these curves indicatesthat phase-transition occurs at a critical pressurenearly to P s8.8=10y2 fmy4. At low pressure0

the local minimum of the Gibbs potential per particlecorresponds to pure nuclear matter, while at pres-sures beyond P the stable state corresponds to pure0

dibaryon matter. In order to construct the EOS weŽ .have drawn Fig. 3 with the pressure of Eq. 17 ,

corresponding to Qs0 and Qs0.5 as functions ofbaryon density. The value P is reached at baryon0

densities of r s2.55r in nuclear matter and r sI 0 II

3.18 r in the dibaryon condensate. Therefore it is a0

first-order phase-transition with a discontinuityjumping 0.62 r in the baryon density. The horizon-0

tal segment PsP between r and r represents0 I II

two-phase coexistence. The effect of the phase-tran-sition is to reduce the compressibility at high densi-ties. For example, the quotient of the thermodynami-cal compressibility evaluated in the dibaryon con-densate to its value in nuclear matter gives at r sB

3.5r the value 0.706. This fact could have some0

effect on the process of forming the shock wave inthe collapse of very massive stars. In Table 1 acomparison of the quotient of thermodynamical com-

Ž .Fig. 2. The dibaryon chemical potential dashed line evaluated atŽ .Qs0.5 and two times the nucleon chemical potential solid line

at Qs0 plotted as functions of the total pressure. The phase-tran-sition for in-vacuum dibaryon mass M s1970 MeV occurs at aD

baryon density of r s2.55 fmy3 in the nuclear matter phase,I

corresponding to a baryon density of r s3.17 fmy3 in the pureII

dibaryon phase.


Fig. 3. Pressure as a function of baryon density for the dibaryonŽ .abundances Qs0, 0.25, 0.33, 0.47 and 0.5 dashed lines . The

arrow shows the increasing Q values. Solid line represents thephysical states of the system, corresponding to nuclear matterŽ .Qs0 below the relative density r r r s2.55 and to pureB 0

Ž .dibaryon matter Qs0.5 above the value r r r s3.17. TheB 0

horizontal segment between these densities represents a coexis-tence region of those phases.

pressibility at Qs0.5 for several values of M atD

the transition point is shown.In addition, we have studied how much is modi-

fied the phase-transition point as the in-vacuumdibaryon mass M is changed. It has been foundD

that the density r is an increasing function of theI

mass M , as it is shown in Table 1. We must takeD

into account that dibaryons have not been observedat densities around the normal saturation density r0

and that at sufficiently high densities our model cannot describe the quark-gluon plasma phase-transition.Therefore we limit the search for the nuclear-di-baryon phase-transition at densities ranged from1.5r to 4r . This requirement constrains the varia-0 0

tion of dibaryon mass to 1940 MeV -M -2000D

MeV.A comparison with the hadronic field theoretical

w xmodel of Ref. 9 shows that in the Hartree approxi-mation a nuclear-dibaryon heterophase appears ratherthan a pure condensate. The Bose-condensate for dX

dibaryon appears at 3 times the saturation density ofnuclear matter. Independent calculations using thequantum field theory of hadrons and dibaryons have

w xbeen developed in Ref. 20 . In this framework thetransition density for nuclear and neutron matter is

w xr rr s2.87 and r rr s2.16 respectively 21 ,D 0 D 0w xsimilar results are found in Ref. 22 : r rr s2.69D 0

and r rr s2.57 for nuclear and neutron matter,D 0

respectively. Therefore our prediction of r rr sD 0

2.55 is consistent with these values.In this work we have studied the properties of a

system composed of symmetric infinite nuclear mat-ter with dibaryons. This has been performed in atheoretical framework which takes into account thefinite size as well as the quark structure of nucleonsand dibaryons. The variation of properties of theparticles with baryon density have also been consid-ered. The nucleon mass shows a monotonic decreas-ing behavior for all the densities and dibaryon abun-dances Q studied here. At Qs0 nucleon swelling isobserved at densities below the normal nuclear mat-ter density. The bag radius increment is less than2%, in accordance with theoretical estimates basedon y-scaling interpretation of quasi-elastic electron

w xscattering 19 . A nuclear-dibaryon matter phase-transition is found at zero temperature. The dibaryoncondensate predicted in this work could be an inter-mediate state preceding the transition to quark mat-ter. If the dibaryon condensation takes place forbaryon densities in the range of 1.5-r rr -4,B 0

then values of M between 1940 MeV and 2000D

MeV are predicted for non-strange dibaryon masses.As a consequence of phase transition the compress-ibility of the system is considerably reduced at highdensities, as it should be expected from astrophysicalscenarios.

Extensions of the present work in order to de-scribe strange dibaryons can be studied by the inclu-sion of the color-electric and color-magnetic interac-tions in the MIT bag model, also multiquark bagsdescribing some exotic states of multibaryons couldbe treated in this framework.

Table 1The baryon density r and the quotient of thermodynamical com-pressibility k sE PrEr evaluated at the transition point as func-B

Ž .tions of the in-vacuum dibaryon mass M . The label I IIDŽ .corresponds to pure nuclear dibaryon matter state

y3 y3w x w x w xM GeV r fm r fm k rkD I II II I

1.94 0.228 0.335 1.301.95 0.283 0.380 1.091.96 0.335 0.429 1.011.97 0.383 0.477 0.981.98 0.427 0.522 0.961.99 0.470 0.566 0.942.00 0.505 0.603 0.93


Acknowledgements

This work is supported in part by a grant ofFundacion Antorchas, Argentina and the grant´PMT-PICT0079 of ANPCYT, Argentina. We havebeen benefited from stimulating discussions with J.E.

Ž .Horvath IAG-USP, Brazil . M.S. is greatly indebtedŽ .to J.L. Goity Jefferson Lab-Hampton University for

kind hospitality at Jefferson Lab.

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11 March 1999


One-loop quantum holography for higher dimensional black holes

Andrei A. Bytsenko 1, Luciano Vanzo 2, Sergio Zerbini 3

Dipartimento di Fisica, UniÕersita di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trento, Italy`

Received 17 November 1998Editor: L. Alvarez-Gaume

Abstract

The one-loop quantum corrections to the free energy associated with scalar field in a higher dimensional static curvedspace-time is investigated making use of the conformal transformation method. For a space-time with bifurcate horizon,horizon divergences are accounted for choosing the Planck length as natural cutoff. The leading term in the high temperaturequantum correction satisfies holographically the ‘‘area law’’, like the tree level Bekenstein-Hawking term. Furthermore it isstressed that only for the asymptotically AdS black holes one may have a microscopic interpretation of the entropy also atquantum level. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 04.62.qv; 04.70.Dy

I. Recently, the issue concerning the microscopicexplanation of the Bekenstein-Hawking formula for

w xthe black hole entropy 1,2 has been vastly dis-cussed in literature. Among the several approaches,we would like to recall the investigations related toŽ . w x2q1 -dimensional BTZ black hole 3 , the stringy

Ž w x.approach see, for example, 4 , the Matrix theoryw x w xapproach 5 , the loop gravity approach 6 , the in-

w xduced gravity approach 7 and the new approachw xappeared in Ref. 8 .

There have also been some attemps to computesemiclassically the quantum corrections to theBekenstein-Hawking classical entropy for the 4-di-mensional Schwarzschild lack hole. However, so farall the evaluations have been plagued by the appear-

1 On leave from Universidade Estadual de Londrina, Brazil.E-mail: [email protected]


w x Žance of divergences, first noticed in Ref. 9 see alsow x.Refs. 10–15 .

In the 4-dimensional black hole case, the physicalorigin of these divergences can be traced back to theequivalence principle according to which, in a staticspace-time with canonical horizons, a system in ther-mal equilibrium has a local Tolman temperature

Ž .given by T x sTr yg x , T being the generic( Ž .00

asymptotic temperature. Since, roughly speaking,very near the horizon a static space-time may beregarded as a Rindler-like space-time, one gets for

Ž .the Tolman temperature T r sTrr, r being thedistance from the horizon. As a consequence, omit-ting the multiplicative constant, the total entropyreads

`3AT

3S; dx T r drs , 1Ž . Ž .H H 22´´

where A is the area of the horizon and ´ is thehorizon cutoff. These considerations suggest the use


( )A.A. Bytsenko et al.rPhysics Letters B 449 1999 168–172 169

< <of the optical metric g sg rg , conformallymn mn 00

related to the original one, and one of the purposesof this paper is to implement this idea, and try to

w xshed some light on the holographic property 16 ofblack hole space-times, which has been discussed

w xrecently in Refs. 17,18 within the AdSrCFT corre-w xspondence 19 .

w xIn particular, Barbon and Rabinovici 18 haveexplained how the internal dimensions remain invisi-ble to the thermodynamics of the 4D conformal fieldtheory living on the boundary of AdS . The dimen-5

sionality of a spacetime is revealed in the free energyof massless fields, which scales as T D in a D-di-mensional space. By allowing the phase transition toan anti-de Sitter black hole, they have shown that theregion where the local temperature exceeds the

ŽKaluza-Klein threshold i.e. T)1rR , where R isc c.the radius of compact internal dimensions , is roughly

inside the event horizon and thus is cut out from theEuclidean section, which is where the free energy iscomputed. This gives one-loop holography, but onlyif one chooses to renormalize away the horizondivergences.

From the point of view of the low energy effec-tive field theory, this is unsatisfactory, because itrequires a tree level bare entropy with no statisticalinterpretation. Instead, we can rely on the correspon-dence between ultraviolet effects in the bulk with

w xinfrared effects in the boundary theory 17 , to arguethat there should be no horizon divergences, becausethere should be no infrared divergences in a fieldtheory on a compact space. Hence, we shall keep thehorizon contribution and show that it scales holo-graphically.

II. To begin with, we consider a scalar field on aŽ .Nq1 -dimensional static space-time with the met-

Ž .ric signature yq..qq , DsNq1.

22 0 i jds sg x dx qg x dx dx ,Ž . Ž . Ž .00 i j

� j4xs x , i , js1, . . . , N . 2Ž .

The restriction to scalar fields may be justified not-ing that the horizon divergences we are going todiscuss here should be independent on the specific

Ž .features of the bulk supergravity theory.

ŽThe one-loop partition function is given by weperform the Wick rotation x syit , thus all differ-0

.ential operators one is dealing with will be elliptic

1 Dw xZs d f exp y fL f d x , 3Ž .H H D2ž /where f is a scalar density of weight y1r2 and LD

is a Laplace-like operator that has the form

L syD qm2 qj R . 4Ž .D D

Here D is the Laplace-Beltrami operator acting inDŽ .D- dimensional space-time, m the mass and j are

arbitrary parameters and R is the scalar curvature.We recall method of the conformal transformation

w x20 . This method is useful because it permits tocompute all physical quantities in an ultrastatic man-

Ž w x.ifold called the optical manifold 21 . The ultra-static Euclidean metric g is related to the staticmn

one by the conformal transformation

2 s Ž x .g x se g x , 5Ž . Ž . Ž .mn mn

1Ž .with s x sy lng . In this manner, g s1 and00 002

Ž .g sg rg Euclidean optical metric .i j i j 00

With regard to the one-loop partition function, itis possible to show that

w xZsJ g,g Z , 6Ž .w xwhere J g,g is the Jacobian of the conformal trans-

formation. Such a Jacobian can be explicitely com-puted, but here we shall need only its structuralform. Using z-function regularization for the deter-minant of the second order differential operator weget

w xlnZs lnZy ln J g,g

d21 < < w xs z s L ll y ln J g,g , 7Ž .Ž . ss0D2 ds

where ll is an arbitrary parameter necessary to2Ž < .adjust the dimensions, the function z s L ll asso-D

ciated with the operator L , which explicitly readsD

ys ysL se L eD D

2 y2 s 2syE yD qj Rqe m q jyj RŽ .t N D D

2syE qL . 8Ž .t N

( )A.A. Bytsenko et al.rPhysics Letters B 449 1999 168–172170

Ž . Ž .In Eq. 8 , j s Ny1 r4N andD

y2 s mnRse Ry2 ND syN Ny1 g E sE s .Ž .N m n

9Ž .

Now we formally have to deal with an ultrastaticspace-time. For a scalar field in thermal equilibriumat finite temperature Ts1rb the partition functionZ can be obtained, within the path integral ap-b

proach, after the Wick rotation ts ix 0 imposing onthe field a b periodicity in t . Thus one obtainsw x20,15,22

b1 <lnZ sy PPz y LŽ .b N22

1 <q 2y2ln2 ll Res z y LŽ . Ž .N2

d bq lim 'dss™0 4p G sŽ .

=` ` 2 2sy3r2 yn b r4 t ytLNt e Tr e dt ,Ý H

0ns1

10Ž .

where PP and Res stand for the principal part and forthe residue of the zeta function. The free energy isrelated to the canonical partition function by meansof the equation

1 1w xF b sy lnZ sy lnZ y ln J g,g .Ž . ž /b b

b b

11Ž .

Since we are considering a static space-time thew xquantity ln J g,g depends linearly on b and accord-Ž .ing to Eq. 11 it gives no contribution to the en-

tropy, which has the form

S sb 2E F , 12Ž .b b b

Žwhere F is the temperature dependent part statisti-b

. Ž .cal sum of F b .Let us apply this formalism to scalar fields in a

D-dimensional static space-time with metric

y12 2 2 2ds sA r dt qA r dr qr dSŽ . Ž . dy1

qdE , 13Ž .Dy dy1

where we are using polar coordinates, r being theradial one and dS and dE is are the met-dy1 Dydy1

Ž . Ž .rics of two dy1 -dimensional and Dydy1 -dimensional Einstein spaces. Particularly interestingare the two D-dimensional space-times relevant in

w xthe CFTrAdS correspondence 17 : X sAdS =1 dq1

M and X sAdS BH =M . The firstDy dy1 2 dq1 Dydy1

contains the periodically identified AdS space andthe second one the Schwarschild AdS black hole,

Ž .and M is a suitable compact Dydy1 -Dy dy1

dimensional manifold. However, we would like toconsider a more general class of metrics, which may

Ž .be defined by the function A r and by the relatedŽ .dy1 -dimensional manifold, namely

Ž .I . The AdS space,

r 2

A r s 1q , dS sdV . 14Ž . Ž .dy1 dy12ž /l

Ž .II . The Schwarzschild black hole,

C MdA r s 1y , dS sdV . 15Ž . Ž .dy1 dy1dy2ž /r

Ž . w xIII . The Schwarzschild-AdS black hole 23,24 ,

r 2 C MdA r s 1q y , dS sdV .Ž . dy1 dy12 dy2ž /l r

16Ž .

Ž . w xIV . The toroidal AdS black hole 25–28 ,

r 2 C MdA r s y , dS sdT . 17Ž . Ž .dy1 dy12 dy2ž /l r

Ž . w xV . The hyperbolic AdS black hole 25–28 ,

A rŽ .dy2

2 dy2 22C My l2 d ž /ž /r d ds y1q y ,2 dy2l r� 0

dS sdH . 18Ž .dy1 dy1

Ž . Ž .In Eqs. 15 – 18 , M is the mass of the black hole,< < 2the cosmological constant L is given by L s1rl ,

C is a normalization constant depending on thedŽ .dq1 y dimensional Newton Constant and dV ,dy1

Ž .dT and dH are the the metric of the d-1 -di-dy1 dy1

mensional sphere, torus and compact hyperbolicmanifold respectively.

( )A.A. Bytsenko et al.rPhysics Letters B 449 1999 168–172 171

Generally speaking, the optical metric reads

dr 2 r 22 2ds sdt q q dSdy12 A rŽ .A rŽ .

1q dE . 19Ž .Dy dy1A rŽ .

When a metric has an event horizon and the blackhole is not extremal, the localization of the horizon isgiven by the simple positive root of g11, namelyŽ .A r s0. Thus one can make use of the near-hori-q

zon approximation, which is valid for large blackhole mass. As a result the optical metric may be

Ž w x.approximated by see, for example, Refs. 15,22

dr 2 r 2 1q2 2ds ,du q q dS q dE ,dy1 Dydy12 2 2r r r

20Ž .where

y1r2d1r2 <rs2 ryr A r ,Ž . Ž . rsrq qdr

t d<us A r . 21Ž . Ž .rsrq2 dr

In this not extremal case, one gets in a standard waythe inverse of the Hawking temperature requiring theabsence of the conical singularity, namely b sH

<y1

4p dA r rdr .Ž .Ž .rsrq

In order to study the quantum properties of matterfields, it is sufficient to investigate the kernel of the

yt LN Ž .operator e and use the Eq. 10 . We shall toassume the high temperature expansion or, in thecase of black hole, the Rindler-like approximation.

In both the circumstances, the leading term of thew xheat-kernel turns to be 20,15

VNytLNTr e , 22Ž .Nr24p t ,Ž .where V is the optical volume of the whole spatialN

Ž .section. From Eq. 10 the corresponding off-shellfree energy reads

VNF ,y . 23Ž .b Db

ŽFor a space-time without event horizon the AdS.space there are no horizon divergences and the free

energy has a leading term byD , i.e. no holographicreduction of the demensionality is present.

The situation is different for black hole space-times. The optical volume, formally given by

`dy1r

V sV V dr , 24Ž .HN dy1 Dydy1 Dr2r A rŽ .q

is divergent, because of the non integrable singular-ity at rsr . Thus one has to introduce a cutoffqparameter, i.e. r s´ . In fact, making use of theqnear-horizon approximation, it is possible to showthat the leading term in the off-shell free energy isw x22

r dy1V V b Dy 1q dy1 Dydy1 H

F ,y . 25Ž .b Dy2 D´ b

The off-shell entropy can be computed by means ofŽ .Eq. 12 and the result is

Ddy1r V V bq dy1 Dydy1 HS , . 26Ž .b Dy2 ž /b´

From a phenomelogical point of view, well knownŽstringy arguments suggest to choose ´, l of theP

. w xorder of the D-dimensional Planck length 9 . Fur-thermore, G , l Dy 2 and, and recalling the relationD P

between the Newton constants in different dimen-sions

V 1Dy dy1s 27Ž .

G GD dq1

and finally going on-shell at bsb , one obtains theH

‘‘area law-like’’ expression also for the one-loopquantum correction:

r dy1Vq dy1S, , 28Ž .

Gdq1

where the proportionality factor depends on speciesof fields. As a consequence, going on-shell, oneobtains the holographic reduction of the dimensional-

w xity, in agreement with the results of Ref. 18 , eventhough the dimensionality reduction mechanism iscompletely different. In fact, in our approach, thedimensionality reduction is due to the presence ofquantum horizon divergences.

( )A.A. Bytsenko et al.rPhysics Letters B 449 1999 168–172172

III. We conclude with some remarks. First, as faras the computation of the quantum black hole en-tropy is concerned, the ‘‘internal’’ D y d y 1-dimensional manifold does not contribute. Second,the class of D-dimensional blac k hole we haveconsidered and for which we have estimated theone-loop quantum correction to the entropy, can bedivided into two subclasses: the first one, exampleŽ . Ž .II Schwarzschild black hole , corresponds tonon-negative cosmological constant and the second

Ž . Ž . Ž . Žone, exam ples I , III y V toroidal,.Schwarzschild-AdS and hyperbolic AdS black holes ,

corresponds to a negative cosmological constant. Forall solutions, one can easily computed a relationshipbetween the r and the Hawking temperature, elimi-qnating the mass of the black hole. This is not a trivialtask, particularly for the hyperbolic AdS black holesŽ w x Ž ..see 25 and Eq. 18 . Omitting the details, it turnsout that for the first type of black holes r ,b , andq H

for the second type r ,by1. As a consequence, theq H

entropy for the Schwarzschild black hole goes like

S,Tydq1 , 29Ž .related to negative specific heat and the instability ofthis black hole. With regard to the asymptoticallyAdS black holes,

S,T dy1 , 30Ž .which, in turn, is related to the positivity of the theirspecific heat and their stability.

As a consequence, only for the stable AdS blackhole it seems to exist, also at quantum level, amicroscopic explanation of the black hole entropy

Žvia the AdSrCFT correspondence. In fact confor-. Ž .mal quantum fields living on the horizon boundary

of the black hole have a statistical entropy withS,T dy1 as leading term.

Acknowledgements

We would like to thank G. Cognola, D. Fursaevand M. Tonin for discussions. A.A. Bytsenko wishes

Ž . Ž .to thank CNPq Brazil and INFN Italy for finan-

cial support. The research of A.A. Bytsenko wasŽ .supported in part by RFFI grant No. 98-02-18380-a

Ž .and by GRACENAS grant No. 6-18-1997 .

References

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Ž .Carlip, Class. Quantum Grav. 15 1998 3609.w x Ž .4 A.W. Peet, Class. Quantum Grav. 15 1998 3291.w x Ž .5 D.A. Lowe, Phys. Rev. Lett. 81 1998 256.w x Ž .6 C. Rovelli, Phys. Rev. Lett. 77 1996 3288; A. Ahtekar, J.

Ž .Baez, A. Corichi, K. Krasnov, Phys. Rev. Lett. 80 1998904.

w x7 V.P. Frolov, D.M. Fursaev, A.I. Zelnikov, Nucl. Phys. B 486Ž .1997 339; V.P. Frolov, D.M. Fursaev, Phys. Rev. D 56Ž .1997 2212.

w x Ž .8 R.J. Epp, R.B. Mann, Mod. Phys. Lett. A 13 1998 1875.w x Ž .9 G. ’t Hooft, Nucl. Phys. B 256 1985 727.

w x10 L. Bombelli, R. Koul, J. Lee, R. Sorkin, Phys. Rev. D 34Ž .1986 373.

w x Ž .11 L. Susskind, J. Uglum, Phys. Rev. D 50 1994 2700.w x Ž .12 J.S. Dowker, Class. Quantum Grav. 11 1994 L55.w x Ž .13 V.P. Frolov, D.M. Fursaev, Phys. Rev. D 54 1996 2731.w x Ž .14 R. Emparan, Phys. Rev. D 51 1995 5719.w x15 G. Cognola, L. Vanzo, S. Zerbini, Class. Quantum Grav. 12

Ž .1995 1927.w x16 G.’t Hooft, Dimensional Reduction in Quantum Gravity,

Salamfest, 1993, gr-qcr9310026; L. Susskind, J. Math. Phys.Ž .36 1995 6377.

w x Ž .17 E. Witten, Adv. Theor. Math. Phys. 2 1998 253.w x18 J.L. Barbon, E. Rabinovici, Extensivity Versus Holography

in Anti-de Sitter Spaces, hep-thr9805143.w x Ž .19 J. Maldacena, Adv. Theor. Math. Phys. 2 1998 231.w x Ž .20 J.S. Dowker, G. Kennedy, J. Phys. A 11 1978 895; J.S.

Ž .Dowker, J.P. Schofield, Phys. Rev. D 38 1988 3327; J.S.Ž .Dowker, J.P. Schofield, Nucl. Phys. B 327 1989 267.

w x Ž .21 G.W. Gibbons, M.J. Perry, Proc. R. Soc. Lond. A 358 1978467.

w x22 A.A. Bytsenko, G. Cognola, S. Zerbini, Nucl. Phys. B 458Ž .1996 267.

w x Ž .23 S. Hawking, D. Page, Commun. Math. Phys. 83 1983 577.w x Ž .24 E. Witten, Adv. Theor. Math. Phys. 2 1998 505.w x Ž .25 L. Vanzo, Phys. Rev. D 56 1997 6475.w x Ž .26 R.B. Mann, Class. Quantum Grav. 14 1997 L109.w x Ž .27 D.R. Brill, J. Louko, P. Peldan. Phys. Rev. D 56 1997

3600.w x28 D. Birmingham, Topological Black Holes in Anti-de Sitter

Space, hep-thr9808032.

11 March 1999


Induced wormholes due to quantum effectsof spherically reduced matter in large N approximation

S. Nojiri a,1, O. Obregon b,2, S.D. Odintsov c,3, K.E. Osetrin c,4

a Department of Mathematics and Physics, National Defence Academy, Hashirimizu, Yokosuka 239, Japanb Instituto de Fisica de la UniÕersidad de Guanajuato, P.O. Box E-143, 37150 Leon Gto., Mexico

c Tomsk State Pedagogical UniÕersity, 634041 Tomsk, Russia


Abstract

Using one-loop effective action in large N and s-wave approximation we discuss the possibility to induce primordialwormholes at the early Universe. An analytical solution is found for self-consistent primordial wormhole with constantradius. Numerical study gives the wormhole solution with increasing throat radius and increasing red-shift function. There isalso some indication to the possibility of a topological phase transition. q 1999 Elsevier Science B.V. All rights reserved.

Ž w x.It is a well known fact that spherical reduction of Einstein gravity see, for example 1 leads to specific 2dw xdilatonic gravity which belongs to the general class of 2d dilatonic gravities 2 . Spherical reduction of 4d matter

leads to 2d dilaton coupled matter theories.w x Ž w x.Recently, 2d conformal anomaly for dilaton coupled scalars has been found in Ref. 3 see also 4–6 .

w x ŽIntegrating this conformal anomaly one finds the anomaly induced effective action 4,5 in s-wave and large N. w xapproximation which was written in Ref. 7 in most complete form. A wide spectrum of physical problems

may be addressed using the above effective action. Let us only mention some of most interesting: anti-evapora-w x w x w xtion of multiple horizon black holes 7,8 . Hawking radiation 9 in dilatonic supergravity 5 , quantum

w xcosmological models in dilatonic supergravity 10 , study of semi-classical energy-momentum tensor in thew xpresence of dilaton 11 , etc.

In the present work we would like to discuss one more of these physical phenomena: the appearance ofspherically symmetric wormholes due to the anomaly induced effective action. The wormholes represent

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]


( )S. Nojiri et al.rPhysics Letters B 449 1999 173–179174

handles of topological origin. They may be considered as bridges joining two different Universes or twow xseparate regions of the same Universe 12 . It is known that wormholes cannot occur as solution of classical

w xgravity-matter theory due to violations of energy conditions in classical relativity 12 . Hence, it is natural toŽexpect their manifestations only at the quantum level or yet in the theory like Born-Infeld D-brane or M-theory,

w x.for review, see 13 .Recently, using quantum scalar stress-energy tensor calculated on spherically symmetric background in Ref.

w x Ž .14 the back reaction problem i.e. self-consistent wormholes in semiclassical gravity has been investigated inw x Ž w x.Ref. 15 see also 16 . It has been indeed numerically found a semiclassical quantum solution representing a

w xwormhole connecting two asymptotically spatially flat regions 15 . This result is extremely interesting and itŽ .should be verified in another models and or approximations, taking into account the fact that the approach

w x w xdeveloped in Ref. 15 is not completely free from some serious drawbacks 16 .In the present work we show the possibility of inducing wormholes in the early Universe, making use of the

w x Ž .effective action method 17 large N and s-wave approximation is used .We will start from the action of Einstein gravity with N minimal scalars

N11Ž .4 4 4 a bSsy d x yg R y2 L q d x yg g E x E x , 1Ž .Ž .( (ÝH HŽ4. Ž4. Ž4. a i b i216p G is1

Žwhere x are scalars, N is the number of scalars in order to apply the large N approach, N is considered to bei.large, N41 , G and L are the gravitational and cosmological constants.

The convenient choice for the spherically symmetric spacetime is the following one

ds2 sg dx mdxn qey2 fdV , 2Ž .mn

where m,ns0,1, g and f depend only on x 0, x1 and dV corresponds to the two-dimensional sphere.mn

Ž . Ž .The action 1 , reduced according to 2 takes the form

N1 2 212 y2f 2f'S s d x yg e y Rq2 =f y2 Lq2e q =x . 3Ž . Ž . Ž .� 4 ÝHred i216p G is1

ŽWorking in large N and s-wave approximation, one can calculate the quantum correction to S effectivered. w x Ž w x.action . Using 2d conformal anomaly for dilaton coupled scalar, calculated in Ref. 3 see also 4–6 one can

w x Žfind the anomaly induced effective action 4,5 with accuracy up to conformally invariant functional for thew x .total effective action, see 17 for a review . There is no consistent approach to calculate this conformally

invariant functional in closed form. However, one can find this functional as some expansion of Schwinger–De-w xWitt type 7 keeping only the leading term. Then, the effective action may be written in the following form

w x 55,7

N 1 112 l 2 l'Wsy d x yg R Ry= f= f RqfRq2lnm = f= f , 4Ž .H l l128p D D

where D is two-dimensional laplacian, m2 is a dimensional parameter. Here, the first term represents thePolyakov anomaly induced action, the second and third terms give the dilaton dependent corrections to the

Ž w xanomaly induced action these terms were found in Ref. 5 and first discussed in connection with quantumw x w x.gravity in Ref. 6 , similar terms with slightly different coefficients were derived in Ref. 4 . The last term

Ž . w x Ž .conformally invariant functional is found in Ref. 7 . Note that it is possible to write the action 3 as some

5 Ž . w xNote that recently the effective action 4 has been rederived in Ref. 18 .

( )S. Nojiri et al.rPhysics Letters B 449 1999 173–179 175

Ž . Ž .non-linear s-model with one loop quantum correction 4 . For this purpose, we rewrite 4 in a local form byintroducing auxiliary fields u and Õ

N12 l 2 l l'Wsy d x yg RÕy= f= f ÕqfRq2lnm = f= fq= u= ÕquR . 5Ž .H l l l128p

2 rThen in the conformal gauge g se g , we findmn mn

12 mn i j'S qWs d x yg G X g E X E X qRF X qT X , 6Ž . Ž . Ž . Ž .Hred i j m2 n

where

1 N Õi y2f� 4X s f ,r ,u ,Õ ,x , F X sy e y qfqÕ ,Ž .a ž /16p G 8p 12

12 ry2f 2 rT X sy y2 Le qe ,Ž . Ž .

16p G

° y2 f y2f ¶e N e N .2y q Õy2lnm y 0 0 . 0Ž . .4p G 4p 4p G 4p

y2 fe N N .y 0 y 0 . 0.4p G 4p 4p

N N .0 y 0 y . 0G s . 7Ž .i j .4p 8p

N .0 0 y 0 . 0.8p.

PPP PPP PPP PPP PPP PPP PPP PPP PPP PPP PPP PPP . PPP PPP.. y2 f¢ ß0 0 0 0 . e.

Working in the conformal gauge

1 2 rg sy e , g s0, 8Ž .". ""2

the equations of motion may be obtained by the variation of GsS qW with respect to g "", g ". and fred

ey2 f N N N2 2 212 2 20s 2E rE fq E f yE f y E ry E r y rq E f y 2E rE fyE fŽ . Ž . Ž .Ž . Ž .Ž . Ž .r r r r r r r r r r24G 12 2 4

N 22y lnm E f qNt 9Ž . Ž .r 04

ey2 f N N N2 22 2 r 2 rq2f 2 20s 2E fy4 E f y2 Le q2e q E rq E f y E f 10Ž . Ž . Ž .Ž .r r r r r8G 12 4 4

ey2 f N N22 2 2 r 2 2 20sy yE fq E f qE rqLe q 2E rE f qE r q lnm E f . 11Ž . Ž . Ž .Ž .Ž .r r r r r r r4G 4 2


Here, t is a constant which is determined by the initial conditions. Below we are interested in the static solution01that is why we replace E ™" E where r is radial coordinate." r2

Ž . Ž .Combining 9 and 10 we get

ey2 f N N N2 2 22 r 2 rq2f0s y E f q2E rE fyLe qe q E r y r E f y E rE fŽ . Ž . Ž .Ž .r r r r r r r4G 12 2 2N 22y lnm E f qNt . 12Ž . Ž .r 04

This equation may be used to determine t from the initial condition, it decouples from the other two equations.0Ž .Hence, Eq. 12 is not necessary in subsequent analysis.

Ž .Below, we consider a subclass of metrics 2 of the following type

ds2 sye2 rdt 2 qdl 2 qey2 fdV 2 , 13Ž .Ž . Ž .where rsr l , fsf l and l is the proper distance. Note that to study exact solutions for matter in metrics

w xof such sort, one can use the methods developed in Ref. 19 .Ž .In order to come to metrics of the form 13 , it is convenient to change the radial coordinate r to l by

dlse rdr . 14Ž .Then we obtain

E se r E , E 2 se2 r E 2 qE rE . 15Ž .Ž .r l r l l l

Ž . Ž .Eqs. 10 and 11 may be rewritten as follows

ey2 f N 2ey2 f ey2 f 122 2 y2f0s yN E fq E rq y qN E f q yN E rE fq yLe q1Ž . Ž .l l l l lž / ž / ž /G 3 G G G

N 2q E r , 16Ž . Ž .l3

ey2 f ey2 f ey2 f

22 2 20sy yN E rq qN 2 rq2lnm E fy E fŽ .Ž .l l l½ 5ž /G G G

ey2 f ey2 f ey2 f

22q qN 2 rq2q2lnm E rE fy yN E r y L . 17Ž . Ž .Ž . l l l½ 5 ž /G G G

Ž . Ž . Ž .Our next task will be the study of Eqs. 16 , 17 for the wormhole metric 13 , where the usual notations areŽ . Ž . Ž . Ž . Ž . Ž . Ž .f l sexp 2 r redshift function and r l sexp yf shape function . Actually, in our notations r l is

always positive, this function gives the wormhole throat.First, we consider purely induced gravity, i.e. N™` case. Then Einstein action can be dropped away. For

this case, the field equations have the form2 21 12 20syE fq E rq E f yE rE fq E r , 18Ž . Ž . Ž .l l l l l l3 3

22 20sE rq 2 rqa E fq 2 rq2qa E rE fq E r , 19Ž . Ž . Ž . Ž .l l l l l

where as2lnm2. These equations admit the next integrals of motion

I se r rX q 2 rqa f

X 20Ž . Ž .Ž .1

2 2X X X X2 rI se r y3 2 rqa f y6rf , 21Ž . Ž . Ž . Ž .Ž .2

here X'E . In our study we take the following initial conditions:l

f X 0 srX 0 s0 22Ž . Ž . Ž .


Ž .For the conditions 22 we have I s I s0, we get the trivial solution0 1

r l ,f l , r l , f l sconst. 23Ž . Ž . Ž . Ž . Ž .This indicates the possibility of inducing wormholes with finite throat.

X X Ž . Ž .For the case f sr s0 from Eqs. 16 – 17 in point ls0 we have0 0

ey2 f 0 N 1XX XX y2f 00s yN f q r q yLe q1 , 24Ž . Ž .0 0ž /G 3 G

ey2 f 0 ey2 f 0 ey2 f 0XX XX0sy yN r q qN 2 r qa f y L . 25Ž . Ž .0 0 0½ 5ž /G G G

Ž . Ž .Let us consider the case when Ls0 and r l , f l are non-decreasing functions near point ls0, then we havethe next restrictions for r and f0 0

2y2f y2f y2f0 0 0e 3 e e1

f Fy ln GN , y yay Ny -2 r Fy ya. 26Ž . Ž .0 02 2 ž /NG G NGN

In other words, for the case Ls0 the throat radius is

'r G GN . 27Ž .0

Ž . Ž . Ž .Let us consider Eqs. 16 – 17 for the case when r l sr sconst. We get two solutions0'< <1. L/0, 2GNLr3y2 G 3 ,22 Ž .(r s 2GNLr3q1" 2GNLr3y2 y3 r 2 L ,Ž .0 ž /

2° ' 'c cosh k lqc sinh k l , k)0Ž .1 2

2~Ž .f l s c qc l , ks0Ž .1 2

2¢ ' 'c cos yk lqc sin yk l , k-0,Ž .1 2

Ž 2 .with ks Lr y1 rG,0

2. Ls0,2

X' Ž . ' 'r s GN , f l s f cos 3rGN lq r f r2 3 f sin 3rGN l .( (ž /ž /0 0 0 0 0

Ž .Hence we found the wormhole as a quantum effect induced object with a constant throat radius and increasingŽ .or decreasing redshift function f l . This wormhole connects two spatial regions of spacetime. This can be

viewed as a kind of infinitely long wormhole.

Fig. 1.


Fig. 2.

Ž .From another side in some above cases we get oscillating behaviour for f l . This fact indicates to someinstability of wormhole configuration. Hence one can expect that kind of topological phase transition may occurand transform the wormhole to some another object, like black hole.

Ž . Ž .Let us discuss now the case when the classical gravity action is included, i.e. Eqs. 16 , 17 but r is notconstant. In this case we perform a numerical study of the equations of motion. Considering for simplicityLs0 and lnm2 s1, we present in Figs. 1 and 2 the typical result of our numerical study. The redshift function

Ž . Ž . Ž .f increases as well as r l . It is interesting that qualitatively r l and f l behave in the same way as in Ref.w x15 . In particularly, the redshift function does not approach to constant value at large l, hence the whole metricis not asymptotically flat. In these figures r s27.1828, f s2.1405=10y31, rX s f X s0 and Ns100.0 0 0 0

Hence, we showed that even for the whole system, i.e. classical gravity plus the quantum effective action dueŽ .to matter, we can expect the appearance of primordial induced wormholes in the early Universe. The

Ž .wormhole which appears as result of quantum fluctuations connects two asymptotically flat regions of theearly Universe. It may be with constant or increasing throat radius. The redshift function maybe increasing oroscillating. It would be very interesting to generalize above study for all types of quantum fields. Then onecould investigate the connection between content of GUTs and inducing of quantum wormholes. For example, itmay happen that SUSY GUTs better support inducing of wormholes as they include more fields then theirnon-SUSY versions.

Acknowledgements

SDO would like to thank S. Hawking and E. Mottola for useful discussion. O.O. was partially supported by aCONACYT Grant 28454-E. KEO was partially supported by a RFBR.

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w x Ž .10 S.J. Gates, T. Kadoyoshi, S. Nojiri, S.D. Odintsov, Phys. Rev. D 58 1998 084026.w x11 R. Balbinot, A. Fabbri, hep-thr9807123; F.C. Lombardo, F.D. Mazzitelli, J.G. Russo, gr-qcr9808048.w x Ž . Ž .12 M.S. Morris, K.S. Thorne, Am. J. Phys. 56 1988 395; M.S. Thorne, K.S. Thorne, U. Yurtsever, Phys. Rev. Lett. 61 1988 1446.w x13 G. Gibbons, hep-thr9801106.w x Ž .14 P. Anderson, W.A. Hiscock, D. Samuel, Phys. Rev. D 51 1995 4337; B.E. Taylor, W.A. Hiscock, P. Anderson, gr-qcr9608036.w x Ž .15 D. Hochberg, A. Popov, S.N. Sushkov, Phys. Rev. Lett. 78 1997 2050.w x Ž .16 V. Khatsymovsky, in: I.L. Buchbinder, K.E. Osetrin Eds. , Proc. II Int. Conference Quantum Field Theory and Gravity, TGPU

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11 March 1999


String theory on AdS =S3 =S3 =S13

Shmuel Elitzur, Ofer Feinerman, Amit Giveon, David TsabarRacah Institute of Physics, The Hebrew UniÕersity, Jerusalem 91904, Israel


Abstract

Spacetime properties of superstrings on AdS =S3 =S3 =S1 are studied. The boundary theory is a two dimensional3Ž .superconformal field theory with a large Ns 4,4 supersymmetry. q 1999 Published by Elsevier Science B.V. All rights

reserved.

1. Introduction

In this note we study the superstring propagationon the curved spacetime manifold

MMsAdS =S3 =S3 =S1 1.1Ž .3

w xfollowing 1 . In string theory on AdS =NN the3

spacetime theory is a two dimensional conformalŽ .field theory CFT on the boundary of AdS . The3

Ž . Ž .left right moving affine SL 2 symmetry on theŽworldsheet arising from the AdS part of the back-3

. Ž .ground is lifted to a left right moving Virasoroalgebra in the spacetime boundary theory. Moreover,any affine GG symmetry in the background NN CFTis lifted to an affine GG algebra in the spacetimetheory.

In Section 2, we will show that type II stringŽ . Ž .theory on MM 1.1 allows the analogue of a chiral

Ž .GSO projection, giving rise to a large Ns 4,4 , 2-dSCFT in spacetime. For simplicity, we usually dis-cuss the left moving sector of the theory. The affine

Ž . Ž . Ž . Ž 3 3 1SU 2 =SU 2 =U 1 arising from the S =S =SŽ ..background in 1.1 is the ‘‘R-symmetry’’ of this

large Ns4 algebra. Hence, while string theory onAdS =S3 =M 4, with M 4 sT 4 or K , gives rise to3 3

w x Ža small Ns4 algebra in spacetime 1 whose R-Ž ..symmetry is SU 2 , the example considered in this

work provides a boundary theory with the largerkind of Ns4 supersymmetry in two dimensions.

Another reason to consider string propagation onŽ . 3MM 1.1 is the following. The geometry AdS =S =3

S3 =R is obtained at the throat limit of two differ-ently oriented coincident sets of fivebranes intersect-ing in one direction, together with a set of infinitely

w x 1stretched strings 2–4 . Such background is the S ™Ž .R limit of MM in 1.1 . Indeed, the study of Killing

spinors on this space already indicates the appear-w xance of large Ns4, 2-d supersymmetry 3 .

In Section 3, we consider more aspects of theŽ .theory 1.1 , and remark on some properties of the

spacetime SCFT.

2. Superstrings on AdS =S3 =S3 =S13

2.1. Worldsheet properties

w xAs in Ref. 1 , we consider a fermionic string onthe Euclidean version of AdS , and study the world-3


( )S. Elitzur et al.rPhysics Letters B 449 1999 180–186 181

Ž .sheet theory in the free field representation of SL 2w x5 . The worldsheet Lagrangian has the form

2Ž2.LLsEfEfy R fqbEgqbE g(

k

2ybbexp y f qLL qLL 2.1Ž .( WZW ferž /k

Ž .The first part of LL is the bosonic part of the SL 2ŽWZW sigma model the target space of which is

parametrized by the coordinates f, g , g with metric2 2 2fds sdf qe dg dg , and b is an auxiliary field

inducing this metric in the sigma model upon inte-.gration . The boundary of AdS is at f™`, where3

Ž .the screening charge term vanishes, and g ,g areassociated with the coordinates parametrizing thetwo dimensional boundary. The two point functions

² Ž . Ž .: < < 2of the fields f, b , g are f z f 0 sylog z ,² Ž . Ž .:b z g 0 s1rz, and all other two point func-tions vanish. LL is the bosonic part of theWZW

Ž . Ž . Ž .SU 2 =SU 2 =U 1 WZW model, and LL in-fer

cludes all other terms which, in particular, haveworldsheet fermions in them. There are three world-

A Ž .sheet fermions c on the SL 2 manifold, threeqa a Ž . Ž .three fermions x and v on SU 2 =SU 2 , and a

Ž .single fermion l on U 1 . They are normalized suchthat

h A BA B² :c z c w s , A , Bs1,2,3,Ž . Ž .

zyw

h A B sdiag q,q,yŽ .

d aba b a b² : ² :x z x w s s v z v w ,Ž . Ž . Ž . Ž .

zyw

a,bs1,2,3

1² :l z l w s 2.2Ž . Ž . Ž .

zyw

Ž . Ž . Ž . Ž .The SL 2 =SU 2 =SU 2 =U 1 WZW sigmamodel has affine bosonic currents j A, k a, ma, E Ywith levels kq2, kX y2, kXX y2, respectively. The

worldsheet Ns1 supercurrent G of this system isw x6

2 iA B A B CG z s h c j y e c c cŽ . ( A B A BCž /k 6

2 ia a b cq x k y e x x x( X a abcž /k 6

2 ia a b cq v m y e v v v qlE Y( XX a abcž /k 6

2.3Ž .

Ž . Ž . Ž . AThe total affine SL 2 =SU 2 =SU 2 currents J ,a a ŽK , M are the upper components with respect to

Ž .the Ns1 worldsheet supersymmetry 2.3 and up to. A a anormalization of the fermions c , x , v , respec-

tively:

i iA A A B C a a a b cJ s j y e c c K sk y e x x ,BC bc2 2

ia a a b cM sm y e v v . 2.4Ž .bc2

Ž . Ž . Ž .They obey affine SL 2 , SU 2 and SU 2 algebrasat levels k, kX and kXX, respectively:

kh A Br2 ih e A BCJ DC DA BJ z J w s q q PPP ,Ž . Ž . 2 zywŽ .zywŽ .

A , B ,C , Ds1,2,3 2.5Ž .

kXd abr2 ie ab K c

ca bK z K w s q q PPP ,Ž . Ž . 2 zywŽ .zywŽ .

a,b ,cs1,2,3 2.6Ž .

kXXd abr2 ie ab M c

ca bM z M w s q q PPPŽ . Ž . 2 zywŽ .zywŽ .2.7Ž .

Ž .Finally, the U 1 affine current E Y is the uppercomponent of l and it satisfies:

1E YE Ys q PPP 2.8Ž .z w 2zywŽ .

( )S. Elitzur et al.rPhysics Letters B 449 1999 180–186182

The central charge c of the Ns1 worldsheet theoryon AdS =S3 =S3 =S1 is3

3 kq2 3 kX y2 3 kXX y2Ž . Ž . Ž .3 3cs q q q qX XX2 2k k k

3 1q q1q 2.9Ž .2 2

Criticality of the fermionic string cs15 togetherŽ .with Eq. 2.9 gives rise to the relation:

1 1 1s q 2.10Ž .X XXk k k

2.2. Spacetime properties

In string theory, affine worldsheet symmetries arelifted into global symmetries in spacetime. As shown

w xin Ref. 1 , a novelty of string theory on AdS is that3

such symmetries on the worldsheet are lifted toinfinite dimensional symmetries in the boundary, 2-d

Ž .spacetime theory. In particular, the affine SL 2worldsheet symmetry is lifted to a Virasoro algebrawhose spacetime generators L correspond to then

zero modes of worldsheet holomorphic operators,Žwhich can be chosen to be presented in the world-

. w xsheet BRST cohomology by 1 :

n ny1Ž .2 3 n y nq1L sy dz 1yn J g q J gŽ .En 2

n nq1Ž .q ny1q J g 2.11Ž .

2

They satisfy the Virasoro algebra:

cst 3w xL , L s nym L q n yn dŽ . Ž .n m nqm nqm ,0122.12Ž .

with a spacetime central charge

c s6kp 2.13Ž .st

w xwhere p is interpreted 1 as the number of infinitelystretched fundamental strings at the boundary ofAdS .3

Ž . Ž . Ž .The affine worldsheet SU 2 = SU 2 = U 1Ž . Ž . Ž .symmetry is lifted to an affine SU 2 =SU 2 =U 1

algebra in spacetime. The modes T a generating then

Ž .first affine SU 2 symmetry in spacetime correspondto the zero modes of the worldsheet holomorphicoperators:

Xka a nT s dz G ,x g zŽ .� 4( En y1r22

s dzg nE

=

Xk1 1a a 3 y q y1K yn x c y c gy c gŽ .( 2 2ž /k

2.14Ž .Ž . Ž . Ž .where G sEdz G z , and G z is given in 2.3 .y1r2

They satisfy the algebraXksta b ab c abT ,T s ie T q nd d 2.15Ž .n m c nqm nqm ,02

a aL ,T synT 2.16Ž .m n nqm

w xwith a spacetime level 1

kX skX p 2.17Ž .st

a Ž .Similarly, the modes R of the second SU 2 alge-n

bra correspond toXXk

a a nR s dz G ,v g z 2.18Ž . Ž .� 4( En y1r22

Ž .They satisfy the same algebra as 2.15 with a space-time level:

kXX skXX p 2.19Ž .st

Ž .Finally, the affine U 1 algebra has modes a corre-n

sponding to

a s dz G ,lg n zŽ .� 4En y1r2

s dzg nE

=2

1 13 y q y1E Yyn l c y c gy c gŽ .( 2 2ž /k

2.20Ž .

They satisfy the algebra

w xa ,a spnd 2.21Ž .n m nqm ,0

Next we perform a chiral GSO projection, thusmaking the spacetime theory supersymmetric. To


construct the spacetime supercharges we introducethe 32 worldsheet spin fields Sa, as1, . . . , 32,

w xwhich satisfy 7 :

1A A bc z S w s G S w q PPPŽ . Ž . Ž .a a b1r2'2 zywŽ .

1a 3qa bx z S w s G S w q PPPŽ . Ž . Ž .a a b1r2'2 zywŽ .

1a 6qa bv z S w s G S w q PPPŽ . Ž . Ž .a a b1r2'2 zywŽ .

110 bl z S w s G S w q PPPŽ . Ž . Ž .a a b1r2'2 zywŽ .

2.22Ž .

where G i, is1, . . . , 10, are 32 dimensional Diracmatrices representing the Clifford algebra:

� i j4 i j i j 2 7G ,G s2 g I , g sdiag 1 ,y1,1 2.23Ž . Ž .

A spacetime supersymmetry generator Q has thew xform 7 :

wyQs dz e S z 2.24Ž . Ž .2E

where w is the scalar field arising in the bosonizedsuperghost system of the fermionic string, and SsÝ32 u Sa is a linear combination of spin fields.as1 a

To be physical, Q has to satisfy two conditions: ithas to be BRST invariant and it has to pass the GSOprojection guaranteeing mutual locality amongstspacetime fermions. The BRST invariance reduces to

Ž . Ž .y3r2the condition that S z have no zyw singu-Ž . Ž .larity when contracted with G w 2.3 . Using the

Ž .OPEs of 2.22 , this becomes the following con-straint on the coefficients u defining Q:a

G u b s0 2.25Ž .ab

where the 32 dimensional matrix G is given by

1 11 2 3 4 5 6Gs G G G q G G GŽ . Ž .( ( Xk k

17 8 9q G G G 2.26Ž . Ž .( XXk

The matrix G satisfies1 1 1

2G s y y I 2.27Ž .X XXž /k k kŽ .Eq. 2.25 has a solution if and only if the criticality

Ž .condition 2.10 is satisfied; in that case, one hasG 2 s0. The hermitian conjugate of G ,

1 1† 1 2 3 4 5 6G s G G G q yG G GŽ . Ž .( ( Xk k

17 8 9q yG G G 2.28Ž . Ž .( XXk

Ž †.2obeys also G s0, and we have2

†� 4G ,G s I 2.29Ž .k

†' 'The matrices kr2 G and kr2 G are subjectedto the algebra of fermionic annihilation and creationoperators. The 32 dimensional space splits into two16 dimensional spaces corresponding to occupationnumber 0 or 1 for that fermion. The space of solu-

Ž .tions of 2.25 is then the 16 dimensional occupation0 subspace.

As in the flat space case, an appropriate GSOprojection corresponds to the requirement:

1011 b 11 iG u su , G s G 2.30Ž .Łab a

is111 Ž .Since G anticommutes with G of Eq. 2.26 , it

preserves its space of solutions reducing the numberŽ .of spacetime supercharges to 8. Under the SO 9,1

rotating the c i, the spin fields transform with theŽ .spinor representation. The condition 2.25 breaks

Ž . Ž . Ž .this symmetry down to SL 2 =SU 2 =SU 2 , un-der which the 8 supersymmetry generators transform

Ž .in the 1rrrrr2,1rrrrr2,1rrrrr2 representation.It is convenient to work with a bosonized form of

the spin fields Sa. A particularly simple bosonizedŽ . Ž .form for the solutions of 2.25 and 2.30 is ob-

tained when defining the bosonic fields according tothe following choice of complex structure on MM:

E H sc 1c 2 , E H sx 1x 2 , E H sv1v 2 ,1 2 3

k k3 3 3iE H s x q v c ,( (X XX4 ž /k k

k k3 3E H s x y v l . 2.31Ž .( (XX X5 ž /k k


The five scalars H are normalized such thatI

² :H z H w syd log zyw 2.32Ž . Ž . Ž . Ž .I J I J

Ž . Ž .The eight solutions of Eqs. 2.25 and 2.30 havethen the form

iŽ .H qH qH qH qH1 2 3 4 52S se ,qqq

iŽ .H yH yH yH yH1 2 3 4 52S se ,qyy

iŽ .yH qH qH yH qH1 2 3 4 52S se ,yqq

iŽ .yH yH yH qH yH1 2 3 4 52S se , 2.33Ž .yyy

iŽ .k H yH qH yH qH1 2 3 4 52S s e( Xqyq k

iŽ .k H yH qH qH yH1 2 3 4 52q e ,( XXk

iŽ .k H qH yH qH yH1 2 3 4 52S s e( Xqqy k

iŽ .k H qH yH yH qH1 2 3 4 52q e ,( XXk

iŽ .k yH yH qH qH qH1 2 3 4 52S s e( Xyyq k

iŽ .k yH yH qH yH yH1 2 3 4 52q e ,( XXk

iŽ .k yH qH yH yH yH1 2 3 4 52S s e( Xyqy k

iŽ .k yH qH yH qH qH1 2 3 4 52q e . 2.34Ž .( XXk

Ž .The four solutions in 2.34 can be obtained, forŽ .instance, by applying the global generators of SU 2

Ž . a a Ž . Ž .=SU 2 , T and R of Eqs. 2.14 and 2.18 , to0 0Ž .the four simple solutions 2.33 .

Notice that in the limit kX or kXX tending toŽ .infinity, both the complex structure 2.31 and the

Ž . Ž .spin fields 2.33 , 2.34 are identical to those usedw xin Ref. 1 , corresponding to string theory on AdS3

=S3 =T 4. Indeed, in the limit of large level anŽ . Ž .3SU 2 affine algebra approaches a U 1 one.Altogether, we have found eight left moving su-

percharges, thus leading to a global Ns4 supersym-metry in the NS sector of the spacetime theory.Indeed, the algebra generated by these eight physicalsupercharges, together with L , L , T a, Ra, is a0 "1 0 0

two dimensional global Ns4 superconformal alge-Ž . Ž .bra with an SU 2 =SU 2 subalgebra; explicitly:

w xL , L s myn L , m ,ns0,"1Ž .m n mqn

a b ab c a b ab cT ,T s ie cT , R , R s ie cR ,0 0 0 0 0 0

a,bs1,2,3a b a aT , R s L ,T s L , R s00 0 n 0 n 0

ni j i jL ,G s yr G ,n r nqrž /2

i , js"1r2, r ,ss"1r2i j1 1a i j a k j a i j a i kT ,G s s G , R ,G s s GŽ . Ž .k k0 r r 0 r r2 2

Gi j ,Gk l s2e i ke jlL y2 rysŽ .� 4r s rqs

=k kik jljl a i k ae s T q e s RŽ . Ž .X XXa 0 a 0ž /k k

2.35Ž .Ž .1r4Here Gs i 2k Q is normalized to satisfy the last

anticommutator, and the labels on Gi j are identifiedr1 1 1with: r' e , i' e and j' e . In the limits1 2 32 2 2

kX™` or kXX

™`, this algebra reduces to the global,Ž Ž .3 .small Ns4 algebra and a U 1 factor .

Ž .Combining the global Ns4 algebra 2.35 withŽ . Ž .the full Virasoro algebra 2.12 and affine SU 2 =

Ž . Ž . Ž . Ž . Ž . ŽSU 2 =U 1 , 2.15 , 2.16 , 2.21 in the world-sheet BRST cohomology and using the freedom of

w x.picture changing 7 leads to the so called ‘‘large’’w xNs4 superconformal algebra 8,9 . This chiral alge-

Žbra is generated by a spin-2 stress tensor whose. Žmodes are L , ngZ , four spin-3r2 fields withn

i j . Žmodes G , rgZq1r2 , seven spin-1 currents withra a .modes T , R , a , and four spin-1r2 fermionsn n n

Ž i j. i jwith modes G . For instance, the modes G ofr r

the four spacetime supercurrents can be obtained byconsidering the products of the integrands in LnŽ . Ž . Ž .2.11 with the eight spin fields 2.33 , 2.34 . Simi-larly, the modes G i j of the four spacetime fermionsr

Ž .– the Ns4 superpartners of the U 1 spacetimeboson – can be obtained by considering the products

Ž .of these eight physical spin fields with the a 2.20 .n


The left movers on the worldsheet are lifted to leftmovers in the spacetime boundary theory, whileworldsheet right movers are lifted to right movers inspacetime. All in all, the spacetime theory is a two

Ž .dimensional SCFT with a large Ns 4,4 supersym-Ž . Xmetry, with a central charge c 2.13 and levels k ,st st

XX Ž . Ž . Ž .k 2.17 , 2.19 . Using 2.10 , we see that thest

spacetime theory obeys the condition

6kX kXXst st

c s 2.36Ž .X XXst k qkst st

w xas required in a unitary large Ns4, 2-d theory 8,9 .

3. Remarks

w xAs shown in Ref. 1 , the properties of physicaloperators on the worldsheet are naturally lifted intoproperties of their corresponding physical states inthe boundary spacetime theory. For instance, the

Ž .SL 2 quantum number j and worldsheet spin s arerelated to the scaling dimension h and spin s in thest

spacetime 2-d theory. Moreover, operators in a rep-Ž . Ž . Ž .resentation R of the affine SU 2 =SU 2 =U 1

worldsheet symmetry are lifted to states with similarŽproperties in spacetime. Instead of being general for

w x.details see 1 we will discuss some particularlyinteresting examples.

Consider first the worldsheet vertex operators1 1ywyw 3 y y1 qV j se c y gc y g cŽ . Ž .2 2

=X XX1 13 y y1 q

X X XX XXc y gc y g c V V VŽ . jm m jm m jm m2 2

3.1Ž .X XX

X X XX XXwhere V , V ,V are vertex operators of thejm m jm m jm mŽ . Ž . Ž .SL 2 =SU 2 =SU 2 WZW model with isospins

X X3 3 3 3 3Ž . Ž . Žj, j sm, j sm , j,k sm ,k sm and j,m sXX XX3 .m ,m sm , respectively. The worldsheet scaling

Ž .dimensions of V j are

1 1 11 1hshs q q j jq1 y q q s1Ž . X XX2 2 ž /k k k

3.2Ž .Žthe last equality is obtained using the criticality

Ž ..condition 2.10 , and they are BRST invariant. Fur-Ž .thermore, the spacetime states corresponding to V j

are primaries of the Virasoro algebra with scaling

dimensions h sh s j, as well as primaries of thest stŽ . Ž . Ž .spacetime affine SU 2 = SU 2 = U 1 algebra.

Ž .Therefore, the physical states corresponding to V jŽ .are chiral primaries of the Ns 4,4 superconformal

algebra. For js1r2, the spacetime upper compo-Ž . Ž . Ž .nents of V 1r2 include a singlet of SU 2 =SU 2

Ž .=U 1 with scaling dimensions h sh s jq1r2st stŽ .s1, which thus correspond to the m,m modes of a

single marginal operator in the spacetime theoryŽ .which preserves Ns 4,4 supersymmetry. On the

worldsheet, this marginal deformation is in the R-Rsector and, therefore, turns on a R-R background.

Together with the manifest marginal deformationchanging the radius of S1, we thus see that the

Ž .superstring on MM 1.1 has a two dimensional mod-uli space. Let us discuss the structure of this modulispace in more details.

The theory studied in Section 2 is a WZW modelon MM ; it has a vanishing R-R background. In partic-ular, its S1 ™R limit is obtained at the throat limit

Ž .of appropriate smearing of strings in the followingw xsystem 2–4 :

1. kX coincident NS-fivebranes stretched, say, in theŽ 0 1 2 3 4 5.x , x , x , x , x , x directions.

2. kXX coincident NS-fivebranes stretched in theŽ 0 1 6 7 8 9.x , x , x , x , x , x directions.

3. p fundamental strings infinitely stretched inŽ 0 1.x , x .

In the type IIB string, this system is S-dual to acorresponding configuration of D5-branes and D-strings. The marginal deformation discussed abovemay very well be the one which connects continu-ously the theories with vanishing R-R backgrounds

Žto those which have them turned on instead of the.WZ term in the NS-NS sector .

It would be interesting to understand the structureŽ .of the 2-d, Ns 4,4 spacetime theory. Although this

is beyond the scope of this work, let us speculateabout the particular case kX skXX s2k. In this case,

Ž .the central charge 2.13 can be any positive integerproduct of three: c s3n, nskX p. A large Ns4st

theory with cs3 is constructed out of a singlescalar field and four fermions; we denote such atheory by TT . A candidate for the spacetime CFT is3

Ž .thus a deformation of an orbifold sigma model likeŽ .nthe symmetric product TT rS . Indeed, this is an3 n

Ž .Ns 4,4 SCFT with a single modulus in the un-twisted sector, corresponding to the radius of the


scalar in TT , and another modulus in the twisted3

sector which we found amongst the S sector, invari-3

ant twist fields 1.Ž .n Ž .States in the TT rS SCFT appear in Ns 4,43 n

Ž . Ž .multiplets and fall into SU 2 =SU 2 representa-tions

lqq lyq Z, , l ,qg , 0F lyqgZ 3.3Ž .ž /2 2 2

Ž .Amongst the operators in a representation 3.3 , theone with the smallest scaling dimension appears inthe Z twisted sector. The scaling dimension of2 lq1

such a twist field can be obtained by standard con-siderations of the symmetric orbifold:

lqq lyq lqd q2

h , s , ds 3.4Ž .orb ž /2 2 2 2 lq1

Ž . Ž .In string theory on AdS =SO 4 =U 1 , world-3

sheet operators corresponding to spacetime, primarystates with lowest scaling dimensions h , h in ast st

Ž .given representation 3.3 areX XX 1 1ywyw 3 y y1 qV j , j se c y gc y g cŽ . Ž .2 2

= 1 13 y y1 qc y gc y g cŽ .2 2

=X XXX X X XX XX XXV V V 3.5Ž .jm m j m m j m m

with

lqq lyqX XXj , j s , 3.6Ž . Ž .ž /2 2

X X XX XXj j q1 q j j q1Ž . Ž .1 1jsy q q 3.7Ž .(2 4 2

and

h sh s j 3.8Ž .st st

We thus find that upon deforming the orbifold SCFTto the spacetime SCFT corresponding to string the-

Ž . Ž .ory on AdS =SO 4 =U 1 , the dimensions of3

1 Ž .Note that for nF2, unitarity implies that 3.1 are not in theŽ w x .physical spectrum see 1 , and references therein, for details ,

even for js1r2. Therefore, there is a perfect agreement with theŽ .2fact that TT rZ does not have a marginal deformation in the3 2

twisted sector.

Ž .smallest h operators in a representation 3.3 arechanged as

2 2 1h q1r2 y h q1r2 s d dq1 3.9Ž . Ž . Ž . Ž .orb st 4

Ž .where d is given in 3.4 .For ds0, the dimensions of such operators are

not changed. Indeed, such operators are in theŽ . Ž .lr2,lr2 representation of SO 4 and their scaling

Ž .dimension is lr2; they are the chiral primaries 3.1discussed above. For d/0, the dimensions arechanged when one deforms away from the orbifold

Ž .point by a relatively small amount 3.9 .Finally, we should mention that the generalization

Ž .of this work to models with a large Ns 4,0 space-time supersymmetry can be done along the lines ofw x10 .

Acknowledgements

We thank E. Rabinovici, A. Schwimmer, N.Seiberg, and especially D. Kutasov for very usefuldiscussions. This work is supported in part by theIsrael Academy of Sciences and Humanities – Cen-ters of Excellence Program. The work of A.G. issupported in part by BSF – American-Israel Bi-Na-tional Science Foundation. S.E. and A.G. thank thetheory division at CERN where part of this work wasdone, and the Einstein Center at the WeizmannInstitute for partial support.

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w x1 A. Giveon, D. Kutasov, N. Seiberg, hep-thr9806194.w x Ž .2 P.M. Cowdall, P.K. Townsend, Phys. Lett. B 429 1998

281, hep-thr9801165.w x3 H.J. Boonstra, B. Peeters, K. Skenderis, hep-thr9803231.w x4 J.P. Gauntlett, R.C. Meyers, P.K. Townsend, hep-

thr9809065.w x Ž .5 M. Wakimoto, Comm. Math. Phys. 104 1986 605.w x Ž .6 Y. Kazama, H. Suzuki, Nucl. Phys. B 321 1989 232.w x7 D. Friedan, E. Martinec, S. Shenker, Nucl. Phys. B 271

Ž .1986 93.w x8 A. Sevrin, W. Troost, A. Van Proeyen, Phys. Lett. B 208

Ž .1988 447.w x Ž .9 K. Schoutens, Nucl. Phys. B 295 1988 634.

w x10 D. Kutasov, F. Larsen, R.G. Leigh, to appear.

11 March 1999


Supersymmetric non-abelian Born-Infeld theory

S. Gonorazky 1, F.A. Schaposnik 2, G. Silva 3

Departamento de Fısica, UniÕersidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina´


Abstract

Using the natural curvature invariants as building blocks in a superfield construction, we show that the use of asymmetric trace is mandatory if one is to reproduce the square root structure of the non-Abelian Dirac-Born-InfeldLagrangian in the bosonic sector. We also discuss the BPS relations in connection with our supersymmetry construction.q 1999 Published by Elsevier Science B.V. All rights reserved.

Ž .Dirac-Born-Infeld DBI type actions arise in thew xstudy of low-energy dynamics of D-branes 1–5

Ž w x .see 6,7 for a complete list of references . In thecase of superstring theory, one has to deal with a

w xsupersymmetric extension of DBI actions 8,9 and,when a number of D-branes coincide, there is a

w xsymmetry enhancement 10 and the Abelian DBIaction should be generalized to the non-Abelian case.

Several possibilities for extending the AbelianBorn-Infeld action to the case of a non-Abeliangauge symmetry have been discussed in the literaturew x11–17 . Basically, they differ in the way the grouptrace operation is defined. In the string context, asymmetric trace operation as that advocated by

w xTseytlin 14 seems to be the appropriate one. Amongits advantages, one can mention:

Ž .i It eliminates unwelcome odd powers of thecurvature, this implying that the field strength F

1 UNLP-FOMEC.2 Investigador CICBA.3 CONICET.

Ž .although possibly large should be slow varying3 w x 2since F ; D, D F . With this kind of Abelian

Ž X .approximation it implies commuting F s one canmake contact with the tree level open string effectiveaction.

Ž .ii It is the only one leading to an action which islinearized by BPS conditions and to equations ofmotion which coincide with those arising by impos-ing the vanishing of the b-function for background

w xfields in the open superstring theory 15–18 .It should be mentioned, however, that there are

some unsolved problems concerning the use of asymmetric trace for the non-Abelian Born-Infeld ac-tion. In particular, some discrepancies between theresults arising from a symmetrized non-Abelian BornInfeld theory and the spectrum to be expected in

w xbrane theories are pointed out in Ref. 16 .w xAs noticed in Ref. 17,19 , the fact that the sym-

metric trace is singled out as that leading to BPSrelations should be connected with the possibility ofsupersymmetrizing the Born-Infeld theory. In thisrespect, we construct in this work the supersymmet-ric version of the non-Abelian Dirac-Born-Infeld ac-


( )S. Gonorazky et al.rPhysics Letters B 449 1999 187–193188

tion and discuss the trace issue and the Bogomol’nyistructure of the resulting bosonic sector.

Our analysis, close to that developed in Refs.w x8,9 , extends to the non-Abelian case the resultspresented for the Abelian Supersymmetric Born-In-

w xfeld theory in Ref. 20 .As it is well-known, the basic object for con-

Žstructing supersymmetric gauge theories is the non-.Abelian gauge vector superfield V which we shall

Ž .write in ds4 Minkowski space in the form

iV x ,u ,u sCq iuxy iuxq uu Mq iNŽ . Ž .

2

imy uu My iN yus u AŽ . m2

i iq iuuu lq Eux y iuuu lq Euxž / ž /2 2

1 1q uuuu Dq IC 1Ž .Ž .2 2

ŽIn the case of SUSY Yang-Mills theory, gener-.alized gauge invariance allows to work in the

Wess-Zumino gauge, for which C,x , M and N areall set to zero, thus remaining a multiplet with thegauge field A , the Majorana fermion field l andm

the auxiliary real field D, all taking values in the Liealgebra of the gauge group which we take for defi-

Ž .niteness as SU N ,

A sAa t a , lslat a , DsDat a , 2Ž .m m

a Ž . Ž .with t the hermitian SU N generators,

w a b x abc ct ,t s if t 3Ž .

tr t at b sNNd ab 4Ž .

It is convenient to define a chiral variable y m inthe form

m m my sx q ius u 5Ž .

so that the usual derivatives D and D can be definedas

E E EmD s q2 i s u , D sy 6Ž . Ž .aa aa m aEu E y Eu

Ž .when acting on functions of y,u ,u and

E E EmD s , D sy y2 i us 7Ž . Ž .aa a ˙˙a †maEu E yEu

†Ž .on functions of y ,u ,u .Generalized gauge transformation will be written

in the form

exp 2 i L sexp 2 i Lat a 8Ž . Ž . Ž .Ž .where L y,u is a chiral left-handed superfield and

† †Ž .L y ,u its right-handed Hermitian conjugate,

†D LsD L s0 9Ž .a a˙

Explicitly,1 1

L y ,u s Ay iB quxquu Fq iG 10Ž . Ž . Ž . Ž .2 2

Here A, B,F,G are real scalar fields and x is aMajorana spinor. Under such a transformation, su-perfield V transforms as

exp 2V ™exp y2 i L† exp 2V exp 2 i L 11Ž . Ž . Ž . Ž . Ž .From V, the non-Abelian chiral superfield W cana

be constructed,

1 aW y ,u s D D exp y2V D exp 2V 12Ž . Ž . Ž . Ž .a a a8 ˙

Ž .In contrast with 11 , under a gauge transformationW transforms covariantly,a

W ™exp y2 i L W exp 2 i L 13Ž . Ž . Ž .a a

Concerning the hermitian conjugate, W , it trans-a

forms as

† †W ™exp y2 i L W exp 2 i L 14Ž . Ž . Ž .a a˙ ˙

Written in components, W readsa

inW y ,u s il yu Dy usms FŽ . Ž . aa a a mn2

yuu =ul 15Ž . Ž .a

with

F sE A yE A q i A , A 16Ž .mn m n n m m n

and

m a a˙ ˙=ul s s E l q i A ,l 17Ž . Ž . Ž .a a a ž /m m˙

As it is well-known, the SUSY extension of Ns1Yang-Mills theory can be constructed from W by

( )S. Gonorazky et al.rPhysics Letters B 449 1999 187–193 189

2 2considering W and its Hermitian conjugate W .Indeed, W 2 reads

W aW sylly i ulDqDulŽ .a

1 n nq usms lF qF usms lŽ .mn mn2

quu yil=uly i =ul lŽ .ž12 mn mn˜qD y F F q iF F 18Ž .ž / /mn mn2

with

1 abF s ´ F 19Ž .mn mna b2

Ž .Or, writing explicitly the SU N generators,

1 1a a b a b a b m n a b� 4W W s t ,t y l l y iul D q us s l Fa mn2 2ža m bc bcd d cyiuul s d E q f A lŽ .m m

1 1a b a b mn a b mn˜q uu D D y F F q iF Fž /ž /mn mn2 2 /20Ž .

where

� a b4 a b b at ,t s t t q t t 21Ž .aŽ .From Eq. 18 and an analogous one for W Wa

one sees that the supersymmetric Yang-Mills La-grangian can be written in the form

12 2 2 2L s tr d u W qd u W 22Ž .Ž .HSYM 24e NN

with an on-shell purely bosonic part giving

1a amnL sy F F 23Ž .SYM mnbos 24e

We are ready now to extend the treatment in Refs.w x8,9,20 , and find a general gauge invariant non-Abelian Ns1 supersymmetric Lagrangian of theDBI type. This Lagrangian will be basically con-

Ž .structed in terms of W, W and exp "2V . It isimportant to note that at this stage the trace operation

Ž Ž ..on internal SU N indices to be used in order todefine a scalar Lagrangian could differ, in principle,

Ž .from the ordinary trace ‘‘tr’’ defined in 4 and usedŽ .in Eq. 22 .

ŽIn order to get a DBI like Lagrangian written as.space-time determinant in the bosonic sector of the

theory, one should include terms which cannot bemn ˜mnexpressed in terms of F F and F F like formn mn

example those containing F 4 'F aF bFgF m. Indeed,m a b g

ignoring for the moment ambiguities arising in thedefinition of a non-Abelian space-time determinant,one has, concerning even powers,

1 2ydet g IqF s Iq FŽ .mn mn 2even powers

21 1 2 4q F yF 24Ž . Ž .ž /4 2

In the Abelian case, the F 4 term in the r.h.s. of Eq.2 ˜Ž .24 can be written in terms of F and FF but this

is not the case in the non-Abelian case. Moreover,odd powers of F which were absent in the formerare present in the latter case.

Let us start at this point our search for a super-symmetric extension of the non-Abelian DBI model.

Ž .To begin with, in order to get higher even powersmn ˜ mnof F F and F F which necessarily arise in amn mn

DBI-like Lagrangian, we shall have to include higherpowers of W and W combined in such a form as torespect gauge-invariance. In the Abelian case, this

2 2was achieved by combining W W with adequate2 2 2 2 w xpowers of D W and D W 8,9,20 . In the present

non-Abelian case, in view of transformation lawsŽ . Ž . Ž .11 , 13 , 14 , the situation is a little more involved.Consider then the possible gauge-invariant super-fields that can give rise to quartic terms. There aretwo natural candidates,

˙2 2 a bQ s d u d u W W exp y2V W W exp 2VŽ . Ž .˙H1 a b

25Ž .

˙2 2 a bQ s d u d u W exp y2V W exp 2VŽ . Ž .H2

=W exp y2V W exp 2V 26Ž . Ž . Ž .˙a b

with purely bosonic components

221 2 ˜trQ s tr F q tr FF 27Ž . Ž .Ž .ž /1 4bos

1 mn rs mn rs˜ ˜trQ s trF F F F q trF F F Fž /2 mn rs mn rs4bos

28Ž .

One can see now that a particular combination ofQ and Q generates the quartic terms one expects in1 2

the expansion of a square root DBI Lagrangian,


provided this last is defined using a symmetric trace.Indeed, one has

1 Ž .tr 2Q qQ1 224 bos

4th ord .21 1mn mn˜(sStr 1y 1q F ,F y F ,FŽ . Ž .mn mn2 16ž /

29Ž .

where

1Str t ,t , . . . ,t s tr t t . . . t ,Ž . Ž .Ý Ž . Ž . Ž .1 2 N p 1 p 2 p NN !

p

30Ž .

It is important to note that products of generatorscannot be performed within the Str operation butonly after Str is expressed as a sum of ordinarytraces; hence the commas between the arguments inŽ .30 .

Another feature in favour of using the symmetrictrace is that it gives the natural way of resolvingambiguities arising in the definition of the DBILagrangian as a determinant. Indeed, one hasw x14,17,19 ,

21 1mn mn˜(Str 1y 1q F ,F y F ,FŽ . ž /mn mn2 16ž /

sStr 1y det g qF 31Ž .Ž .( mn mnž /with the r.h.s. univoquely defined through the Strprescription.

Ž .Eq. 29 is one of the main steps in our deriva-tion: it shows that in order to reproduce the quarticterm in the expansion of a DBI-type square root, onehas to choose a particular combination of the normaltrace. But this combination of normal traces corre-sponds precisely to the symmetric trace, originally

w xproposed by Tseytlin 14 for the DBI theory in orderto make contact with the low energy effective theoryarising from superstring theory. It is worthwhile to

˜Ž .notice that the r.h.s. of 29 vanishes for Fs"iF.This will guarantee, at least at the quartic order weare discussing up to now, that the supersymmetricLagrangian will reduce to SUSY Yang-Mills when

Ž .the Bogomol’nyi bound in the Euclidean version isw xsaturated, as it should be 16,17,21 .

The analysis above was made for the purelybosonic sector. It is then natural to extend it byconsidering the complete superfield combination

2Q qQ with a trace that again acommodates in1 2

the form of a symmetric trace

1 a2trQ q trQ sStr W ,W ,exp y2VŽ . Ž .Ž1 2 a3

bW exp 2V ,exp y2V W exp 2V 32Ž . Ž . Ž . Ž ..b

Note that in this expression the factorŽ . Ž .exp y2V W exp 2V takes values in the algebra ofb

Ž . Ž .SU N . Moreover, Eq. 32 should be understood as

1 a b c d a a b2trQ q trQ sStr t ,t ,t ,t W WŽ . Ž .1 2 a3

=c

exp y2V W exp 2VŽ . Ž .˙ž /b

=d

bexp y2V W exp 2VŽ . Ž .Ž .33Ž .

Now, in order to construct higher powers of F 2

˜and FF, necessary to obtain the DBI Lagrangian, wew xdefine, extending the treatment in Ref. 20 , super-

fields X and Y,

1 2 aXs D exp y2V W W exp 2VŽ . Ž .ž /a16 ž ˙

qexp y2V D2 exp 2V W aWŽ . Ž .Ž a

=exp y2V exp 2V 34Ž . Ž . Ž .. /i

2 aYs D exp y2V W W exp 2VŽ . Ž .ž /až ˙32

yexp y2V D2 exp 2V W aWŽ . Ž .Ž a

=exp y2V exp 2V 35Ž . Ž . Ž .. /Both fields transform like W under generalizeda

gauge transformations

X™exp y2 i L Xexp 2 i L ,Ž . Ž .

Y™exp y2 i L Yexp 2 i L 36Ž . Ž . Ž .

and their us0 component give, as in the Abeliancase, the two basic invariants

1 1mn mn˜< <X s F F , Y s F F 37Ž .us0 us0mn mn4 8

Ž .Inspired in the result 32 obtained in order to˜reproduce the adequate quartic power in F and F,

we propose the following supersymmetric non-


Abelian Lagrangian as a candidate to reproduce theDBI dynamics in its bosonic sector,

L sL q C d4u Str W a ,W ,exp y2VŽ .Ý H žS SYM nm ažn ,m

=W exp 2V ,exp y2VŽ . Ž .b

= b n mW exp 2V , X ,Y qh.c. 38Ž . Ž ./ /Ž n.In this expression, objects like Str A, X in the

Ž .sum, should be understood as Str A, X, X, . . . , X .The arbitrary coefficients C remain to be deter-nm

mined. One should retain at this point that expressionŽ .38 gives a general Lagrangian corresponding to thesupersymmetric extension of a bosonic gauge invari-ant Lagrangian depending on the field strength F

mn ˜ mnthrough the algebraic invariants F F and F F ,mn mn

in certain combinations constrained by supersymme-Ž .try. The Abelian version of 38 was engineered in

w xRefs. 8,9,20 , so that the Dirac-Born-Infeld La-grangian could be reproduced by an appropriatechoice of coefficients C . The same happens in thenm

non-Abelian case: a particular choice of C corre-nm

sponds to the non-Abelian Born-Infeld theory,1C s0 0 8

m nq2-jmC s y1 qŽ . Ýny2 m 2 m nq1yjž /j

js0

C s0 , 39Ž .n 2 mq1

1q sy0 2

nq1y1 2ny1 !Ž . Ž .q s for nG1n 4n nq1 ! ny1 !Ž . Ž .

40Ž .

With this choice one has for the purely bosonicŽ .part of Lagrangian 38 ,

L sStr 1y ydet g qF 41Ž .Ž .(S mn mnž /bos

This is the non-Abelian Dirac-Born-Infeld La-w xgrangian in the form originally proposed in Ref. 14 .

As in the Abelian case, there are other choices ofcoefficients C which also give consistent causalm n

supersymmetric gauge theories with non-polynomialgauge-field dynamics. In particular, the alternative

Ž .proposal for a SO N DBI action recently presentedw xin Ref. 22 should correspond to one of such choices.

We have then been able to construct a Ns1Ž Ž ..supersymmetric Lagrangian Eq. 38 within the

superfields formalism, with a bosonic part expressedŽ .in terms of the square root of det g qF . Wemn mn

have employed the natural curvature invariants asbuilding blocks in the superfield construction arriv-ing to a Lagrangian which, in its bosonic sector,

mn ˜mndepends only on the invariants F F and F Fmn mn

and can be expressed in terms of the symmetric traceof a determinant. Odd powers of the field strength Fwere excluded in our treatment due to the fact that itis not possible to construct a superfield functional of

3Ž . Ž .W W and DW DW containing F terms in itshigher u component.

As mentioned above, the trace structure of thenon-Abelian Born-Infeld theory was fixed in Ref.w x17,19 , by demanding the action to be linearized by

ŽBPS-like configurations instantons, monopoles, vor-.tices . In the present work we have seen that the

symmetric trace naturally arises in the superfieldformalism in the route to the construction of thesquare root Dirac-Born-Infeld Lagrangian. This con-fluence of results is nothing but the manifestation ofthe well-known connection between supersymmetryand BPS relations. Then, in order to complete ourwork, we shall now describe the BPS aspects in themodel.

For definiteness we shall concentrate on instantonconfigurations in ds4 dimensional space-time. Inthe Wess-Zumino gauge, the Ns1 supersymmetry

Ž .vector multiplet is A ,l, D , with l a Majoranam

fermion. Now, in order to look for BPS relations, weshould consider a Ns2 supersymmetric modelwhich includes, apart from these fields, those belong-ing to a chiral scalar multiplet. Indeed, in analogywith what was done to obtain the Ns1 general

Ž .supersymmetric Lagrangian 38 , one can construct ageneral Ns2 SUSY Lagrangian by adding to thevector multiplet a chiral multiplet as in the Ns2SUSY Yang-Mills Lagrangian case. We shall notdetail this construction here but just consider therelevant Ns2 SUSY transformation laws in orderto derive BPS relations.

A complete Ns2 vector multiplet can be acom-modated in terms of the fields described above in the

Ž .form A ,l,f, D,F with l now a Dirac fermion,m


Ž .ls l ,l , f a complex scalar, fsMq iN, D1 2

and F auxiliary real fields. The gaugino supersym-Ž Ž .metric transformation law reads we call js j ,j1 2

.the Ns2 transformation parameter

dl s G mn F qg D jŽ .i mn 5 i

q i´ Fqg m= Mqg N j 42Ž . Ž .Ž .i j m 5 j

where

imn m nw xG s g ,g 43Ž .

4

Instanton configurations correspond to DsFsfsŽ .0 so that 42 simplifies to

dlsG mn F j 44Ž .mn

or

1 mn ˜dls G F q ig F j 45Ž .ž /mn 5 mn2

In order to look for BPS relations one imposes asusual dls0 thus obtaining

˜ ˜F q iF j s0 , F y iF j s0 46Ž .ž / ž /mn mn 1 mn mn 2

Ž .In Euclidean space, Eqs. 46 become

˜ ˜F qF j s0 , F yF j s0 47Ž .ž / ž /mn mn 1 mn mn 2

with j and j two Euclidean Weyl fermion inde-1 2

pendent parameters. As usual, these conditions leadto instanton or anti-instanton self-dual equations

˜F s"F 48Ž .mn mn

each one of its solutions breaking half of the super-symmetries.

The fact that Yang-Mills self-dual equations arisealso when the dynamics of the gauge field is gov-erned by a non-Abelian Born-Infeld Lagrangian was

w xalready observed in Refs. 15–17 . In the context ofsupersymmetry, this can be understood followingw x21 where it is shown how the supersymmetry trans-

Žformation law for the gaugino and for the Higgsinow x.in the case of the example discussed in Ref. 21 ,

Ž .together with the algebraic equation of motion forthe auxiliary fields, make the BPS relations remainunchanged irrespectively of the specific choice forthe gauge field Lagrangian. Moreover, one can seethat the Ns2 SUSY charges for a general non-poly-nomial theory, obtained via the Noether construction,

coincide, on shell, with those arising in Maxwell orYang-Mills theories.

In summary, using the superfield formalism, wehave derived a supersymmetric non-Abelian Dirac-Born-Infeld Lagrangian which shows the expected

Ž .BPS structure, namely that of the normal Yang-Mills theory. In our construction, we have seen thatthe natural superfield functionals from which super-symmetric non-Abelian gauge theories are usuallybuilt, combine in the adequate, square root DBI formin such a way that the symmetric trace is singled outas the one to use in defining a scalar superfieldLagrangian. It should be stressed that the fact thatthe purely bosonic Lagrangian depends on the basic

2 ˜invariants F and FF and not on odd powers of Fis not the result of the choice of a symmetric tracebut the consequence of using W and DW as buildingblocks for the supersymmetric Lagrangian. Finally,let us mention that not only the supersymmetric DBILagrangian but a whole family of non-polynomialLagrangians are then included in our main result, Eq.Ž .38 and all of them are linearised by BPS configura-tions which coincide with those of the normalYang-Mills theory.

Acknowledgements

G.S. would like to thank Dominic Brecher forhelpful e-mail correspondence. We all would like tothank Adrian Lugo and Carlos Nunez for helpful´ ´ ˜comments and discussions. This work is partially

Ž .supported by CICBA, CONICET PIP 4330r96 ,Ž .ANPCyT PICT 97 No:03-00000-02285 , Fundacion´

Antorchas, Argentina and a Commission of the Euro-pean Communities contract No:C11)-CT93-0315.

References

w x Ž .1 E.S. Fradkin, A.A. Tseytlin, Phys. Lett. B 163 1985 123.w x Ž .2 A.A. Tseytlin, Nucl. Phys. B 276 1986 391.w x Ž .3 R.G. Leigh, Mod. Phys. Lett. A 4 1989 2767, 2073.w x Ž .4 C.G. Callan, J.M. Maldacena, Nucl. Phys. B 513 1998 198.w x Ž .5 G. Gibbons, Nucl. Phys. B 514 1998 603.w x Ž .6 J. Polchinski, in: C. Efthimiou, B. Greene Eds. , Fields,

Ž .Strings and Duality TASI 96 , World. Sci., 1997, p. 293.w x7 W. Taylor IV, hep-thr9801182, Lectures on D-Branes, Gauge

Ž .Theory and M atrices .


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w x Ž .10 E. Witten, Nucl. Phys. B 460 1996 335.w x11 A. Abouelsaood, C.G. Callan, C.R. Nappi, S.A. Yost, Nucl.

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Ž .1991 2635.w x Ž .14 A.A. Tseytlin, Nucl. Phys. B 501 1997 41.w x Ž .15 J.P. Gauntlett, J. Gomis, P.K. Townsend, JHEP 01 1998

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11 March 1999


The KW theorem for the SKP hierarchy

Jingsong He a, Yi Cheng b,a

a Department of Mathematics, UniÕersity of Science and Technology of China, Hefei, Anhui 230026, Chinab Nonlinear Science Center, UniÕersity of Science and Technology of China, Hefei, Anhui 230026, China


Abstract

Ž .The supersymmetric Kadomtsev-Petviashvili SKP hierarchy is first introduced by Manin and Radul. In this letter, bythe factorization Ln sL L PPP L with L sDqu qu Dy1 qu Dy2 q PPP , js1,2 PPP n, being the indepen-n ny1 1 j j,0 j,y1 j,y2

Ž .dent super-pseudodifferential operators SC DOs , we construct the supersymmetric Miura transformation for the SKPhierarchy, which leads to decomposition of the second Poisson brackets based on Ln to a direct sum. Each term in the sumcontains the second brackets for L . q 1999 Published by Elsevier Science B.V. All rights reserved.j

1. Introduction

It is well known that the Kdv hierarchy associated with a scalar nth order differential operator

En ny1LsE qu E q PPP qu Equ , Es , 1.1Ž .ny1 1 0 E x

has the following remarkable property. There exists a Miura transformation relating the equations to a modifiedequation. By Miura transformation the second Poisson brackets of the nth Kdv type equation is transformed to a

Ž .vastly simpler one essentially just ErE x on an appropriate space of the modified variables. This is what wew x w xcalled the KW theorem 1 . A short proof of this theorem was then given by Dickey 2 . On the other hand, the

Ž .nth Kdv hierarchy can be reduced from the Kadomtsev-Petviashvili KP hierarchy. The latter is based on thew xpseudodifferential operator 3

LsEqu Ey1 qu Ey2 q PPP , 1.2Ž .1 2

and is written in the Lax representation

w xL s B , L , ms1,2, PPP , 1.3Ž .t mm

where B sLm and Lm denote respectively the differential and residual part of Lm. The coefficients u ,m q y jŽ .js1,2, PPP , are understood as functions of infinitely many variables ts t ,t , PPP with t sx. As a natural1 2 1

generalization, the Miura transformation and KW theorem for the KP hierarchy has proven by one of the authorsw x Ž . w xof Ref. 4 see its Proposition 3.1 , which also is relating other important questions 5–7 in integrable system

theory.


( )J. He, Y. ChengrPhysics Letters B 449 1999 194–200 195

In the last decade there has been increasing interest in supersymmetrization of the integrable systems.Ž .Recently, for the supersymmetric Kdv SKdv hierarchy associated with a nth supersymmetric differential Lax

operator

LsDn qu Dny1 q PPP qu Dqu , 1.4Ž .ny1 1 0

w xa bi-hamiltonian structure and a supersymmetric version of the KW theorem are presented 8–10 . Of course,w xthe SKdv hierarchy can be reduced from the SKP hierarchy 11 , which defined by following supersymmetric

pseudodifferential Lax operator

LsDqu qu Dy1 qu Dy2 q PPP . 1.5Ž .0 y1 y2

In view of the existence of the KW theorem for the KP hierarchy, it is natural to ask whether there is aŽ .supersymmetric version of SKP hierarchy we call it KW theorem for the SKP hierarchy or not. In this letter,

we will provide a affirmative answer by a direct computation.We organize this letter as follows. In Section 2, we review the bi-hamiltonian structure and the KW theorem

for the SKdv hierarchy. The KW theorem for the SKP is proved in Section 3. Finally Section 4 includesconcluding remarks.

2. The SKdv hierarchy

w x Ž .For simplicity, we will adopt the notation of Refs. 8,9 . In a 1N1 supersymmetric space with coordinateŽ .x,u , we consider a homogeneous supersymmetric differential Lax operator for SKdv hierarchy of the form

LsDn qu Dny1 q PPP qu Dqu , 2.1Ž .ny1 1 0

where DsErEuqu ErE x is a supercovariant derivative and satisfies D2 sErE x, u is the Grassmann variable2 < < Ž .and u s0. The homogeneity condition requires parity u 'nq j mod2 . We will define the second Poissonj

brackets on functionals of the form

w xF L s f u , 2.2Ž . Ž .HB

Ž .where f u is a homogeneous differential polynomial of the u and H sHd xdu is the Berezin integral: ifj BŽ . Ž . Ž . Ž . Ž .u su qu Õ , and f u sa u,Õ qu b u,Õ , then H f u sHb u,Õ d x.i i i B

Ž . w xA super-pseudodifferential operator SC DO P can be expressed as 10m

iPs P D , 2.3Ž .Ý iy`

and P sÝmP Di, P sÝy1 P Di. The multiplication of SC DOs is described by so-called supersymmetricq 0 i y y` iw xanalogue of the Leibniz rule 11 , i.e.

`< <Ž .k F kyik w i x kyiD Fs y F D , 2.4Ž . Ž .Ý ky1

is0

in which the ‘‘superbinomial coefficients’’ are defined as

°0 for i-0 or k ,i ' 0,1 mod 2 ,Ž . Ž . Ž .1k ~ ks 2.5Ž .2

ky i for iG0 and k ,i / 0,1 mod 2 .Ž . Ž . Ž .1¢ž /ky iŽ .2

( )J. He, Y. ChengrPhysics Letters B 449 1999 194–200196

For a given SC DO PsÝm P Di we set its superresidue as sres PsP , and its supertrace as StrPsH sres P.y` i y1 Bw xFurthermore, one can check that, for any two SC DO P and Q, Str P,Q s0, if we define the graded

w x Ž . < P < <Q < Ž . < P < <Q <commutator of P and Q as P,Q 'PQy y1 QP. This then followed by StrPQs y StrQP.w xTo define the Poisson bracket we will still need the gradient dF of a a arbitrary functional F L . It is given

by

ny1 d fk yky1dFs y D , 2.6Ž . Ž .Ýd ukks0

where we define variational derivative of function f as

w xk`d f E< < Ž .u iqi iq1 r2ks y , 2.7Ž . Ž .Ý w xiž /d u E uk kis0

w i x Ž i . � 4 w xwith u s D u . Finally the second Poisson brackets F,G of F and G, is defined as 8,9k k

< < < < < <F q G q L q1� 4F ,G s y Str LdF LdGyL dFL dG , 2.8Ž . Ž . Ž . Ž .q q

Ž .with the help of L. It has been shown that the second Poisson brackets in 2.8 is antisymmetric and satisfies thew xsuperJacobi identity 9 , hence it is indeed defines the second Hamiltonian structure for the SKdv hierarchy.

Ž .Ž . Ž .Let us factorize Ls DyF DyF PPP DyF . This yields a expression for each u as differentialn ny1 1 iw xpolynomials in the F , which is called the supersymmetric version of the Miura transformation 8 . Particularlyj

w xthe second Poisson brackets based on L of the SKdv hierarchy can be expressed as 8

n d f d gi� 4F ,G s y D , 2.9Ž . Ž .ÝH ž /dF dFB i iis1

if we set the fundamental Poisson brackets of F asj

iF X ,F Y s y d Dd XyY , 2.10Ž . Ž . Ž . Ž . Ž .� 4i j i j

Ž . Ž . Ž . Ž . Ž .where Xs x,u ,Ys y,v , and d XyY sd xyy d uyv . This is the KW theorem for the SKdvhierarchy.

3. The KW theorem for the SKP hierarchy

We are now in a position to start our discussion on the KW theorem for the SKP hierarchy. To this end, weintroduce a homogeneous superpseudodifferential Lax operator

Ln sDn qu Dny1 q PPP qu Dqu qu Dy1 qu Dy2 q PPP . 3.1Ž .ny1 1 0 y1 y2

Ž .It is interesting to note that the second Poisson brackets in 2.8 also provides the second hamiltonian structuren Ž . w xfor L in 3.1 10 ,

< < < < < n <F q G q L q1 n n n nn� 4F ,G s y sres L dF L yL dFL dG . 3.2� 4Ž . Ž . Ž . Ž .Ž .Ž . HL q q

B

w n xObviously the gradient dF of a functional F L is rewritten as

ny1d f d fk yky1dFs s y D 3.3Ž . Ž .ÝndL d uky`


For a fixed integer n, we factorize Ln to be the following multiplication form

Ln sL L PPP L L , 3.4Ž .n ny1 2 1

where

L sDqu qu Dy1 qu Dy2 q PPP , js1,2, PPP ,n ,j j ,0 j ,y1 j ,y2

Ž .are SC DOs having the same form as 1.5 and independent of each other. Comparing coefficients of the sameŽ . npowers in both side of 3.2 , all coefficients u of L can be expressed as differential polynomials in u ,j j,k

nny ju s y u , u sR u , jsny2,ny3, PPP 3.5Ž . Ž . Ž .Ýny1 j ,0 j j j ,k

j

We call these expressions the supersymmetric Miura transformation for the SKP hierarchy.We now discuss the main result of this letter. The second Poisson brackets based on Ln for the SKP

hierarchy is expressed as

n d f d f d g< < < <j F q G q1q1n� 4F ,G sy y sres y L L yL L 3.6Ž . Ž . Ž .Ž . Ý HL j j j j½ 5ž / ž /ž /dL dL dLB j j jjsy` q q

We call this result the KW theorem for ths SKP hierarchy. We would like to emphsize that the above-monen-Žtioned FsH f and GsH g are functionals in u , and also can be regarded as founctionals in u jsB B k j,k

.1,2, PPP ,n;ksy`, PPP ,0 via the supersymmetric Miura transformation for the SKP hierarchy.We first express d frdL in terms of d frdLn. According to the two understanding of F mentioned in above,j

the variation of F can be calaulated in two ways:ny1 n 0d f d f

dFs d u s d u . 3.7Ž .Ý Ý ÝH k j ,kd u d uB k j ,kksy` js1 ksy`

It impliesn d f< < < <F qnq1 Fny Str dL dF s y sres dL . 3.8Ž . Ž . Ž . Ž .Ý H jž /dLB jjsy1

Ž .Basing on 3.4 we have

dLn sdL L L L PPP L PPP L L Ln ny1 ny2 ny3 j 3 2 1

qL dL L L PPP L PPP L L Ln ny1 ny2 ny3 j 3 2 1

qL L dL L PPP L PPP L L Ln ny1 ny2 ny3 j 3 2 1

qL L L dL PPP L PPP L L Ln ny1 ny2 ny3 j 3 2 1...qL L L L PPP dL PPP L L Ln ny1 ny2 ny3 j 3 2 1

...qL L L L PPP L PPP dL L Ln ny1 ny2 ny3 j 3 2 1

qL L L L PPP L PPP L dL Ln ny1 ny2 ny3 j 3 2 1

qL L L L PPP L PPP L L dLn ny1 ny2 ny3 j 3 2 1

n

s L L L L PPP L dL L PPP L L L , 3.9Ž .Ý n ny1 ny2 ny3 jq1 j jy1 3 2 1js1


Ž n . Ž .then the Str dL dF of 3.8 becomes

nnStr dL dF s sres L L L L PPP L dL L PPP L L L dFŽ . ÝH n ny1 ny2 ny3 jq1 j jy1 3 2 1ž /B js1

nŽ < <.Ž .jq d F nyjs sres y dL L PPP L dFL L PPP L . 3.10Ž . Ž .ÝH j jy1 1 n ny1 jq1ž /B js1

Ž n . Ž .Taking Str dL dF back into 3.8 , we get

d f Ž < <.Ž .nq1 jq d F nyjs y y L L PPP L dFL PPP LŽ . Ž . jy1 jy2 1 n jq1dLj

< <Ž .F nqj qjq1s y L L PPP L dFL PPP L , 3.11Ž . Ž .jy1 jy2 1 n jq1

< < < <where dF s F qn. Furthermore, there is a very useful relation

d f d f< <F q1L s y L . 3.12Ž . Ž .jq1 jdL dLjq1 j

Ž .We now start to prove 3.6 by a direct calculating its right hand side. By above equations, the right handŽ .side of 3.6 reads as

n d f d f d g< < < <j F q G q1q1r.h.ssy y sres y L L yL LŽ . Ž .ÝH j j j j½ 5ž / ž /ž /dL dL dLB j j jjs1 q q

n d f d f d g< < < <j F q Gsy y y sres L L y L LŽ . Ž .ÝH j j j j½ 5ž / ž /ž /dL dL dLB j j jjs1 y y

n d f d g< < < < < < Ž .j F q G F q nqj qjq1sy y y sres L y L L PPP L L PPP L LŽ . Ž . Ž .ÝH j jy1 jy2 1 n jq1 jn½ 5ž /dL dLB y jjs1

n d f d g< < < < < < Ž .j F q G F q nqj qjq1q y y sres y L L PPP L L PPP L LŽ . Ž . Ž .ÝH j jy1 1 n jq1 jn½ 5ž /dL dLB y jjs1

n< < < <j F q Gsy y yŽ . Ž .ÝH ½

B js1

d g< < Ž . < < < <F q nqj qjq1 F q G q1=sres y y L PPP L dFL PPP L LŽ . Ž . Ž .jy1 1 n j jy 5½ 5dLj

n< < < <j F q Gq y yŽ . Ž .ÝH ½

B js1

d f d g< < Ž .F q nqj qjq1=sres y L L PPP L L PPP L LŽ . j jy1 1 n jq1 jn 5ž /½ 5dL dLy j


n< < < <j F q Gsy y yŽ . Ž .ÝH ½

B js1

d g< < Ž . < < < < < <F q nqj qjq1 F q G q1 G q1=sres y y L PPP L dFL PPP L y LŽ . Ž . Ž .Ž .jy1 1 n j jy1y 5½ 5dLjy1

n d f d g< < < < < < Ž .j F q G F q nqj qjq1q y y sres y L L PPP L L PPP L LŽ . Ž . Ž .ÝH j jy1 1 n jq1 jn½ 5ž /½ 5dL dLB y jjs1

n< < < < < < Ž . < < Ž . < < < <jF q G F q nqj qjq1 G q nqj qjq1 F q G q1sy y sres y y y yŽ . Ž . Ž . Ž . Ž .ÝH ½½

B js1

= L L PPP L dFL L PPP L L L PPP L dGL L PPP LŽ . 5jy1 jy2 1 n ny1 j jy1 jy2 1 n ny1 jy 5n

< < < < < < Ž . < < Ž . jF q G F q nqj qjq1 G q nqj qjq1q y sres y y yŽ . Ž . Ž . Ž .ÝH ½½B js1

= L L PPP L dFL L PPP L L L PPP L dGL L PPP LŽ . 5j jy1 1 n ny1 jq1 j jy1 1 n ny1 jq1y 5< < < < < <Ž . < <Ž . < < < <1F q G q1 F nq1 q1q1 G nq1 q1q1 F q G q1s y y y y yŽ . Ž . Ž . Ž . Ž .½H

B

n< <Ž . < <Ž .F nqn qnq1 G nqn qnq1=sres dFL L PPP L dGL L PPP L y y y yŽ . Ž . Ž . Ž .n ny1 1 n ny1 1y

=sres L L PPP L dF L L PPP L dGŽ . 5n ny1 1 n ny1 1y

< < < <F q G qnq1s y sres L L PPP L dFL L PPP L dG½Ž . Ž .H n ny1 1 n ny1 1 yB

ysres L L PPP L dF L L PPP L dG5Ž .n ny1 1 n ny1 1y

< < < < < <F q G q L q1 n n n nn� 4s y sres L dF L yL dFL dG s F ,G .½ 5Ž . Ž . Ž .Ž . Ž .H Lq q

B

Ž .In fact, Eq. 3.6 can be written in the following compact formn

jn� 4 � 4F ,G sy y F ,G , 3.13Ž .Ž . Ž .LŽ . ÝL j

1

where

d f d f d g< < < <F q G q1q1� 4F ,G s sres y L L yL L . 3.14Ž . Ž . Ž .L Hj j j j j½ 5ž / ž /ž /dL dL dLB j j jq q

0d f d fk yky1s y D . 3.15Ž . Ž .ÝdL d uj j ,kksy`


Ž .If we let L sDyF , Eq. 3.14 yieldsj j

d f d g� 4F ,G sy D . 3.16Ž . Ž .L Hj ž /dF dFB j j

Ž . w xSo it is easy to reobtain the KW theorem for SKdv hierarchy from Eq. 3.13 8 :

n n d f d gj jn� 4 � 4F ,G sy y F ,G s y D . 3.17Ž .Ž . Ž . Ž .LŽ . Ý Ý HL j ž /dF dFB j j1 js1

Here Ln sDn qu Dny1 PPP qu Dqu is the supersymmetric differential Lax operator for the Skdvny1 1 0

hierarchy.

4. Concluding remarks

In this letter we obtains the supersymmetric Miura transformation for the SKP hierarchy by means of theŽ . nfactorization 3.4 of SC DO L to a multiplication form, which leads to decomposition of the second Poisson

Ž .bracket for Lax operator L js1,2, PPP ,n . In view of many applications of Miura transformation and thejw xsecond Poisson bracket for KP hierarchy and constrained KP hierarchy 4–7,12 , we expect to discuss in future

w xthe analogous object in SKP and superconstrained KP 10,13–15 hierarchy.

Acknowledgements

This work was supported by the NSF of China. He Jingsong would like to thank Professor J.M.Figueroa-w xO’Farrill for answering my question about Refs. 8,9 and sending me his papers.

References

w x Ž .1 B.A. Kupershmidt, G. Wilson, Invent. Math. 62 1981 403.w x Ž .2 L.A. Dickey, Commun. Math. Phys. 87 1982 127.w x3 L.A. Dickey, Soliton Equations and Hamiltonian Systems, Advanced Series in Math. Phys. vol. 12, Singapore, 1991.w x Ž .4 Yi Cheng, Commun. Math. Phys. 171 1995 661.w x Ž .5 Yi Cheng, Yi-shen Li, Phys. Lett. A 157 1991 22.w x Ž .6 Yi Cheng, J. Math. Phys. 33 1992 3774.w x Ž . Ž .7 Yi Cheng, Lett. Math. Phys. 33 1995 159; 36 1996 35.w x Ž .8 J.M. Figueroa-O’Farrill, E. Ramos, Phys. Lett. B 262 1991 265.w x Ž . Ž .9 J.M. Figueroa-O’Farrill, E. Ramos, Commun. Math. Phys. 145 1992 43; Mod. Phys. Lett. A 10 1995 2767.

w x10 Jin-Chang Shaw, Ming-Hsien Tu, solv-intr9805002.w x Ž .11 Yu.I. Manin, A.O. Radul, Commun. Math. Phys. 98 1985 65.w x Ž .12 H. Aratyn, E. Nissimov, S. Pacheva, Int. J. Mod. Phys. A 12 1997 1265.w x Ž .13 H. Aratyn, C. Rasinariu, Phys. Lett. B 391 1997 99.w x14 H. Aratyn, E. Nissimov, S. Pacheva, solv-intr9801021.w x15 H. Aratyn, E. Nissimov, S. Pacheva, solv-intr9808004.

11 March 1999


Exploring softly broken SUSY theoriesvia Grassmannian Taylor expansion

D.I. Kazakov 1

BogoliuboÕ Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Moscow Region, Russia

Received 5 January 1999Editor: P.V. Landshoff

Abstract

We demonstrate that soft SUSY breaking introduced via replacement of the couplings of a rigid theory by spurionsuperfields has far reaching consequences. Substituting these modified couplings into renormalization constants, RGequations, solutions to these equations, fixed points, finiteness conditions, etc., one can get corresponding relations for thesoft terms by a simple Taylor expansion over the Grassmannian variables. This way one can get new solutions of the RGequations. Some examples including the MSSM, SUSY GUTs and the Ns2 Seiberg-Witten model are given. q 1999Published by Elsevier Science B.V. All rights reserved.

PACS: 11.10Gh; 11.10Hi; 11.30PbKeywords: Soft supersymmetry breaking; Renormalization; Renormalization group

1. Introduction

w xIn a recent paper 1 , which is based on thew xprevious publications 2,3 we have shown that

renormalizations in a softly broken SUSY theoryfollow from those of an unbroken SUSY theory andcan be performed in the following straightforwardway:

One takes renormalization constants of a rigidtheory, calculated in some massless scheme, substi-

(tutes instead of the rigid couplings gauge and)Yukawa their modified expressions, which depend

on a Grassmannian Õariable, and expand oÕer thisÕariable.


This gives renormalization constants for the softterms. Differentiating them with respect to a scaleone can find corresponding renormalization groupequations.

Thus the soft term renormalizations are not inde-pendent but can be calculated from the known renor-malizations of a rigid theory with the help of thedifferential operators. Explicit form of these opera-tors has been found in a general case and in someparticular models like SUSY GUTs or the MSSMw x1 . The same expressions were obtained also in Ref.w x4 .

In this letter we demonstrate that this procedureworks at all stages. One can make the above men-tioned substitution on the level of the renormaliza-tion constants, RG equations, solutions to these equa-


( )D.I. KazakoÕrPhysics Letters B 449 1999 201–206202

tions, approximate solutions, fixed points, finitenessconditions, etc. Expanding then over a Grassmannianvariable one obtains corresponding expressions forthe soft terms. This way one can get new solutions ofthe RG equations and explore their asymptotics, orapproximate solutions, or find their stability proper-ties, starting from the known expressions for a rigidtheory.

Below we give some examples and in particularconsider the MSSM with low tanb , where analyticalsolutions are known. We show how one can easilyobtain solutions to the RG equations for the softmass terms much simpler than are known in theliterature. In a finite SUSY GUT finiteness condi-tions for the soft terms appear as a trivial conse-quence of a finiteness of a rigid theory. Anotherexample is the Ns2 SUSY model, where the exactŽ .non-perturbative Seiberg-Witten solution is known.Here one can extend the S-W solution to the softterms.

2. Soft SUSY breaking and renormalization

Consider an arbitrary Ns1 SUSY gauge theorywith unbroken SUSY. The Lagrangian of a rigidtheory is given by

1 12 a 2 aLL s d u TrW W q d u TrW WH Hrigid a a2 24 g 4 g

j2 2 i Vq d u d u F e FŽ .H i j

2 2q d u WWq d u WW , 1Ž .H Hwhere W a is the field strength chiral superfield andthe superpotential WW has the form

1 1i jk i jWWs l F F F q M F F . 2Ž .i j k i j6 2

To perform the SUSY breaking, which satisfiesthe requirement of ‘‘softness’’, one can introduce agaugino mass term as well as cubic and quadraticinteractions of scalar superpartners of the matter

w xfields 2

M1 1i jk i jyLL s llq A f f f q B f fsoft - breaking i j k i j6 22

i2 ) jqh.c. q m f f , 3Ž . Ž .j i

where l is the gaugino field and f is the loweri

component of the chiral matter superfield.Ž .One can rewrite the Lagrangian 3 in terms of

Ns1 superfields introducing the external spurion2 2w xsuperfields 2 hsu and hsu , where u and u

w xare Grassmannian parameters, as 3

12 2 aLL s d u 1y2 Mu TrW WŽ .Hsoft a24 g

12 2 aq d u 1y2 Mu TrW WŽ .H a24 g

k j2 2 i k 2 Vq d u d u F d y m hh e FŽ . Ž .H ž /i ki j

12 i jk i jkq d u l yA h F F FŽ .H i j k6

1 i j i jq M yB h F F qh.c. 4Ž .Ž . i j2

Ž . Ž .Comparing Eqs. 1 and 4 one can see that Eq.Ž . Ž .4 is equivalent to 1 with modification of the rigidcouplings g 2,li jk and M i j, so that they becomeexternal superfields dependent on Grassmannian pa-

2 2 2rameters u and u . The scalar mass term m hh

modifies fields F and F . These modifications of thecouplings and fields are valid not only for the classi-cal Lagrangian but also for the quantum one. 2 As

w xhas been shown in Ref. 1 the following statement isvalid:

Ž .If a rigid theory 1, 2 is renormalized Õia intro-duction of renormalization constants Z , definedi

within some minimal subtraction massless scheme,Ž .then a softly broken theory 4 is renormalized Õia

˜introduction of renormalization superfields Z whichi

are related to Z by the coupling constants redefini-i

tion

2 2 ˜˜ ˜Z g ,l,l sZ g ,l,l , 5Ž .˜Ž . ž /i i

where the redefined couplings are

2 2g sg 1qMhqMhq2 MMhh ,˜ ž /2 2hsu , hsu , 6Ž .

2 Throughout the paper the existence of some SUSY invariantregularization is assumed.

( )D.I. KazakoÕrPhysics Letters B 449 1999 201–206 203

i j1i jk i jk i jk n jk 2 i nk 2l sl yA hq l m ql mŽ . Ž .ž n n2

ki jn 2ql m hh , 7Ž . Ž ./n

n n1 2 2l sl yA hq l m ql mŽ . Ž . jiži jk i jk i jk n jk ink2

n2ql m hh . 8Ž . Ž .k /i jn

Thus a softly broken SUSY gauge theory isequivalent to an unbroken one in external spurionsuperfield as far as the renormalization properties are

Ž . Ž .concerned. Substitutions 6 – 8 can be made notonly in the renormalization constants, but at everystage of the renormalization procedure, since the RGfunctions and RG equations are derived from renor-malization constants applying the differential opera-tors. The key point is that one can consider anunbroken theory in external superfield which isequivalent to replacing of the couplings by external

Ž . Ž .superfields according to Eqs. 6 – 8 . Then one canexpand over Grassmannian parameters.

In what follows we would like to simplify thenotations and consider numerical rather than tenso-rial couplings. When group structure and field con-tent of the model are fixed, one has a set of gauge� 4 � 4g and Yukawa y couplings. It is useful toi k

consider the following rigid parameters

g 2 y2i k

a ' , Y ' .i k2 216p 16p

Ž . Ž .Then Eqs. 6 – 8 look like

a sa 1qM hqM hq2 M M hh , 9Ž .˜ ž /i i i i i i

Y sY 1qA hqA hq A A qS hh , 10Ž .Ž .ž /k k k k k k k

where to standardize the notations we have redefinedthe A parameter A™Ay in a usual way and havechanged the sign of A to match it with the gauge softterms. S stands for a sum of m2 soft terms, one fork

each leg in the Yukawa vertex.Now the RG equation for a rigid theory can be

written in a universal form

� 4a sa g a , a s a ,Y , 11Ž . Ž .˙i i i i i k

Ž .where g a stands for a sum of correspondingi

anomalous dimensions. In the same notation the softŽ .terms 9,10 take the form

a sa 1qm hqm hqS hh , 12Ž .˜ Ž .i i i i i

� 4 � 4where m s M , A and S s 2 M M , A A qS .i i k i i i k k k

3. Grassmannian Taylor expansion

We demonstrate now how the RG equations forthe soft terms appear via Grassmannian Taylor ex-

Ž .pansion from those for the rigid couplings 11 .Ž . Ž .Indeed, let us substitute Eq. 12 into Eq. 11 and

expand over h and h. One has to be careful, how-Ž .ever, since as it follows from the soft Lagrangian 4

gauge couplings are involved in chiral Grassmannintegrals and expansion over h or h makes sense upto F-terms only. On the contrary, the Yukawa cou-plings Y, being a product of y and y, are generalsuperfields, so the expansion is valid for D-terms aswell. Having this in mind one gets

a sa g a , 13Ž . Ž .˜ ˜ ˜i i i

Consider first the F-terms. Expanding over h one has

<a m qa m sa m g a qa g a , 14Ž . Ž . Ž .˙ ˙ ˜ Fi i i i i i i i i

or

Egim sg a s a m . 15Ž . Ž .˙ ˜ Ýi i j jF E ajj

This is just the RG equation for the soft terms Miw xand A which was written in Ref. 1 in the formk

m sD g a . 16Ž . Ž .˙ i 1 i

Proceeding the same way for the D-terms one getsafter some algebra

Eg Egi i˙ <S sg a s2m a m q a SŽ .˜ Ý ÝDi i i j j j jE a E aj jj j

E 2g iq a a m m . 17Ž .Ý j k j kE a E aj kj,k

Substituting S sm m qS one has the RG equa-i i i i

tion for the mass terms

Eg E 2gi iS s a m m qS q a a m m .Ž .Ý Ýi j j j j j k j kE a E a E aj j kj j,k

18Ž .

One can also obtain the RG equation for theindividual soft masses out of field renormalization.


Consider the chiral Green function in a rigid theory.It obeys the following RG relation

t X X² : ² :F F s F F exp g a t dt . 19Ž . Ž .Ž .0 Hi i i i iž /0

Making the substitution

2² : ² :F F ™ F F 1qm hh , a™a,˜Ž .i i i i i

Žand expanding over hh since it stands under the full.Grassmann integral only D-term contributes one has

t X X2 2m sm q dt g a t . 20Ž . Ž .Ž .˜Hi i0 i D0

Differentiating this relation with respect to t leads toP2m sD g a , 21Ž . Ž .2 ii

where D stands for a second order differential2Ž . w xoperator 17 introduced in Ref. 1 .

As mentioned above one can make the sameexpansion not only in equations, but also in solu-tions. Let us start with the simplest case of puregauge theory with one gauge coupling. Then one hasin a rigid theory

aX 2da Q

s log . 22Ž .H X 2ž /b a LŽ .˜ 2ŽMaking a substitution a™a and LsL 1qcu q˜

.... one has

daX Q2

a

s log . 23Ž .H X 2ž /˜b aŽ . L

Expansion over h gives

b aŽ .Mscg a , g a s . 24Ž . Ž . Ž .

a

One can make the same expansion for any ana-lytic solution in a rigid theory. Below we considerthree particular examples, namely the MSSM, thefinite SUSY GUT and the Seiberg-Witten Ns2SUSY model.

4. Examples

4.1. The MSSM

Consider the MSSM in low tanb regime. One hasthree gauge and one Yukawa coupling. The one-loop

w xRG equations are 5332a syb a , b s ,1,y3 , is1,2,3, 25Ž .˙ Ž .i i i i 5

16 13Y sY a q3a q a y6Y , 26Ž .Ž .t t 3 2 1 t3 15

Ž . Ž .with the initial conditions: a 0 sa , Y 0 sYi 0 t 0Ž 2 2 . w xand ts ln M rQ . Their solutions are given by 5X

a Y E tŽ .0 0a t s , Y t s , 27Ž . Ž . Ž .i t1qb a t 1q6Y F tŽ .i 0 0

wherec rb 13 16i iE t s 1qb a t , c s ,3, ,Ž . Ž . Ž .Ł i 0 i 15 3

i

t X XF t s E t dt .Ž . Ž .H0

To get the solutions for the soft terms it is enough˜to perform substitution a™a and Y™Y and ex-˜

pand over h and h. Expanding the gauge coupling inŽ . Ž .27 up to h one has hereafter we assume M sMi0 0

a M a b a M t0 0 0 i 0 0a M s yi i 21qb a t 1qb a tŽ .i 0 i 0

a M0 0s ,

1qb a t 1qb a ti 0 i 0

orM0

M t s . 28Ž . Ž .i 1qb a ti 0

Performing the same expansion for the Yukawa cou-pling and using the relations

˜ ˜dE dE dFsM t , sM tEyF ,Ž .0 0dh dt dh

h h

w xone finds a well known expression 5

A t dE0A t s qMŽ .t 0 ž1q6Y F E dt0

6Y0y tEyF . 29Ž . Ž ./1q6Y F0

To get the solution for the S term one has to makeexpansion over h and h. This can be done with thehelp of the following relations

2 ˜d E d dE2 2sM t ,0 ž /dhdh dt dt

h ,h

2 ˜d F dE2 2sM t .0dhdh dt

h ,h

( )D.I. KazakoÕrPhysics Letters B 449 1999 201–206 205

This leads to

22S yA A yM 6Y tEyFŽ .Ž .0 0 0 0 0S t s qŽ .t 21q6Y F 1q6Y FŽ .0 0

2d t dE 6Y dE02 2qM y t ,0 ž /dt E dt 1q6Y F dt0

30Ž .

which is much simpler than known in the literaturew x5 , though coinciding with it after some cumber-some algebra.

Ž .With analytic solutions 29,30 one can analyzeasymptotics and, in particular, find the infrared quasi

w xfixed points 6 which correspond to Y ™`0

EFPY s , 31Ž .

6F

t dE tEyFFPA sM y , 32Ž .0 ž /E dt F

2 2 2tEyF d t dE t dEFP 2S sM q y .0 ž / ž /F dt E dt F dt

33Ž .

However, the advantage of the Grassmannian expan-sion procedure is that one can perform it for fixed

Ž .points as well. Thus the FP solutions 32,33 can bedirectly obtained from a fixed point for the rigid

Ž .Yukawa coupling 31 by Grassmannian expansion.This explains, in particular, why fixed point solu-tions for the soft couplings exist if they exist for the

w xrigid ones and with the same stability properties 7 .

4.2. SUSY GUTs

One can consider not only fixed points, but alsomore complicated configurations like renormaliza-tion invariant trajectories which lead to reduction of

w x w xthe couplings 8 or fixed lines or surfaces 9 , orw xfiniteness relations 10 . The same procedure is valid

here as well.Let us consider, for example, construction of a

Ž .finite theory free from ultraviolet divergences inthe framework of SUSY GUTs. It is achieved in arigid theory by a proper choice of the field content

and of the Yukawa couplings being the functions ofw xthe gauge one 10,11

Y sY a scŽk .aqcŽk .a 2 q ..., 34Ž . Ž .k k 0 1

where coefficients cŽk . are calculated within pertur-i

bation theory.To achieve complete finiteness, including the soft

terms, one has to choose the latter in a proper wayw x12 . To find it one just have to modify the finiteness

Ž .relation for the Yukawa coupling 34 as

Y sY a , 35Ž . Ž .˜k k

and expand over h and h. This gives:

dlnYkA sM , 36Ž .k dlna

and after the rearrangement of terms

d dlnYk2 2S sM a , 37Ž .k da da

w xwhich coincides with the relations found in Ref. 12 .

4.3. Ns2 SUSY

Consider now the Ns2 supersymmetric gaugetheory. The Lagrangian written in terms of Ns2

w xsuperfields is 13 :

112 2 2˜LLs IImTr d u d u tC , 38Ž .H 24p

˜Ž .where Ns2 chiral superfield C y,u ,u is defined˜by constraints D Cs0 and D Cs0 anda a˙ ˙

topological4p uts i q . 39Ž .2 2pg

˜The expansion of C in terms of u can be writtenas

Ž1. a Ž2.˜ ˜'C y ,u ,u sC y ,u q 2 u C y ,uŽ . Ž .Ž . a

˜a˜ Ž3.qu u C y ,u ,Ž .a

m m m m Žk .˜ ˜ Ž .where y sx q ius uq ius u and C y,u areNs1 chiral superfields.


The soft breaking of Ns2 SUSY down to Ns1can be achieved by shifting the imaginary part of t :

4p2˜t™tstq i u M . 40Ž .˜ 2g

This leads to

1 M 22 Ž1.D LLs Tr d u C , 41Ž . Ž .H2 2g

which is the usual mass term for Ns1 chiral super-field C Ž1. normalized to 1rg 2.

Now one can use the power of duality in Ns2SUSY theory and take the Seiberg-Witten solutionw x14

da daDts , 42Ž .

du du

where

i 1yua u s uy1 F 1r2,1r2,2; ,Ž . Ž .D ž /2 2

2(a u s 2 1qu F y1r2,1r2,1; .Ž . Ž . ž /1qu

Assuming that renormalizations in Ns2 theoryfollow the properties of those in Ns1 one canapply the same expansion procedure. Substituting

˜2Ž . Ž . Ž .Eq. 40 into 42 with u™usu 1qM u and˜ 0˜ 2expanding over u , one gets an analog of S-W

solution for the mass term:XX XXa aD

IIm u y tX Xž /a aDMsM . 43Ž .0

IImt

Ž 2 2 .In perturbative regime u;Q rL ™` one hasi'w x Ž .13 as 2u , a s a 2lnaq1 , which leads toD p

4p 12 2s lnQ rL q3 ,2 pg

2 2MsM r lnQ rL q3 .0

This procedure can be continued introducing softNs1 SUSY breaking via u dependent t superfield.Thus one can achieve soft SUSY breaking along thechain

Ns2 ´ Ns1 ´ Ns0

preserving the properties of the exact solution. This

will lead to a sequence of new solutions for the softŽ .terms like Eq. 43 .

5. Conclusion

We conclude that the Grassmannian expansion insoftly broken SUSY theories happens to be a veryefficient and powerful method which can be appliedin various cases where the renormalization procedurein concerned. It demonstrates once more that softlybroken SUSY theories are contained in rigid onesand inherit their renormalization properties.

Acknowledgements

Financial support from RFBR grants a 96-02-17379a and a 96-15-96030 and DFG grant a 436RUS 113r335 is kindly acknowledged.

References

w x1 L.A. Avdeev, D.I. Kazakov, I.N. Kondrashuk, Nucl. Phys. BŽ .510 1998 289, hep-phr9709397.

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phr9709364.w x5 A. Brignole, L.E. Ibanez, C. Munoz, Nucl. Phys. B 422´˜ ˜

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Ž .J.H. Schwarz, Phys. Lett. B 141 1984 349.w x11 A.V. Ermushev, D.I. Kazakov, O.V. Tarasov, Nucl. Phys. B

Ž . Ž .281 1987 72; D.R.T. Jones, Nucl. Phys. B 277 1986 153;Ž .C. Lucchesi, O. Piguet, K. Sibold, Phys. Lett. B 201 1988

Ž .241; D.I. Kazakov, Mod. Phys. Lett. A 9 1987 663.w x Ž .12 D.I. Kazakov, Phys. Lett. B 421 1998 211, hep-

phr9709465.w x13 See e.g. L. Alvarez-Gaume, S.F. Hassan, Fortsch. Phys. 45

Ž .1997 159, hep-thr9701069.w x Ž .14 N. Seiberg, E. Witten, Nucl. Phys. B 426 1994 19.

11 March 1999


Impacts of the CP-odd phase of the chargino mass matrixin light chargino-pair production at LEP II

S.Y. Choi a,1, J.S. Shim b,2, H.S. Song c,3, W.Y. Song c,4

a Department of Physics, Yonsei UniÕersity, Seoul 120-749, South Koreab Department of Physics, Myong Ji UniÕersity, Yongin 449-728, South Korea

c Center for Theoretical Physics and Department of Physics, Seoul National UniÕersity, Seoul 151-742, South Korea

Received 15 August 1998; revised 30 November 1998Editor: H. Georgi

Abstract

One CP-odd rephase-invariant phase appears in the chargino mass matrix in the Minimal Supersymmetric StandardModel. We investigate in detail the phenomenological impacts of the CP-noninvariant phase in the pair production of lightcharginos in eqey annihilation. The values of the chargino masses and the mixing angles, determining the size of the winoand higgsino components in the chargino wave functions, are so sensitive to the phase that the constraints on thesupersymmetry parameters based on the conventional assumptions for the parameters are recommended to be re-evaluatedincluding the CP-noninvariant phase. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

Ž . w xThe Minimal Supersymmetric Standard Model MSSM 1 is a well-defined quantum theory of which theŽ .Lagrangian form is completely known, including the general R-parity preserving, soft supersymmetry SUSY

breaking terms.The full MSSM Lagrangian has 124 truly independent parameters – 79 real parameters and 45 CP-nonin-

w xvariant complex phases 2 . The number of parameters in MSSM is too large compared to 19 in the StandardŽ .Model SM . Therefore, many studies on possible direct and indirect SUSY effects have been made by making

w xseveral assumptions and investigating the variation of only a few parameters 3–5 . Recently, it has, however,w xbeen shown 6 that limits on sparticle masses and couplings are very sensitive to the assumptions and need to

be re-evaluated without making any of the simplifying assumptions that have been standard.w xDespite the large number of phases in the model as a whole, just one CP-odd rephase-invariant phase 7 ,

w xstemmed from the chargino mass matrix, takes part in chargino production 9 . In light of this aspect, the

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]


( )S.Y. Choi et al.rPhysics Letters B 449 1999 207–213208

analyses with the general parameter set in the chargino system are not so much more difficult than those withparameters assumed real. The CP-noninvariant phase may be constrained indirectly by the electron or neutron

w xelectric dipole moment 7 and may be small, but the constraints on its actual size depends strongly on thew xassumptions in those analyses. Moreover, as has been recently shown 8 , cancellations between different terms

contributing to the dipole moments can allow for values of the phases very different from zero even the sparticlemasses are relatively light. Consequently, unless there exist any concrete demonstrations for a small value of thephase, it will be more reasonable to take the CP-odd phase as a free parameter.

Charginos are produced in eqey collisions, either in diagonal or in mixed pairs. However, the second" w xchargino x is generally expected to be significantly heavier than the first state. At LEP2 10 , and potentially˜2

q y Ž w x. "even in the first phase of e e linear colliders see e.g. Ref. 11 , the chargino x may be, for some time, the˜1

only chargino state that can be studied experimentally in detail. In the present note, we will focus on thediagonal pair production of the light chargino x " in eqey collisions,˜1

eqey™xqxy˜ ˜1 1

and investigate in detail the phenomenological impacts of the CP-odd complex phase in the determinations ofthe relevant SUSY parameters in the production process.

The production of the light chargino-pair is completely described by the chargino mass x " , the cosines of˜1

two rotation angles determining the size of the wino and higgsino components in the charginos, and thesneutrino mass 5. First of all, in Section 2, we recapitulate the elements of the mixing formalism andquantitatively discuss the dependence of the chargino masses and the cosines of the two rotation angles on theCP-noninvariant phase. In Section 3 the cross section for chargino production along with the light chargino mass

< <is mapped over the parameter space, especially for the gaugino mass M and the higgsino mass parameter m ,2

by varing the CP-noninvariant phase. Then, we examine the phenomenological impacts of the CP-odd phase inconstraining the parameter space. Conclusions are given in Section 4.

2. Chargino masses and mixing angles

˜ " ˜ "In the MSSM, the spin-1r2 partners of the W boson and charged Higgs boson, W and H , mix toform chargino mass eigenstates x " . The mass eigenvalues m " and the mixing angles and phases are˜1,2 x1,2

˜y ˜yŽ . w xdetermined by the elements of the chargino mass matrix in the W , H basis 1

'M 2 m cosb2 WMM s , 1Ž .C 'ž /2 m sinb mW

which is built up by the fundamental SUSY parameters; the gaugino mass M , the higgsino mass parameter m,2

and the ratio tanbsÕ rÕ of the vacuum expectation values of the two neutral Higgs fields which break the2 1

electroweak symmetry. In CP-noninvariant theories, the gaugino mass M and the Higgs mass parameter m can2

be complex. However, by reparametrizations of the fields, M can be assumed to be real and positive without2w xloss of generality 7 so that the only non-trivial phase is attributed to m:

< < iFmms m e . 2Ž .

5 w xSneutrino inter-generational mixing is neglected. This will be a phenomenologically acceptable assumption because present bounds 12on the mixing from flavor-changing neutral current processes such as m™ eg strongly support that sneutrinos are almost degenerate, or elsemix only very weakly.

( )S.Y. Choi et al.rPhysics Letters B 449 1999 207–213 209

Ž .In these theories the complex chargino mass matrix Eq. 1 is diagonalized by two unitary matrices U and U ,L R

which can be parameterized in the following way:

cosf eyi bLsinfL LU s ,L i bLž /ye sinf cosfL L

ig cosf eyi bR sinf1 R Re 0U s , 3Ž .R ig i bž /2 Rž /0 e ye sinf cosfR R

and which render U MM U † diagonal. The two chargino mass eigenvalues are given byR C L

212 2 2< <"m s M q m q2m .D , 4Ž .x 2 W C2˜1,2

with D involving F :C m

22 22 4 2 2 2< < < < < <D s M y m q4m cos 2bq4m M q m q2 M m sin2bcosF . 5( Ž .Ž . ž /C 2 W W 2 2 m

The rotation angles f and f satisfy the relations for two cosines 6, cos2f and cos2f .L R L R

2 < < 2 2M y m y2m cos2b2 Wcos2f sy ,L

DC

2 < < 2 2M y m q2m cos2b2 Wcos2f sy . 6Ž .R

DC

� 4The four nontrivial phase angles b ,b ,g ,g are not independent but can be expressed in terms of the phaseL R 1 2w xangle F ; for their expressions we refer to the Appendix of the work 4 .m

Ž . Ž .Note that cosF in Eqs. 4 and 6 for the chargino masses, cos2F and cos2F appears along with am L R

unique combination factor:2tanb

< < < <M m sin2bsM m .2 2 21q tan b

This is a reflection of the fact that the CP-noninvariant phase angle F can be absorbed by field re-definitions ifm

at least one of the chargino mass matrix elements vanishes. In particular, when tanb is very small or very large,i.e. one of the two Higgs vacuum expectation values Õ and Õ is relatively very small 7, the effects of the phase1 2

angle F diminish. Keeping in mind that the CP-odd phase effects are very small for large tanb , we presentm

numerical analyses for a fixed value of tanbs2 in the following.Ž . Ž .The light and heavy chargino masses are presented in Fig. 1 a and b as a function of the cosine of the

CP-noninvariant phase angle F for a representative set of parameters. The parameters are chosen in them

< < < <higgsino-dominated region M 4 m , the gaugino-dominated region M < m and in the intermediate region2 2< <M ; m for tanbs2 as2

< <gaugino-dominated region: M , m s 80 GeV, 240 GeV ,Ž .Ž .2

< <higgsino-dominated region: M , m s 220 GeV, 100 GeV ,Ž . 7Ž . Ž .2

< <intermediate region: M , m s 100 GeV, 100 GeV .Ž .Ž .2

The two masses are very sensitive to the phase angle F in all scenarios; the light chargino mass is morem

sensitive to the phase angle than the heavy chargino mass, and the sensitivity is most prominent in the

6 � 4The expressions for sin2f ,sin2f are not presented because they are not involved in the diagonal pair production of the light orL R

heavy charginos as shown in the next Section.7 2 2Two vacuum expectation values Õ and Õ can not be larger than Õs Õ q Õ f250 GeV.(1 2 1 2


Ž . Ž . Ž . Ž .Fig. 1. a the light chargino mass, b the heavy chargino mass, c cos2f , and d cos2f as a function of the cosine of the CP-violatingL RŽ .phase angle F for the representative set of SUSY parameters in Eq. 7 : solid line for the gaugino-dominated case, dashed line for them

higgsino-dominated case, and dot-dashed line for the intermediate case.

Ž . Ž .intermediate scenario. Fig. 1 c and d exhibit cos2f and cos2f as a function of cosF . Similarly, both ofL R m

them depend more strongly on the CP-violating phase angle F in the intermediate scenario than in them

gaugino-dominated and higgsino-dominated scenarios.

3. Production cross-section

The process eqey™xyxq is generated by the three mechanisms: s-channel g and Z exchanges, and˜ ˜1 1

t-channel n exchange. The transition matrix element, after a Fierz transformation of the n-exchange amplitude,˜ ˜e2

q y q y q y y m qT e e ™x x s Q Õ e g P u e u x g P Õ x , 8Ž . Ž . Ž .Ž . Ž .˜ ˜ ˜ ˜Ž .1 1 a b m a 1 b 1s

can be expressed in terms of four bilinear charges, classified according to the chiralities a ,bsL, R of theassociated lepton and chargino currents

DZ 1 3 12 2Q s1q s y s y y cos2f ,Ž . Ž .L L W W L2 4 42 2s cW W

D DZ n1 3 12 2Q s1q s y s y y cos2f q 1qcos2f ,Ž .Ž . Ž .L R W W R R2 4 42 2 2s c 4 sW W W

DZ 3 12Q s1q s y y cos2f ,Ž .R L W L4 42cW

DZ 3 12Q s1q s y y cos2f . 9Ž .Ž .R R W R4 42cW


The first index in Q refers to the chirality of the e" current, the second index to the chirality of the xy˜ab 1� 4Ž� 4. Ž .current. The bilinear charges Q ,Q Q ,Q with right left chargino chirality depend only onR R L R R L L L

Ž .cos2f cos2f so that the dependence of the production amplitude on the CP-noninvariant phase F is easilyR L m

readable. The n exchange affects only the LR chirality charge while all other amplitudes are built up by g and˜Ž 2 . Ž 2Z exchanges. D denotes the sneutrino propagator D ssr tym , and D the Z propagator D ssr symn n n Z Z Z˜ ˜ ˜

.q im G ; the non-zero width can in general be neglected for the energies considered in the present analysis soZ Z

that the charges are real.Certainly, the production cross section strongly depends on the chargino mass that is very sensitive to the

CP-noninvariant phase as shown in Section 2. Let us show the chargino pair cross section for a fixed charginomass of 90 GeV such that we can clarify what extent the variation of the cross section simply reflects thedependence of the chargino mass of the phase F . For a given tanb , the chargino mass is determined by them

< <three parameters, M , m , and cosF . Here, we investigate the dependence of the production cross section on2 m

< < Ž . Ž .cosF by restricting the ratio M r m to three values, 1r3 gaugino-dominated , 3 higgsino-dominated and 1m 2Ž .intermediate , while varing their absolute values.

Ž .Fig. 2 a shows the production cross section as a function of cosF for a fixed sneutrino mass of 200 GeVm

and a fixed chargino mass of 90 GeV at a c.m. energy of 200 GeV. The solid line is for the gaugino-dominatedcase, the dashed line for the higgsino-dominated case and the dot-dashed line for the intermediate case. Like thechargino masses and the rotation angles, the cross section for a fixed chargino mass is most sensitive to the

Ž .CP-noninvariant phase in the intermediate case. On the other hand, Fig. 2 b shows the production cross sectionas a function of the sneutrino mass for a fixed chargino mass of 90 GeV at a c.m. energy of 200 GeV in theintermediate case. The solid line is for cosF sy1, the dashed line for cosF s0 and the dot-dashed line form m

cosF s1. We note that the production cross section grows up very sharply with cosF for small sneutrinom m

masses, and the sneutrino mass for the largest destructive interference in the production amplitude shifts to alower value. Prior or simultaneous determination of m will be therefore necessary to determine the phase andn

the other SUSY parameters.Ž < <.Let us consider the dependence of the excluded M , m parameter space on the phase, taking the chargino2

Ž .mass lower bound to be 90 GeV. The excluded region is explicitly presented in Fig. 3 a for three differentvalues of cosF with tanbs2; cosF s"1,0. As cosF increases, the covered parameter space is enlarged,m m m

but it will be not fully covered in actual experiments. Actually, the searchable parameter regime relies on thenumber of produced charginos and the reconstruction efficiency of the chargino signals determined from their

Ž .Fig. 2. The cross section for the production of light charginos a as a function of cosF with m s200 GeV for a given chargino mass ofm n'Ž . Ž .90 GeV, and b as a function of the sneutrino mass in the intermediate scenario at s s200 GeV; in a the solid line for theŽ .gaugino-dominated case, the dashed line for the higgsino-dominated case, and the dot-dashed line for the intermediate case, and in b the

solid line for cosF sy1, the dashed line for cosF s0, and the dot-dashed line for cosF s1.m m m


Ž . Ž < <. Ž ."Fig. 3. a the contours of the light chargino mass m s90 GeV on the M , m parameter space and b the minimal cross section sx 2 min˜1 'with m s200 GeV as a function of the light chargino mass at s s200 GeV for the three values of cosF ; the line with circle symbols forn m˜cosF sy1, the line with plus symbols for cosF s0 and the line with square symbols for cosF sq1.m m m

w x wdecay patterns 12 . In the case that the light chargino decays into the lightest neutralino, usually considered toŽ .xbe the lightest supersymmetric particle LSP , a fermion and an anti-fermion, the difference between the

chargino mass and the LSP mass plays a crucial role. Since the LSP mass also depends strongly on thew xCP-noninvariant phase F as well as an extra phase 6 , the determination of the upper limits of the charginom

cross section, which can be excluded at high energy colliders such as LEP II, will be rather involved, but ofcourse doable.

In the present work, we assume the mass difference between the light chargino and the LSP to be largeenough, and simply investigate how the CP-noninvariant phase affects the minimal production cross section as a

Ž .function of the chargino mass. Fig. 3 b shows the dependence of the minimal cross section s as a functionmin

of the chargino mass at a c.m. energy of 200 GeV for cosF s"1,0. Here, tanb and the sneutrino mass arem

taken to be 2 and 200 GeV, respectively. There exists an abrupt change of the minimal cross section aroundŽ < <."m s50 GeV, especially for a non-negative cosF . Scanning the M , m parameter space, we find that thisx m 2˜1

behaviour is due to the fact that the chargino mass and the cross section behave quite differently in theparameter space. On the other hand, the minimal cross section is of the order of pb in size except for thechargino mass close to 100 GeV and it increases with cosF . So, the experimental bound on the chargino massm

with a negative m parameter will be reasonably conservative. Neverthless, since the relation between theobservable quantities and the SUSY parameters is sensitive to the CP-noninvariant phase, the experimentalconstraints on the SUSY parameters based on the conventional assumptions for the parameters should bere-evaluated including the CP-noninvariant phase.

4. Conclusions

We have analyzed how the parameters of the chargino system, the masses of the charginos x " and the size˜of the wino and higgsino components in the chargino wave functions, parametrized in terms of the two anglesf and f , are affected by the CP-noninvariant phase F in the chargino mass matrix. In addition, we haveL R m

studied the dependence of the production cross section of the light chargino-pair in eqey collisions on thephase and have determined the dependence of the minimal cross section as a function of the chargino mass for

w xdifferent cosF values to check the validity of the reported experimental bounds on the chargino mass 12 .m

The chargino masses and the production cross section of the light chargino-pair in eqey collisions aresensitive to the CP-noninvariant phase. The sensitivities are most prominent around tanbs1 and in the


< <intermediate scenario with M ; m . The minimal cross section depends quite non-trivially on the chargino2

mass. Nevetheless, we find that the actual size of the cross section at LEP II is of the order of pb in size exceptfor the chargino mass close to the kinematical limit and increases with the cosine of the phase angle so that thereported chargino mass bounds for a negative m parameter can be considered to be authentic. However, weemphasize that it is crucial to include the possible CP-noninvariant phase F in probing parameters at LEP IIm

because of the strong dependence of the observable quantities on the CP-noninvariant phase.

Acknowledgements

This work was supported by Korean Science and Engineering Foundation in part through KOSEF-DFG largecollaboration project, Project No. 96-0702-01-01-2 and in part through the Center for Theoretical physics, SeoulNational University.

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Munich-Annecy-Hamburg 1991r93, DES 92-123AqB, 93-123C.w x Ž . Ž .8 T. Ibrahim, P. Nath, Phys. Lett. B 418 1998 98; Phys. Rev. D 57 1998 478; hep-phr9807591; M. Brhlik, G.J. Good, G.L. Kane,

hep-phr9810457.w x Ž .9 J. Ellis, J. Hagelin, D. Nanopoulos, M. Srednicki, Phys. Lett. B 127 1983 233; V. Barger, R.W. Robinett, W.Y. Keung, R.J.N.

Ž . Ž .Phillips, Phys. Lett. B 131 1983 372; D. Dicuss, S. Nandi, W. Repko, X. Tata, Phys. Rev. Lett. 51 1983 1030; S. Dawson, E.Ž . Ž .Eichten, C. Quigg, Phys. Rev. D 31 1985 1581; A. Bartl, H. Fraas, W. Majerotto, Z. Phys. C 30 1986 441.

w x Ž .10 G. Altarelli, T. Sjostrand, F. Zwirner Eds. , Proc. Workshop on Physics at LEP II, Report No. CERN-96-01,¨w x Ž .11 E. Accomando et al., Phys. Rept. 299 1998 1.w x Ž .12 Particle Data Group, C. Caso et al., Eur. Phys. J. C 3 1998 1.

11 March 1999


ž /Partial duality in SU N Yang-Mills theory

Ludvig Faddeev a,2, Antti J. Niemi a,b,3,4

a Helsinki Institute of Physics, P.O. Box 9, FIN-00014 UniÕersity of Helsinki,Helsinki, Finlandb The Mittag-Leffler Institute, AuraÕagen 17, S-182 62 Djursholm, Sweden¨


Abstract

Ž .Recently we have proposed a set of variables for describing the infrared limit of four dimensional SU 2 Yang-MillsŽ .theory. Here we extend these variables to the general case of four dimensional SU N Yang-Mills theory. We find that the

Ž . Ž .SU N connection A decomposes according to irreducible representations of SO Ny1 , and the curvature two form Fm mn

Ž .is related to the symplectic Kirillov two forms that characterize irreducible representations of SU N . We propose a generalclass of nonlinear chiral models that may describe stable, soliton-like configurations with nontrivial topological numbers.q 1999 Elsevier Science B.V. All rights reserved.

w xRecently 1 we have proposed a novel decompo-Ž .sition of the four dimensional SU 2 Yang-Mills

connection Aa . In addition of a three componentm

unit vector na, it involves an abelian gauge field Cm

and a complex scalar fsrq is . The fields C andm

Ž . Ž .f determine a U 1 multiplet under SU 2 gaugetransformations in the direction na,

Aa sC na qe E nbnc qrE na qse abc E nbncm m abc m m m

1Ž .

1 Permanent address: St. Petersburg Branch of Steklov Mathe-matical Institute, Russian Academy of Sciences, Fontanka 27, St.Petersburg, Russia.

2 Supported by Russian Fund for Fundamental Science. E-mail:[email protected] [email protected]

3 Permanent address: Department of Theoretical Physics, Upp-sala University, P.O. Box 803, S-75108, Uppsala, Sweden.

4 Supported by NFR Grant F-AArFU 06821-308. E-mail:[email protected]

In four dimensions this decomposition is complete inthe sense that it reproduces the Yang-Mills equations

w xof motion 1 . This is already suggested by thenumber of independent fields: if we account for theŽ . Ž .U 1 invariance, 1 describes Dq2 field degrees of

Ž .freedom. For Ds4 this equals 3 Dy2 , the num-ber of transverse polarization degrees of freedom

Ž .described by a SU 2 connection in D dimensions.Furthermore, if we properly specify the component

Ž . Ž .fields, 1 reduces to several known SU 2 fieldconfigurations. As an example, if we set A sC s0 m

a Žrsss0 and specify n to coincide with the sin-. a Ž .gular radial vector x rr, the parametrization 1

Ž .yields the singular Wu-Yang monopole configura-w xtion 2 . As another example, if we specify a rotation

symmetric configuration with C sCx i, and C , C,i 0a a Ž .f to depend on r and t only, and set n sx rr, 1

w xreduces to Witten’s Ansatz for instantons 3 .Ž . Ž .Here we wish to generalize 1 to SU N We

Ž .shall argue that exactly in four dimensions the SU N


( )L. FaddeeÕ, A.J. NiemirPhysics Letters B 449 1999 214–218 215

Yang-Mills connection admits the following decom-position

Aa sC i ma q f abc E mbmc qr i j f abc E mbmcm m i m i i m i j

qs i j dabc E mbmc 2Ž .m i j

With is1, . . . , Ny1 we label the Cartan subalge-bra. We shall construct the Ny1 mutually orthogo-

a Ž 2nal unit vector fields m with as1, . . . , N y1siw Ž .x .Dim SU n in the following so that they describe

Ž .N Ny1 independent variables. The combination

Aa sC i ma q f abc E mbmc 3Ž .m m i m i i

Ž . w xis the SU N Cho connection 4 , under the Ny1independent gauge transformations generated by theLie-algebra elements

a sa maT a 4Ž .i i i

i Ž .the vector fields C transform as U 1 connectionsm

C i ™C i qE am m m i

Ž .Ž 2 .Consequently these fields describe Dy2 N y1Žindependent variables. The f sr q is are N Ni j i j i j

.y1 independent complex scalars, mapped onto eachŽ . Ž .other by the U 1 gauge transformations 4 . As a

Ž i .consequence C ,f can be viewed as a collectionm i j

of abelian Higgs multiplets. We shall find that thefields r and s decompose according to the trace-i j i j

less symmetric tensor, the antisymmetric tensor, theŽ .vector and the singlet representations of SO Ny1 .

Ž .In D dimensions 2 then describes

N Ny1 q Dy2 Ny1 qN Ny1Ž . Ž . Ž . Ž .s2 N 2 q Dy4 Nq 2yDŽ . Ž .

independent variables. In exactly Ds4 this coin-Ž .Ž 2 .cides with Dy2 N y1 which is the number of

independent transverse variables described by aŽ . aSU N Yang-Mills connection A .m

We also observe that for Ds3, the number ofŽ .independent variables in 2 coincides with the di-

Ž .mension of the SU N gauge orbit, independently ofthe Yang-Mills equations of motion.

Ž .We note that a decomposition of the SU Nconnection has been recently considered by Periwalw x5 . It appears that his results are different from ours.

We proceed to the justification of the decomposi-Ž .tion 2 . For this we need a number of group theoret-

ical relations. Some of these relations seem to be

Ž .novel, suggesting that further investigations of 2could reveal hitherto unknown structures.

Ž .The defining representation of the SU N Liealgebra consists of N 2 y1 traceless hermitian N=Nmatrices T a with

1 i1a b ab abc c abc cT T s d q f T q d T22 N 2

normalized to

1a b a b abT ,T 'Tr T T s dŽ . Ž . 2

abc Ž .The f are completely antisymmetric and realabc Žstructure constants. The d are the completely

.symmetric coefficients

1 abc a b c a b c c b� 4d sTr T T ,T 'Tr T T T qT TŽ .Ž . Ž .2

and we note that in the defining representation wecan select the Cartan subalgebra so that

Ny1 2 Ny2Ž .i jk i jld d s d 5Ž .Ý k lNi , js1

For any four matrices A, B, C, D we have

w x w x� 4 � 4Tr A , B C , D q A ,C B , DŽw x � 4q A , D B ,C s0.

and

w x w x � 4 � 4Tr A , B C , D q A ,C B , DŽ� 4 � 4y A , D B ,C s0.

Ž a.bc abcHence, if we introduce the matrices FF s fŽ . Ž a.bcwhich define the adjoint representation and DD

sdabc, we have

w a b x abc cFF , DD syf DD 6Ž .and

2a b ab e e ad bc ac b dw xDD , DD s f FF q d d yd d 7Ž . Ž .cd cd N

Note that the DDa are traceless.We conjugate the Cartan matrices H of the defin-i

Ž .ing representation by a generic element ggSU N .This produces set of Lie algebra valued vectors

m smaT a sgH gy1 8Ž .i i i

an over-determined set of coordinates on the orbitŽ . Ž .Ny1 aSU N rU 1 : By construction the m depend oni

( )L. FaddeeÕ, A.J. NiemirPhysics Letters B 449 1999 214–218216

Ž .N Ny1 independent variables, they are orthonor-mal,

m ,m smama sdŽ .i j i j i j

and it is straightforward to verify that

w xm ,m s0 9Ž .i j

� 4 i jkm ,m sd m 10Ž .i j k

Tr m E m s m ,E m s0 11Ž .Ž . Ž .i m j i m j

We can also show that

yf acdmc f debme sd ab ymamb 12Ž .i i i i

is a projection operator. This can be verified e.g. byusing explicitly the defining representation of

Ž .SU N . The result follows since the weights ofŽ .SU N have the same length and the angles between

Ž .different weights are equal. Notice that 12 can alsoŽ .be represented covariantly, using the SU N permu-

tation operatord d b ay m ,T m m ,T sT mT ym mm 13Ž .i i i i

We now consider the Maurer-Cartan one-form

dggy1 sv T adx mam

We find

w xdm s v ,m 14Ž .i i

We now recall that each Cartan matrix H can beiŽ .used to construct a symplectic Kirillov two-form

Ž . Ž .Ny1on the orbit SU N rU 1 ,i y1 y1 iV sTr H g dg , g dg , dV s0 15Ž .Ž .i

i Ž .The V are related to the representations of SU N ;According to the Borel-Weil theorem each linearcombination of V i

n V i 16Ž .Ý ii

with integer coefficients corresponds to an irre-Ž .ducible representation of SU N .

Ž .When we combine the projection operator 12Ž .with the relation 14 and the Jacobi identity, we find

that these two-forms V i can be represented in termsof the m ,i

i w xV sTr m dm ,dm s m ,dm ndm 17Ž . Ž .Ž .i k k i k k

Explicitly, in components

V s f abcmaE mbE mc 18Ž .i ,mn i m k n k

Next, we proceed to consider a generic Lie alge-a a Ž .bra element ÕsÕ T . We define the infinitesimal

adjoint action d i of the m on Õ byi

i w xd Õs Õ ,m 19Ž .i

Ž .Using 12 and by summing over i we find that thisŽ .yields up to a sign a projection operator to a

subspace which is orthogonal to the maximal torusand is spanned by the m ,i

2id ÕsyÕqm m ,Õ 20Ž . Ž . Ž .i i

Ž .We shall also need a local basis of Lie-algebraŽ .valued one-forms in the subspace to which 20

Ž .projects. For this we first use 10 to conclude thatfor the symmetric combination

dm ,m q dm ,m sd dm 21� 4 � 4 Ž .i j j i i jk k

Ž .and using 5 we invert this into

Ndm s d dm ,m q dm ,m� 4 � 4Ž .k k i j i j j i2 Ny2Ž .

Ž .Consequently the symmetric combination 21 yieldsŽ .the SO Ny1 vector one-form

X i sX i dx m sE maT adx mm m i

Ns d dm ,m q dm ,m 22� 4 � 4 Ž .Ž .i jk j k k j2 Ny2Ž .

Ž .The antisymmetric combination yields a SO Ny1antisymmetric tensor one-form

Y i j sY i jdx m syY jidx ms dm ,m y dm ,m� 4 � 4m m i j j i

23Ž .

sdabc E mamb yE mamb T cdx m 24Ž .Ž .m i j m j i

Ž .Finally, we define the SO Ny1 symmetric tensorone-form

i j i j m ji mZ sZ dx sZ dx s dm ,mm m i j

s f abc E mamb T cdx m 25Ž .m i j

Ž .Under SO Ny1 this decomposes into the tracelesssymmetric tensor representation and the trace i.e.singlet representation, but we use it as is.

Ž . Ž . Ž .Observe that that 22 , 24 and 25 are the onlyinvariant one-forms that can be constructed using the

Ž .variables m and natural SU N invariant concepts.i

In particular, X i, Y i j and Z i j are orthogonal to mi,hence they determine a basis in the corresponding

( )L. FaddeeÕ, A.J. NiemirPhysics Letters B 449 1999 214–218 217

Ž . Ž .Ny1subspace SU N rU 1 . We can identify them asŽ .a basis of roots in SU N .

Ž .The dimension of the SO Ny1 vector represen-Ž .tation 22 is Ny1. The dimension of the antisym-

1Ž . Ž .Ž .metric tensor representation 24 is Ny1 Ny2 .2

The sum of these coincides with the dimension ofŽ .25 ,

1 1Ny1q Ny1 Ny2 s N Ny1Ž . Ž . Ž .2 2

Ž . Ž .Moreover, we find that the U 1 generators 19 mapX i and Y i j into Z i j and Õice Õersa,m m m

d iX j sZ i j 26Ž .d i Y jk sdik lZ jl ydi jlZk l 27Ž .and

1i jk k jd Z sy d X qd XŽ .i j i kN

1 nq d d yd d yd d XŽ .jk l l i n ji l lk n k i l l jn4

1 i l k l jlq d Y qd Y qd Y 28Ž .Ž .jk l ji l k i l4

We note that this determines a natural complexstructure.

We have now constructed four different sets ofŽ . Ž .SU N Lie-algebra valued forms in m from the m .i

Each of these four sets induces an irreducible repre-Ž .sentation of SO Ny1 , they decompose into the

vector, the antisymmetric tensor, and the tracelesssymmetric tensor plus scalar representations of

Ž .SO Ny1 . The cotangent bundle to the co-adjointŽ . Ž .Ny1orbit SU N rU 1 is spanned by the one-forms

Ž . i i j i jin m X , Y and Z .Ž i i j i j.By construction m , X ,Y ,Z yields a com-i

Ž .plete set of basis states for the SU N Lie algebra,Ž .and can be used to decompose generic SU N con-

nections. For this we need appropriate dual variablesthat appear as coefficients. We first note that the

a Ž .connection A is a SU N Lie-algebra valued one-m

Ž .form, and the SO Ny1 acts on it trivially. Conse-quently the variable which is dual to m must be ai

Žone-form which transforms as a vector under SO N. iy1 . We call it C . The variables which are dual toŽ i i j i j.the X ,Y ,Z are zero-forms, and in order to form

invariant combinations they must decompose underŽ . ithe action of SO Ny1 in the same manner as X ,

Y i j and Z i j. Since we have also found a naturalcomplex structure which is determined by the d i,this suggests that we denote these dual variables by

f i j sr i j q is i j. Here r i j is dual to the Z i j and canbe decomposed into a traceless symmetric tensor and

Ž . i ja singlet under SO Ny1 . The s is dual to theX i and Y i j. It decomposes into a vector and an

Ž . Ž Žantisymmetric tensor under SO Ny1 . The SO N. . Ž . ay1 invariant SU N connection A then decom-m

poses into

w x � 4AsCPmq 1qr dm ,m qs dm ,m 29Ž . Ž .Exactly in four dimensions this contains the correct

Ž .number of independent variables for a general SU Nconnection.

The ensuing curvature two form FsdAqAA isobtained by a direct substitution. Of particular inter-est is the structure of F in the direction m , thei

maximal torus,

F a sma E C i yE C i ymaV i q ,... 30Ž .Ž .mn i m n n m i mn

i Ž .Here the V are the Kirillov two forms 17 , theterms that we have not presented explicitly depend

Ž .on f andror are in the direction of SU N ri jŽ .Ny1 ŽU 1 . If we evaluate the curvature two form for

Ž . Ž .the SU N Cho connection 3 , we find exactly theŽ . .terms in 30 . Consequently F is a generating func-

tional for the Kirillov two forms. In particular, for aflat connection we have

dC i sV i

w xIn Ref. 1 , using Wilsonian renormalization groupŽ . Ž .arguments we suggested that for SU 2 1 the fol-

lowing action

12 24Ss d x E n q n ,dnndn 31Ž . Ž .Ž .H m 2e

may be relevant in the infrared limit of Yang-MillsŽ .theory. This is interesting since 31 describes knot-

Ž .like configurations as solitons. The self linking ofthese knots is computed by the Hopf invariant

Qs FnA 32Ž .Hwhere

FsdAs n ,dnndnŽ .The present construction suggests a natural gener-

Ž . Ž .alization of 31 to SU N ,

1 224Ss d x E m q E m ,E m 33Ž .Ž . Ž .H m i m i n i2ei

( )L. FaddeeÕ, A.J. NiemirPhysics Letters B 449 1999 214–218218

Ž . Ž Ž .which reduces to 31 for Ns2. Since p SU N r3Ž .Ny1.U 1 sZ we expect that in the general case ofŽ .SU N we also have solitons. It would be interesting

to understand their detailed structure.Ž .As an example we consider SU 3 , the gauge

Ž .group of strong interactions. There are two SU 3Lie-algebra valued vectors ma which we denote byi

ma and na respectively. We have m2 sn2 s1 anda a Ž .m n s0, and 2 becomes

Aa sB ma qC na q f abc E mbmc q f abc E nbncm m m m m

34Ž .

qr f abc E mbmc qr f abc E mbncm m m m n m

qr f abc E nbnc 35Ž .nn m

qs E ma qs E na qs dabc E mbnc 36Ž .m m m nn m m n m

Ž . Ž . Ž .Here 34 is the SU 3 Cho connection, and 35 ,Ž . a36 are the components of A in the direction of them

Ž . Ž . Ž .orbit SU 3 rU 1 =U 1 . These components trans-form into each other under the action of the opera-tors

w x w xd s Ø ,m , d s Ø ,n .m n

In the Gell-Mann basis the vectors m and n satisfy

1 1w x � 4 � 4m ,n s0; m ,m s n; n ,n sy n;' '3 3

1� 4m ,n s m'3

Notice that n is represented in terms of m, a unitvector with six independent field degrees of freedom

Ž . Ž . Ž .to parametrize SU 3 rU 1 =U 1 .Ž . Ž . Ž .The action of d and d on SU 3 rU 1 =U 1m n

can be diagonalized. We find that d corresponds tom

the adjoint action of the Cartan element l and d to3 n

the adjoint action of the Cartan element l . This8Ž . Ž . Ž .decomposes the SU 3 rU 1 =U 1 components of

a Ž . 2 2A into U 1 multiplets. In particular, d qd is am m n

projection operator onto the basis one-forms ofŽ . Ž . Ž .SU 3 rU 1 =U 1 . Finally, the Kirillov symplectic

two-forms are

V s f abcma E mb E mc qE nb E ncŽ .m m n m n

and

V s f abcna E mb E mc qE nb E ncŽ .n m n m n

Ž .and the action 33 is

1 22 24 abc b cSs d x E m q E n q f E m E mŽ . Ž . Ž .H m m m n2ž em

1 2abc b cq f E n E nŽ .m n2 /en

In conclusion, we have presented a group theoret-Ž .ical decomposition of four dimensional SU N con-

nection Aa , which we argue is complete in the sensem

w xdescribed in Ref. 1 . This decomposition involves anumber of natural group theoretical concepts, and in

Ž .particular relates the SU N curvature two form toŽ .the Kirillov symplectic two forms on the SU N

coadjoint orbits. Curiously, we also find that thecomponents of A in the direction of these orbits canbe decomposed according to irreducible representa-

Ž .tions of SO Ny1 , with a natural complex struc-ture. Our construction suggests a new class of non-linear models generalizing the model first proposed

w xin Ref. 6 . These models may have interesting prop-erties, including the possibility of solitons with non-

w xtrivial topological structures 7 .

Acknowledgements

We thank W. Kummer, J. Mickelsson and M.Semenov-Tyan-Shansky for discussions and com-ments. We also thank the the Erwin Schrodinger¨Institute for hospitality.

References

w x1 L. Faddeev, A.J. Niemi, hep-thr9807069.w x Ž .2 T.T. Wu, C.N. Yang, in: H. Mark, S. Fernbach Eds. , Proper-

ties of Matter Under Unusual Conditions, Interscience, NewYork, 1969

w x Ž .3 E. Witten, Phys. Rev. Lett. 38 1977 121.w x Ž . Ž .4 Y.M. Cho, Phys. Rev. D 21 1980 1080; D 23 1981 2415;

Ž .Phys. Rev. Lett. 44 1980 1115.w x5 V. Periwal, hep-thr9808127.w x6 L. Faddeev, Quantisation of solitons, preprint IAS Print-75-

Ž .QS70, 1975; M. Pantaleo, F. De Finis Eds. , Einstein andseveral contemporary tendencies in the field theory of elemen-tary particles in Relativity, quanta and cosmology, vol. 1,Johnson Reprint, 1979.

w x Ž .7 L. Faddeev, A.J. Niemi, Nature 387 1997 58.

11 March 1999


A bound on violations of Lorentz invariance

R. Cowsik a,b,1, B.V. Sreekantan c

a Indian Institute of Astrophysics, Bangalore 560 034, Indiab Tata Institute of Fundamental Research, Homi Babha Road, Mumbai 400005, India

c National Institute of AdÕanced Studies, Bangalore 560 012, India


Abstract

w Ž .Recently Coleman and Glashow Phys. Lett. B 405 1997 249; Harvard University Theoretical Physics PreprintŽ .x98rAO76 pvt. comm have developed a model which allows the introduction of a small violation of Lorentz invariance.

Observational signatures arise because this interaction also violates flavor conservation and allows the radiative decay of themuon, m™eqg , whose branching ratio increases as bg 4 where g is the Lorentz factor of the muon with respect to thereference frame in which the dipole anisotropy of the universal microwave radiation vanishes. In this paper we place a boundon the Lorentz invariance violating parameter, b, of b-10y25 based on observations of horizontal air showers withn G5=106. With such small values of b the proposed radiative decay of the muon will not affect the functioning of thee

muon collider. q 1999 Elsevier Science B.V. All rights reserved.

To test by experiments the limits of validity ofLorentz invariance or indeed any of the fundamentalprinciples of physics we need a theoretical modelwhich assumes a specific form for the violation andmakes predictions of physical phenomena which can

w xbe searched for by the experiments 2–5 . The recentmodel of Coleman and Glashow incorporates tinydepartures from Lorentz invariance which does not

w xrespect flavor conservation also 1 . One of the signa-tures of such a flavor non conservation is the transi-tion m™eqg whose rate increases rapidly withthe energy of the muon as measured in a preferredframe such as the one in which the 2.7 K universalmicrowave background does not have any dipole


anisotropy. Following their suggestion we calculatethe possible contributions of such a process to theflux of ‘‘horizontal air showers’’ and m-less showerswhich provide useful estimates for the possiblestrength of such an interaction and also provide agood bound on such violations.

The idea on which the bound on flavor violatinginteractions is derived becomes clear by noting thatthe primary cosmic rays consist mainly of nucleiwhich interact strongly when they are incident on thetop of the earth’s atmosphere. The amount of shield-ing provided by the atmosphere in the vertical direc-tion above the earth is about 1000 g cmy2 andincreases as the secant of the zenith angle u upto;808. The total grammage in the horizontal direc-tion is about 36500 g cmy2 . The primary cosmicrays interact in the atmosphere and create a ‘nuclearactive’ cascade. Since the atmosphere is tenuous


( )R. Cowsik, B.V. SreekantanrPhysics Letters B 449 1999 219–222220

with a scale height h f7=105 cm, pions andkaons in the cascade decay producing the cosmic-raymuonic component. Nuclear interactions of pionsand kaons with the atmosphere compete with theirdecay and become increasingly dominant as the par-ticle energy increases, so that the spectrum of themuonic component at high energies is steeper thanthat of the nuclear active component by a factorEy1. Also the muon component at high energiesincreases as ;secu , as the scale height of theatmosphere also has this dependence. Since the inter-action mean free path of the hadronic components is; 70 g cmy2 , after reaching their maximum devel-opment, they are absorbed with an absorption meanfree path of ; 100 g cmy2 . In contrast the muonssuffer only electromagnetic interactions and propa-gate with hardly any reduction in flux. Now note thatas we move away from the vertical towards thehorizontal direction, with increasing secu the nu-clear active components get severely absorbed butthe high energy muonic component increases as ;

secu ! Thus at large angles we have a nearly purebeam of high energy muons, traversing distances ofthe order of few times the scale height h ;hsecu .u

Now should the muon decay radiatively, the decayproducts e and g will induce an electromagneticcascade which can easily be observed signalling theviolation of flavor conservation, as described in themodel of Glashow and Coleman. Indeed as the en-ergy of the muon increases the observability of theeg-cascade increases as it penetrates deeper, spreadswider and produces more observable electrons andphotons. The electromagnetic cascade has a verybroad peak at about 500 g cm y2 from the point ofinitiation for an electron or g of energy E;104

GeV and the depth of the maximum increases loga-rithmically with energy. The total number of elec-trons at the peak of an electromagnetic cascade isapproximately equal to the energy of the initiatingelectron or gamma ray in GeV units. Thus any arrayof particle detectors deployed to detect extensive airshowers will be able to detect such showers gener-ated by the radiative decay of the muon. There willbe negligible amount of nuclear active particles andmuons in these showers. The background due toshowers induced by the primary cosmic ray nucleibecome negligible as we go to large zenith angles.Thus ‘m-less’ showers appearing in near horizontal

directions constitute a signal of the new processdescribed by Coleman and Glashow.

To quantify these ideas we note that the spectrumof muons at high energies near the earth may beparametrized as

k secui y2 y1 y1 y1m E s cm s sr GeV 1Ž . Ž .b q1iE

with

k s10, b s2.7 for 103 GeV-E-105 GeV1 1

2Ž .and

k s104 ,2

b s3.3 for 105 GeV-E-3=107 GeV 3Ž .2

Here b and b are the power law exponents of the1 2

primary cosmic ray spectrum at energies of 10 to 30times the energy of the muon.

w xAccording to Coleman and Glashow 1 the totaldecay probability per unit time, G , of a muon ofLorentz factor g is given by:

1qbg 4 1 bg 3

GsG qG s s q 4Ž .w rgt gt to o o

Here t f2.2=10y6 s is the life-time of the muono

and b is a very small parameter describing theviolation of Lorentz invariance and flavor conserva-tion. For a muon to decay close to the earth, say at a

Ž y2 .distance d of about 5 km ;700 g cm from theair shower array, it has to survive decay during itsflight though the atmosphere upto this point i.e. adistance of few times h , the scale height in thatu

direction. Thus the number of muons decaying in the5 km stretch is given by

s E fk secu Eyby1exp yjh Grc G drc , 5Ž . Ž . Ž .u

where j is a number of the order of 2 to 3. Notingthat G is a small number and that at high energies

Ž .G;G , the exponential in Eq. 5 may be set to unityrŽ .and Eq. 5 is rewritten as

s E ;k secu Eyby1 G drcŽ . r

k secu b dmy3m 2yb 2ybf E 'k bh E , 6Ž .

cto

where h s dmy3securct GeVy3 f 5 = 104m o

y3 ² :GeV , for secu f7. The products of the radia-

( )R. Cowsik, B.V. SreekantanrPhysics Letters B 449 1999 219–222 221

tive decay of the muon generate an extensive airshower which contains a large number of electrons,n , near the maximum, related to the muon energye

through the simple relation

n fEre , 7Ž .e

where ef 1 GeV for an electromagnetic shower ofprimary energy in the range 104–106 GeV. Thenumber spectrum of particles that will be seen by anair shower array is given by

f n fe 3yb k bh n2yb 8Ž . Ž .e e

Or the number of showers F, of size larger than ne

is given by

`X XF n s f n dn 9Ž . Ž . Ž .He e e

ne

e 3yb 2k bh2 3yb 52F n s n for n G10 10Ž . Ž .2 e e eb y32

3yb1e k bh1 5 3yb 3ybŽ .1 1F n s 10 ynŽ .1 e e3yb1

qF 105 for n -105 11Ž . Ž .2 e

We compare the integral number spectrum ofw xhorizontal air showers obtained by Nagano et al. 6

with the Akeno array in Fig. 1 for bs10y23 andbs10y25. The theoretically derived spectrum is veryflat, ;ny0.3, in contrast with the observed spectrume

of horizontal air showers which shows ;ny2 be-e

havior. Note that b;3=10y23 is excluded even bythe lower energy data at n ;105 and that the bounde

b-10y25 12Ž .is obtained when we consider the fluxes of horizon-tal air showers quoted by Nagano et al for n ;5=e

106. Clearly these bounds are considerably morestringent than those derived by looking at the depthintensity curves for muons and as such small values

Fig. 1. The integral flux of horizontal air showers given by Nagano et al. is compared with the expectation from the Coleman-Glashowprocess for the two values of b, 10y23 and 10y25 respectively.

( )R. Cowsik, B.V. SreekantanrPhysics Letters B 449 1999 219–222222

of branching ratio for radiative decay will not haveany detrimental effects on the functioning of muon

Ž .colliders Coleman and Glashow 1998 . It is interest-ing to note that in the Coleman Glashow model thislimit translates to

N1ycNF6=10y21 13Ž .

Acknowledgements

It is a pleasure to acknowledge the kind interest ofProf. Sheldon Glashow in the limit derived in thisletter.

References

w x Ž .1 S. Coleman, S.L. Glashow, Phys. Lett. B 405 1997 249;Harvard University Theoretical Physics Preprint 98rAO76Ž .pvt. comm .

w x2 C.M. Will, Theory and Experiment in Gravitational Physics,Cambridge University Press, Cambridge, 1993

w x3 M.I. Haugan, C.M. Will, Physics Today, vol. 40, May 1987.w x4 E. Fischbach, M.P. Haugan, D. Tadic, H.Y. Chang, Phys. Rev.

Ž .D 32 1985 154.w x5 G.L. Greene, M.S. Dewey, E.G. Kessler Jr., E. Fischbach,

Ž .Phys. Rev. D 44 1991 2216.w x6 M. Nagano, H. Yoshii, T. Hara, N. Hayashida, M. Honda, K.

Kamata, S. Kawaguchi, T. Kifune, Y. Matsubara, G. Tana-Ž .hashi, M. Teshima, J. Phys. G: Nucl. Phys. 12 1986 69.

11 March 1999


Exact BPS monopole solution in a self-dual background

Choonkyu Lee a, Q-Han Park b,1

a Department of Physics and Center for Theoretical Physics, Seoul National UniÕersity, Seoul 151-742, South Koreab Department of Physics, Kyunghee UniÕersity, Seoul 130-701, South Korea

Received 7 October 1998; revised 27 October 1998Editor: M. Cvetic

Abstract

Ž .An exact one monopole solution in a uniform self-dual background field is obtained in the BPS limit of the SU 2Yang-Mills-Higgs theory by using the inverse scattering method. q 1999 Elsevier Science B.V. All rights reserved.

Ž .There has been much theoretical interest concerning magnetic monopole solutions in an SU 2 Yang-Mills-w xHiggs theory after ’t Hooft and Polyakov 1 made the initial discovery of such structure in the seventies.

Ž . w x w xEspecially, in the Bogomolny-Prasad-Sommerfend BPS limit 2,3 , the ADHMN method 4,5 can be used toconstruct exact static multi-monopole solutions satisfying the first-order Bogomolny equations

F sye D F , 1Ž .i j i jk k

w x w x Ž a a a a .where F sE A yE A q i A , A 'e B and D FsE Fq i A , F with A 'A t r2,F'F t r2 .i j i j j i i j i jk k k k k i iŽ . ŽBPS monopoles refer to solutions of Eq. 1 , with the asymptotic fields approaching the Higgs vacuum as is

.necessary for any finite-energy configuration . At large distances, they feature the field B characteristic of aiŽ .system of localized magnetic monopoles and also the gauge-invariant magnitude of the Higgs field given as

g< <F r fÕy , for large r 2Ž . Ž .

4p r

Ž .where gs4p n ns1,2,... is the strength of the magnetic charge. Note that studies of BPS monopoles aredirectly relevant in nonperturbative investigations of certain supersymmetric gauge theories.

Ž .In this letter, we shall discuss a new solution of Eq. 1 which becomes possible if we assume a more generalŽ .asymptotic configuration than the Higgs vacuum. As a particular solution of Eq. 1 , we have the uniform

Ž .self-dual field described by up to arbitrary gauge transformation1 3 3A sy r=B t r2, fsy ÕqB Pr t r2. 3Ž . Ž . Ž .i 0 02 i

If the magnetic field strength B were zero, this would reduce to the usual Higgs vacuum. In this work, we will0Ž . Ž .look for a solution of Eq. 1 which describes a static monopole in the asymptotic uniform field background of



( )C. Lee, Q.-H. ParkrPhysics Letters B 449 1999 223–229224

Ž .the form 3 with B /0. For sufficiently weak B , the corresponding, everywhere regular, solution was first0 0w x Ž Ž . Ž . .discussed in Ref. 6 see Eqs. 3.35 – 3.37 of that article . From the latter, we know that the Higgs field in an

appropriate gauge takes the form

1 Õr1

F r sy Õ cothÕry q B Pr 2cothÕry rPtr2Ž . ˆ02 2ž /ž /Õr sinh Õr

r1y B Ptr2y B Pr rPtr2 , 4Ž .Ž .ˆ ˆ0 02 sinhÕr

Ž w x.where rsrrr. Note that, with B s0, this reduces to the well-known Prasad-Sommerfield expression 2 .ˆ 0

This is a perturbative solution, i.e., valid only to the first order in B , and therefore we still have no guarantee0Ž .for the existence of the corresponding, globally well-defined, exact solution with a finite background field B0

Ž . Ž Ž .. Ž .to the full nonlinear system 1 . The full solution see Eq. 28 , which reduces to the perturbative result 4 forsmall B , will be found below with the help of the inverse scattering method. However, as we shall see, there0

arises some unusual feature when one tries to extend the solution to the whole 3-dimensional space.Ž .As we make the choice B sB z with B )0 , an obvious starting point for the solution, suggested by theˆ0 0 0

w xsymmetry consideration, will be the following cylindrical ansatz 7 :

h r , z y1 h r , zŽ . Ž .2 1a i a a i i aA syw z q r q z W r , z qr W r , z w ,Ž . Ž .ˆ ˆ ˆ ˆ ˆ ˆi 1 2r r

F a sf r , z r a qf r , z z a , 5Ž . Ž . Ž .ˆ ˆ1 2

Ž . Žwhere r,w, z refer to cylindrical coordinates, and we have introduced normalized basis vectors in ordinary. Ž . Ž . Ž .3-space and isospin space rs cosw,sinw,0 , ws ysinw,cosw,0 and zs 0,0,1 . Performing a judiciousˆ ˆ ˆ

Ž . Ž . w xsingular gauge transformation with Eq. 5 , it is also possible to write the ansatz in an alternative form 8Ž a a .here note that A 'A t r2 :i i

t 1 0 yW21A 'cosw A qsinw A syW s ,r 1 2 2 2 ž /yW 02 2

h t 2 h t 3 1 yh ih1 2 2 1A 'ysinw A qcosw A sy y s ,w 1 2 ž /yih hr 2 r 2 2 r 1 2

t 1 0 yW f yif1 2 11 1A syW s , Fs . 6Ž .3 1 2 2ž / ž /yW 0 if yf2 1 1 2

Ž .Using either form, one finds from the Bogomolny equation in Eq. 1 that the functions f ,f ,h ,h ,W and W1 2 1 2 1 2

should satisfy the coupled equations

1 1E f yW f sy E h yW h , E f yW f s E h yW h ,Ž . Ž .r 1 2 2 z 1 1 1 z 1 1 1 r 1 2 2

r r

1 1E f qW f sy E h qW h , E f qW f s E h qW h ,Ž . Ž .r 2 2 1 z 2 1 1 z 2 1 1 r 2 2 1

r r

1E W yE W sy h f yh f . 7Ž . Ž .r 1 z 2 1 2 2 1

r

( )C. Lee, Q.-H. ParkrPhysics Letters B 449 1999 223–229 225

w x Ž .By making a judicious gauge choice, it was shown in Refs. 8,9 that the solution to Eq. 7 can always bewritten as

E c E f E cz z rf s , f sy , h syr ,1 2 1f f f

E f 1rh sr , W syf , W s h , 8Ž .2 1 1 2 1f r

Ž . Ž . w x Žwith the two real functions fs f r, z and csc r, z which must satisfy the Ernst equations 10 here,12 2 2 .= 'E qE q Ez r rr

2 2 < < 2 < < 2f = cy2=fP=cs0, f = fy =f q =c s0. 9Ž .If we here define the real symmetric, 2=2 unimodular matrix g by

1 1 cgs , 10Ž .2 2ž /c c q ff

Ž . Ž w xEq. 9 can further be changed into the chiral equation or Yang’s equation 11 for axially symmetric.monopoles

y1 y1E r E g g qE r E g g s0. 11Ž . Ž .Ž .r r z z

w xNote that, for the Prasad-Sommerfield one-monopole solution, we have 12

r 1fs , cs zcoshÕzyrsinhÕz cothÕr 12Ž . Ž .

F F

where F'rrsinhÕrqrcoshÕz cothÕryzsinhÕz.Ž .In order to incorporate the effect of the background field on the result 12 , we may use the inverse scattering

w x Ž .method with the above chiral equation 9,13 . It is based on the fact that Eq. 11 can be viewed as thecompatibility conditions of the linear system

y1 y12 r r E g g yl E g g2l Ž . Ž .z rD C' E y E Cs C1 z l2 2 2 2ž /l qr l qr

y1 y1r r E g g ql E g g2lr Ž .Ž .r zD C' E q E Cs C 13Ž .2 r l2 2 2 2ž /l qr l qr

Ž . Ž . Ž .for a 2=2 matrix CsC r, z;l . Now, for some initial solution gsg r, z of Eq. 11 , suppose that we0Ž . Ž . Ž . Ž .know a corresponding solution C r, z;l of Eq. 13 , with the boundary condition C r, z;ls0 sg r, z0 0 0

Ž . Ž . Ž . Ž . Ž . Ž .satisfied. Then, the dressed functions, C r, z;l sx r, z;l C r, z;l and g r, z sx r, z;ls0 g r, z ,0 0Ž . Ž . Ž .give new solutions of Eqs. 11 and 13 , provided that x r, z;l satisfies

y1 y1 y1 y1r r E g g yl E g g r r E g g yl E g gŽ . Ž .Ž . Ž .z r z 0 0 r 0 0D xs xyx ,1 2 2 2 2l qr l qr

y1 y1 y1 y1r r E g g ql E g g r r E g g ql E g gŽ . Ž .Ž . Ž .r z r 0 0 z 0 0D xs xyx , 14Ž .2 2 2 2 2l qr l qr

Ž .and also the condition originating from the hermiticity of g and g0

†y1 y12x r , z ;l sg r , z x r , z ;yr rl g r , z . 15Ž . Ž . Ž . Ž .Ž . 0


Ž .The function x r, z;l , needed in generating N-monopole solutions, may have only simple poles in theŽ w x.complex l-plane see Refs. 9,12 , viz.,

N R r , zŽ .kx r , z ;l s1q 16Ž . Ž .Ý

lym r , zŽ .kks1

Ž .with the poles m r, z explicitly given byk

2 2(m r , z sw yzq w yz qr , 17Ž . Ž . Ž .k k k

Ž .where w are arbitrary constants. The residues R r, z are also found readily and then the resulting expressionk kŽ . Ž . Ž .for x r, z;l may be used to secure the following formula for the new solution gsg r, z of Eq. 11 :ph

'g sgr det g ,ph

N Ny1 y1 j ig s g y m m G g m m g , a,bs1,2 18Ž . Ž . Ž . Ž . Ž . Ž .Ž . i jÝ Ýab 0 i j 0 c d 0ab ac db

is1 js1

k k y1 k i j 2w Ž . x Ž . Ž . Ž .where m sM C r, z;m M are constants and G sm g m r r qm m .b c 0 k cb c i j a 0 ab b i j

For our problem, we may apply the above dressing method on the initial solutions which correspond toŽ .uniform self-dual fields. By a direct integration of the Ernst Eq. 9 , we have a particular solution

2B r1rf 0 00 2g s ; f sexp Õzq z y 'exp ÕZ , 19Ž . Ž .0 0 ž /ž /0 yf 2 20

Ž . Ž .and the corresponding fields, if used in Eq. 6 , yield precisely the uniform field configuration given in Eq. 3 .Ž .The minus sign in the component of g is introduced in order to make det g in Eq. 18 to be positive definite0

C 1 00w x Ž . Ž .9 . Given the matrix g as in 19 , we may then solve the linear equations 13 for C s . All0 0 2ž /0 C0

together, we have here four equations for C 1 and C 2, which may be integrated by noticing that two equations0 0

from the four in fact implyl

1E q ÕqB z q E C s0,Ž .z 0 r 0r

l2E y ÕqB z q E C s0. 20Ž . Ž .z 0 r 0

r

Ž . Ž . Ž .For a solution C r, z;l which satisfies the boundary condition C r, z;ls0 sg r, z , we have found0 0 0

through this analysis the following expression:y11 l B0 1 11 2 2 2C r , z ;l s sexp yÕ zq y z y r ql zq l 'K . 21Ž . Ž .Ž .0 2 42 ž /2 2C r , z ;lŽ .0

Ž Ž .. Ž .Then, for the one monopole case i.e., Ns1 in Eq. 16 , the dressing method yields the 2=2 matrix g r, zwith

m2 f 2K 4M 2 qr 2M 20 2 1

g sy11 2 2 4 2 2m f M yK M fŽ .0 1 2 0

r 2 qm2 f K 2M MŽ . 0 1 2g sg s12 21 2 2 4 2 2m M yK M fŽ .1 2 0

m2 f M 2 qr 2K 4M 2 f 30 1 2 0

g sy , 22Ž .22 2 2 4 2 2m M yK M fŽ .1 2 0


2 2 Ž .(where msyzqr, r' z qr , and f is given in Eq. 19 . Finally, a new solution can be constructed0Ž .directly from Eq. 18 . However, in order to compare with previously known results in the limiting case, we

' '1r 2 1r 2y1make a gauge transformation of g through g ™hg h where hs . Note that this isph ph ph ž /' 'y1r 2 1r 2Ž .indeed a gauge transformation which leaves the chiral Eq. 11 covariant. This gives rise to the identification:

m1rfs g yg yg qgŽ .11 21 12 222 r

m fcs g yg . 23Ž . Ž .11 222 r

Explicit evaluation then gives the expressionsr r

˜fs , F' qrcoshÕZcoshÕRyzsinhÕZ,˜ sinhÕRF1

cs zcoshÕZyrsinhÕZ cothÕR , 24Ž . Ž .F

1B z B 2 20 0Ž . Ž . Ž Ž ..where R'r 1q and Z'zq z y r see Eq. 19 .22Õ 2 Õ

Ž . Ž .Note that, with B s0 i.e., in the zero background field limit , our expressions 24 reduce to the known0Ž . Ž .results 12 ; in this sense, Eq. 24 provides a deformation of the Prasad-Sommerfield solution by allowing the

Ž . Ž . Ž .background magnetic field. If the functions f ,f ,h ,h ,W ,W , calculated using Eqs. 8 and 24 , are1 2 1 2 1 2Ž . Ž .inserted into Eq. 5 , we have an exact solution to the Bogomolny equations 1 which are regular at rs0 and

also on the z-axis. Explicitly, for the Higgs field, we findB 10a aF r syÕ 1q z cothÕRy r cos L r , z qzsin L r , zŽ . Ž . Ž .ˆ ˆ½ 5ž /Õ Õr

B r0 aq r sin L r , z yzcos L r , z 25Ž . Ž . Ž .ˆ ˆ2sinhÕRŽ .with the function L r, z defined through

z 1qcoshÕZ coshÕR yrsinhÕZ sinhÕRŽ .tan L r , z s . 26Ž . Ž .

r coshÕZqcoshÕRŽ .This leads to the gauge-invariant Higgs field magnitude

2 2 2B z 1 B r0 02 2< <F r sÕ 1q cothÕRy q . 27Ž . Ž .2ž /Õ Õr 4sinh ÕR

The small B -limit of this expression can easily be shown to coincide with the gauge-invariant magnitude0Ž .obtained using the perturbative solution 4 ; up to gauge transformation, the solution we have above is what we

Ž .were after. Also, the appearance of the function Z z,r above can be ascribed to a gauge artifact. AfterŽ .performing an appropriate complicated gauge transformation with the above solution, we have succeeded in

1aŽ . Ž w x.casting our full solution, including A r , into the form with R'r 1q B Pri 02Õ

1 1 r aa a aF r syr Õ 1q B Pr cothÕRy y B yB Pr r ,Ž . Ž .ˆ ˆ ˆ0 0 0ž /Õ Õr 2sinhÕR

r 1 Õr rˆjaA r sye 1y 1q B Pr qe B yB Pr rŽ . Ž . ˆ ˆji ai j 0 ai j 0 0 jž /r Õ sinhÕR 2sinhÕR

r 1ycoshÕRaq r e r B . 28Ž . Ž .ˆ ˆi lm l 0 mž /2 sinhÕR


w Ž . Ž . x Ž .We have verified that Eq. 7 , and hence Eq. 1 , is satisfied by this expression . From Eq. 27 we note that theHiggs zero or the monopole center, orginally at the origin for B s0, gets displaced along the z-axis for0

< < 2 .nonzero B . Evidently, F may have zeros only along the line rs0. In fact, a detailed analysis shows that0

zeros of the Higgs field occurs in a rather nontrivial way depending on the strength of the background field B0Ž .See below . Nevertheless, at large distances where ÕR41, we find

1< <F r fÕqB zy , ÕR41 29Ž . Ž . Ž .0 r

which is the expected behavior if an ns1 monopole is situated near the origin in the presence of theŽbackground field B sB z. But, at points on the plane zsy2ÕrB which is, for small B , on the far left ofˆ0 0 0 0

B0. Ž . < Ž . < Ž .our monopole , Rsr 1q z ™0 and F r in Eq. 27 diverges, therefore, our solution possesses a2Õ

surface singularity.It turns out that Higgs field has a single zero at zs0 only for the vanishing background case, B s0. For0

B /0, Higgs field has a couple of zeros when B is small and has no zero at all when B exceeds a critical0 0 0

value, B )Bc f0.3Õ2. This makes it difficult to address notions such as the monopole center or the monopole0 0

number in terms of Higgs zeros as in the case of the vanishing B . In order to help understand the situation0Ž .better, it would be useful to consider the background self-dual solution itself as given in Eq. 3 . It has the plane

zsyÕrB as the zero of the background Higgs field. Note that, if there exists some extended region where the0

Higgs field becomes very small, the topological character usually related to a magnetic monopole gets rathermurky. Our solution has a monopole deep in the right half-space z)yÕrB and shows a plausible behavior in0

the very right half. On the other hand, the plane zsyÕrB as the zeros of the background Higgs field has0Ž . Ž .disappeared. In our solution 25 or 28 , we have instead an isolated zero near the point rs0, zsyÕrB0

immersed in the region of small, but non-zero, Higgs field and there is no distinctive long-range tail associated< Ž . <with this zero. Even in the other half-space where the Higgs field was aligned in the opposite direction, F r is

Ž . < Ž . <well approximated by 29 if z is not too close to y2ÕrB . However, the divergence of F r encountered at0

zsy2ÕrB is nontrivial; above all, it is not an gauge artifact. Thus, our monopole solution cannot be0Ž w x.extended beyond this singular plane. If one is concerend with only restricted physical problems as in Ref. 6 ,

this ill-behavior of our solution in the ‘wrong’ Higgs vacuum region might not be taken too seriously. But ouropinion is that this singularity issue deserves further investigation in the future.

w xA couple of comments are in order. We note that the well-known trick 14 may be used on our solution tow xobtain the corresponding dyon solution which solves the generalized Bogomolny equations 15

B sycosb D F , E sysinb D F 30Ž .i i i i

in the background of a uniform magnetic and electric field. Also, our approach can be applied to the problem offinding exact instanton solutions in nonvanishing background fields as well. In this regard, it would be

w x w xinteresting to extend the ADHM construction 4 and the Nahm equation 5 in the presence of backgroundfields.

Acknowledgements

Useful discussions with D. Bak, K. Lee and H. Min are acknowledged. This work was supported in part byŽthe Korea Science and Engineering Foundation through the SRC program of the SNU-CTP and 97-07-02-02-

.01-3 and the Basic Science Research program under BSRI-97-2418 and BSRI-97-2442.


References

w x Ž . Ž .1 G. ’t Hooft, Nucl. Phys. B 79 1974 294; A.M. Polyakov, JETP Lett. 20 1974 194.w x Ž .2 M.K. Prasad, C.M. Sommerfield, Phys. Rev. Lett. 35 1975 760.w x Ž .3 E.B. Bogomol’nyi, Sov. J. Nucl. Phys. 24 1976 861.w x Ž .4 M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu.I. Manin, Phys. Lett. A 65 1978 185; N.H. Christ, E.J. Weinberg, N.K. Stanton, Phys.

Ž . Ž .Rev. D 18 1978 2013; E. Corrigan, D. Fairlie, P. Goddard, S. Templeton, Nucl. Phys. B 140 1978 31; E. Corrigan, P. Goddard, S.Ž .Templeton, Nucl. Phys. B 151 1979 93.

w x Ž . Ž .5 W. Nahm, Phys. Lett. B 90 1980 413; N.S. Craigie et al. Eds. , Monopoles in Quantum Field Theory, World Scientific, Singapore,1982.

w x Ž .6 D. Bak, C. Lee, K. Lee, Phys. Rev. D 57 1998 5239.w x Ž .7 N.S. Manton, Nucl. Phys. B 135 1978 319.w x Ž . Ž . Ž .8 P. Forgacs, Z. Horvath, L. Palla, Phys. Rev. Lett. 45 1980 505; Ann. Phys. 136 1981 371; Phys. Lett. B 99 1981 232.´ ´w x Ž .9 P. Forgacs, Z. Horvath, L. Palla, Nucl. Phys. B 221 1983 235.´ ´

w x Ž .10 F.J. Ernst, Phys. Rev. 167 1968 1175.w x Ž .11 C.N. Yang, Phys. Rev. Lett. 38 1977 1377.w x Ž .12 M.A. Lohe, Nucl. Phys. Phys. B 142 1978 236.w x Ž .13 P. Forgacs, Z. Horvath, L. Palla, Monopoles in quantum field theory N. Craigie et al. Eds. , World Scientific, Singapore, 1982, p. 21.´ ´w x Ž .14 E. Weinberg, Phys. Rev. D 20 1979 936.w x Ž .15 S. Coleman, S. Parke, A. Neveu, C.M. Sommerfend, Phys. Rev. D 15 1977 544.

11 March 1999


Superheavy dark matter and parametric resonance

Masato Arai a, Hideaki Hiro-Oka b, Nobuchika Okada c,2, Shin Sasaki a

a Department of Physics, Tokyo Metropolitan UniÕersity, Hachioji, Tokyo 192-0397, Japanb Institute of Physics, Center for Natural Science, Kitasato UniÕersity Sagamihara, Kanagawa 228-8555, Japan

c Theory Group, KEK, Tsukuba, Ibaraki 305-0801, Japan

Received 15 August 1998; revised 10 January 1999Editor: H. Georgi

Abstract

We propose a new scenario to produce the superheavy dark matter based on the inflationary universe. In our scenario, theinflaton couples to both a boson and a stable fermion. Although the fermion is produced by the inflaton decay after inflation,almost energy density of the inflaton is transmitted into the radiation by parametric resonance which causes the explosivelycopious production of the boson. We show that the fermion produced by the inflaton decay can be the superheavy darkmatter, whose abundance in the present universe coincides with the critical density. We also present two explicit models asexamples in which our scenario can be realized. One is the softly broken supersymmetric theory. The other is the ‘‘singletmajoron model’’ with an assumed neutrino mass matrix. The latter example can simultaneously explain the neutrinooscillation data and the observed baryon asymmetry in the present universe through the leptogenesis scenario. q 1999Published by Elsevier Science B.V. All rights reserved.

The existence of the dark matter in the presentuniverse is the commonly accepted consequence from

w xobservations 1 . In addition, most of the inflationw xmodels 2 , which can solve the flatness and horizon

problems, naturally predict the density parameterVs1. On the other hand, the big-bang nucleosyn-thesis implies that the contribution of baryons to thematter density at the present universe is at most 10%.Therefore, the present universe is dominantly ful-filled by the dark matter; V G0.9.DM

In general, there are two possibilities for the typeof the dark matter. One is that the dark matter is a

1 E-mail: [email protected] JSPS Research Fellow.

thermal relic, and the other is that it is a non-thermalrelic. Most of discussions have been performed forthe first type. In this case, the present abundance ofthe dark matter can be estimated, and, especially, wecan derive an upper bound on its mass, m -500DM

w xTeV by the unitarity argument 3 . According to thisargument, if the dark matter is superheavy 3, itshould be a non-thermal relic not to over-close thepresent universe. However, in this case, we shouldmake it clear what is the mechanism to produce thesuperheavy dark matter in order to discuss its presentabundance.

3 The word ‘‘superheavy’’ means that mass of the dark matteris larger than 500 TeV.


( )M. Arai et al.rPhysics Letters B 449 1999 230–236 231

Recently, some production mechanisms of thenon-thermal superheavy dark matter were proposedw x4 , by which the dark matter is produced throughgravitational effect, broad parametric resonance andso on. Furthermore, some candidates for the super-heavy dark matter were also considered in the con-

w xtext of the string theory, M-theory 5 and supersym-w xmetric theory with discrete gauge symmetry 6 .

In this letter, we propose another scenario basedon the inflationary universe in order to produce thesuperheavy dark matter whose abundance in thepresent universe coincides with the critical density.In our scenario, the inflaton very weakly couples toboth a boson and a stable fermion whose masses aremuch smaller than the inflaton mass. Although thefermion is produced by the inflaton decay afterinflation, almost energy density of the inflaton israpidly transmitted into the radiation of the boson by

w xparametric resonance 7,8 which causes the explo-sively copious production of the boson. If the bosoncouples to ordinary particles in the standard modelwith not so weak coupling constants, the universecan be thermalized as soon as the parametric reso-nance occurs. The production of the fermion is effec-tively over when the parametric resonance occurs.We will show that the fermion can be the superheavydark matter, whose abundance in the present uni-verse coincides with the critical value.

We also present two explicit models as examplesin which our scenario can be realized. One exampleis the softly broken supersymmetric theory. Note thatit is natural for a boson to simultaneously couple tosome bosons and fermions in the context of super-symmetric theories. The other is the ‘‘singlet ma-

w xjoron model’’ 9 with an assumed neutrino massmatrix. This example have other phenomenologicaland cosmological implications. In this model, we cansimultaneously explain the solar and the atmosphericneutrino deficits and the baryon asymmetry in thepresent universe through the leptogenesis scenariow x10 .

As mentioned above, in our scenario, the inflatonfield couples to both the fermion and boson. Let usconsider the inflaton potential of the form

21 2 2Vs l f ys , 1Ž .Ž .8

where f is the inflaton field, and it is regarded as a

real field, for simplicity. We also consider interac-tion terms of the form,

2 2 †LL syg f x xyg fcc , 2Ž .int B F

where x and c are the boson and fermion fieldsŽ .which couple to the inflaton with positive and real

coupling constants g and g , respectively. Assum-B F

ing s<m , where m ;1019 GeV is the Planckp l p l

mass, the chaotic inflation occurs with the initial² : Ž .value of inflaton fields, f ; several =m . Notep l

that l;10y12 is implied by the anisotropy of thew xcosmic microwave background radiation 11 , and

'g , g F l is required by naturalness.B F² :Since vacuum lies at f ss , let us shift the

inflaton fields such as f™fqs . Then, we canŽ . Ž .rewrite Eqs. 1 and 2 as

l l l2 2 3 4Vs s f q sf q f 3Ž .

2 2 8

and

LL syg 2 f 2 q2sfqs 2 x †xŽ .int B

yg fqs cc , 4Ž . Ž .F

respectively. Masses of the inflaton, boson x and2 2 2'fermion c are given by m s l s , m sg s qf x B

m2 and m sg s , respectively. Here, m is tree0 c F 0

level mass of the boson field. In the followingdiscussion, we assume m ,m <m .x c f

Now, let us consider behavior of an amplitude ofthe inflaton field after the end of inflation. In thisepoch, the inflaton field is coherently oscillating, butits amplitude is decreasing due to both the expansionof the universe and the inflaton decay. When the

² :amplitude becomes f <s , the inflaton potentialŽ .of Eq. 3 is reduced to only the mass term, and thus

we can regard the coherent oscillation as the har-monic oscillator with frequency m ; f sf

Ž .F cos m t .0 f

Regarding the inflaton as the background field,equation of motion for the Fourier modes of x fieldis approximately given by

d22 2x q k q2 g sF cos m t x s0 , 5Ž .Ž .Ž .k B 0 f k2dt

where k is the spatial momentum, and we neglectedmass term m2 and g 2 f 2 term because of ourx B

( )M. Arai et al.rPhysics Letters B 449 1999 230–236232

² :assumption m <m and f <s , respectively.x f

Here, note that we also neglected a friction term3Hdx rdt introduced by the expansion of the uni-k

verse, and this approximation will be justified later.The above equation is well known as the Mathieu

w x Ž .equation 12 , and narrow parametric resonanceoccurs if the condition 4 g 2 sF <m2 is satisfied.B 0 f

In the following discussion, we take F ;s and0

g 2 ;l as a rough estimation. The amplitude of xB k

with momentum in the first resonance band k;

m r2, whose contribution is dominant, is exponen-f

Ž .tially growing up such as x Aexp m t , wherek

g 2 sFB 0m;4 ;m . 6Ž .fmf

Note that the Hubble parameter is given by

1r22 2ls F s0H; ; m , 7Ž .f2 ž /ž / mm plp l

and H<m is satisfied because of the assumptions<m . Then, the friction term 3Hdx rdt can bep l k

Ž .neglected in Eq. 5 . Almost energy density of theinflaton field is rapidly transmitted into radiation of

Žx fields by this parametric resonance. If x field not.so weakly couples to ordinary fields in the standard

model, the universe can be thermalized as soon asthe parametric resonance occurs. Then, we expect

Ž .that the reheating temperature T is estimated asRH

V f;s ;T 4 . 8Ž . Ž .RH

Next, let us consider the energy density of thefermion which is produced from the time of the end

Ž .of inflation t till the time when the parametricEOI

resonance occurs and the universe is thermalizedŽ .t . At t , the energy density of the producedPR PR

fermion is estimated as

3a tŽ .EOIyG t ytŽ .PR EOI� 4r t sV t 1ye ,Ž . Ž .c PR EOI ž /a tŽ .PR

9Ž .

Ž . 4 Ž .where V t ;lm is the potential energy den-EOI p lŽ .sity of the inflaton at t , a t is the scale factor ofEOI

the universe, and G is the width of the inflatondecay process f™cc which is given by

g 2F

Gs m . 10Ž .f8p

We regard time dependence of the scale factor be-tween t and t as same as the matter dominatedEOI PR

w x Ž . 2r3 Ž .universe 7 , namely, a t A t and H t ;Ž . y1 Ž .2r3 t . The Hubble parameters, H t andEOIŽ .H t , are given byPR

1r2V fsmŽ .p l 'H t ; ; l m , 11Ž . Ž .EOI p l2ž /mpl

1r2V fss sŽ .

H t ; ; m . 12Ž . Ž .PR f2ž / mm plp l

Ž . Ž . 4If G<H t <H t , the energy density isPR EOI

approximately given by2

G H tŽ .PR2r t ; V tŽ . Ž .c PR EOI3 ž /H t H tŽ . Ž .PR EOI

2 2; G H t m . 13Ž . Ž .PR p l3

The present energy density of the fermion c isgiven by

34m Tc 0r t ;r t = = , 14Ž . Ž . Ž .c 0 c PR ž / ž /T mRH c

where t is the present time, and T ;2=10y130 0

GeV is the present temperature of the microwavebackground radiation. If this energy density is com-

Ž y47parable to the critical density r ;8=10crit4.GeV , the fermion can be the dark matter. UsingŽ . Ž .Eqs. 8 – 14 , we can get resultant mass ratio of the

dark matter to the inflaton 5,

m rm ;10y3 . 15Ž .c f

If we take, for example, s;1018 GeV, the fermioncan be the superheavy dark matter with mass m ;c

109 GeV.

4 We can see that the condition is satisfied in our final resultwith our assumption s < m .p l

5 y9 Ž .We get the small coupling constant g ;10 from Eq. 15 .F

It is easy to check that this coupling constant is too small for thesuperheavy dark matter to be in thermal equilibrium with otherfields. This fact is consistent with our assumption.


Note that the existence of the parametric reso-nance is crucial in our scenario. If there is noparametric resonance, according to the old scenarioof reheating, the reheating temperature is estimatedas

T 2RH

GsH; , 16Ž .mpl

where G is the decay width of the inflaton. How-ever, in this scenario, the energy density of thefermion in the early universe is comparable with thatof radiation, and thus the present universe is over-closed, if not g is unnaturally small compared withF

g .B

Here, we give a comment on our analysis in thisletter. In general, there is a possibility that the broadresonance occurs before the amplitude of the inflatonis dumped into the narrow resonance band. However,it is non-trivial problem to definitely decide whetherthe broad resonance can affect or not, because of its

w xstochastic behavior as analyzed in Ref. 8 . Then, inour analysis we ignore the effect of the broad reso-nance for simplicity. If the energy of the inflaton istransmitted enough by the broad resonance, the pe-riod of reheating is shorten and total number of c

produced by the inflaton decay is smaller than that ofour estimation. In this case, the resultant mass of thesuperheavy dark matter becomes larger than that ofour estimation in order for its present abundance tocoincide with the critical value. It is always possibleto understand our result as the lower bound on thesuper heavy dark matter mass. Some numerical cal-culations are needed for correct evaluation.

In the following, we discuss two explicit modelsas examples in which our scenario of the superheavydark matter production can be realized. The firstexample is the softly broken supersymmetric theory.Let us consider superpotential of the form

1 12 2 2Ws m Z q m X ql ZX , 17Ž .Z X Z2 2

where Z and X are superfields with masses m andZ

m , respectively, and l is the dimension-less cou-X Z

pling constant. In the following discussion, we as-sume m <m and l <1, and identify the scalarX Z Z

Ž .component of Z and fermion component of X cX

with the inflaton and the dark matter 6, respectively.The scalar potential is given by

22 † 2 † 2 †Vsm Z Zqm X Xql X XŽ .Z X Z

q4l2 Z†Z X †XŽ . Ž .Z

q l m ZX †2 q2l m Z X †X qh.c. .Ž .� 4Z Z Z X

18Ž .

Note that m ;1013 GeV is required by the anisot-Zw xropy of the microwave background radiation 11 .

The inflaton field Z 7 is coherently oscillatingafter the end of inflation and this oscillation is justthe harmonic oscillator with frequency m ; ZsZ

Ž .F sin m t . In the following, we consider the case0 Z2 Ž † .Ž † .in which the fourth term, 4l Z Z X X , in Eq.Z

Ž .18 is dominant compared with the second line. Forexample, this case is realized, if we introduce thesoft supersymmetry breaking terms which cancel theterms in the second line. Regarding the inflaton asthe background field, the equation of motion for theFourier mode of the scalar X is given by

X XX q A y2 qcos2 z X s0 , 19Ž . Ž .k k k

2 2 Ž .2where A sk rm q2 q, qs l F rm , zsmt,k Z Z 0 Z

and the prime denotes differential with respect to z.This is nothing but the Mathieu equation, and theŽ .narrow parametric resonance occurs if we fix pa-rameters as l <1 and q<1, which means l <Z Z

y6 8 Ž10 we take F ;m , see the following discus-0 p l.sion .

Since the oscillating inflaton is regarded as theharmonic oscillator from the beginning, the reso-nance occurs just after the end of inflation. Thus, it

Ž . Ž .is naturally expected that H t ;H t and theEOI PR

reheating temperature is given by T 4 ;m2F 2 ;RH Z 02 2 Ž .m m . Following the discussion from Eq. 9 to Eq.Z p l

6 The fermion can be stable, if we introduce a discrete symme-try such as R-parity.

7 We use the same notation both superfields and their scalarcomponents in this letter.

8 This coupling constant is also too small for the superheavydark matter to be in thermal equilibrium with other fields.


Ž .14 , the present energy density of the fermion com-ponent of X is described as

34m Tc 0X2 2r ; G m m = = ;r ,c p l Z crit3X ž / ž /T mRH c X

20Ž .

where Gsl2 m r8p is the inflaton decay widthZ Z

with respect to the channel Z™c c . We can getX X

the mass of the dark matter as m ;10y7rl24105

c ZX

GeV. This result depends on the parameter l .Z

In the above discussion we assumed that theuniverse can be thermalized as soon as the paramet-ric resonance occurs. This can be possible if we

Žintroduce the soft supersymmetry breaking term A-.term such as

LL sAXHH , 21Ž .soft

where A is a parameter of mass dimension one, andH and H is the down- and up-type Higgs doublets inthe supersymmetric standard model, respectively.Since the scalar X couples to the standard modelparticles by this soft terms, the thermalization of theuniverse is realized.

Next, let us discuss the second example. Thereare some current data suggesting masses and flavor

w xmixing of neutrinos. The solar neutrino deficit 13w xand the atmospheric neutrino anomaly 14 seem to

be indirect evidences for the neutrino mass andflavor mixing from the viewpoint of neutrino oscilla-

w xtion. The ‘‘singlet majoron model’’ 9 is a simpleextension of the standard model to give Majoranamasses to neutrinos. It is well known that this model

w xincludes the see-saw mechanism 15 , which cannaturally explain the smallness of the neutrino massescompared with other leptons and quarks.

In addition, the model has a cosmological impli-cation. Since, in the model, the lepton number isbroken and CP is also violated by non-zero CP-phasesin general, we can explain the observed baryonasymmetry in the present universe through the lepto-

w xgenesis scenario 10 .Assuming a typical matrix for the neutrino mass

matrix, the singlet majoron model can be an exampleto which our scenario can apply. We can also simul-taneously explain the observations of the solar andatmospheric neutrino deficits and the baryon asym-

metry in the universe through the leptogenesis sce-nario.

We introduce the Yukawa interaction of the form

i ki j j k l c lLL syg n f n yg n f n qh.c. , 22Ž .Y Y L d R M R s R

where f is the neutral component of the Higgsd

doublet in the standard model, f is the electroweaks

singlet Higgs field, and i, j,k,ls1,2,3 are generationindices. The Dirac and Majorana mass terms appearby non-zero vacuum expectation values of theseHiggs fields. The neutrino mass matrix is given by

0 mD, 23Ž .Tm MD

i j² :where m sg f is the Dirac mass term, andD Y dk l² :Msg f is the Majorana mass term.M s

We assume typical Yukawa coupling constantsand mass matrices as follows:

M 0 010 a 00 M 00 b 0m s , Ms .2D

id0 b be 0 0 M3

24Ž .

Here, a, b and d are real parameters, and M , M1 2

and M are real masses of the heavy Majorana3

neutrinos. Note that, since the right-handed neutrinoof the first generation decouples to the left-handedneutrinos, it can be stable if its mass is smaller thanthe singlet Higgs mass. This is the case we consider.

In this model, we can identify the inflaton and theŽ . Ž .boson x in Eqs. 1 and 2 with the singlet Higgs

f and the standard model Higgs doublet, respec-s

tively. Then, our scenario can be realized, and thusthe right-handed neutrino of the first generation canbe the superheavy dark matter. The mass of the darkmatter is three orders of magnitude smaller than thesinglet Higgs mass, M ;10y3 =m , according to1 f s

Ž .our result of Eq. 15 .Next, we consider the problems of the solar and

the atmospheric neutrino deficits. By the see-sawŽ .mechanism, the mass matrix of Eq. 23 is approxi-

mately diagonalized as

y1 Tm M m 0D D , 25Ž .0 M


where the mass matrix for the light neutrinos,m My1 mT , is given byD D

m My1 mTD D

2a rM abrM abrM2 2 2

2 2abrM b rM b rMs .2 2 2

2 2 2 2 idabrM b rM b rM qb e rM2 2 2 3

26Ž .

Note that M is absent in this matrix since the1

right-handed neutrino is decoupled to left-handedneutrinos. For simplicity, we assume arb'e<1,M r2 M 'g<1 and d<1. Neglecting the ele-2 3

ments higher than the second order with respect to e ,g and d , we can get

2 0 e eby1 T e 1 1m M m ; . 27Ž .D D M2 e 1 1q2g

The unitary matrix which diagonalizes this matrixis the physical Kobayashi-Maskawa matrix in thelepton sector at low energies. Moreover, assuminge<g , we can get hierarchical mass eigenvalues

2 Ž .such as b rM 0, g , 2qg by diagonalizing this2

matrix. The mass squared differences and mixingangles corresponding to the solar and atmosphericneutrino oscillation data are given by

22 2b e2 2 2

Dm ; g , sin 2u ;4 <1 ,( ( ž /ž /M g2

22b2 2

Dm ;4 sin 2u ;1 . 28Ž .[ [ž /M2

By the assumption e<g , our description for thesolar neutrino data corresponds to the small angle

w xMSW solution 16 . By appropriately choosing thevalues of the free parameters, we can reproduce thesolar neutrino data Dm2 ;5=10y6 eV 2 for the(

w xsmall angle MSW solution 17 and the atmospheric2 y3 2 w xneutrino data Dm ;5=10 eV 14 .[

On the other hand, we can also explain the baryonasymmetry in the present universe through the lepto-genesis scenario. Since the right-handed neutrinos inthe second and the third generations couple to the

Ž .left-handed neutrinos and m of Eq. 24 includesD

CP-phase d , the original scenario of the leptogenesis

w xcan work. According to the original work 10 , wecan calculate the net lepton number production rate,which is proportional to

1 T† †Ds Im m m m mŽ . Ž .D D D D23 322 †p Õ m mŽ .D D 22

=f M 2rM 2 . 29Ž .Ž .3 2

Ž 2 2 .Here, f M rM ;M r2 M sg for M <M ,3 2 2 3 2 3

and Õs246 GeV is the vacuum expectation value ofthe standard model Higgs doublet. Using the Dirac

Ž . Ž .mass matrix of Eq. 24 and Eq. 28 , D is given by

2(Dm M( 2D; d . 30Ž .2p Õ

In the following, to consider the observed baryon tophoton ratio n rn ;10y10, we use the rough esti-B g

w xmation n rn ;Drg 1 , where g ;100 is theB g ) )

effective degrees of freedom of the radiation in theearly universe.

Choosing appropriate values for the parametersM and d , we can simultaneously explain the correct2

abundance of the superheavy dark matter, neutrinooscillation data and the observed baryon ratio. Whenwe take ds0.1, for example, we can get the massspectrum as follows:

M ;109 - M ;1010 - M ;10111 2 3

- m ;1012 GeV . 31Ž .fs

Here, we took the vacuum expectation value of thesinglet Higgs as ss1018 GeV.

In summary, we proposed a new scenario toproduce the superheavy dark matter. In our scenario,the inflaton couples to both the boson and the stablefermion whose masses are much smaller than theinflaton mass. The fermion is produced by the decayof inflaton after the end of inflation, but this produc-tion is effectively over when the parametric reso-nance occurs and the copious bosons are explosivelyproduced. We showed that the abundance of thefermion in the present universe can be comparable tothe critical value, and the fermion can be the super-heavy dark matter. In addition, we presented twoexplicit models as examples in which our scenariocan be realized. One is the softly broken supersym-metric theory. In this example, the dark matter hasmass of larger than 105 GeV. The other example is


the ‘‘singlet majoron model’’ with the assumed neu-trino mass matrix. In this model, the right-handedneutrino of the first generation can be the superheavydark matter with mass, for example, 109 GeV. Fur-thermore, this model can simultaneously explain theneutrino oscillation data and the observed baryonasymmetry in the universe through the leptogenesisscenario.

Acknowledgements

Ž .One of the authors N.O. would like to thankNoriaki Kitazawa for useful discussion and com-ments, and also for his hospitality during the author’svisit to Yale University where some parts of thiswork were completed. This work was supported inpart by a Grant-in-Aid for Scientific Research fromthe Ministry of Education, Science and Culture andResearch Fellowship of Japan Society for the Promo-tion of Science for Young Scientists.

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w x2 For recent reviews, see for example A. Linde, astro-phr9601004; M.S. Turner, astro-phr9704040; R.H. Bran-denberger, astro-phr9711106; G. Lazarides, hep-phr9802415; D.H. Lyth, A. Riotto, hep-phr9807278, andreferences therein.

w x Ž .3 K. Griest, M. Kamionkowski, Phys. Rev. Lett. 64 1990615.

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w x5 K. Benakli, J. Ellis, D.V. Nanopoulos, hep-phr9803333.w x6 K. Hamaguchi, Y. Nomura, T. Yanagida, hep-phr9805346.w x Ž .7 J.H. Traschen, R.H. Brandenberger, Phys. Rev. D 42 1990

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Ž . Ž .1996 1011; Phys. Rev. D 56 1997 3258.w x9 Y. Chikashige, R.N. Mohapatra, R.D. Peccei, Phys. Lett. B

Ž .98 1981 265.w x Ž .10 M. Fukugita, T. Yanagida, Phys. Lett. B 174 1986 45; M.

Ž .Luty, Phys. Rev. D 45 1992 455.w x Ž .11 J.E. Lidsey et al., Rev. Mod. Phys. 69 1997 373.w x12 L.D. Landau, E.M. Lifshitz, Mechanics, Pergamon, Oxford,

1960; V. Arnold, Mathematical Methods of Classical Me-chanics, Springer, New York, 1978.

w x Ž .13 K.S. Hirata et al., Phys. Rev. D 44 1991 2241; A.I. AbazovŽ .et al., Phys. Rev. Lett. 67 1991 3332; P. Anselmann et al.,

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11 March 1999


Anisotropy of ultra high energy cosmic raysin the dark matter halo model

V. Berezinsky a,b, A.A. Mikhailov c

a ( )INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi AQ , Italyb Institute for Nuclear Research, Moscow, Russia

c Institute of Cosmophysical Research and Aeronomy, 31 Lenin AÕe., 677891 Yakutsk, Russian Federation

Received 20 November 1998Editor: R. Gatto

Abstract

The harmonic analysis of anisotropy of Ultra High Energy Cosmic Rays is performed for the Dark Matter halo model. Inthis model the relic superheavy particles comprise part of the Dark Matter and are concentrated in the Galactic halo. TheUltra High Energy Cosmic Rays are produced by the decays of these particles. Anisotropy is caused by the non-centralposition of the Sun in the Galactic halo. The calculated anisotropy is in reasonable agreement with the AGASA data. Formore precise test of the model a comparison of fluxes in the directions of the Galactic Center and Anticenter is needed.q 1999 Elsevier Science B.V. All rights reserved.

The spectrum of Ultra High Energy Cosmic RaysŽ .UHECR is measured now up to a maximum energy

Ž . 20 w xof 2–3 =10 eV 1,2 . More than 1000 particlesare detected at energies higher than 1=1019 eVw x2–5 . The detailed energy spectrum was recently

w xpresented in Ref. 6 . No steepening of the spectrumhas been observed in the energy range between 1018

and 2=1020 eV. If extragalactic, the UHE protonsŽ .must have the Greisen-Zatsepin-Kuzmin GZK cut-

w x 19 w xoff 7 at energy E s6=10 eV 8 . A similar1r2

cutoff should exist if primaries are extragalactic nu-w x w xclei 8–10 or photons 11,12 .

Recently it was suggested that UHECR can begenerated by the decay of superheavy relic particlesw x13–16 . These particles can be effectively produced

w xin the post-inflationary Universe 14,17,18 and canconstitute now a small or large part of Cold Dark

Ž .Matter DM . As any other form of Cold DM theserelic particles are concentrated in the halo of our

galaxy, and thus UHECR produced by their decaysw xdo not exhibit the GZK cutoff 14 . Realistic particle

candidates for this scenario and possible mechanismsto provide the long lifetime for superheavy particles

w xare discussed in Refs. 13,14,19,20 .The halo model discussed above has three signa-

w xtures 14,21,22 : the excess of high energy photonsin the primary radiation, direct signal from a nearby

Ž .clump of DM e.g. Virgo Cluster , and anisotropycaused by the asymmetric position of the Sun in theGalactic halo. These signatures allow to confirm orto reject the DM halo hypothesis by the data ofexisting arrays.

w xAs calculations show 22,23 , anisotropy revealsitself very strongly in the direction of the GalacticCenter. This prediction can be reliably examined bythe Pierre Auger detector in the southern hemispherew x24 . At present there is no detector which canobserve this direction. In this article we present the


( )V. Berezinsky, A.A. MikhailoÕrPhysics Letters B 449 1999 237–239238

Table 1Anisotropy

ISO model NFW model

R A A a R A A ac 1 2 s 1 2

Ž . Ž . Ž . Ž .5 kpc 0.43 0.46 0.11 0.13 2508 30 kpc 0.41 0.45 0.10 0.13 2508

Ž . Ž . Ž . Ž .10 kpc 0.32 0.35 0.06 0.07 2508 45 kpc 0.37 0.41 0.09 0.11 2508

Ž . Ž . Ž . Ž .50 kpc 0.15 0.15 0.01 0.01 2508 100 kpc 0.33 0.36 0.07 0.08 2508

calculations of anisotropy for the arrays in the north-ern hemisphere, taking as an example the geographi-cal position of the Yakutsk and AGASA arrays.

We shall assume that primary photons dominatein the decays of SH particles as QCD calculationsw x25 imply. Then the flux of UHE photons in the

Ž .direction z ,f per unit solid angle can be written as

Ž .r zmaxI z sK dr r R , 1Ž . Ž . Ž .H X0

where r and R are the distances to a decayingX-particle from the Sun and the Galactic Center,respectively, z is the angle between the line ofobservation and the direction to the Galactic Center,f is the azimuthal angle in respect to Galactic planeŽ . Ž .the flux depends only on z , r R is the massX

Ž .density of superheavy particles X at a distance Rfrom the Galactic Center, K is an overall constant,

2 2 2Ž . (r z s R yr sin z qr cosz , r s8.5 kpc ismax h ( ( (

a distance between the Sun and Galactic Center, Rh

is the size of the halo, in our calculations we shallŽuse R s50 kpc the values of 100 and even 500h

.kpc result in similar anisotropy ; the distance R fromthe Galactic Center to the decaying particle is given

2Ž . 2 2by R z sr qr y2 rr cosz .( (

We shall use two distributions of DM in the halo:Ž . w xone given by the Isothermal Model ISO 26 ,

r0r R s , 2Ž . Ž .21q RrRŽ .c

and the other following the NFW numerical simula-w xtion 27

r0r R s . 3Ž . Ž .2RrR 1qRrRŽ .s s

In the ISO model we shall use for R the values 5c

kpc, 10 kpc and 50 kpc. For the NFW model thecalculations are performed for R equal to 30 kpc, 45s

w xkpc and 100 kpc. The NFW distribution 27 is givenin terms of the virial radius r , the rotational200

velocity at the virial distance Õ , the constant d200 c

and the dimensionless Hubble constant h. We ap-plied this distribution to our Galaxy using the follow-

Ž .ing parameters: the local density of DM r r sD M (

0.3 GeVrcm3, Õ s200 kmrs and hs0.6. As a200

result we obtain R f45 kpc.sŽ .The flux 1 was expressed first in terms of

galactic coordinates, longitude lsf and latitude b,which is given by cosbscoszrcosf, and thentransferred into equatorial coordinates, declination d

and right ascension a . We calculated the amplitudes

Fig. 1. Amplitude and phase of the first harmonic of anisotropyfor the AGASA and Yakutsk arrays. Solid lines are for the ISO

Ž .distribution of DM and dots NFW for the NFW numericalsimulation. BM and TW show the results of this paper and Ref.w x Ž .28 , respectively. The three dots of BM from left to right aregiven for R s30, 45 and 100 kpc, respectively; five dots of TWs

for R s10, 20, 30, 50 and 100 kpc. The AGASA data are takensw xfrom Ref. 28 .

( )V. Berezinsky, A.A. MikhailoÕrPhysics Letters B 449 1999 237–239 239

Žof the first and the second harmonics A and A ,1 2.respectively and the phase a of the first harmonic

for the geographical position of the Yakutsk andAGASA arrays. These quantities are the standardones used for measured anisotropy. The results are

Žgiven in Table 1 predictions for the AGASA array.are shown in brackets . Depending on the parameters

of the DM distribution, the anisotropy varies from10% to 45%. The phase of the first harmonic af2508 is close to the RA of Galactic Center, af2658.

After this paper was submitted for publication wereceived the preprint by Medina-Tanco and Watsonw x28 , where similar calculations were performed. Theresults of both calculations are displayed in Fig. 1 forE)4=1019 eV together with the data of AGASAŽ . Ž .AG and Yakutsk YK arrays. The AGASA anisot-

w xropy is taken from analysis of Ref. 28 . Our calcula-Ž . w xtions BM agree well with that of Ref. 28 . Both

agree with the data of AGASA array and do notcontradict to the Yakutsk data.

Acknowledgements

The authors are grateful to P. Blasi, M. Hillas, B.Hnatyk, M. Nagano, P. Sokolsky, A. Vilenkin andA. Watson for discussions and correspondence.

References

w x Ž .1 N. Hayashida et al., Phys. Rev. Lett. 73 1994 3491.w x Ž .2 D.J. Bird et al., Ap. J. 424 1994 491.w x Ž .3 G. Cunningham et al., Ap. J. 236 1980 L71.w x Ž .4 S. Yoshida et al., Astroparticle Physics 3 1995 105.w x Ž .5 B.N. Afanasiev et al., in: M. Nagano Ed. , Extremely High

Energy Cosmic Rays: Astrophysics and Future Observations,ICRR, University of Tokyo, 1996, p. 32.

w x Ž .6 M. Takeda et al., Phys. Rev. Lett. 81 1998 1163.w x Ž .7 K. Greisen, Phys. Rev. Lett. 16 1966 748; G.T. Zatsepin,

Ž .V.A. Kuzmin, Pisma Zh. Exp. Teor. Fiz. 4 1996 114.w x8 V.S. Berezinsky, S.V. Bulanov, V.A. Dogiel, V.L. Ginzburg,

V.S. Ptuskin, Astrophysics of Cosmic Rays, chapter 4, Else-vier, 1990.

w x9 V. Berezinsky, S. Grigorieva, G.T. Zatsepin, in: Proc. 14thInt. Cosm. Ray Conf., vol. 2, Munich, 1975, p. 711.

w x Ž .10 L.N. Epele, E. Roulet, J. High Energy Phys. 9810 1998009. astro-phr9808104.

w x Ž .11 V.S. Berezinsky, Soviet Phys. Nucl. Phys. 11 1970 399.w x Ž .12 R.J. Protheroe, P.L. Biermann, Astroparticle Physics 6 1996

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the Desert, Castle Rindberg, 1997; Yadernaya Fisika 61Ž .1998 1122.

w x14 V. Berezinsky, M. Kachelriess, A. Vilenkin, Phys. Rev. Lett.Ž .79 1997 4302.

w x15 P.H. Frampton, B. Keszthelyi, N.J. Ng, astro-phr9709080.w x Ž .16 M. Birkel, S. Sarkar, Astrop. Phys. 9 1998 297.w x Ž .17 D.J.H. Chung, E.W. Kolb, A. Riotto, Phys. Rev. D 59 1999

023501.w x Ž .18 V. Kuzmin, I. Tkachev, JETP Lett. 68 1998 271.w x19 K. Benakli, J. Ellis, D. Nanopoulos, heprphr9803333.w x20 K. Hamaguchi, Y. Nomura, T. Yanagida, Phys. Rev. D 58

Ž .1998 103503.w x Ž .21 S.L. Dubovsky, P.G. Tinyakov, JETP Lett. 68 1998 107.w x Ž .22 V. Berezinsky, P. Blasi, A. Vilenkin, Phys. Rev. 58 1998

103515.w x23 A. Benson, A. Smialkowski, A.W. Wolfendale, 1998, sub-

mitted to Astrop. Phys.w x Ž . Ž .24 J.W. Cronin, Nucl. Phys. B Proc. Suppl. 28B 1992 213.w x Ž .25 V. Berezinsky, M. Kachelriess, Phys. Lett. B 434 1998 61.w x26 A.V. Kravtsov, A.K. Klypin, J.S. Bullock, J.R. Primack.

astro-phr9708176 to be published in Ap. J.w x27 J.F. Navarro, C.S. Frenk, S.D.M. White, Astroph. J. 462

Ž .1996 563; F.F. Navarro astro-phr9801073.w x28 G.A. Medina-Tanco, A.A. Watson, 1998, submitted to As-

trop. Phys.

11 March 1999


Zee mass matrix and bi-maximal neutrino mixing

Cecilia Jarlskog a,1, Masahisa Matsuda a,2, Solveig Skadhauge a,3,Morimitsu Tanimoto b,4

a Department of Mathematical Physics, LTH, Lund UniÕersity, S-22100 Lund, Swedenb Science Education Laboratory, Ehime UniÕersity, 790-8577 Matsuyama, Japan


Abstract

We investigate neutrino masses and mixings within the framework of the Zee mass matrix, with three lepton flavors. It isshown that the bi-maximal solution is the only possibility to reconcile atmospheric and solar neutrino data, within thisansatz. We obtain two almost degenerate neutrinos, which are mixtures of all three neutrino flavors, with heavy masses

2 2 2Ž .( (, Dm . The predicted mass of the lightest neutrino, which should consist mostly of n and n , is ,Dm r 2 Dm .atm m t ( atm

q 1999 Elsevier Science B.V. All rights reserved.

PACS: 14.60.Pq; 14.60.Lm; 12.60.Fr

In the past few years stronger experimental signals, than ever before, have been seen for neutrino oscillations.w xRecent atmospheric neutrino experiments 1,2 indicate oscillations among neutrino flavors with large mixing

w xangles 3 . The simplest solution to the solar neutrino deficit problem, observed by the Super-Kamiokandew x w xexperiment 4 as well as other experiments 5 , is again neutrino oscillations. Indeed, the field of neutrino

Ž . w xoscillations is expected to enter into a new era, with the start of long baseline LBL neutrino experiments 6–8 .These experiments will hopefully solve the present neutrino anomalies.

w x 2Since the CHOOZ experiment 9 excludes oscillation of n ™n with a large mixing angle for Dm G9=m e

10y4 eV 2, large mixing between n and n is the simplest interpretation of the atmospheric n deficit. Them t m

difference between the quark-mixing and lepton-mixing matrices is striking. The Cabibbo-Kobayashi-Maskawaw xmixing matrix V 10 in the quark sector is described by small mixing angles among different flavors but inC K M

w xthe leptonic sector 11 at least one large mixing angle seems to be needed. It will be important to understandwhy these patterns are so different.

1 E-mail: [email protected] Permanent address: Department of Physics and Astronomy, Aichi University of Education, Kariya, Aichi 448, Japan. E-mail:

[email protected] E-mail: [email protected] E-mail: [email protected]


( )C. Jarlskog et al.rPhysics Letters B 449 1999 240–252 241

Another issue, to be understood, is why the neutrino masses are so small? The most popular answer to thew xlatter question is given by the see-saw mechanism 12 which introduces heavy right-handed Majorana neutrinos

with masses of the order 1010 y1016 GeV. This attractive model has been extensively studied in the literature.However, it is important to consider also other possible scenarios with small neutrino masses, specially

Ž . w xextensions of the standard model SM at a low energy scale. The Zee model 13 is such an alternative and hasw xbeen studied in the literatures for almost twenty years 14–18 . In this paper, we will discuss the present status

of the Zee mass matrix, in the light of recent experimental results.w xIn the Zee model 13 neutrino masses are generated by radiative corrections, and hence the model may

provide an explanation of the smallness of neutrino masses. In this model, the following Lagrangian is added tothe SM;

c y T yX XLLs f C is C h qmF is F h qh.c.Ž .Ý l l l L 2 l L 1 2 2

Xl , l se ,m ,t

c c c cy ys2 f n m ye n h q2 f n t ye n hŽ . Ž . Ž . Ž .em e L L L m L et e L L L t L

c c y q 0 0 q yq2 f n t ym n h qm F F yF F h qh.c. , 1Ž . Ž . Ž .Ž .mt m L L L t L 1 2 1 2

Ž .T Ž q 0.T "where C s n ,l , F s F ,F , is1,2. The Higgs potential is omitted here. The charged Zee boson, h ,l L l L i i iŽ .is a singlet under SU 2 . We need at least two Higgs doublets in order to make the Zee mechanism viable,L

since the antisymmetric coupling to the Zee boson is the cause of ByL violation, and hence of Majoranamasses. Note that only F couples to leptons, as in the SM. The mass matrix, generated by radiative correction1

w xat one loop level 13–18 , is given by

0 m mem et

m 0 m , 2Ž .em mt� 0m m 0et mt

wheremÕ22 2 2 2m s f m ym F M , MŽ .Ž .em em m e 1 2Õ1

mÕ22 2 2 2m s f m ym F M , M 3Ž .Ž . Ž .et et t e 1 2Õ1

mÕ22 2 2 2m s f m ym F M , MŽ .Ž .mt mt t m 1 2Õ1

and

1 1 M 212 2F M , M s ln . 4Ž .Ž .1 2 2 2 2 216p M yM M1 2 2

The parameter Õ is the vacuum expectation value of the neutral component of the Higgs doublet F . M1Ž2. 1Ž2. 1

and M are the masses of the physical particles defined by the fields2

HqshqcosfyFqsinf , Hqshqsin fqFqcosf , 5Ž .1 2

where Fq is the charged Higgs boson that would have been a physical particle in the absence of the hq.Finally, the mixing angle f is defined by

'4 2 mMWtan2fs . 6Ž .

222 2 y1'(g M yM y 4 2 g mMŽ . Ž .1 2 W

( )C. Jarlskog et al.rPhysics Letters B 449 1999 240–252242

Due to the antisymmetry of the coupling matrix, f X syf X , the Zee model requires all diagonal elements in Eq.l l l lŽ .2 to vanish at one loop level. Small corrections will however be obtained at higher orders in perturbationtheory. Hereafter, we refer to the above matrix, with vanishing diagonal elements, as the Zee mass matrix.

Ž .The parameters m ,m ,m in the Zee mass matrix are not described by the eigenvalues m is1,2,3 dueem et mt i

to the traceless property of the matrix, leaving only two independent observable parameters. In the same way itw xis impossible to represent the mixing matrix U by using m . In literature 14–18 , it has been assumed that therei

is a hierarchy such as m <m ,m and the neutrino masses and mixings are discussed under suchem et mt

assumptions. This hierarchy is natural if the coupling constants f X are of the same order of magnitude.l l

However, we would like to explore all possibilities of masses and mixings in the Zee model by relaxing thisw xassumption. Instead we use recent atmospheric neutrino data 1,2 as input to determine patterns in the Zee mass

matrix that are viable.The neutrino mass matrix M is generally, for Majorana particles, constructed byn

M sUM diagU T 7Ž .n

T w xdue to its symmetric property M sM . The mixing matrix U, called the MNS mixing matrix 11 , is defined inn n

the basis where the mass matrix of charged leptons is diagonal, with masses m . Furthermore,e,m ,t

m 0 01

diag 0 m 0M s . 8Ž .2� 00 0 m3

For Majorana neutrinos there are three phases in the matrix U and this is generally given by

Ž .i b yi dyac c s c e s e1 3 1 3 3

id id i b i ays c yc s s e c c ys s s e e s c eUs , 9Ž .Ž .1 2 1 2 3 1 2 1 2 3 2 3

id id i b i a� 0s s yc c s e yc s ys c s e e c c eŽ .1 2 1 2 3 1 2 1 2 3 2 3

w xwhere c 'cosu and s 'sinu . The Zee mass matrix exhibits no CP violation 19 and we will thereforei i i i

neglect the phases in our investigation. The diagonal elements in M are given byn

1,1 m c2c2 qm s2c2 qm s2 ,Ž . 1 1 3 2 1 3 3 3

2 2 2 22,2 m s c qc s s qm c c ys s s qm s c , 10Ž . Ž . Ž . Ž .1 1 2 1 2 3 2 1 2 1 2 3 3 2 3

2 2 2 23,3 m s s yc c s qm c s qs c s qm c c .Ž . Ž . Ž .1 1 2 1 2 3 2 1 2 1 2 3 3 2 3

Ž .These should be zero in general small in the Zee model and we arrive at the following relations:

cos2u y tan2u1 3m sy m , m sym ym . 11Ž .2 1 3 1 22 2sin u y tan u1 3

The second equality is obvious by the traceless property of the Zee mass matrix. Two of three masses have thesame sign and the remaining one has opposite sign, which implies that one of the fields has opposite CP parityas compared to the other two. The dependence of mass eigenvalues on u and u is shown in Fig. 1. Inserting1 3Ž . Ž .11 into 10 gives the relation

1 2cos2u cos2u cos2u s sin2u sin2u 3cos u y2 sinu . 12Ž .Ž .1 2 3 1 2 3 32

This equation means that the three mixing angles are not independent. For typical values of u , u , u the1 2 3

structure of the mixing matrix is discussed later, using this equation.Before entering into the analysis of the Zee mass matrix, we give a short survey of recent neutrino

experiments. Our approach is to assume that oscillations account for the solar and atmospheric neutrino data,


Fig. 1. The dependence of mass eigenvalues on the mixing angles u and u . The lines represents special relations among the three masses1 3Ž . Ž .as denoted in the figure. In the right hand side the domains are running oppositely, starting with a in the bottom and ending with f in the

Ž . Ž . Ž . Ž . Ž .top. In region a we have m )y m )y m , b m )y m )y m , c y m ) m ) m , d y m ) m ) m , e y m ) m ) m ,1 2 3 1 3 2 3 1 2 3 2 1 2 3 1Ž .and f y m ) m ) m . Here the sign of m is taken to be positive.2 1 3 1

thus pinning down two mass squared differences, which is the maximal number of mass differences in thew x w xmodel we are investigating. If the results of LSND 20 would be confirmed by KARMEN 21 or any other

experiment, the model examined in this paper would no longer be relevant. The deficit of n in recent Superm

w xKamiokande data 1 , is interpreted as oscillation of n ™n with nearly maximal mixing angle in a two flavorm t

w xanalysis 3,22 . These results yield

Dm2 , 0.5y6 =10y3 eV 2 , sin2 2u )0.82 90%C. L . 13Ž . Ž . Ž .atm atm

w xFurther, the deficit in the solar neutrino experiments suggests the following best-fit solutions 23 as

1 MSW small angle solution; Dm2 ,5.4=10y6 eV 2 , sin2 2u ,6=10y3 ,Ž . ( (

2 MSW large angle solution; Dm2 ,1.8=10y5 eV 2 , sin2 2u ,0.76 ,Ž . ( (

3 ‘‘just-so’’ vacuum solution; Dm2 ,6.5=10y11 eV 2 , sin2 2u ,0.75 .Ž . ( (

It is noted that the large angle MSW solution seems to be excluded by the simultaneous fits to all the availablew x w xdata 23 . As there are still theoretical uncertainties about this 24 , we will keep this possibility in our

considerations. The combined results of atmospheric and solar neutrino experiments suggest that there exist two2 2 Ž .hierarchical mass squared differences Dm 4Dm . Further, the component U in Eq. 9 should be small, asatm ( e3

w x 2 < < 2suggested by the CHOOZ experiment 9 . The value sin 2u -0.18 implies U -0.22 for Dm G9=CHOOZ e3y4 2 w x Ž .10 eV 25 . For the MSW small angle solution, case 1 , together with the atmospheric results, a possible

mixing matrix has the form

1 e e1 2

U , , 14Ž .e c s1 3� 0e ys c4

'where c,s,1r 2 . This mixing matrix might be realized in the case;

< 2 < < 2 < 2 < 2 < 2Dm , Dm ,Dm , Dm ,Dm , 15Ž .32 31 atm 21 (

Ž .which demands m 4m Gm or m Gm 4m i, js1,2 or 2,1 or3 i j i j 3

< 2 < < 2 < 2 < 2 < 2Dm , Dm ,Dm , Dm ,Dm , 16Ž .32 21 atm 31 (


Ž .implying m 4m Gm or m Gm 4m i, js1,3 or 3,1 . Another solution for MSW small angle mixing is2 i j i j 2

e 1 e1 2XU , . 17Ž .c e s1 3� 0ys e c4

This type of mixing suggests

< 2 < < 2 < 2 < 2 < 2Dm , Dm ,Dm , Dm ,Dm 18Ž .32 31 atm 21 (

or

< 2 < < 2 < 2 < 2 < 2Dm , Dm ,Dm , Dm ,Dm 19Ž .31 21 atm 32 (

Ž .and we obtain the solution m 4m Gm or m Gm 4m i, js2,3 or 3,2 for the latter case.1 i j i j 1

For the ‘‘just-so’’ and MSW large angle solutions a typical mixing matrix is

c s e c s e1 1

ys c e or . 20Ž .e e 12 2 3� 0� 0 ys c ee e 1 43 4

Neither of these is compatible with the maximal mixing pattern of the atmospheric n ™n oscillation, and mustm t

be discarded for this reason. We are left with one possibility;

c ’ sXe

X XU , 21Ž .ycs cc s3X Xž /ss ysc c

X X 'for interpreting large angle solar neutrino solutions, where c,s,c ,s ,1r 2 . In the limit es0 this isw x w xknown as the bi-maximal mixing matrix 26 . Nearly bi-maximal mixing is discussed in Ref. 27 . Taking

< 2 < < 2 < 2 < 2 < 2Dm , Dm ,Dm , Dm ,Dm 22Ž .32 31 atm 21 (

Ž .with m 4m Gm or m Gm 4m , i, js1,2 or 2,1 , it has been shown that the mixing matrix U is3 i j i j 3 3

consistent with vacuum solution for both solar and atmospheric neutrino anomalies within the experimentalw xuncertainties 28 . The MSW large angle solution could also be accommodated here, but then the allowed

parameter space is rather small.Furthermore the degenerate case m ,m ,m is another possibility for all the above cases. Using the1 2 3

definition;

d dqem sm , m sm 1y , m sm 1y , 23Ž .a 0 b 0 c 0ž / ž /2 2

we can assign a,b,c to each of 1,2,3 according to the mass relation obtained above. Only in this degenerate caseŽ .is it possible to have a mass m sOO 1 eV for the neutrinos, as is suggested by the hot dark matter argument.0

Ž y3 y2 . Ž y5 .Setting m s1eV the parameters d and e take the values OO 10 y10 and OO 10 respectively to0

reproduce Dm2 ,Dm2 ,Dm2 and Dm2 ,Dm2 .ab ac atm bc (

We now return to the analysis of the Zee mass matrix. In accordance with the data of atmospheric neutrinoŽ .experiments we require u ,pr4. Now Eq. 12 implies that we have the following possibilities; u ,0 or2 3'u ,0 or u ,arctan 1r2 . The latter case will be commented on later. We will only consider the case of1 3

u ,0 together with u ,0, due to the experimental constraints on U . To begin with we concentrate on the1 3 e3

solution u ,0.3


If we take the extreme limit u spr4 and u s0, the mixing matrix becomes2 3

c s 01 1

s c 11 1y ' ' '2 2 2U , . 24Ž .atm

s c 11 1� 0y' ' '2 2 2

It is noted that the extreme case does not change the qualitative structure of the model and is consistent withw xcombined SK and CHOOZ data in a three flavor analyses 3,22 . In this limit we obtain the constraints

m c2 qm s2 s0 , m s2 qm c2 qm s0 25Ž .1 1 2 1 1 1 2 1 3

Ž .from Eq. 10 . Then the parameters satisfy the constraint

m12tan u sy )0 , 26Ž .1 m2

and we arrive at

c s c s1 1 1 1° ¶0 y m ym m ymŽ . Ž .1 2 1 2' '2 2c s1 1 1 2 2y m ym 0 y m s qm c ymŽ . Ž .M s 1 2 1 1 2 1 32n '2

c s1 1 1 2 2m ym y m s qm c ym 0Ž . Ž .1 2 1 1 2 1 32¢ ß'2

° ¶< < < <m m m m1 2 1 20 " .( (

2 2

< <m m1 2s , 27Ž ." 0 ym ym( 1 22

< <m m1 2. ym ym 0( 1 2¢ ß2

Ž . Ž .where the upper lower sign corresponds to the case m -0 m )0 . The mixing matrix becomes1 1

° ¶< < < <m m2 10( (< < < < < < < <m q m m q m1 2 1 2

< < < <m m 11 2yUs . 28Ž .( (< < < < < < < < '2 m q m 2 m q m 2Ž . Ž .1 2 1 2

< < < <m m 11 2y( (¢ ß< < < < < < < < '2 m q m 2 m q m 2Ž . Ž .1 2 1 2

As the two mass squared differences are hierarchical we have to stay close to the lines in Fig. 1. Due to thew xsymmetry we only have to survey the left part of the parameter region, i.e u g 0,pr4 . Changing from the left1


to right side merely corresponds to an interchange of m lm and has no physical effect. We now consider the1 2

three possible values for u .1Ž .I The case with u ,0. The Zee mass matrix then requires the following mass relations to hold;1

< < < < < < 2 2 2 2 2m , m 4 m , Dm ,Dm sDm , Dm sDm . 29Ž .2 3 1 21 31 atm 23 (

In this case we get the mixing matrix;

1 e 01

e 1 11y

Zee ' ' '2 2 2U s , 30Ž .1

e 1 11� 0y' ' '2 2 2

< < Ž Ž ..with e , m rm . This corresponds to the small angle MSW solution U Eq. 14 . Nevertheless due to the(1 1 2 1Ž .mass relations in Eq. 29 the predicted probability of n ™n oscillation,m t

Dm2 L Dm2 L Dm2 Li j atm atm) ) 2 2 2 2 4 2P n ™n sy4 U U U U sin ,4U U sin se sin , 31Ž . Ž .Ým t m i t j m j t i m1 t 1 14E 4E 4Ei)j

is tiny in contradiction with the SK experiment. Here and in the following we neglect terms with Dm2 in(

Ž .P n ™n . Thus, the Zee mass matrix and large angle solution are not compatible, for u ,0. Settingm t 1

u ,pr2 would correspond to m lm compared to the current case. It also causes an interchange of the two1 1 2Zee Ž .first columns in U , which yields the matrix in Eq. 17 . However this must be discarded for the same reason.1 'Ž .II Here we take u ,arctan1r 2 and obtain1


This case requires the mixing matrix to be

2 1 0( (3 3

1 1 1Zee y( ( (U s . 33Ž .6 3 22

1 1 1� 0y( ( (6 3 2

Giving the angles:8 42 2 2 2 2 2sin 2u s4U 1yU , , sin 2u s4U U , . 34Ž .Ž .CHOOZ e2 e2 atm m2 t 29 9

Hence this is also incompatible with experiments.Ž .III In this third case u ,pr4, whereby1


The mixing matrix is

1 1° ¶0' '2 2

11 1Zee yU s . 36Ž .2 23 '2

11 1y2 2¢ ß'2


Ž .This corresponds to the solution given by U in Eq. 21 . The oscillation probability reads3

Dm2 Latm2P n ™n ,sin , 37Ž . Ž .m t 4Ein good agreement with experiments. Therefore we have a unique solution compatible with large angle inn ™n within the ansatz of the Zee mass matrix. Due to the traceless property this impliesm t

2Dm(2< < < < < <(m , m , Dm , m , , 38Ž .1 2 atm 3 2(2 Dmatm

This is the case of ‘‘pseudo-Dirac’’ since m ,ym . The probabilities for other oscillation processes are as1 2

follows:

Dm2 L(2P n ™n ,1ysin ,Ž .e e 4EDm2 L(1 2P n ™n ,P n ™n , sin . 39Ž . Ž . Ž .e m e t 2 4E

Thus the solar neutrinos are converted into an equal amounts of n and n .m t

The mixing patterns U Zee corresponds to the small angle MSW solution, and the matrix U Zee corresponds to1 3

the large angle MSW or ‘‘just-so’’ solution. It is interesting to notice that the solution of three degenerateŽ . Ž .masses with OO 1 eV is not allowed due to the relation m sym ym and Eq. 26 . This model requires3 1 2

naturally a hierarchical structure for mass matrix in the case of u spr4 and u s0. In conclusion only the2 3Ž .bi-maximal solution, given in Eq. 36 , is feasible, within the framework of the Zee mass matrix, whereas the

solutions like U Zee and U Zee are not allowed.1 2

Above, we have analyzed the Zee mass matrix at the limit u s0. It is also important to discuss the case of3

nonzero u in order to obtain three flavor angles and to have predictions for future experiments. We3

parameterize this asp p

u s qd , u s qd , u sd 40Ž .1 1 2 2 3 34 4Ž . 2with d is1,2,3 <1 and we will neglect terms of order d . The traceless property of the Zee mass matrixi i

Ž .requires the relation d ,8d d , by using Eq. 12 , and yields the approximate mixing matrix3 1 2

1yd 1qd° ¶1 1c c d3 3 3' '2 2

1qd yd qd 1yd yd yd 1qd1 2 3 1 2 3 2y cU , , 41Ž .3Zee '2 2 2

1qd qd yd 1yd qd qd 1yd1 2 3 1 2 3 2y c3¢ ß'2 2 2

2(where c s 1yd and the mass matrix is described as3 3

1y2d yd 1y2d qd° ¶1 2 1 20 y ' '2 2

1y2d yd1 2y 0 y4dM , m . 42Ž .1n 1'2

1y2d qd1 2y4d 01¢ ß'2


Ž . Ž .The ratio of lightest mass to heavy mass is approximately given by d as shown in Eqs. 27 , 42 . Which1Ž .restrictions do the present experiments give for the parameters d ,d ,d ? Taking Eq. 13 and the lower limit for1 2 3

large angle solution of solar neutrino as sin2 2u )0.65 we obtain the following constraints;(

1y2d 2 1y2d 2Ž . Ž .2 32 22 < < < <sin 2u ,4 U U , G0.82 ,atm m3 t 3 2

1y2d 2 1y2d 2Ž . Ž .1 32 22 < < < <sin 2u ,4 U U , G0.65 . 43Ž .( e1 e2 2

< < < <These inequalities with d ,8 d d leads to3 1 2

1 0.82 0.65Ž .2d F 1y . 44Ž .1Ž2. 2 2ž /128d 1y2d Ž .2Ž1. 2 1

We obtain the upper limit for d to be3

< <d F0.28 . 45Ž .3

2 2 < <If the data will be improved as sin 2u )0.95, sin 2u )0.95 we get the upper limit d -0.1. A betteratm ( 3Žestimate can nevertheless be deduced by noticing the following. The eigenvalues of M are m ,y 1yn 1

. 2 Ž .4d m ,4d m . In order to adjust the mass squared difference of Dm using Eq. 38 , d should be chosen as1 1 1 1 ( 1Ž y3 . Ž y8 .OO 10 for MSW or OO 10 for ‘‘just-so’’. This demands also d to be tiny due to the relation d ,8d d .3 3 1 2

Ž y3 . Ž y8 .The value d ,OO 10 or OO 10 is well within the present experimental upper limit. In this case we expect3w xno n ™n oscillations at the LBL experiments 6–8 since,m e

Dm2 Latm2 2P n ™n ,2d sin ,0 . 46Ž . Ž .m e 3 4E

'Ž .We now briefly discuss the remaining solution: u sarctan 1r 2 qd , m ,m ,ym r2, when requir-3 3 1 2 3w xing u spr4qd in the Zee mass matrix. Here we take u spr4qd to obtain the ‘‘maximal’’ case 29 .2 2 1 1

The mixing matrix reads

1yd 1qd 1° ¶1 1

' ' '3 3 3

1qd yd 1yd qd 1yd yd 1qd qd 1qd1 2 1 2 1 2 1 2 2y y yU, 47Ž .' ' '2 22 3 2 3 3

1qd qd 1yd yd 1yd qd 1qd yd 1yd1 2 1 2 1 2 1 2 2y y y¢ ß' ' '2 22 3 2 3 3

and

' '0 1y 2 3 y1 d 1y 2 3 q1 dŽ . Ž .2 2

' 'M ,y m . 481y 2 3 y1 d 0 1y2 3 d Ž .Ž .n 12 2� 0' '1y 2 3 q1 d 1y2 3 d 0Ž . 2 2

' ' Ž .Here d ,y2 2 d d r 3 , from Eq. 12 , and therefore terms with d are neglected in the above matrices.3 1 2 32 y4 2 w xThis ‘‘maximal’’ case is only allowed for Dm -9=10 eV due to results of the CHOOZ experiment 9 .


w xHowever, a three flavor analysis 22 without CHOOZ data shows that the ‘‘maximal’’ solution lies in theregion

1.5=10y3 FDm2 ,Dm2 ,3m2 F6.5=10y3 eV 2 49Ž .32 31 1

at 90%C.L. Combining results from CHOOZ and SK experiments tends to exclude this ‘‘maximal’’ solution,Ž .which predicts large effects for P n ™n in LBL experiments.m e

A comment on neutrino-less double beta decay is now in order. Although we get the heaviest mass< < < <m , m ,0.02;0.08eV for bi-maximal mixing, neutrino-less double beta decay is forbidden in the Zee1 2

model. The effective neutrino mass in the decay

² : < 2 <m s U m 50Ž .Ýn ei n ii

Ž .is exactly zero due to the condition that 1,1 entry in the mass matrix is zero. Generally, a theory which hasŽ .zero entry in 1,1 leads to vanishing neutrino-less double beta decay, even if the neutrino is a Majorana particle

w x30 .Returning to the Zee model which motivated the use of the Zee mass matrix, we find that the requirement of

nearly bi-maximal mixing necessarily yields

d y1 11

y1 d dM , . 51Ž .1 1n � 01 d d1 1

Ž .By using the relation m ,m , in Eq. 2 , we obtainem et

f m2em t 2, ,2.9=10 52Ž .2f met m

and

2'm f 2 Dmet et atm 2 7s , ,10 or 10 53Ž .2m f Dmmt mt (

2 y3 2 2 y5 y10 2 Ž .for Dm ,10 eV and Dm ,10 or 10 eV . The magnitude of f may be estimated using Eq. 3atm ( em

and 0.02-m -0.08 eVem

100 GeV 1y4f , 2;7 =10 54Ž . Ž .em ž /m tanb

where tanbsÕ rÕ and we have taken M s200 GeV and M s300 GeV. We see that the relation of2 1 1 2

couplings must be of the form

f 4 f 4 f 55Ž .em et mt

in order to agree with experiments. This indicates that most likely an extension of the Zee model is needed toŽ . w xgive the anti-hierarchy above. This might be realized by assigning an approximate conserved U 1 31 charge

L'L yL yL , 56Ž .e m t

which will strongly suppress f . Although this cannot explain the factor of ;102 between f and f , it canmt em et7 Ž .account for the more demanding factor of 10 in the ‘‘just-so’’ case. With this extension the matrix in Eq. 51

can be derived from the Zee model. The tiny values on the diagonal will be caused by higher order effects, as< < < < < <mentioned before. Also it is remarkable that this will give an inverse mass hierarchy as m < m , m . This3 1 2


w xpattern is substantially different from those studied in the previous analyses of the Zee model 18,32 . In Ref.w x < < < < < < < < < < Ž .18 , the case m < m , m is taken and the heavy masses m , m adjusted to be 1–5 eV as the1 2 3 2 3

candidates of hot dark matter and the mass squared differences are assigned to be Dm2 sDm2 , Dm2 ,32 atm 312 2 w xDm ,Dm . The addition of one sterile neutrino to the original Zee model is studied in Ref. 32 , giving a31 LSND

< < < <possible explanation of all positive neutrino oscillation experiments, still keeping the assumption of f , fem et

< <, f .mt

The current constraints on the new parameters f , f , f ,m in the Zee model are discussed in detail in Ref.em et mt

w x18 . Unfortunately there are not enough experimental data to determine these parameters. Here we will onlyw xmention about the radiative decays of neutrino and charged leptons induced by Zee boson exchange 15 . In the

Ž .present case the possible radiative decays are n ™n qg . Under the assumption of Eq. 55 the amplitude is1Ž2. 3

given as

e2 n mA n ™n qg s m ym f U U n is q e g n , 57Ž . Ž . Ž .1 3 1 3 em m1 m3 3 mn 5 12 232p M

where

1 cos2f sin2fs q . 58Ž .2 22 M MM 1 2

Here the CP property of n and n is taken to be the same. The decay width and lifetime are1 3

23 2m ym a f U UŽ .1 3 em m1 m32 5< <G n ™n qg s A , m t n ™n qgŽ . Ž .1 3 1 1 32 2ž /8p 2 32p M

2 5y2' '2 2 2 3.2=1045)4=10 years 59Ž .ž /ž /U U m eVŽ .m1 m3 1

w xwhere we have used the present limits obtained in Ref. 18

f 2em y4-7=10 G . 60Ž .F2M

The amplitude for radiative decay of charged leptons also induced by Zee boson exchange at one loop levelis

eX m nA m™eqg s f f u p s q e 1qg u p . 61Ž . Ž . Ž . Ž . Ž .mt et e mn 5 m2 2768p M

For t™eg and t™mg the parameters f f should be replaced with yf f and f f , respectively. Themt et mt em em et

branching ratio is

2a f f 1mt et y27Br m™eqg s -4=10 62Ž . Ž .22ž /3072p GM F

Ž . Ž . Ž .by using the upper limit 60 and Eqs. 52 , 53 . Thus the radiative decays of neutrinos and charged leptons,induced by the Zee boson, are negligible.

In conclusion we have searched for solutions within the framework of the Zee mass matrix, that hasvanishing or in general very small diagonal elements, by taking maximal mixing angle u ,pr4 between n2 m

Ž .and n . The solution we have found is given by Eq. 51 . It corresponds to the bi-maximal solution, whicht


requires large mixing angles for both solar- and atmospheric neutrinos. The two heaviest neutrinos n and n ,1 2

which are approximately degenerate, and the lightest neutrino n are given by3

11 1

n , n y n q n ,1 e m t2 2'2

11 1

n , n q n y n , 63Ž .2 e m t2 2'2

1 1n , n q n .3 m t' '2 2

The solution thus found requires a large hierarchy for the couplings of the Zee boson to leptons, in the formf 4 f 4 f in contrast to ‘‘natural’’ expectations. It is therefore desirable to impose an approximateem et mt

L yL yL symmetry in the Zee model in order to explain this hierarchy. The model will undergo severe testse m t

in the future neutrino experiments.

Acknowledgements

Ž . Ž .One of the authors M.M expresses his thanks to the Royal Swedish Academy of Sciences RSAS for theŽ .support by a grant. M.T is thankful to the High Energy Group in CFIFrIST Portugal for their hospitality. His

work is supported by the Grant-in-Aid for Science Research, Ministry of Education, Science and Culture, JapanŽ .No.1014028, No.10640274 .

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w x Ž .2 MACRO Collaboration, M. Ambroiso et al., Phys. Lett. B 434 1998 451.w x Ž .3 Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81 1998 1562.w x Ž .4 Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81 1998 1158.w x Ž . Ž .5 GALLEX Collaboration, P. Anselmann et al., Phys. Lett. B 327 1994 377; B 388 1996 384; SAGE Collaboration, J.N.

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w x Ž .6 Y. Suzuki, in: K. Enqvist et al. Eds. , Proc. of Neutrino 96, Helsinki, June 1996, World Scientific, Singapore, 1997, p. 237.w x Ž .7 S.G. Wojcicki, in: K. Enqvist et al. Eds. , Proc. of Neutrino 96, Helsinki, June 1996, World Scientific, Singapore, 1997, p. 231.w x8 ICARUS Collaboration, P. Cennini et al., LNGS-94r99-I, May 1994; F. Pietropaola, invited talk at Neutrino 98, June 1998,

Takayama, Japan.w x Ž .9 The CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B 420 1998 397.

w x Ž . Ž .10 N. Cabibbo, Phys. Rev. Lett. 10 1963 531; M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 1973 652.w x Ž .11 Z. Maki, M. Nakagaa, S. Sakata, Prog. Theor. Phys. 28 1962 870.w x Ž .12 M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwenhuizen, D. Freedmann Eds. , Supergravity, Proceedings of the Workshop,

Ž .Stony Brook, New York, 1979, North-Holland, Amsterdam, 1979, p. 315; T. Yanagida, in: O. Sawada, A. Sugamoto Eds. ,Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, Tsukuba, Japan, 1979, KEK Report No.79-18, Tsukuba, 1979, p. 95.

w x Ž . Ž .13 A. Zee, Phys. Lett. B 93 1980 389; B 161 1985 141.w x Ž .14 L. Wolfenstein, Nucl. Phys. B 175 1980 92.w x Ž .15 S.T. Petcov, Phys. Lett. B 115 1982 401.w x Ž . Ž . Ž .16 J. Liu, Phys. Lett. B 216 1989 367; W. Grimus, H. Neufeld, Phys. Lett. B 237 1990 521; B.K. Pal, Phys. Rev. D 44 1991 2261;

Ž .W. Grimus, G. Nardulli, Phys. Lett. B 271 1991 161.w x Ž .17 A.Yu. Smirnov, Z. Tao, Nucl. Phys. B 426 1994 415.


w x Ž .18 A.Yu. Smirnov, M. Tanimoto, Phys. Rev. D 55 1997 1665.w x Ž .19 S.M. Bilenky, S.T. Petcov, Rev. Mod. Phys. 59 1987 671.w x Ž . Ž . Ž .20 LSND Collaboration, C. Athanassopoulos et al., Phys. Rev. Lett. 75 1995 2650; 77 1996 3082; 81 1998 1774; Phys. Rev. C 54

Ž . Ž .1996 2685; J.E. Hill, Phys. Rev. Lett. 75 1995 2654.w x21 KARMEN Collaboration, B. Armbruster et al., hep-exr9809007; B. Zeitnitz, invited talk at Neutrino 98, June 1998, Takayama, Japan.w x22 G.L. Fogli, E. Lisi, A. Marrone, G. Scioscia, hep-phr9808205.w x Ž .23 J.N. Bahcall, P.I. Krastev, A.Yu. Smirnov, Phys. Rev. D 58 1998 6016.w x24 Private communication with J.N. Bahcall; The large angle MSW solutions are excluded only if one fits the SK electron recoil energy

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11 March 1999


Loop transfer matrix and gonihedric loop diffusion

T. Jonsson 1, G.K. SavvidyNational Research Center Demokritos, Ag. ParaskeÕi, GR-15310 Athens, Greece

Received 30 November 1998Editor: L. Alvarez-Gaume

Abstract

We study a class of statistical systems which simulate 3D gonihedric system on euclidean lattice. We have found theŽ .exact partition function of the 3D-model and the corresponding critical indices analysing the transfer matrix K P , P whichi f

describes the propagation of loops on a lattice. The connection between 3D gonihedric system and 2D-Ising model is clearlyseen. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

w xIn the articles 3 the authors formulated a modelof random surfaces with an action which is propor-tional to the linear size of the surface. The model hasa number of properties which make it very close tothe Feynman path integral for a point-like relativisticparticle. This is because in the limit when the surfacedegenerates into a single world line, the action be-comes proportional to the length of the path 2.

In addition to the formulation of the theory in thecontinuum space the system allows an equivalentrepresentation on Euclidean lattices where a surface

w xis associated with a collection of plaquettes 6 . Inthese lattice spin systems the interface energy coin-

1 Permanent address: University of Iceland, Dunhaga 3, 107Reykjavik, Iceland.

2 The problems of spiky instability and the convergence of thew xpartition function have been studied in Refs. 3–5 .

cides with the linear-gonihedric action for randomsurfaces. This gives an opportunity for analytical

w x w xinvestigations 7–9 and numerical simulations 9,10of the corresponding statistical systems.

Additional understanding of the physical be-haviour of the system comes from the analysis of the

w xtransfer matrix 7 which describes the propagationof the closed loops-strings P in time direction withan amplitude which is proportional to the sum of the

Ž .length l P of the string and of the total curvatureŽ .k P . In this article we shall study the physical

picture of string propagation which was suggested inw xthe transfer matrix approach 7 .

The partition function of the system is defined asw x3

Z b s exp yb A M ,� 4Ž . Ž .Ýgonihedric� 4M

< <A M s l P pya , 1Ž . Ž .Ý i j i j² :ij


( )T. Jonsson, G.K. SaÕÕidyrPhysics Letters B 449 1999 253–259254

² :where l sa is the length of the edge ij which isi j

equal to the lattice spacing a on the cubic lattice, a i j

is the dihedral angle between two neighbouring pla-quettes of the singular surface M sharing a common

² : 3 Ž . � 4edge ij , a s0,pr2,p . In 1 M denote thei j

set of closed singular surfaces on the three-dimen-sional toroidal lattice T 3 of size N=N=N. Asingular surface M is a collection of plaquettes in T 3

such that eÕery link is contained in 0, 2 or 4plaquettes and eÕery plaquette of the lattice T 3 can

w xbe occupied only once 6 . The surfaces are closedbecause only an even number of plaquettes meet at agiven lattice link. The singular surfaces of interfacewhich describe the states of arbitrary three-dimen-sional spin system can be viewed as a set of surfaces� 4 w xM 13,6 .

w xIn Ref. 7 it has been proven that the partitionŽ .function 1 can be represented in the form

Z bŽ .s K P , P PPP K P , PŽ . Ž .Ý b 1 2 b N 1

� 4P , P , . . . , P1 2 N

s trK N , 2Ž .b

Ž .where K P , P is the transfer matrix of size g=g ,b 1 2Ž Ž . Ž . w x.defined as see formulas 15 , 16 of Ref. 7

K gonihedric P , PŽ .b 1 2

sexp yb k P q2 l P n P qk P ,� 4Ž . Ž . Ž .1 1 2 2

3Ž .

where P and P are closed polygons on a two-di-1 2

mensional toroidal lattice T 2 of size N=N and g isthe total number of polygon-loops on a toroidal

2 { }lattice T . Closed polygons P 'P are associatedwith the collection of links on T 2 with the restrictionthat only an eÕen number of links can intersect at agiÕen Õertex of the lattice and that the links can beoccupied only once. 4

Ž .The transfer matrix 3 can be viewed as describ-ing the propagation of the polygon-loop P at time t1

3 Usually we take as1.4 The polygon-loops P appear as the intersection of the singu-

lar surfaces M with the planes between coordinate planes in T 3

w x7 .

to another polygon-loop P at the time tq1 5. The2Ž .functional k P is the total curvature of the

polygon-loop P which is equal to the number ofŽcorners of the polygon the vertices with self-inter-. Ž .section are not counted and l P is the length of P

which is equal to the number of its links. The lengthŽ . Ž Ž .functional l P n P is defined as see formula 121 2

w x .of 7

l P n P s l P q l P y2 l P lP , 4Ž . Ž . Ž . Ž . Ž .1 2 1 2 1 2

where the polygon-loop P n P 'P jP _P lP1 2 1 2 1 2

is a union of links P jP without common links1 2

P lP . The operation n maps two polygon-loops1 2

P and P into a polygon-loop PsP n P 6. The1 2 1 2Ž .length functional l P n P defines a distance be-1 2

tween two polygon-loops P and P 7. The ampli-1 2Ž .tude decreases when the distance 4 between two

configurations increases, this follows from the in-equality

< <l P n P G l P y l P , 5Ž . Ž . Ž . Ž .1 2 1 2

thereforegonihedric < <K P , P Fexp y2b l P y l P .� 4Ž . Ž . Ž .b 1 2 1 2

6Ž .

The expression for the transition amplitudeŽ .K P , P in the polygon-loop space P is veryb 1 2

close in its form with the transfer matrix for therandom walks

< <� 4K X ,Y sexp yb XyY , 7Ž . Ž .b

which depends only on the distance between initial

5 Layer-to-layer transfer matrices for three-dimensional statisti-cal systems, whose elements are the product of all Boltzmannweight functions of cubes between two adjacent layers have been

w xconsidered in the literature 11,12 . Using Yang-Baxter and Tetra-hedron equations one can compute the spectrum of the transfer

w xmatrix in a number of interesting cases 12 . In the given case theŽ .transfer matrix 3 has geometrical interpretation which helps to

compute the spectrum.6 Note that the operations j and l do not have this property.

These operations acting on a polygon-loops can produce linkconfigurations which do not belong to P . The symmetric differ-ence of sets P n P is an important concept in functional analysis1 2w x15 .

7 Algebraically one can construct many functionals of that kind,but what is important here is that this distance functional appears

Ž .naturally from the geometrical action 1 of the original theory.

( )T. Jonsson, G.K. SaÕÕidyrPhysics Letters B 449 1999 253–259 255

w xand final position of the point particle 14 . The aimof this work is to study spectral properties of the

Ž .transfer matrix K P , P and critical behaviour ofb 1 2Ž .the statistical system 2 .

Ž .The eigenvalues of the transfer matrix K P , Pb 1 2

define all statistical properties of the system and canbe found as a solution of the following integralequation in the loop space P 8

K P , P C P sL b C P , 8Ž . Ž . Ž . Ž . Ž .Ý b 1 2 2 1� 4P2

Ž .where C P is a function on loop space. The HilbertŽ .space of complex functions C P on P will be

2Ž .denoted as HsL P . The eigenvalues define theŽ .partition function 2

Z b sLN q ...q LN , 9Ž . Ž .0 g

and in the thermodynamical limit the free energy isequal to

1yb f b s lim ln Z b . 10Ž . Ž . Ž .3NN™`

The correlation lengths are defined by the ratios ofŽ . Ž . Ž .eigenvalues L b rL b as j b s 1ri 0 i

Ž Ž . Ž ..yln L b rL b , and grow if the eigenvaluesi 0Ž . Ž .L b approach the eigenvalue L b at somei 0

critical temperature b . By the Frobenius-Perron the-cŽ . Ž .orem L b is simple and we have L b )0 0

Ž . Ž .L b GL b G ... Finite time propagation ampli-1 2

tude of an initial loop P to a final loop P for thei f

time interval tsMrb can be defined as

K P , P sLyM K P , P PPPŽ .Ž . Ýi f 0 b i 1� 4P , P , . . . , P1 2 My1

=K P , P , 11Ž .Ž .b My1 f

where we have introduced natural normalization tothe biggest eigenvalue L and MFN. The trace of0

Ž . Žthe operator K P , P is equal to TrKs 1qi f

8 We shall use the word ‘‘loop’’ for the ‘‘polygon-loop’’.

Ž .M Ž .M .L rL q PPP q L rL and depends on the1 0 g 0

ratio L rL .i 0

2. 3D gonihedric system and 2D-Ising model

We consider below a transfer matrix which is lessŽ .complicated than the original matrix 3 and depends

Ž . 9only on the distance functional l P n P1 2

2K P , P sexp y2b l P n P yN .Ž . Ž .� 4b 1 2 1 2

12Ž .

The largest eigenvalue appearing in the equation

exp y2b l P n P q2bN 2 PC PŽ . Ž .� 4Ý 1 2 2� 4P2

sL b C P 13Ž . Ž . Ž .1

can be found because the corresponding eigenfunc-Ž . w xtion is a constant function C P s1 16 . Therefore0

L s exp y2b l P n P q2bN 2 . 14Ž . Ž .� 4Ý0 1 2� 4P2

Ž .The sum 14 does not depend on P . To prove this1

we note that the loop PsP n P runs over all1 2

loops in P as P runs over P , i.e. the mapping2

P ™P n P is one to one for any P . This change1 1 2 2

of the variable proves that

L s exp y2b l P q2bN 2 , 15� 4Ž . Ž .Ý0� 4P

so L is the partition function of the 2D-Ising0w xferromagnet. Indeed 1 ,

Z 2 D s exp y2b l P q2bN 2 seyb f Ž b .PN 2,� 4Ž .ÝIsing

� 4P

16Ž .Ž .where f b is the free energy of the 2D-Ising

model. Therefore

L sZ 2 D slN q ...q lNN , 17Ž .0 Ising 0 2

where l are the eigenvalues of the transfer matrixiw xof 2D-Ising model 1,2 . Thus the largest eigenÕalue

( )of the 3D-system 12 is equal to the partition func-tion of the 2D-Ising ferromagnet.

9 It is convenient to subtract the vacuum energy 2 N 2 in theexponent.


The free energy of the 2D-Ising ferromagnet inw xthe thermodynamical limit is given by 1,2

1 dj dh2p 24yb f b s ln 1qwŽ . Ž .H 22 0 2pŽ .2 6y2 w yw cosjqcosh , 18Ž . Ž . Ž .

where wseyb . Therefore the free energy of thethree-dimensional system which is defined by the

Ž . Ž .transfer matrix 12 is given by f b and coincideswith the one of 2D-Ising ferromagnet.

From this result we can deduce that the criticalŽ .temperature of the three-dimensional system 12 is

equal to the one for the 2D-Ising ferromagnet 2b sc'Ž .ln 2 y1 , that the specific heat exponent as0and from the hyperscaling law n ds2ya that ns2r3.

w xWe recall 7 that in the alternative approximationŽ .when the intersection term 2k P lP is ignored in1 2

Ž .the original transfer matrix 3

K P , P sexp yb k P n PŽ . Ž .�0b 1 2 1 2

2q2 l P n P y2 N 19Ž . Ž .41 2

w xthe free energy can also be computed 7

dj dh2p 21 4yb f b s ln 1qwŽ . Ž .H0 2 20 2pŽ .

y4w8v 2 1yv 2Ž .q4w4 1yv 2 cosj coshŽ .y2 w2 y2w6v 2 qw6Ž .

= cosjqcosh , 20Ž . Ž .

where v 2 sw is the contribution from the curvatureŽ .term k P and exhibits the same critical behaviour

as the 2D-Ising ferromagnet. Thus the original goni-Ž . Ž .hedric system 2 , 3 is bounded by two close

statistical systems

yb f b Fyb f b Fyb f b , 21Ž . Ž . Ž . Ž .0 gonihedric

because

K P , P FK gonihedric P , P FK P , P .Ž . Ž . Ž .0b 1 2 b 1 2 b 1 2

22Ž .w xThis confirms the conjecture 7 that 3D gonihedric

system should have statistical properties close to the

ones of 2D-Ising ferromagnet. Earlier numerical sim-w xulations 10 support this dimensional ‘‘reduction’’.

It is also consistent with the analytical estimate ofthe entropy factor of the random surfaces on a cubic

w x 10lattice 8

3. Loop space and eigenfunctions

To find the rest of the eigenfunctions it is conve-nient to introduce some notation. An invariant prod-uct in P can be defined as

² < :P P s l P n P y l P n P , 23Ž . Ž .Ž .1 2 1 2 1 2

2 2where PsT n PsT _P is the complement oper-ation and T 2 is the loop which contains all links of

2Ž Ž 2 . 2 .the toroidal lattice T l T s2 N . The invariantŽ .product 23 is also equal to

² < : 2P P s2 N y2 l P n P 24Ž . Ž .1 2 1 2

² < : ² < :and is odd with respect to P, P P sy P P .1 2 1 2

In particular the energy functional of the system isequal to

² < : 2E s 0 P s2 N y2 l P . 25Ž . Ž .P

Ž .The product 23 is invariant under the simultaneousrotations of the loops P and P , defined as1 2

P™P nd , 26Ž .² <where d is an arbitrary loop because P nd1

: ² < :P nd s P P . This group of transformations2 1 2

in the loop space P is Abelian because for itsŽ . Ž .representations on H R C P sC P nd , we haved

2R , R s0, R s1. 27Ž .d d d1 2

² < :The product P P is invariant also under transla-1 2

tions. The group of translations is defined as a rigid

10 � 4The model becomes trivial if the loops P are not restrictedto be closed. In the model where all subsets of links from T 2 areallowed as loop configurations the transfer matrix is the 2 N 2 thtensor product of 1D-Ising model transfer matrix K 1 D. Using thisb

Ž y2 b .2 N 2y1 Žobservation one can derive inequality L F 1q e 1y1y2 b . 2 De rZ .Ising


translation of the loop P in x and y directions onT 2 by same units of lattice spacing aP™Pqa e qa e . 28Ž .x x y y

Together these two groups form a Non-Abelian groupwhich acts on the loop space P . With this notation

Ž .the transfer matrix 12 takes the form² < :K P , P sexp b P P 29� 4Ž . Ž .b 1 2 1 2

Ž .and the integral equation 13 the form

e b²P1 < P2:PC P sL b C P . 30Ž . Ž . Ž . Ž .Ý 2 1� 4P2

Ž . Ž .Note that 30 is invariant under rotations 26 . Thissuggests that we search for eigenfunctions of the

Ž .operator 29 in the form of invariant polynomials in2Ž .HsL P . These polynomials can be constructed

² < :in the form of powers of the invariant P Q asfollows:

C Ž0. P s1,Ž .Q

Ž1. ² < :C P s P Q ,Ž .Q

Ž2. ² < :2C P s P Q yr ,Ž .Q 2

...Žn. ² < :n ² < :ny2

C P s P Q yr P Q y ... 31Ž . Ž .Q n

where the coefficients r are chosen so that C Ž i. isn QŽ j. ² < :2orthogonal to C if i / j, r sÝ 0 P rQ 2 �P 4

� Žn.Ž .4Ý 1 ,... and so on. The set of functions C P in�P 4 Q

H form an invariant subset H of the level n. Then

index Q numerates functions inside the level. Weintroduce general Legandre loop polynomials as

L x sx n yr x ny2 y ... 32Ž . Ž .n n

Ž² < :. Žn.Ž .so L P Q sC P . This set of functions in Hn QŽ .is appropriate for solving the integral Eq. 30 and in

its form is very similar with the ones which are usedfor the random paths with curvature-dependent ac-

w xtion 14 .Ž1.Ž .Let us begin by proving that C P is theQ

Ž . Ž .eigenfunction of 30 . Because see the Appendix² < :n ² < : ² < :Q P P P Q sm Q Q 33Ž .Ý 1 2 n1 1 2

� 4P

where1

nq1² < :m s 0 P , m s0 34Ž .Ýn1 2 k 122 N � 4P

we haveb²P1 < P2: ² < : ² < :e P P Q sL b P Q 35Ž . Ž .Ý 2 1 1

� 4P2

where

1 EL s L . 36Ž .1 02 Eb2 N

For the ratio L rL one can get1 0

y1 Ey L rL s ln L su b 37Ž . Ž . Ž .1 0 02 Eb2 N

which is the internal energy of the 2D-Ising system.Thus the second eigenÕalue of the three-dimensionalsystem coincides with internal energy of the two-di-mensional Ising system.

Ž .Using Eq. 33 one can also address the questionof the degeneracy of this eigenvalue. The scalarproduct of the loop functions can be defined as

C PC s C P C P 38Ž . Ž . Ž .ÝQ Q Q Q1 2 1 2� 4P

Ž .then for the zero and first level functions 31 wehave

Ž0. Ž1. Ž1. Ž1. ² < :C PC s0, C PC sm Q Q . 39Ž .Q Q Q Q 11 1 21 2 1 2

² < :The rank of the matrix M s Q Q definesQ Q 1 21 2

the number e of linearly independent functions on1

the first level H and thus the degeneracy of the1

eigenvalue L 11 The number e is bigger or equal1 1

to the number of functions C Ž1. which are orthogo-Qi

nal to each other. The orthogonality condition fol-Ž .lows from 39

² < : 2Q Q s0, i , js1,2, . . . , 2 N .i j

The last equation can be rewritten also in the formŽ . 2 2l Q nQ sN and its solutions provide 2 N lin-i j

early independent functions on the first level. Thesolutions will be presented in a separate place.

4. Approximate eigenfunctions

Above we have found two exact eigenfunctionsC Ž0., C Ž1. and the corresponding eigenvalues of theQ Q

Ž .transfer matrix 12 . Since the sums

² < :n ² < :kQ P P P Q , kG2 40Ž .Ý 1 2� 4P

² < :depend not only on invariant Q Q , but also on1 2

the shape of Q and Q it is difficult to find out1 2

11 The degeneracy of the zero level is one e s1.0


exact expressions for the rest of the eigenfunctionsand in the following we shall search for approximatesolutions. At high temperature the sum depends onlyweakly on the shape of QXs and the leading be-

w xhaviour involves only the length 16 . In this approx-imation we have

² < :n ² < :2 ² < :2Q P P P Q fm Q Q qm ,Ý 1 2 n2 1 2 n0� 4P

41Ž .

where

1nq2 n² < : ² < :m s 0 P yr 0 P ,Ž .Ýn2 2222 N yrŽ . � 4P2

r2m syn0 222 N yrŽ . 2

=2nq2 n2² < : ² < :0 P y 2 N 0 PŽ .Ý ž /

� 4P

Ž .and a similar formulas for ks3,4... in 40 . Usingthese expressions one can find eigenvalues as corre-sponding derivatives of the Ising partition function

1 EL b f L L b , 42Ž . Ž . Ž .n n 02 ž /EbL 2 NŽ .n

Ž .like it was in 36 .

5. Appendix

Ž .The equality 33 can be proven by using the factŽ .that the l.h.s. of 33 is equal to

² < :n ² < :0 P P P QÝ� 4P

² < :nq1 ² < :ns 0 P y2 l Q 0 PŽ .Ý Ý� 4 � 4P P

² < :nq4 0 P P l PlQ . 43Ž . Ž .Ý� 4P

Ž . Ž .The last term in 43 is a linear function of l Q . IfŽ . ŽQsQ jQ and Q lQ s0 then l PlQ s l P1 2 1 2

. Ž .lQ q l PlQ and we have1 2

² < :n0 P P l PlQ sh l Q . 44Ž . Ž . Ž .Ý n1� 4P

The normalization constant can be computed at thepoint QsT 2

1n² < :h s 0 P l P . 45Ž . Ž .Ýn1 22 N � 4P

Ž .The linearity of the last term in 43 with respect toŽ .l Q follows also from the decomposition of the

Ž .functional l PlQ into the sum over the links of QŽ . Ž . Ž .l PlQ sÝ x P where x P s1 if igPi g Q i i

² < :nand zero otherwise. By homogenuity Ý 0 P P�P 4Ž .x P is independent of i. Hence the summationi

Ž .over igQ gives 44 . We have to compute also theŽ .sum in 40 for ks2, which by the transformation

Ž .similar to 43 can be expressed in terms of thetwo-set correlation function

² < :n0 P P l PlQ P l PlQ . 46Ž . Ž . Ž .Ý 1 2� 4P

The last correlation function is a sum over igQ1

and jgQ of the two-link correlation function2

² < :n0 P Px P x P . 47Ž . Ž . Ž .Ý i j� 4P

In the approximation when the two-link correlationŽ .function 47 factorises, then the correlation function

Ž . Ž . Ž . Ž .46 is equal to h l Q l Q qh l Q lQ andn2 1 2 n0 1 2Ž .we recover the expression 41 .

For the high powers we have to compute a many-set correlation function

² < :n0 P P l PlQ sh l Q q PPPŽ . Ž .Ý Ł Ła na aa a� 4P

48Ž .which can be expressed as a function of lengths inthe approximation when many-link correlation func-tion

² < :n0 P Px P x P x P PPP , 49Ž . Ž . Ž . Ž .Ý i j k� 4P

Ž .factorises. Because the link variable x P is ai

quadratic function of Ising spins it follows thatŽ .many-link correlation function 49 is nothing else

than many-spin correlation function of the 2D-Isingmodel.

Acknowledgements

Ž .One of the authors T.J. is indebted to the Na-tional Research Center Demokritos for hospitality.This work was supported in part by the EEC Grantno. ERBFMBICT972402.


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w x Ž .12 A.B. Zamolodchikov, Commun. Math. Phys. 79 1981 489;Ž .V.V. Bazhanov, R.J. Baxter, J. Stat. Phys. 69 1992 453.

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11 March 1999


Cooper pairing at large N in a two-dimensional model

Alan Chodos a, Hisakazu Minakata b, Fred Cooper c,d

a Center for Theoretical Physics, Yale UniÕersity, P.O. Box 208120, New HaÕen, CT 06520-8120, USAb Department of Physics, Tokyo Metropolitan UniÕersity, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan

c Theoretical DiÕision, Los Alamos Scientific Laboratory, Los Alamos, NM 87455, USAd Physic Department, Boston College, Higgins Hall, 140 Commonwealth AÕenue, Chestnut Hill, MA 02167-3811, USA


Abstract

We study a two-dimensional model of fermi fields c that is closely related to the Gross-Neveu model, and show that to² :leading order in 1rN a cc condensate forms. This effect is independent of the chemical potential, a peculiarity that we

expect to be specific to two dimensions. We also expect the condensate to be unstable against corrections at higher orders in1rN. We compute the Green’s functions associated with the composite cc , and show that the fermion acquires a Majoranamass proportional to the gap, and that a massless Goldstone pole appears. q 1999 Elsevier Science B.V. All rights reserved.

w xRecently several papers have appeared 1–3 dealing with the properties of QCD at high density. The basicprocedure is to approximate QCD by a direct four-quark interaction term, justifying this either by appeal toinstanton effects or to one-gluon exchange. In the presence of a chemical potential, this theory admits acondensate of quark-quark pairs, very similar to the Cooper pairs that are well-known in the BCS theory ofsuperconductivity. This phenomenon, which has been dubbed ‘‘color superconductivity’’, may or may not beaccessible to experiments on heavy-ion collisions that will be performed over the next few years.

In this paper we shall examine similar phenomena in the context of a one-plus-one dimensional model that isŽ . w xa close relative to the Gross-Neveu GN model 4 . From its inception, it has been recognized that the GN

model exhibits many of the same features as QCD, such as asymptotic freedom and spontaneous chiralsymmetry breaking. Moreover, the four-fermi interaction is the model - there is no need to regard it as anapproximation to an underlying gauge theory. Unlike in higher dimensions, in two dimensions this interaction isrenormalizable, and we shall find, just as in the usual GN model, that coupling constant renormalizationremoves all the divergences that we shall encounter. Another advantage is that by judiciously introducing aflavor index i, 1F iFN, one can insure that the mean-field approximation that is commonly used in analyzingthe condensate in QCD is in the case of the two-dimensional model justified as the leading contribution inpowers of 1rN.

There are, however, a couple of peculiarities associated with two dimensions that make this GN-like modelw xqualitatively different from QCD. The first is the Coleman-Mermin-Wagner theorem 5 , which forbids

spontaneous symmetry breaking of a continuous symmetry. Whereas in the original GN model the broken² :symmetry is a discrete one, and hence not in conflict with the theorem, in this case the formation of a cc


( )A. Chodos et al.rPhysics Letters B 449 1999 260–266 261

condensate breaks fermion number, a continuous symmetry. The same problem arises in the chiral GN modelw x4 , where the symmetry is continuous, and in a variety of other two-dimensional models where the spontaneousbreaking of a continuous symmetry is predicted in leading order in 1rN. This means that instabilities must arise

w xin higher order that vitiate the prediction of a condensate. However, as Witten has pointed out 6 , the 1rNexpansion may still be an excellent guide to the physics of the model, except for the formation of the condensateŽwhat happens in these models is that the condensate ‘‘almost’’ forms, in the sense that the pair-pair correlation

.function decays in the infrared only like a power instead of exponentially, and the power vanishes as N™` .The second peculiarity is, as we shall show below, the chemical potential has nothing to do with the

formation of the condensate. In higher dimensions the chemical potential is crucial, because it gives rise to theFermi surface at which the gap equation has an infrared singularity as the gap goes to zero. It is this feature thatinsures that the gap equation will have a solution for arbitrarily weak coupling. In two dimensions, however, theFermi surface has dimension zero, and the infrared singularity exists whether or not there is a chemicalpotential. In fact, the gap equation turns out to be completely independent of the chemical potential. Thisbehavior will be exhibited explicitly below.

The model we consider is defined by the following Lagrangian:

Ž i. Ž i. 2 Ž i. Ž j. Ž i. Ž j.LLsc iEuc q2 g c g c c g c . 1Ž .5 5

Ž i. Ž .c is a two-component spinor with a flavor index that takes on N values. Repeated flavor indices in Eq. 1are summed. Because of the unconventional arrangement of flavor indices in the second term, the model does

Ž . Ž . Ž i. i j Ž j. i jnot have SU N symmetry, but it does possess O N symmetry, c ™q c where q is a real N=NŽ . Ž . ² :orthogonal matrix. The Lagrangian 1 also has a U 1 symmetry, which we shall find is broken by a cc

Ž .condensate, whereas the O N symmetry is kept intact.Our representation for the g-matrices is: g 0 ss ; g 1 syis ; g ss , and it is then easy to check that1 2 5 3

1Ž i. Ž j. Ž i. Ž j. †Ž i. †Ž i. Ž j. Ž j.c g c c g c sy e c c e c c . 2Ž .Ž . Ž .5 5 a b a b gd g d2

w xFollowing the usual Hubbard-Stratonovich procedure, in the form introduced by Coleman 7 , we add to LL theterm

1† 2 †Ž i. †Ž i. 2 Ž j. Ž j.y B yg e c c Bqg e c c 3Ž .Ž . Ž .ab a b gd g d2g

which does not affect the physics because B and B† are simply auxiliary fields. We then have

1Ž i. 0 Ž i. † †Ž i. †Ž i. † Ž i. Ž i.LLsc iEuymg c y B BqB e c c yB e c c 4Ž .Ž . ab a b a b a b2g

where we have also introduced a chemical potential m. In anticipation of taking the large N limit, N™` withN2 †lsg N fixed, we rewrite the second term as y B B.l

Ž . Ž † .Thus the classical or tree-level term in V B B is of order N. As we perform a perturbation expansion, weeffŽ . † Ž .observe that additional factors of N arise in 2 ways: i the ByB propagator is proportional to 1rN; and ii

each closed fermion loop gives a factor of N from summing on the flavor index. If we examine the computationŽof higher-loop corrections to the effective potential i.e. the summation over all one-particle-irreducible

† . Ždiagrams with zero-momentum B and B external legs we see that the one-loop term is of order N one† . 0fermion loop and no ByB propagator whereas anything else is of order N or lower. Hence the leading

contribution is just what we get from keeping the tree and one-loop graphs. Moreover, in a path-integralapproach, one first integrates out the fermions. Then, because the exponent is proportional to N, one can employthe stationary phase approximation in the integral over B and B† to evaluate the integrand at the solution of theequations

E V E Vs s0 . 5Ž .†E B E B

( )A. Chodos et al.rPhysics Letters B 449 1999 260–266262

The task of integrating out the fermions is complicated slightly by the cc and c †c † terms. We observe that

Dc eiŽc , MMc .sdet1r2MM , 6Ž .Hwhere MM is any anti-symmetric matrix. We write the fermion part of our Lagrangian as

1 † T † † † †LLs c Acyc A c qc BBc qc BB c 7Ž .Ž .2

where

As iE q is E ym d i j ; 8aŽ . Ž .ab0 3 x

AT s yiE y is E ym d i j ; 8bŽ . Ž .ab0 3 x

BBsBe d i j s iB s d i j ; 8cŽ . Ž .aba b 2

BB† syiB† s d i j . 8dŽ . Ž .ab2

Now we perform a translation,

csxqac † 9aŽ .sxqc †a T 9bŽ .

1 † y1 T †Ž .where as BB A ; this factorizes the x and c path integrals into the product of two path integrals, and2

after some manipulations and discarding an overall factor that is independent of BB and BB†, we obtain

Ž1. † y1i G ŽB , B . 1r2 y1 T †effe sdet 1q4 A BB A BB 10Ž . Ž .

where G Ž1. denotes the one-loop contribution to the effective action.eff

Note that the matrix Ay1 is given by

2 w x i j i kPŽ xyy.d k k ys k qm d eab0 3 1y1A x , y sy 11Ž . Ž .H 2 2 22p k qmq ie sgnk ykŽ . Ž .0 0 1

where kPxsk x 0 qk x1, and the i e prescription is introduced in the proper way to take account of the0 1Ž T .y1chemical potential. A is the same expression but with x and y interchanged.

To obtain the effective potential, we take B and B† to be constant. It is then convenient to write everythingin momentum space, and after some algebra, using G syV Hd2 x, we obtaineff eff

† 2 †NB B iN d k 4B BV s q tr ln 1yHeff 2l 2 k qmq i e sgnk qk s k ymq i e sgnk yk sŽ . Ž .2pŽ . 0 0 1 3 0 0 1 3

12Ž .

where the trace is over the spinor indices only. Setting B†BsMr4 and ks4l, and observing thatŽ .V Ms0 s0, we can writeeff

1 1 dVM XV s dM , 13Ž .H XeffN N dM0

where

21 dV 1 i d k 1s q tr . 14Ž .H 2 22N dM k 2 2p Myk q k s qm y i eŽ . Ž .0 1 3


Here the trace is just summation over s s"1.3

We note that this integral is logarithmically divergent. We shall deal with this by renormalizing k ; but firstŽ .2 2we shall do the k integral. Let Mq k s qm sv with v)0. We have0 1 3

` `1 1 p iIs dk s dk sy 15Ž .H H0 02 k yvq ie k qvy ie vk yv q ie Ž . Ž .y` y` 0 00

so

`1 dV 1 1 1s y dk tr . 16Ž .H 1 ž /N dM k 8p vy`

We renormalize by requiring that at ms0,

21 E V 4effs . 17Ž .†

†N kE BE B B BsM r4 R0

which is the same as

21 1 E V E V2s qM . 18Ž .0 2ž /k N E M E MR M 0

The solution to this is

`1 1 1 dk1s q qd X 19Ž .H

k k 4p vy`R 0

where v 2 sk 2 qM , and where for our choice of renormalization prescription, d Xsy1r4p . Some other0 1 0

choice of prescription would yield a different pure number for d X.We then have

`1 dV 1 1 1 1s y dk tr y qd X . 20Ž .H 1N dM k 8p v vy`R 0

The gap equation is just the statement that dVrdM vanishes:

`1 1 1 1d Xq s dk tr y . 21Ž .H 1

k 8p v vy`R 0

Note that

1 1 1tr s q

2 2v ( (Mq k qm Mq k ymŽ . Ž .1 1

which is even in k . Therefore1

1 1d Xq s J , where J is the integral 22Ž .

k 4pR

` 1 1 2Js dk q y . 23Ž .H

2 2 20 (M qk( (Mq kqm Mq kymŽ . Ž . 0


Ž .If we evaluate J, we can find the particular M that obeys Eq. 22 , thereby solving the gap equation. We canalso obtain the more general expression

1 dV 1 1s y Jqd X

N dM k 4pR

Ž . Ž .as a function of M, and integrate it to obtain V M via Eq. 13 .We find, after some mild computational exertions, that

J M syln MrM 24Ž . Ž .0

and

1 1 M 1V M s qd X Mq ln MrM y1 qq . 25Ž . Ž . Ž .0 ž /ž /N k 4p NR

As advertised, these expressions are independent of m. The solution to the gap equation for our choice of d X is

MsM eŽ1y4p rk R . . 26Ž .0

Ž .There is no critical lower bound to k below which no solution exists. We see from Eq. 23 that this is dueR

to the fact that as M™0, the expression for J diverges logarithmically. This infrared singularity is present in 2dimensions independent of the value of m. In higher dimensions, we expect this singularity to be present at theFermi surface, ksNmN , and to disappear as m™0.

Ž .We see from Eq. 26 that as M is increased for fixed M, k becomes smaller. This is an indication that the0 R

coupling k is asymptotically free, just as in the original GN model. In fact, it is not hard to show from theRŽ .renormalization condition 19 with cutoff L that the beta function has the form

yk 2

b k s ,Ž .2p

Ž .so that the gap, Eq. 26 , obeys

E E2 M yb k Ms0.Ž .0 Rž /E M Ek0 R

As a consequence of the Coleman-Mermin-Wagner theorem we expect that terms which are higher order in1rN will destabilize the leading order result, i.e. will give rise to contributions that dominate the ones we havefound for sufficiently large M.

w x †We may also 4,8 , compute the Green’s functions associated with the fields B and B . To do this, we needŽ .the effective action, not just the effective potential. This can be read off from Eq. 10 :

y4N i4 † y1 y1 †˜w xG s d x B x B x y tr ln 1q4 A BA B . 27Ž . Ž . Ž .Heff ž /k 2

˜ THere we have defined Ass A s . Because the gap equation is independent of m, we shall simplify our2 2

task somewhat by setting ms0; then AT syA, and

As iE q is E 28Ž .0 3 1

AsyiE q is E . 29Ž .0 3 1

˜ ˜ 2 2Note that AAsAAsE yE .0 1


To obtain the desired Green’s functions, we perform the following sequence of steps:Ž .a We write

BsB qBX , 30aŽ .0

B† sB qBX† . 30bŽ .0

Ž 2 .where, as above, B is a solution of the gap equation Ms4B .0 0Ž . X X† X X†b We expand G to second order in B and B . The linear terms in B and B will cancel because of theeff

gap equation. The coefficients of the quadratic terms will be the inverses of the Green’s functions that weseek.

Ž . X X X†c We observe that by introducing the real and imaginary parts of B : B sf q if , B sf y if , the1 2 1 2

off-diagonal terms will disappear; i.e. there will be no mixed f f terms. In fact, what we find is:1 2

2d p 1 iŽ .2 2 yi pP xyyG ,y4N d xd y f x f y e y F pŽ . Ž . Ž .H Heff 1 1 q2½ k 22pŽ .

2d p 1 iyi pPŽ xyy.qf x f y e y F p 31Ž . Ž . Ž . Ž .H2 2 y2 5k 22pŽ .

where

2d k k p qk yk p qk "MŽ . Ž .0 0 0 1 1 1F p s2 . 32Ž . Ž .H" 2 2 22 22p yk qk qM y k qp q k qp qMŽ . Ž . Ž .0 1 0 0 1 1

Ž .Now F p is logarithmically divergent, but so is 1rk , and using the gap equation it is easy to see that the"

divergence cancels, along with all residual dependence on k and d X.RŽ . Ž .d The integrals defining F p can be done explicitly, with the result that"

1 i 1 1qbGG p ' y F p s log 33Ž . Ž . Ž .11 q ž /k 2 4pb 1yb

1 i b 1qbGG p ' y F p s log . 34Ž . Ž . Ž .22 y ž /k 2 4p 1yb

2 2 2 2 2(Here bs p r p y4M , and p sp yp .Ž . 0 1

We see that both GG and GG give rise to a branch point at p2 s4M, and become complex for p2 )4M,11 222 Ž . 2 2whereas they are real for p -4M. Furthermore, GG p has a simple zero in p at p s0, which means22

the corresponding Green’s function has a pole. Together, these results suggest that the fermion acquires a2'mass m s M , while the pole at p s0 is evidence for the existence of a would-be Goldstone boson thatF

² :reflects the condensation of cc .In this note we have analyzed a two-dimensional model that exhibits the formation of Cooper pairs in leading

Ž .order in 1rN. This condensation occurs for all values of the coupling as long as the bare coupling is positivefor any value of the chemical potential m, including ms0. The coupling itself is asymptotically free. At ms0,we have computed the two-point functions associated with the composite fields cc and c †c †, and have found

² :that c acquires a Majorana mass 2 B , where B s cc . We also find evidence for a massless pole, which0 0² :indicates the spontaneous breaking of fermion number at large N. However, we expect that the cc

condensate will be unstable against higher order corrections in 1rN, so as not to violate the Coleman-Mermin-Wagner theorem. Bearing this in mind, we speculate that our GN-like model at large N could serve as a

w xtheoretical laboratory for a one-dimensional superconductor 9 .


Acknowledgements

The research of AC is supported in part by DOE grant DE-FG02-92ER-40704. In addition, AC and HM aresupported in part by the Grant-in-Aid for International Scientific Research No. 09045036, Inter-UniversityCooperative Research, Ministry of Education, Science, Sports and Culture of Japan. This work has beenperformed as an activity supported by the TMU-Yale Agreement on Exchange of Scholars and Collaborations.AC wishes to acknowledge the hospitality of Tokyo Metropolitan University, where part of this work wasbegun. FC and HM are grateful for the hospitality of the Center for Theoretical Physics at Yale.

References

w x Ž . Ž .1 D. Bailin, A. Love, Phys. Rep. 107 1984 325; M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 422 1998 247; R. Rapp, T.Ž .Schafer, E.V. Shuryak, M. Velkovsky, Phys. Rev. Lett. 81 1998 53.¨

w x2 M. Alford, K. Rajagopal, F. Wilczek, hep-phr9802284 and hep-phr9804403; J. Berges, K. Rajagopal, hep-phr9804233; T. Schafer,¨nucl-thr9806064; T. Schafer, F. Wilczek, hep-phr9811473.¨

w x3 N. Evans, S.D.H. Hsu, M. Schwetz, hep-phr9808444 and hep-phr9810514; T. Schafer, F. Wilczek, hep-phr9810509.¨w x Ž .4 D.J. Gross, A. Neveu, Phys. Rev. D 10 1974 3235.w x Ž . Ž .5 S. Coleman, Comm. Math. Phys. 31 1973 259; N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 1966 1133.w x Ž .6 E. Witten, Nucl. Phys. B 145 1978 110.w x Ž .7 R.L. Stratonovich, Doklady Akad. Nauk. S.S.S.R. 115 1957 1097; S. Coleman, Aspects of Symmetry, Cambridge Press, 1985, p. 354.w x Ž . Ž .8 S. Coleman, R. Jackiw, H.D. Politzer, Phys. Rev. D 10 1974 2491; L. Abbott, J. Kang, H. Schnitzer, Phys. Rev. D 13 1976 2212;

Ž . Ž .C.M. Bender, F. Cooper, G.S. Guralnik, Ann. Phys. NY 109 1977 165.w x Ž .9 For a discussion of Cooper pairing in 1y D electron systems, see for example J. Solyom, Advances in Physics 28 1979 201.

11 March 1999


ž /Abelian monopoles and action density in SU 2 gluodynamicson the lattice

B.L.G. Bakker a, M.N. Chernodub b, M.I. Polikarpov b, A.I. Veselov b

a Department of Physics and Astronomy, Vrije UniÕersiteit, De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlandsb ITEP, B. Cheremushkinskaya 25, Moscow 117259, Russia

Received 1 November 1998; revised 18 December 1998Editor: P.V. Landshoff

Abstract

Ž .We show that the extended Abelian magnetic monopoles in the Maximal Abelian projection of lattice SU 2Ž .gluodynamics are locally correlated with the magnetic and the electric parts of the SU 2 action density. These correlations

are observed in the confined and in the deconfined phases. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 11.15.H; 12.10; 12.15; 14.80.H

1. Introduction

w xThe monopole confinement mechanism 1 in lat-tice gluodynamics seems to be confirmed by many

w xnumerical calculations 2 . Monopoles in the Maxi-Ž . w xmal Abelian MaA projection 1,3 are condensed in

w xthe confinement phase of gluodynamics 4 , theircurrents satisfy the classical equations of motion for

Ž . w x Ž .the dual Abelian Higgs model 5 and the SU 2string tension is reproduced by the monopole cur-

w xrents 6 . The confining string connecting the staticw xquark-antiquark pair is clearly seen 7 . The next

problem to solve is to build the qualitative andquantitative model for this flux tube, or more gener-ally the effective infrared Lagrangian for gluodynam-ics. The first steps in this direction are done alreadyw x8,9 . In brief the main results of the numerical studyof the confinement problem are: the vacuum ofgluodynamics behaves as the dual superconductor,the abelian monopoles playing the role of the Cooper

pairs and the confining string is an analogue of theAbrikosov-Nielsen-Olesen string.

On the other hand in the continuum theory theAbelian monopoles arise as singularities in the gauge

w xtransformations 10 . The definition of the Abelianmonopoles is projection–dependent, monopoles de-fined in different projections are different ingeneral 1. Therefore it is not clear whether thesemonopoles are ‘‘physical’’ objects. The first argu-ment in favour of the physical nature of the Abelian

w xmonopoles was given in Ref. 12 : it was found thatŽ .the total action of SU 2 fields is correlated with the

total length of the monopole currents, so there existsa global correlation. Recently it was shown that theAbelian monopoles in the MaA projection are locally

w xcorrelated with the non-Abelian action density 14 .

1 However, there exists a gauge invariant definition of thew xmonopole current in any chosen Abelian projection, see Refs. 11 .


( )B.L.G. Bakker et al.rPhysics Letters B 449 1999 267–273268

Really it means that monopoles are the physicalŽobjects not the artifacts of the singular gauge trans-.formation , since by definition we call the object

w xphysical if it carries the action. In Ref. 15 thecorrelation of monopoles, ZZ strings and the action2

density was discussed. The investigation of the corre-lations of monopoles, the topological density and the

w xaction density was performed in Refs. 13,16 .Thus monopoles are important dynamical vari-

ables for the confinement problem and the detailedstudy of their anatomy is interesting. At present wehave no idea what is the general class of the gaugefields which generate the monopole currents in theMaA projection 2. But since the elementary

w xmonopoles carry nonabelian magnetic action 14they are related with some nonabelian objects. Thenumerical study of the effective infrared Lagrangian

w xof lattice gluodynamics shows 9 that to approachthe continuum limit we have to consider also the

Ž . w xextended blocked monopoles 18 . In the presentpublication we continue the study of correlations ofthe monopole currents and the action density started

w xin Ref. 14 . We investigate the extended monopolecurrents, and also study the correlations of the elec-tric part of the action with the monopole currents.

w xThe couplings of the monopole action 9 obeysscaling, it means that these couplings do not dependseparately on the monopole size in the lattice unitsand bare coupling, but only on the physical size ofthe monopole. This fact in turn means that thecouplings lie on the renormalised trajectory, and weknow the values of the coupling and the size of the

Ž .monopoles in the continuum limit a™0 . The cal-culations presented in the present paper are done justfor that sizes of monopoles and for that values of thebare coupling which correspond to the initial partŽ .2.2-b-2.5 of the renormalised trajectory of

w xRefs. 9 . In that sense our results correspond to thecontinuum limit.

There are two different types of extendedŽ . w xmonopoles type-I and type-II monopoles 18 . As

we already discussed the type-II extended monopolesare important dynamical variables in lattice gluody-

2 w xIt is known 17 that instantons induce Abelian monopolecurrents in the Abelian gauge but it seems that they are not theonly sources of Abelian monopoles.

w xnamics 9,12 . The type-I extended monopoles play anon-trivial role for the dynamics of the phase transi-

w x Ž .tions in electroweak theory 19 and in the U 1w xAbelian Higgs model 20 .

The paper is organised as follows. In Section 2we introduce two quantities h E and h M, whichdefine the correlation of the magnetic and electric

Ž . Ž .parts of the SU 2 action with the extendedmonopole charge. In Section 3 we describe the re-sults of numerical calculations. We discuss the re-sults in Section 4.

2. Correlations of monopoles with action densities

If the Abelian monopole carries the non-Abelianaction, then the action density near the monopolecurrent should be larger than the action density farfrom the monopole trajectory. One of the quantitieswhich can show this effect is the relative excess ofthe mean action density in the region near the

w xmonopole current 14 . The total action can be di-vided into electric and magnetic parts. The relative

Ž .excess of the magnetic electric action density isdefined as:

SM ŽE .ySmM ŽE .h s . 1Ž .S

1² :Here Ss S ' 1y Tr U is the expectation² :Ž .P P2

value of the lattice plaquette action. The quantity SMm

is the action averaged over the plaquettes closest toŽ . Mthe monopole current j x . The definition of S is:n m

1MS s S , 2Ž .Ým P6¦ ;Ž .PgE C xn

where the summation is over the plaquettes P whichŽ . Ž .are the faces of the cubes C x ; a cube C x isn n

Ž .dual to the monopole current j x . For the staticn

Ž . Ž . Ž .Abelian monopole j x /0, j x s0 is1,2,3 ,0 i

and the boundaries of the cubes dual to the monopolecurrent are formed by the space-like plaquettes P ,i, j

i, js1,2,3. Therefore only the magnetic part of the1 2 MŽ .SU 2 action density, TrF , contributes to S .i j m2

E Ž .The quantity S in Eq. 1 is:

1ES s S , 3Ž .Ým P24¦ ;Ž Ž ..PgPP C xn

( )B.L.G. Bakker et al.rPhysics Letters B 449 1999 267–273 269

Ž Ž ..where PP C x is the set of all plaquettes P whichn

satisfy the following two conditions: all plaquettes PŽ .i have one, and only one, common link l with them

Ž . Ž .cube C x ; ii they are lying in the planes, definedn

by the vectors m and n . There are 24 such plaquettesˆ ˆŽ .corresponding to a cube C x . For the staticn

monopole current these plaquettes lie in the planesŽ .0,i , is1,2,3; therefore only the electric part of

1 2Ž .SU 2 action density, TrF , contributes to the0 i2

quantity SE.m

Thus, our definition of electric, SE, and magnetic,SM, parts corresponds to the electric and magneticparts of the action density only for a static monopole.For non-static monopoles it is convenient to keepthese notations.

Ž .In the naive continuum limit the expressions 2Ž .and 3 and the plaquette action S have the following

form:2

1M ˜² :S s Tr n x F x , 4Ž . Ž . Ž .ž /m m mn24

21E ² :S s Tr n x F x , 5Ž . Ž . Ž .Ž .m m mn6

1 2² :Ss TrF , 6Ž .mn24

1˜ Ž .where F s ´ F , and n x is the unit vec-mn mna b a b m2

Ž .tor in the direction of the current: n x sm

Ž . < Ž . < Ž . Ž . Ž .j x r j x if j x /0, and n x s0 if j xm m m m m

Ž . Ž .s0. For a static monopole Eqs. 4 and 5 give the1 a 2Ž .normalised average of the chromomagnetic, B ,i3

1 a 2Ž .and chromoelectric, E , action density at thei3

Ž .point where the monopole is located. Eq. 6 gives1 a 2² : ²Ž .the normalised total action density: S s Bi6

Ž a.2:q E .i

3. Numerical results

Below we present the quantities h M and h E cal-culated on symmetric, 244, and asymmetric, 243 P4

Ž . 3lattices in standard SU 2 lattice gluodynamics . Inw xall these cases, we find in the MaA projection 3

3 To check the finite volume corrections we also performedcalculations on the smaller lattices: 164, 204, 163P4 and 163P4. Itoccurs that the results obtained on these small lattices coincidewithin the statistical errors with the results obtained on 244 and243P4 lattices

that the quantities h M ,E are different from zero forŽall values of b. We also considered the F di-12.agonalization of the F lattice field strength tensor ,12

Ž .Polyakov diagonalization of the Polyakov line and‘‘random’’ Abelian gauges. The ‘‘random’’ Abeliangauge means no gauge fixing at all: we take a fieldconfiguration, apply a random gauge transformationand then treat the phases of the diagonal elements of

Ž .the SU 2 gauge field as the Abelian gauge field.To fix the MaA projection we use the overrelax-

w xation algorithm of Ref. 21 . The number of gaugefixing iterations is determined by the following crite-

w xrion 22 : the iterations are stopped when the matrixŽ .of the gauge transformation V x becomes close to

1 y6� Ž .4the identity matrix: max 1y Tr V x F10 .x 2

We also check that a more accurate gauge fixingdoes not change our results. By performing a suffi-cient number of iterations between measurements wehave made sure that the configurations on which weperformed our measurements are statistically inde-pendent.

Ž . M ,EFig. 1 a shows the quantities h in the MaAprojection for the lattice 244. The quantity h M is4–6 times larger than the quantity h E for all consid-ered values of bs4rg 2. Thus the excess of thechromomagnetic action near the monopole positionis larger than the excess of the chromoelectric action.

The correlations increase with increasing b. Forsmall b monopoles are present almost everywhere,so the action averaged over the cubes containingmonopoles differs very little from the action aver-aged over all cubes. The density of monopoles de-creases with increasing b , thus the increase of thecorrelator as b™` means that at large b theAbelian monopoles disappear mainly in the regions

Ž .with a small SU 2 action density.Note, that the monopole current j is derivedm

from the plaquettes E C which contribute to SM,m

thus the fact that h M /0 is rather natural. Theplaquettes which contribute to SE are not directlyrelated to the monopole current and the fact thath E /0 probably means that there exist some struc-tures in the vacuum of gluodynamics which generatemonopole currents and carry electric and magneticaction.

Our numerical simulations show that the quanti-ties h E,M for the F , Polyakov and random Abelian12

gauges coincide with each other within the numerical


Ž . M Ž . E Ž . 4Fig. 1. a The quantities h boxes and h triangles versus b on the lattice 24 for the MaA projection. In all figures the error bars areŽ . Ž . 3much smaller than the sizes of the symbols used; b the same as in a , but now for the lattice 24 P4.

errors. For all studied values of the coupling constantb the values of the quantities h M ,E calculated inthese gauges are more than 10 times smaller than

those for the MaA gauge. This fact probably indi-cates that the Abelian monopoles in the F and12

Polyakov Abelian projections carries much less in-

Ž . M 4 Ž .Fig. 2. a The correlator h in the MaA gauge for a 24 lattice versus b for type-II extended monopoles of sizes 2 triangles , 3Ž . Ž . Ž . Ž .diamonds and 4 circles ; b The same as in a , but now for type-I monopoles.


formation about the properties of the non-Abelianvacuum than the Abelian monopole in the MaAprojection 4.

The finite-temperature analysis of the correlatorsh M ,E is performed on an asymmetric lattice. Wefound that at finite temperature the correlators in theMaA projection turned out to be much larger thanthe correlations in the F , Polyakov and random12

gauges. We show the quantities h M and h E in theŽ .MaA gauge in Fig. 1 b . These calculations are

performed on a 243 P4 lattice. It is seen that theŽconfinement–deconfinement phase transition which

.occurs at bsb s2.3 has no observable influencec

on the behaviour of the correlators h M ,E.The correlation between electric and magnetic

action in the vicinity of Abelian monopoles is small.We measured the correlation of the product of theelectric and the magnetic action with the monopolecurrents. We find that the correlator

² E M :S x S xŽ . Ž .EMh s y1 7Ž .E M² : ² :S x S xŽ . Ž .

vanishes within the statistical error for the studiedregion of the bare coupling b on the lattice 244.This occurs not only when the averages are takenover the full lattice, but also if only the cubesassociated with monopoles are included in the aver-age. This result is independent of the gauge fixingcondition. Therefore the magnetic and electric fluctu-ations around the Abelian monopole in the MaAgauge are independent.

We also study the correlations of extendedw xmonopoles 18 with the electric and magnetic ac-

tion. There are two types of extended monopolesw x18 : type I corresponds to the plaquettes of sizel= l; type II uses all 1=1 plaquettes that tile thefaces of an l 3 sized cube associated with a monopolecurrent. We measured the correlations of the mag-netic and the electric action densities with the ex-

4 w x MIn Ref. 16 the correlator h is studied under the smoothingprocedure. It was found that for elementary monopoles in differentgauges this correlator is of the same order. The smoothing proce-dure removes short-range fluctuations, therefore the result of Ref.w x16 probably indicates that the small correlation of the monopoleswith the action density in, say, the Polyakov gauge, is due toultraviolet vacuum fluctuations.

Fig. 3. The correlator h M plotted as a function of the lineardimension l of the extended monopoles, the lattice size is 244.

tended monopoles of sizes ls2, 3 and 4. It turns outthat for the whole range of the bare coupling b

studied, the quantity h M for type-II monopoles islarger than that for the type-I monopoles. In Fig.Ž . M2 a,b we show the dependence of the quantity h

on b for type-II monopoles and type-I monopoles.In order to show a similarity between different typesof monopoles we plot the quantity h M for the type-Iand type-II monopoles versus linear size of the ex-

Ž .tended monopoles l Fig. 3 . The figure clearlyshows that the larger the size of the monopole thesmaller the correlation h M is. This fact is not unex-pected since with increasing monopole size the partof the lattice which belongs to a monopole getslarger and therefore the averaged action associatedwith the monopole gets closer to the total averagedaction. If the correlations h M and h E are physicalquantities then they should depend on the physicalmonopole size, bs la, where a is the physical lat-tice spacing. We are planning to study this depen-dence in our next publication.

4. Discussion and conclusions

We discussed the local correlations of the electricŽ .and magnetic parts of the SU 2 action with Abelian


monopoles in various Abelian projections. We haveshown that monopoles in the Maximal Abelian pro-jection are correlated with the electric and magneticparts of the action density at zero and at finitetemperature. The same result is obtained also fortype-I and type-II extended monopoles. The correla-tors h M ,E for the type-II monopoles are alwayslarger than the correlators for the type-I monopoles.

Ž .Thus, for the description of the vacuum of SU 2gluodynamics the type-II monopoles are more suit-able variables than the type-I monopoles, in agree-

w xment with Refs. 9,12 .The correlation of the monopoles with the electric

part of the action density is smaller than the correla-tion with the magnetic part of the action density. Thecorrelations of the Abelian monopole with both parts

Ž .of the SU 2 action density in the Polyakov, F and12

random gauges are of the same order; all of them aremuch smaller than the correlations in the MaA gauge.

We note here that the existence of the correlationof the electric and the magnetic action densities withthe Abelian monopoles can be understood from thefact that the Abelian monopoles are correlated with

w xthe topological charge density 13,23,24 . Indeed thiscorrelation means that the monopole currents areaccompanied by a non-zero density of the topologi-cal charge. This charge is non-zero if and only ifboth the electric and magnetic action densities arenon-zero.

We conclude that the Abelian monopoles in theMaximal Abelian projection are physical objectswhich carry both magnetic and electric parts of the

Ž .SU 2 action density.

Acknowledgements

The authors are grateful to T. Suzuki and Yu.A.Simonov for useful discussions. M.N.Ch. and A.I.V.feel much obliged for the kind reception given tothem by the staff of the Department of Physics andAstronomy of the Free University at Amsterdam.This work was partially supported by the grantsINTAS-96-370, INTAS-RFBR-95-0681, RFBR-96-02-17230a, RFBR-97-02-17491a and RFBR-96-15-96740. The work of M.N.Ch. was supported by theINTAS Grant 96-0457 within the research program

of the International Center for Fundamental Physicsin Moscow.

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11 March 1999


Nonequilibrium photons as a signatureof quark-hadron phase transition

Da-Shin Lee a,1, Kin-Wang Ng b,2

a Department of Physics, National Dong Hwa UniÕersity, Hua-Lien, Taiwan, ROCb Institute of Physics, Academia Sinica, Taipei, Taiwan, ROC


Abstract

We study the nonequilibrium photon production in the quark-hadron phase transition, using the Friedberg-Lee typesolitons as a working model for quark-hadron physics. We propose that to search for nonequilibrium photons in the directphoton measurements of heavy-ion collisions may be a characteristic test of the transition from the quark-gluon to hadronicphases. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 12.38.Mh; 25.75.-q; 11.15.Tk

Quark-confinement is an unresolved problem inQCD physics. Nevertheless, it is generally believedthat quarks are deconfined at high energies. One ofthe primary goals of relativistic heavy-ion-collisionexperiments such as BNL-AGS and CERN-SPS is tocreate a hot central region and a dense fragmentation

Ž .region in which quark-gluon plasma QGP may beformed. One of the ways for testing the occurrenceof QGP is to observe the spectrum of the emittedphotons. An advantage of measuring the direct pho-tons is that photons do not suffer from strong final-state interactions as hadrons do, so they can be usedto monitor the initial stages of the colliding heavy

w xions 1 .


In relativistic heavy-ion collisions, a large numberof p 0 and h are produced by soft QCD processes.Their two-photon decays constitute the major sourceof photon emission. Other photon sources arise fromradiative decays of other mesons and baryon reso-nances, as well as hadron scattering processes. Also,if QGP is formed, photons will be produced viaquark-antiquark annihilations or quark-gluon Comp-ton scatterings. It is an experimental challenge toseparate out the photon yield into a part arising fromthe decays of produced p 0 and h and the ‘‘singlephoton’’ part from other sources. For instance, thepart from decays of mesons can be subtracted byreconstructing the distribution of the two-photon in-variant mass of all photon pairs. If this can be done,then the excess single photon spectrum can be aprobe of the properties of the nuclear matter or QGP,or even a discriminator between the two phases.


( )D.-S. Lee, K.-W. NgrPhysics Letters B 449 1999 274–280 275

w xHowever, recent studies 2–4 have shown that thetwo single photon spectra are similar, and distin-guishing between them might need further high-pre-cision direct photon measurements or other comple-mentary measurements such as dilepton productionand Jrc suppression.

In this Letter, we give an attempt to utilize thenonequilibrium nature of the photon emission associ-

Ž .ated with the quark-hadron phase transition QHPTto test the formation of QGP. In essence, we calcu-late the particle production from the release of latentheat of the phase transition, adopting the formalismof nonequilibrium field theory that is well-suited to

wstudy the dynamics of nonequilibrium processes 5–x7 . To model the QHPT, we use the Friedberg-LeeŽ .FL phenomenological nontopological soliton modelw x8 , which is simple and adequate enough for ourpresent considerations. In fact, FL model has been

w xapplied to fit the hadronic properties 9 , as well as tow xdiscuss the QHPT of the early Universe 10,11 . We

believe that the particle production considered hereis generic to all Landau-Ginzberg type models ofQHPT. Our main result will be the photon spectrumproduced from the QHPT, and the experimental sig-nature will be briefly discussed.

We begin with the simplest FL model whosew xLagrangian is given by 8

nF1m mLL s i c g E c q E sE syU sŽ .Ýs i m i m2

is1nF

y fsc c , 1Ž .Ý i iis1

where c represents the quark field, n s2 is thei F

number of light flavors, s is a real scalar field, f isŽ .a coupling constant, and the potential U s is given

bya b c

2 3 4U s s s y s q s qB , 2Ž . Ž .2! 3! 4!

where a, b, and c are positive parameters, and B isŽ .the bag constant. Note that U s has a local mini-

mum at ss0, and an absolute minimum at ssscŽ .which denotes the true vacuum with U s s0, sep-c

arated by a potential barrier. A typical potential isŽ .shown in Fig. 1. Due to the nonlinearity of U s ,

the model bears nontopological soliton solutionswhich are identified with hadrons. Essentially, insidea hadron is a perturbative vacuum with s,0 deco-

Ž . y2 y1Fig. 1. Scalar potential U s with as1.6 fm , bs69 fm ,and cs500, and f is the initial value.i

rated with localized free quarks, whereas outside isthe true vacuum.

The simple picture of the production of nonequi-librium photons from the QHPT is the following.Suppose that a thermalized QGP is produced initiallyin a relativistic nucleus-nucleus collision, which hasa perturbative vacuum with ss0. It then expandsand cools, and undergoes hadronization in a phasetransition. Here we assume that the transition isweakly first-order as consistent with lattice QCD

w xcalculations 12 . As such, transition to a state withŽ . Žs,f see Fig. 1 by quantum tunneling or thermali.activation will occur through nucleation of hadronic

bubbles in the QGP. Due to the plasma expansion, itis conceivable that the s field inside a bubble mightget trapped at s,f as the interaction rate of s isi

small compared to expansion rate. When the expan-sion slows down, s will start to oscillate and relaxvia production of the particles, to which the s

particle is coupled, to the true vacuum. This nonequi-librium stage will take place for a time-scale ofabout 10 fm, as revealed by hydrodynamical simula-

Ž w xtions for example, see Ref. 3 and references.therein . Typically, during this stage the dominant

channel for the relaxation is the production of stronginteracting particles such as s quanta, gluons andother mesons. These particles will be immediatelytrapped and thermalized in the nuclear matter. Per-haps the s quanta are short-lived to decay intophotons before thermalization. If this happens, the

( )D.-S. Lee, K.-W. NgrPhysics Letters B 449 1999 274–280276

decay photons would be a source for nonequilibriumphotons. However, our current interest is the directphoton production driven by the oscillations of the s

field due to parametric amplification. We will ignorethe hydrodynamic expansion and simplify the dy-namics by adopting a ‘‘quenched’’ phase transitionfrom an initial state in local thermodynamic equilib-rium at a temperature slightly above the criticaltemperature cooled instantaneously to zero tempera-ture. This ‘‘quench’’ approximation is far from adefinitive description of the dynamics. Nevertheless,it captures the qualitative features and allows a sim-

w xple but concrete calculation 6,7 .At tree level the s field is inert to electromag-

netic interaction, but it can couple to photon througha quark loop. After integrating out the quark field,we obtain an one-loop effective Lagrangian includ-ing strong interacting final states,

1 1 1m mn mnLLs E sE syU s y F F y G GŽ .m mn mn2 4 4

g1 2 m mnq m B B y sys F FŽ .V m c mn2 4sc

hmny sys G F , 3Ž . Ž .c mn4sc

where F sE A yE A is the photon, and G smn m n n m mn

E B yE B is a quark-antiquark vector meson withm n n m

mass m . It is straightforward to calculate the cou-V

pling constant g,2.6=10y3. But the couplingconstant h would depend on the wave function at the

Ž .origin of the vector meson, C 0 . Assuming m ,2V V< Ž . < 2GeV and taking a conservative value for C 0 ,V

y3 w x y20.1 fm 9 , we find h,1.8=10 . Note thateffective strong interacting vertices such as s ggŽ .where g denotes a gluon and spp should alsoappear at one-loop level. However, we have omitted

Ž .them in the effective Lagrangian 3 because theireffects to the particle production can be fully repre-sented by the s strong self-couplings. Since we areconcerned with photon production only, we integrateout the vector meson and obtain

1 1m mnLLs E sE syU s y F FŽ .m mn2 4

gmny sys F FŽ .c mn4sc

h2mn ay E sE s F F , 4Ž .m a n2 28m sV c

where we have dropped off higher derivative terms.At this stage, we should emphasize that the aboveobtained effective Lagrangian for the processes such

X X Ž Xas s ™2g as well as 2s ™2g here s is theX .shifted field with sss qs is clearly understoodc

in perturbation theory. It is not certain whether thiseffective Lagrangian can also describe the nonequi-librium situation. In other words, the effective ver-tices that account for the above mentioned processesmay be modified in the strongly out-of-equilibriumsituation. The fuller study of off-equilibrium effec-tive vertices by integrating out the quark fields andvector meson is a challenging task that lies beyondthe scope of this paper, but certainly deserves to betaken up in the near future. However, in this work,

Ž .we will use this effective Lagrangian 4 to computethe nonequilibrium photon production.

Following the nonequilibrium closed time pathw xformalism 5–7 , the nonequilibrium effective La-

grangian is given by

q q y yLL sLL s , A yLL s , A , 5Ž .neq m m

Ž . Ž .where q y denotes the forward backward timebranches. We then split s " into a mean field andthe quantum fluctuating fields:

s " x ,t sf t qx " x ,t , 6Ž . Ž . Ž . Ž .

with the tadpole conditions,

² " :x x ,t s0. 7Ž . Ž .

This tadpole conditions will be imposed to all ordersin the corresponding expansion to obtain thenonequilibrium equations of motion from the firstprinciples approach.

To derive the nonequilibrium evolution equationsthat consistently take into account quantum fluctua-tion effects from the strong s self-interaction, weadopt the following Hartree factorization for x im-

w xplemented for both " components 5–7 :

4 ² 2: 2 3 ² 2:x ™6 x x qconstant, x ™3 x x . 8Ž .

The expectation value will be determined self-con-sistently. Before proceeding any further, it is worthnoting that although the above Hartree factorization


is uncontrolled in this effective field theory thatw xinvolves a single scalar field 5–7 , our justification

of using this approximation is based on the fact thatit provides a non-perturbative framework that allowsus to treat the strong s dynamics self-consistently.

After doing this factorization, the Lagrangian thenbecomes

q q y yLL f t qx , A yLL f t qx , AŽ . Ž .m m

21 1q q 2 q2s Ex yU t x y M t xŽ . Ž . Ž .x2 2½g

1 q qmn q qmny F F y f t ys F FŽ .Ž .mn c mn4 4sc

gq qmn qy x F Fmn4sc

h22 q0 i q0˙y f t F FŽ . i2 28m sV c

h2q q0 i qa˙y f t E x F FŽ . a i2 24m sV c

h2q q qmn qay E x E x F Fm a n2 2 58m sV c

� 4y q™y , 9Ž .where

¨U t sf tŽ . Ž .b c c

2q ay f t q f t q S t f tŽ . Ž . Ž . Ž .2 6 2

b2² :y x t ,Ž .

2c c

2 2M t saybf t q f t q S t ,Ž . Ž . Ž . Ž .x 2 2

² 2: ² 2:S t s x t y x 0 . 10Ž . Ž . Ž . Ž .² 2:Ž .Here, we have performed a subtraction of x t

² 2:Ž .at ts0 absorbing x 0 into the finite renormal-ization of a. Besides, since we are interested in theprocesses of direct photon production driven by a

Ž .time dependent f t field, in which the photons donot appear in the intermediate states, to avoid thegauge ambiguities, we can work on the Coulombgauge, and concentrate only on physical transverse

w xgauge fields, A 7 .T

With the above Hartree-factorized Lagrangian inthe Coulomb gauge, we perform a perturbative ex-pansion in the weak couplings g and h2. However,the strong s dynamics is treated non-perturbativelyw x Ž .7 . Following from the tadpole conditions 7 , weobtain the following full one-loop equation of mo-

Ž .tion of f t given by

b c c2f t q ay f t q f t q S t f tŽ . Ž . Ž . Ž . Ž .

2 6 2

g2² < < :y E A tŽ .m T2sc

2h d2¨ ˙ ˙² < < :y f t qf t A tŽ . Ž . Ž .T2 2 dt4m sV c

s0. 11Ž .The Heisenberg field equations can be read off fromthe quadratic part of the Lagrangian in the form

2d2 2y= qM t x t s0,Ž . Ž .x2dt

2g h2˙ ¨1q f t ys q f t A tŽ . Ž . Ž .Ž .c T2 2s 4m sc V c

2g h˙ ˙ ¨ ˙q f t q f t f t A tŽ . Ž . Ž . Ž .T2 2s 2m sc V c

g2y 1q f t ys = A t s0. 12Ž . Ž . Ž .Ž .c T

sc

Now we decompose the fields x and A into theirTŽ . Ž .Fourier mode functions U t and V t respec-k l k

tively,

3d ki kP xx x ,t s a U t e qh.c. ,Ž . Ž .H k k32 2p vŽ .( x k

d3k elkA x ,t sŽ . Ý HT 3(ls1,2 2 2p vŽ . A k

= i kP xb V t e qh.c. , 13Ž . Ž .lk l k

where a and b are destruction operators, and ek l k l k

are linear polarization unit vectors. The frequenciesv and v can be determined from the initialx k A k


states and will be specified below. Then the modeequations are

2d2 2qk qM t U t s0,Ž . Ž .x k2dt

2 2g h d2˙1q f t ys q f tŽ . Ž .Ž .c 2 2 2½ s 4m s dtc V c

2g h d˙ ˙ ¨q f t q f t f tŽ . Ž . Ž .2 2s dt2m sc V c

g2qk 1q f t ys V t s0, 14Ž . Ž . Ž .Ž .c lk5sc

with the vacuum expectation values given by

d3kL 22² : < <x t s U t ,Ž . Ž .H k32 2p vŽ . x k

d3kL2² < < :E A t sŽ . ÝHm T 32 2p vŽ .l A k

=2 22˙< < < <V t yk V t , 15Ž . Ž . Ž .lk lk

where we set the cutoff scale L,m , and theV

expectation value of the number operator for theasymptotic photons with momentum k is given byw x7

1˙ ˙² :N t s A k ,t PA yk ,tŽ . Ž . Ž .k T T2k

2qk A k ,t PA yk ,t y1Ž . Ž .T T

12 22˙< < < <s V t qk V t y1.Ž . Ž .Ý lk lk22k

l

16Ž .

This gives the spectral number density of the pho-Ž . 3tons produced at time t, dN t rd k. To solve the

Ž .evolution equations 11,14 , we propose the follow-ing initial conditions for the mode functions at the

w xtime of ‘‘quench’’ 7 :

˙U 0 s1, U 0 syiv ,Ž . Ž .k k x k

c2 2 2v sk qaqbf q f ;x k i i2

˙V 0 s1, V 0 syiv , v sk , 17Ž . Ž . Ž .lk lk A k A k

where the initial mode functions are chosen to be atzero temperature inside the hadronic bubbles with

Ž .the mean field f t displaced initially away from theŽ Ž . .equilibrium position i.e., f 0 sf /0 . The abovei

specified initial conditions are physically plausibleand simple enough for us to investigate a quantita-tive description of the dynamics.

Let us use the FL model parameters: as1.6fmy2 , bs69 fmy1, cs500, and fs9.57. Thisimplies that s s0.36 fmy1. This set of parametersc

w xhas been used for fitting hadronic mass spectrum 9 ,Ž . Ž .and with this potential 2 see Fig. 1 the phase

transition is weakly first-order at finite temperaturew x y110 . We choose the initial amplitude f s0.07 fmi

Ž .to solve Eq. 11 .In Fig. 2, the time evolution of the mean fieldŽ .f t is shown. It can be seen that f is oscillating

about a mean value of about 1.1 fmy1, which issignificantly different from the classical value s .c

Ž .Also, f t oscillates with a frequency v ,8.8f

fmy1. This is due to the quantum fluctuation effectscoming from the s field and the gauge field A toT

Ž . Ž .the motion of f t 11 . The back reactions of theŽ .quantum fields also account for the damping of f t

with time. In Fig. 3, we plot the time dependence ofthe photon number density integrated over momen-

Ž . Ž .tum k, N t . As expected, N t oscillates with theŽ .same frequency as f t . Although photons are pro-

duced efficiently during the first half of an oscilla-tion, almost all of them are re-absorbed to the back-ground during the second half of the oscillation.

Ž .Fig. 2. Time evolution of the mean field f t .


However, it is important to note that the time aver-Ž .age of N t is indeed increasing with time, that is to

say, photons are effectively produced from f oscil-w xlations due to parametric amplification 6 .

It is useful to calculate the time-averaged invari-ant photon production rate, kdRrd3k, where

1 dN tŽ .TdRs dt , 18Ž .H

T dt0

over a period from the initial time to time T. Theresults are shown in Fig. 4 with Ts3,5,10 fm. Alsoshown are the thermal photon production rate from aquark-gluon plasma and a hadron gas taken from

w xRef. 13 . It is apparent that the f oscillations withfrequency v produce non-thermal photons whosef

spectrum peaks around two photon momenta, ksv r2 and ksv , which correspond to the unstablef f

Ž .bands or resonant bands arising from the fact thatŽ .the mode equations of A 14 consistently dependT

Ž . w xon the time dependent mean field f t 7 . Clearly,the photon production mechanism is that of paramet-

Ž .ric amplification. From the mode Eqs. 14 , we caneasily recognize that the 4.4 fmy1 peak is resultedfrom the coupling s F 2 while the 8.8 fmy1 peak is

Ž .2 2 y1from the interaction Es F . Note that the 4.4 fmpeak has a peak value of the production rate beingcomparable to that of thermal photons, while inter-estingly the 8.8 fmy1 peak is almost two orders ofmagnitude larger than the thermal photons. Althoughthese results more or less depend on the choice ofthe FL model parameters, we do not find a change of

Fig. 3. Time evolution of produced total number density ofnonequilibrium photons from quark-hadron phase transition.

Fig. 4. Spectral production rate of nonequilibrium photons fromŽ .quark-hadron phase transition, calculated from Eq. 18 with

T s3,5,10 fm, drawn with solid lines. Dashed lines are the ratesw xfor hadron gas and quark-gluon plasma taken from Ref. 13 .

two orders of magnitude. So these high-energy non-thermal photons can be a distinct signature of QGPformation.

In summary, we have computed the nonequilib-rium photon production during the quark-hadronphase transition at high-temperature in the Friedberg-Lee model of quark-hadron physics. Under the‘‘quench’’ approximation, the invariant productionrate for nonequilibrium photons driven by the oscil-lation of the s field due to parametric amplificationis given, which is two orders of magnitude largerthan that from a thermal quark-gluon plasma forphoton energies around 2 GeV. These high-energynon-thermal photons can be a potential test of theformation of quark-gluon plasma in relativisticheavy-ion-collision experiments. Of course, in orderto compare the results with experimental data ondirect photons, a more realistic dynamics of thephase transition should be considered and thenonequilibrium photon production rate should beconvolved with the expansion of the plasma. Thiswork is in progress.

Acknowledgements

We would like to thank D. Boyanovsky and S.-P.Li for their useful discussions. The work of D.S.L.Ž .K.W.N. was supported in part by the National


Science Council, ROC under the Grant NSC88-Ž .2112-M-259-001 NSC88-2112-M-001-042 .

References

w x1 For a review, see, for example, C.-Y. Wong, Introduction toHigh-Energy Heavy-Ion Collisions, World Scientific, Singa-pore, 1994.

w x Ž .2 D.K. Srivastava, B. Sinha, Phys. Rev. Lett. 73 1994 2421.w x Ž .3 J. Sollfrank et al., Phys. Rev. C 55 1997 392.w x Ž .4 G.Q. Li, G.E. Brown, Nucl. Phys. A 632 1998 153.w x Ž .5 D. Boyanovsky, H.J. de Vega, Phys. Rev. D 47 1995 2343;

Ž .D. Boyanovsky, D.-S. Lee, A. Singh, Phys. Rev. D 48 1995800; D. Boyanovsky, H.J. de Vega, R. Holman, Phys. Rev. D

Ž .51 1995 734.

w x6 D. Boyanovsky, H.J. de Vega, R. Holman, Phys. Rev. D 49Ž .1994 2769; D. Boyanovsky, H.J. de Vega, R. Holman,

Ž .D.-S. Lee, A. Singh, Phys. Rev. D 51 1995 4419; D.Boyanovsky, M. D’Attanasio, H.J. de Vega, R. Holman,

Ž .D.-S. Lee, Phys. Rev. D 52 1995 6805.w x7 D. Boyanovsky, H.J. de Vega, R. Holman, S. Prem Kumar,

Ž . Ž .Phys. Rev. D 56 1997 3929; D 56 1997 5233, andreferences therein.

w x Ž .8 R. Friedberg, T.D. Lee, Phys. Rev. D 15 1977 1694; 16Ž . Ž .1977 1096; 18 1978 2623.

w x9 L. Wilets, Non-topological Solitons, World Scientific, Singa-pore, 1989.

w x Ž .10 K.-W. Ng, W.-K. Sze, Phys. Rev. D 43 1991 3813.w x11 W.N. Cottingham, D. Kalafatis, R. Vinh Mau, Phys. Rev.

Ž .Lett. 73 1994 1328.w x Ž .12 A. Ukawa, Nucl. Phys. A 638 1998 339c.w x Ž .13 J. Kapusta, P. Lichard, D. Seibert, Phys. Rev. D 44 1991

2774.

11 March 1999


Non-perturbative couplings and color superconductivity

Nick Evans a,1, Stephen D.H. Hsu b,2, Myckola Schwetz c,3

a Department of Physics, Boston UniÕersity, Boston, MA 02215, USAb Department of Physics, UniÕersity of Oregon, Eugene, OR 97403-5203, USA

c Department of Physics and Astronomy, Rutgers UniÕersity, Piscataway, NJ 08855-0849, USA

Received 15 November 1998; revised 14 January 1999Editor: H. Georgi

Abstract

Quark matter at sufficiently high density exhibits color superconductivity, due to attractive gluonic interactions. At lowerdensities of order L3 , it has been proposed that instanton generated vertices may play an important role in the Cooper pairQCD

formation. We study the renormalization group flow to the Fermi surface of the full set of couplings generated by gluonicand instanton interactions. In earlier work we showed that if the gluonic interactions dominate at the matching scale, theirrunning determines the scale of the Cooper pair formation D. Here we consider all possibilities, including the one in whichthe instanton interactions dominate all others at the matching scale. In the latter case we find that a number of additional

Ž .induced couplings including the gluonic ones reach their Landau poles almost simultaneously with the instanton vertex.Presumably all contribute to the Cooper pair formation. The most important consequence of including all the couplings is alarge increase in the gap size D. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Quark matter at high density exhibits color superconductivity through the dynamical generation of a Cooper² T Ž . Ž .: Ž w x .pair c yp Cg c p see 1 and references contained therein . A simple way to understand this5

w x Ž .phenomena is through an effective field theory description 2,3 of the physics near the Fermi surface FS . Anattractive coupling between the quarks, such as that provided by one gluon exchange, will eventually run to aLandau pole as the Wilsonian cutoff approaches the FS. Of course, this is just a modern reformulation of an

w xinsight first gained from the study of laboratory superconductors 4 . At very high densities the effective theorymay be matched to QCD in a naive fashion with perturbative effects like one gluon exchange dominating the

w x Ždynamics 5 . However, when the Fermi momentum is reduced to of order L assuming that this density isQCD.just above the transition from the low density chiral symmetry breaking phase it is possible that additional

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]


( )N. EÕans et al.rPhysics Letters B 449 1999 281–287282

attractive interactions exist generated by non-perturbative dynamics. An example of such a vertex is thatŽ .generated by instantons we assume there are two quark flavors throughout this paper

a a a ayk u T u d T d yu T d d T u . 1Ž .ž /R L R L R L R L

w xRecent investigations 6 examined the role of this interaction in color superconductivity under the assumptionthat it is the unique important coupling for the dynamics. This assumption, while an interesting starting point forstudying the effects of these non-perturbative couplings, is somewhat ad hoc. Quantum loops generated by theinstanton vertex produce additional interactions with different Lorentz structure which, since the couplings areof order one, we would expect to be of equal importance. In this paper we wish to study the renormalizationgroup flow of the full set of possible couplings that close under renormalization to determine which couplingsare important to the Cooper pair formation.

QCD at finite density may be described by the following lagrangian with chemical potential m

1 amn aLLsy F F q c iDu qmg c . 2Ž .Ž .mn i 0 i4

We make a guess as to the form of the effective theory close to the Fermi surface; the obvious guess based onthe dynamics of non-relativistic systems is that the theory is one of weakly interacting quarks: these are thedressed ‘‘quasi-particles’’ of solid state physics language. Rather than treating the gluons and instantons asdynamical degrees of freedom we will integrate them out leaving a potentially infinite sum over local, higherdimension fermion operators. The locality of these operators requires that gluons be screened at long-distances,presumably by effective electric and magnetic mass terms induced by the medium. In the effective theory closeto the Fermi surface we will for simplicity assume that the QCD dynamics can be matched to momentumindependent four fermion interactions. This can be justified as a reasonable approximation by studying the

Ž .coupled renormalization group RG equations flow with components of higher angular momentum and notingan approximate decoupling of the equations. We assume that the typical gauge propagator has momentum of

Ž . Ž .order m. Since the Fermi surface breaks the O 3,1 invariance of the theory to O 3 we must treat spatial andŽ .temporal interactions independently. The full set of couplings we consider are up to parity transformations

g u g u u g u q u m dŽ .1 L 0 L L 0 L

g u g u u g u q u m dŽ .2 L i L L i L

g u g u u g u q umdŽ .3 L 0 L R 0 R

g u g u u g u q umdŽ .4 L i L R i R

g u g u d g d5 L 0 L R 0 R

g u g u d g d6 L i L R i R

3Ž .g u g u d g d7 L 0 L L 0 L

g u g u d g d8 L i L L i L

g u g d d g u9 L 0 L L 0 L

g u g d d g u10 L i L L i L

g u u d d11 L R L R

g u g g u d g g d12 L 0 i R L 0 i R

g u d d u13 L R L R

g u g g d d g g u14 L 0 i R L 0 i R

( )N. EÕans et al.rPhysics Letters B 449 1999 281–287 283

where the indicated groupings of couplings imply closure under RG flow. This is still not the full possible set ofcouplings consistent with the space-time symmetries but it is the full set including both gluon and instantonvertices that closes under RG evolution. Henceforth we will include signs from the contraction of spacelike gi

matrices in the coupling constants.ŽHere we will consider the 3 channel which has been shown to be the most attractive channel for both the

w x w x .gluonic 1,2 and instanton 6 vertices with the color group matrix structure

d d yd d . 4Ž .ca db cb da

At high densities, where single gluon exchange dominates the perturbative effects, the appropriate matchingconditions are

4pa mŽ .s1g sg sg sg sgs g sg sg sg syg1 3 5 7 2 4 6 83 2 5m Ž .g syg sk g sg sg sg s013 11 9 10 12 14

w xwhere k is the instanton-generated four-fermion coupling considered in Ref. 6 . The sign of k is a result of thew x Ž .non-perturbative dynamics and is traditionally 6 taken positive so 1 is capable of driving chiral symmetry

w xbreaking at ms0. Since we are addressing the consistency of 6 we will do likewise.Ž .The Fermi surface in 2 picks out momenta of order m. It is therefore natural to study the theory as we

approach the Fermi surface in a Wilsonian sense. We parameterize four momenta in the following fashion

p m s p , p s k ,kq l , 6Ž . Ž . Ž .0 0

where k lies on the Fermi surface and l is perpendicular to it. We study the Wilsonian effective theory ofmodes near the Fermi surface, with energy and momenta

< < < <k , l -L , L™0 . 7Ž .0

In this limit the four fermion operators are all irrelevant operators excepting those with the particular threew xmomentum structure corresponding to quarks with momenta k and yk scattering to momenta q and yq 2,3 .

As a result of this truncation of the theory the only diagrams allowed by the momentum structure are theŽ . 4bubble diagrams found at large N in the familiar O N model . To display the basic behavior, consider a

theory with just the simple interaction

Gcccc . 8Ž .Ž w x.The interaction generates cooper pair formation through the exact gap equation see Bailin and Love 1

d4 pLUV mDs iG G C p G , 9Ž . Ž .H m40 2pŽ .

m Ž .where G is the coupling, G any associated Dirac structure and C k a 4=4 off diagonal component of theŽ C .8=8 propagator associated with the fermion vector c ,c

1 1C p s D . 10Ž . Ž .y1˜pu ymgŽ . D pu ymg Dy pu qmgŽ . Ž .0 0 0

The gap integral would be log divergent near the FS were it not for the condensate D, which cuts off the

4 w xThe expansion parameter analogous to 1rN is Lrm 3 .


Ž .contribution of modes with very low energies. Thus no matter how small the attractive coupling G there willalways be a solution for some D. Approximating the IR cutoff, and keeping the lowest order in D we have

d4 p 1 1LUVD;yiG D . 11Ž .H 4 pu ymg pu qmgŽ . Ž .D 2pŽ . 0 0

Neglecting factors form the gamma matrix structure, the Cooper pair condensate is of the form

cyDsL e , 12Ž .NGUV

where c is a constant.An alternative understanding of this condensate formation is found by resumming bubble graphs to calculate

the renormalization group flow of the four quark vertex. Note that because of the restricted momentum structureof the vertices one loop b functions are exact. The one loop graph contributes a logarithmic divergence to therunning in the presence of a Fermi surface:

4d p i i2yG . 13Ž .H m n4 p g qmg y ie yp g qmg y ie2pŽ . m 0 n 0 jli k

< < < < 2 2Performing the gamma matrix algebra and taking the limit k , l ™0, k ™m , this becomes0

12G y g g q g g I , 14Ž . Ž . Ž . Ž . Ž .i j i j0 0 a a3k l k l

where the log divergent part of the integral I is given by

dk d2 k dl 1 i L0 IR1Is , N ln . 15Ž .H4 4 ž /k q ly ie k y lq ie 4 LŽ . Ž .2pŽ . 0 0 UV

2 Ž .3 2 2Ns d kr 2p sm r2p , assuming the density of states at the Fermi surface is given by the lowest orderHŽ .approximation. The running effective coupling neglecting the gamma matrix structure again is given by

G LŽ .UVG L s , 16Ž . Ž .IR 11q G L NtŽ .UV4

Ž . Ž < < .where ts ln L rL . Here we have moved from an effective theory with cutoff L k , l -L , to aIR UV UV 0 UVŽ .new effective theory with cutoff L . As we approach the Fermi surface L ™0 , the coupling G runsIR IR

logarithmically. The Landau pole of the coupling corresponds to the scale D of Cooper pair formation.Ž .In this fashion we can calculate the one loop beta functions for the vertices in 3 as a function of t. There

are in principle 142 entries to the RG matrix but many entries are trivially zero. The RG equations simplifydrastically if one introduces the linear combinations of couplings

G sg qg G sg qg G sg qg1 1 2 7 7 8 11 11 12

G sg y3g G sg y3g G sg y3g2 1 2 8 7 8 12 11 12

G sg yg G sg qg G sg qg3 3 4 9 9 10 13 13 1417Ž .

G sg q3g G sg y3g G sg y3g4 3 4 10 9 10 14 13 14

G sg yg5 5 6

G sg q3g6 5 6


Fig. 1. The RG running of the temporal and spatial one gluon derived couplings between two left handed quarks and that between a lefthanded and a right handed quark in the absence of instanton effects.

and the RG equations then simplify to

1 1 22 2 2 2 2˙ ˙ ˙G sy NG G sy N G qG qG qG G sy N G G qG GŽ .Ž .1 1 7 7 9 11 13 11 7 11 9 133 3 3

2 2 2 2 2˙ ˙ ˙G syNG G syN G qG qG qG G sy2 N G G qG GŽ .Ž .2 2 8 8 10 12 14 12 8 12 10 14

2 2 22˙ ˙ ˙ 18G sy NG G sy N G G qG G G sy N G G qG G Ž .Ž . Ž .3 3 9 7 9 11 13 13 7 13 9 113 3 3

˙ ˙ ˙ ˙G sG s0 G sy2 N G G qG G G sy2 N G G qG GŽ . Ž .4 6 10 8 10 12 14 14 10 12 8 14

2 2G sy NG5 53

We proceed further by numerical solution of the RG equations. At very high density the instanton vertex isexpected to be exponentially suppressed relative to the one gluon exchange. Neglecting the instanton vertexremoves any flavor dependence from the problem and there are just four independent couplings, the spatial andtemporal couplings between two left handed quarks and between a left and a right handed quark. We show theRG running in Fig. 1. The coupling between two left handed quarks reaches its Landau pole first as found in

w xRef. 2 .We may check the effect of the instanton by including it but with a smaller value than the one gluon

exchange at the matching scale. We show this case in Fig. 2 with ks0.01 g. The gluonic couplings reach theirLandau pole before the instanton vertex becomes large, though the instanton vertex is eventually driven to

Ž .Fig. 2. The influence of a small instanton coupling g and g on the running of the gluonic couplings between an up quark and a down11 13Ž .quark g and g are in the left left channel, g and g in the left right channel .7 8 3 4


Ž .Fig. 3. A toy model in which the instanton couplings g and g are the only non-zero coupling at the matching scale. The other six11 13Ž .couplings grow from zero g , g , g have positive values, g , g , g negative and reach Landau poles at approximately the same scale7 10 12 8 9 14

as the instanton vertices.

infinity by feedback from the larger gluon interaction. When we include the full RG flow the instanton vertexdoes eventually catch the gluonic coupling but only at very large values of the coupling 410, which increasefor smaller values of the instanton coupling at matching. The scale of the Landau pole, and hence the size of theCooper pairing gap, is determined by the running of the gluonic couplings as expected.

We next turn to the effects of the instanton vertex when it is the dominant interaction. It is interesting to seewhat happens in the case where the instanton vertex is treated as the sole non-zero coupling at the matching

Ž .scale. This scenario is shown in Fig. 3. The 8 couplings g yg all reach values of order one simultaneously7 14

and presumably play an equal role in Cooper pair formation. This is not a surprise since, as can be seen fromŽ .18 , the instanton couplings G yG only run after the generation of couplings G yG . Note that for a11 14 7 10

Ž .very thin shell around the FS L <m this running is implicitly included in a gap equation analysisUV

performed with just the instanton coupling at the matching scale, since the gap equation is exact in this limit.What this analysis shows is that as expected the instanton coupling, which generates the other vertices at oneloop, drives them rather quickly to large values. The analysis we have performed retains only the relevant

Ž .operators near the FS. The effects of the intermediate running including irrelevant operators which takes usclose to the FS is assumed to be reflected in our boundary conditions at the matching scale. Our results suggest

Ž 3 .that in QCD at low densities of order L , where the instanton coupling k is of order one, the correctQCD

analysis would include all 8 couplings with approximately equal values at matching.If we do assume that all eight interactions are equal at the matching scale and have the natural choices of sign

suggested by Fig. 3 then the RG equations simplify. The couplings G taken with initial conditions 0 remain 0odd

Ž .Fig. 4. The RG flow when all 8 couplings interacting with the instanton vertex g y g are taken equal at matching. The couplings7 14

evolve together.


for the full running. The couplings G with initial conditions "4G run with the same absolute values andeven

share the same Landau pole. In terms of the original 8 couplings this means that they all run together and reachtheir Landau poles simultaneously. All possible interactions appear to play an equal role in the Cooper pairformation. We show the running in Fig. 4.

w xThe Cooper pair formation with only an instanton interaction at matching has been studied in Ref. 6 and forthe two flavor case the condensate that forms is in the anti-symmetric color 3, is an anti-symmetric singlet in

Žflavor, and an anti-symmetric singlet of spin as is the case when the condensate is driven by gluonicw x.interactions 1 . Since for the instanton case, as can be seen from Fig. 3, all the couplings g yg are7 14

essentially equal at the Landau pole we may deduce that for the case where this equality is enforced at thematching the same condensate forms. The important difference is that the scale of the Landau pole is increasedsharply by the inclusion of the full set of couplings at matching. Typically, whatever the matching condition

< <taken on the couplings, the value of t at the pole is significantly smaller than for the pure instanton case. TheCooper pair, if non-perturbative effects are sufficiently large to play a role, is therefore expected to beconsiderably larger than the estimates including only the instanton vertex.

Acknowledgements

The authors would like to thank M. Alford, M. Peskin, K. Rajagopal, T. Schafer and F. Wilczek for usefuldiscussions and comments. After this work was completed we became aware of work by Schafer and WilczekŽ .IASNS-HEP-98-90 which addresses similar issues. Their analysis includes a simplification of the RGequations through the use of flavor symmetries and Fierz identities. This work was supported in part under DOEcontracts DE-FG02-91ER40676, DE-FG06-85ER40224 and DE-FG02-96ER40559

References

w x Ž . Ž . Ž . Ž .1 D. Bailin, A. Love, Nucl. Phys. B 190 1981 175; B 190 1981 751; B 205 1982 119; Phys. Rep. 107 1984 325.w x2 N. Evans, S.D.H. Hsu, M. Schwetz, hep-phr9808444.w x Ž . Ž . Ž .3 G. Benfatto, G. Gallavotti, J. Stat. Phys. 59 1990 541; Phys. Rev. C 42 1990 9967; R. Shankar, Physica A 177 1991 530; Rev. Mod

Ž . Ž .Phys. 66 1993 129; J. Polchinski, in: J. Harvey, J. Polchinski Eds. , Proceedings of the 1992 TASI, World Scientific, Singapore, 1993.w x4 See, for example, N. Ashcroft, N.D. Mermin, Solid State Physics, Saunders College Publishing, 1976.w x Ž . Ž .5 Freedman and McLerran, Phys. Rev. D 16 1977 1130; D 16 1977 1147, 1169.w x Ž .6 R. Rapp, T. Schafer, E.V. Shuryak, M. Velkovsky, Phys. Rev. Lett. 81 1998 53; M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B

Ž .422 1998 247; M. Alford, K. Rajagopal, F. Wilczek, hep-phr9804403.

11 March 1999


Nonequilibrium chiral perturbation theoryand pion decay functions

A. Gomez Nicola a,1, V. Galan-Gonzalez b,2´ ´ ´a Departamento de Fısica Teorica, UniÕersidad Complutense, 28040 Madrid, Spain´ ´

b Theoretical Physics, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK

Received 1 December 1998Editor: R. Gatto

Abstract

We extend chiral perturbation theory to study a meson gas out of thermal equilibrium. Assuming that the system isinitially in equilibrium at T -T and working within the Schwinger-Keldysh contour technique, we define consistently thei c

time-dependent temporal and spatial pion decay functions, the counterparts of the pion decay constants, and calculate themto next to leading order. The link with curved space-time QFT allows to establish nonequilibrium renormalisation. Theshort-time behaviour and the applicability of our model to a heavy-ion collision plasma are also discussed in this work.q 1999 Elsevier Science B.V. All rights reserved.

PACS: 12.39.Fe; 11.10.Wx; 05.70.Ln; 12.38.MhKeywords: Nonequilibrium thermal field theory; Chiral perturbation theory; Pion decay constants; Heavy-ion collisions; Quantum fieldtheory in curved space-time

1. Introduction

The chiral phase transition plays a fundamental role in the description of the plasma formed after aŽ .relativistic heavy-ion collision RHIC , where it is imperative to use meson effective models to describe QCD.

Ž . Ž .Two of the most successful approaches are the O 4 linear sigma model LSM , valid only for N s2 lightfŽ .flavours, and Chiral Perturbation Theory ChPT , based on derivative expansions compatible with the QCD

Ž . w xsymmetries, and whose lowest order action is the nonlinear sigma model NLSM 1 . In ChPT, the perturbativeŽ .parameter is prL , with p a meson energy like masses, external momenta or temperature and L , 1 GeV.x x

Ž 2 2 .Every meson loop is OO p rL and all the infinities coming from them can be absorbed in the coefficients ofx

w xhigher order lagrangians 1,2 .In thermal equilibrium at finite temperature T , the chiral symmetry is believed to be restored at T ,c

w x150–200 MeV 3 . In fact, near T , the mean-field LSM is well known to undergo a second-order phasec



( )A. Gomez Nicola, V. Galan-GonzalezrPhysics Letters B 449 1999 288–298´ ´ ´ 289

transition. The NLSM is equally valid for reproducing the phase transition, provided one works in the large-Nw x w x 2 2limit 4 . Strictly within ChPT, the low-temperature meson gas has been studied 5,6 on expansion in T rL ,x

predicting the correct behaviour of the observables as T approaches T .c

The equilibrium assumption is not realistic if one is interested in the dynamics of the expanding plasmaformed after a RHIC, where several nonequilibrium effects could be important. One of them is the formation of

Ž .disoriented chiral condensates DCC , regions in which the chiral field is correlated and has nonzerow xcomponents in the pion direction 7 . As the plasma expands, long-wavelength pion modes —propagating as if

they had an effective negative mass squared— can develop instabilities growing fast as the field relaxes to thew xground state, an observable consequence being coherent pion emission 8 . This issue has been extensively

studied in the literature, mostly within the LSM assuming initial thermal equilibrium at T )T , either encodingi cw xthe cooling mechanism in the time dependence of the lagrangian parameters 8–11 or describing the plasma

w xexpansion in proper time and rapidity 12 . This phenomenon has also been studied using Gross-Neveu modelsw x w x13 . Another important nonequilibrium observable is the photon and dilepton production 14 , to which the

w xanomalous meson sector could significantly contribute 10 .In this work we will construct an effective ChPT-based model to describe a meson gas out of thermal

equilibrium, as an alternative to the LSM approach. Our only degrees of freedom will be then the Nambu-Gold-Ž .stone bosons NGB and we will consider the most general low-energy lagrangian compatible with the QCD

Ž . Žsymmetries. We will restrict here to N s2 where the NGB are just the pions and to the chiral limit masslessf.quarks , which is the simplest approximation allowing to build the model in terms of exact chiral symmetry.

One of the novelties of our approach is to exploit the analogy between ChPT and the physical regime where thesystem is not far from equilibrium and then a derivative expansion is consistent.

2. The NLSM and ChPT out of equilibrium

We will take the system in thermal equilibrium for tF0 at a temperature T -T and for t)0 we let thei c

lagrangian parameters be time-dependent. We are also assuming that the system is homogeneous and isotropic.The generating functional of the theory can then be formulated in the path integral formalism, by letting the time

w xintegrals run over the Schwinger-Keldysh contour C displayed in Fig. 1 15–18 . We will eventually lett ™y` and t ™q`, although we will show that our results are independent of t and t . We remark that,i f i f

even in that limit, the imaginary-time leg of C has to be kept, since it encodes the KMS equilibrium boundaryw xconditions 17–19 . With these assumptions, our low-energy model will be the following nonequilibrium NLSM

f 2 tŽ .4 † mw xS U s d x tr E U x ,t E U x ,t 1Ž . Ž . Ž .H m4C

4 3 Ž . Ž . Ž . Ž . y1where H d x'H dtHd x, U x,t gSU 2 is the NGB field, satisfying U x,t q ib sU x,t with b sT ,C C i i i i iŽ . Ž .and f t is a real function which in equilibrium and to the lowest order see Section 4 would be fs f ,93p

Ž . Ž . Ž .MeV the pion decay constant i.e, f tF0 s f. Note that f t cannot be analytic at ts0 and, in particular, it

Fig. 1. The contour C in complex time t. The lines C and C run between t q ie and t q ie and t y ie and t y ie respectively, with1 2 i f f i

e ™0q.

( )A. Gomez Nicola, V. Galan-GonzalezrPhysics Letters B 449 1999 288–298´ ´ ´290

w xcould be discontinuous, like the meson mass in quenched LSM approaches 8–11 . This is a consequence of thenature of our approach, since the system is driven off equilibrium instantaneously. An alternative, which we

Ž . w xwill not attempt here, is to choose f t analytic ; t, having equilibrium only at ts t 20 . Thus, the temporali

evolution of our results will start at ts0q, an infinitesimally small response time. As we will see below, ourapproach is consistent because the discontinuities at ts0 appear to NLO.

We will parametrise the field U as

1 1r22 2 aU x ,t s f t yp x ,t Iq it p x ,t ; as1,2,3, 2Ž . Ž . Ž . Ž . Ž .½ 5af tŽ .

2 a a aŽ . aŽ .where p sp p , p the pion fields satisfying p t q ib sp t and I and t are the identity and Paulia i i i aŽ .matrices. Note that with the choice 2 we recover the canonical kinetic term in the action after expanding U in

powers of p . Other choices amount to a time-dependent normalisation of the pion fields and should not haveŽ . a a Ž . Ž .any effect on the physics see Section 4 . For instance, if we redefine p sp f 0 rf t , the action for the˜

aŽ . 2Ž . 2Ž . Žp x,t fields is just the equilibrium NLSM multiplied by the time-dependent scale factor f t rf 0 see˜.below .

Ž . Ž Ž . Ž . †.Our action 1 is manifestly chiral invariant U x ™LU x R . Notice that we work in the chiral limit andhence there are no explicit symmetry-breaking pion mass terms in the action. The conserved axial and vector

w xcurrents for the chiral symmetry can be derived by applying the standard procedure 1,2 , so that the axialcurrent reads

2f tŽ .a a † †A x ,t s i tr t U E UyUE U 3Ž . Ž .Ž .m m m4

Let us now discuss how to establish a consistent nonequilibrium ChPT. The new ingredient we need is theŽ .temporal variation of f t . We will then consider

22˙˙ ¨ f tf t p f t pŽ .Ž . Ž .

,OO , , ,OO , 4Ž .2 3 4 2ž / ž /Lf t f t f t LŽ . Ž . Ž .x x

and so on, the rest of the chiral power counting being the same as in equilibrium. Therefore, in our approach wetreat the deviations of the system from equilibrium perturbatively, following the ChPT guidelines. Thus, we will

Ž .expand our action 1 to the relevant order in pion fields and take into account all the contributing Feynmandiagrams. The loop divergences should be such that they can be absorbed in the coefficients of higher order

Ž .lagrangians, which in general will require the introduction of new time-dependent counterterms see below .Ž .Notice also that according to 4 , we can always describe the short-time nonequilibrium regime, just by

Ž . q y1expanding f t around ts0 . In fact, for times tF f , that is equivalent to a chiral expansion, since thenp

˙ q qŽ . Ž . Ž . Ž .f 0 trf 0 sOO prL and so on. Nonetheless, we stress that the conditions 4 do not imply working atx

short-times, but just to remain close enough to equilibrium.Ž . Ž .To leading order in p fields, the action 1 , after using 2 , reads

f tŽ .1 4 a 2 a 2w xS p sy d xp x ,t Iqm t p x ,t with m t sy , 5Ž . Ž . Ž . Ž . Ž .H0 2 f tŽ .C

where we have partial integrated in C. Thus, the leading order nonequilibrium effect of our model can bewritten as a time-dependent pion mass term, which, as commented before, is a common feature of nonequilib-

w x 2Ž .rium models 9–12 . Notice that m t can be negative, so that our model accommodates unstable pion modes,


whose importance we have discussed before. Note also that this mass term does not break the chiral symmetry,Ž .i.e, the axial current is classically conserved. Indeed, to leading order we have, from 3 ,

LOa a a˙A x ,t syf t E p x ,t qd f t p x ,t , 6Ž . Ž . Ž . Ž . Ž . Ž .m m m0

m 2 awhich satisfies E A s0 using Iqm t p s0, the equations of motion to the same order. Had weŽ .m

included the pion mass term m —explicitly breaking the symmetry— the instabilities threshold, to leadingp2Ž . 2order, would have been m t -ym instead.p

It is very interesting to rephrase our model as a NLSM in a curved space-time background g , which readsmn

w x21,22 ,2f 0Ž .

4 mn †'w x w xS U s d x yg g tr E U x E U x qj S U, R 7Ž . Ž . Ž .Ž .Hg m n R4 C

plus U independent terms, where gsdet g and the last term accounts for possible couplings between the pionŽ . Ž Ž . 2 w x.fields and the scalar curvature R x like R x f for a free scalar field f 21 . Now, notice that our

Ž . Ž . Ž Ž .nonequilibrium model 1 is obtained by writing U x in the p parametrisation discussed before i.e, with f t˜Ž . Ž .. Ž . Ž .replaced by f 0 in 2 , choosing js0 minimal coupling and a spatially flat Robertson-Walker RW

2 2Ž .w 2 2 x Ž .space-time in conformal time, whose line element is ds sa h dh ydx , with the scale factor a h sŽ . Ž .f h rf 0 . Our effective theory is then not only suitable for a RHIC environment, but also in a cosmological

framework. Notice also that if we take j/0, the lowest order S term we can construct has the form of anR

effective mass term breaking explicitly the chiral symmetry. In fact, it is not difficult to see that we could cancel2Ž . Ž . w xthe m t term in 5 by choosing js1r6, which is the value rendering the theory scale invariant 21 . This is

just a consequence of the lagrangian chiral and conformal symmetries being incompatible in a curvedw xbackground 22 or, equivalently, at nonequilibrium. In other words, for js0 —which is our choice, since we

w x 2Ž .want to preserve chiral symmetry, as in Ref. 22 — we may interpret the m t term, in the chiral limit, as theminimal coupling with the background yielding chiral invariance.

The above equivalence turns out to be very useful to renormalise our model, consistently with ChPT. In fact,Ž . Ž 4.all the one-loop divergences arising from 7 can be absorbed in the coefficients of the OO p action S , which4

consists of the Minkowski terms with indices raised and lowered with g plus new chiral-invariant couplingsmn

w xof pion fields with the curvature 22 . In the chiral limit, those new terms read

R 4 mn mn †'w xS p s d x yg L R x g qL R trE U x E U xŽ . Ž . Ž .Ž .H4 11 12 m nC

1 4 a 2 2 2 a 4sy d xp f t E y f t = qm t p qOO p 8Ž . Ž . Ž . Ž . Ž .H 1 t 2 12C

w xwhere R is the Ricci tensor, L and L are two new low-energy constants 22 and we have given themn 11 12Ž .two-pion contributions in the parametrisation 2 , after partial integration, with our RW metric, where

2˙¨ f tf t Ž .Ž .f t s12 2 L qL yL ,Ž . Ž .1 11 12 123 4f t f tŽ . Ž .

2˙¨ f tf t Ž .Ž .f t s4 6L qL qL ,Ž . Ž .2 11 12 123 4f t f tŽ . Ž .

¨ ˙ ˙f t f t q f t f tŽ . Ž . Ž . Ž .1 1 12 ¨m t sy q f t . 9Ž . Ž . Ž .1 12f tŽ .Ž .for t)0 and f tF0 s0. The above terms are the only ones in S containing two pions and they willi 4

renormalise purely nonequilibrium infinities —which are time-dependent and vanish for tF0—. It is important


to bear in mind that to cancel the one-loop new divergences only L needs to be renormalised, whereas11r w xL sL 22 . We will come back to this point below.12 12

w xNext, we will concentrate on the Green functions time-ordered along C 17,18 . Unless otherwise stated, weŽ .will be using the parametrisation 2 in the remaining of this work. The two-point function defines the pion

abŽ . ² aŽ . aŽ .: abŽ . ab Ž .propagator G x, y syi T p x p y , which to leading order G x, y sd G x, y , by isospinC 0 0

invariance, and

I qm2 x 0 G x , y syd x 0 yy0 d Ž3. xyy 10Ž . Ž . Ž . Ž .� 4 Ž .x 0 C

) Ž . - Ž .with KMS equilibrium conditions G x,t y ib ; y sG x,t ; y , the advanced and retarded propagators0 i i 0 iŽ X. Ž X X.being defined as customarily along C. Notice that G x, x sG t,t , xyx due to the nonequilibrium lack of

time translation invariance. Therefore, we will define, as customarily, the ‘‘fast’’ temporal variable ty tX andŽ X. Ž . Ž X. 2 < < 2the ‘‘slow’’ one t' tq t r2, so that F q ,v ,t and F v ,t,t , with v s q , will denote, respectively,0 q q q

Ž . Ž X.the fast and mixed in which only the spatial coordinates are transformed Fourier transforms of F x, x . NoteŽ .that F q ,v ,t depends separately on q and v because of the thermal loss of Lorentz covariance and has0 q 0 q

Ž .the extra nonequilibrium t-dependence. Then, in the mixed representation, 10 becomes2d

X X2 2qv qm t G v ,t ,t syd ty t 11Ž . Ž . Ž .Ž .q 0 q C2dt

Ž . 2Ž . w xThe general solution of 11 is only known explicitly for some particular choices of m t 21,17,18 .Formally, we can write it as a Schwinger-Dyson equation as

G v ,t ,tX sGeq v ,ty tX q dzm2 z Geq v ,tyz G v , z ,tX 12Ž . Ž .Ž . Ž . Ž . Ž .H0 q 0 q 0 q 0 qC

eqŽ X. Ž . 2Ž .with G v ,ty t the equilibrium solution of 11 , i.e, with m t s0.0 qŽ . ) Ž . - Ž .Another object of interest for our purposes is the Lehman spectral function r x, y sG x, y yG x, y

w x eqŽ . Ž . Ž 2 . w x16 , which in equilibrium to leading order is r q sy2p isgn q d q 19 . Note that, by construction,0 0) Ž . - Ž . Ž . Ž . Ž . Ž .G x, y sG y, x , so that r x, y syr y, x and r q ,v ,t syr yq ,v ,t . The normalisation of r0 q 0 q 0

isX

q`1 dr v ,t ,tŽ .0 qq r q ,v ,t s sy1, 13Ž . Ž .H 0 0 0 q

X2p i dty` ts t

Ž . Ž .which can be readily checked by using 12 and r v ,t,t s0.0 q

3. Next to leading order propagator

Ž .We will now obtain the NLO correction to the propagator. For that purpose, we need the action in 1 up tofour-pion terms:

2¨ ˙1 f t f tŽ . Ž .21 14 a m b 2 6w x w xS p sS p q d x E p E p p p q p y qOO p 14Ž . Ž . Ž .H0 m a b2 22 2½ 5ž /f tf t f tŽ .Ž . Ž .C

Ž . Ž . Ž .plus the two-pion terms in 8 . The two diagrams contributing are, respectively, a and b in Fig. 2.

Ž . Ž . .Fig. 2. Diagrams contributing to the NLO pion propagator a,b and axial-axial correlator c . The black dot in b represents the interactionR Ž .coming from S in 8 .4


Ž X . XLet us concentrate on G i.e, t,t gC in Fig. 1 and t and t positive, which is the relevant case for our11 1Ž .purposes, as commented above. We will use dimensional regularisation DR , so that evaluating the above

Ž . Žd .Ž .diagrams, using 10 with d 0 s0, and after some algebra, we obtain in the mixed representation to NLO

2i b v Ti q iX X X X1) )G t ,t sG t ,t 1y f t q f t q coth cos v tq tŽ . Ž . Ž . Ž . Ž .Ž .11 0,11 1 1 q2 22v 2 12 fq

t X X) ) ) )˙ ˙˜ ˜ ˜ ˜ ˜ ˜ ˜q i d t D t ,v G t ,t G t ,t qD t G t ,t G t ,tŽ . Ž . Ž . Ž . Ž .Ž .H 1 q 0 0 2 0 0½0

Xt X X- - - -˙ ˙˜ ˜ ˜ ˜ ˜ ˜ ˜y d t D t ,v G t ,t G t ,t qD t G t ,t G t ,tŽ . Ž . Ž . Ž . Ž .Ž .H 1 q 0 0 2 0 00

t X X- ) - )˙ ˙˜ ˜ ˜ ˜ ˜ ˜ ˜y d t D t ,v G t ,t G t ,t qD t G t ,t G t ,t 15Ž . Ž . Ž . Ž . Ž . Ž .Ž .H 1 q 0 0 2 0 0 5Xt

- Ž X . ) Ž X .and G t,t sG t ,t , where we have suppressed for simplicity the v dependence of the propagators, the11 11 q

˜dot denotes drdt,

2¨ ˙ ˙˜ ˜ ˜1 f t f t f tŽ . Ž . Ž .2 ¨ ˙˜ ˜ ˜ ˜D t ,v s 6 y5 yv G t y2G t q4 G tŽ . Ž . Ž .Ž .1 q q 0 0 02 ž /˜ ˜ ˜˜ f t f t f tf t ž /Ž . Ž . Ž .Ž .

˙ ˙˜ ˜f t f tŽ . Ž .1 12 ¨˜ ˜ ˜q iv f t y f t y i q f t , 16Ž . Ž . Ž . Ž .q 2 1 12˜f tŽ .

˜G tŽ .0˜D t s , 17Ž . Ž .2 2 ˜f tŽ .

Ž 0. Ž . Ž .and G z 'G z, z is the equal-time correlation function. We observe that 15 is t and t independent,0 0 i fŽ .which is a good consistency check. Notice also that by replacing the equilibrium propagators in 15 , we recover

T 2X Xeq ) eq )G ty t sG ty t 1y 18Ž . Ž . Ž .11 0,11 2ž /12 f

w x Žwhich agrees with 4 note that we have derived it for the contour C, including both imaginary-time and.real-time thermal field theory and is finite in the chiral limit, where there is no tadpole renormalisation in DR

w x2 . However, out of equilibrium, the NLO propagator is in general divergent, even in the chiral limit, and theŽ .infinities have to be absorbed in the two-pion counterterms in 8 .

4. The nonequilibrium pion decay functions

In a thermal bath, the concepts of LSZ and asymptotic states are subtle, and so is then the extension oflow-energy theorems like PCAC. Thus, pion decay constants are more conveniently defined through the thermal

ab Ž . ² aŽ . bŽ .:axial-axial correlator A x, y s T A x A y . At T/0 the loss of Lorentz covariance in the tensorialmn C m ms Ž . t Ž .structure of A implies that one can define two independent and complex f spatial and f temporal , theirmn p p


real and imaginary parts being related respectively with the pion velocity and damping rate in the thermal bathw x w x23 . Nevertheless, to one-loop in the chiral limit one has 5,4,23

2T22s t 2f T s f T s f 1y , 19Ž . Ž . Ž .p p 2ž /Tc

' Ž .with T s 6 f , 228 MeV. Despite it being just the lowest order in the low temperature expansion, 19c p

predicts the right behaviour and a reasonable estimate for the critical temperature, although, strictly speaking,Ž . w x s t s, t w xf T is not the order parameter 4 . To higher orders, f / f and Im f /0 23 .p p p p

Let us then analyze Aab in our nonequilibrium model. The relevant quantity, as far as f is concerned, is themn p) - ab ab Ž . Žspectral function r sA yA , with A sd A . We readily realise that r q ,q,t syr yq ,ymn mn mn mn mn mn 0 nm 0

.q,t . Then, from rotational symmetry,

r q ,q ,t sq q r q ,v ,t qd r q ,v ,t 20Ž . Ž . Ž . Ž .i j 0 i j L 0 q i j d 0 q

Ž . Ž . 3 Ž . Ž .with r q syr yq and r q ,q,t sq r q ,v ,t . Therefore, r is characterized, in principleL,d 0 L,d 0 j0 0 j S 0 q mn

by the four functions r , r , r and r . However, they are related through the A conservation Ward IdentityL d S 00 m

Ž . x mn Ž . y mn Ž .WI E r x, y sE r x, y s0, which also holds in our model. Thus, we getm n

i0 2q r q ,v ,t yv r q ,v ,t y r q ,v ,t s0,Ž . Ž . Ž .˙00 0 q q S 0 q 00 0 q2

i0 2q r q ,v ,t yv r q ,v ,t q r q ,v ,t qr q ,v ,t s0, 21Ž . Ž . Ž . Ž . Ž .˙S 0 q q L 0 q S 0 q d 0 q2

w xwhere the dot denotes ErEt . Thus, only two components of r are independent, as in equilibrium 4 , wheremn

there are no time derivatives in the above equation.2 Ž 0. Ž 2 .At Ts0 one has r s2p f sgn q d q , since there exist NGB states. That is not the case at T/0,L p

w x Ž .where the pion dispersion relation is not in general a d-function 4 . In fact, to define properly f T requirespq w xtaking the v ™0 limit, in which a zero-energy excitation still exists 4 , although to NLO there is no need toq

take that limit. Extending these ideas to nonequilibrium, we will define the time-dependent pion decay functionsŽ .PDF as

`1 d2 Xsf t s lim dq q r q ,v ,t s lim i r v ,t ,t 22Ž . Ž .Ž . Ž .Hp 0 0 L 0 q L qq q X2p dtv ™0 v ™0y` ts tq q

`1s tf t f t s lim dq r q ,v ,t s lim r v ,t ,t 23Ž . Ž . Ž .Ž . Ž .Hp p 0 S 0 q S qq q2p v ™0 v ™0y`q q

`i dX1sf t g t sy lim dq q r q ,v ,t s lim r v ,t ,t 24Ž . Ž . Ž .Ž . Ž .Hp p 0 0 S 0 q S q2q q X2p dtv ™0 v ™0y` ts tq q

sŽ . t Ž .The functions f t and f t are the nonequilibrium counterparts of the spatial and temporal pion decayp p

Ž .constants respectively, whereas g t vanishes in equilibrium. However, the above PDF are related through thep

Ž .WI. Integrating in q in 21 , we get0

1 ds t sf t g t s f t f t , 25Ž . Ž . Ž . Ž . Ž .p p p p2 dt

w xso that only two PDF are independent, as in equilibrium 23 . Let us now check the consistency of ourŽ . LO Ž X. Ž . Ž X. Ž X. Ž .definitions to leading order. From 6 , r v ,t,t s if t f t r v ,t,t and r s0, so that, using 13 yieldsL q 0 q d

3 w x Ž 0 . L 2 2 2 Ž 0 . TOur r and r correspond in in the notation of 4 , to sgn q r q rv q and sgn q r respectively.L d A 0 q A


sŽ .2 2Ž . Ž .f t s f t , i.e, the PDF coincides with f t to leading order, as it should be. Similarly, we findp LOt s ˙Ž . Ž . Ž . Ž . Ž .f t s f t s f t and g t s f t , so that our definitions are consistent to leading order.p LO p LO p LO

Ž 3. Ž .To NLO, we need the axial current up to OO p . From 3 ,

˙1 f tŽ .LOa a a 2 2 a 2 a 5A x ,t s A x ,t y p E p yp E p yd p p qOO p 26Ž . Ž . Ž . Ž .m m m m m0ž /2 f t f tŽ . Ž .LO Ž .with A in 6 . Thus, according to our chiral power counting, we have three types of NLO corrections to A .m mn

The first is the NLO correction to the pion propagator we have evaluated in Section 3, coming from the productŽ . Ž . Ž 3. .of the OO p terms above. The second is the product of the OO p with the OO p , represented by diagram c in

Ž .Fig. 2, and the third comes from the modification in A due to the action 8 , which amounts to prefactorsm

w Ž .x w Ž .x Ž . Ž1q f t and 1q f t in A and A respectively. Then, evaluating r to NLO, after using 15 we take,1 2 0 j mnX . Ž . Ž .without loss of generality, both t,t gC and positive and 22 – 23 , we finally arrive to1

2s 2f t s f t 1q2 f t y f t y2 iG t 27Ž . Ž . Ž . Ž . Ž . Ž .p 2 1 0

2t 2f t s f t 1q f t y2 iG t 28Ž . Ž . Ž . Ž . Ž .p 2 0

Ž . 4for t)0. This is the main result of this work. It provides the NLO relationship between the PDF and f t .sŽ . t Ž .Notice that f t / f t to NLO, unlike the equilibrium case, due to the effect of nonequilibrium renormalisa-p p

w sŽ .x2 w t Ž .x2 2Ž .w Ž . Ž .xtion. However, note that f t y f t s f t f t y f t , which is finite, since it depends only on L ,p p 2 1 12

which does not renormalise. This is indeed an interesting consistency check, because the one-loop infinitiesŽ . sŽ . t Ž .appearing in G t can then be absorbed in L , rendering both f t and f t finite. We also remark that we0 11 p p

q Ž . Ž . Ž . Ž . Ž .did not need to take v ™0 in 22 – 23 to arrive to 27 – 28 there are still NGB to NLO . Note also thatqs, t Ž . Ž .both f are real to this order. We have performed the following consistency checks on 27 – 28 : first, thep

Ž . Ž . eq 2equilibrium result 19 is recovered for the contour C simply by replacing G syiT r12 and f s f s0.0 1 2Ž . Ž . Ž .Second, by calculating g t from r , through 24 , we check explicitly that the WI 25 holds and, third, wep S

have calculated A in the p parametrisation, arriving to the same result.˜mn

Ž . Ž . Ž .Therefore, 27 – 28 allow to express nonequilibrium observables like decay rates, masses, etc to one loopŽ .in ChPT, in terms of the physical f t , which could be measured, for instance, in nonequilibrium lepton decaysp

Ž .p™ ln . At this stage one can follow different approaches. Exact knowledge of f t would require to solvel

self-consistently the plasma hydrodynamic equations or, equivalently, Einstein equations for the metric.Ž . Ž . Ž .Alternatively, one can treat f t as external —so that 27 – 28 provide the system response— and study

Ž . w x Ž .simple choices consistently with 4 20 . In what follows, we shall take f t arbitrary and expand it nearts0q, analysing thus the short-time evolution.

Ž .For short times, the particular form of f t is not important and we can parametrise the nonequilibriumŽ . qdynamics in terms of the values of f t and its derivatives at ts0 . As we discussed in Section 2, this

Ž . Žapproach is justified for times t- t with t ,1rf 0 , 2 fmrc compare to the typical plasma timemax max p

w x. Ž .scales 5–10 fmrc 12 . The general solution of 11 with KMS conditions at t can be constructed in terms ofi

two independent solutions to the homogeneous equation, which have to be continuous and differentiable ; tgCw xso that the solution is uniquely defined 17 . Therefore, they have to match the equilibrium solution and its first

Ž . qtime derivative at ts0. With these conditions and expanding both f t and the solutions near ts0 we find tothe lowest order

i b vi q11 2 2 4 4G v ,t ,t sy coth 1ym t qOO m t 29Ž . Ž .Ž .0 q 2v 2q

2 ¨ q q 2Ž . Ž .for t)0, with m syf 0 rf 0 . For m -0 we see the unstable modes threshold, making the pioncorrelation function grow with time. The effect of those modes is not important for short times though, where

4 Ž . Ž . Ž . Ž . Ž . Ž .G t , f t and f t depend implicitly on f t , through 10 and 9 .0 1 2


Ž .the exponential growth of the correlator is not appreciable. Observe that in 29 the time dependence factorises,so that the momentum dependence is the same as in equilibrium and then we can integrate it in DR, yielding the

Ž . Ž . Ž .finite answer 18 . Then, from 27 – 28 we get

2 2T T2 2 i is , t s , t 2 2 2 3 2f t s f 1y q2 Hty m 1y yH t qOO p rL 30Ž . Ž .Ž .p R x2 2½ 5ž /T Tc c

˙ q qŽ . Ž .for t)0, where Hs f 0 rf 0 , with the renormalised constants

2s 2 q 2 2f s f 0 q4 L y6L m yL HŽ . Ž .R 12 11 12

2t 2 q 2 2f s f 0 q4 y L q6L m qL H 31Ž . Ž . Ž .R 12 11 12

Ž Ž .and where the H and m parameters which are OO p and play the role of the Hubble constant and the˙ q ¨ q. Ž . Ž .deceleration parameter in the Universe expansion also get renormalised, in terms of f 0 , f 0 and so1,2 1,2

on, but those are subleading contributions. Thus, for short times, all the effect of the S terms, which is T4 iŽ q.independent, is to redefine f 0 , since there are no infinities coming from G in DR. We insist that this is justp 0

Ž .the effect of truncating the series in t and it is not true in general. Notice that 31 implies necessarily a nonzeroŽ q. Ž .jump D fs f 0 y f see our comments in Section 2 so that the divergent part of L can be absorbed in11

2Ž q. s, t s, tf 0 rendering a finite D f sD f . In fact, that effect is very small compared to the other contributions inR p

Ž . sŽ . t Ž . r r y3 w x30 , and so it is the difference f t y f t , since L , L ,10 22 . Notice also that for the particular casep p 11 122 2 2Ž q. 2 2 s tH sm )0, renormalising as f 0 s f q24m L we get f s f and D s0.11 p p fp

Ž .Finally, we will estimate some physical effects related to f t . For that purpose, we will ignore, forp

Ž .simplicity, the effect of L and L and, based upon 19 , define the plasma effective temperature as11 122Ž . 2w 2Ž . 2 x Ž .f t s f 1yT t rT . Therefore, we can also define a critical time as T t sT and a freezing timep c c cŽ . Ž . Ž .T t s0. Thus, we will impose 0-T t -T and then, through our short-time results for f t , determinef c p

Ž Ž . .either t or t , depending on the initial conditions T t is just quadratic in time to this order . Notice that wec fŽ .are following a similar approach as in equilibrium when one extrapolates 19 until TsT . Let us then takec

< < < < 2 2typical values T , H , m ,100 MeV and retain only the leading order in x'T rT , consistently with thei i c2 2 Ž .Ž . 2chiral expansion. Then, if Hs0, the system cools down until t m ,yx 1qx t ,0.2 fmrc if m -0,f f

2 2 2 Ž .whereas for m )0 it is heated until t m s1 t ,2 fmrc , independent of T . For H)0, there is coolingc c i< < Ž . 2 < < Ž . Žuntil t H ,xr2 t ,0.2 fmrc . Finally, for H-0 and m )0 there is heating until t H , 1y3 xr4 r2 tf f c c

. 2 < < Ž .,2 fmrc , whereas if m -0, there is heating until a maximum t H , 1qxr2 r2 and then cooling downm< < Ž . Ž 2 .until t H ,1qx t ,2.3 fmrc . We observe that the effect of the unstable modes m -0 is always to coolf f

down the system and that the freezing time for H-0 is much longer than that for H)0. Some of these timescales are indeed longer than those to which our short-time approximation remains valid, but they have to be

Ž .understood as estimates, similarly to estimating T at equilibrium through 19 , even though the low T approachc

is less reliable near T,T .cw x Ž . Ž . Ž .Comparing with 12 , naively identifying the LSM order parameter Õ t , f t in proper time , we see ap

similar short-time evolution, although our estimates for the time evolution duration are somewhat lower. Thisw x < <was expected, since the initial values in Ref. 12 correspond to T , 200 MeV and H , 400 MeV, which arei

w x Ž .too high for our low-energy approach. An important remark is that in typical simulations like 12 , Õ t reachesŽ .a stationary value, about which it oscillates thermalisation . It is clear that we cannot predict that type of

behaviour only within our short-time approach, quadratic in time, but only estimate the time scales involved—similarly as to why ChPT cannot see the phase transition—. Therefore, in view of the above estimates, webelieve that our ChPT model may be useful for studying the different nonequilibrium observables evolution,from a stage where some cooling has already taken place onwards. In principle, we could approach closer to Tc

by considering enough orders in our ChPT, although in practice, beyond one-loop, some resummation method,like large N, will need to be implemented.


5. Conclusions and Outlook

We have extended chiral lagrangians and ChPT out of thermal equilibrium. The chiral power countingŽ . Ž .requires all time derivatives to be OO p and to lowest order our model is a NLSM with f™ f t . This model

accommodates unstable pion modes and corresponds to a spatially flat RW metric in conformal time with scaleŽ . Ž . Ž .factor a t s f t rf 0 and minimal coupling. We have exploited this analogy to establish the renormalisation

procedure, which allows to construct the fourth order lagrangian absorbing all the loop divergences, which ingeneral will be time-dependent.

We have applied our model to study the time-dependent pion decay functions, extending the equilibrium piondecay constants. In general there are two independent PDF, as in equilibrium, and to NLO in ChPT they alreadydiffer, unlike equilibrium, due to renormalisation. We have obtained them to NLO in terms of the equal-timecorrelation function, analysing their lowest order short-time coefficients and their dependence with T , andi

discussing the relevant time scales involved within the context of a RHIC plasma.Among the aspects of our model which are worth to be studied further are the long-time evolution, by

Ž .choosing suitable parametrisations for f t , including the analytic approach, and the behaviour of the two-pointcorrelation function at different space points, which would allow us to investigate the formation of regions of

Ž . w xunstable vacua DCC 20 . Other applications and extensions, to be explored in the future include photonŽ 0 .production in the pion sector by gauging the theory and including p anomalous decay , the quark condensate

Ž .time dependence by including the mass explicit symmetry-breaking terms , the N s3 case, large Nf

resummation and proper time evolution.

Acknowledgements

We are grateful to T. Evans and R. Rivers for countless and fruitful discussions, as well as to R.F.Alvarez-Estrada, A. Dobado and A.L. Maroto for providing useful references and comments. A.G.N wishes tothank the Imperial College group for their kind hospitality during his stay there and has received financialsupport through a postdoctoral fellowship of the Spanish Ministry of Education and CICYT, Spain, projectAEN97-1693.

References

w x Ž . Ž . Ž . Ž .1 S. Weinberg, Physica A 96 1979 327; J. Gasser, H. Leutwyler, Ann. Phys. NY 158 1984 142; Nucl. Phys. B 250 1985 465.w x2 A. Dobado, A. Gomez Nicola, A. Lopez-Maroto, J.R. Pelaez, Effective lagrangians for the Standard Model, Springer, 1997, and´ ´ ´

references therein.w x Ž .3 F. Wilczek, Int. J. Mod. Phys. 7 1992 3911.w x Ž .4 A. Bochkarev, J. Kapusta, Phys. Rev. D 54 1996 4066.w x Ž .5 J. Gasser, H. Leutwyler, Phys. Lett. B 184 1987 83.w x Ž .6 P. Gerber, H. Leutwyler, Nucl. Phys. B 321 1989 387.w x Ž . Ž .7 A. Anselm, Phys. Lett. B 217 1989 169; A. Anselm, M. Ryskin, Phys. Lett. B 226 1991 482.w x Ž .8 K. Rajagopal, F. Wilczek, Nucl. Phys. B 399 1993 395.w x Ž .9 D. Boyanowsky, H.J. de Vega, R. Holman, Phys. Rev. D 51 1995 734.

w x Ž .10 D. Boyanowsky et al., Phys. Rev. D 56 1997 3929.w x Ž .11 A.J. Gill, R.J. Rivers, Phys. Rev. D 51 1995 6949.w x Ž .12 F. Cooper, Y. Kluger, E. Mottola, J.P. Paz, Phys. Rev. D 51 1995 2377; M.A. Lampert, J.F. Dawson, F. Cooper, Phys. Rev. D 54

Ž .1996 2213.w x Ž .13 A. Barducci et al., Phys. Lett. B 369 1996 23.w x Ž . Ž .14 R. Baier et al., Phys. Rev. D 56 1997 2548, 4344; M. Le Bellac, H. Mabilat, Z. Phys. C 75 1997 137.w x Ž . Ž .15 J. Schwinger, J. Math. Phys. 2 1961 407; L.V. Keldysh, Sov. Phys. JETP 20 1965 1018.w x Ž .16 K.C. Chou, Z.B. Sou, B.L. Hao, L. Yu, Phys. Rep. 118 1985 1.


w x Ž .17 G. Semenoff, N. Weiss, Phys. Rev. D 31 1985 689.w x Ž .18 D. Boyanowsky, D.S. Lee, A. Singh, Phys. Rev. D 48 1993 800.w x19 M. Le Bellac, Thermal Field Theory, Cambridge U.P., 1996.w x20 A. Gomez Nicola, work in progress.´w x21 N. Birell, P. Davies, Quantum Fields in Curved Space, Cambridge U.P., 1982.w x Ž .22 J.F. Donoghue, H. Leutwyler, Z. Phys. C 52 1991 343.w x Ž .23 R.D. Pisarski, M. Tytgat, Phys. Rev. D 54 1996 R2989.

11 March 1999


Pion form factors with improved infrared factorization

N.G. Stefanis a,1, W. Schroers a,3, H.-Ch. Kim b,4

a Institut fur Theoretische Physik II, Ruhr-UniÕersitat Bochum, D-44780 Bochum, Germany¨ ¨b Department of Physics, Pusan National UniÕersity, Pusan 609-735, South Korea

Received 5 January 1999Editor: P.V. Landshoff

Abstract

Ž 2. Ž .We calculate electromagnetic pion form factors with an analytic model for a Q which is infrared IR finite withouts

invoking a ‘‘freezing’’ hypothesis. We show that for the asymptotic pion distribution amplitude, F 0 ) agrees well with thep g g

Ž .data, whereas the IR-enhanced hard contribution to F and the soft nonfactorizing part can jointly account for the data.p

q 1999 Elsevier Science B.V. All rights reserved.

PACS: 12.38.Bx; 12.38.Cy; 12.38.Lg; 13.40.Gp

The issue of computing exclusive processes, likeelectromagnetic form factors, within QCD is of fun-damental interest because it reveals the basic struc-ture of hadrons. But in contrast to inclusive pro-cesses, there is much uncertainty about the applica-bility of perturbative QCD at laboratory momenta. Itseems that the more interpretations this subject in-spires and grounds the more unsettled it is.

This paper attempts to bring together some crucialresults about pion form factors and combine themwith novel theoretical developments concerning the

Ž .infrared IR regime of QCD, in order to benchmarkthe status of our current understanding. Such anapproach appears attractive since the incorporationof nonperturbative power corrections in the perturba-

1 E-mail: [email protected] Also Fachbereich Physik, Universitat Wuppertal, D-42097¨

Wuppertal, Germany.3 E-mail: [email protected] E-mail: [email protected]

tive domain may improve both the IR insensitivity ofexclusive observables and the self-consistency of

w xcalculations entrusted to perturbative QCD 1–10 .To this end, the conventional representation of the

running strong coupling ‘‘constant’’ is given up infavor of an analytic model, recently proposed by

w xShirkov and Solovtsov 1 , which incorporates asingle power correction to remove the Landau singu-larity. Bearing in mind that the definition of as

beyond two loops cannot be uniquely fixed, one mayregard the ambiguity in the IR modification of therunning coupling as resembling the freedom ofadopting a particular non-IR-finite renormalization

w xscheme 11 .The Shirkov-Solvtsov model employs Lehmann

analyticity to bridge the regions of small and largemomenta by changing the one-loop effective cou-pling to read

24p 1 LŽ1. 2a Q s q , 1Ž .Ž .s 2 22 2b L yQln Q rLŽ .0


( )N.G. Stefanis et al.rPhysics Letters B 449 1999 299–305300

where L'L is the QCD scale parameter. ThisQCD

approach provides a nonperturbative regularization atlow momenta and leads to a universal value of the

Ž1. 2Ž .coupling constant at zero momentum a Q s0sŽ .s4prb ,1.396 for three flavors , defined only0

by group constants, i.e., avoiding the introduction ofŽ .external parameters, like an effective gluon mass to

‘‘freeze’’ the coupling at low momentum scales.Ž .Note that this limiting value i does not depend

on the scale parameter L – this being a consequenceŽ .of renormalization group invariance – and ii ex-

tends to the two-loop order, and beyond, i.e.,Ž2. 2 Ž1. 2 2Ž . Ž . Ž . Ža Q s0 sa Q s0 'a Q s0 . In thes s s

.following the bar is dropped. Hence, in contrast tostandard perturbation theory, the IR limit of thecoupling constant is stable, i.e., does not depend onhigher-order corrections, and is therefore universal.As a result, the running coupling also shows IRstability. This is tightly connected to the nonpertur-

Ž .bative contribution Aexp y4prb which ensures0

analytic behavior in the IR domain by eliminatingthe ghost pole at Q2 sL2 .QCD

At very low momentum values, say, below 1GeV, L in this model deviates from that used inQCD

minimal subtraction schemes. However, since we areprimarily interested in a region of momenta which ismuch larger than this scale, the role of this renormal-ization-scheme dependence is only marginal. In ourinvestigation we use LanŽ3.s242 MeV which corre-QCD

Ž3.MSsponds to L s200 MeV.QCD

This analytic model for the strong running cou-pling is very suitable for calculations of exclusiveamplitudes, mainly for two reasons: Firstly, it en-sures IR safety of the factorized short-distance partwithout invoking the additional assumption of satura-tion of color forces by using a gluon mass – exten-sively used up to now in form-factor calculationsŽ w x.see, for example, 12 . Furthermore, the Sudakov

w xform factor 13 does not have to serve as an IRprotector against a singularities. Hence, the extras

constraint of using the maximum between the longi-tudinal and the transverse scale, as argument of a ,s

w xproposed in Ref. 14 , becomes superfluous. This is aserious advantage relative to previous analyses be-cause now one is able to choose the unphysical

w xconstants 15 , which parametrize different factoriza-tion and renormalization schemes, in such a way as

Žto optimize calculated observables for a more de-

w x.tailed discussion of this point, we refer to 11 .Second, and more important, the non-logarithmic

Ž .term in Eq. 1 enters all anomalous dimensions, viz.the cusp anomalous dimension, which gives rise to

w xthe Sudakov form factor 16–20 , as well as thequark anomalous dimension which governs evolu-tion. As a result, the suppression due to transverse

w xmomenta, intrinsic 21 and those generated by radia-w xtive corrections 14 , is counteracted, and hence there

is no reduction of the form-factor magnitude.We are going to show in this work that the

enhancement effect originating from the power cor-rection to the running coupling is enough for theasymptotic pion distribution amplitude to contributeŽ .at leading order to the spacelike electromagnetic

Ž 2 .form factor of the pion, F Q , a hard part that canp

account for almost half of the form-factor magnitudew xrelative to the existing data 22,23 .

On the other hand, the transition form factor,Ž 2 .0 )F Q , is only slightly changed, as compared top g g

w xthe result given in Ref. 24 , and matches the recentw xhigh-precision CLEO data 25 as good as the dipole

w x 2fit. Also the older CELLO data 26 at lower Q areŽ .well reproduced see below .

In both cases, no adjustment of the theoreticalpredictions to the experimental data is involved.

Therefore, there appears to be no need to reani-mate endpoint-concentrated pion distribution ampli-

Ž .tudes, proposed by Chernyak and Zhitnitsky CZw x27 , in order to make contact with the experimental

w xdata – as recently attempted in Ref. 28 . We are lessenthusiastic about using such distribution amplitudes

Ž .because of the following serious disadvantages: i Itw xhas been recently shown 24,29 that distribution

amplitudes of the CZ-type lead to a pg transitionform factor which significantly overestimates theCLEO data just mentioned. Our reasons for scepti-

w xcism parallel the arguments given in Ref. 24 andwill not be repeated here. The excellent agreement

Ž .between theory QCD and measurement for thisprocess, already at leading order, when the asymp-totic pion wave function is used, cannot be overem-phasized. Note in this context that the calculation of

w xCao et al. 30 , which predicts for the CZ waveŽ 2 .0 )function smaller values of F Q than the data,p g g

uses for modeling the k distribution in the pion anHansatz that strongly suppresses the endpoint region.Hence, in effect, their wave function, though claimed

( )N.G. Stefanis et al.rPhysics Letters B 449 1999 299–305 301

to be the CZ one, excludes this region and yieldstherefore a result even smaller than the one predictedby the asymptotic distribution amplitude in the rangeof Q2 where there are data. Independently, investi-

w xgations 31,32 based on QCD sum rules come to aŽ 2 .0 )comparably good description of F Q at notp g g

too low Q2 on the basis of local duality withoutpresuming the asymptotic form of the pion distribu-tion amplitude, but favoring again a shape close tothat.

Ž .ii Distribution amplitudes of the CZ-type yield aŽ 2 .direct-overlap, i.e., soft contribution, to F Qp

which turns out to be of the same large order ofw xmagnitude 33–35,24 as that resulting from the con-

volution with the hard-scattering amplitudew x 236,37,27 , or is even larger, at currently probed Qvalues. Inclusion of this contribution into the pionform factor leads eventually to a total result whichoverestimates the existing data considerably – evenallowing for some double counting of hard and softtransverse momenta near the transition region.Though the present quality of the high-Q2 dataw x22,23 on the spacelike pion form factor is quitepoor, the trend seems to be indicative.

Ž .iii The underlying QCD sum rules analysis ofw x27 suffers with respect to stability – as outlined by

w xRadyushkin 34 . As a result, the duality intervalincreases with moment order N, meaning that theŽ .nonperturbative condensate contributions grow withorder relative to the perturbative term. A directconsequence of this is that the moments of the piondistribution amplitude for Ns2,4, extracted fromthese sum rules, are artificially enhanced. Such largemoment values can only be reproduced by a double-humped endpoint-concentrated distribution ampli-tude and correspond to the basic assumption thatvacuum field fluctuations have infinite size, orequivalently that vacuum quarks have exactly zero

w xvirtuality 34 .Ž .iv The characteristic humps in the endpoint

Ž .regions xs0,1 are not generic, but merely theresult of truncating the eigenfunctions expansion ofGegenbauer polynomials at polynomial order twowhile keeping the normalization fixed to unity. In-cluding higher and higher order polynomials, thehumps become less and less prominent and the cen-

Ž .tral region xs1r2 gets enhanced. To this point,we mention that an independent QCD sum rules

w x Ž .analysis 38 gives the constraint f xs1r2 s1.2p

Ž ."0.3, which is close to the value f xs1r2 sas

3r2, and definitely violated by the CZ amplitude.Physically, the source of the endpoint enhance-

ment of CZ-type distribution amplitudes can be un-derstood as follows. If the vacuum quark virtuality iszero, an infinite number of such quanta can migratefrom the vacuum to the pion state at zero energycost. This happens exactly in the kinematic regionxs0 or x'1yxs0 and leads to a strong en-hancement of that region at the expense of depletingthe amplitude for finding configurations in which thequark and the antiquark, or more precisely, the struckand the spectator partons, share almost equal mo-mentum fractions around xs1r2. In the pion, con-figurations close to the kinematic boundary containone leading parton, which picks up almost all of theinjected momentum, and an infinite number of weepartons ;1rx with no definite transverse positionsrelative to the electromagnetic probe, which consti-tute a soft cloud. In this regime, gluons have verysmall virtualities and therefore it is inconsistent toassume hard-gluon rescattering, i.e., the factorizationof a short-distance part in the exclusive amplitudebecomes invalid. This region of momenta has to betreated separately on the basis of the Feynman mech-anism just described, but a theoretical approach fromfirst principles, though of paramount importance, isstill lacking. On the other hand, if the vacuumvirtuality is sizable, say, of the order of L orQCD

w xeven larger 39 , then an energy gap might exist thatprohibits the diffusion of vacuum quarks into thepion state, and hence Feynman-type configurationsare insulated from those for which hard-gluon ex-change applies. This gap may be the result of nonlo-

w xcal condensates 34,40,41 , which have a finite fluc-tuation size, or alternatively being induced in theform of an effective quark mass acquired through the

w xinteraction with an instanton background 42,43 . Butthe general result is the same: the shape of the piondistribution amplitude gets strongly enhanced in thecentral region and resembles closely the asymptoticone.

In view of these drawbacks, a potentially goodagreement between theoretical estimates employingCZ-type distribution amplitudes for the pion – as

w xrecently reported in Ref. 28 – and experimentalmeasurements is entirely circumstantial.


In the present work, we are going to show thattaking together the soft form-factor contribution dueto the overlap of the initial and final pion wave

w xfunctions 24 , and the hard, i.e., factorizing part, wecan obtain a result in qualitative agreement with theexisting data and complying with the power countingrules. Actually, including also NLO contributions to

Ž w xthe hard-scattering amplitude see 44 , and previous. Ž 2 .references therein , F Q gets additionally en-p

hanced to account for approximately half of thew xform-factor magnitude 11 , modulo the large uncer-

tainties of the existing data.However, we cannot and do not exclude that the

true pion distribution amplitude may deviate fromthe asymptotic one, but this deviation should bewithin the margins allowed by the experimental datafor the pg transition form factor. Hence the truepion distribution amplitude may well be a hybrid ofthe type F s90%F q9%F q1%C3r2. Thistrue as CZ 4

mixing ensures a broader shape of the pion distribu-tion amplitude, with the fourth-order, ‘‘Mexicanhat’’-like, Gegenbauer polynomial, being included inorder to cancel the dip at xs1r2. The shapes

w xderived from instanton-based approaches 42,43 areof this type. For such distribution amplitudes, evolu-tion already at LO must be taken into account thattends to reduce the importance of the endpoint re-gion leading to a decrease of the magnitude of the

w xform factors towards the data 24,44 . On the otherhand, for the asymptotic solution, evolution entersonly at NLO and is a tiny effect which is ignored inthe present exploratory investigation.

The starting point of our analysis is the expressionfor the pion form factor in the transverse configura-tion space after employing factorization to separate a

Žshort-distance, i.e., hard-scattering part where thew x .the terminology of 15,13,14 is adopted :

F Q2Ž .p

`2d b1 Xouts d xd y PP y ,b , P ,m ,mŽ .H H p F R2

0 y` 4pŽ .=T x , y ,b ,Qrm ,QrmŽ .H R F

=PP in x ,b , P ,m ,m , 2Ž . Ž .p F RX Xq y 2 2' Ž .Here P sQr 2 sP , Q sy P yP , and mR

sC j Q and m sC rb are, respectively, the2 F 1

renormalization and factorization scales, with

CS' w x Ž .C ,C 2 sC 15 jsx, x, y, y , the constants C ,1 2 2 1

C being integration constants of order unity, so that2Ž .uncalculated higher-order corrections are smallw x15,18 . Finally, b is the variable conjugate to thetransverse gluon momentum, and denotes the trans-verse distance between quark and antiquark.

The hard-scattering amplitude T is the amplitudeH

for a quark and an antiquark to scatter collinearly viaa hard-gluon exchange with wavelengths limited byb, and is given in leading order by

T x , y ,b ,Qrm ,bQŽ .H R

an 2 's8C a m K xy bQ . 3Ž .Ž . Ž .F s R 0

Ž .In Eq. 2 , PP describes the valence qq amplitudep

w xwhich includes gluonic radiative corrections 14 aswell as the primordial transverse size of the bound

w xstate 21 :

PP x ,b ,Q,m ,mŽ .p F R

sexp ys x ,b ,Q,m ,m ys x ,b ,Q,m ,mŽ . Ž .F R F R

m dmR

y2 g g m PP x ,b ,m . 4Ž . Ž . Ž .Ž .H q FmmF

The pion distribution amplitude at the factorizationpoint is approximately given by

PP x ,b ,m sC rb ,f x ,m sC rb S x ,b ,Ž . Ž . Ž .F 1 p F 1

5Ž .Ž .where S x,b parametrizes the intrinsic transverse

Ž .size of the pion see below . In the collinear approxi-mation, one has

f d2 k2p HmF2f x ,m s C x ,k , 6Ž . Ž .Ž . Hp F p H316p2 2 N( c

where f s130.7 MeV and N s3. Integrating onp c

both sides of this equation over x normalizes f top

unity, i.e., H1d x f x ,m2 s1 because the rhs isŽ .0 p Ff qpfixed to by the leptonic decay p™m n form

2 2 N' c

any factorization scale.The Sudakov functions can be written in terms of

the momentum-dependent cusp anomalous dimen-w xsion to read 17–20

s j ,b ,Q,m ,mŽ .F R

dmm sC j QR 21s G g , g m , 7Ž . Ž .Ž .H cusp2 C1 mm sF b


Fig. 1. Spacelike pion form factor calculated with f withinas

different factorization schemes, except for the long-dashed linew xwhich shows the soft contribution 24 , in comparison with the

w xexisting experimental data 22,23 . The lower solid line shows theIR-enhanced result obtained in our analysis, whereas the upperone stands for the sum of the long-dashed and the lower solid line.

w xThe dot-dashed line reproduces the calculation of 28 , which doesnot include an intrinsic k -dependence.H

C j Q2where gs ln is the cusp angle, i.e., thež /m

emission angle of a soft gluon and the bent quarkline after injecting at the cusp point the externalŽ .large momentum by the off-mass-shell photon, and

C j Q2G g , g m s2ln A g mŽ . Ž .Ž . Ž .cusp ž /m

qB g m . 8Ž . Ž .Ž .The functions A and B are defined by

A g mŽ .Ž .E

1s 2 G g m qb g KK C , g mŽ . Ž . Ž .Ž . Ž .cusp 12 E g

2an ana g m a g mŽ . Ž .Ž . Ž .s s1sC q KC ,F F2 ž /p p

9Ž .

and

B g mŽ .Ž .1sy KK C , g m qGG j ,C , g mŽ . Ž .Ž . Ž .1 22

a an g m C 2 e2g Ey1Ž .Ž .s 12s ln , 10Ž .3 2ž /p 4C2

respectively, and the K-factor in the MS scheme isgiven by the expression

p 267 10 g EKs y C y n T qb ln C e r2Ž .A f F 0 118 9ž /6

11Ž .where C sN s3, n s3, T s1r2, g being theA c f F E

Euler-Mascheroni constant. The functions KK, GG arew xcalculable within perturbative QCD 15 . Both

Ž Ž .. anŽ 2 .anomalous dimensions, G g m sC a m rcusp F sanŽ 2 .p and g g m sya m rp , will be evalu-Ž .Ž .q s

w xated using the analytic model of 1 in next-to-lead-Žing logarithmic order for more technical details, we

w x.refer to 11 .w xFor simplicity, we follow 21 and model the

Ž .distribution of primordial intrinsic transverse mo-mentum in the pion wave function in the form of aGaussian normalized to unity:

16p 2 fp 2C x ,k s f x b g xŽ . Ž . Ž .p H 2 2 N( c

= 2 2exp yg x b k , 12Ž . Ž .H

Ž .where g x s1rxx and the quark masses are ne-2Ž .glected. For f x s6 xx, one has b s0.883as

w y2 x ² 2 :1r2GeV which corresponds to k s350 MeV.HBefore we proceed with the presentation of our

results, exposed in Fig. 1, let us at this point interjectsome comments regarding the role of the scalesentering the calculation of the pion form factor.

1C1Whenever j- , all Sudakov exponential fac-bQC2

w xtors are set equal to unity 14 . For all values of b,there is a hierarchy of scales according to Lan

<QCD

m sC rbFm sC j QFQ. The limit m ,mF 1 R 2 R F

Ž .Fig. 2. Dependence on the critical transverse distance b of thehard contribution to the scaled pion form factor, calculated withf within our IR-finite factorization scheme, for three differentas

Ž .values of the momentum transfer: Q s2 GeV solid line , Q s51 2Ž . Ž .GeV dashed line , and Q s10 GeV dashed-dotted line .3


Fig. 3. Pion-photon transition form factor calculated with f andasŽ .IR enhancement lower solid line . The other solid line shows the

w xasymptotic behavior. The data are taken from 25,26 .

can be interpreted as the minimum virtuality scale ofŽexchanged quanta or equivalently as the maximum.transverse separation in T below which propaga-H

tors cannot be treated within perturbation theory andare therefore absorbed into f . In the present analy-p

sis we use m sC rb, m sC j Q with jsF 1 R 21 'x, x, y, y, and C sexp y 2g y1 , C s1r 2 ,Ž .1 E 22

CS Žthe latter corresponding to the value C s1 cf.2w x.15 . For this choice of the scheme constants, thelogarithmic term in the K factor is eliminated. Notethat for technical reasons, the argument of a in Ts H(is taken to be m sC jj Q, where j is either x orR 2

y.In order to show how the contribution to the pion

form factor is accumulated in b space, we show inFig. 2 the dependence of the scaled pion form factoragainst b L . One observes a fast rise of theQCD

displayed curves as Q increases. Indeed, already forthe smallest value shown, Qs2 GeV, the formfactor accumulates half of the whole contribution inthe region b,0.5rL . For still larger Q values,QCD

the form factor levels off already around bs0.3rL for Qs5 GeV, and bs0.25rL forQCD QCD

Qs10 GeV. This behavior of the curves uncovershow the IR stability and self-consistency of theperturbative treatment in the present scheme is im-proved, as compared to previous, conventional, ap-

w xproaches 14,21,28 .Ž .A similar expression to Eq. 2 holds also for

F 0 ) , the main difference being that the latterp g g

contains only one pion wave function, and further-more the associated short-distance part, T , does notH

depend on a in LO. The only dependence on thes

strong coupling constant at leading order entersthrough the anomalous dimensions of the cusp andthe quark wave function. The result of this calcula-tion is displayed in Fig. 3. Notice that in this case,we use m sC j Q, where jsx or jsx.R 2

In summary, we have shown that modifying a ins

the IR region by a nonperturbative power correction,which removes the unphysical Landau singularity,may play a crucial role in the practical calculationof exclusive processes because it improves the IRstability of computed observables based on perturba-tion expansions without introducing external parame-ters to ‘‘freeze’’ the running strong coupling. Fur-thermore, we have given quantitative evidence thatin this way it is possible to get an enhanced hard

Ž 2 .contribution to F Q , relying exclusively on thep

asymptotic form of the pion distribution amplitude,so that, though this contribution comprises Sudakovcorrections and a primordial k -dependence, it isHnot suppressed. Together with the soft part of

Ž 2 .F Q , this contribution can account for the trendp

Ž .of the existing admittedly low-accuracy data with-out employing endpoint-concentrated pion distribu-tion amplitudes. The same treatment yields for F 0 )p g g

a theoretical prediction which is in good agreementwith the data.

Acknowledgements

It is a pleasure to thank Peter Kroll for usefuldiscussions and remarks. The work of H.-Ch.K. wassupported in part by the Research Institute for BasicSciences, Pusan National University under GrantRIBS-PNU-98-203.

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version.

11 March 1999


Semi-exclusive processes: new probes of hadron structure

Stanley J. Brodsky a, Markus Diehl b, Paul Hoyer c, Stephane Peigne d´ ´a Stanford Linear Accelerator Center, Stanford, CA 94309, USA

b Deutsches Elektronen-Synchroton DESY, D-22603 Hamburg, Germanyc Nordita, BlegdamsÕej 17, DK-2100 Copenhagen, Denmarkd LAPTHrLAPP, F-74941 Annecy-le-Vieux Cedex, France


Abstract

We define and study hard ‘‘semi-exclusive’’ processes of the form AqB™CqY which are characterized by a largemomentum transfer between the particles A and C and a large rapidity gap between the final state particle C and theinclusive system Y. Such reactions are in effect generalizations of deep inelastic lepton scattering, providing novel currentswhich probe specific quark distributions of the target B at fixed momentum fraction. We give explicit expressions for photo-and leptoproduction cross sections such as g p™p Y in terms of parton distributions in the proton and the pion distributionamplitude. Semi-exclusive processes provide opportunities to study fundamental issues in QCD, including odderon exchangeand color transparency, and suggest new ways to measure spin-dependent parton distributions. q 1999 Elsevier Science B.V.All rights reserved.

In this letter we shall study a new class of hard‘‘semi-exclusive’’ processes of the form AqB™CqY, characterized by a large momentum transfer

Ž .2ts p yp and a large rapidity gap between theA C

final state particle C and the inclusive system Y.Ž .Here A, B and C can be hadrons or real or virtual

photons. The cross sections for such processes fac-torize in terms of the distribution amplitudes of Aand C and the parton distributions in the target B.Because of this factorization semi-exclusive reac-tions provide a novel array of generalized currents,which not only give insight into the dynamics ofhard scattering QCD processes, but also allow exper-imental access to new combinations of the universalquark and gluon distributions.

The hard QCD processes which have been mostlystudied to date can be divided into two main cate-gories:1. InclusiÕe processes such as DIS, ep™eqX. In

the limit of large photon virtuality Q2 and energyn in the target rest frame, the cross section can beexpressed in terms of universal quark and gluon

Ž 2 . Ž 2 .distributions q x,Q , g x,Q in the target,where x is the fraction of target momentumcarried by the struck parton and Q2 the factoriza-tion scale.

2. ExclusiÕe processes such as ep™ep. For largeQ2 the form factor of the proton can be expressedin terms of its distribution amplitude, given by thevalence Fock state wave function in the limit of


( )S.J. Brodsky et al.rPhysics Letters B 449 1999 306–312 307

vanishing transverse separation between thew xquarks 1 .

More recently it has been shown that the QCDscattering amplitude for deeply virtual exclusive pro-cesses like Compton scattering g ) p™g p and me-son production g ) p™Mp factorizes into a hardsubprocess and soft universal hadronic matrix ele-

w xments 2,3 . For example consider exclusive mesonq Ž Ž ..electroproduction such as ep™ep n Fig. 1 a .

Ž .Here one takes as in DIS the Bjorken limit of large2 Ž .photon virtuality, with x sQ r 2m n fixed, whilepB

Ž .2the momentum transfer ts p yp remains small.p n

These processes involve ‘skewed’ parton distribu-tions, which are generalizations of the usual partondistributions measured in DIS. The skewed distribu-

Ž .tion in Fig. 1 a describes the emission of a u-quarkfrom the proton target together with the formation ofthe final neutron from the d-quark and the protonremnants. As the subenergy s of the scattering pro-ˆcess g ) u™pqd is not fixed, the amplitude in-volves an integral over the u-quark momentum frac-tion x.

Ž . ) qFig. 1. a Factorization of g p™p n into a skewed partonŽ .distribution SPD , a hard scattering H and the pion distribution

Ž . Ž) . qamplitude f . b Semi-exclusive process g p™p Y. Thep

d-quark produced in the hard scattering H hadronizes indepen-Ž .dently of the spectator partons in the proton. c : Diagram for the

cross section of a generic semi-exclusive process. It involves ahard scattering H, distribution amplitudes f and f and aA C

Ž .parton distribution PD in the target B.

An essential condition for the factorization of thedeeply virtual meson production amplitude of Fig.Ž .1 a is the existence of a large rapidity gap between

the produced meson and the neutron. In fact, thisfactorization remains valid if the neutron is replacedwith a hadronic system Y of invariant mass M 2

<Y

W 2, where W is the c.m. energy of the g ) p process.For M 2

4m2 the momentum kX of the d-quarkY pŽ .in Fig. 1 b is large with respect to the proton

remnants, and hence it forms a jet. This jet hadronizesindependently of the other particles in the final stateif it is not in the direction of the meson, i.e., if themeson has a large transverse momentum qX sDH Hwith respect to the photon direction in the g ) p c.m.Then the cross section for an inclusive system Y canbe calculated as in DIS, by treating the d-quark as afinal state particle.

The large D furthermore allows only transver-Hsally compact configurations of the projectile A to

Ž .couple to the hard subprocess H of Fig. 1 b , as itw xdoes in exclusive processes 1 . Hence the above

discussion applies not only to incoming virtual pho-2 Ž 2 .tons at large Q , but also to real photons Q s0

and in fact to any hadron projectile.Let us then consider the general process AqB

™CqY, where B and C are hadrons or real pho-tons, while the projectile A can also be a virtualphoton. In the semi-exclusive kinematic limit

L2 , M 2 , M 2<M 2 , D2

<W 2 1Ž .QCD B C Y H

we have a large rapidity gap

W 2

< <y yy s log 2Ž .C d 2 2D qMH Y

between C and the parton d produced in the hardŽ Ž ..scattering see Fig. 1 c . The cross section then

factorizes into the form

dsAqB™CqYŽ .

dt dxS

ds2s f x ,m Ab™Cd , 3Ž . Ž .Ž .Ý br B S dtb

Ž X.2 Ž 2 .where ts qyq and f x ,m denotes thebr B S

distribution of quarks, antiquarks and gluons b in the

( )S.J. Brodsky et al.rPhysics Letters B 449 1999 306–312308

target B. The momentum fraction x of the struckS

parton b is fixed by kinematics to the value

ytx s , 4Ž .S 2M y tY

and the factorization scale m2 is characteristic of thehard subprocess Ab™Cd.

Ž . ŽIn the kinematic limit 1 the subenergy ss qqˆ.2x p of the hard process, the momentum transfer t,S

and the fraction x of the light-cone momentum ofF

projectile A carried by particle C are respectivelygiven by

D2H 2ss W , 5Ž .ˆ 2 2D qMH Y

D2 qx M 2 D2H B Y H

yts s , 6Ž .1yx 1yx rxB B S

D2 qM 2H Y

1yx s , 7Ž .F 2W

where we notice that x is close to 1. We also haveF

the relation

D2 qx M 2H B Y

x - x s - 1 , 8Ž .B S 2 2D qMH Y

with x s0 in the case where the projectile A is aB

hadron or real photon.It is conceptually helpful to regard the hard scat-

Ž .tering amplitude H in Fig. 1 c as a generalizedcurrent of momentum qyqX sp yp , which inter-A C

acts with the target parton b. For Asg ) we obtaina close analogy to standard DIS when particle C isremoved. With qX

™0 we thus find yt™Q2, M 2Y

2 Ž .™W , and see that x in 4 goes over into x sS B2 Ž 2 2 .Q r W qQ . The possibility to control the value

X Žof q and hence the momentum fraction x of theS.struck parton as well as the quantum numbers of

particles A and C should make semi-exclusive pro-cesses a versatile tool for studying hadron structure.The cross section further depends on the distribution

Ž Ž ..amplitudes f , f cf. Fig. 1 c , allowing newA C

ways of measuring these quantities. The use of thisnew current requires a sufficiently high c.m. energy,since according to 1 we need to have at least one

intermediate large scale. We note that the possibilityof creating effective currents using processes similarto the ones we discuss here was considered already

w xbefore the advent of QCD 4 .It is instructive to compare our semi-exclusive

Ž . 2 2limit 1 for electroproduction, Q ;W , with thex ™1 limit of semi-inclusive DIS. After beingF

Ž .struck by the virtual photon the u-quark in Fig. 1 b2 2 2 Ž .has a virtuality s;Q when D ;M , cf. Eq. 5 .ˆ H Y

Since the time scale 1rQ of the photon interaction is'then similar to the time scale 1r s of the furtherˆ

interactions of the struck quark, these processes can-not be physically separated. Hence the hard subpro-

Ž .cess H of Fig. 1 b is compact. On the other hand, ifD2

<M 2 the virtual photon time scale is muchH Y

shorter than that of quark fragmentation, and Hfactorizes into g ) u™u times u™pqd. This is thephysics of semi-inclusive DIS and also of lepton pair

Ž q y .production p p™m m Y in the limit x ™1Fw xwhen D is integrated over 5 .H

Pion photoproduction at large transverse momen-w xtum D was studied in Ref. 6 for x -1, i.e., inH F

the case of no rapidity gap. In this case the struckquark emits a gluon at a short time-scale 1rD , butHthe pion is predominantly produced via a standardnon-perturbative fragmentation process.

Next we consider in more detail the specificsemi-exclusive process g Ž) .p™pqY shown in Fig.Ž . Ž .1 b . We work in the kinematic limit 1 and for

simplicity take a single intermediate scale, D2 ;M 2.H Y

The virtuality Q2 of the photon can scale as

0 photoproduction°22 ~M DIS, x ™0Q ; 9Ž .Y B

2¢W DIS, x finiteB

Ž . Ž . 2Note that according to Eqs. 5 and 6 yt;D isHof intermediate scale, and s;W 2 is very large, soˆthat we have L2

<yt<s in the hard scattering.ˆQCDŽ Ž ..The target parton b in Fig. 1 c attached to the

Ž .hard amplitude H can at lowest order in a besqeither one of the valence quarks u,d of the p .

These two contributions add incoherently in the crosssection, weighted by the respective parton distribu-

Ž .tion f . In Fig. 2 a we show one the four dia-br p

grams which contribute to H in the case bsu. The


Ž . Ž) . qFig. 2. a One of the diagrams of the hard scattering g u™p dŽ .to leading order in a . b One of the leading order diagrams fors

the pion transition form factor in g )g ™p .

three other diagrams are obtained by different order-ings of the photon and gluon vertices on the u- andd-quark lines.

Exclusive processes can be sensitive to infraredend-point contributions, where the momentum of oneof the valence quarks in a hadron wave function

Ž .vanishes. In Fig. 2 a it may be seen that the gluonpropagator goes on-shell for z™1, since its momen-tum then equals that of the final d-quark. It isimportant to note that the gluon four-momentumdoes not vanish in this limit, since it still has a largetransverse component yD . As a consequence theHHHHHu-quark propagator does not become singular at this

Ž X .2 Ž . 2end-point, in other words zq yq szty 1yz Qis ‘‘protected’’ by the large momentum transfer ytin the z™1 limit. This suggests that the hard ampli-tude is no more sensitive to end-point contributions

Ž Ž ..than the pion transition form factor Fig. 2 b . Onthe other hand, in exclusive meson production atlarge Q2 and small t both internal propagators in

Ž .Fig. 2 a go on-shell at zs1, which is what makesthe amplitude with transversally polarized virtual

w xphotons infrared sensitive 3 .Ž .We also note that for z/0,1 all propagators in

the hard scattering subprocess have at least a virtual-ity of order Q2 or yt, whichever is larger. Thisensures that the scattering amplitude H is compact,and that the photon couples coherently to both va-

Žlence quarks i.e., all four diagrams contribute at.leading order . In contrast to ordinary DIS and semi-

inclusive processes, the contribution of the partondistribution f will thus not necessarily beu r p

weighted by e2, the square of the electric charge ofu

the struck quark.

In the case of photoproduction one finds in thelimit of 1 with Q2 s0 that the g u™pqd subpro-

w xcess cross section is 6ds

qg u™p dŽ .dt

22128p e yeŽ .u d2s aas 227 s ytŽ .ˆ

=

2 2f z f zŽ . Ž .1 1p p

dz q dz ,H H½ 5z 1yz0 0

10Ž .w xwhere we choose the convention of Refs. 1,6 for

Ž .the pion distribution amplitude, namely Hdz f z spq'f r 12 with f s93 MeV. The g d™p u subpro-p p

cess has an identical cross section. The result for thephysical process g p™pqY is then, according to

Ž .Eq. 3 ,ds

qg p™p YŽ .dt dxS

dsqs u x ,y t qd x ,y t g u™p dŽ . Ž . Ž .S S dt

11Ž .Ž . Ž .with the notation q x ,y t s f x ,y t . ThereS qr p S

are several interesting aspects of this result.Ø Both the target u- and d-quark contributions are

weighted by the total charge e ye sq1 of theu d

produced pq. An analogous formula holds ofcourse if the pq is replaced with another pseu-doscalar. For neutral meson production, g p™

M 0 Y with M 0 sp 0, K 0, h, . . . , the expressionŽ .10 Õanishes; more exactly one finds that the

Ž .2cross section is suppressed by ytrs comparedˆw xwith the charged meson case 6 . This illustrates

that the new type of current probe we are consid-ering weights the target parton distributions dif-ferently from the photon current, as is also thecase for weak currents.

Ž .Ø The cross section Eq. 10 has a power-law be-havior, dsrdtA1rs3 at fixed trs. This is theˆ ˆbasic signature that the amplitude factorizes into ameson distribution amplitude and a hard scatter-ing subprocess.

Ž . 2Ø At fixed t the expression 10 goes like 1rs ,ˆwhich is characteristic of two spin 1r2 quarkexchanges in the t-channel. Notice that with


L2<yt<s the hard scattering takes placeˆQCD

in the perturbative Regge regime: if a deviationfrom this s-behavior were to be observed experi-ˆmentally it would indicate that the quark ex-change reggeizes, i.e., that contributions fromhigher order ladder diagrams are important. For adiscussion of QCD expectations and experimental

Ž .evidence on how meson Regge trajectories a tw xbehave at large yt we refer to 7 . Let us empha-

size that perturbative Reggeization would notchange the power-law behavior in s at fixed trsˆ ˆmentioned above.

Ž .Ø The pion distribution amplitude f z enters inp

precisely the same way as it does in the pion) 0 w xtransition form factor for g g™p 1,6

2 2'48 e ye f zŽ .Ž . 1u d p2F Q s dz . 12Ž .Ž . Hpg 2 zQ 0

Ž . Ž .A comparison of Figs. 2 a and b suggests thatthis may again be interpreted as the result of

)replacing the g probe with an effective udcurrent of hardness t. Hence the relation betweenobservables,

2 < < 2ds 16p yt F ytŽ .pgq 2g u™p d s aa ,Ž . s 2dt 9 s13Ž .

which holds at lowest order in a , may have as

broader range of validity. Note that in order toobtain 13 we have used isospin symmetry,

Ž . Ž . Ž .q 0f z sf z , which also implies f z sp p p

Ž .f 1yz .p

Ž . 2 2In the limit 1 with Q ;W , i.e., at finite x ,B

the semi-exclusive electroproduction cross section is

ds ep™epqYŽ .2dQ dx dt dxB S

22a 1yy 512p x f zŽ .1B p2s aa dzHs2 4p 27 zQ x s Q xˆ 0B S

=

2xBu x e q 1y eŽ .S u dž /½ xS

2xBqd x e q 1y e 14Ž . Ž .S d už / 5xS

where ysnrE is the momentum fraction of thee

projectile electron carried by the virtual photon, andŽ . Ž .we have used again f z sf 1yz . We makep p

the following remarks.Ž .Ø The semi-exclusive cross section in Eq. 14 cor-

responds to longitudinal photon exchange. Thecontribution from transverse photons is sup-pressed, as in the exclusive case g ) p™Mp at

2 w xlarge Q and small yt 3 .Ø For D2

<M 2 we have x ™x according toH Y S BŽ .Eq. 8 . We find that the parton distributions are

then multiplied by the corresponding quark chargesquared in the cross section. This is a conse-quence of the fact that, as discussed above, thehard subprocess factorizes in this limit into avirtual photon interaction and a quark fragmenta-

Ž .tion process. We note that Eq. 14 was derivedfor D2 ;M 2 and acquires corrections when D2

H Y H<M 2.Y

Ø The subprocess is of hardness Q2, which thus isthe relevant scale for the quark parton distribu-tions. If one takes into account higher orders ina the target parton couples to a ladder. Itss

hardness, which then becomes the appropriatescale, will be between yt and Q2.

2 2 Ž .In the intermediate range Q ;M of Eq. 9Y

transverse and longitudinal photon polarizations con-tribute with comparable strength, and the structure ofthe cross section is richer than in the two extremecases just discussed.

We conclude with a number of more generalremarks and suggestions for future work.1. Vector mesons. In addition to pseudoscalar mesons

one can also consider vector meson production.Ž . Ž .We find that exact analogs of Eqs. 10 and 14

hold if the vector meson is longitudinally polar-ized. Transverse vector mesons are suppressed in

Ž .2the cross section by ytrs for photo- andˆytrs for electroproduction. For symmetry rea-ˆsons there is no interference between differentmeson polarizations.

2. Particle production ratios. Systematic compar-isons of semi-exclusive photoproduction of vari-ous particles can give useful information onparton distributions and distribution amplitudes.

Ž .The hard subprocess 10 cancels in the ratio ofŽ . q yphysical cross sections 11 for p and p .

Ž q. Ž y.Hence ds p rds p directly measures the


Ž . Ž .uqd r dqu parton distribution ratio. Si-Ž q. Ž y.milarly, the ds K rds K ratio measures

the strange quark content of the target withoutuncertainties due, e.g., to fragmentation functions.Conversely, the parton distributions drop out in

Ž q. Ž q.the ratio ds r rds p of longitudinally po-L

larized r mesons to pions, allowing a comparisonof their distribution amplitudes. Since the normal-ization of both f and f is fixed by the leptonicr p

decay widths such a comparison can reveal differ-ences in their z-dependence. In the intermediate

2 Ž . 2Q -range of Eq. 9 the relative size of Q and tcan furthermore be tuned to change the depen-dence of the hard subprocess on z. This may beused to get further information on the shape ofŽ .f z .

3. Hard odderon and pomeron exchange. As wenoted above, the lowest order quark exchangecontribution to g p™p 0 Y is strongly suppressed.At higher orders in a there is a contributions

Ž Ž ..from scattering on gluons bsg in Fig. 1 c .Due to charge conjugation at least two gluons

Žneed to be emitted from the hard scattering dsŽ ..gg in Fig. 1 c . Altogether three gluons are thus

exchanged in the t-channel, corresponding to hardodderon exchange. Semi-exclusive production ofp 0, h or other neutral pseudoscalars may thus beparticularly sensitive to odderon effects sincecompeting production mechanisms are sup-

w xpressed. See Ref. 8 for a recent discussion ofodderon physics.An analogous argument suggests that g p™r 0 Yis a good process for studying hard two-gluonladders, i.e., pomeron exchange at large t. In thiscontext neutral vector meson production has infact been studied in a kinematic region very

w xsimilar to ours 9,10 . Real photon production,g p™g Y, may also be interesting in this respectw x11 , since compared to two-gluon exchange thequark exchange contribution is again suppressedby a power of ytrs.

4. Spin and transÕersity distributions. Polarizationof the target B can naturally be incorporated inour framework. A longitudinally polarized targetselects the usual spin-dependent parton distribu-

Ž .tions Dq x . It also appears possible to measureS

the quark transverse spin, or transversity distribu-tion in photoproduction of r mesons on transver-

sally polarized protons. In this case only theinterference term between transversally and longi-tudinally polarized r mesons should contribute.Since very little experimental information on thetransversity distribution is available, this questionmerits further study.

5. Color transparency. The factorization of the hardŽ .amplitude H in Fig. 1 c from the target remnants

is a consequence of the high transverse momen-tum which selects compact sizes in the projectileA and in the produced particle C. In the case of

w xnuclear targets this color transparency 12 im-Ž .plies according to Eq. 3 that all target depen-

dence enters via the nuclear parton distribution.Thus tests of color transparency can be madeeven in photoproduction, e.g., through

pq D qYŽ .Hg A™ 15Ž .½ p D qYŽ .H

in the semi-exclusive kinematic region specifiedŽ .by Eq. 1 .

Color transparency has so far been studied mainlyin exclusive processes where the target scatters elas-

) w x Ž .tically, such as g A™r A 13 , pA™ppq Ay1w x ) Ž . w x14 and g A™pq Ay1 15 . Semi-exclusiveprocesses provide a possibility to study color trans-parency in processes where the target dissociates into

w xa heavy inclusive system Y 16 . This puts lessstringent requirements on the energy resolution ofthe apparatus, but it also requires a higher beamenergy to ensure the existence of a rapidity gap.

Acknowledgements

It is a pleasure to acknowledge discussions withB. Pire and O. Teryaev.

We are grateful for the hospitality of the Euro-pean Centre for Theoretical Studies in Nuclear

Ž .Physics and Related Areas ECT) , where part ofthis work was done. MD and PH also wish to thankfor the hospitality of CPhT, Ecole Polytechnique.SJB is supported in part by the Department of En-ergy, contract DE-AC03-76SF00515, and MD, PHand SP are supported in part by the EUrTMRcontract EBR FMRX-CT96-0008.


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A. Brandenburg, V.V. Khoze, D. Muller, Phys. Lett. B 347¨Ž .1995 413, hep-phr9410327.

w x Ž .6 C.E. Carlson, A.B. Wakely, Phys. Rev. D 48 1993 2000; A.Afanasev, C.E. Carlson, C. Wahlquist, Phys. Lett. B 398Ž . Ž .1997 393, hep-phr9701215; Phys. Rev. D 58 1998054007, hep-phr9706522.

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w x8 P.V. Landshoff, O. Nachtmann, hep-phr9808233.w x Ž .9 L. Frankfurt, M. Strikman, Phys. Rev. Lett. 63 1989 1914;

H. Abramowicz, L. Frankfurt, M. Strikman, Surveys HighŽ .Energy Phys. 11 1997 51, hep-phr9503437; J.R. Forshaw,

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at HERA, contribution to the International Europhysics Con-Ž .ference on High Energy Physics EPS97 , Jerusalem, Israel,

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tional Europhysics Conference on High Energy PhysicsŽ .EPS97 , Jerusalem, Israel, August 1997.

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11 March 1999


W production in an improved parton-shower approach

Gabriela Miu 1, Torbjorn Sjostrand 2¨ ¨Department of Theoretical Physics, Lund UniÕersity, Lund, Sweden


Abstract

Ž " 0 ) .In the description of the production properties of gauge bosons W , Z , g at colliders, the lowest-order graphnormally is not sufficient. The contributions of higher orders can be introduced either by an explicit order-by-ordermatrix-element calculation, by a resummation procedure or by a parton-shower algorithm. Each approach has its advantagesand disadvantages. We here introduce a method that allows the parton-shower algorithm to be augmented by higher-orderinformation, thereby offering an economical route to a description of all event properties. It is tested by comparing with thep spectrum of W bosons at the Tevatron. q 1999 Elsevier Science B.V. All rights reserved.H

The W " and Z0 bosons have been extensivelystudied at colliders, in order to test the standard

w xmodel 1 . In recent years they have also made theirdebut as backgrounds to other processes of interest:top studies, Higgs searches, and so on. Here it isoften the association of the WrZ with one or severaljets that is the source of concern. Such higher-ordercorrections to the basic processes also serve as testsof QCD. It is therefore of some interest to improvethe accuracy with which gauge boson production canbe described.

In this letter we will take the W " production athadron colliders as a test bed to develop some ideasin this direction. Specifically, we will discuss how to

X "improve the lowest-order description qq ™W byXa merging of the first-order matrix elements qq ™


gW " and qg™qX W " with a leading-log partonshower. However, the formalism is valid for allcolourless massive vector gauge bosons within andbeyond the standard model: g ) , Z0, ZX 0, WX ", andso on. It also applies e.g. in eqey™g Z0. One couldin addition imagine extensions to quite differentprocesses, such as Higgs production by gg™h0, butthis would require further study.

The outline of the letter is the following. First wediscuss various approaches to W production, andtheir respective limitations. Then we zoom in on theshower method and introduce a matrix-element-motivated method to improve it. Finally we comparewith data, specifically the W transverse-momentumspectrum at the Tevatron, and draw some conclu-sions.

In essence, one may distinguish three alternativedescriptions of W production:1. Order-by-order matrix elements. By a systematic

expansion in powers of a , a quite powerfuls

machinery is obtained. For instance, there are


( )G. Miu, T. SjostrandrPhysics Letters B 449 1999 313–320¨314

calculations of the total Drell-Yan cross section tow xsecond order 2 and for the associated production

of a W and up to four partons at the Born levelw x3 . A main problem is that there is no smoothtransition between the different event classes, asone parton becomes soft or two partons collinear.The method is therefore better suited for exclu-sive questions than for an inclusive view of Wevent properties.

2. Resummed matrix elements. Here the effects ofmultiple parton emission are resummed, in im-

w xpact-parameter or transverse-momentum space 4 .Inclusive quantities such as the p spectrumH W

can be well described in such an approach, but itshould be noted that a nonperturbative input isrequired. The standard formalism does not givethe exclusive set of partons accompanying the W,however, and internally does not respect correctkinematics.

3. Parton showers. The parton-shower approachgenerates complete events, with correct kinemat-ics. An arbitrary number of partons is obtained,with a smooth and physical transition betweenevent classes ensured by the use of Sudakov formfactors. On the other hand, the shower approach

Žis formally only to leading-log accuracy althoughmany detailed choices are made to maximize

.agreement with next-to-leading-log results , andthe description of the rate of exclusive partonconfigurations may be poor.

Finally, note that the perturbative partonic stage isnot observable in experiment, but instead the hadronicjet one. Traditional hadronization descriptions, such

w xas string fragmentation 5 , are intended to be univer-sal if applied at some low cut-off scale Q ;1 GeV0

of the perturbative phase. This perfectly matches theshower approach, but causes problems in the use ofmatrix elements.

Given their complementary strengths, it is naturalto attempt a marriage of the matrix-element andparton-shower methods, where the rate of well-sep-arated jets is consistent with the former while thesubstructure of jets is described by the latter. Thesimpler solution, matching, is to introduce a transi-tion from one method to the other at some intermedi-

w xate scale 6–8 . Such an approach is convenient fordescriptions of exclusive jet topologies, but tend tosuffer from discontinuities between event classes and

around the transition scale. More ambitious is themerging strategy, where matrix-element informationis integrated into the shower in such a way as toobtain a uniform and smooth description. This ap-proach so far has only been implemented for the

q y q yŽ .merging to O a of e e ™qq with e e ™qqgsw x Ž .9,7 . We will here introduce a corresponding O as

merging in hadronic W production. Further detailsw xmay be found in Ref. 10 .

Since we neglect the decay of the W, alternativelyimagine it decaying leptonically, all QCD radiationoccurs in the initial state. We will base our approach

w xon the initial-state shower algorithm of 11 , as im-w xplemented in Pythia 12 . The principle of back-

wards eÕolution implies that a shower may be recon-structed by starting at the large Q2 scale of the hardprocess and then gradually considering emissions atlower and lower virtualities, i.e. earlier and earlier in

Ž .the cascade chain and in time .The starting point is the standard DGLAP evolu-

w xtion equation 13 ,

d f x ,t d xXa tŽ . Ž .1b s Xs f x ,t P z ,Ž . Ž .ÝH X a a™ bcd t x 2pxa

1Ž .

with f the distribution function of parton species i,i

x the momentum fraction carried by the parton,Ž 2 2 . Ž .ts ln Q rL the resolution scale, and P zQCD a™ bc

the AP splitting kernels for parton b obtaining afraction zsxrxX of the a momentum. Normally theevolution is in terms of increasing t, but in thebackwards evolution t is instead decreasing. Thenthe DGLAP equation expresses the rate at whichpartons b of momentum fraction x are ‘unresolved’into partons a of fraction xX, in a step d t backwards.The corresponding relative probability is d P rd tsbŽ . Ž .1rf d f rd t . The probability that b remains re-b b

solved from some initial scale t down to t- tmax max

is thereby obtained by a Sudakov form factor

S x ,t ;tŽ .b max

1 d f x ,tXŽ .t bmax Xsexp y d tH X Xž /f x ,t d tŽ .t b

d xXa tXŽ .t 1 smax Xsexp y d t ÝH H Xž x 2pt xa

( )G. Miu, T. SjostrandrPhysics Letters B 449 1999 313–320¨ 315

=f xX ,tXŽ .a

P zŽ .X a™ bc /f x ,tŽ .b

a tXŽ .t smax Xsexp y d tHž 2pt

=xX f xX ,tXŽ .1 a

d z P z . 2Ž . Ž .ÝH X a™ bc /xf x ,tŽ .x ba

From this expression it is a matter of standard MonteCarlo techniques to generate the complete branching

w xa™bc 11 ; e.g., the t distribution of the branchingŽ .is ydS x,t;t rd t. Given parton a, one may inb max

turn reconstruct which parton branched into it, andso on, down to the starting scale Q . In each branch-0

ing, the t scale gives the t value of the branchingmax

to be considered next, i.e. the Q2 values are assumedstrictly ordered.

The definition of the Q2 and z variables is notunambiguous. Referring to the notation of Fig. 1,and to the branching 3™1q4, the Q2 scale in our

w xalgorithm 11 is associated with the spacelike virtu-ality of the produced parton 1, Q2 syp2, while z is1

given by the reduction of squared invariant mass ofŽ .2 Žthe contained subsystem, z s p q p r p q1 2 3

.2 2p . In the limit of collinear kinematics, Q s0,2

one recovers the momentum fraction zsp rp . The1 3

z definition couples the two sides of the events, sothat the order in which the branchings 3™1q4 and5™2q6 are considered makes some difference forthe final configuration. The rule adopted is thereforeto reconstruct branching kinematics strictly in orderof decreasing Q2, i.e. interleaving emissions on thetwo sides of the event.

XNow let us compare the step from qq ™W toXqq ™gW between the matrix-element and parton-

shower languages. Since only one branching is to beconsidered, the comparison has to be with a trun-cated shower, e.g. where only the branching 3™1

q4 occurs in Fig. 1. The 2™2 process thus isXŽ . Ž . Ž . Ž .q 3 qq 2 ™g 4 qW 0 , for which

2 2p qp mŽ .1 2 W2ss p qp s s ,Ž .ˆ 3 2 z z2 2 2ts p yp sp syQ , 3Ž . Ž .3 4 1

1yz2 2 2ˆusm ysy tsQ y m .ˆ ˆW Wz

XThe matrix element for qq ™gW can be writtenw xas 14

2 2 2ˆds s a 4 t qu q2m sˆ ˆ ˆ0 s Ws . 4Ž .ˆd t s 2p 3 ˆˆ tuME

XHere s is the cross section for qq ™W, s s0 02 2 2 2 2Ž . < < Ž .Xp a r3sin u m V d 1ym rx x s in theem W W qq W 1 2

Ž 2 .narrow-width limit, with d 1 y m rx x s ¨W 1 2Ž 2 .Hd z d 1ym rzx x s in the 2™2 process kine-W 3 2Žmatics. The details of the s factor are not relevant0

for the point we want to make, so the presentation is. Ž .intentionally sketchy. Now rewrite Eq. 4 in terms

2 Ž .of z and Q , using Eq. 3 :

ds s z a 4ˆ 0 ss2 2 2p 3dQ mWME

1qz 2 m4 y2 z 1yz Q2 m2 q2 z 2 Q4Ž . Ž .W W= 2 2 2zQ 1yz m yzQŽ .Ž .W

22 a 4 1qz 1 dsQ ™0 ˆs™ s s . 5Ž .0 2 22p 3 1yz Q dQ PS1

We here easily recognize the splitting kernel forq™qg, i.e. the matrix element reduces to the thenormal shower expression in the collinear limit, as itshould be. Some extra but trivial work is necessaryto include the convolution with parton distributions,

Ž 2 .which involves f x ,Q in lowest order and1 1Ž 2 . Ž .f x ,Q for the O a processes.3 3 s

Fig. 1. Schematic picture of an initial-state parton shower, extending from both sides of the event in to the W.


In order to study how the shower populates thephase space, it is straightforward to translate backthe above expression,

2 4ds s a 4 s qmˆ ˆ0 s Ws . 6Ž .ˆd t s 2p 3 ˆ ˆˆ t tquŽ .ˆPS1

To this we should add the other possible showerhistory, where the gluon is emitted by a branching5™2q6 instead; after all, the matrix-element ex-pression contains both amplitudes. The collinear sin-gularity Q2 ™0 here corresponds to emission alongdirection 2 rather than direction 1. In that case the

ˆroles of t and u are interchanged, and the crossˆ ˆ<ˆsection dsrd t is easily obtained. The totalˆ PS2

shower rate is given by the sum,2 4ds ds ds s a 4 s qmˆ ˆ ˆ ˆ0 s W

s q s .ˆ ˆ ˆd t d t d t s 2p 3 ˆˆ tuPS PS1 PS2

7Ž .Thus the singularity structure of the parton-showerand matrix-element rates agree, giving a ratio

ˆ ˆ2 2 2dsrd t t qu q2m sŽ .ˆ ˆ ˆWMEX ˆR s,t s sŽ .ˆqq ™ gW 2 4ˆdsrd t s qmŽ .ˆ ˆ WPS

ˆ2 tus1y 8Ž .2 4s qmˆ W

constrained to the range1

X ˆ-R s,t F1 . 9Ž . Ž .ˆqq ™ gW2

The same exercise may be carried out for qg™

qX W:

d s

ˆd t ME

2 2 2 ˆs a 1 s qu q2m tˆ ˆ0 s Ws

s 2p 2 ysuˆ ˆˆs z a 10 s

s 2 2p 2mW

22 4 2 2 2 2 4z q 1yz m q2 z Q m qz QŽ .Ž . W W= 2 2zQ mW

2 a 1 1Q ™0 s 22= ™ s z q 1yzŽ .Ž .0 22p 2 Q

d ss , 10Ž .2dQ PS

2 2 ˆds s a 1 s q2m tquŽ .ˆ ˆ ˆ0 s Ws , 11Ž .ˆd t s 2p 2 ysuˆ ˆˆPS

ˆ 2 2 2 ˆdsrd t s qu q2m tŽ .ˆ ˆ ˆ WMEX ˆR s,t s sŽ .ˆqg ™ q W 2 2ˆdsrd t ˆŽ .ˆ s q2m tquŽ .ˆ ˆPS W

u uy2m2ˆ ˆŽ .Ws1q , 12Ž .22 4sym qmˆŽ .W W

'5 y1X ˆ1FR s,t F -3. 13Ž . Ž .ˆqg ™ q W '2 5 y2Ž .

XNote that, unlike the qq ™gW process, there is noaddition of two shower histories when comparingwith matrix elements, since here also the latter con-tains two separate terms corresponding to qg and gqinitial states, respectively.

XThe qq ™gW process receives contributions fromtwo Feynman graphs, t-channel and u-channel, andthe shower thus exactly matches this set, althoughobviously it does not include interference betweenthe two. The qg™qX W process is different, sinceonly its u-channel graph is covered by the parton-shower formalism, while the s-channel one has nocorrespondence. Since this latter graph is free fromcollinear singularities, the shower is not misbehavingin any regions of phase space because of this omis-sion, but it is interesting to speculate that the larger

Ž . Ž .X Xˆ ˆvalue for R s,t than for R s,t partlyˆ ˆqg ™ q W qq ™ gWŽmay have its origin here remember that a larger

Ž . .ˆR s,t means a smaller shower emission rate .ˆBased on the above exercise, the standard parton-

shower approach may be improved in two steps. Thefirst is to note that, since the shower so closelyagrees with the correct matrix-element expression —much better than one might have had reason toexpect — it is safe to apply the shower to all ofphase space, i.e. to have Q2 fs rather than themax

more traditional shower-generator limit Q2 fm2max W

w x12,8 . The older choice was inspired in part by thefear of a completely erroneous behaviour for Q2

4

m2 , in part by the typical factorization scale usedW

for parton distributions in W cross-section formulae.Such a scale choice can be motivated by double-counting arguments. Most easily this is seen in pureQCD processes, where a 2™3 process such asgg™qqg could be obtained starting from severaldifferent 2™2 processes, and classification by the


Ž .hardest most virtual subgraph is necessary to re-solve ambiguities. Correspondingly, a W productiongraph could be reclassified once some parton hasQ2 )m2 . That is, in general, one would have toW

consider QCD processes where the emission of a WŽis allowed as a ‘parton shower’ correction. The

s-channel graph in qg™qX W is an example of this.kind. Doublecounting is not an issue, however, once

we decide to represent the full W cross section byŽqq™W. Remember that the shower does not change

.total cross sections.The second step is to use standard Monte Carlo

techniques to correct branchings in the shower by theŽ .ˆrelevant ratio R s,t , to bring the shower parton-ˆ

emission rate in better agreement with the matrix-element one. This correction is applied to the branch-ing closest to the hard scattering, i.e. with largestvirtuality, on both sides of the event, i.e. for 3™1q4 and 5™2q6 in Fig. 1. By analogy with results

w xfor time-like showers 7 , one could attempt to for-mulate more precise rules for when to apply correc-tions, but this one should come close enough and is

Žtechnically the simplest solution. For instance, whileour cascade is ordered in Q2 rather than in p2 , theHemission with largest p2 normally coincides withHthe largest Q2 one, so either criterion for when to

.apply a correction would give very similar results.For a q™qg shower branching, where the correction

Ž . Ž .Xˆ ˆfactor R s,t s R s,t F 1, a candidateˆ ˆqq ™ gW

branching selected according to the Sudakov factorŽ . Ž .în Eq. 2 is then accepted with a probability R s,t .ˆ

In case of failure, the evolution downwards in Q2 isŽcontinued from the scale that failed the ‘veto algo-

.rithm’, ensuring the correct form of the Sudakov .Ž .ˆFor a g™qq branching, the fact that R s,t sˆ

Ž .X ˆR s,t G1 means that the procedure aboveˆqg ™ q W

cannot be used directly. Instead the normal g™qqbranching rate is enhanced by an ad hoc factor of 3,

Ž .ând the acceptance rate instead given by R s,t r3-ˆ1.

Even with this injection of matrix-element infor-mation into the parton shower, it is important torecognize that the shower still is different. The hard-est emission is given by the matrix-element expres-sion times the related Sudakov form factor, thusensuring a smooth p spectrum that vanishes in theHlimit p ™0. By the continued shower historyHŽ .without any matrix-element corrections , further

emissions pick up where the Sudakov factor sup-presses the hardest one, giving a total p spectrumHof emitted partons that is peaked at the lower pHcut. This total spectrum is similar to the matrix-ele-ment one, but deviates from it in that the showerincludes kinematical and dynamical effects of gradu-ally having partons at larger and larger x values andpossibly of different species at each softer emission.In some respects, it thus provides a more sophisti-cated approach to resummation for the properties ofthe recoiling W. It also gives exclusive final states,

Žincluding the possibility for the emitted partons such.as 4 and 6 in Fig. 1 to branch in their turn.

The parton shower redistributes the W’s in phasespace but does not change the total W cross section.It is thus feasible to use a higher-order calculation ofthis cross section as starting point, although we didnot do it here. If higher orders enhance the total

Žcross section by a factor K with K a function e.g.X.of rapidity relative to the lowest-order qq ™W one,

Ž . Ž .the implication of Eqs. 4 and 10 is that the one-jetrate is enhanced by the same factor. If instead an-

X Ž .ôther factor K is wanted here, the respective R s,tˆweight could then be modified by a factor K XrK.Note, however, that it is more difficult to introducesuch a K XrK factor consistently, since it presup-poses a common definition between showers andmatrix elements of what constitutes a jet.

As a first check, we want to confirm that theshower algorithm works as intended, reproducing thematrix-element expressions where it should. Thus theshower is artificially modified so as only to generateone branching at a time. In order to eliminate theinfluence of the Sudakov and the change of kinemat-ics by previously considered emissions, the shower isrestarted from each Q2 actually selected above thecut, but returning to the original kinematics for

Xqq ™W. Furthermore parton distributions and as

are frozen, so as to avoid any scale-choice mis-matches. The resulting W transverse momentumspectrum is shown in Fig. 2, classified by the twopossible branchings. In the old scheme, with Q2 smax

m2 , the drop of the p spectrum at p fmW H W H W W

is easily visible. Already the modification to Q2 ssmaxŽ .the ‘intermediate’ curves brings a marked improve-

Ž .ˆment, and the further introduction of the R s,tˆŽ .weighting ‘new’ results in good agreement be-

tween the shower and the matrix elements.


Fig. 2. The p distribution in pp collisions at 1.8 TeV. Parton distributions and a are frozen and only one emission at a time is allowedH W sX XŽ . Ž .in the shower. Events are classified either as a qq ™gW or b qg™q W.

Ž .ˆThe R s,t factors are further studied in Fig. 3. ItˆŽ .ˆis seen that R s,t is close to unity for most of theˆ

Ž .branchings note the logarithmic scale . Also that

Ž .ˆR s,t ™1 as p ™0, in accordance with theˆ Hdemonstrated agreement of the parton shower andmatrix elements in the collinear limit. At large pH

Ž . Ž . Ž .ˆFig. 3. R s,t distributions in pp collisions at 1.8 TeV. a The inclusive distribution. b The average value as a function of p . Eventsˆ H WX Xare classified either as qq ™gW or qg™q W.


Ž .ˆvalues, the R s,t factors enhance the importance ofˆX Xthe qg™q W process relative to the qq ™gW one

by about a factor of 2. When the two processes arenot separated, the partial cancellation of having oneŽ .ˆR s,t a bit above unity and the other a bit belowˆ

leads to a rather modest net correction to p spectra.HWe now present results for the new parton shower,

in all its complexity, i.e. with normal completeshowers augmented by the two-step correction pro-cess described above. In Fig. 4 the shower pH W

distribution is compared with experimental data fromw xthe D0 collaboration 15 . The agreement is good for

large p , but the shape at small p is ratherH W H W

different, with less activity in the shower than indata. These results are for the default Gaussian pri-mordial k spectrum of width 0.44 GeV, as wouldHbe the order expected from a purely nonperturbativesource related to confinement inside the incominghadrons. By now several indications have accumu-

w xlated that a larger width is needed, however 16 ,although the origin of such an excess is not at allunderstood. One hypothesis is that some radiation isoverlooked by an imperfect modelling of the pertur-bative QCD radiation around or below the Q cut-off0

scale. Whatever the reason, we may quantify thedisagreement by artificially increasing the primordial

k width. Fig. 4 shows that an excellent agreementHcan be obtained, at all p values, with a 4 GeVH W

width. The p distribution is essentially un-H W

changed at large values, i.e. only the region p QH W

20 GeV is affected. In order to put the 4 GeVnumber in perspective, it should be noted that this isintroduced as a ‘true’ primordial k , i.e. carried byHthe parton on each incoming hadron side that initi-ates the initial-state shower at the Q scale. If such a0

parton has an original momentum fraction x and0Žthe parton at the end of the cascade that actually

.produces the W has fraction x, the W only receivesa primordial k kick scaled down by a factor xrx .H 0

In the current case, and also including the fact thattwo sides contribute, this translates into a rms widthof 2.1 GeV for the primordial k kick given to theHW. This number actually is not so dissimilar fromvalues typically used in resummation descriptionsw x4 .

In summary, we see that it is possible to obtain agood description of the complete p spectrum byH W

fairly straightforward improvements of a normal par-ton-shower approach. Corresponding improvementscan also be expected for the production of jets inassociation with the W. Especially, the good match-ing offered to hadronization descriptions in this ap-

w x ŽFig. 4. Transverse momentum spectrum of the W; full parton-shower results compared with data from the D0 Collaboration 15 . The error.bars include both statistical and systematic uncertainties.


proach allows a complete simulation of the finalŽstate, including the addition of a possibly process-

.dependent underlying event. This way it should bepossible to address e.g. the ratio of eventswithrwithout a jet accompanying the W, where CDFand D0 have obtained partly conflicting results, how-

w xever using different jet definitions 17 .We end by reiterating that the formalism pre-

sented here is universal, in the sense that the formu-lae in this paper are not unique for the W, but sharedby all vector gauge bosons, after an appropriatereplacement of m and the constants in the sW 0

Ž .ˆprefactor. Specifically, the reweighting factors R s,tˆneed only be modified to reflect the mass of thecurrent resonance. The method therefore should offeran accurate and economical route to the prediction ofkinematical distributions for a host of new particles,to be searched for at the Tevatron and the LHC. It isalso likely that similar approaches can be developedfor other classes of processes.

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11 March 1999


Weak phases g and a from Bq, or B 0 and B decayss

Michael Gronau a,1, Dan Pirjol b,2

a Physics Department, Technion – Israel Institute of Technology, 32000 Haifa, Israelb Floyd R. Newman Laboratory of Nuclear Studies, Cornell UniÕersity, Ithaca, NY 14853, USA


Abstract

Ž .An improved flavor SU 3 method is presented for determining the weak angle g of the unitarity triangle using decayq q q 0 q q 0Ž .rates for B ™Kp , B ™K K and B ™p h or B ™Kp and B ™Kp , their CP-conjugate modes and thes

" " 0 Ž . qŽ 0.CP-averaged rate for B ™p p . Rescattering color-suppressed contribution in B B ™Kp is subtracted away. TheŽ .only significant SU 3 breaking effects are accounted for in the factorization approximation of tree amplitudes. The weak

angle a is obtained as a byproduct. q 1999 Elsevier Science B.V. All rights reserved.

The determination of the angles of the unitaritytriangle is an important goal of physics studies atexisting and future B meson facilities. It is expectedto provide tests of the CKM mechanism of CPviolation in the Standard Model and to shed light onpossible new physics. In particular, the determination

Ž ) .of the weak phase gsArg V has stimulated aub

great deal of effort, both on the theoretical andexperimental side. A variety of ways have been

w xproposed to extract this angle 1 , ranging fromw xtheoretically clean methods 2 applied to DK modes

w x3 hampered by some very small branching ratios, toapproximate methods applied to Kp modes most of

w xwhich have already been observed 4 . In the latterŽ .case one usually uses approximate flavor SU 3 sym-

w xmetry of strong interactions 5 to relate B™Kp toB™pp amplitudes.


In a somewhat simplified version of this idea,w xGronau, Rosner and London 6 suggested to deter-

mine g through a triangle construction for Bq decayamplitudes into K 0pq, Kqp 0 and pqp 0, and forthe corresponding charge-conjugate decays. It was

w xlater noted 7 that higher order electroweak penguinŽ .EWP contributions upset this triangle construction.Various attempts were made to eliminate the uncer-

w xtainties due to EWP amplitudes 1 . Recently Neu-w xbert and Rosner 8 included the EWP amplitudes in

Bq™Kp in a model-independent manner, by relat-ing them to corresponding current-current ampli-

Žtudes. One often refers to such amplitudes as ‘‘tree’’amplitudes, since they are of lowest order in elec-

.troweak couplings. In their revised triangle con-w xstruction the authors of 8 must rely, however, on

the dynamical assumption that the amplitude forBq™K 0pq is dominated completely by a QCDpenguin contribution, and involves no term propor-

ig w xtional to e 9 . This assumption is equivalent tow xneglecting certain final state rescattering effects 10 .

Present experimental limits on such effects from


( )M. Gronau, D. PirjolrPhysics Letters B 449 1999 321–327322

Ž . w xSU 3 related B™KK decays 11,12 are not yetsufficiently strong for ignoring them. Indirect evi-dence against such effects could also be obtainedfrom future limits on the CP asymmetry in B"™

Kp ".In view of the possibility that rescattering effects

could give rise to a small, however non-negligible,contribution in Bq™K 0pq with phase g , thus up-

w xsetting the construction of Ref. 8 , we propose in thepresent Letter to combine the processes B"™Kp

with future information from B"™K "K and B"

" Ž Ž . .™p h decays h is an SU 3 octet . Alterna-8 8

tively, to avoid the question of hyhX mixing, the

same procedure can be applied by combining B0 ™Ž .Kp and B ™Kp decays. Using a simple SU 3s

relation between these pairs of processes, we willshow that one can avoid uncertainties due to finalstate rescattering in Bq™K 0pq and due to a

0 0 0 Ž .color-suppressed amplitude in B ™K p . SU 3breaking effects occurring in these relations will beshown to contribute only a very small uncertainty in

Ž .g . SU 3 breaking, in the relation between tree am-plitudes of B™Kp and B™pp , will be ac-counted for in the factorization approximation. Elec-troweak penguin effects will be included in a

w xmodel-independent way 13 .w xUsing the notations of 13 , we write the neutral

and charged B decay amplitudes into Kp states inŽ .terms of graphical SU 3 amplitudes

0 q y < Ž s. < igA B ™K p s l e yTyPŽ . Ž .u uc

qlŽ s. yP qP EW , 1Ž .Ž .t c t 1

0 0 0 Ž s. ig' < <2 A B ™K p s l e yCqPŽ . Ž .u uc

Ž s. EW'ql P q 2 P , 2Ž .Ž .t c t 2

q 0 q < Ž s. < igA B ™K p s l e AqPŽ . Ž .u uc

qlŽ s. P qP EW , 3Ž .Ž .t c t 3

q q 0 Ž s. ig' < <2 A B ™K p s l e yTyCyAyPŽ . Ž .u uc

Ž s. EW'ql yP q 2 P ,Ž .t c t 4

4Ž .

where l XŽq .sV )

X V X , the amplitudes T ,C, A, P in-q q b q q

clude unknown strong phases, and P EW are the1y4

respective EWP contributions to these decays. TheŽ . Ž .amplitudes 1 – 4 satisfy the two triangle relations

w x8,14

q q 0 q 0 q'2 A B ™K p qA B ™K pŽ . Ž .

0 0 0 0 q y's 2 A B ™K p qA B ™K pŽ . Ž .

q q 0 iŽgqf . yig' < <s 2 l A B ™p p e r 1yd e .Ž . Ž .EW

5Ž .

Ž . < Ž s.Here we denote lsV rV , d sy 3r2 l ru s ud EW tŽ s. < Ž Ž . Ž .l k , 0.66 k ' c q c r c q c s y8.8 =u 9 10 1 2

y3 .10 , while f is an unknown strong phase. Thesecond term in the brackets represents the sum ofEWP contributions to the amplitudes on the left-

w xhand-sides 8,13 . The correction factor r sŽ . < Ž . < Žd . Žd . < Ž . <y1f rf 1q 3r2 k l rl exp ia f1.22 ac-K p t u

Ž .counts for factorizable SU 3 breaking effects andŽ q q 0.EWP contributions to the amplitude A B ™p p

respectively. Numerically this factor is dominated bythe former contribution.

Ž .Each of the two amplitude triangles 5 cannot beused by itself, together with the corresponding rela-

Ž .tion for the CP-conjugate amplitudes A B™ f '2 ig Ž .e A B™ f , to allow a determination of g . The

Ž . Ž .reason is that in general all four amplitudes in 1 – 4involve two terms with different weak phases. g canbe determined only when one of these terms can beneglected in one of the amplitudes as assumed in

w x Ž . Ž .Refs. 6,8 . Although the first terms in 2 and 3 arelikely to be significantly smaller than the secondterms, we will not neglect them in the forthcomingdiscussion. For definiteness, we present in detail theversion of our method applied to neutral B decays.A similar brief treatment of Bq decays precedes theconclusion.

Ž q q 0.Normalizing amplitudes by A B ™p p , wedefine reduced amplitudes

< 0 q y <1 A B ™K pŽ .x s ,qy q q 0' < <A B ™p pŽ .2 lr

< 0 0 0 <1 A B ™K pŽ .x s , 6Ž .00 q q 0< <lr A B ™p pŽ .

( )M. Gronau, D. PirjolrPhysics Letters B 449 1999 321–327 323

0 y q< <1 A B ™K pŽ .x s ,˜yq q q 0' < <A B ™p pŽ .2 lr

0 0 0< <1 A B ™K pŽ .x s . 7Ž .˜00 q q 0< <lr A B ™p pŽ .

Ž . 0The triangle relation 5 for B decays and its CP-conjugate are given by

x eif1 qx eif X1 s1yd eyig , 8Ž .00 qy EW

˜ ˜Xif if ig1 1x e qx e s1yd e , 9Ž .˜ ˜00 yq EW

X ˜ ˜Xwhere f ,f ,f ,f contain both strong and weak1 1 1 1

phases. These triangles are represented in Fig. 1. The˜angles f and f are functions of cosg defined by1 1

second order equations

1yd cosg cosf qd sing sinfŽ .EW 1 EW 1

x 2 yx 2 q 1qd 2 y2d cosgŽ .00 qy EW EWs , 10Ž .

2 x00

˜ ˜1yd cosg cosf yd sing sinfŽ .EW 1 EW 1

x 2 yx 2 q 1qd 2 y2d cosg˜ ˜ Ž .00 yq EW EWs . 11Ž .

2 x00

As mentioned, a major simplification occurs whenŽ 0the color-suppressed amplitude yCqP in A Buc

0 0. Ž .™K p 2 is neglected relative to the dominantpenguin contribution. In this limit, the angle between

0 0 0 0˜' 'Ž . Žthe amplitudes 2 A B ™K p and 2 A B ™0 0 ˜. ŽK p in Fig. 1 is 2g . This implies cos2gscos f1

.yf which determines g . A similar argument ap-1w xplies in charged B decays 8 when the annihilation

Fig. 1. Relative orientation of B0 amplitude triangles. C and DŽ .are the tips of the B triangles not shown for clarity .s

Ž q 0 q.amplitude AqP is neglected in A B ™K pucŽ .3 . In general, without neglecting these terms, thetwo triangles in Fig. 1 involve an arbitrary relativeangle which prohibits a determination of g .

In order to avoid these dynamical assumptionsand to establish another constraint on the relativeangles between the above two triangles, let us con-sider together with B0 ™Kp also the following Bs

w xdecay amplitudes 13

y q < Žd . < ig X XA B ™K p s l e yT yPŽ .Ž .s u uc

< Žd . < yi b X X EWq l e yP qP ,Ž .t c t 1

12Ž .X X0 0 Žd . ig' < <2 A B ™K p s l e yC qPŽ .Ž .s u uc

X X EWŽd . yi b '< <q l e P q 2 P .Ž .t c t 2

13Ž .

Ž .In the SU 3 symmetric limit the reduced amplitudesappearing in these expressions are equal to those

Ž . Ž . X Xappearing in Eqs. 1 and 2 , TsT ,CsC , P sucX X EW X EW EW X EW ŽP , P sP , P sP , P sP . We willuc ct ct 1 1 2 2

consider below uncertainties due to this approxima-.tion. In the first case this follows simply from

Ž . Ž .U-spin. The amplitudes 12 and 13 satisfy a trian-Ž .gle relation similar to 5

0 0 y q'2 A B ™K p qA B ™K pŽ .Ž .s s

X q q 0's 2 r A B ™p p . 14Ž . Ž .

This relation is exact, even accounting for EWPX ŽŽ 2 2 .contributions. The factor r s M y M F -B K B Ks s

Ž 2 .. ŽŽ 2 2 . Ž 2 ..M r M y M F M parametrizes theK B p Bp p

Ž .leading factorizable SU 3 breaking effects.Ž .The SU 3 relations between the terms of definite

Ž . Ž . Ž . Ž .CKM factors in 1 , 2 and in 12 , 13 , respec-tively, allow a simple geometrical interpretation.

Ž . Ž .Drawing the amplitudes 2 , 13 -scaled by l and2 ig˜Ž Ž . Ž ..their CP-conjugates A B ™ f ' e A B ™ f ,

such that all amplitudes originate in a common point,the other ends of the four amplitudes form a quad-

Žrangle as shown in Fig. 2. The point of origin is notshown in this figure. In Fig. 1 it is chosen as the

.point O. This quadrangle is not determined by ratemeasurements alone, since it involves the unknown

Ž . Ž .relative angle between the triangles 5 , 14 and


Fig. 2. Quadrangle formed by the tips of the triangles for B0 and0 0 0 0 0 0˜' 'Ž . Ž .B decays As 2 A B ™ K p , Bs 2 A B ™ K p , Cs ls

0 0 0 0˜' 'Ž . Ž .2 A B ™ K p and Ds l 2 A B ™ K p . The point X iss s

determined by the intersection of the two circles of radius x r r00

, x r r centered at C and D respectively.˜00

their charge-conjugates, which depends on g through˜f ,f . We will show now that the quadrangle pro-1 1

vides another condition on g which fixes this phase.Consider the four sides of the quadrangle in Fig. 2

EW'Ž . Ž .given in the SU 3 limit by with p'P q 2 Pct 2

l22 igÕsV 1y 1ye p ,Ž .cb ž /2

l22 igxsV 1q e p ,cb ž /2

l2

zsV 1q p ,cb ž /2

i bq2g yi bŽ .< <ysl V e ye p . 15Ž .Ž .t d

Ž s. Ž 2 . < < igWe used l s yV 1 y l r2 y l V e andt cb ubŽd . Ž 2 . < < igl slV y 1yl r2 V e , as required by thet cb ub

unitarity of the CKM matrix. Since all four sides ofthe quadrangle are proportional to a single hadronic

EW'amplitude p'P q 2 P , its shape is determinedct 2

exclusively by CKM parameters. In fact this quad-< < < <rangle is an isosceles trapezoid, x s z , whose

sides Õ and y are parallel. The angle between thesides x and z is 2g . This condition on g , together

Ž . Ž .with 10 , 11 illustrated in Fig. 1, are sufficient forŽdetermining this phase up to discrete ambiguities to

.be discussed below .

Fig. 2 can also be used to measure a . For this wewill select a point X on the median of the trapezoidŽ .the line bisecting the sides Õ and y perpendicularlywith the property that its distances to the vertices ofthe trapezoid are in the following ratio

< Ž s. < < <AX BX l 1 Vt t srs s s s s22"4 .Žd . < << <CX DX l Vl l t dt

16Ž .< <The value of V rV is taken from a recent globalt s t d

w xanalysis of the unitarity triangle 15 . It is easy to seethat the angle through which the side y is seen from

Ž .the point X is 2a . See Fig. 2 .Ž .The conditions 16 can be applied to determine

the point X in Fig. 2 in the following way. First, weŽ .note that the points C and D are fixed by Eq. 14

and its charge-conjugate. Then, recall that the set ofpoints X, for which the ratio of the distances to twogiven points A and C takes a fixed value r, is acircle given by

2 < <Cr yA r AyCXy s . 17Ž .2 2r y1 r y1

2 < < < < < < 2 < < ŽUsing r 41, A ;20 C and C r 4 A see dis-.cussion below , where A and C are the coordinates

of these points with respect to the origin O shown in< < <Fig. 1, the circle is approximated by XyC s Ay

< < < Ž .C rr, A rr. The second condition 16 , applied to< < < <B and D, has a similar form, XyD s ByD rr,

< < < < < <B rr. The two circles of equal radii, A rrf B rr,with centers at C and D, intersect at X and deter-mine this point up to a possible two-fold ambiguity.

In order to demonstrate the algebraic solution forthe two weak phases, let us introduce also reducedamplitudes for B decayss

< y q <1 A B ™K pŽ .sy s ,yq X q q 0' < <A B ™p pŽ .2 r

0 0< <1 A B ™K pŽ .sy s , 18Ž .X00 q q 0< <r A B ™p pŽ .

q y< <1 A B ™K pŽ .sy s ,˜qy X q q 0' < <A B ™p pŽ .2 r

0 0< <1 A B ™K pŽ .sy s . 19Ž .˜ X00 q q 0< <r A B ™p pŽ .


These amplitudes satisfy the triangle relations

y eif 2 qy eif X2 s1 ,00 yq

˜ ˜Xif if2 2y e qy e s1 , 20Ž .˜ ˜00 qy

where tiny EWP contributions to the amplitudeŽ q q 0. ŽA B ™p p are neglected. Their effects will be

˜.estimated below . The phases f and f are deter-2 2Ž .mined from rate measurements through 20

1qy2 yy200 yq

cosf s ,2 2 y00

1qy2 yy2˜ ˜00 qy˜cosf s . 21Ž .2 2 y00

The angle g is extracted as the root of the equa-Ž .tion Arg xrz s2g . Denoting the position of the

intersection point Y of AC and BD with r eif, anexplicit form for this equation is

˜2 x x sin f yf sin2g˜ Ž .00 00 1 1

2 ˜y2 r yx x cos f yf cos2g˜ Ž .00 00 1 1

sx 2 qx 2 y2 r 2 . 22Ž .˜00 00

This determines g when combined with Eqs.Ž .Ž .10 11 . The angle a is given directly by the angleCXD,

˜if if2 2< <1 y e yy e˜00 00sinas . 23Ž .

2 x rr00

In order to evaluate the precision of this method,let us first consider the magnitudes of the amplitude

Ž . Ž .ratios appearing in the triangle relations 8 , 9 andŽ .20 . Using the measured decay rates of B™Kp

w xand B™pp 4 , one estimates from the dominantw x < Ž s. Ž s.Žterms 9,12 x ,x ,x ,x , l P rl T˜ ˜00 00 yq qy t c t u

. <q C , 4. Similarly, y , y , 1, y , y ,˜ ˜qy yq 00 00< <CrT ,0.2. We also note that since BXrCXs< Ž s. < < Žd . <l rl l sr, the isosceles triangle AXB in Fig.t t

2 is about 20 times larger than the one with angle2a . These estimates justify the approximations made

Ž .below Eq. 17 . The errors in the distances of thecenter points of the two circles from O, and theerrors in the radii of these circles are each of orderx rr 2 ,0.01 and can be neglected.00

We now discuss the theoretical errors in the deter-mination of g and a . An intrinsic source of uncer-

w xtainty is the parameter d ,0.63"0.11 8 , whereEW

the 5% shift from 0.66 accounts for factorizableŽ .SU 3 -breaking corrections, and the error is domi-

nated by the present poorly known ratio of CKM< <matrix elements V rV . Its effect on the extrac-ub cb

w xtion of g was examined in detail in Ref. 8 , and wehave nothing new to add to that discussion.

Ž .We will focus instead on the SU 3 breakingeffects introduced by the additional amplitudes con-sidered in this method. They show up as differencesbetween the amplitudes contributing to B0 ™K 0p 0

X0 0Ž . Ž . < < < < Ž .2 and B ™K p 13 , c / c with c'CyPs uc

and analogous inequalities holding for the corre-< X < < < Žsponding penguin amplitudes, p / p with p'

EW' .P q 2 P . One expects these amplitudes to dif-ct 2

fer by at most 30%. A smaller uncertainty exists inthe factor r

X, for which the deviation from unity can< < < <be taken from quark models. Fixing p and c and

< X X < < X X <allowing p rr and c rr to vary within 40%, thepoints C and D in Fig. 2 can vary within smallcircles of radius D y ,0.4 y ,0.08. Therefore,00 00

Ž .the corrections to g due to SU 3 breaking areŽexpected to be small. This is due to our judicious

choice of origin about which the triangles in Fig. 1are rotated, this point being adjacent to the color-

.suppressed amplitude for B decay. To estimate thesŽ .absolute value of the error in g arising from SU 3

breaking, we consider the most unfavorable case of asimultaneous shift of y and y by D y s0.08.˜00 00 00

This translates into an error in g of Dg,D y rx00 00

,0.02 which is about 18.Another source of theoretical uncertainty is con-

nected with the neglect of EWP contributions inŽ q q 0.A B ™p p . We have recently shown that when

these effects are included, the relation between thisw xdecay amplitude and its CP conjugate is 13,16

q q 0 2 ij ˜ y y 0A B ™p p se A B ™p p ,Ž . Ž .xsina

tanjs , 24Ž .1qxcosa

Ž . Ž .where xsy 3r2 k sinarsin aqg . Numericallythe angle 2j is seen to be very small, under 28. Thisuncertainty will affect only the relative orientation of

˜the two B triangles, shifting the angles f and fs 2 2˜ Žby an amount Df syDf sj . The effect of2 2

Ž .these EWP on the B™Kp triangles 5 enters onlythrough the factor r, to which they contribute at the


.level of 1% . The corresponding error in the posi-tions of C and D is of order 0.2j,0.003 which iswell under the uncertainty arising from the othersources discussed above.

On the other hand, the smallness of the CXDŽ .triangle implies that SU 3 breaking effects will have

a larger impact on the extraction of a from thismethod. Another uncertainty is introduced through

Ž .the ratio r 16 , which is known with an error ofabout 20%. This affects the determined position ofthe point X through the radii of the circles centeredat C and D. The radii of these circles are of theorder of x rr,0.2, implying an error in the posi-00

tion of the point X of the order of 0.04, which isŽ .half of the uncertainty arising from SU 3 breaking.

The estimates given above indicate that the error insuch a determination is at the level of 30%.

A similar method can be applied to the determina-tion of g from Bq™K 0pq and Bq™Kqp 0 de-cays. In this case uncertainties due to rescattering inBq™K 0pq can be eliminated by considering in

q q 0 q qaddition the decays B ™K K and B ™p h ,8Ž .where h is an SU 3 octet. Their amplitudes are8

w xgiven by 5,13

q q 0 Žd . ig< <A B ™K K s l e AqPŽ . Ž .u uc

< Žd . < yi b EWq l e P qP ,Ž .t c t 3

25Ž .q q'6 A B ™p hŽ .8

< Žd . < igs l e yTyCy2 Ay2 PŽ .u uc

< Žd . < yi b EWq l e y2 P qP , 26Ž .Ž .t c t 5

Ž . Ž .and are closely related to 3 and 4 . Their relativeŽ . Ž .orientation with respect to 3 and 4 can be fixed as

0 Ž .in the B case with the help of the exact trianglerelation

3q q 0 q qA B ™K K q A B ™p h(Ž . Ž .82

1q q 0s A B ™p p . 27Ž . Ž .'2

Ž .This triangle relation replaces Eq. 14 in the case ofneutral B decays. Instead of the quadrangle of Eqs.Ž . Ž q15 , one now constructs a quadrangle from A B

0 q q q 0. Ž .™K p , l A B ™K K and their charge-con-jugates, the four sides of which are all proportional

to P qP EW. The extraction of g and a follows inct 3

a similar way. This set of processes is experimentally0 0 qmore accessible than B ™K p , however B ™s

pqh involves a certain amount of model-depen-8X w xdence related to hyh mixing 17 .

We note that information about rescattering inBq™K 0pq can be obtained only from the U-spin

q q 0 w xrelated process B ™K K 11,18 , without theneed for Bq™pqh. While in principle this can be

w x Ž .used to determine g 16 , SU 3 breaking effects inw xthis procedure prohibit a precise determination 19 .

In conclusion, we have presented a new methodfor extracting the weak angle g using combined B0

and B decays, or combining Bq™Kp with Bq™sq 0 q qK K and B ™p h. This method represents an

w ximprovement of the method suggested in Ref. 8 inthat color-suppressed contributions in B0 decay, orrescattering effects in case of Bq decay, are elimi-

Ž .nated with the help of SU 3 flavor symmetry. TheŽ .additional SU 3 breaking corrections were shown to

be negligible. Under ideal experimental conditions,this method would allow a substantial improvementin the precision of determining g . In reality, Bs

decay modes involving neutral pions, which requireB flavor tagging, pose a particularly difficult experi-s

mental challenge. Alternatively, the use of chargedB decays involves a theoretical difficulty due tohyh

X mixing which must be resolved.

Acknowledgements

D. P. would like to thank Peter Gaidarev andTung-Mow Yan for useful discussions. We are grate-ful to Kaustubh Agashe for pointing out an error inan early version of the paper. This work is supportedby the National Science Foundation and by the UnitedStates – Israel Binational Science Foundation underResearch Grant Agreement 94-00253r3.

References

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Ž .Italy, 21y25 April 1997, Nucl. Phys. Proc. Suppl. 65 1998Ž .245; R. Fleischer, Int. J. Mod. Phys. A 12 1997 2459.


w x Ž .2 M. Gronau, D. Wyler, Phys. Lett. B 265 1991 172; D.Ž .Atwood, I. Dunietz, A. Soni, Phys. Rev. Lett. 78 1997

Ž .3257; M. Gronau, Phys. Rev. D 58 1998 037301; M.Ž .Gronau, J.L. Rosner, Phys. Lett. B 439 1998 171; J.-H.

Ž .Jang, P. Ko, Phys. Rev. D 58 1998 111302.w x3 CLEO Collaboration, M. Athanas et al., Phys. Rev. Lett. 80

Ž .1998 5493.w x4 J. Alexander, Rapporteur’s talk presented at the 29th Interna-

tional Conference on High-Energy Physics, Vancouver, BC,Canada, July 1998; CLEO Collaboration, M. Artuso et al.,presented at the Vancouver Conference, op cit., CornellUniversity report CLEO CONF 98-20, ICHEP98 858, unpub-lished.

w x5 M. Gronau, O. Hernandez, D. London, J.L. Rosner, Phys.´Ž .Rev. D 50 1994 4529.

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w x Ž .7 N.G. Deshpande, X.-G. He, Phys. Rev. Lett. 74 1995 26;w Ž . xE: 74 1995 4099 ; O.F. Hernandez, M. Gronau, D. Lon-´

Ž .don, J.L. Rosner, Phys. Rev. D 52 1995 6374.w x8 M. Neubert, J.L. Rosner, hep-phr9809311, to be published

in Phys. Rev. Lett.w x9 This assumption can be avoided in order to obtain more

limited information about g . See M. Neubert, J.L. Rosner,CERN Report CERN-THr98-273, hep-phr9808493, to bepublished in Phys. Lett. B.

w x Ž .10 B. Blok, M. Gronau, J.L. Rosner, Phys. Rev. Lett. 78 1997Ž .3999, 1167; J.M. Soares, Phys. Rev. Lett. 79 1997 1166;

Ž .M. Neubert, Phys. Lett. B 424 1998 152; D. Delepine, J.M.´Ž .Gerard, J. Pestieau, J. Weyers, Phys. Lett. B 429 1998 106;´

Ž .D. Atwood, A. Soni, Phys. Rev. D 58 1998 036005; A.J.Ž .Buras, R. Fleischer, T. Mannel, Nucl. Phys. B 533 1998 3;

Ž .R. Fleischer, Phys. Lett. B 435 1998 221.w x11 A. Falk, A.L. Kagan, Y. Nir, A.A. Petrov, Phys. Rev. D 57

Ž .1998 4290.w x Ž .12 M. Gronau, J.L. Rosner, Phys. Rev. D 57 1998 6843; D 58

Ž .1998 113005.w x13 M. Gronau, D. Pirjol, T.M. Yan, Cornell University report

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w x19 M. Gronau, D. Pirjol, work in progress.

11 March 1999


Vector-meson electroproductionfrom generalized vector dominance

Dieter Schildknecht a,b,1, Gerhard A. Schuler a,2, Bernd Surrow c

a Theoretical Physics DiÕision, CERN, CH-1211 GeneÕa 23, Switzerlandb Fakultat fur Physik, UniÕersitat Bielefeld, D-33501 Bielefeld, Germany¨ ¨ ¨

c Experimental Physics DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland

Received 20 October 1998; revised 5 January 1999Editor: R. Gatto

Abstract

Including destructively interfering off-diagonal transitions of diffraction-dissociation type, we arrive at a formulation ofGVD for exclusive vector-meson production in terms of a continuous spectral representation of dipole form. The transversecross-section, s ) , behaves asymptotically as 1rQ4, while R 's ) rs ) becomes asymptoticallyT,g p™ V p V L ,g p™ V p T,g p™ V p

constant. Contributions violating s-channel helicity conservation stay at the level established in low-energy photoproductionand diffractive hadron–hadron interactions. The data on r 0-meson production from the Fermilab E665 Collaboration and onf- and r 0-meson production from HERA are found to be in agreement with these predictions. q 1999 Elsevier Science B.V.All rights reserved.

The key role played by the vector mesons in thedynamics of hadron photoproduction on nucleons, atenergies sufficiently above the vector-meson produc-tion thresholds, became clear in the late sixties andearly seventies. Indeed, the total photoproduction

Ž 2 .cross-section on protons, s W , was found to beg p

1 Supported by the BMBF, Bonn, Germany, Contract 05 7BI92Pand the EC-network contract CHRX-CT94-0579.

2 Heisenberg Fellow; supported in part by the EU FourthFramework Programme ‘‘Training and Mobility of Researchers’’,Network ‘‘Quantum Chromodynamics and the Deep Structure of

ŽElementary Particles’’, contract FMRX-CT98-0194 DG 12-.MIHT .

related to forward vector-meson photoproduction,0 < Ž 2 . w x 3ds rd t W , extrapolated to ts0 1 ,g p™ V p

ap2 's W s 16pŽ . Ýg p 2( g0 VVsr ,v ,f ,Jrc

=

1r20ds2< W , 1Ž . Ž .g p™ V pž /d t

and to the total cross-sections for the scattering oftransversely polarized vector mesons on protons,

3 Ž .A precision evaluation of 1 requires a correction for theŽ .small ratio of real to imaginary forward scattering amplitudes to

Ž .be inserted in the right-hand side of 1 .


( )D. Schildknecht et al.rPhysics Letters B 449 1999 328–338 329

w xs , obtained 2 by applying the additive quarkV p

model for hadron–hadron interactions

ap2 2s W s s W . 2Ž . Ž . Ž .Ýg p V p2g0 VVsr ,v ,f ,Jrc

2 Ž . Ž .The factor aprg in 1 and 2 denotes the strengthVŽ .of the coupling of the virtual photon to the vector

meson V, as measured in eqey annihilation by theintegral over the vector-meson peak:

ap 1q ys s s d s , 3Ž . Ž .ÝH e e ™ V ™ F2 2g 4p aV F

or by the partial width of the vector meson:

a 2 mVq yG s . 4Ž .V ™ e e 212 g r4pŽ .V

Ž . Ž .The sum rules 1 and 2 are based onŽ .i the direct couplings of the vector mesons to the

photon and onŽ .ii subsequent strong-interaction diffractive scat-

tering of the vector mesons on the proton.Ž . Ž .Relations 1 and 2 accordingly provide the

theoretical basis for applying concepts of strong-in-teraction physics, such as Regge-pole phenomenol-ogy, to the interaction of the photon with nucleons.

w xCompare 3 for a recent analysis of the experimentaldata for the total photoproduction cross-section interms of Regge phenomenology.

Ž .The sum rule 1 is an approximate one. Thefractional contributions of the different vector mesons

w x 4to the total cross-section, s , were found to be 4g p

r s0.65 , r s0.08 , r s0.05 , 5Ž .r v f

adding up to approximately 78% of the total cross-section. An additional contribution of r ,1–2%Jrc

has to be added for the Jrc vector meson. ToŽ .saturate the sum rule 1 , the contributions of the

leading vector mesons have to be supplemented bymore massive contributions also coupled to the pho-

4 w xCompare also the review 5 .

ton, as observed in eqey annihilation. From theŽ .point of view of generalized vector dominance GVD

w x Ž . Ž .4 , the sum rules 1 and 2 appear as an approxi-mation that is reasonable for the Q2 s0 case ofphotoproduction, while breaking down with increas-ing space-like Q2, the role of r 0, v and f beingtaken over by more massive states.

Ž . Ž .Relations 1 and 2 implicitly contain the propa-gators of the different vector mesons. Being evalu-ated for real photons at Q2 s0, no explicit propaga-

Ž . Ž .tor factors appear in 1 and 2 , and the photonvector-meson transition with subsequent vector-me-son propagation is reduced to a multiplication of thevarious cross-sections by coupling constants charac-teristic of the vector-meson photon junctions. It was

w x 5pointed out a long time ago 6 that an experimen-tal study of vector-meson electroproduction wouldprovide an additional and particularly significant testof the underlying photoproduction dynamics.

The presence of the vector-meson propagators inthe respective production amplitudes for the variousvector mesons would be explicitly tested in vector-meson electroproduction. In addition, vector-mesonproduction by virtual photons, at values of Q2

4m2 ,V

would allow to test the expected dominance of theproduction by longitudinal photons over the produc-

Ž .tion by transverse ones. Moreover, the approximatehypothesis of helicity conservation with respect tothe centre-of-mass frame of the reaction g ) p™V p,the hypothesis of ‘s-channel helicity conservation’Ž . w xSCHC , introduced in Ref. 6 by generalizing ex-

w xperimental results from photoproduction 8 to elec-troproduction, would become subject to experimentaltests.

w xMore recently, it was conjectured 9–14 thatvector-meson electroproduction at large values of Q2

Ž .was calculable in perturbative QCD pQCD andwould provide experimental tests of it. We willcomment on the results from the pQCD approachbelow.

Ž .Expressing the cross-section for forward t,0production of vector mesons on nucleons by trans-versely polarized virtual photons in terms of the

5 w xSee also Ref. 7 .

( )D. Schildknecht et al.rPhysics Letters B 449 1999 328–338330

respective real-photon cross-sections, we have theŽ .vector-meson dominance model VDM prediction

w x60dsT 2 2W ,QŽ .

)d t g p™V p

4 0m dsV 2s W . 6Ž . Ž .22 2 d t g p™V pQ qmŽ .V

For longitudinally polarized virtual photons, as aconsequence of the coupling of the vector meson Vto a conserved source as required by electromagnetic

w xcurrent conservation, the result 60dsL 2 2W ,QŽ .

)d t g p™V p

4 2 0m Q dsV 2 2s j W 7Ž . Ž .V2 22 2 d tm g p™V pVQ qmŽ .V

Ž . Ž .was obtained. Both relations 6 and 7 containSCHC. The parameter j denotes the ratio of theV

imaginary forward scattering amplitudes for the scat-tering of longitudinally and transversely polarizedvector mesons and may in principle depend on thevector meson V under consideration and on theenergy W. The value of j s1 corresponds to theV

conjecture of helicity independence of vector-mesonnucleon scattering in the high-energy limit.

Ž . Ž .The predictions 6 and 7 for vector-meson pro-duction by virtual photons are based on the idealiza-tion that the propagation of the single vector mesonV is responsible for the Q2 dependence of thediffractive electroproduction of that vector meson V.This idealization is by no means true in nature.Time-like photons also couple to the continuum ofhadronic states beyond r 0, v, f, etc., resulting fromeqey annihilation into quark–antiquark pairs, andvector-meson forward scattering need not necessarilybe ‘diagonal’ in the masses of the ingoing andoutgoing vector mesons. The process of diffractiondissociation, corresponding in the present context to‘off-diagonal’ transitions such as r 0 p™r

X 0 p etc., isin fact well known to exist in hadron–hadron interac-tions, as explicitly observed in proton–proton scat-

w xtering 15 .The modification of the vector-meson electropro-

duction cross-section resulting from the inclusion of

off-diagonal transitions of the diffraction-dissocia-w xtion type was investigated in Ref. 16 . For definite-

w xness, in Ref. 16 , the calculation of vector-mesonproduction was based on a spectrum of an infiniteseries of vector-meson states coupled to the photonin a manner that assures duality 6 to quark–anti-quark production in eqey annihilation. Under thefairly general assumption of a power law for the

Ž .diffraction-dissociation amplitudes at zero t interms of the ratios of the masses of the diffractivelyproduced vector states

2 pq1m1w x w xT V p™V p sc T V p™V pN 0 ž /mN

Ns1,2,3, . . . , 8Ž . Ž .an intuitively very simple and satisfactory result wasobtained.

The sum of the poles in the transverse amplitudefor g ) p™V p was shown to sum up approximatelyto a single pole, the pole mass m of the vectorV

meson V being changed, however, to a value ofŽ .m different from m . Prediction 6 , taking intoV ,T V

account off-diagonal transitions as embodied in GVD,w x 7thus becomes 16

0dsT 2 2W ,QŽ .)d t g p™V p

4 0m dsV ,T T 2s W . 9Ž . Ž .22 2 d t g p™V pQ qmŽ .V ,T

For destructive interference among neighbouringw xvector-meson states, incorporated in Ref. 16 through

an alternating-sign series of vector-meson states, onefinds

m -m . 10Ž .V ,T V

Ž .The precise value of m in 9 depends on theV ,T

details of the strong amplitude, i.e. on the strength c0Ž .and the exponent p of the power-law ansatz 8 for

Ž .spin-conserving diffraction dissociation.

6 w xCompare 17 for a recent examination of the validity ofq yduality to qq production in e e annihilation at energies below 3

GeV.7 Ž .The simple result 9 is an approximation that coincides with

the full GVD result at Q2 s0 and Q2 ™`, but may vary by;10% at intermediate Q2 values.


Ž .The alternating-sign assumption leading to 10w xwas originally motivated 18 by GVD investigations

on the total photo-absorption cross-section. For aw xrecent analysis in this context, see e.g. 19 . Destruc-

tive interference among neighboring states, charac-teristic for alternating signs, was independently intro-

w xduced in Ref. 20 , and more recently in a QCD-basedw xanalysis in Ref. 14 in order to explain the interfer-

ence pattern observed in eqey annihilation intopqpy in the r

X,rXX region between 1 GeV and 2GeV.

Diagonal vector-meson dominance would requirethe mass spectrum observed in diffractive pqpy

photoproduction to resemble the one seen in eqey

annihilation. Experimentally, there are significantdifferences, constructive versus destructive interfer-

w xence effects 20,14 . From the point of view of GVD,these differences qualitatively support the presenceand importance of off-diagonal transitions: whenpassing from eqey annihilation to photoproduction,each specific vector-meson photon coupling is to bereplaced by a sum of contributions due to the transi-

Ž .tion of the spacelike photon to various vector-me-son states subsequently scattered on the proton. This,in general, will lead to the observed differences inthe mass spectra.

The destructive interference pattern specificallyobserved in eqey™pqpy together with the some-what different pattern seen in g p™pqpyp , supportour ansatz for vector-meson production with alternat-ing signs and off-diagonal contributions.

Ž .With 9 , the asymptotic behaviour of the trans-Žverse forward-production cross-section in off-diago-

.nal GVD becomes0dsT 2 2W ,Q ™`Ž .

)d t g p™V p

4 0m dsV ,T 2s W . 11Ž . Ž .4 d tQ g p™V p

2 Ž . Ž .While the power of Q in 9 and 11 remainsŽ .unchanged with respect to 6 , the normalization of

the asymptotic cross-section relative to photoproduc-tion is affected by the fourth power of m . Con-V ,T

Ž .cerning sum rule 1 : it is unaffected by the introduc-tion of off-diagonal terms, since the initial photonremains, when passing from the left-hand side to

Ž . Ž .right-hand side of 1 . In relation 2 , off-diagonal

terms with destructively interfering amplitudes implymultiplication of each s by a specific correctionV p

w xfactor somewhat smaller than unity 16 .Ž . Ž .The result 9 or rather the underlying amplitude

Ž . w xwith the constraint 10 in Ref. 16 was obtained bystraightforward summation of an alternating series.In view of the ensuing extension to longitudinalphotons, we note that the transverse amplitude may,to a good approximation, be represented by a sum ofdipole terms 8 by combining neighbouring terms inthe series. Switching to an equivalent continuumformulation, we obtain the following representationof the transverse amplitude as an integral over dipoles

A ) W 2 ,Q2 ,ts0Ž .T ,g p™ V p

dm22 2sm A W ,ts0 .Ž .HV ,T g p™ V p22 2 2m Q qmV ,T Ž .

12Ž .Note that the modified pole mass m of the dis-V ,T

crete formulation has turned into an effective thresh-Ž .old in 12 . Upon integration and squaring we imme-

Ž .diately recover 9 .The impact of off-diagonal transitions on the re-

Ž .sult for longitudinally polarized virtual photons 7w xwas not explored in Ref. 16 . The representation

Ž .12 for the transverse production amplitude as acontinuous sum over dipole contributions, abstractedfrom the assumed destructive interference betweenproduction amplitudes from neighbouring states, iswell suited for a generalization to longitudinal pho-tons. Taking into account the coupling of the photonto a conserved source as transmitted to the hadronicamplitude, we have

A ) W 2 ,Q2 ,ts0Ž .L ,g p™ V p

2 2Q dm2sj m (HV V ,L 2 22 2 2mm Q qmV ,L Ž .

=A W 2 ,ts0 . 13Ž . Ž .g p™ V p

Ž .In deriving 13 , we have taken j to be m-inde-V

pendent. We expect the threshold mass of the longi-

8 Ž .Although always possible, given the result 9 of the series,the dipole approximation of two neighbouring terms in the series

Ž .is most natural for the choice ps0 in 8 , the value supported byw xdiffraction-dissociation data 15 .


tudinal amplitude, m , to be larger than m , i.e.V ,L T

m2 -m2 -m2 . As a consequence of the alternat-V ,T V ,L V

ing signs, this is certainly true if the occurrence of anadditional inverse mass, associated with the extra

2()Q factor in A , is the only differenceL ,g p™ V p

between the m-dependence of A ) andT,g p™ V p

A ) . A priori, the transverse and longitudinalL ,g p™ V pŽ .strong-interaction diffraction-dissociation ampli-

w xtudes T V p™V p may possess different m-de-Tr L NŽ Ž ..pendences p /p in 8 , thus affecting the ratioL T

m2 rm2 .V ,L V ,TŽ .Integration of 13 yields

A ) W 2 ,Q2 ,ts0Ž .L ,g p™ V p

2 3p m mV ,L V ,Lsj yV 2 2 2 22 Q (Q Q qmŽ .V ,L

2m mV ,L V ,L 2y arctan A W ,ts0Ž .g p™ V p2 2Q (Q2(Q

2 2 2™ j A W ,ts0 for Q ™0Ž .V g p™ V p3 mV ,L

p m2V ,L 2 2™ j A W ,ts0 for Q ™` .Ž .V g p™ V p22 Q

14Ž .

The above predictions for transverse and longitu-dinal production amplitudes are valid for high-en-

Ž 2 Ž 2 2 . . Ž .ergy xsQ r W qQ <1 forward t,0 pro-duction. It would be preferable to compare the pre-dictions with forward-production data, thus eliminat-ing the influence of a possible Q2 dependence of theslope of the t-distribution. No reliable data for for-ward production have been extracted from the exper-iments so far. Accordingly, in order to be able tocompare at all with data available at present, weignore a possible Q2 dependence of the t-distribution

Ž . Ž 2 .by putting b 0 rb Q ,1, where b is the slopeŽ < <. Ž .parameter in the t-distribution, exp yb t . From 9 ,

the transverse production cross-section integratedover t then becomes

s ) W 2 ,Q2Ž .T ,g p™ V p

m4V ,T 2s s W . 15Ž . Ž .g p™ V p22 2Q qmŽ .V ,T

Ž . Ž .From 14 and 15 we obtain for the longitudinal-to-transverse ratio RV

s )L ,g p™ V p2 2R W ,Q sŽ .V)sT ,g p™ V p

22 2 2Q qm p mŽ .V ,T V ,L2s j V4 22m QV ,T

23 2m m mV ,L V ,L V ,Ly y arctan22 2 2 2Q( (Q Q qm QŽ .V ,L

Q24 2 2™ j for Q ™0V9 2mV ,L

p 2 m4V ,L2 2™ j for Q ™` . 16Ž .V 44 mV ,T

The approach to the large-Q2 limit is rather slow,Ž .4but note the enhancement factor m rm inV ,L V ,T

Ž .16 . For completeness, we quote also the total vir-tual-photon cross-section and its asymptotic limit

s ) W 2 ,Q2Ž .g p™ V p

's ) qe s )T ,g p™ V p L ,g p™ V p

ss ) 1qe R W 2 ,Q2Ž .Ž .T ,g p™ V p V

m4 p 2 m4V ,T V ,L2 2™ 1qe j s WŽ .V g p™ V p4 4ž /4Q mV ,T

Q2 ™` . 17Ž .Ž .We note that our simple ansatz for diffraction

dissociation does not lead to the change of the Wdependence of r 0-meson production with increasingQ2 for which there is some experimental indicationw x21,22 . Such an effect can be incorporated intoGVD by modifying the W dependence of diffractiondissociation. In essence this amounts to replacing

Ž 2 . Ž .A W ,ts0 on the right-hand-sides of 12g p™ V pŽ . Ž .and 13 by an appropriate Regge ansatz in terms

of W 2rm2. We have found that this modificationmay lead to the change in the W dependence with

2 w x 0increasing Q indicated by the data 21,22 on r

production. A detailed discussion is beyond the scopeof the present note.

A remark on SCHC is appropriate at this point.From photoproduction measurements at lower ener-

w xgies it is known 5 that SCHC is not strictly valid. ItŽ .is violated at non-zero t at the level of approxi-

mately 10%. In vector dominance this amount of


helicity-flip contributions is traced back to helicity-flip contributions in diffractive hadron reactionswhich occur at approximately the same rate. Extrapo-lating the results from lower to the higher HERAenergies quantitatively may seem somewhat prob-lematic. If helicity-flip contributions are associatedwith non-pomeron exchange, one may expect suchcontributions to be diminished with increasing en-ergy, at least as long as Q2 is kept fixed. Thesituation may be different, if W and Q2 are in-creased at the same rate, keeping xfQ2rW 2 smalland constant. In this case very massive vector statesbecome important with increasing Q2 and contribu-tions different from helicity-conserving pomeron ex-change may still be present in the limit of very highenergies. If an average over a large range in Q2 isperformed, one may accordingly expect helicity flipcontributions to be present at HERA energies at arate similar to the one found at lower energies.

In the comparison of our predictions with experi-ment, we proceed in two steps. In a first step weconsider the experimental evidence for the validity

Ž .of SCHC, before we turn to a comparison of 15 –Ž .17 with HERA and Fermilab data. The validity ofSCHC is not only of interest in itself, due to thepresence of the longitudinal degree of freedom of thevirtual photon in electroproduction, but is as well aprerequisite for the determination of R , as long asV

data are lacking for a direct separation of s )T,g p™ V p

and s ) .L ,g p™ V p

A recent measurement by the ZEUS collaborationw x23 of the full set of 15 density matrix elements

Ž 0 .determining the vector-meson r and f decayw xdistribution 24 can be analyzed in terms of

helicity-conserving and helicity-flip amplitudes. Us-ing parity invariance as well as natural-parity ex-change, the number of independent helicity ampli-tudes determining the density matrix elements can bereduced to ten. This number is reduced to five, ifnucleon helicity-flip amplitudes are assumed to van-ish. The normalized density matrix elements, accord-ingly, depend on four ratios of amplitudes, if onetakes the amplitudes to be purely imaginary.

Ž . ) 0In the notation M l ,l for g p™r p, theseg r

ratios of helicity amplitudes can be chosen asŽ . Ž . Ž . Ž . Ž .M 00 rM qq , M q0 rM qq , M qy rŽ . Ž . Ž .M qq , and M 0q rM 00 . The first one of the

above ratios measures the relative magnitude of lon-

gitudinal and transverse helicity-conserving transi-tions. The other three ratios quantify helicity-non-conservation.

w xThe four ratios can be determined 25 from themeasured r 0 density-matrix elements. According tow x25 , violations of SCHC are small and of the orderof magnitude measured in photoproduction at the

w xlower energy of W s 9.4 GeV, where 5Ž . Ž . Ž .M q0 rM qq s 0.14 " 0.02 and M qy rŽ .M qq sy0.05"0.02. As a consequence, the

longitudinal-to-transverse production ratio

< < 22 M 0qŽ .1q2 2< <M 00 < <Ž . M 00Ž .

R sr 2 2 2< < < < < <M qq M q0 M qyŽ . Ž . Ž .1q q2 2< < < <M qq M qqŽ . Ž .

18Ž .is found to be well approximated by

< < 2 04M 00 1 rŽ . 00R , , , 19Ž .r 2 04e< < 1yrM qqŽ . 00

i.e. by the SCHC expressions for R . Here r 04r 00

denotes the r 0 density matrix element solely deter-mined by the dependence on the polar-angle of the

0 q y w xr ™p p decay distribution. The result 25 forŽ .the four ratios of the helicity amplitudes in 18 ,

obtained from the nine measured density matrix ele-Ž .ments, thus justifies the widely used procedure 19

of determining R from the r 0 decay distributionr

under the assumption of helicity conservation. Theexperimental results on R to be given below arer

based on this procedure.We now turn to the Q2 dependence and compare

Ž . Ž .predictions 15 – 17 with experimental data fromw x )HERA 22,26 at an average g p c.m. energy of

Ž . Ž 0. 9Ws80 GeV 50 GeV for f r production . Fora given vector meson V, our predictions depend onfour parameters, the two effective vector-mesonmasses m and m , the ratio j of the longitudi-V ,T V ,L V

nal-to-transverse strong-interaction amplitudes, andŽ . 2the photoproduction cross-section, i.e. 15 at Q s0.

The solid lines in Figs. 1–3 show the result of asimultaneous four-parameter fit to the data for

9 At HERA energies, we may take the polarization parametere s1.


Ž Ž .. Ž Ž ..) )Fig. 1. Data for s in a and for s in b fromg p ™ f p g p ™ r p

Ž .HERA compared with the GVD prediction 17 . Solid lines:Ž .Four-parameter fit with the values 20 of the fit parameters.

Ž .Dashed line: Two-parameter fit with the values 21 of the fitparameters.

s ) and R , performed separately for the r 0g p™ V p V

and the f meson. The data are well described by thefits, with the parameters

j s1.06 , m2 s0.68 m2 , m2 s0.71 m2 ,r r ,T r r ,L r

j s0.90 , m2 s0.41 m2 , m2 s0.57 m2 ,f f ,T f f ,L f

20Ž .

and s s11.1 mb, s s1.2 mb. The sta-g p™ r p g p™ f p

tistical errors in the parameters are small comparedwith the estimated systematic ones.

The quality of the fits strongly supports the under-lying picture: the propagation of hadronic spin-1

Ž . Ž .Fig. 2. a GVD prediction 16 for the longitudinal-to-transverseŽ .ratio R . Solid line: Four-parameter fit with the values 20 of thef

fit parameters. Dashed line: Two-parameter fit with the valuesŽ . Ž .21 of the fit parameters. b Data for f production by trans-versely polarized photons, s ) , extracted from the mea-T,g p ™ f p

sured values of s ) by using the two-parameter R fitg p ™ f p f

Ž . Ž .shown in a . Dashed line: GVD prediction 15 with the two-Ž . Ž .parameter fit values 21 . Dotted line: VDM prediction, i.e. 15

with m ' m .f,T f


Fig. 3. As Fig. 2, but for r 0-meson production including HERAdata for R extracted from the r-decay distribution using SCHC.r

states and destructive interference govern the Q2

dependence of exclusive electroproduction of vectormesons at small x and arbitrary Q2. Both the asymp-

4 Ž .totic 1rQ behaviour of the cross section, see 17 ,Ž .and the flattening of R , see 16 , are clearly visibleV

Ž .in the data. Moreover, the fitted values 20 are inaccordance with theoretical expectation. The value ofj ,1, i.e. helicity independence of diffractive vec-V

tor-meson scattering, is very gratifying indeed. Themass parameters, m and m , show the theoreti-V ,T V ,L

cally expected ordering m2 -m2 -m2 .V ,T V ,L V

The values of R obtained in the fit seem some-V

what low with respect to the central values of thedata at large Q2. This is of course merely a conse-quence of the fact that the large-Q2 R data hardlyV

contribute to the overall x 2, owing to their largeerrors. Varying the four fit-parameters within onestandard deviation from their best-fit values, we findthat a considerable spread in R is allowed. In otherV

words, with current data a precision determination ofour parameters is not yet possible. In fact, a two-

2 Žparameter fit results in a similar x dashed lines in.Figs. 1–3 as the four-parameter fit. In the two-

parameter fit, obtained by fixing j s1 and m2 sV V ,L2 Ž1.5 m corresponding to an asymptotic value RV ,T V.™5.5 , we find

m2 s0.62 m2 , m2 s0.40 m2 , 21Ž .r ,T r f ,T f

and s s11 mb, s s1.0 mb. With re-g p™ r p g p™ f pŽ . Ž .spect to the results of the fits given in 20 and 21

it may be worth quoting the estimate 0.41 m2 Qm2V V ,T

2 w xQ0.74 m from Ref. 16 , based on a reasonableV

choice of the diffraction-dissociation parameters inŽ .8 .

In Figs. 2b and 3b, we show the transverse cross-section, s ) . The data in Figs. 2b and 3bT,g p™ V p

were extracted from the data on s ) in Fig. 1g p™ V p

with the help of our two-parameter fit 10 for R .V

Figs. 2b and 3b demonstrate the dramatic differenceat large Q2 between the data and the GVD predic-

Ž .tion 15 with m -m on the one hand, and theV ,T VŽ . Ž .VMD prediction 6 , or rather 15 with m 'm ,V ,T V

on the other hand.Comparing the dotted VMD predictions in Figs.

2b and 3b for the transverse cross-section s )T,g p™ V p

with the data for s ) in Fig. 1a and 1b, oneg p™ V p

notices that the dotted curves would approximatelydescribe the data for s ) s s ) qg p™ V p T ,g p™ V p

s ) . This, at first sight paradoxical, coinci-L,g p™ V p

dence of fits of s ) qs ) , entirelyT,g p™ V p L ,g p™ V p

based on the transverse VMD formula, was in factw xobserved previously 21,27,28 in fits that vary the

10 No other procedure to extract s ) suggests itself, asT,g p ™ V p

the number of data points for R is very small, and the Q2 valuesV

for s ) and R are not identical.g p ™ V p V


Ž 2 2 .power of Q qm at fixed mass m . ImplicitlyV V

the fits obviously assume s ) s0, and, dis-L ,g p™ V p

regarding the information from vector-meson decayindeed seem to confirm s ) s0. This con-L ,g p™ V p

clusion is inconsistent, however, with the results ofthe above analysis of the r 0 density matrix elements.This analysis establishes that longitudinal r 0 mesons

Žare almost exclusively produced by longitudinal vir-.tual photons. The mentioned approximate coinci-

dence of fits based on the VMD formula fors ) with the data for s ) qT,g p ™ V p T,g p ™ V p

s ) appears as a numerical accident.L ,g p™ V p

We turn to a comparison of our results with ther 0-production data from the Fermilab E665 Collabo-

w xration 29 taken at energies 10-W-24 GeV andat four-momentum transfers 0.15-Q2 -10 GeV 2.Fig. 4 shows the E665 data for R and for s takenr Tr

w xfrom Tables 7 and 13 of Ref. 29 . The theoreticalŽ .curves are based on the parameters from 21 , i.e.

j s1, m2 s1.5 m2 , m2 s0.62 m2 and sr r ,L r ,T r ,T r g p™ r p

s11 mb.Recent theoretical work on the electroproduction

of vector mesons has been concentrated on attemptsw xto deduce the cross-sections from perturbative 9–13

w xand non-perturbative 14 QCD. For the productioncross-section by transversely polarized vectormesons, the pQCD calculations typically lead to astrong asymptotic decrease, as 1rQ8, modifiedsometimes by additional corrections to become 1rQ7.

w x 2It may be argued 11 that the region of Q Q30GeV 2 explored at present, in which experiments finda fall-off rather like 1rQ4, is not sufficiently asymp-totic for pQCD to yield reliable results. Furtherexperiments at still larger values of Q2 will clarifythe issue.

As for the longitudinal-to-transverse ratio, R ,V

pQCD calculations led to the same result of a linearrise in Q2 as the simple VDM predictions, compareŽ . Ž .6 and 7 . Such a linear rise is always obtained, ifelectromagnetic current conservation is the onlysource of the Q2 dependence of R . For large Q2,V

this linear rise is in conflict with experimental re-sults. A behaviour of the cross-section for Q2

4m2 ,r

for both the production of longitudinally as well astransversely polarized r 0 mesons, somewhat closer

w xto the experimental data, was obtained in Ref. 12 ;the calculation was based on open qq productionand parton-hadron duality. It is interesting to note

Fig. 4. As Fig. 3, but for E665 data.

that the resulting cross-sections have a VDM form 11

multiplied by correction factors depending on thescaling variable x. The asymptotic form for RV

w xderived in Ref. 12 has recently been reproduced ina calculation based on r 0- meson wave-functionsw x w x13 . In Ref. 13 , also pQCD calculations of thehelicity-flip amplitudes have been presented. Whilethe magnitude of the helicity-flip amplitudes is ap-

11 Ž . Ž . w xCompare 37 and 38 in Ref. 12 .


w xproximately reproduced, a detailed comparison 25shows that the x 2 of these predictions is not betterthan the x 2 for a representation of the densitymatrix elements under the assumption of SCHC. Thecoincidence of the relative magnitude of thehelicity-flip amplitudes at large Q2 with thehelicity-flip amplitudes in photoproduction anddiffractive hadron physics remains unexplained inthe pQCD approach.

In summary, we have investigated electroproduc-tion of vector mesons in GVD. We have shown thatdestructive interference between neighbouring vectorstates naturally leads to the spectral representationsŽ . Ž . Ž . )12 and 13 of the zero-t amplitudes for g qpT,L

™V qp. Both predictions, the asymptotic 1rQ2T,L

fall-off of the transverse amplitude and the approachof R towards a constant value, are in good agree-V

ment with the experimental data. The expected hier-archy, m2 -m2 -m2 , of the pole masses m ,V ,T V ,L V V ,T

m and the helicity independence of the strong-in-V ,LŽ .teraction amplitudes reflected in j ,1 stronglyV

support the GVD picture: the propagation of hadronicvector states determines, for arbitrary Q2, the Q2

dependence of vector-meson production by virtualphotons in the diffraction region of x,Q2rW 2

<1.Moreover, SCHC is experimentally violated at theorder of magnitude of 10%, a value also observed indiffractive hadron-hadron scattering and photopro-duction at lower energies.

Returning to our starting point, the photoproduc-Ž . Ž .tion sum rules 1 and 2 , the present analysis

strengthens their dynamical content, which is to re-duce photoproduction to vector-meson-induced reac-tions. More generally, in conjunction with the exper-imental observation of states with masses up to about

w x20 GeV 30 in diffractive production in DIS at smallx and up to large Q2, the present investigation

w xsupports the point of view 31 that propagation anddiffractive scattering of hadronic vector states is thebasic dynamical mechanism in DIS at small valuesof the scaling variable.

Acknowledgements

It is a pleasure to thank Sandy Donnachie, TeresaMonteiro and Gunter Wolf for useful discussions.¨

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11 March 1999


Mixing and decay constants of pseudoscalar mesons: the sequel

Th. Feldmann a, P. Kroll a,b, B. Stech c

a Fachbereich Physik, UniÕersitat Wuppertal, D-42097 Wuppertal, Germany¨b Centre for the Subatomic Structure of Matter, UniÕersity of Adelaide, SA 5005, Australiac Institut fur Theoretische Physik, UniÕersitat Heidelberg, D-69120 Heidelberg, Germany¨ ¨

Received 21 December 1998; revised 14 January 1999Editor: P.V. Landshoff

Abstract

We present further tests and applications of the new h–hX mixing scheme recently proposed by us. The particle states are

decomposed into orthonormal basis vectors in a light-cone Fock representation. Because of flavor symmetry breaking themixing of the decay constants can be identical to the mixing of particle states at most for a specific choice of this basis.Theoretical and phenomenological considerations show that the quark flavor basis has this property and allows, therefore, fora reduction of the number of mixing parameters. A detailed comparison with other mixing schemes is also presented. q 1999Published by Elsevier Science B.V. All rights reserved.

w xIn a recent reinvestigation 1 of processes involv-ing h and h

X mesons we pointed out that a propertreatment of the h–h

X system requires a sharp dis-tinction between the mixing properties of the mesonstates from the mixing properties of the decay con-stants. While the particle state mixing involves theglobal wave functions, the decay constants probe thequark distributions at zero spatial separation. Con-ventionally, h and h

X are expressed as superpositionsŽ .of an SU 3 flavor octet and a flavor singlet state

corresponding to an orthogonal transformation withmixing angle u . The decay constants of h and h

X

defined by their matrix elements with singlet andoctet axial vector currents will in general not showthe same mixing since flavor symmetry breakingmanifests itself differently at small and large dis-

Ž .tances. Because SU 3 breaking is solely caused bythe current quark masses a simpler picture can be

expected for properly defined decay constants in thequark-flavor basis. Indeed, a dramatic simplificationis achieved by taking two orthogonal basis states 1

which are assumed to have in a Fock state descrip-tion the parton composition

'< : < :h sC uuqdd r 2 q . . . ,q q

< : < :h sC ss q . . . 1Ž .s s

Ž .Here C denote light-cone wave functions of thei

corresponding parton states, and the dots stand for

1 In principle, the two-state basis should be extended by statesof higher energy, for instance by adding a cc state. Energyconsiderations indicate, however, that the mixing with these statesis small. The small charm components in the h and h

X isw xdiscussed in Ref. 1 .


( )Th. Feldmann et al.rPhysics Letters B 449 1999 339–346340

< :higher Fock states which also include gg compo-nents. These higher Fock states play no explicit rolein the following discussions where we are mainlyinterested in mesonic states and decay constants. The

Ž .physical meson states are related to the basis 1 byan orthogonal transformation

< :h< :h qsU f ,Ž .Xž /< : < :ž /h hs

cosf ysinfU f s , 2Ž . Ž .ž /sinf cosf

where f is the mixing angle. Ideal mixing corre-sponds to the case fs0. It is to be emphasized thatour definition of meson states is given in terms ofparton degrees of freedom without introducingmodel-dependent concepts like constituent quarks.

Ž .Our central ansatz 1,2 has important conse-quences for the weak decay constants which probethe short-distance properties of the quark-antiquarkFock states. To see this in detail, let us define the

2 Ž .decay constants by f s131 MeVp

² < i < : i0 J P ' i f p , 3Ž .m5 P m

where Psh,hX. Here J i denotes the axial-vectorm5

currents with quark content isq,s. The decay con-stants are related to the quark-antiquark wave func-tions at the origin of configuration space. Because ofthe fact that light-cone wave functions do not dependon the hadron momentum we can define two basicdecay constants f and f arising from h and h ,q s q s

respectively,2dx d kH'f s2 6 C x ,k . 4Ž . Ž .Hi i H316p

Ž .Here x denotes the usual light-cone q momentumfraction of the quark and k its transverse momen-Htum with respect to its parent meson’s momentum.

Ž .Eq. 4 is exact, only the quark-antiquark Fock statecontributes to the decay constant, higher Fock states

Ž . Ž .do not contribute. Using Eqs. 1 – 3 , one immedi-ately observes that our ansatz for the Fock decompo-sition naturally leads to decay constants in the

2 We stress that occasionally used decay constants ‘‘ f , f X’’ areh h

ill-defined quantities.

quark-flavor basis which simply follow the pattern ofstate mixing:

f q f sh h

sU f diag f , f . 5Ž . Ž .q s q sX Xf fž /h h

The conventional octet-singlet basis states are ob-tained from the quark-flavor basis states by perform-ing a rotation with the ideal mixing angle. Thephysical states are then related to the octet-singletbasis states by

< :< : hh 8sU u , 6Ž . Ž .Xž /< : < :ž /h h1

'with usfyarctan 2 . The corresponding Fock de-compositions of these octet-singlet basis states, fol-

Ž .lowing from Eqs. 1,2,6 , read

< :C q2C uuqddy2 ssq s< :h s8 '3 6

' < :2 C yC uuqddqssŽ .q sq q . . .'3 3

' < :2 C yC uuqddy2 ssŽ .q s< :h s1 '3 6

< :2C qC uuqddqssq sq q . . . 7Ž .'3 3

Obviously, it is unavoidable that the so-defined octetŽ . Ž . Žsinglet meson state contains an SU 3 singlet oc-.tet admixture, except for identical wave functions

C sC , an equality which holds in the flavor sym-q s

metry limit only. Only then one would find pureŽ .octet and singlet states in Eq. 7 . Certainly, these

results are based on our central ansatz, namely that h

and hX can be decomposed into two orthogonal states

where one state has no ss and the other no qqcomponent. One may alternatively start from the

Ž .assumption that h and h defined in Eq. 6 have8 1

Fock decompositions with only parton octet or sin-glet combinations in the quark-antiquark sector, re-spectively. Rotating these states back by the ideal

Ž . Ž .mixing angle, the resulting h h state has an ss qqq s

component, unless the octet and singlet wave func-tions are equal. However, from both, theoretical andphenomenological considerations performed in Refs.w x1–3 , the quark-flavor basis is to be favored.

( )Th. Feldmann et al.rPhysics Letters B 449 1999 339–346 341

One may also define decay constants throughmatrix elements of octet and singlet axial-vector

Ž . Ž .currents, analogously to Eq. 3 . Using Eqs. 1,2 ,one easily sees that these decay constants cannot be

Ž . w xexpressed as U u diag f , f . Rather one has8 1

f 8 f 1f cosu yf sinu ,h h 8 8 1 1s 8Ž .

8 1 ž /f sinu f cosuX Xž /f f 8 8 1 1h h

where we use the new and general parametrizationw xintroduced in Ref. 2 . The parameters appearing in

Ž .Eq. 8 are related to the basic parameters f, f andq

f , characterizing the quark flavor mixing scheme assw xfollows 1 ,

' '2 f 2 fs qu sfyarctan , u sfyarctan ,8 1f fq s

f 2 q2 f 2 2 f 2 q f 2q s q s2 2f s , f s . 9Ž .8 13 3

The decay constants f i do not follow the pattern ofP

state mixing in the octet-singlet basis; only in theŽ .SU 3 symmetry limit one would have u susu .F 8 1

This is a consequence of the non-trivial Fock decom-Ž .position in Eq. 7 . The difference between u and8

Ž .u following from Eq. 9 leads to the same formula1w xas derived within chiral perturbation theory 2 . In

our approach the quantities u and u are parame-8 1

ters determined by the fundamental quantities 3 f,< :f and f . They are not to be used as h sq s

< : < :cosu h ysinu h etc.8 8 1 1

Let us now briefly review the determination of thew xmixing parameters performed in Ref. 1 . For this

purpose we considered the divergences of axial-vec-Ž .tor currents which incorporate the U 1 anomalyA

Ž .q su,d,si

asm ˜E q g g q s2m q ig q q GG. 10Ž .i m 5 i i i 5 i 4p

Ž .Sandwiching Eq. 10 between the vacuum and themeson states and using the definition of the decay

3 We remark at this point, that the flavor singlet axial-vectorcurrent is not conserved in QCD. Consequently, the singlet decayconstant f , and, hence, f and f too, are renormalization scale1 q s

dependent, although only mildly since the corresponding anoma-2 w xlous dimension is of order a 2 . Varying the scale m betweensŽ .M and M , the value of f m changes by 3% only, an effecth h 1c

which we discard.

Ž . Ž . Xconstants 3 together with 5 one obtains the h–h

mass matrix in the quark flavor basis. Its elementsare composed of the gluonic matrix elements

as ˜² < < :0 GG h and matrix elements of the quark massi4p

Ž .terms contained in 10 , which can be expressed bythe pion and the K meson masses using flavorsymmetry and its breaking to first order. The known

Žeigenvalues of this mass matrix the masses of h andX.h give then the value of the gluonic matrix ele-

ments and, in particular, the value 42.48 for themixing angle f.

Alternatively, the mixing parameters can be deter-Ž .mined from phenomenology without using SU 3 F

relations. The mixing angle f can be determined byconsidering appropriate ratios of decay widths orcross sections, in which either the h or the hq s

components are probed. The decay constants f andq

f can be evaluated from the h,hX™gg decay widths,s

relying on the chiral anomaly prediction. The analy-sis of a number of decay and scattering processesleads to the phenomenological set of parametersf rf s1.07, f rf s1.34, fs39.38 which we willq p s p

use in the following. We like to point out that thephenomenological values for the mixing angle f

from different experiments are all consistent witheach other within a rather small uncertainty. The

Žresulting differences between u , u and u al-8 1Ž . .though only caused by SU 3 breaking effects areF

Ženormous, see Table 1. We also list in this table in.anti-chronological order the parameter values ob-

tained in previous approaches.It is possible to take decompositions of h and h

X

where either the octet or the singlet state is pure –in the sense that admixtures of the orthogonalquark-antiquark combination are absent – but notboth. Note that one may, for instance, write the weak

Ž .decay constants in Eq. 8 in the form

f 8 f 1 f f sin u yuŽ .h h 8 1 8 1sU u .Ž .88 1 ž /0 f cos u yuX X Ž .ž /f f 1 8 1h h

11Ž .

Then it is tempting to introduce a new basis by

< :< : hh 8sU u . 12Ž . Ž .X 8ž /< : < :ž /h h1


Table 1Comparison of different determinations of mixing parameters. The values given in parentheses are not quoted in the original literature buthave been evaluated by us from information given therein. Crosses indicate approaches where the difference between u , u and u has1 8

been ignored

u u u f rf f rf method8 1 8 p 1 p

Ž . w xy12.38 y21.08 y2.78 1.28 1.15 qs–scheme theo. 1Ž . w xy15.48 y21.28 y9.28 1.26 1.17 qs–scheme phen. 1

w x– y20.58 y48 1.28 1.25 ChPT 2w xy21.48 x x 1.19 1.10 GMO formula 4

w xy15.58 – – – – phenomenology 5w x w x w x w x w xy19.78 y12.28 y30.78 0.71 0.94 model 6w x w x w x w x w xy12.68 y19.58 y5.58 1.27 1.17 model 7

Ž . w xy 238y178 x x 1.2y1.3 1.0y1.2 phenomenology 8–11w x w x w x w x Ž . w xy98 y208 y58 1.2 1.1 U 1 anomaly 12A

The elements of the second matrix on the r.h.s. ofŽ .Eq. 11 can now be viewed as the decay constants

of h , h through octet or singlet axial vector cur-˜ ˜8 1

rents, respectively. This matrix is still non-diagonalbut triangular. The new basis has the special feature

Ž .that the anomaly contributes to the singlet h mass˜ 1

alone, i.e. one has

as ˜² < < :0 GG h s0. 13Ž .˜ 84p

This property is related to the fact that the ratioa X as s˜ ˜² < < : ² < < : w x0 GG h r 0 GG h is given by ycotu 1 .84p 4p

It allows to determine a Gell-Mann–Okubo formulafor the mass of the h basis state. Transforming the˜ 8

Žmass matrix found in the quark flavor basis see. Ž .above to the new basis 12 one finds

f 2 M 2 q2 f 2 2 M 2 yM 2Ž .q p s K p2 2f m s . 14Ž .˜ ˜8 88 3

This formula reminds of the suggestion put forwardw x Ž w x.in Ref. 4 see also 13 , namely to use the product

2 2 2 Ž .f M rather than M to determine the SU 3 F

breaking effects in the Gell-Mann–Okubo formula 4.

4 Note however that the decay constants used in the analysis ofw x4 are not defined as proper matrix elements of weak currents.

Ž . Ž .Insertion of the relations in Eq. 9 into Eq. 14yields

4M 2 yM 2 M 2 yM 2K p K p2m , yD 15Ž .˜ ˜88 GMO3 3

Ž 2 2 . Ž 2 .with D s4 f y f r 3 f . The deviation fromGMO q s 8

the standard Gell-Mann–Okubo relation D canGMOw xalso be derived in chiral perturbation theory 14,8 .

The above discussion clearly shows: An analysisŽ .which implicitly uses Eq. 13 provides for an esti-

mate of the parameter u rather than the angle u .8

Indeed, previous treatments along these lines ob-tained mixing angles close to y208, which is consis-

Ž .tent with our value of u see Table 1 .8

As a first test of our mixing approach we ana-lyzed the hg and h

Xg transition form factors in Ref.

w x1 . A good description of the experimental data hasbeen found from the phenomenological set of param-

w xeters. For details we refer to 1,3 . Let us now turn tofurther tests and applications of our results which

w xhave not been discussed in Ref. 1 .

RadiatiÕe decays of S-waÕe quarkoniaŽ3 .We define the ratio of decay widths R S sn

w3 X x w3 x 3G S ™h g rG S ™hg where S representsn n n

one of the quarkonia Jrc ,c X,F , . . . According tow x15 the photon is emitted by the c quarks whichthen annihilate into lighter quark pairs through theeffect of the anomaly. Thus, the creation of the


corresponding light mesons is controlled by the ma-as ˜² < < : w xtrix element 0 GG P , leading to 14p

3Xkh g3 2R S scot u 16Ž .Ž .n 8 ž /khg

22 2 2 2 2 Ž .(where k s M ym ym y4m m r 2 MŽ .12 1 2 1 2

denotes the final state’s three-momentum in the restframe of the decaying particle. The experimental

w xvalue of R in the Jrc case 16 has already beenused in the phenomenological determination of the

w xbasic mixing parameters in Ref. 1 . Using the phe-nomenological value of u quoted in Table 1, we8

Ž X. Ž .predict R c s5.8 and R F s6.5. The predictionŽ X . q5.4for R c agrees with the experimental value 2.9 ,y1.8

which still has uncomfortably large errors howeverw x17 . For the radiative F decays only upper boundsexist at present.

x decays into two pseudoscalarsc J

Because these are energetic decays the currentŽ .quarks produced will be in an almost pure SU 3

singlet state. However, flavor symmetry violationcan occur in the hadronization process. The ratio ofx decay widths into different pairs of pseudoscalarc J

Ž .mesons can be written Js0,2

2 2 Jq1w xG x ™P P C kc J 1 2 12 12s . 17Ž .ž / ž /w xG x ™P P C kc J 3 4 34 34

For the coefficients C two limiting cases can bei j

considered. If the mesons are formed at hadronicdistances the influence of the different decay con-stants will be a minor one and one expects to a goodaccuracy C sC X X sC 0 0 , C X s0 and, fromhh h h p p hh' Žq y 0 0isospin symmetry, C s 2 C where we in-p p p p' .cluded the statistical factor 2 . If, however, themeson generation starts already at a time at whichthe inter-quark distances are very small, the decayamplitudes are obtained from the convolution of ahard scattering process with the corresponding wave

w xfunctions 18 . Assuming equal shapes of the wavefunctions, an assumption which is not in conflictwith present experimental information, differences inthe decay amplitudes are then solely due to thedifferent decay constants and the mixing angle.One finds: C s f 2 cos2fq f 2 sin2fs1.41 C 0 0 ,hh q s p p

2 2 2 2 ŽX X 0 0C s f sin f q f cos fs1.53 C and withh h q s p p

2' '. ŽXthe statistical factor 2 included C s 2 f yhh q2 . 0 0 q yf sin f cosf s y0.45 C and C ss p p p p' w0 02 C . Numerically we obtain G x ™p p c0Ž2.

x w 0 0 x Ž . whh rG x ™ p p s 1.9 1.7 , G x ™c0Ž2. c0Ž2.X X x w 0 0 x Ž . wh h rG x ™ p p s 1.9 1.3 , G x ™c0Ž2. c0Ž2.

X x w 0 0 x Ž . whh rG x ™ p p s 0.2 0.1 and G xc0Ž2. c0Ž2.q yx 0 0 x™p p rG x ™p p s2.w c0Ž2.

From the differences between the two limitingcases it appears that x decays are less suited forc J

testing h–hX mixing parameters, but, taking the mix-

ing parameters from other processes, they will pro-vide interesting information on meson formation inthese reactions. Experimentally, only the ratio

w x w 0 0 xG x ™ hh rG x ™ p p is knownc 0Ž2. c 0Ž2.w x q1.1 Ž q0.9. Ž 0 016,19 : 0.76 0.76 for the p p branchingy0.5 y0.6

ratio we combined the data with the one for theq y .p p channel . At present, the large experimental

errors prevent any definite conclusion.ŽSimilar relations as given here modified accord-.ing to the correct charge factors should hold for

two-photon annihilations into pairs of pseudoscalarmesons.

g ) g ) ™h,hX transition form factorsThese form factors offer, in principle, a way to

measure the angle u . Allowing both the gluons to1

be virtual where at least one of the virtualities q21

and q2 is supposed to be very large, one may easily2

work out the leading-twist result for these formw x Ž .factors 20 . In this approximation one has isq,s

f xŽ .i2 2)F q ,q syC a f dxŽ . Hh g 1 2 i s i 2 2i x q q 1yx qŽ .1 2

18Ž .

where f is the h distribution amplitude, and C ai i i'Ž .numerical factor C s 2 , C s1 . Combining bothq s

the form factors into those for the physical mesonsand assuming the equality of the two distribution

Žamplitudes which at least holds in the formal limitq2 ™` since both the distribution amplitudes evolvei

.into the asymptotic one , one arrives at

2 2 ')F q ,q 2 f cosfy f sinfŽ .hg 1 2 q s

s sytanu .12 2 'X)F q ,q 2 f sinfq f cosfŽ .h g 1 2 q s

19Ž .


Of course, at finite values of momentum transfer onemay expect corrections from differences between thetwo distribution amplitudes and from transverse mo-mentum. Such corrections can be worked out follow-

w xing, for instance, Ref. 3 .For a measurement of these form factors one may

Ž X.consider the process pp™ jetq jetqh h wherethe mesons are supposed to be produced in the

w x w xcentral rapidity region 21 . According to Close 22these form factors may also be of relevance in

Ž X.pp™pph h , provided the Pomeron couples tom Ž X.quarks Ag andror gg™h h is the elementary

process in the Pomeron-Pomeron interaction. Theexplicit extraction of the form factors from suchmeasurements may, however, be very difficult. Kil-

w x Ž X.ian and Nachtmann 23 discuss g-Odderon-h h

form factors which appear in diffractive e-p scatter-ing processes. We find that the ratio of these formfactors is given by cotu at large momentum trans-8

fer.

( X)Z™h h g decayThe treatment of these processes is rather aca-

demic since the expected branching ratios are farŽbelow the present experimental bounds The Z™pg

w x.decay has e.g. been discussed in Ref. 24 . Never-theless, they provide an additional example of reac-tions which are sensitive to the angle u . The ratio of1

the decay widths are given by2 3X

X Xw xG Z™h g F kh g Z h gR Z s s 20Ž . Ž .ž /w xG Z™hg F khg Z hg

where F is the time-like form factor for PgPg Z

transitions mediated by the Z boson. That formfactor can be calculated along the same lines as the

) w xPgg transition form factor 1,3 . Since the value ofM 2 is very large it suffices to consider the asymp-Z

totic limit of the form factor only. In terms of theŽ .octet and singlet decay constants, defined in Eq. 8 ,

the result reads:

6 C f 8 q6 C f 18g Z P 1g Z P2F M s 21Ž .Ž .Pg Z Z 2MZ

2 'Ž . Ž . Žwhere C s 1y4sin u r 6 6 and C s 28g Z W 1g Z2 '. Ž .y4sin u r 3 3 . The weak coupling of the flavorW

Ž 2 .octet current is strongly suppressed by 1y4sin u .WŽ . 2Hence, R Z ,cot u .1

The same transition form factors F appear inPg ZŽ X. q yh h ™gm m decays, but the momentum transfer

is very small. In analogy to the P™gg decays theamplitudes in this case involve the inverse decayconstants, and the h to h

X ratio is sensitive to theŽ X .angle u . To measure these form factors in h h ™8

gmqmy decays one has to extract the g-Z interfer-ence term from suitably chosen asymmetries as dis-

w xcussed in detail in Ref. 25 .

RadiatiÕe transitions between light Õector andpseudoscalar mesons

The relevant coupling constants are defined byw x26

² < EM < : < 2P p J V p ,lŽ . Ž . q s0P m V

syg e pn p r´ s l . 22Ž . Ž .V Pg mnrs P V

w xIn Ref. 11 these coupling constants are expressed interms of meson masses and decay constants by ex-

Ž 2 .ploiting the chiral anomaly prediction at q s0and vector meson dominance. However, the differ-ence between u and u has not been considered.8 1

Translating the expressions for g correctly to theP Vg

w xquark-flavor scheme, following otherwise Ref. 11 ,we arrive at the formulas and values listed in Table2.

The numerical result for the coupling constantsdepend on the actual values of the vector mixingangle f , which is expected to amount to only a fewV

Ž w x.degrees see e.g. 5,11 . The values in Table 2 arecalculated for f s0. For the vector meson decayV

Table 2Various coupling constants g from theory and experimentV Pg

w x y116 . The numerical values are quoted in units of GeVmVŽ . < <Ž . < <Ž .P V g in units g theo. g exp.V Pg V Pg V Pg

2f pV

3 cosfh r 1.52 1.85"0.344 fq

X 3 sin fh r 1.24 1.31"0.124 fq

cosf cosf 2 sin f sinfV Vh v y 0.56 0.60"0.154 f 4 fq s

X sinf cosf 2 cosf sinfV Vh v q 0.46 0.45"0.064 f 4 fq s

cosf sin f 2 sin f cosfV Vh f q 0.78 0.70"0.034 f 4 fq s

X sinf sin f 2 cosf cosfV Vh f y 0.95 1.01"0.254 f 4 fq s


w xconstants we take 27 f s210 MeV, f s195 MeV,r v

f s237 MeV. The predictions agree rather wellf

with experiment. Indeed the relations

X Xg g grh g vh g fhgs s s tanfs0.82 23Ž .

Xg g grhg vhg fh g

w xare well confirmed by experiment. In Ref. 11 re-sults of similar quality could only be achieved byusing a value of usu su sy178 which deviates8 1

from the values obtained from other applicationssubstantially.

h and hX admixtures to the pion

Ž w x.As is well-known see e.g. 28 an accurate pre-Ž X.scription of the decays of h h to three pions can

only be achieved by taking isospin violation intoaccount. This effect is usually parametrized in termsof h and h

X admixtures to the pion,

0 < : X < X:p sf qe h qe h 24Ž .3

where f denotes the pure isospin-1 state. A3

straightforward generalization of our mixing schemeyields for the strength of h and h

X admixtures in thepion

m2 ym2dd uu

escosf ,2 22 M yMŽ .h p

m2 ym2dd uuX

e ssinf , 25Ž .2 2X2 M yMŽ .h p

where the difference m2 ym2 can be estimateddd uuŽ 2 2 2 2 .0 " " 0from 2 M yM qM yM to amount toK K p p

0.0104 GeV 2. A possible difference in uy and dyquark decay constants is ignored in the derivation ofŽ . X25 . The expressions for e and e look rathersimple in the quark flavor scheme and are intimatelyconnected to physical quantities. Inserting our phe-nomenological number for the mixing angle f weobtain es0.014 and e

X s0.0037. By exploiting thew x Xproperties of the mass matrix 1 the ratio ere

Ž .following from 25 can also be expressed in termsof u and u8

2X 'e cosuq 2 sinusytanu . 26Ž .8 ž /'e 2 cosuysinu

The numerical value, following from our phe-nomenological set of parameters, is 0.26. In contrast,

Ž w x.the conventional approach see e.g. 28 , using usu ,y208 gives the much smaller value 0.17.8

The values of the parameters e and eX have

recently been shown to be of importance for theinvestigation of CP-violation in B™pp decayssince it breaks the isospin triangle relation for theamplitudes of the three processes Bq™pqp 0, B0

0 0 0 q y w x X™p p and B ™p p 29 . The value of e

w x Ž X .used in Ref. 29 e s0.0077 is substantially largerthan our value.

SummaryWe discussed the mixing properties of the h and

hX meson state vectors and of their decay constants

and showed that there is, at most, only one basiswhere the mixing of the decay constants can followthe pattern of state mixing. Chiral perturbation the-ory as well as phenomenological analyses favor thisproposition for the quark-flavor basis. In general,e.g. in the familiar octet-singlet basis, one needs twoangles in order to parametrize the decay constants.However, when using our quark-flavor mixingscheme, these new angles are fixed by the basicparameters f, f , f , leading to a number of impor-q s

tant consequences for many reactions. The results arequite different from conventional mixing schemes inwhich the subtleties discussed here are not consid-ered and where, as a consequence of that, the mixingparameters often show a strong process dependence.

The improved knowledge of the mixing parame-ters is also of importance for the analysis of B

X Ž w x.decays, like B™Kh see e.g. 30 or B™pp

w x29 . Further interesting applications of our approachrefer to h and h

X production processes in highenergy hadron collisions where exotic form factors

) ) Ž X.such as the g g h h form factors play an impor-tant role.

Acknowledgements

T.F. was supported by Deutsche Forschungsge-meinschaft. P.K. thanks the Special Research Centrefor the Subatomic Structure of Matter at the Univer-sity of Adelaide for support and the hospitality ex-tended to him.


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161.

11 March 1999


Coulomb final state interactions for Gaussian wave packets

Urs Achim Wiedemann a, Daniel Ferenc b,c, Ulrich Heinz b,c

a Physics Department, Columbia UniÕersity, New York, NY 10027, USAb CERNrTH, CH-1211 GeneÕa 23, Switzerland

c Institut fur Theoretische Physik, UniÕersitat Regensburg, D-93040 Regensburg, Germany¨ ¨

Received 30 November 1998; revised 14 January 1999Editor: P.V. Landshoff

Abstract

Two-particle like-sign and unlike-sign correlations including Coulomb final state interactions are calculated for Gaussianwave packets emitted from a Gaussian source. We show that the width of the wave packets can be fully absorbed into thespatial and momentum space widths of an effective emission function for plane wave states, and that Coulomb final stateinteraction effects are sensitive only to the latter, but not to the wave packet width itself. Results from analytical andnumerical calculations are compared with recently published work by other authors. q 1999 Published by Elsevier ScienceB.V. All rights reserved.

1. Introduction

To analyze the geometry and dynamics of theŽ .collision region, two-particle correlations C q,K of

like-sign and unlike-sign hadrons have been studiedextensively in relativistic heavy ion collisions at

w x w xAGS 1 and CERN SPS 2,3 energies. They show< <q-dependent structures at relative pair momenta q

w x Ž .-100 MeV. These originate mainly 4 from iŽ < <final state interactions which for pions at q -100

MeV are dominated completely by the Coulomb. Ž .force and ii the quantum statistics of identical

particles.Practical attempts to reconstruct space-time infor-

mation from two-pion correlation data in momentumspace so far exploit mostly the quantum statistical

Ž .Hanbury Brown – Twiss HBT effect between iden-w xtical bosons 3,5,6 . This requires a prior subtraction

of final state Coulomb interaction effects from themeasured correlation functions, with proper account

w xfor the finite size of the emission region 4,7–11 .Up to now this is done directly in the experimentalanalysis, either by taking experimental unlike-sign

w xpion correlations to correct the like-sign ones 3 , orby a model calculation for the Coulomb effect ex-

w xpected for a finite size emission region 1 .This approach was recently questioned by Merlitz

w xand Pelte 12–14 . From a numerical analysis basedon Gaussian wave packets emitted from a Gaussian

w xsource, they concluded that 12 ‘‘the expectedCoulomb distortion in the momentum correlation. . . becomes unobservable’’ and that therefore ‘‘ex-perimental data, which are published after Coulombcorrection, are wrong for small momentum differ-ences’’. If correct, this conclusion would invalidate asubstantial part of the existing work on the analysisof two-particle correlation data since it implies that

Ž .either i any attempt to base a space-time interpreta-tion of identical two-particle correlations on Coulomb

Ž .corrected data is ill-founded or that ii any attempt


( )U.A. Wiedemann et al.rPhysics Letters B 449 1999 347–353348

to describe the particle emitting source in heavy ioncollisions by a set of Gaussian wave packets isinconsistent.

This dramatic perspective has led us to reconsiderthe calculation of two-particle correlations for Gauss-ian wave packets. Following Merlitz and Pelte wedescribe the particle emitting source in terms of adistribution of wave packet centers and a characteris-tic wave packet width s of the emitted particles.The emitted Gaussian wave packets are propagatedinto the detector under the influence of mutualCoulomb final state interactions. We derive analyti-cal expressions which show that the wave packetsize s can always be absorbed by a redefinition ofthe model parameters characterizing the source size.For Gaussian source models, two-particle correlationmeasurements cannot differentiate between ‘‘sourcesize’’ and ‘‘wave packet width’’. In a limiting casewe further prove analytically the equivalence of theGaussian wave packet formalism and the usuallyadopted plane wave calculations irrespectiÕe of thesize of the waÕe packet width. These analytical calcu-lations show quite generally that the problem pointedout by Merlitz and Pelte does not exist. Their resultsdisagree with our numerical calculations as well aswith analytical formulae which we derive withoutapproximations from the same starting point as the

w xcalculation presented in Ref. 12 .

2. Unlike-sign pion correlations

Our starting point is a set of Gaussian one-particlew xwave packets 16–19

1 2 2yŽ xyr . rŽ2 s .q i p P xˇ ˇi if x ,t s e , 1Ž . Ž .i 0 3r42psŽ .which are centered at initial time ts t at phase-space0

Ž .points r , p . We expand the time evolution of theˇ ˇi iŽ .corresponding two-particle state C x , x ,t si j 1 2 0

Ž . Ž .f x ,t f x ,t in terms of plane waves f ,i 1 0 j 2 0 p , p1 2

d3p d3p1 2C x , x ,t s AA p , p ,tŽ . Ž .Hi j 1 2 i j 1 23 32p 2pŽ . Ž .

=f x , x ,t , 2Ž . Ž .p , p 1 21 2

f x , x ,t seyi E t f X f rŽ . Ž . Ž .p , p 1 2 2 K qr21 2

iqPr

2yiŽE qE . t 2 i KP X1 2'e e e , 3Ž .

which we write in terms of center of mass coordi-1 1Ž . Ž .nates Xs x qx , Ks p qp , and relative1 2 1 22 2

Ž . Ž .coordinates rs x yx , qs p yp . The prob-1 2 1 2

ability PP for detecting at time t™` the twoi j

particles prepared in the state C with momenta pi j 1w xand p is given by 112

< ) < 2PP p , p sPP q,K s lim AA p , p ,t ,Ž . Ž . Ž .i j 1 2 i j i j 1 2t™`

4Ž .

lim AA p , p ,tŽ .i j 1 2t™`

ˆ ˆyi H Ž tyt . yi H Ž tyt .0 0 0² < :s lim e f t e C tŽ . Ž .p , p 0 i j 01 2t™`

² < pair: ² < rel:s f C V f C . 5Ž .2 K i j q qr2 i j

Ž .Here we separated the state C x , x ,t into rela-i j 1 2 0

tive and center of mass wave functions,

1 2 2ˇ ˇpair yŽ XyX . rs q2 i K P Xi j i jC X s e ,Ž .i j 3r42psŽ .6Ž .

i1 2 2rel yŽ ryr . rŽ4s .q qˇ ˇi j i jPrC r s e , 7Ž . Ž .2i j 3r42psŽ .

ˇ ˇwhere X , K ,r ,q are the corresponding center-ˇ ˇi j i j i j i j

of-mass and relative coordinates constructedŽ .from the wave packet centers. In 5 we have also

introduced the Møller scattering operator V sqˆ ˆi H Ž tyt . yi H Ž tyt .0 0 0lim e e for the final state inter-t ™`

1ˆ ˆ ˆŽ .action hamiltonian HsH qV r , H sy D y0 0 X4m1

D . This Møller operator maps the plane waverm

f onto the solution of the Lippmann-Schwingerqr2

equation for the corresponding stationary scatteringproblem. For two-particle Coulomb interactions thisis the Coulomb scattering wave

V f r sF coul rŽ . Ž .Ž .q qr2 qr2

iqPr1

ph 22sG 1y ih e eŽ .

=F ih ;1;iz , 8Ž . Ž .y

me21z s qr"qPr , hs , 9Ž . Ž ." 2 4p q

( )U.A. Wiedemann et al.rPhysics Letters B 449 1999 347–353 349

2 < < < <where e r4p s a s 1r137, r s r , q s q ,Ž .F ih;1;iz is the confluent hypergeometric func-y

Ž .tion, and h is the Sommerfeld parameter. Eq. 8applies for pairs with opposite charges; for like-signpairs one replaces h¨yh. With the help of Eq.Ž . Ž .8 , the calculation of the amplitudes 5 is reducedto a six-dimensional integral.

2.1. Gaussian source model

We consider a toy model of simultaneous particleemission at time ts t for which initially the wave0

packet centers are distributed with Gaussians ofwidths R and D in coordinate and momentum space,respectively. It can be specified by the followingnormalized distributions of relative distances andpair coordinates:

r 2 q 2ˇ ˇ1y y

2 2S r ,q s e ,Ž .ˇ ˇ 4 R 4Drel 34p RDŽ .ˇ 2 ˇ 2X K1

y yˇ ˇ 2 2S X , K s e . 10Ž .Ž . R Dpair 3p RDŽ .

With the choice R2 sR2r2, D2 sm T , and s 2 ss

2 s 2, this model coincides with the one considered0w xby Merlitz and Pelte 12 . We calculate the unlike-

sign two-particle correlator via the two-particle spec-Ž . Ž .trum PP p , p of Eq. 4 , averaged over the distri-i j 1 2Ž .butions 10 and normalized to the corresponding

spectrum for pairs of non-interacting particles:

I int q,KŽ .qyC q,K s , 11Ž . Ž .nonintI q,KŽ .

3 3 3 ˇ 3 ˇI q,K s d r d q d X d K S r , pŽ . ˇ ˇ ˇ ˇŽ .H i j i j i j i j rel i j i j

= ˇ ˇS X , K PP q,K . 12Ž . Ž .ž /pair i j i j i j

Ž .For the non-interacting case Eq. 12 must be evalu-ated with PP nonint which is obtained by replacing ini j

Ž .the amplitude 5 the relative Coulomb waveV f by the plane wave f . Since the centerq qr2 qr2

of mass coordinate is not affected by the two-particlefinal state interaction, the integrations over

ˇ ˇŽ . Ž .S X , K drop out in the ratio 11 , and thepair i j i j

qyŽ .correlator C q,K becomes independent of thepair momentum K. One finds

23 qD 2qy 4DC q sG yh eŽ . Ž . ž /4p R

=r 2 i

3 y q rPq2d r e F ih ;1;izŽ .2H 8 R y

=r X i

XX3 y y r Pq2d r e 2H 8 R

=

2D 1X) 2X Ž .y y ryr

2ž /F ih ;1;iz e ,Ž . 4 16 Ry

13Ž .1 2y ph 2ph2Ž . < Ž . < Žwhere G h s G 1q ih e s2phr e y

.1 is the Gamow factor. It is important that thiscorrelator depends only on the parameter combina-tions

s 2 12 2 2 2R sR q , D sD q . 14Ž .22 2s

This shows that for Gaussian models of particleemission, the wave packet width s can be absorbedin a redefinition of the model parameters. There is nomeasurement which allows to determine s indepen-dent of R and D. Of course, the specific s-depen-

Ž .dence in 14 still constrains the values which R andD can take. In particular, R,D always satisfy theuncertainty relation RDG"r2, rendering the expo-

Ž .nent of the last term in 13 always negative.In the absence of final state interactions, the

dependence of the correlator on the parameter com-Ž . w xbinations 14 has been noted repeatedly 16,18,20 .

The momentum spectra and correlations are entirelydetermined by the ‘‘effective’’ emission functionw x18,20

S r , p s d3r d3p S r,p S rIr,pIp ,Ž . Ž . Ž .ˇ ˇ ˇ ˇ ˇ ˇHeff w .p .

15Ž .which is a folding integral between the distribution

Ž .of wavepacket centers S r, p and the Wigner den-ˇ ˇsity S of a single particle wave packet. Forw.p.

Gaussian source models, S depends on R and Deffw xonly 20 . The same holds true in the presence of

final state interactions where the correlator can beŽ .written for arbitrary model distributions S r, p as aˇ ˇ

quite involved expression depending on S onlyeffŽ Ž . w x. Ž .see Eq. 60 in Ref. 11 . Our result 13 is an


explicit representation of this general relation, ob-Ž .tained for the Gaussian source models 10 . It allows

further analytical and numerical studies:

2.2. Limiting cases

qyŽ .In two interesting limits the correlator C qcan be further simplified analytically. Using

2 2XD2 3r2 Ž3.Ž . Žlim D r4p exp y ryr sd rŽ .D™` 4X.yr we find

1qylim C q sŽ . 3r22D™` 4p RŽ .

=r 2

23 y coul2 < <d r e F r . 16Ž . Ž .H 4 R qr2

w xThis expression, first written down by Koonin 23 ,is the usually adopted starting point for plane wave

w xcalculations 1–3,7,8,10,21–23 ; it was shown byw xBaym and Braun-Munzinger 10 to be well approxi-

mated by a semi-classical approach. The limit D™`

can be taken for arbitrary values of the wave packetwidth s and is equivalent to the limit D™` whichdescribes an emission function S without momentum

qyŽ .dependence. In this limit, the correlator C q forsimultaneously emitted Gaussian wave packets coin-cides exactly with the starting point of conventional

w xplane wave calculations 10,11 , irrespectiÕe of thesize s of the waÕe packet. Rescaling s then simplyamounts to a change of the effective spatial size R ofthe source.

What happens if the source size becomesŽ .large? Changing in Eq. 13 the integration variables

X X' 'r ™ R r , r ™ R r , and replacing theCoulomb wave function by its leading con-

icoul ' 'tribution, F R r ™ exp R r P q qŽ . žq r 2 2

'R 1ih ln qryqPr qO , we findŽ .ž / ž //2 'R

lim Cqy q s1 . 17Ž . Ž .R™`

This is expected: as the source becomes larger, theaverage spatial separation between particles in-creases and their Coulomb attraction decreases, lead-ing to a flat correlator in the limit of infinite sourcesize.

2.3. Numerical results

One may wonder whether a large but realisticeffective source size R can come sufficiently close

Ž .to the limiting case R™` of 17 to support thew xclaim of Merlitz and Pelte 12,13 that the Coulomb

repulsion becomes effectively unobservable. To studyŽ .this question we have calculated the correlator 13

numerically, after doing the azimuthal integrationsŽ < <.qs q :

Cqy qŽ .23 q

D 22 4Ds4p G yh eŽ . ž /4p R

=

i2 qr xr` 1 22 y

2r dr e dx eH H8 R0 y1

=r X 2

`iX 2 X y

2F ih ;1; qr 1yx r dr eŽ . H 8 Rž /2 0

=

)

i iX1 Xy qr ydy e F ih ;1; qr 1yyŽ .2H ž /2y1

=X2 2 2' (I 2 B rr 1yx 1yyž /0

=eyB 2Ž r 2y2 r rX x yqrX 2 . . 18Ž .Here I is the modified Bessel function and B2 s0

2 2Ž .D r4 q 1r 16 R . The numerical results forqyŽ .C q are shown in Fig. 1 for the model parame-

ters Rs3.5 fm, Ds84 MeV and different values

Fig. 1. Two-particle correlator of unlike-sign pion pairs for theŽ .Gaussian model 10 . The correlation depends only on the model

2 2 2 2 2parameter combinations R s R q s r2 and D s D qŽ 2 .1r 2s . The results for Ds84 MeV agrees well with Koonin’s

Ž .expression 16 obtained in the limit D™`.


of the wave packet width s . For ss2.5 fm, thesevalues correspond to the model parameters chosen in

w xRef. 12 .2One sees that, with increasing source size R s

R2 qs 2r2, the resulting Coulomb correlation in-deed becomes smaller than the Gamow factor, but itremains clearly observable even for a very largewave packet size ss14 fm. We also compare in

qyŽ . Ž .Fig. 1 the full correlator C q of Eqs. 13r18 toŽ .the limit 16 , calculated for the same value of R. At

least for the model parameters studied here, bothexpressions agree almost exactly. There seems to beno possibility to distinguish on the basis of final statecorrelations between the Gaussian wave packet for-malism and the plane wave calculation based on the

Ž .approximate expression 16 which was used in otherw xstudies 1,3,10 . As long as the Gaussian wave packet

width s is included consistently in the definition of2 2(the source size Rs R qs r2 , both formalisms

lead qualitatively and quantitatively to the same re-sult. We conclude that the differing results derived in

w xRef. 12 with the help of a numerical simulation ofthe time evolution of wave packets are incorrect. 1

3. Like-sign pion correlations

For pairs of pions of identical charge the Coulombfinal state effects are superimposed on the quantumstatistical effects resulting from the symmetrizationof the two-particle wave function. Paralleling thecalculation of Section 2. with Bose-Einstein sym-metrized Gaussian wavepackets C and plane wavesi j

f , one arrives at the symmetrized asymptoticp , p1 2

two-particle amplitude

lim AA BE p , p ,tŽ .i j 1 2t™`

pair rel² < : ² < :s f C V f CK i j q qr2 i j

rel² < :q V f C , 19Ž .q yqr2 i j

from which the two-particle momentum-space proba-BE Ž .bility PP p , p is again calculated according toi j 1 2

1 We can only speculate on the origin of this discrepancy. Thew xauthors of Ref. 12 use a particular basis expansion. The error

introduced by this step is uncontrolled and may accumulate overlarge times in the numerical propagation of the wavepackets. Ourapproach avoids explicit propagation of the wavepackets by solv-ing this part of the problem analytically.

Ž . BE Ž .Eq. 4 . Averaging PP according to 12 over thei j

model distribution gives the numerator of the two-BE ,intŽ .particle correlator which we call I q,K . We

normalize it by the method of ‘‘mixed pairs’’: anŽ .uncorrelated mixed pair is described by an unsym-

metrized product state without Coulomb interactionand leads, after averaging over the model distri-

nonintŽ . Ž Ž ..bution, to I q,K see Eqs. 11r12 . TakingŽ . Ž .both distinguishable states f x ,t f x ,t andi 1 0 j 2 0

Ž . Ž .f x ,t f x ,t into account we havei 2 0 j 1 0

I BE ,int q,KŽ .qqC q,K sŽ . nonint nonintI q,K q I yq,KŽ . Ž .

sCdir q qCex q . 20Ž . Ž . Ž .Ž .For the Gaussian model 10 , the center of mass

coordinate is affected neither by two-particle finalstate interactions nor by two-particle Bose-Einstein

Ž .symmetrization. The correlator 20 hence does notdepend on the pair momentum K. It splits into twocontributions. The ‘‘direct term’’ can be obtained

qyŽ . Ž .from C q in Eq. 13 by changing the sign of theSommerfeld parameter, yh™h. The ‘‘exchangeterm’’ is given by

Cex qŽ .23 q

2r iD 2 3 y q rPq4D2sRe G h e d r eŽ . 2H 8 Rž /½ 4p R

r X 2 iXX3 y q r Pq

2=F yih ;1;iz d r eŽ . 2H 8 Ry

2D 1X) 2X Ž .y y ryr

2ž /= F yih ;1;iz e .Ž . 4 16 Rq 521Ž .

This integral can be simplified to a 4-dimensionalŽ .expression similar to 18 . The limits D™` and

dirŽ .R™` of the first term C q are obtained fromŽ .16r17 by replacing yh™h. The correspondinglimits for the exchange term are given by

r 21ex 3 y

2lim C q s d r eŽ . H 4 R3r22D™` 4p RŽ .

= < coul < 2cos rPq F r , 22Ž . Ž . Ž .qr2

lim Cex q s0 . 23Ž . Ž .R™`


As in the case of unlike-sign correlations, the corre-qqŽ .lator C q depends only on the parameter combi-

nations R and D, but not explicitly on the wavepacket width s . In the limit D™`, the Gaussianwave packet formalism again coincides with the

w x ŽKoonin formula 23 now with the additional sym-Ž . .metrization factor 1qcos qPr under the integral

which is the starting point of most conventionalplane wave calculations.

In Fig. 2 we compare numerically the full correla-qqŽ . Ž .tor C q from Eq. 20 , for Ds84 MeV, with

the limit D™`, both calculated for the same value2 2 2 Žof R sR qs r2. Due to spherical symmetry of

< < .the source the correlator depends only on qs q .For small values of s one observes a small, butsignificant difference. The reason is that even in theabsence of Coulomb final state interactions, the HBT

Žradius parameter which gives the q-width of the2.correlator is not exactly given by the source size R

w xbut rather by 18

12 2R sR y . 24Ž .HBT 24D

2For large values of R or s , the term R dominatesthis expression, and the difference between R2

HBT2 Ž .and R disappears see Fig. 2 . In fact, when com-

qqŽ . 2puting the limit D™` of C q using RHBT2instead of R , the agreement with the full correlator

Fig. 2. Two-particle correlator of like-sign pion pairs for theŽ .Gaussian model 10 . The correlation depends only on the model

2 2 2 2 2parameter combinations R s R q s r2 and D s D qŽ 2 .1r 2s . The full correlator is well approximated by the standard

Koonin expression for a static momentum-independent source of2 2Ž .radius squared R y1r 4D .

qqŽ .C q becomes almost exact even for small valuesŽ .of s see inset in Fig. 2 . We note that even for the

smallest value studied here, ss2.5 fm, the term2 2Ž .1r 4D contributes only f5% to R . This smallHBT

difference is clearly visible in the exchange termŽ . qqŽ .21 of C q , whereas its influence on the direct

dir Ž qy.term C and hence on the correlator C is foundnumerically to be an order of magnitude smaller.This illustrates that the two-particle correlator Cqq

of identical pions is more sensitive than Cqy to asmall change in the Gaussian width of the phasespace density.

Ž .The modification 24 of the radius parameter tobe used as input in the plane wave calculation can

w xalso be obtained from the Koonin expression 23 or,Ž . w xmost explicitly, from Eq. 65 of Ref. 11 . The

consistency of Koonin’s expression with the fullcorrelator in the present model calculation is a non-trivial check of the so-called smoothness approxima-

w xtion used in Ref. 11 to derive Koonin’s expressionfrom a general treatment of two-body final stateinteractions.

To sum up: as long as the Gaussian wave packetwidth s is included consistently in the definition ofthe source size, both the plane wave calculationsw x1–3,7,8,10,21–23 and the Gaussian wave packet

w xformalism 16–19 lead to qualitatively and quantita-tively equivalent results. While the present studyproved this only for Gaussian source models, weexpect it to be true quite generally since we know

w x Ž .that 11 the relation 15 between the effectiveemission function and the Wigner density of singleparticle wave packets holds for arbitrary model dis-

w xtributions and that 15 two-particle momentum cor-relations are mostly sensitive to the Gaussian charac-teristics of the source in space-time.

Acknowledgements

This work was supported by the U.S. Departmentof Energy under Contract No. DE-FG02-93ER40764,by DFG, BMBF and GSI. We thank P. Braun-Munzinger, H. Feldmeier, M. Gyulassy, J. Hufner,¨and J. Stachel for stimulating discussions, and D.

w xPelte for an open debate of Refs. 12–14 .


References

w x1 E877 Collaboration, J. Barrette et al., Phys. Rev. Lett. 78Ž .1997 2916.

w x Ž .2 NA35 Collaboration, T. Alber et al., Z. Phys. C 73 1997443.

w x3 NA49 Collaboration, H. Appelshauser et al., Eur. Phys. J. C¨Ž .2 1998 661.

w x4 M. Gyulassy, S.K. Kauffmann, L.W. Wilson, Phys. Rev. CŽ .20 1979 2267.

w x5 G. Roland for the NA49 Collaboration, Nucl. Phys. A 638Ž .1998 91c.

w x6 U.A. Wiedemann, B. Tomasik, U. Heinz, Nucl. Phys. A 638´ˇŽ .1998 475c.

w x Ž .7 S. Pratt, Phys. Rev. Lett. 53 1984 1219; Phys. Rev. D 33Ž .1986 72.

w x Ž .8 M.G. Bowler, Phys. Lett. B 270 1991 69.w x9 D. Ferenc, in: Symposium on Quantum Correlations in High

Energy Nuclear Collisions, Hiroshima, Japan, April 1995,Ž .Nuclear Physics Research 40 1996 59.

w x Ž .10 G. Baym, P. Braun-Munzinger, Nucl. Phys. A 610 1996286c.

w x Ž .11 D.V. Anchishkin, U. Heinz, P. Renk, Phys. Rev. C 57 19981428.

w x Ž .12 H. Merlitz, D. Pelte, Phys. Lett. B 415 1997 411.w x13 H. Merlitz, D. Pelte, nucl-thr9805028.w x14 H. Merlitz, D. Pelte, nucl-thr9806049.w x Ž .15 U. Heinz, in: M.N. Harakeh et al. Eds. , Correlations and

Clustering Phenomena in Subatomic Physics, NATO ASIŽ .Series B 359 1997 137, Plenum, New York.

w x Ž .16 H. Merlitz, D. Pelte, Z. Phys. A 357 1997 175.w x Ž .17 U.A. Wiedemann, Phys. Rev. C 57 1998 3324.w x18 U.A. Wiedemann, P. Foka, H. Kalechofsky, M. Martin, C.

Ž .Slotta, Q.H. Zhang, Phys. Rev. C 56 1997 R614.w x Ž .19 T. Csorgo, J. Zimanyi, Phys. Rev. Lett. 80 1998 916; J.¨ ˝ ´

Zimanyi, T. Csorgo, hep-phr9705432.´ ¨ ˝w x Ž .20 U. Heinz, in: S. Costa et al. Eds. , CRIS’98: Measuring the

size of things in the Universe: HBT interferometry and heavyion physics, World Scientific, Singapore, 1998, hep-phr9806512.

w x21 Yu.M. Sinyukov, R. Lednicky, J. Pluta, B. Erazmus, S.V.Ž .Akkelin, Phys. Lett. B 432 1998 248.

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11 March 1999


The muon anomalous magnetic moment in QED:three-loop electron and tau contributions

Andrzej Czarnecki a, Maciej Skrzypek b

a Physics Department, BrookhaÕen National Laboratory, Upton, NY 11973, USAb Institute of Nuclear Physics, ul. Kawiory 26a, 30-055 Cracow, Poland

c CERN, Theory DiÕision, CH-1211 GeneÕa 23, Switzerland


Abstract

Ž 3.We present an analytic calculation of electron and tau OO a loop effects on the muon anomalous magnetic moment.Computation of such three-loop diagrams with three mass scales is possible using asymptotic and eikonal expansions. Anevaluation of a new type of eikonal integrals is presented in some detail. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

Ž .The new measurement of the muon anomalous magnetic moment, a s g y2 r2, in the experiment E821m m

in Brookhaven has motivated many recent theoretical studies. In the Standard Model, a receives contributionsm

from electromagnetic and weak interactions, as well as from loop effects involving hardrons. All three types ofeffects have been studied recently. QED contributions are known to the four-loop level, and even somefive-loop diagrams have been evaluated. The next largest contribution is due to hadronic loops and is the mostdifficult one to evaluate. There has been recently significant progress in both evaluation of the light-by-light

w x w x w xdiagrams 1 and hadronic vacuum polarization effects 2 . Electroweak two-loop effects are also known 3 .The present paper is devoted to the only 3-loop QED contribution to a that has not been evaluatedm

analytically so far: from a diagram with electron and t lepton loop insertions in the photon propagator, shownin Fig. 1. Because of the three mass scales present in this diagram, m , it cannot at present be evaluated in ae,m ,t

closed form. However, we present an approach based on asymptotic and eikonal expansions, which takesadvantage of the wide separations between those scales and permits an evaluation with an arbitrary accuracy.

We begin with a brief summary of the present knowledge of QED contributions to the electron and muonŽ w x.anomalous magnetic moments, a for more details see e.g. 4 .e,m


( )A. Czarnecki, M. SkrzypekrPhysics Letters B 449 1999 354–360 355

Fig. 1. Three-loop contribution to muon anomalous magnetic moment with the electron and tau loops inserted in the photon propagator.

Ž .a Electron. To match the present experimental precision, one needs four terms in the expansion of a in thee

fine structure constant a ,n4 a

a s A q . . . 1Ž .Ýe n ž /pns1

where ellipses indicate contributions of loops containing the heavy leptons m and t ; A have been known1,2w xsince the early years of QED 5 . An analytical evaluation of A required the efforts of many groups and took3

w xalmost 40 years; it has been completed only recently 6 . A is known only numerically.4

The perturbative series for a is very well behaved, and, together with the most recent experimental valuesew x w xfor electron and positron 7 , allows one to deduce a very precise value of the fine structure constant 8,4 ,

ay1 s137.03599959 38 13 . 2Ž . Ž . Ž .Ž . Ž .b Muon. The value of a found from electron gy2, Eq. 2 , can be applied to compute the QED

contribution to the muon anomalous magnetic moment a . Because of the presence of electron loops,m

higher-order QED contributions to a are enhanced with respect to a . At present five terms of the expansion inm e

a are needed:n5 a

QEDa s C 3Ž .Ým n ž /pns1

with

C sA s0.5,1 1

C sA qa m rm qa m rm s0.765 857 388 44 ,Ž .Ž . Ž .2 2 1 e m 2 m t

C sA qCgg e qCgg t qC vac . pol . e qC vac . pol . t qC vac . pol . e,t s24.050 509 2 ,Ž . Ž . Ž . Ž . Ž . Ž .3 3 3 3 3 3 3

C sA q127.55 41 s126.04 41 ,Ž . Ž .4 4

C s930 170 , 4Ž . Ž .5

w xwhere a describe contributions of two-loop diagrams with electron and tau loops, respectively 9,10 . We will1,2

discuss them in detail later on.w xIn C we have contributions from light-by-light scattering diagrams with e and t loops 11 , and vacuum3

w xpolarization diagrams with either e, or t 12 , or both types of loops. An analytical evaluation of this latter

( )A. Czarnecki, M. SkrzypekrPhysics Letters B 449 1999 354–360356

vac. pol.Ž .contribution of mixed eyt diagram, C e,t , is the main purpose of this paper and will be presented3w xbelow. Numerically it can be evaluated using the kernel from 13 :

C vac . pol . e,t s0.0005276 2 . 5Ž . Ž . Ž .3

w xFor C one uses the difference between the muon and electron coefficients found in Ref. 14 . For C only a4 5w xnumerical estimate of the presumably dominant contributions is known 15 .

w xThe present estimate of the total QED contribution to a is 4m

aQED s116584705.6 2.9 =10y11. 6Ž . Ž .m

2. Two-loop diagrams with electron and tau loops

Before calculating the contribution of the three-loop diagram of Fig. 1, we would like to discuss a method ofŽ .evaluating the two-loop diagrams shown if Fig. 2 we discuss only the Pauli formfactor, relevant for a . It willm

serve us as an example to illustrate the main points of our calculational techniques. Of course, full analyticalw xresults for these two-loop contributions are known 9,10 . It is instructive to note how they were obtained. First,

a closed analytical formula was found, with full functional dependence on the ratio of masses m rm or m rme m m t

w x10 . That formula, containing dilogarithms, was found awkward to use, because of cancellations and difficultiesw xin estimating the accuracy in the numerical evaluation. In Ref. 9 expansions of the exact result in powers of

small mass ratios were given. Such expansions avoid evaluation of special functions and their accuracy can beprecisely assessed.

Here we demonstrate how such expansions can be obtained without knowledge of the exact result. We firstconsider the case of the t loop insertion. In this case a well known method of heavy mass expansion is

Ž w x.applicable for a review see 16 . Let us denote the loop momenta by p and p , for the momentum flowingt m

inside the t loop and for that in the virtual photon respectively. There are two regions of integration, withcharacteristic scales of momenta p ;p ;m and p ;m , p ;m . In the case of the first region, which wet m t t t m m

can call the hard contribution, we can safely regard the muon mass and external momentum as small withrespect to the integration momenta, and expand the integrand in these small parameters. The resulting integral

Ž .corresponds to a vacuum diagram shown in Fig. 3 a . In the second region we cannot neglect the external muonmomentum, but now, since p <p , we can expand the t loop propagators in Taylor series in p , so that them t m

Ž .integral factorizes into a product of two one-loop diagrams, shown in Fig. 3 b . This nice factorization of therelevant integration regions is only possible in regularization schemes, which do not introduce additional massscales, such as dimensional regularization.

Fig. 2. Two-loop contributions of lepton loops in the photon propagator.


Fig. 3. Graphic representation of characteristic momentum scales in the two-loop diagram with a t loop. Solid and dashed lines denote,respectively, massive and massless propagators.

After evaluation of these two sub-diagrams and renormalization of the electric charge of the muon, we find aŽ . w xfinite result. Several of the terms we calculated agree with formula 12 in Ref. 9 :

2 4 6 ` 3 2 2 nq2l l ln l 9 131 4 l 8n q28n y45 lŽ .4 6a lsm rm s q q l y l q ln lyŽ . Ý2 m t 245 70 19600 99225 315 nq3 2nq3 2nq5Ž . Ž . Ž .ns3

` 2 nq2nlq2ln l .Ý

nq3 2nq3 2nq5Ž . Ž . Ž .ns3

The other two-loop diagram that we have to consider is the electron loop insertion in the photon propagatorŽ .Fig. 2 . Here the situation is somewhat more involved, since there are now three integration regions.

Ž . ŽIntroducing the obvious notation for the integration momenta, p , we have: p ;p ;m , p ;m ande,m e m m e e. Ž .p ;m , and p ;p ;m . The first two regions correspond to known cases of the ‘‘large momentumm m e m e

w x Ž . Ž .expansion’’ 16 . They are depicted in Fig. 4 a,b . Fig. 4 a denotes a simple Taylor expansion of the electronŽ .propagators in m , justified if both integration momenta are large. In Fig. 4 b we have illustrated one of thee

two cases, where one of the electron lines cannot be expanded in m , but where the integration factorizes andeŽ .we have a product of one-loop diagrams. More exotic is the third case, Fig. 4 c , where the only scale of

Ž . Ž .Fig. 4. Integration regions contributing to the two-loop diagram with an electron loop. a : Taylor expansion in m . b : region of softeŽ . Želectron loop momentum. c : region of both momenta soft double line denotes muon propagator expanded in the square of its virtual

.momentum .


integration is m , since all dependence on the muon mass and external momentum factorizes. Sub-diagrams ofew xthis type have been encountered in a different context in eikonal expansions 17,18 . Integrals that arise in the

present case are somewhat different and we give here some details of their evaluation.Since now the integration momentum p is much smaller than the external muon momentum p, we canm

expand muon propagators in p2. As a result we have to compute integrals of the formm

1 dD p dD pm eJ a ,a s . 7Ž . Ž .H a1 2 aD 1 222 2 2 2p p 2 p Pp p qm p qp qmŽ .Ž . Ž .Ž .m m e e m e e

First, we use Feynman parameters to combine the last two propagators and integrate over p . We gete

1 dD pmyeyeJ a ,a s G e x 1yx , 8Ž . Ž . Ž . Ž .H a e1 2 aD r2 1 22 2 2p p 2 p Pp p qmŽ .Ž . Ž .m m m x

2 2 Ž .with m 'm rx 1yx . Next, using again Feynman parameters, we combine the first and the last term in thex e

denominator. Finally, using

`by11 1 l

s dl , 9Ž .Ha b aqbB a ,ba b Ž . 0 w xaqbl

and integrating over p , we getm

a a a a2 2 2 224y2 a ya y4eG G a q y2q2e G 2ya y ye G y1qa q qe 1 21 1 1 mž / ž / ž / ž / e2 2 2 2J a ,a sŽ .1 2 aa 22 mm2 G a G 2y ye G y2q2 a qa q2eŽ . Ž .2 1 2ž /2

a22 4y2 a ya y4e1 2yp G a q y2q2e m1 ež /2s .a q1 a y1 a a2 2 2 22 a q2 a y3q2 e a1 2 22 G G a q qe G 2y ye sinp a q qe m1 1 mž / ž /ž / ž /2 2 2 2

Ž .We should mention that the integrand in Eq. 7 could also contain products pPp . It is possible to replace themeŽ w x.by combinations of products of p Pp and p Pp using traceless products see e.g. 17 . However, in them e m

present case there are at most two powers of p in the numerator and we can use the following simple formulas:e

22 2 2 2 2 2p p p p D p Pp yp p pPp p PpŽ . Ž . Ž .e m e m e m m m e2 2pPp ™ q p Pp y , pPp ™ .Ž . Ž .e m e2 22D D pmDy1 pŽ . Ž .m

Adding the contributions of the three integration regions, we find, after renormalization, that the terms weŽ . w xobtained agree with formula 11 of 9 :

25 p 2 1 52 2 3a ksm rm sy q ky lnkq 3q4lnk k y p kŽ .Ž .1 e m 36 4 3 4

2p 44 14 8 1092 4 6 6q q y lnkq2ln k k q k lnky k

3 9 3 15 225

3 2` 2 nq3 8n q44n q48nq9Ž .2 nq4q lnky k .Ý 2 22n 2nq1 2nq3Ž . Ž . n 2nq1 2nq3Ž . Ž .ns2


3. Three-loop diagram with e and t loop insertions

We now proceed to the actual focus of our work, the contribution of the three-loop diagram with e and t

w xloop insertions, shown in Fig. 1. So far its contribution to a has been evaluated only numerically 12 .m

Using the techniques described above, we can easily obtain an expansion of the Pauli part of this diagramwith arbitrary accuracy. There are now three integration momenta and we have to consider five integrationregions, combinations of the conditions described in the context of two-loop diagrams. Using the notation pt ,m ,e

for the integration momenta in the three loops, the regions we have to consider are:Ø p ;p ;p ;m ,t m e t

Ø p ;m ; p ;p ;m ,t t m e m

Ø p ;m ; p ;m ; p ;m ,t t m m e e

Ø p ;m ; p ;p ;m ,t t m e e

Ø p ;p ;m ; p ;m .t m t e e

Calculations in each of these regions are analogous to the cases described in the previous section. For presentpurposes it is more than sufficient to retain the first three terms in the m2rm2 expansion and two terms inm t

m2rm2. After renormalization we finde t

m2 4 m 1m mvac . pol .C e,t , ln yŽ .3 2 ž /135 m 135m et

m4 229213 p 2 37 m 1 m m m 3 mm t t t m mq y q y ln y ln ln q ln4 2ž /12348000 630 11025 m 105 m 4900 mm mm m et e

m6 1102961 4p 2 398 m 8 m m m 524 mm t t t m mq y q y ln y ln ln y ln6 2ž /75014100 2835 297675 m 945 m 297675 mm mm m et e

2 m2 4p 2 m3e e

q y s0.0005276 2 , 10Ž . Ž .2 215 45m m mt t m

in agreement with the numerical evaluation. The error in the result is due to the t–lepton mass uncertainty. TheŽ 2 . Ž 2 .leading-logarithmic term of this expansion corresponds to simply replacing a q s0 by a m in them

two-loop diagram with a t loop. We have included the last term, with odd powers of m and m , even thoughe m

it is not relevant numerically. It illustrates typical contributions of the eikonal expansion, the only source ofterms non-analytical in masses squared.

Ž .We have checked Eq. 10 by comparing it with an analytical integration, using the kernel function given inw xRef. 13 . Terms quadratic in masses m are in complete agreement. The odd powers of m have not beent ,m ,e e

w xincluded in Ref. 13 .Ž .With formula 10 the complete QED contribution to a is now known analytically. In this particular casem

Ž .this result does not noticeably improve the accuracy of the QED prediction, since the error in Eq. 10 comesfrom the t lepton mass measurement. However, the technique presented here might facilitate other calculations.We have seen that a combination of large momentum, heavy mass, and eikonal expansions eliminates the needfor numerical calculations and enables us to construct arbitrarily accurate expansions without knowledge of theexact result.

Acknowledgements

We thank the CERN Theory Group for support and hospitality during our visit, where most of this projectwas completed. This work was supported in part by the U.S. Department of Energy under grant number


DE-AC02-98CH10886, Polish Government grants numbers KBN 2P03B08414, KBN 2P03B14715, and theMaria Skłodowska-Curie Joint Fund II PAArDOE-97-316.

References

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11 March 1999


On recent data of branching ratiosž / ž /for c 2S ™Jrcpp and c 2S ™Jrch

Y.F. Gu a,1, X.H. Li b,2

a Institute of High Energy Physics, Beijing 100039, Chinab Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, UniÕersity of California, Berkeley, CA 94720, USA


Abstract

Ž . q y 0 0Recent data on branching ratios for c 2S decays to Jrcp p , Jrcp p and Jrch are reviewed. An alternativetreatment of data is proposed to get rid of the logical inconsistency which occurs in original computational procedure.q 1999 Published by Elsevier Science B.V. All rights reserved.

Ž .There is a group of exclusive c 2S decays intoJrc whose branching ratios are of importance to

Žboth charmonium and other e.g. B and relativistic.nuclear collision physics, but were mostly measured

w xtwo decades ago 1 . Recently, new results wereŽ .reported on the branching ratios of c 2S ™

q y Ž . 0 0 Ž .Jrcp p , c 2S ™ Jrcp p , and c 2S ™w xJrch by the E760 experiment 2 , and on the ratio

Ž Ž . q y. Ž Ž . q y.of B c 2S ™Jrcp p rB c 2S ™m m byw xthe E672rE706 experiment 3 . With these new en-

Ž .tries, 9 branching ratios of the c 2S are reanalyzedŽ . w xby the Particle Data Group PDG 1 . In this letter,

we comment on these recent data and propose, amongother things, an alternative way of treating the datato get rid of the logical inconsistency which occurs

w xin the computational procedure of Ref. 2 .

1 E-mail: [email protected] On leave from IHEP, Beijing. E-mail: [email protected]

Ž .In determining the branching ratios of c 2S ™ f ,where f is Jrcpqpy, Jrcp 0p 0, or Jrch, the

w xauthors of Ref. 2 used the expression

´ NJrc X fB c 2S ™ f sŽ .Ž .

´ Nf Jrc X

=B c 2S ™Jrc X , 1Ž . Ž .Ž .where ´ and ´ include the geometrical accep-f Jrc X

tance and efficiencies for triggering and selection ofŽ 0 .the exclusive f with p or h™gg and inclusive

Ž q y. Ž .Jrc X with Jrc™e e decays of the c 2S ,

Table 1Ž .c 2S branching ratios determined by solving a linear equation of

Ž .the form 2

Channel 1990 1991q yŽ .c 2S ™ Jrcp p 0.136"0.099 0.156"0.0740 0Ž .c 2S ™ Jrcp p 0.118"0.087 0.084"0.041

Ž .c 2S ™ Jrch 0.025"0.021 0.017"0.009


( )Y.F. Gu, X.H. LirPhysics Letters B 449 1999 361–363362

Table 2Ratios from the E760 measurement as input data of fit

Ratio 1990 1991 Combined0 0 q yŽ Ž . . Ž Ž . .G c 2S ™Jrcp p rG c 2S ™Jrcp p 0.868"0.171 0.539"0.089 0.609"0.079

Ž Ž . . Ž Ž . .G c 2S ™Jrch rG c 2S ™Jrc X 0.080"0.033 0.056"0.018 0.062"0.016

Table 3Ž .c 2S branching ratios by a simultaneous least-square fit

Channel 1990 1991 Combined PDG96 E760q yŽ .c 2S ™Jrcp p 0.318"0.025 0.323"0.025 0.318"0.025 0.324"0.026 0.283"0.0290 0Ž .c 2S ™Jrcp p 0.196"0.025 0.178"0.022 0.186"0.021 0.184"0.027 0.184"0.023

Ž .c 2S ™Jrch 0.027"0.003 0.027"0.003 0.027"0.003 0.027"0.004 0.032"0.010Ž .c 2S ™gx 0.086"0.008 0.087"0.008 0.087"0.008 0.087"0.008c1Ž .c 2S ™gx 0.078"0.008 0.078"0.008 0.078"0.008 0.078"0.008c2

respectively; N and N are the numbers off Jrc X

exclusive and inclusive events selected, respectively.´ NJrc X fThe authors claimed that, by determining

´ Nf Jrc X

Ž .and using the branching ratio of c 2S ™Jrc X inthe PDG, they are able to make measurements ofŽ Ž . q y. Ž Ž . 0 0.B c 2S ™Jrcp p and B c 2S ™Jrcp p

with errors comparable to the world average. How-ever, one notes that the method used by the authorsto treat their data is questionable.

Ž Ž .As a matter of fact, the PDG value of B c 2S.™Jrc X , 0.57"0.04, on which the authors rely

Ž Ž .for determining the branching ratios of B c 2S ™q y. Ž Ž . 0 0. Ž Ž .Jrcp p , B c 2S ™Jrcp p , and B c 2S

.™Jrch , is not an independent, direct measure-ment, but is a value from a constrained fit to 7

Ž .branching ratios for the c 2S of 13 significantŽ Ž . q y.measurements including B c 2S ™ Jrcp p ,

Ž Ž . 0 0. Ž Ž . . w xB c 2S ™Jrcp p and B c 2S ™Jrch 4 ,which are what the authors attempt to measure. Onefinds here an apparent logical inconsistency.

A correct way to make the measurements self-consistent would be to solve a linear equation of theform

xsa xqb , 2Ž . Ž .Ž Ž . q y. Ž Ž .where x ' B c 2S ™ Jrcp p q B c 2S ™

0 0 . Ž Ž . .Jr c p p q B c 2 S ™ Jr c h , a '

´ N3 Jrc X f i Ž Ž . .Ý , and b'0.273B c 2S ™gx qis1 c1´ Nf i Jrc X

Ž Ž . .0.135B c 2S ™gx using the PDG data whichc2

Ž .are irrelevant to x. The solutions of Eq. 2 for xŽ Ž . .with the addition of b give B c 2S ™Jrc X s

0.315"0.226 and 0.292"0.136 for the two dataŽ .sets 1990 and 1991 of the E760 experiment, respec-

Ž .tively, which can now be applied to Eq. 1 consis-tently. Table 1 gives the computed branching ratiosfor three exclusive decays. The same results can beachieved by using an iterative method of solving Eq.Ž .2 . However, the results thus obtained are muchworse than the PDG world averages or fit values.Here the substantial errors are mainly due to the

1enlargement of the uncertainty by a factor of f9.1y a

Such an approach has practically nothing to recom-mend it.

We would thus propose that, as a natural andprobably the best way to deal with the E760 data, its

´ NJrc X fmeasured ratios be used directly as the´ Nf Jrc X

Ž Ž .ratios of two branching ratios, G c 2S ™. Ž Ž . . Ž q y 0 0 .f rG c 2S ™Jrc X fsp p , p p or h .

With these new entries, in addition to the previousw x13 measurements 4 , an overall fit as used by the

PDG is redone with minor simplification 3. In order

3 We ignore the one constraint and fit to determine 5 parame-ters without the other fit modes. A check using 13 measurements

w xcontained in Ref. 4 shows no essential difference in fit quality,results and errors between the PDG fit and our simplification,

Ž Ž . .except for B c 2S ™ Jrch s0.027"0.003 obtained by usw xcompared with 0.027"0.004 in Ref. 4 .

( )Y.F. Gu, X.H. LirPhysics Letters B 449 1999 361–363 363

to make the input data of E760 independent, twouncorrelated ratios presented in Table 2,

G c 2S ™Jrcp 0p 0 rG c 2S ™JrcpqpyŽ . Ž .Ž .Ž .´ q y N 0 0Jrcp p Jrcp p

s 3Ž .0 0 q y´ NJrcp p Jrcp p

and

G c 2S ™Jrch rG c 2S ™Jrc XŽ . Ž .Ž . Ž .´ NJrc X Jrch

s , 4Ž .´ NJrch Jrc X

are in fact used in the fit instead of three ratios ofŽ Ž . . Ž Ž . .ŽG c 2 S ™ Jrc f rG c 2 S ™ Jrc X f s

q y 0 0 .p p , p p and h . Table 3 summarizes the fitw xresults. The 1996 PDG fit values 4 and the E760

w xreporting values 1 are also included for comparison.Using the combined fit values in Table 3, along with

Ž . Žthe 1998 PDG averages for B x ™g Jrc , B xc1 c2. Ž 0 . Ž . 4™g Jrc , B p ™gg , B h™neutral modes ,

Ž q y. 5and B Jrc™e e , one may also compute

4 Ž .We use the new fit value for B h™ neutral modes given inw xRef. 1 , 0.715"0.006, instead of the previous fit value, 0.708"

Ž Ž .0.008, which is still used by the PDG in calculating B c 2S ™. w xJrc neutral modes in Ref. 4 .

5 Ž q y .We use the new fit value for B Jrc ™ e e given in Ref.w x w x Ž . y21 or 4 , 6.02"0.19 =10 , instead of the previous fit value,Ž . y25.99"0.25 =10 , which is still used by the E760 in calculat-

Ž Ž . q y . w xing B c 2S ™ e e in Ref. 2 .

Ž Ž . . Ž Ž .B c 2S ™Jrc X to be 0.57"0.04, B c 2S ™. Ž Ž .Jrc neutrals to be 0.235"0.026, and B c 2S ™

q y. Ž . y3e e to be 8.4"0.8 =10 .We do not recommend using the 1998 PDG fit

Ž .values of branching ratios for the c 2S decays tow x Ž .Jrc and anything 1 . The ratio of the two c 2S

Ž Ž . q y.decay partial widths, B c 2S ™ Jrcp p rŽ Ž . q y.B c 2S ™m m , measured by the E672rE706

w x Ž Ž .experiment 3 was mistaken for B c 2S ™q y. Ž Ž . q y. w xJrcp p rB c 2S ™Jrcm m in Ref. 1 . In

addition, the E760 data in their present form areinappropriate for such a fit by the above-mentionedargument.

Acknowledgements

The authors wish to thank their colleagues on theBES collaboration for useful discussions. This workwas supported by the National Natural ScienceFoundation of China under Contract No. 19290400,the Chinese Academy of Sciences under ContractNo. K10, the U.S. Department of Energy underContract No. DE-AC03-76SF00098.

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11 March 1999


0 ž / ž / )0 ž /Measurement of inclusive r , f 980 , f 1270 , K 14300 2 2X ž / 0and f 1525 production in Z decays2

DELPHI Collaboration

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1 On leave of absence from IHEP Serpukhov.


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2 Now at University of Florida.

( )P. Abreu et al.rPhysics Letters B 449 1999 364–382366

F. Mazzucato ai, M. Mazzucato ai, M. Mc Cubbin v, R. Mc Kay a, R. Mc Nulty v,G. Mc Pherson v, C. Meroni aa, W.T. Meyer a, A. Miagkov ap, E. Migliore as,

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C. Vollmer q, G. Voulgaris c, V. Vrba l, H. Wahlen az, C. Walck ar, C. Weiser q,D. Wicke az, J.H. Wickens b, G.R. Wilkinson i, M. Winter j, M. Witek r, G. Wolf i,

J. Yi a, O. Yushchenko ap, A. Zalewska r, P. Zalewski ay, D. Zavrtanik aq,E. Zevgolatakos k, N.I. Zimin p,x, G.C. Zucchelli ar, G. Zumerle ai

a Department of Physics and Astronomy, Iowa State UniÕersity, Ames, IA 50011-3160, USAb Physics Department, UniÕ. Instelling Antwerpen, UniÕersiteitsplein 1, BE-2610 Wilrijk, Belgium,

and IIHE, ULB-VUB, Pleinlaan 2, BE-1050 Brussels, Belgium,and Faculte des Sciences, UniÕ. de l’Etat Mons, AÕ. Maistriau 19, BE-7000 Mons, Belgium´

c Physics Laboratory, UniÕersity of Athens, Solonos Str. 104, GR-10680 Athens, Greeced Department of Physics, UniÕersity of Bergen, Allegaten 55, NO-5007 Bergen, Norway´

e Dipartimento di Fisica, UniÕersita di Bologna and INFN, Via Irnerio 46, IT-40126 Bologna, Italy`f Centro Brasileiro de Pesquisas Fısicas, rua XaÕier Sigaud 150, BR-22290 Rio de Janeiro, Brazil,´

and Depto. de Fısica, Pont. UniÕ. Catolica, C.P. 38071 BR-22453 Rio de Janeiro, Brazil,´ ánd Inst. de Fısica, UniÕ. Estadual do Rio de Janeiro, rua Sao Francisco XaÕier 524, Rio de Janeiro, Brazil´ ˜

g Comenius UniÕersity, Faculty of Mathematics and Physics, Mlynska Dolina, SK-84215 BratislaÕa, SloÕakiah College de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, FR-75231 Paris Cedex 05, France`

i CERN, CH-1211 GeneÕa 23, Switzerlandj Institut de Recherches Subatomiques, IN2P3 - CNRSrULP - BP20, FR-67037 Strasbourg Cedex, France

k Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greecel FZU, Inst. of Phys. of the C.A.S. High Energy Physics DiÕision, Na SloÕance 2, CZ-180 40 Praha 8, Czech Republic

m Dipartimento di Fisica, UniÕersita di GenoÕa and INFN, Via Dodecaneso 33, IT-16146 GenoÕa, Italy`n Institut des Sciences Nucleaires, IN2P3-CNRS, UniÕersite de Grenoble 1, FR-38026 Grenoble Cedex, France´ ´

o Helsinki Institute of Physics, HIP, P.O. Box 9, FI-00014 Helsinki, Finlandp Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, RU-101 000 Moscow, Russian Federation

q Institut fur Experimentelle Kernphysik, UniÕersitat Karlsruhe, Postfach 6980, DE-76128 Karlsruhe, Germany¨ ¨r Institute of Nuclear Physics and UniÕersity of Mining and Metalurgy, Ul. Kawiory 26a, PL-30055 Krakow, Polands UniÕersite de Paris-Sud, Lab. de l’Accelerateur Lineaire, IN2P3-CNRS, Bat. 200, FR-91405 Orsay Cedex, France´ ´ ´ ´ ˆ

t School of Physics and Chemistry, UniÕersity of Lancaster, Lancaster LA1 4YB, UKu LIP, IST, FCUL - AÕ. Elias Garcia, 14-1o, PT-1000 Lisboa Codex, Portugal

v Department of Physics, UniÕersity of LiÕerpool, P.O. Box 147, LiÕerpool L69 3BX, UKw ( )LPNHE, IN2P3-CNRS, UniÕ. Paris VI et VII, Tour 33 RdC , 4 place Jussieu, FR-75252 Paris Cedex 05, France

x Department of Physics, UniÕersity of Lund, SolÕegatan 14, SE-223 63 Lund, Swedenÿ UniÕersite Claude Bernard de Lyon, IPNL, IN2P3-CNRS, FR-69622 Villeurbanne Cedex, France´

z UniÕ. d’Aix - Marseille II - CPP, IN2P3-CNRS, FR-13288 Marseille Cedex 09, Franceaa Dipartimento di Fisica, UniÕersita di Milano and INFN, Via Celoria 16, IT-20133 Milan, Italy`

ab Niels Bohr Institute, BlegdamsÕej 17, DK-2100 Copenhagen Ø, Denmarkac NC, Nuclear Centre of MFF, Charles UniÕersity, Areal MFF, V HolesoÕickach 2, CZ-180 00 Praha 8, Czech Republic

ad NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlandsae National Technical UniÕersity, Physics Department, Zografou Campus, GR-15773 Athens, Greece

af Physics Department, UniÕersity of Oslo, Blindern, NO-1000 Oslo 3, Norwayag Dpto. Fisica, UniÕ. OÕiedo, AÕda. CalÕo Sotelo srn, ES-33007 OÕiedo, Spain

ah Department of Physics, UniÕersity of Oxford, Keble Road, Oxford OX1 3RH, UKai Dipartimento di Fisica, UniÕersita di PadoÕa and INFN, Via Marzolo 8, IT-35131 Padua, Italy`

aj Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UKak Dipartimento di Fisica, UniÕersita di Roma II and INFN, Tor Vergata, IT-00173 Rome, Italy`

al Dipartimento di Fisica, UniÕersita di Roma III and INFN, Via della Vasca NaÕale 84, IT-00146 Rome, Italyàm DAPNIArSerÕice de Physique des Particules, CEA-Saclay, FR-91191 Gif-sur-YÕette Cedex, France

an ( )Instituto de Fisica de Cantabria CSIC-UC , AÕda. los Castros srn, ES-39006 Santander, Spainao Dipartimento di Fisica, UniÕersita degli Studi di Roma La Sapienza, Piazzale Aldo Moro 2, IT-00185 Rome, Italy`

ap ( )Inst. for High Energy Physics, SerpukoÕ P.O. Box 35, ProtÕino Moscow Region , Russian Federationaq J. Stefan Institute, JamoÕa 39, SI-1000 Ljubljana, SloÕenia

and Laboratory for Astroparticle Physics, NoÕa Gorica Polytechnic, KostanjeÕiska 16a, SI-5000 NoÕa Gorica, SloÕenia,and Department of Physics, UniÕersity of Ljubljana, SI-1000 Ljubljana, SloÕenia

ar Fysikum, Stockholm UniÕersity, Box 6730, SE-113 85 Stockholm, Swedenas Dipartimento di Fisica Sperimentale, UniÕersita di Torino and INFN, Via P. Giuria 1, IT-10125 Turin, Italy`


at Dipartimento di Fisica, UniÕersita di Trieste and INFN, Via A. Valerio 2, IT-34127 Trieste, Italy,ànd Istituto di Fisica, UniÕersita di Udine, IT-33100 Udine, Italy`

au UniÕ. Federal do Rio de Janeiro, C.P. 68528 Cidade UniÕ., Ilha do Fundao, BR-21945-970 Rio de Janeiro, Brazilãv Department of Radiation Sciences, UniÕersity of Uppsala, P.O. Box 535, SE-751 21 Uppsala, Sweden

aw ( )IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, AÕda. Dr. Moliner 50, ES-46100 Burjassot Valencia , Spainax Ïnstitut fur Hochenergiephysik, Osterr. Akad. d. Wissensch., Nikolsdorfergasse 18, AT-1050 Vienna, Austria¨

ay Inst. Nuclear Studies and UniÕersity of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Polandaz Fachbereich Physik, UniÕersity of Wuppertal, Postfach 100 127, DE-42097 Wuppertal, Germany


Abstract

0 Ž . Ž . )0Ž .DELPHI results are presented on the inclusive production of the neutral mesons r , f 980 , f 1270 , K 1430 and0 2 2XŽ . 0f 1525 in hadronic Z decays. They are based on about 2 million multihadronic events collected in 1994 and 1995, using2

the particle identification capabilities of the DELPHI Ring Imaging Cherenkov detectors and measured ionization losses inthe Time Projection Chamber. The total production rates per hadronic Z0 decay have been determined to be: 1.19"0.10 for

0 Ž . Ž . )0Ž . XŽ .r ; 0.164"0.021 for f 980 ; 0.214"0.038 for f 1270 ; 0.073"0.023 for K 1430 ; and 0.012"0.006 for f 1525 .0 2 2 20 Ž . Ž .The total production rates for all mesons and differential cross-sections for the r , f 980 and f 1270 are compared with0 2

the results of other LEP experiments and with models. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

The production of several orbitally excited mesonsŽ . Ž .w x "Ž .w xsuch as f 980 and f 1270 1,2 , a 980 3 ,0 2 0

) 0Ž .w x XŽ .w xK 1430 4,5 and f 1525 6 has been measured2 2

by DELPHI and OPAL using the large statisticsaccumulated by these experiments at the Z0 peak. Asignificant rate of production of Ls1 excitedmesons in the hadronization was clearly established.Orbitally and radially excited mesons in the heavyquark sector were also observed by the LEP experi-

w xments 7–13 to be produced with significant rates.The results obtained on the production of orbitally

excited mesons in the light quark sector have usuallyw x w xbeen compared with the string 14 or cluster 15

models implemented in the QCD-based Monte Carlow x w xgenerators JETSET 16 and HERWIG 17 respec-

tively. In most cases, after proper tuning of a numberof adjustable parameters, a reasonable description ofthe experimental data was obtained, thus allowinguseful information to be obtained about the nature of

Ž w x.the fragmentation process see, for example, 18 .However in some cases a significant disagreement

w xwith these models was observed 5 . This is not verysurprising, since the underlying physics of hadroniza-

tion is not fully understood and such models cannotsupply sufficiently reliable guidance on possible dif-ferences in production mechanisms of differentmesons and baryons or on their dependences on spinand orbital momentum dynamics. Studies of theproduction properties of the orbitally excited statesare thus of special interest in view of the possiblydifferent dynamics of their production.

This paper describes new DELPHI measurements0 Ž . Ž . ) 0Ž . 3 XŽ .of r , f 980 , f 1270 , K 1430 and f 15250 2 2 2

production in Z0 hadronic decays at LEP1. Theprevious DELPHI results on the inclusive production

0 Ž . Ž .properties of the r , f 980 and f 1270 mesons0 2w x1 were based on data collected in 1991 and 1992and were obtained without the use of particle identi-fication. The previous DELPHI results on the

) 0Ž . XŽ . w xK 1430 and f 1525 production 5,6 were ob-2 2

tained using the 1994 data sample, with particleidentification coming from the RICH detectors only.The present results, superseding the previous DEL-PHI measurements, are based on a data sample of 2million hadronic Z0 decays collected during 1994

3 Unless otherwise stated, antiparticles are implicitly included.


and 1995 and make use of the particle identificationcapabilities provided by the Ring Imaging CherenkovŽ .RICH detectors and by measured ionization losses

Ž .d Erd x in the Time Projection Chamber TPC .

2. Experimental procedure

2.1. EÕent and particle selection

Detailed descriptions of the DELPHI detector andw xits performance can be found elsewhere 19,20 .

The charged particle tracks were measured in the1.2 T magnetic field by a set of tracking detectors.The average momentum resolution for charged parti-cles in hadronic final states, D prp, was usuallybetween 0.001 and 0.01, depending on which detec-tors were included in the track fit.

A charged particle was accepted in this analysis ifits momentum, p, was greater than 140 MeVrc, itsmomentum error, D p, was less than p, its polarangle with respect to the beam axis was between 258

and 1558, its measured track length in the TPC wasgreater than 50 cm, and its impact parameter withrespect to the nominal crossing point was within 5

Ž .cm in the transverse xy plane and 10 cm along theŽ .beam direction z-axis .

Hadronic events were then selected by requiringat least 5 charged particles, with total energy of thecharged particles greater than 15 GeV and at least 3GeV in each hemisphere of the event, defined withrespect to the beam direction. In addition, the polarangle of the sphericity axis was required to liebetween 408 and 1408.

The sample selected with the above cuts consistedof 1.13 million events. The contamination fromevents due to beam-gas scattering and to gg interac-tions was estimated to be less than 0.1% and thebackground from tqty events less than 0.2% of thetotal number accepted.

After the event selection, in order to ensure abetter signal-to-background ratio for the resonancesin the pqpy, Kqpy and KqKy invariant massspectra, tighter requirements were imposed on thetrack impact parameters with respect to the nominalcrossing point: they had to be within 0.3 cm in thetransverse plane and 2 cm along the beam direction.

Charged particles were used only from the barrelregion of the detector and were further required tohave hits in the Vertex Detector. Any particle identi-fied by the RICH was required to have a tracksegment in the Outer Detector.

Charged particle identification was provided bythe barrel RICH detectors for particles with momen-tum above 700 MeVrc, while the ionization lossmeasured in the TPC could be used for momentaabove 100 MeVrc. The corresponding identificationtags were based on the combined probabilities de-rived from the measured average Cherenkov angleand the number of observed photons in the RICH,and from the measured d Erd x in the TPC. Tightcuts were applied to achieve the highest possible

Ž w xidentification purity see 21 and references thereinwhere further details of particle identification rou-

.tines can be found . The identification performancewas evaluated by means of the detector simulation

w xprogram DELSIM 20 . In DELSIM, about 3 millionhadronic decays of the Z0 satisfying the same selec-tion criteria as the real data were produced using the

w xJETSET generator 16 with the DELPHI defaultw xparameters 18 obtained before the measurements

reported in this paper. Subsequent references to JET-SET always mean this tuning, which is described in

w xdetail in Ref. 18 . The particles were followedthrough the detector, and the simulated digitizationsobtained were processed with the same reconstruc-tion programs as the experimental data. Good agree-ment between the data and simulation was observed.

2.2. Fit procedure and treatment of detector re-sponse

Particle identification inefficiencies, detector im-perfections such as the limited geometrical accep-tance and electronic inefficiencies, particle interac-tions in the detector material, and the different kine-matical cuts imposed for charged particle and eventselection, were accounted for by applying the ap-

w xproach first described in Ref. 1 , developed in Refs.w x5,22,23 and outlined in brief below.

In the present analysis, a vector a of parameterswas used in the definition of the anticipated distribu-

Ž .tion function, f M,a , of the invariant mass M. Theparameters a were then determined by a least squaresfit of the function to the data.


Ž .The function f M,a was composed of threeparts:

f M ,a s f S M ,a q f B M ,a q f R M ,a , 1Ž . Ž . Ž . Ž . Ž .corresponding to the signal, background, and reflec-tion contributions respectively.

SŽ .The signal function, f M,a , described the reso-nance signals in the corresponding invariant massdistributions. For the pqpy mass distributions ithad the form

f S M ,a sa PS 0 M PBW 0 M ,a ,aŽ . Ž . Ž .1 r r 2 3

qa PS M PBW M ,a ,aŽ . Ž .4 f f 5 60 0

qa PS M PBW M ,a ,a , 2Ž . Ž . Ž .7 f f 8 92 2

where the relativistic Breit–Wigner functions BW0 Ž . Ž .for the r , f 980 and f 1270 are multiplied by0 2

Ž .the functions PS M to account for the distortion ofthe resonance Breit–Wigner shapes by phase space

Ž w x . q yeffects see 1 for details. For each of the K p

and KqKy mass distributions only one Breit–) 0Ž .Wigner term, representing the K 1430 and2

XŽ . SŽ .f 1525 respectively, contributed to f M,a .2BŽ .The background term, f M,a , was taken to be

of the form

f B M ,a sBG M PP M ,a , 3Ž . Ž . Ž . Ž .Jetset

Ž .where BG M represented the background shapeJetsetŽgenerated by JETSET presumed to describe the

. Ž .gross features of the real background and P M,as 1 q a M q a M 2 q a M 3 q a M 4 was a10 11 12 13

Ž .polynomial of order 4 or sometimes of order 3introduced to account for possible deviations of

Ž .BG M from the real background. All pairs ofJetset

charged particles which do not come from the reso-nances considered and reflections in the invariantmass spectra were included in the definition of

Ž .BG M . This parameterization of the back-Jetset

ground was different from the analytical form usedw xin a previous DELPHI analysis 1,5,22,23 .

RŽ .The third term, f M,a , represented the sum ofŽ .all the reflection functions RF :i

isnRf M ,a s a RF M , 4Ž . Ž . Ž .Ý i i

is14

with different numbers n of the reflection functionsfor each of the mass distributions under considera-

tion. Two types of reflection function contributing toŽ .Eq. 4 were considered. Reflections of the first type

arise from particle misidentification, for examplewhen resonances in the Kqpy and KqKy systemsdistort the pqpy mass spectra. Due to the efficientparticle identification of the combined RICH andTPC tags and to the high identification purity pro-vided by the tight cuts, the influence of reflections ofthis type was found to be much smaller than in the

w xprevious DELPHI analysis 1 , which was performedwithout particle identification. Reflections of the sec-ond type arise from resonances and particles decay-ing in the same system, for example from K 0 ™S

pqpy or v™pqpyX in the pqpy mass spectra,or from charmed particle production. The reflectionsfrom charmed particle decays are of special impor-tance for the tensor mesons, as discussed in Section3.

Ž . Ž .The functions RF M in Eq. 4 were determinedi

from events generated according to the JETSETmodel. The contributions of the reflections to the raw

RŽ . Žmass spectra defined by the function N a see Eq.mŽ . .5 below were then obtained by passing theseevents through the detector simulation. This alsotook proper account of the influence of particlemisidentification.

Ž .In each mass bin, m, the number of entries N amŽ .predicted by the function f M,a , representing a

sum of contributions from the resonance signals,Ž w x.background and reflections see 23 , is given by

GN a sC S A f a , 5Ž . Ž . Ž .Ým m m n n nn

Mnq1f a s f M ,a d M , 6Ž . Ž . Ž .Hn

Mn

where GsS, B or R, and M is the lower edge ofn

the n-th histogram bin in the distribution of thevariable M. The coefficients A characterize then

detector acceptance and the losses of particles due tothe selection criteria imposed, and the C take intom

account the contamination of the sample by particlesfrom V 0 decays, wrongly associated charged parti-cles, secondary interactions, etc. The smearing ma-trix S represents the experimental resolution. Them n

A , C and S were estimated separately for then m m n


resonance signals, background and reflection contri-butions using the detector simulation program DEL-SIM. Due to differences in the detector performanceand data processing in different running periods, theevents generated by DELSIM for these periods weretaken with weights corresponding to the relativenumber of events in the real data. The distortion ofthe smearing matrix by residual Bose–Einstein corre-lations was also accounted for by means of the

w xprocedure described in Ref. 23 .The best values for a were then determined by a

Ž .least squares fit of the predictions of Eq. 5 to themeasured values, N , by minimizing the functionm

22 2x s N yN a rsŽ .Ž .Ý m m mm

22q a ya r Da , 7Ž .Ž . Ž .Ý i i ii

2 2Ž . Ž .where s sN qs N and s N is the errorm m m m

on N due to the finite statistics of the simulationm

used to evaluate A , C and S . The second sum inn m m nŽ .Eq. 7 constrains some of the fitted parameters a toi

the values a "Da taken from external sources,i i

such as the normalization of the reflection functionsto the particle production rates taken from this andother LEP experiments, and the masses and widths

w xtaken from the PDG tables 24 . The errors obtainedfrom the fits thus include the corresponding system-atic components.

The fits were made in the mass ranges from 0.3 to1.8 GeVrc2 for the pqpy, from 1.1 to 2.1 GeVrc2

for the Kqpy and from 1.2 to 2.2 GeVrc2 for theKqKy mass spectra.

The resonance production rates were calculated as

1 1S² :N s f M ,a d M , 8Ž . Ž .H² :Br R

Žwhere the factor 1rBr with the branching ratios, Br,w x.from 24 takes into account the unobserved decay

modes and the integration limits are the same as the² :fit ranges. The factor R , which is almost indepen-

dent of the mass M, takes account of the imperfec-tion of the detector simulation when the stronger cuts

Ž w xon impact parameters are applied see Ref. 1,23 for.details . It is very close to unity.

3. Results

0 ( ) ( )3.1. r , f 980 and f 1270 production0 2

The measured raw pqpy invariant mass dis-tributions are shown for the individual x spŽ q y.p p p rp intervals in Fig. 1 together with thebeam

0 Ž .results of the fits. The r and f 980 resonance0

signals are clearly seen in all x intervals. ThepŽ .relatively broad f 1270 resonance is only just visi-2

ble in the pqpy spectra for x F0.4 but is clearerp

after subtracting the background and reflection con-tributions.

The contribution of reflections is also shown inFig. 1. As discussed in the previous section, goodparticle identification reduces the reflection resultingfrom particle misidentification to a very low level. Inparticular, it is seen from Fig. 1 that the reflection

) 0Ž . 0from the K 892 under the r signal is almostŽ .negligible about 2–3% . This is in stark contrast

with the previous DELPHI analysis of 1991 andw x1992 data 1 , performed without the use of particle

) 0Ž .identification, where the K 892 reflection contri-bution resulted in a strong peak in the r 0 massregion, comparable in magnitude with the r 0 signal.

The dominant contribution of the reflections isdue to resonances and particles decaying into thepqpyX systems. Their influence is mainly concen-

0 Ž .trated in the low mass region. In the r and f 9800

mass regions, the contribution of reflections is rela-tively small, their mass dependence is rather smoothand therefore they do not distort the resonance sig-nals in a significant way. However this is not the

Ž .case for the f 1270 for x G0.2, where the reflec-2 p

tions from the quasi-two-body D0 decays, such as0 )yŽ . q q 0D ™K 892 p , with the p from the D decay

and py from K )y forming the pqpy system,Ž .give a large contribution exactly in the f 12702

mass region. The influence of these reflections wasaccounted for as discussed in Section 2.2. In addi-tion, possible systematic uncertainties for theŽ .f 1270 for x G0.2 arising from these reflections2 p

Žwere accounted for in the systematic errors see.Section 3.4 .

0 Ž . Ž .In the fits, the r , f 980 and f 1270 masses0 20 Ž .and the r and f 1270 widths were constrained by2

Ž .the second term in Eq. 7 using the PDG valuesw x Ž . 224 . The f 980 width was fixed at 50 MeVrc . As0


Fig. 1. The pqpy invariant mass spectra for various x ranges as indicated. Each plot consists of an upper and lower part. In the upperp

part: the raw data are given by the open points; the upper histogram is the result of the fit; the lower histogram is the sum of the backgroundand reflection contributions. In the lower part: the open points represent the data after subtraction of the background and reflections; the

0 Ž . Ž .histograms show the contribution of reflections and result of the fit for the r , f 980 and f 1270 contributions. The histograms in the0 2

lower part are multiplied by the factor indicated.

can be seen from Fig. 1 and Table 1, the fits describethe data very well in all measured x intervals, apartp

from the lowest x region, where x 2rndff2 for 44p0 Ž . Ž .degrees of freedom. The r , f 980 and f 12700 2

Ž .differential production cross-sections, 1rs Ph

dsrd x , where s is the total hadronic cross-sec-p h

tion, are presented in Table 1 and Fig. 2.The rather high value of x 2rndf in the lowest x p

region, comes mainly from a few isolated bad pointsand reflects difficulties in extracting resonance rates

at low momenta. Partly this is due to a poor determi-nation of the opening angle between the low momen-tum particles and to the fact that a significant frac-tion of the particle pairs is contaminated by particlesfrom V 0 decays and secondary interactions and bywrongly associated charged particles. For x F0.05,p

the influence of the residual Bose–Einstein correla-tions, whose treatment in JETSET is not perfect,becomes very important. For these reasons, no at-tempt was made to measure meson resonance rates


Table 10 Ž . Ž . Ž .Differential r , f 980 and f 1270 cross-sections 1rs P0 2 h

ds rd x for the indicated x intervals. The errors obtained fromp p

the fits and the systematic errors are combined quadratically. Thecorresponding values of x 2rndf for the fits are also given

0 2Ž . Ž .x interval r f 980 f 1270 x rndfp 0 2

0.05–0.1 6.15"0.72 0.84"0.16 1.23"0.37 90r440.1–0.2 2.16"0.23 0.35"0.06 0.47"0.12 48r440.2–0.3 0.92"0.10 0.13"0.03 0.18"0.05 58r440.3–0.4 0.45"0.05 0.075"0.017 0.10"0.04 65r440.4–0.6 0.13"0.02 0.029"0.006 0.042"0.016 46r440.6–0.8 0.027"0.005 0.006"0.003 0.012"0.006 47r440.8–1.0 0.003"0.002 – – 31r46

below x s0.05 and thus this analysis is restrictedp

to x G0.05.p0 Ž . Ž .The measured r , f 980 and f 1270 rates per0 2

hadronic event in the x G0.05 range, obtained byp

integrating the x spectra, were determined to bep

² 0:r s0.692"0.034 fit 9Ž . Ž .x G 0.05p

² :f 980 s0.104"0.009 fit 10Ž . Ž . Ž .x G 0.050 p

² :f 1270 s0.148"0.022 fit , 11Ž . Ž . Ž .x G 0.052 p

where the errors were obtained from the fits and, asexplained in Section 2.2, include a systematic com-

Ž . Ž . 0 Ž . Ž . Ž . Ž .Fig. 2. Differential cross-sections 1rs dsrd x for inclusive a r , b f 980 and c f 1270 production, obtained with theh p 0 2Ž . Ž .1994–1995 data open points , in comparison with the previous DELPHI results based on 1991–1992 data triangles , ALEPH results for

0 Ž . Ž . Ž . Ž .the r squares and OPAL results for the f 980 and f 1270 stars . The curÕes represent the expectations of the tuned JETSET0 2

model.


Ž . Ž . Ž .ponent. The values 9 , 10 and 11 agree with thecorresponding values of 0.698"0.035, 0.102"

0.009 and 0.145"0.022, obtained by fitting theoverall mass spectrum in the x G0.05 range.p

) 0( )3.2. K 1430 production2

The measured raw Kqpy invariant mass distri-bution for x G0.04 is shown in Fig. 3 together withp

) 0Ž .the results of the fit. The small K 1430 signal is2

seen in the data and its contribution is well describedby the fit, with x 2rndf s 39r44. In the fit, the

) 0Ž .K 1430 mass and width were constrained by the2Ž . w xsecond term in Eq. 7 using the PDG values 24 .

As seen from Fig. 3, the overall contribution ofreflections, where charmed particle decays play thedominant role, is quite large. However their mass

) 0Ž .dependence in the K 1430 mass region is rather2

smooth and so they do not significantly distort the

resonance signal. Both the shape and the normaliza-tion of the reflections in the Kqpy mass spectrumare well reproduced by the fit. This is seen from avery good description of the sharp peak from thetwo-body D0 ™Kypq decay and of the broaderstructure centered around 1.62 GeV caused by the

0 )yŽ . q qquasi-two-body D ™K 892 p , with the p

from the D0 decay and Ky from K )y forming theKypq system. A fit with the contribution of the D0

reflection left free resulted in an overall D0 produc-tion rate of 0.392"0.044, consistent within errorswith the present average LEP value of 0.454"0.030w x24 . This strengthens our confidence in the result

) 0Ž .obtained. The K 1430 signal for x G0.04 shown2 p

in Fig. 3 corresponds to the production rate of

² ) 0 :K 1430 s0.060"0.018 fit 12Ž . Ž . Ž .x G 0.042 p

per hadronic event.

Fig. 3. The Kqpy invariant mass spectrum for x G0.04. In the upper part: the raw data are given by the open points; the upperp

histogram is the result of the fit; the lower histogram is the sum of the background and reflection contributions. In the lower part: the open) 0Ž .points represent the data after subtraction of the background and reflections; the full histogram is the result of the fit for the K 14302

contribution; the dashed histogram shows the contribution of reflections. The histograms in the lower part are multiplied by a factor of 5.


X( )3.3. f 1525 production2

The measured raw KqKy invariant mass distribu-tion for x G0.05, shown in Fig. 4, exhibits somep

structures around 1.5–1.6 and 1.6–1.75 GeVrc2. Asw xdiscussed in Ref. 6 , they could be due to the

XŽ . Ž .f 1525 and f 1700 . However, the structure around2 J

1.5–1.6 GeVrc2 is rather complicated, indicatingthat other states can possibly contribute to this massregion. Thus a contribution of the relatively narrowŽ .f 1500 , the Crystal Barrel candidate for the scalar0

w x Ž .glueball 25 , and of the tensor meson f 1565 ,2

revived recently in the analysis performed by thew xOBELIX collaboration 26 , cannot be excluded.

Fig. 4 shows that the contribution of reflections inthe mass range 1.40–1.75 GeVrc2 is quite signifi-cant, but with a mass dependence that is comfortablysmall. The reflections are found to be due mainly tocharmed particle decays in the KqKyX system.

However, contrary to the situation in the Kqpy

mass spectrum discussed in the previous section, the0 q y Ž Ž 0expected D ™ K K signal with G D ™

q y. Ž 0 y q. w x.K K rG D ™K p s0.113"0.006 24 issmall and poorly observed in the data. The largercontribution of this signal in the fit might be due toan overestimation of the background on account of

Ž 2resonances in the mass region from 1.4 GeVrc to2 .1.8 GeVrc as discussed above which were not

included in the fit.In this situation, a precise determination of the

XŽ .f 1525 production rate is rather difficult. As seen2

from Fig. 4, the fit of the KqKy mass spectrumXŽ .with the contribution of only one f 1525 reso-2

nance, performed in order to obtain a rough estimateof its rate, is not quite satisfactory in the mass regionbetween 1.45 and 1.9 GeVrc2, although the value ofx 2rndfs59r44 obtained for the full mass range

XŽ .shows that the fit is acceptable. The f 1525 signal2

Fig. 4. The KqKy invariant mass spectrum for x G0.05. In the upper part: the raw data are given by the open points; the upperp

histogram is the result of the fit; the lower histogram is the sum of the background and reflection contributions. In the lower part: the openX Ž .points represent the data after subtraction of the background and reflections; the full histogram is the result of the fit for the f 15252

contribution; the dashed histogram shows the contribution of reflections. The histograms in the lower part are multiplied by a factor of 5.


for x G0.05 shown in Fig. 4 corresponds to ap

production rate of

² X :f 1525 s0.0093"0.0038 fit 13Ž . Ž . Ž .x G 0.052 p

per hadronic event.

3.4. Systematic uncertainties

The systematic uncertainties were estimated in thew xsame way as in previous DELPHI analyses 5,23 by

determining the contributions arising from:1. variations of the charged particle selections;2. uncertainty in particle identification efficiencies;3. treatment of residual Bose–Einstein correlations;4. errors in the branching ratios assumed;5. overall normalization of reflections;6. assumption that the relative contribution of reflec-

tions in different x intervals, if not taken fromp

the LEP experiments, is the same as in JETSET;7. extrapolation procedure used for determination of

the total rate from that measured in the restrictedx range;p

8. uncertainty in the resonance line-shape, back-ground parameterization and choice of the binsize of the mass spectra and mass range used inthe fit.The contribution of the first four factors was

approximately the same for all resonances. The rela-Žtive systematic error from the first factor including

² : Ž ..uncertainty in the factor R in Eq. 8 , affectingmostly the overall normalization of the total rates,was found to be about "2%, significantly smallerthan in previous DELPHI analyses, reflecting a bet-ter understanding of the detector. The uncertainty inparticle identification efficiencies was estimated tobe around "3% as follows from a more detailed

w xanalysis given in Ref. 21 . This agrees with the) 0Ž .estimate obtained from the remaining K 892 re-

0 Žflection contribution under the r signal Section.3.1 . The systematic uncertainties arising from im-

perfect treatment of the residual Bose–Einstein cor-relations in JETSET is difficult to estimate. They

w xwere evaluated as in Ref. 1 by comparing theresonance rates obtained when the treatment ofBose–Einstein correlations was included in JETSETwith those obtained when they were ignored. Thisgave a rather small relative error of about "2%,because the lowest x region, where the residualp

Bose–Einstein correlations are expected to be mostsignificant, was not used in our analysis. The errors

Ž .in the branching ratios, Br, in Eq. 8 were takenw xfrom the PDG tables 24 and amounted to "2% for

Ž . ) 0Ž .the f 1270 , "2.4% for the K 1430 and "3.5%2 2XŽ .for the f 1525 .2

The overall normalization of reflections and theirŽrelative contributions in different x intervals fac-p

.tors 5 and 6 were accounted for by normalizing thecontributions of the different reflections to the corre-sponding production rates measured at this and otherLEP experiments and by using the constraints in the

Ž .second term in Eq. 7 . Their uncertainties are thusincluded in the errors obtained from the fit. Therelative contributions of reflections in the differentx intervals, if not measured, were taken from JET-p

SET. This may result in additional systematic uncer-tainties for the differential cross-sections. Since JET-

0 Ž .SET describes the shape of the r , f 980 and0Ž . Ž .f 1270 momenta spectra very well Fig. 2 , the2

corresponding relative systematic errors are small.However, in view of the significant contribution ofthe reflections from the quasi-two-body D0 decays in

Ž .the f 1270 mass region and some difference be-2

tween Monte Carlo modelling of the pqpy massspectrum from charmed particle decays and theDELPHI data, systematic errors of "10% and "15%

Ž .were assigned to the f 1270 rates in the 0.2-x -2 p

0.4 and 0.4-x -0.8 regions respectively. ThispŽ .gave a relative error of "3% for the total f 12702

rate. No additional systematic uncertainty due to thetreatment of reflections was found to be necessary

) 0Ž .for the K 1430 . In contrast, an error of "10%2XŽ .was assigned to the f 1525 total rate in view of2

some discrepancy between the JETSET expectationand the data for the D0 ™KqKy decay, thus indi-cating possible biases in the calculated reflectioncontributions to the KqKy mass spectrum.

0 Ž . Ž . ) 0Ž .The overall r , f 980 , f 1270 , K 14300 2 2XŽ .and f 1525 rates in the full x range were ob-2 p

Ž . Ž .tained from 9 – 13 by normalizing the JETSETexpected rates in the x ranges under considerationp

to the data measurements in the same ranges andthen taking the overall rates from the correspondingJETSET predictions. Good agreement between the

0 Ž . Ž .measured r , f 980 and f 1270 x -spectra and0 2 pŽ .JETSET predictions Fig. 2 allowed the extrapola-

tion error to be taken as "10% of the difference


between the extrapolated and measured values. Thisgave systematic errors of "4% for the r 0 andŽ . Ž .f 980 , and "3% for the f 1270 total rates. Simi-0 2

larly, a systematic error of "2% was assigned to the) 0Ž . XŽ .K 1430 and f 1525 total rates, with the as-2 2

sumption that JETSET describes the shapes of theirx -spectra equally well.p

The last factor accounts for uncertainties in theresonance parameterizations and fits, apart from thevariation of resonance masses and widths above andbelow their nominal values taken from the PDGtables and accounted for in the errors on the fits 4.The influence of variations of the bin size of themass spectra and of the mass range used in the fit on

0 Ž . Ž . ) 0Ž .the total r , f 980 , f 1270 and K 1430 rates0 2 2

was found to be small. Variations of the backgroundparameterization, using different polynomialsŽ . Ž .P M,a in the background term 3 , also had negli-

gible effects on the total rates. However, the influ-ence of these two factors was found to be more

XŽ .significant for the f 1525 and resulted in a system-2

atic error of "11% for its total rate. Systematiceffects in the resonance parameterization and uncer-tainties in the line-shape of resonances far from thepole position, gave an error of "3% to the r 0 totalrate. It was increased to "5% in view of possibleinterference between the fitted resonances or reso-nances and background, not accounted for in ouranalysis. This error was increased to "7% for theŽ . Ž .f 980 and f 1270 , and to "10% for the0 2) 0Ž .K 1430 , in view of the small rates and low2

signal-to-background ratios for these resonances andŽ .due to a significant coupling of the f 980 to KK0

below threshold. The corresponding error for theXŽ .f 1525 , including the above mentioned "11%, was2

increased to "20% because of the rather compli-cated structure of the KqKy mass distribution in the

XŽ .f 1525 mass region.2

The overall systematic uncertainties for the reso-nance total rates not accounted for in the errors onthe fits were therefore estimated to be "7.1% for the

0 Ž . Ž .r , "9.1% for the f 980 , "9.4% for the f 1270 ,0 2) 0Ž ."11.3% for the K 1430 and "23% for the2

4 Ž .This does not apply to the f 980 , with the width fixed at 500

MeV and for which the results are therefore model-dependent, inw xview of the uncertainty on its width 24 .

XŽ .f 1525 . The correctness of these estimates of the2

systematic uncertainties can be assessed to someextent by comparing the present and previous DEL-

Ž .PHI results see next section , obtained using differ-Ž 0 Ž .ent data samples especially for the r , f 980 and0

Ž ..f 1270 and with a different method. Such a com-2

parison shows that the above estimates of the sys-tematic errors are quite reasonable.

0 Ž . Ž . ) 0Ž .The overall r , f 980 , f 1270 , K 14300 2 2XŽ .and f 1525 rates in the full x range, obtained2 pŽ . Ž .from 9 – 13 by applying the extrapolation proce-

dure just described, were

² 0:r s1.192"0.059 fit "0.085 syst 14Ž . Ž . Ž .² :f 980 s0.164"0.015 fit "0.015 systŽ . Ž . Ž .0

15Ž .² :f 1270 s0.214"0.032 fit "0.020 systŽ . Ž . Ž .2

16Ž .

² ) 0 :K 1430 s0.073"0.022 fit "0.008 systŽ . Ž . Ž .2

17Ž .² X :f 1525 s0.012"0.005 fit "0.003 syst ,Ž . Ž . Ž .2

18Ž .

where the second errors represent our estimates ofthe systematic uncertainties.

4. Discussion

0 Ž . Ž . Ž .The total r , f 980 and f 1270 rates 14–160 2

can be compared with the previous values of 1.21"

0.15, 0.140"0.034 and 0.243"0.062 respectively,w xdetermined by DELPHI 1 from the 1991 and 1992

data samples without the use of particle identifica-tion 5. The corresponding differential cross-sections,Ž .1rs Pdsrd x , for these two data sets are alsoh p

compared in Fig. 2. In general, the agreement be-tween the old and new results, both for the total ratesand for the x -spectra, is very satisfactory. Thisp

shows that the rather complicated procedure of ac-

5 These rates are obtained from the values measured in thew xrestricted x ranges 1 using the same extrapolation procedure asp

in the present paper.


counting for the significant reflections, which wasused in this paper and was most essential for thereliable determination of the r 0 rate in the previous

w xDELPHI analysis 1 without the use of particleidentification, was basically correct. The largest dif-ference between the differential cross-sections in thepresent and previous analyses is observed for theŽ .f 1270 at the largest x values. This is understand-2 p

) 0Ž .able, since the reflections from the K 1430 and2

D0, most significant at large x values, were notpw xaccounted for in Ref. 1 .

Ž . 0The DELPHI result 14 on the total r rateagrees within errors with the value of 1.45"0.21

w xmeasured by ALEPH 27 . The x -spectra measuredp

by the two experiments are also consistent with eachŽ .other Fig. 2a , although the x -spectrum measuredp

by ALEPH appears to be slightly harder than that0 Ž .measured by DELPHI. The total r rate 14 can

also be compared with the rate 2.40"0.43 of their" w xisospin partners r recently measured by OPAL 3 .

² 0: ² ":The ratio of the rates, 2 r r r s0.99"0.20,is close to unity, as expected.

Ž . Ž . Ž .The total f 980 and f 1270 rates, 15 and0 2Ž . w x16 , can be compared with the OPAL values 2 of0.141"0.013 and 0.155"0.021 respectively. The

Ž .DELPHI and OPAL results on the f 980 total rate0Ž .agree quite well. This is also true for the f 9800

Ž . Ž .x -spectra Fig. 2b . The f 1270 x -spectra mea-p 2 pŽsured by DELPHI and OPAL agree in shape Fig.

.2c but differ in the absolute normalization, reflect-ing the difference in the respective total rates of 1.3standard deviations.

0 Ž .Fig. 2 presents a comparison of the r , f 9800Ž .and f 1270 x -spectra with the expectations of the2 p

w xtuned JETSET model. The tuning 18 was madebefore this measurement, but using the previous

0 Ž . Ž .DELPHI results on the r , f 980 and f 1270 .0 2

Since the previous and present results are very closeto each other, good agreement of the tuned JETSETmodel with the present DELPHI data is not surpris-ing. It is still worth noting the good description of

0 Ž . Ž .the r , f 980 and f 1270 x shapes by JETSET.0 2 p0 Ž . Ž .The shapes of the r , f 980 and f 1270 x -spec-0 2 p

tra for x F0.4 appear to be approximately thep

same. For x )0.4, there is some indication that thepŽ . Ž .f 980 and especially the f 1270 x -spectra are0 2 p

harder than the r 0 x -spectrum. This is seen frompŽ . 0 Ž . 0Fig. 5, where the ratios f 980 rr and f 1270 rr0 2

are shown as a function of x . The observed in-p

crease of these ratios with x is consistent with thep

JETSET expectations.) 0Ž . Ž .The total K 1430 production rate 17 agrees,2

within errors, with our previous result of 0.079"w x0.040 5 , obtained on a smaller data sample and

with particle identification by the RICH only. It isalso in good agreement with the DELPHI estimate of

) "Ž . q0.07 w xthe K 1430 production rate of 0.05 1 .2 y0.05) 0Ž . Ž .However the total K 1430 production rate 172

Ž . Ž . 0 Ž . Ž . 0Fig. 5. The ratios of the production rates a f 980 rr and b f 1270 rr as a function of x . The curÕes represent the expectations of0 2 p

the tuned JETSET model.


differs by 1.8 standard deviations from the corre-sponding OPAL value of 0.238"0.088, obtained byextrapolation of the rate of 0.19"0.07 for x F0.3E

w xmeasured by OPAL 4 to the full x range.EXŽ . Ž .The total f 1525 production rate 18 can be2

compared with the previous DELPHI result of 0.020w x"0.008 6 , again obtained from a smaller data

sample and when only RICH detectors were used forXŽ .particle identification. The f 1525 rate was also2

w xmeasured in Ref. 6 assuming a branching ratioŽ XŽ . q y.Br f 1525 ™K K s35.6%, compared with the2

w xvalue of 44.4% 24 in the present analysis. The) 0Ž . XŽ .values for the total K 1430 and f 1525 rates2 2

predicted by the tuned JETSET model, 0.168 and0.024 respectively, are twice the size of those mea-sured.

It is interesting to compare the total productionŽ . Ž . Ž .rates 16 , 17 and 18 of the tensor mesons

Ž . ) 0Ž . XŽ .f 1270 , K 1430 and f 1525 with the respec-2 2 20 ) 0Ž .tive rates of the vector mesons r , K 892 and f.

0 Ž . ) 0Ž .For the r , the value 14 was used. The K 892w xand f total rates were taken from 5 . This gives:

f 1270 rr 0 s0.180"0.035 19Ž . Ž .2

K ) 0 1430 rK ) 0 892 s0.095"0.031 20Ž . Ž . Ž .2

f X 1525 rfs0.115"0.058. 21Ž . Ž .2

) 0Ž . ) 0Ž . XŽ .The K 1430 rK 892 and f 1525 rf ra-2 2

tios are similar within large errors, but smaller thanŽ . 0the f 1270 rr ratio by 1.8 and 1.0 standard devia-2

tions respectively. Although the observed differences) 0Ž . ) 0Ž . XŽ .between the K 1430 rK 892 , f 1525 rf and2 2

Ž . 0f 1270 rr ratios are not very significant, they2w xmight indicate, as has been suggested in Ref. 28 ,

that this is a simple consequence of the difference inparticle masses and the mass dependence of theproduction rates.

This suggestion is supported by Fig. 6, where the² Ž .:total rates, N part , measured by DELPHI for the

Fig. 6. The production rates of the scalar, vector and tensor mesons measured by DELPHI as a function of their mass squared. The dashed0 ) 0Ž . Ž . Ž . ) 0Ž .lines represent the results of separate fits to exponentials of the r , K 892 , f 980 and f rates and the f 1270 , K 1430 and0 2 2

X Ž . 0 Ž .f 1525 rates. The full lines represent the results of separate fits to three exponentials with the same slope of the r and f 1270 , the2 2) 0Ž . ) 0Ž . X Ž .K 892 and K 1430 rates and of the f and f 1525 rates. The results of the fits are described in the text.2 2


0 ) 0Ž . Ž . Ž . ) 0Ž .r , K 892 , f 980 , f, f 1270 , K 14300 2 2XŽ .and f 1525 are plotted as a function of their mass2

squared, M 2. Antiparticles are not included in the) 0Ž . ) 0Ž . 0K 892 and K 1430 rates. Both the r ,2) 0Ž . Ž .K 892 , f 980 and f data points and the0Ž . ) 0Ž . XŽ .f 1270 , K 1430 and f 1525 data points are2 2 2

Ž 2 .well described x rndfs0.07r2 and 0.01r1 byyB M 2 Žexponentials of the form Ae dashed lines in

.Fig. 6 , with the respective slope parameters 5.43"

0.25 and 4.13"0.63. The slopes are consistent witheach other within two standard deviations. It can be

" "Ž . Xnoted that the v, r r2, a 980 r2 and h produc-0Žtion rates measured by other LEP experiments see

w x .3 and references therein are also consistent with0 ) 0Ž . Ž .the exponential describing the r , K 892 , f 9800

and f data points. Thus it appears, as already notedw xin Ref. 29 , that the production rates of particles

with similar masses, such as the r 0 and v or theŽ . "Ž . Xf 980 , a 980 and h , are very similar.0 0

Fig. 6 also shows that the mass dependence of theproduction rates is almost the same for the pairs r 0

Ž . ) 0Ž . ) 0Ž .and f 1270 , K 892 and K 1430 , f and2 2XŽ .f 1525 . These three sets of data points are well2

Ž 2 . yB M 2fitted x rndfs0.5r2 to the exponential AeŽ .full lines in Fig. 6 , with three different normaliza-tion parameters A but the same slope parameter B,with a fitted value of 1.74"0.15. Thus the relationbetween the production rates of tensor and vectormesons indeed appears to be very similar for differ-ent particles if the mass dependence of these produc-tion rates is taken into account.

Ž .The comparison of the f 980 production rate0

with those of other mesons should be treated withŽ .some caution, since the results for the f 980 are0

model-dependent, to a certain extent, due to theŽ . Ž .uncertainty on the f 980 width. If the f 980 is a0 0

3conventional qq meson in the lowest 1 P multiplet0

with J P C s0qq and its mixing isosinglet partner isŽ .the f 1370 , then in analogy with the tensor-to-vec-0

Ž .tor meson ratios, the production rate of the f 9800

should presumably be compared with the productionŽ . 3rate of the v 1600 , the member of the 1 D multi-1

plet with J P C s1yy. However, the inclusive pro-Ž .duction rate of the v 1600 is not known. The ratio

"Ž .of the rates of the a 980 recently measured by0w x Ž . Ž .OPAL 3 and the f 980 15 is 1.64"0.69, com-0

patible with a value of 2, in analogy with the"Ž . Ž .r 770 rv 782 ratio.

The total production rates of the tensor mesonsŽ . ) 0Ž . XŽ .f 1270 , K 1430 and f 1525 are found to be2 2 2

rather small in absolute value, when compared withthe vector meson production rates. This agrees, atfirst sight, with common expectations that the pro-duction of orbitally excited states is suppressed.

w xHowever, recently it was noticed 30 that the pro-duction rates of orbitally excited mesons are notsmaller, but much larger relative to the states with noorbital momentum if compared at the same masseswith the universal mass dependence of the produc-tion rates for the pseudoscalar and vector mesons

w xand the octet and decuplet baryons 29 .Another indication for the excess of orbitally

excited mesons can be seen from Table 2, where acomparison of the data with the recently proposed

w xthermodynamical model 31 is presented. This modelprovides a very good description of the total produc-tion rates for the pseudoscalar and vector mesonsand for the octet and decuplet baryons, both for

q yw x w xe e 31 and for pp and pp 32 collisions. This isillustrated in Table 2 by a very good agreementbetween the model prediction and the data for ther 0. However, comparison of the model predictionswith the present DELPHI results for the total produc-tion rates of orbitally excited mesons indicates thatthe model underestimates their yields by about the

XŽ .same factor of 1.6–2.1, except for the f 1525 ,2

where the experimental uncertainties are quite large.w xAs suggested in Ref. 30 , the large excess of

orbitally excited mesons might be related to theirgluonic excitation, since this can introduce angularmomentum and therefore the states resulting fromquarkonium-gluonium mixing might be produced athigher rates.

Table 20 Ž . Ž . ) 0Ž .Comparison of the measured r , f 980 , f 1270 , K 14300 2 2

X Ž .and f 1525 total production rates with the predictions of the2w xthermodynamical model 31 .

Particle DELPHI results Model predictions0r 1.19"0.10 1.17"0.05Ž .f 980 0.164"0.021 0.0772"0.00760Ž .f 1270 0.214"0.038 0.130"0.0152) 0Ž .K 1430 0.073"0.023 0.0462"0.00412

X Ž .f 1525 0.012"0.006 0.0107"0.00072


5. Summary

The DELPHI results on inclusive production of0 Ž . Ž . ) 0Ž . XŽ .the r , f 980 , f 1270 , K 1430 and f 15250 2 2 2

in hadronic Z0 decays at LEP have been presented.They are based on a data sample of about 2 millionhadronic events, using the particle identification ca-pabilities of the RICH and TPC detectors, and super-sede the previous DELPHI results, with which theyare consistent. The following conclusions can bedrawn.Ø The total r 0 production rate per hadronic Z0

decay amounts to 1.19"0.10. The r 0 momen-tum spectrum is well reproduced by the JETSETmodel tuned to previous DELPHI data. The totalr 0 rate and its momentum spectrum are consis-tent with the ALEPH measurements.

Ž . Ž .Ø The total f 980 and f 1270 production rates0 2

per hadronic Z0 decay are 0.164"0.021 andŽ .0.214 " 0.038 respectively. The f 980 and0

Ž .f 1270 momentum spectra are well described by2

the tuned JETSET model. The shapes of theŽ . Ž .f 980 and f 1270 momentum spectra are simi-0 2

lar to that for the r 0 for x F0.4. For higher xp p

values there is some indication that the ratiosŽ . 0 Ž . 0f 980 rr and especially f 1270 rr may in-0 2

crease with x , in agreement with JETSET ex-pŽ . Ž .pectations. The total f 980 and f 1270 rates0 2

and their momentum spectra are consistent withthe OPAL measurements.

) 0Ž . XŽ .Ø The total K 1430 and f 1525 production2 2

rates per hadronic Z0 decay amount to 0.073"

0.023 and 0.012"0.006 and are about half thesize of the rates predicted by the tuned JETSET

) 0Ž .model. The total K 1430 rate is smaller by2

1.8 standard deviations than the value 0.238"

0.088 measured by OPAL for x F0.3 and ex-E

trapolated by us to the full x range.EŽ . 0 ) 0Ž . ) 0Ž .Ø The ratios f 1270 rr , K 1430 rK 8922 2

XŽ .and f 1525 rf are 0.180 "0.035, 0.095 "2

0.031 and 0.115"0.058 respectively. They ap-pear to be somewhat different. However, the rela-tionships between the production rates of the

Ž .tensor and vector mesons for the f 1270 and20 ) 0Ž . ) 0Ž . XŽ .r , K 1430 and K 892 , f 1525 and f2 2

are found to be very similar when the massdependence of the production rates is accountedfor.

The DELPHI and OPAL results, despite someinconsistency between their measurements of the

) 0Ž .K 1430 rate, show a rather significant production2

rate for orbitally excited states in Z0 hadronic de-cays. It appears, in agreement with the conclusions

w xdrawn in Ref. 30 , that the production rates oforbitally excited tensor mesons are at least as largeas those of states with no orbital momentum, if themass dependence of their production rates is ac-counted for. It is also indicated that the measuredrates of orbitally excited mesons are higher than

w xfollows from the thermodynamical model 31 , whichis quite successful in describing the total productionrates of other particles.

Acknowledgements

We are greatly indebted to our technical collabo-rators, to the members of the CERN-SL Division forthe excellent performance of the LEP collider, and tothe funding agencies for their support in building andoperating the DELPHI detector. We acknowledge inparticular the support of Austrian Federal Ministry ofScience and Traffics, GZ 616.364r2-IIIr2ar98;FNRS–FWO, Belgium; FINEP, CNPq, CAPES,FUJB and FAPERJ, Brazil; Czech Ministry of Indus-try and Trade, GA CR 202r96r0450 and GA AVCRA1010521; Danish Natural Research Council; Com-

Ž .mission of the European Communities DG XII ;Direction des Sciences de la Matiere, CEA, France;`Bundesministerium fur Bildung, Wissenschaft,¨Forschung und Technologie, Germany; General Sec-retariat for Research and Technology, Greece; Na-

Ž .tional Science Foundation NWO and FoundationŽ .for Research on Matter FOM , The Netherlands;

Norwegian Research Council; State Committeefor Scientific Research, Poland, 2P03B06015,2P03B03311 and SPUBrP03r178r98; JNICT–Junta Nacional de Investigacao Cientıfica e˜ ´Tecnologica, Portugal; Vedecka grantova agentura´MS SR, Slovakia, Nr. 95r5195r134; Ministry ofScience and Technology of the Republic of Slovenia;CICYT, Spain, AEN96-1661 and AEN96-1681; TheSwedish Natural Science Research Council; ParticlePhysics and Astronomy Research Council, UK; De-partment of Energy, USA, DE–FG02–94ER40817.


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1 On leave of absence from IHEP Serpukhov.



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A. Zalewska r, P. Zalewski ay, D. Zavrtanik aq, E. Zevgolatakos k, N.I. Zimin p,x,G.C. Zucchelli ar, G. Zumerle ai

a Department of Physics and Astronomy, Iowa State UniÕersity, Ames, IA 50011-3160, USAb Physics Department, UniÕ. Instelling Antwerpen, UniÕersiteitsplein 1, BE-2610 Wilrijk, Belgium,

and IIHE, ULB-VUB, Pleinlaan 2, BE-1050 Brussels, Belgium,and Faculte des Sciences, UniÕ. de l’Etat Mons, AÕ. Maistriau 19, BE-7000 Mons, Belgium´

c Physics Laboratory, UniÕersity of Athens, Solonos Str. 104, GR-10680 Athens, Greeced Department of Physics, UniÕersity of Bergen, Allegaten 55, NO-5007 Bergen, Norway´

e Dipartimento di Fisica, UniÕersita di Bologna and INFN, Via Irnerio 46, IT-40126 Bologna, Italy`f Centro Brasileiro de Pesquisas Fısicas, rua XaÕier Sigaud 150, BR-22290 Rio de Janeiro, Brazil,´

and Depto. de Fısica, Pont. UniÕ. Catolica, C.P. 38071, BR-22453 Rio de Janeiro, Brazil,´ ánd Inst. de Fısica, UniÕ. Estadual do Rio de Janeiro, rua Sao Francisco XaÕier 524, Rio de Janeiro, Brazil´ ˜

g Comenius UniÕersity, Faculty of Mathematics and Physics, Mlynska Dolina, SK-84215 BratislaÕa, SloÕakiah College de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, FR-75231 Paris Cedex 05, France`

i CERN, CH-1211 GeneÕa 23, Switzerlandj Institut de Recherches Subatomiques, IN2P3 - CNRSrULP - BP20, FR-67037 Strasbourg Cedex, France

k Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greecel FZU, Inst. of Phys. of the C.A.S. High Energy Physics DiÕision, Na SloÕance 2, CZ-180 40 Praha 8, Czech Republic

m Dipartimento di Fisica, UniÕersita di GenoÕa and INFN, Via Dodecaneso 33, IT-16146 GenoÕa, Italy`n Institut des Sciences Nucleaires, IN2P3-CNRS, UniÕersite de Grenoble 1, FR-38026 Grenoble Cedex, France´ ´

o Helsinki Institute of Physics, HIP, P.O. Box 9, FI-00014 Helsinki, Finlandp Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, RU-101 000 Moscow, Russian Federation

q Institut fur Experimentelle Kernphysik, UniÕersitat Karlsruhe, Postfach 6980, DE-76128 Karlsruhe, Germany¨ ¨r Institute of Nuclear Physics and UniÕersity of Mining and Metalurgy, Ul. Kawiory 26a, PL-30055 Krakow, Polands UniÕersite de Paris-Sud, Lab. de l’Accelerateur Lineaire, IN2P3-CNRS, Bat. 200, FR-91405 Orsay Cedex, France´ ´ ´ ´ ˆ

t School of Physics and Chemistry, UniÕersity of Lancaster, Lancaster LA1 4YB, UKu LIP, IST, FCUL - AÕ. Elias Garcia, 14-1o, PT-1000 Lisboa Codex, Portugal

v Department of Physics, UniÕersity of LiÕerpool, P.O. Box 147, LiÕerpool L69 3BX, UKw ( )LPNHE, IN2P3-CNRS, UniÕ. Paris VI et VII, Tour 33 RdC , 4 place Jussieu, FR-75252 Paris Cedex 05, France

x Department of Physics, UniÕersity of Lund, SolÕegatan 14, SE-223 63 Lund, Swedenÿ UniÕersite Claude Bernard de Lyon, IPNL, IN2P3-CNRS, FR-69622 Villeurbanne Cedex, France´

z UniÕ. d’Aix - Marseille II - CPP, IN2P3-CNRS, FR-13288 Marseille Cedex 09, Franceaa Dipartimento di Fisica, UniÕersita di Milano and INFN, Via Celoria 16, IT-20133 Milan, Italy`

ab Niels Bohr Institute, BlegdamsÕej 17, DK-2100 Copenhagen Ø, Denmarkac NC, Nuclear Centre of MFF, Charles UniÕersity, Areal MFF, V HolesoÕickach 2, CZ-180 00, Praha 8, Czech Republic

ad NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlandsae National Technical UniÕersity, Physics Department, Zografou Campus, GR-15773 Athens, Greece

af Physics Department, UniÕersity of Oslo, Blindern, NO-1000 Oslo 3, Norwayag Dpto. Fisica, UniÕ. OÕiedo, AÕda. CalÕo Sotelo srn, ES-33007 OÕiedo, Spain

ah Department of Physics, UniÕersity of Oxford, Keble Road, Oxford OX1 3RH, UKai Dipartimento di Fisica, UniÕersita di PadoÕa and INFN, Via Marzolo 8, IT-35131 Padua, Italy`

aj Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UKak Dipartimento di Fisica, UniÕersita di Roma II and INFN, Tor Vergata, IT-00173 Rome, Italy`

al Dipartimento di Fisica, UniÕersita di Roma III and INFN, Via della Vasca NaÕale 84, IT-00146 Rome, Italyàm DAPNIArSerÕice de Physique des Particules, CEA-Saclay, FR-91191 Gif-sur-YÕette Cedex, France

an ( )Instituto de Fisica de Cantabria CSIC-UC , AÕda. los Castros srn, ES-39006 Santander, Spainao Dipartimento di Fisica, UniÕersita degli Studi di Roma La Sapienza, Piazzale Aldo Moro 2, IT-00185 Rome, Italy`

ap ( )Inst. for High Energy Physics, SerpukoÕ P.O. Box 35, ProtÕino Moscow Region , Russian Federationaq J. Stefan Institute, JamoÕa 39, SI-1000 Ljubljana, SloÕenia

and Laboratory for Astroparticle Physics, NoÕa Gorica Polytechnic, KostanjeÕiska 16a, SI-5000 NoÕa Gorica, SloÕenia,and Department of Physics, UniÕersity of Ljubljana, SI-1000 Ljubljana, SloÕenia

ar Fysikum, Stockholm UniÕersity, Box 6730, SE-113 85 Stockholm, Swedenas Dipartimento di Fisica Sperimentale, UniÕersita di Torino and INFN, Via P. Giuria 1, IT-10125 Turin, Italy`

at Dipartimento di Fisica, UniÕersita di Trieste and INFN, Via A. Valerio 2, IT-34127 Trieste, Italy,ànd Istituto di Fisica, UniÕersita di Udine, IT-33100 Udine, Italy`

au UniÕ. Federal do Rio de Janeiro, C.P. 68528 Cidade UniÕ., Ilha do Fundao, BR-21945-970 Rio de Janeiro, Brazil˜


av Department of Radiation Sciences, UniÕersity of Uppsala, P.O. Box 535, SE-751 21 Uppsala, Swedenaw ( )IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, AÕda. Dr. Moliner 50, ES-46100 Burjassot Valencia , Spain

ax ¨Institut fur Hochenergiephysik, Osterr. Akad. d. Wissensch., Nikolsdorfergasse 18, AT-1050 Vienna, Austria¨ay Inst. Nuclear Studies and UniÕersity of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Poland

az Fachbereich Physik, UniÕersity of Wuppertal, Postfach 100 127, DE-42097 Wuppertal, Germany


Abstract

Data collected at the Z resonance using the DELPHI detector at LEP are used to determine the charged hadronmultiplicity in gluon and quark jets as a function of a transverse momentum-like scale. The colour factor ratio, C rC , isA F

directly observed in the increase of multiplicities with that scale. The smaller than expected multiplicity ratio in gluon toquark jets is understood by differences in the hadronization of the leading quark or gluon. From the dependence of thecharged hadron multiplicity on the opening angle in symmetric three-jet events the colour factor ratio is measured to be:

Ž . Ž . Ž .C rC s2.246"0.062 stat. "0.080 syst. "0.095 theo. q 1999 Elsevier Science B.V. All rights reserved.A F

1. Introduction

The gauge symmetry underlying the Lagrangianof an interaction directly determines the relativecoupling of the vertices of the participating elemen-tary fields. A comparison of the properties of quarkand gluon jets, which are linked to the quark andgluon couplings, therefore implies a direct and intu-itive test of Quantum Chromodynamics, QCD, thegauge theory of the strong interaction.

Hadron production can be described via a so-calledparton shower, a chain of successive bremsstrahlungprocesses, followed by hadron formation which can-not be described perturbatively. As bremsstrahlung isdirectly proportional to the coupling of the radiatedvector boson to the radiator, the ratio of the radiatedgluon multiplicity from a gluon and quark source isexpected to be asymptotically equal to the ratio of

w xthe QCD colour factors: C rC s9r4 1 . As theA F

radiated gluons give rise to the production of hadrons,the increased radiation from gluons should be re-flected in a higher hadron multiplicity and also in astronger scaling violation of the gluon fragmentation

w xfunction 2,3 .It was however noted already in the first paper

comparing the multiplicities from gluons and quarks

2 Now at University of Florida.

w x1 that this prediction does not immediately apply tothe observed charged hadron multiplicities at finiteenergy as this is also influenced by differences of thefragmentation of the primary quark or gluon. Thesedifferences must be present because quarks are va-lence particles of the hadrons whereas gluons arenot. This is most clearly evident from the behaviourof the gluon fragmentation function to chargedhadrons at large scaled momentum where it is sup-pressed by about one order of magnitude compared

w xto the quark fragmentation function 3 . This sup-pression also causes a higher multiplicity to be ex-pected from very low energy quark jets compared togluon jets. Moreover, as low momentum, largewavelength gluons cannot resolve a hard radiatedgluon from the initial quark-antiquark pair in theearly phase of an event, soft radiation and corre-spondingly the production of low energy hadrons is

w xfurther suppressed 4–6 compared to the naive ex-w xpectation. In a previous publication 2 it has been

shown that a reduction of the primary splittings ofgluons compared to the perturbative expectation isindeed responsible for the observed small hadronmultiplicity ratio between gluon and quark jets.

If heavy quark jets are also included in the com-parison, a further reduction of the multiplicity ratio isevident due to the high number of particles from thedecays of the primary heavy particles.

Furthermore, the definition of quark and gluonjets in three-jet events in eqey annihilation uses jet


algorithms which combine hadrons to make jets.Low energy particles at large angles with respect tothe original parton direction are likely to be assignedto a different jet. As gluon jets are initially widerthan quark jets this presumably leads to a loss ofmultiplicity for gluon jets and a corresponding gainfor quark jets.

The effects discussed lead to a ratio between thecharged hadron multiplicities from gluon and quarkjets being smaller than the ratio between gluon radia-tion from gluons and from quarks. So far theseeffects have mainly been neglected in experimental

w xand more elaborate theoretical investigations 7 .However, as we will show in this paper, at currentenergies these non-perturbative effects are still im-portant and need to be considered in a proper test of

w xthe prediction 1 that the radiated gluon-to-quarkmultiplicity ratio is equal to the colour factor ratio.

The stronger radiation from gluons is expected tobecome directly evident from a stronger increase ofthe gluon jet multiplicity with the relevant energyscale as compared to quark jets. In this way the sizeof the non-perturbative terms can also be directlyestimated from the quark and gluon jet multiplicityat very small scales. A scale dependence of quarkand gluon properties was first demonstrated in Ref.w x8 with the jet energy as the intuitive scale. Thisresult was later confirmed by other measurementsw x9–11 and has recently been extended to a trans-

w xverse momentum-like scale 12 .A study of the total charged multiplicity of sym-

metric three-jet events as function of the internalscales of the event avoids some of the complicationsmentioned above. A novel precision measurement ofthe colour factor ratio C rC can be performed byA F

combining these data with a Modified Leading LogŽ .Approximation MLLA prediction of the three-jet

w xevent multiplicity 13 which includes coherence ofsoft gluon radiation.

This letter is based on a data analysis which isw xsimilar to that presented in previous papers 2,8 . We

therefore have restricted the experimental discussionin Section 2 to the relevant differences with respectto these papers. More detailed information can also

w xbe found in Refs. 14,15 . In Section 3.1 the ratio ofthe slopes of the mean hadron multiplicities in gluonand quark jets with scale is shown to be determinedby the colour factor ratio C rC . In order to de-A F

scribe the data with the perturbative QCD expecta-tions it is necessary to introduce additional non-per-turbative offsets. This analysis is intended to bemainly qualitative and in many aspects it is similarto previous analyses. Then in Section 3.2 a precisionmeasurement of the colour factor ratio from symmet-ric three-jet events is discussed and an estimate forthe difference of non-perturbative contributions tothe quark and gluon jet multiplicity is given. Finallywe summarize and conclude.

2. Data analysis

The analysis presented in this letter uses the fullhadronic data set collected with the DELPHI detec-

Ž w x.tor described in Ref. 16 at Z energies in the years1992 to 1995. The cuts applied to charged andneutral particles and to events in order to selecthadronic Z decays are identical to those given in Ref.w x w x2 for the qqg analysis and to 8 for the qqg

analysis. For the comparison of gluon and quark jets,three-jet events are clustered using the Durham algo-

w xrithm 17 . In addition it was required that the angles,u , between the low-energy jets and the leading jet2,3

Ž Ž ..are in the range from 1008 to 1708 see Fig. 1 a .Within this sample, events are called symmetric if u2

and u are equal within some analysis-dependent3

tolerance. The leading jet is not used in the gluon orquark jet analysis.

The identification of gluon jets by anti-tagging ofheavy quark jets is identical to that described in

w xRefs. 2,8 . Quark jets are taken from qqg eventswhich have been depleted in b-quark events using animpact parameter technique. In order to achieve mul-tiplicities of pure quark and gluon jet samples, thedata have been corrected using purities from simu-

w xlated events generated with JETSET 7.3 18 withw xparameters set as given in Ref. 19 . This is justified

by the good agreement between data and simulation.Furthermore the model independent techniques de-

w x Žscribed in Ref. 2 for symmetric events see Fig.Ž ..1 b give results largely compatible with those ob-

w xtained with the simulation correction 15 . The ef-fects of the finite resolution and acceptance of thedetector and of the cuts applied are corrected for by

w xusing a full simulation of the DELPHI detector 16 .


Fig. 1. Definition of event topologies and angles used throughout this analysis. The length of the jet lines indicates the energies. In theŽ . Ž Ž ..symmetric Y events see Fig. 1 b u fu .2 3

The correction for the remaining b-quark eventsin the qqg sample does not influence the slope of themeasured multiplicity with scale, but only leads to ashift of its absolute value.

In the simulation, quark and gluon jets are identi-fied at ‘‘parton level’’. The partons entering thefragmentation of a three-jet event are clustered intothree jets using the Durham algorithm. Then for eachparton jet, the number of quarks and antiquarks aresummed where primary quarks contribute with weightq1 and antiquarks with the weight y1. Other quarksand gluons are assigned the weight 0. These sumsare expected to yield q1 for quarks jets, y1 foranti-quark jets and 0 for gluon jets. The small amountof events not showing this expected pattern ofŽ .q1,y1,0 was discarded. Finally, the parton jetswere mapped to the jets at the hadron level byrequiring the sum of angles between the parton andhadron jets to be minimal. Events exceeding a maxi-mum angle between the parton and jet directionswere also rejected. At large opening angles the influ-ence of these rejections is found to be about 3%increasing at low opening angles.

The gluon jet purities vary from 95% for lowenergy gluons to 46% for the highest energy gluons.The few bins with lower purities have been excludedfrom the analysis. The quark purities range from43% to 81%.

For the analysis of the multiplicity of symmetricthree-jet events, all events were forced to three jetsusing the Durham algorithm without a minimal y .cut

The angles between the jets were then used to rescalethe jet momenta to the centre-of-mass energy as

w xdescribed in Ref. 8 . Symmetric events were se-lected by demanding that u be equal to u within2 3

2e . Here e is half the angular bin width of u taken1

to be 38. The analysis has been performed for eventsof all flavours as well as for b-depleted events. Inboth cases the measured multiplicity was correctedfor track losses due to detector effects and cutsapplied. The correction factor was calculated as ratioof generated over accepted multiplicity using simu-lated events. It varies smoothly, from 1.25 at smallu to 1.32 at large u .1 1

3. Results

3.1. Comparison of multiplicities in gluon and quarkjets

In order to determine a scale dependence, thescale underlying the physics process needs to bespecified. The actual physical scale is necessarilyproportional to any variation of an outer scale likethe centre-of-mass energy. As usually only the rela-tive change in scale matters, this outer scale cantherefore be used instead of the physical scale. Forthis analysis the situation is different. The jets enter-ing the analysis stem from Z decays and thus from afixed centre-of-mass energy. So the relevant scaleshave to be determined from the properties of the jetsand the event topology. From the above discussionthe scale has to be proportional to the jet energybecause this quantity scales with the energy in thecentre-of-mass system for similar events. Studies of


hadron production in processes with non-trivialtopology have shown that the characteristics of theparton cascade prove to depend mainly on the hard-

w xness of the process producing the jet 4,20 :

ksE sinur2 . 1Ž .jet

E is the energy of the jet and u its angle to thejet

closest jet. This scale definition corresponds to thebeam energy in two-jet events. It is similar to thetransverse momentum of the jet and also related to

y as used by the jet algorithms. It is also used as( cut

the scale in the calculation of the energy dependenceof the hadron multiplicity in eqey annihilationw x21,22 to take into account the leading effect ofcoherence. It should, however, be noted that severalscales may be relevant in multi-jet events. Henceusing k is an approximation. A similar scale, namelythe geometric mean of the scales of the gluon jet

Ž .with respect to both quark jets while using Eq. 1for the quark jets, has recently been used in a study

w xof quark and gluon jet multiplicities 12 .As stated in the introduction we want to gain

information on the relative colour charges of quarksand gluons from the rate of change of the multiplici-ties with scale. Assuming the validity of the pertur-bative QCD prediction, the ratio of the chargedmultiplicities of gluon and quark jets, N rN ,gluon quark

Žhas to approach a constant value approximately the.colour factor ratio at large scale. This trivially im-

plies that the ratio of the slopes of quark and gluonjet multiplicities also approaches the same limit. Thisfact is a direct consequence of de l’Hopital’s ruleˆw x23 and is also directly evident from the linearity ofthe derivative:

at large scale: N k sC N kŽ . Ž .gluon quark

dN rdkgluon™ sC , 2Ž .

dN rdkquark

i.e. the QCD prediction for the ratio of multiplicitiesapplies equally well to the ratio of the slopes of themultiplicities. In fact it is to be expected that theslope ratio is closer to the QCD prediction than themultiplicity ratio as it should be less affected bynon-perturbative effects.

This effect has been cross-checked using thew xHERWIG model 24 which allows the number of

Ž .colours to be changed and thus by SU n group

relations, the colour factor ratio C rC . The predic-A F

tions of HERWIG are found to follow directly theŽ .expectation of the right hand side of Eq. 2 . This

has also been confirmed in a recent theoretical calcu-w xlation of this quantity 25 in the framework of the

dipole model.Ž .Fig. 2 a shows the multiplicity in quark and

gluon jets as a function of the hardness scale k . Forboth multiplicities an approximately logarithmic in-crease with k is observed which is about twice asbig for gluon jets as for quark jets, thus alreadystrikingly confirming the QCD prediction.

A stronger increase of the gluon jet multiplicityw xwas already noted in a previous paper 8 , where the

jet energy was chosen as scale. Meanwhile thisobservation has been confirmed also by other mea-

w xsurements 9–11 and has been extended to differentw x Ž .scales 12 . Fragmentation models not shown pre-

dict an increase of the multiplicities which is in goodagreement with the data.

In order to obtain quantitative information fromŽ .the data shown in Fig. 2 a , the following ansatz was

fitted to the data:

² : qN k sN qN k ,Ž . Ž .q 0 pert

² : gN k sN qN k Pr k 3Ž . Ž . Ž . Ž .g 0 pert

Here N q, g are non-perturbative terms introduced to0

account for the differences in the fragmentation ofthe leading quark or gluon as discussed in detail inthe introduction. These terms are assumed to beconstant. N is the perturbative prediction for thepert

w xhadron multiplicity as given in Ref. 21 :

N kŽ .pert

cbsK a k exp 1qO aŽ . (Ž . ž /s sž /a k( Ž .s

4Ž .

1 2 n Cf Fbs q 1y ,ž /4 3 b C0 A

32C p 2( Acs , b s11y n .0 f

b 30

A first and a second order a have been used withs

this expression with the number of active flavours,n , equal to five. An alternative prediction has beenf


Ž . Ž . Ž .Fig. 2. a Average charged particle multiplicity for light quark and gluon jets as function of k fitted with Eq. 3 ; b ratio of the gluon toŽ .quark jet multiplicity; the full line shows the ratio of the functions fitted to the data in a , the dashed curve is the ratio of the slopes of the

Ž .fits in a . All curves are extrapolated to the edges of the plot by the dotted lines. Also included are measurements of the multiplicity ratio ofw xsome other experiments 10,11 . The grey band shown with the slope ratio indicates the error estimated by varying all fit parameters within

their errors.

w xgiven in Ref. 26 using the limited spectrum ap-proach:

1yBzN sK G B I z , 5Ž . Ž . Ž .pert 1qBž /2

33q2r9n k 2f

Bs , zs log g ,0233y2n Lf

2g s C a k .Ž .(0 A s

p

Here a first order a has always been used with ns fw xtaken as three 27 . G is the Gamma-function and IB

the modified Bessel-function. K is a non-perturba-tive scale factor. The QCD scale parameter L enters

Ž 2 2 . w xinto the definition of a k rL 22 . The numeri-s

cal values of K and L are not expected to be theŽ . Ž .same in Eqs. 4 and 5 as different approximations

are used. Finally:

CA 2r k s 1yr g yr g 6Ž . Ž .Ž .1 0 2 0CF

with1 n 2n Cf f F

r s 1q y ,1 2ž /6 C CA A

r 25 3 n n C1 f f Fr s y y2 2ž /6 8 4 C CA A

w xis the perturbative prediction 28 for the multiplicityratio in back-to-back gluon to back-to-back quark

Ž .jets. The terms proportional to r r correspond to1 2Ž .the NLO NNLO prediction. Numerically they cor-

respond to corrections of about 8% and 1% respec-tively. The smallness of the higher order correctionsindicates that the perturbative series of the gluon-to-quark multiplicity ratio converges rapidly.

The fits represent the data well. The fit range hasnot been extended to too small scales as here acontribution of initial two-jet events might bias themultiplicities to lower values. Parameters of the fitsfor this specific choice of scale and jet selection aregiven in Table 1. No estimate of systematic error isgiven as this analysis is intended to be mainly quali-tative. The fit parameters should not be compared


Table 1Results of the fits of the quark and gluon jet multiplicities as afunction of k

Ž . Ž .Parameter N from Eq. 4 N from Eq. 5pert pert

w xL GeV 0.032 " 0.011 0.011 " 0.004K 0.005 " 0.001 0.12 " 0.02C rC 2.12 " 0.10 2.15 " 0.10A F

qN 2.82 " 0.14 3.12 " 0.200gN 0.73 " 0.21 1.43 " 0.3102x rn.d.f. 0.61 0.65

directly to those parameters usually obtained fromoverall events in eqey annihilation. The normaliza-tion factor, K , differs strongly due to the differencesin the multiplicity in jets and overall events. Further-more, the introduction of non-perturbative offsetsleads to a strong reduction of the values of theeffective scale parameter L. This is also observed ifthe eqey multiplicity is fitted including an offsetterm, which could be reasonable in this case also.

Using an identical scale definition for quark andgluon jets also allows the gluon-to-quark jet multi-plicity ratio to be directly evaluated as function of

Ž .this scale. Fig. 2 b shows this ratio as calculatedfrom data and the fits as function of the hardnessscale as well as the ratio of the slopes of the fits. Theratio of the multiplicities increases from about 1.15at small scale to about 1.4 at the highest scales

w xmeasured. The measurement 10 performed inŽ . 3F 1S ™g gg decays at small scale , and of ‘‘in-

w xclusive’’ gluons 11 at large scale, agree quite wellwith the expectation from the fits. The corresponding

w xhardness scale for the data at the highest scale 11has been estimated from the average gluon energy

w xand the angle cuts given in Ref. 11 . The goodagreement of the ‘‘inclusive’’ gluon measurementalso implies that angular ordering effects are relevantin this case.

The ratio of the slopes for the different fits isalmost 2 corresponding to a colour factor ratio ofC rC s2.12"0.10, well compatible with the QCDA F

expectation.The fits further indicate that for very small scale

the multiplicity of quark jets is bigger than that of

3 Half of the gg invariant mass is taken as the equivalent scale.

gluon jets. Consequently the constant terms con-tributing to the multiplicity due to the primary gluon

Žor quark fragmentation are larger for quarks see.Table 1 . The difference of these terms is about 2.

w xTaking the scale choice made in Ref. 12 leads toabout a 20% increase of the measured colour factorratio and a corresponding increase in the differenceof the non-perturbative constants to 4.2.

It is instructive here to estimate a lower limit forthe difference of the non-perturbative terms from thebehaviour of the gluon and quark fragmentation

w xfunctions 2 . Due to leading particle effects thefragmentation function of the quark outreaches thefragmentation function of the gluon at high values ofx . Taking the shape of the gluon fragmentationE

function as unbiased by the leading particle effectand assuming the overall multiplicity of gluon jetsroughly as twice as big as of quark jets, one gets anestimate for the lower limit of additional multiplicityin quark jets by integrating the difference betweenthe quark and the halved gluon fragmentation func-tion in the x -region where the fragmentation func-E

tion of the gluon is below that of the quark. Thisyields N q yN g G0.61"0.02 from Y and N q yN g

0 0 0 0w xG0.58"0.05 from so-called Mercedes events 2 . It

should be noted here, that the leading particle effectstill influences the multiplicity at even lower scaledhadron energies. The region of small hadron energycontributes most to the multiplicity. Therefore theestimated limit presumably is much smaller than theactual value of N .0

At first sight a difference of the constant terms ofthe order of ;2 units in charged multiplicity looksunexpectedly large. However, these constants alsoinclude the effects of the jet clustering. Furthermore,stable hadron production to a large extent proceedsvia resonance decays, so that the observed differencemay only correspond to a difference of about oneprimary particle. The larger constant term for quarkscompared to gluons explains the different behaviourof the ratio of multiplicities and the slope ratio in

Ž .Fig. 2 b .The observed behaviour would be expected from

non-perturbative effects of the fragmentation in theleading quark or gluon. In the cluster fragmentationmodel, an additional gluon to quark-antiquark split-ting is needed in the fragmentation of a gluon com-pared to that of a quark.


3.2. Precise determination of C rC from multi-A F

plicities in three-jet eÕents

The analysis presented so far, as in most othercomparisons of quark and gluon jet multiplicities,has the disadvantage of relying on the association ofŽ .maybe low energy particles to jets. Clearly thisinvolves severe ambiguities and specifically does notconsider coherent soft gluon radiation from the ini-tial qqg ensemble. This can be avoided and a precisemeasurement can be obtained by studying the depen-dence of the total charged multiplicity in three-jetevents as function of the quark and gluon scales. In

w xfact there is a definite MLLA prediction 13 for thismultiplicity N :qq g

a s) )N s 2 N Y qN Y P 1qOO 7Ž .Ž .Ž .qq g q qq g g ž /ž /p

with the scale variables:)p p Eq q

)Y s ln s ln ,(qq 2 L2 L

Hp p p p pŽ . Ž .q g q g 1)Y s ln s ln , 8Ž .g 2) 2 L2 L p pŽ .q q

) )Ž . Ž .N Y and N Y describe the scale dependenceq qq g g

of the multiplicity for quark or gluon jets, respec-tively. L is a scale parameter and the p are theq,q, g

four-momenta of the quarks and the gluon. Thethree-jet multiplicity depends on the quark energy,E) , in the centre-of-mass system of the quark-anti-quark pair and on the transverse momentum scale ofthe gluon, pH . For comparison with data, this is1

w xexpressed in Ref. 29 as a dependence on the mea-sured multiplicity in eqey events, N q y, and thee e

Ž .colour factor ratio as given in Eq. 6 . In addition,we again choose to add a constant term, N , to0

account for differences in the fragmentation of quarksand gluons as discussed above. Thus, omitting cor-rection terms:

1) H Hq y q yN sN 2 E qr p N p yN .Ž . � 4Ž . Ž .qq g e e 1 e e 1 02

9Ž .

Although at first sight this appears to be the incoher-ent sum of the multiplicity of the two quark jets andthe gluon jet, this formula includes coherence effectsin the exact definition of the scales of the N q ye e

w xterms 27 . Nevertheless, subtracting the non-per-turbative term N within the curly brackets gives a0

physical interpretation for N as the additional multi-0

plicity in quark jets due to the leading particle effect,which is contained in the measured N q y and has toe e

be subtracted to get the gluon contribution to themultiplicity.

Ž .In principle Eq. 9 still requires the determinationof the quark-antiquark and gluon scales indepen-

Ždently. However, in symmetric Y-type events seeŽ ..Fig. 1 b both scales can be expressed as functions

of the opening angle u only by initially assuming1

that the gluon jet is not the most energetic one.) 2 2E AE E sin u r2 for this type of event is almostq q 3

Ž Ž ..constant see upper full curve in Fig. 3 a at fixedcentre-of-mass energy. However, pH , increases ap-1

proximately linearly with the opening angle as it isproportional to the gluon transverse momentum. Asthe multiplicity change corresponding to the changeof E) corresponds only to about y2, the u depen-dence of the three-jet multiplicity therefore mainlymeasures the scale dependence of the multiplicity ofthe gluon jet.

ŽIn a fraction of the events which strongly in-.creases with opening angle the gluon jet is the most

energetic jet. This can be corrected for in differentŽ .ways when fitting Eq. 9 to the data using Monte

Carlo simulation. Assuming an approximately loga-rithmic increase of the multiplicity with scale, whichis well supported by the data, the average scale at agiven opening angle can be expressed as the geomet-ric mean of the cases where the gluon initiates themost energetic jet and where it does not. These

Ž .corrected scales are shown as the points in Fig. 3 a .The correction first increases with the opening anglebut then decreases again and vanishes for fully sym-metric events. Alternatively, the fraction of eventswhen the gluon initiates the most energetic jet can be

Ž .considered separately in Eq. 9 .To obtain information on the colour factor ratio

C rC , the scale dependence of the three-jet multi-A F

plicity has to be compared to the multiplicity in alleqey events. This has been chosen to be taken fromthe DELPHI measurements with hard photon radia-tion for energies below the Z mass and at 184 GeVw x30 and the LEP combined measurements at the

w xintermediate energies 31 . For studies of systematicerrors, data from lower energy eqey experiments


Ž . ) HFig. 3. a Variation of the scales 2 E and p as function of the opening angle u in symmetric three-jet events. The functions are the1 1Ž .analytic expectation. The points include a correction calculated with JETSET 7.3 for the cases where the gluon forms the most energetic

Ž .jet. The lines matching the points are polynomials fitted to obtain continuous values. b Charged hadron multiplicity as a function of theŽ . Ž . Ž .centre-of-mass energy of the qq-pair fitted with the perturbative predictions Eqs. 4 or 5 . c Charged hadron multiplicity in symmetric

Ž .three-jet events as a function of the opening angle. The dashed curve is the prediction using the ansatz Eq. 9 setting C rC to its defaultA FŽ .value and omitting the constant offset, N . The full curve is a fit of the full ansatz Eq. 9 to the data treating C rC and N as free0 A F 0

Ž . 2parameters. d Stability of the result for C rC against variation of the smallest opening angle used in the fit as well as x rN and theA F d f

x 2 probability of these fits. The dash-dotted horizontal line in the upper half shows the QCD expectation for C rC with the dotted linesA FŽ . Ž . w xrepresenting variations of "10%. The DELPHI data of Figs. 3 b and c will be made available in the DurhamrRAL database 38 .

w x32 have also been used. The DELPHI multiplicitiesin events with hard photon radiation have been ex-

w xtracted as described in Refs. 8,14 , but using the fullstatistics now available. Small energy dependent cor-

Ž . q yrections 2y4% to the e e multiplicities wereapplied to correct for the varying contribution of bquarks. The multiplicities obtained were fitted with

Ž . Ž .the perturbative predictions, Eqs. 4 or 5 , see Fig.Ž .3 b . Both calculations describe the data equally

well. The parameters of the fits are given in theupper part of Table 2.

The measured, fully corrected multiplicity in allsymmetric three-jet events as function of the opening

Ž .angle is shown in Fig. 3 c . A strong increase of themultiplicity from values of around 18 for smallopening angle to about 29 at opening angles of 1208

Ž .corresponding to fully symmetric events is ob-served. Omitting the non-perturbative term, N , in0


Table 2q y Ž .Result of the fits of the e e multiplicity upper part and theŽ .three-jet event multiplicity lower part

w x w xParameter N from 21 N from 26 Relevantpert pert

L 0.275 " 0.070 0.061 " 0.015 data fromq yK 0.026 " 0.003 0.606 " 0.062 e e and qqg

2x rn.d.f. 1.180 1.183

C rC 2.251 " 0.063 2.242 " 0.062 data fromA F

N 1.40 " 0.10 1.40 " 0.10 symmetric02x rn.d.f. 0.998 1.004 3 jet events

Ž .Eq. 9 and setting C rC to its expected valueA F

predicts a similar increase in multiplicity over thisŽ Ž ..angular range dashed curve in 3 c . The prediction

is however higher by about three units of chargedmultiplicity. This discrepancy is expected from thepreviously obtained result due to differences in thefragmentation of the leading quark or gluon.

At small angles the difference between the pri-mary QCD expectation and the measurement in-creases. Studies using Monte Carlo models haveshown that this is mainly due to genuine two jetevents which have been clustered as symmetricthree-jet events. The models indicate that this contri-bution becomes small for angles above 308.

Fitting the full ansatz 9 to the three-jet multiplic-ity data at angles uG308, using the two parameteri-

Ž . Ž .zations in Eqs. 4 and 5 of the multiplicity ineqey events with their parameters fixed as given inTable 2 but varying C rC and N , yields:A F 0

CAs2.251"0.063 10Ž .

CF

CAs2.242"0.062. 11Ž .

CF

The result confirms with great precision the QCDw xexpectation 1 that the ratio of the radiated multi-

plicity from gluon and quark jets is given by thecolour factor ratio C rC . This result also impliesA F

that the proportionality of the number of gluons tow x Ž .hadrons 1 e.g. Local Hadron Parton Duality LPHD

w x33 applies extremely precisely if only the radiatedgluons from a quark or gluon are considered.

The offset term N is bigger if only b-depleted0

events are used. The central result for C rC , how-A F

ever, remains unchanged within errors. This is due tothe fact that C rC is measured from the change ofA F

multiplicity in three-jet events with opening angleand not from the absolute multiplicity.

Ž .The correctness of the ansatz Eq. 9 and the biasintroduced by two-jet events at small u , were fur-1

ther checked by varying the lowest angle used in thefit. The resulting value for C rC , the x 2rN andA F d f

2 Ž .the x probability of the fit are shown in Fig. 3 d .It is observed that for u )308 satisfactory fits are1

obtained. For this angular range the fitted value ofC rC is stable within errors.A F

Systematic uncertainties of the above result forthe colour factor ratio due to uncertainties in thethree-jet multiplicity data as well as in the parameter-ization of the eqey charged multiplicity and in thetheoretical predictions are considered. To obtain sys-tematic errors interpretable like statistical errors, halfthe difference in the value obtained for C rC whenA F

Ža parameter is modified from its central value see.below is quoted as the systematic uncertainty. All

relative systematic errors are collected in Table 3.Results for C rC obtained from the individualA F

data sets corresponding to the different years ofdata-taking as well as from b-depleted events werefound to be fully compatible within the statisticalerror. To estimate uncertainties in the three-jet multi-plicity the following cuts which are sensitive tomisrepresentation of the data by the Monte Carlosimulation have been varied.1. Cut on the minimal particle momentum: the cut

on the minimal particle momentum has been low-ered from 400 MeV to 200 MeV and raised to600 MeV.

2. Minimum angle of each jet with respect to thebeam axis: this cut has been increased from 308 to408 to test for a possible bias due to the limitedangular acceptance.

3. Minimum number of particles per jet: the mini-mum number of particles per jet has been in-creased from 2 to 4 in order to reject eventswhich may not have a clear three-jet structure.

4. Correction for gluon in leading jet: both methodsof correction were compared to account for glu-ons in the most energetic jet. Furthermore therequirements for the mapping of the parton to thehadron level for defining the gluon jet have beenvaried.


Table 3Systematic uncertainties on C rC as derived from three-jet event multiplicitiesA F

Source Sys. error Combined Combined Total

Experimental uncertainties1. Min. particle momentum " 0.42%2. Min. angle of jet w.r.t. beam " 0.38%3. Min. number of tracks per jet " 0.02%4. Corr. for gluon in jet 1 " 0.11% " 0.58%5. jet algorithms " 1.39%

q y6. e e data sets " 0.90%7. Fit function " 0.02%8. binning and range of fit " 3.08% " 3.21% " 3.55%

Theoretical uncertainties9. Variation of n " 1.51%f

10. Calculation in 1str2nd order " 3.95%11. Setting C fixed " 0.08% " 4.23% "5.52%A

To check the stability of the result for differentchoices of jet algorithms the results obtained for alarge sample of events generated with JETSET havebeen compared with:5. Alternative jet algorithms: the angular ordered

Durham algorithm, LUCLUS without particle re-w xassignment, JADE and Geneva 34 were applied

alternatively to Durham on a large statistics MonteCarlo sample. The results for Durham, angularordered Durham and LUCLUS agree reasonably.The spread among the results was taken as error.The JADE and Geneva algorithm which are

w xknown to tend to form so-called junk jets 34show stronger deviations.The following systematic uncertainties arise from

uncertainties in the experimental input other thanfrom the three-jet multiplicities and from choicesmade for the fits of N q y. These uncertainties aree e

considered as experimental systematic uncertainties.'Ž .q y6. Input of parameterization of N s : to esti-e e

mate the influence of an uncertainty in N q y,e e

different choices of input data were compared:Ø DELPHI multiplicities for 184 GeV and from

Z decays with hard photons combined with'LEP data for 90 GeV - s - 180 GeV;

Ø DELPHI multiplicities from Z decays withhard photons;

Ø eqey data taken at low centre-of-mass ener-Ž .gies TASSO, TPC, MARK-II, HRS, AMY ;

Ø all available eqey data between 10 GeV and

Ž184 GeV TASSO, TPC, MARK-II, HRS,.AMY, LEP combined, DELPHI .

7. Choice of prediction used for fit: the fit functions4 and 5 were used alternatively. For consistencyhere n s5 and a second order a was used.f s

8. Variation of the fitted range: the lower limit ofthe angular range used in the fit was variedbetween 248 and 368 as well as changing half thebin width, e , from 2.58 to 58.Finally, systematic errors due to uncertainties in

the theoretical prediction were considered.9. Variation of n : the number of active quarks, nf f

w x22 , relevant for the hadronic final state is un-certain. n therefore has been varied from 3 to 5.f

Ž .10. Order of calculation LO - NNLO : the predic-Ž . Ž Ž ..tion r k Eq. 6 has been calculated for back-

to-back quarks or gluons. As the jets are wellseparated it is expected to apply for this analysisalso. When the gluon recoils with respect to thequarks the prediction is exact. In addition coher-

Ž .ence effects angular ordering are taken intoaccount in the definition of the scales E) andpH .1

As the coupling for the triple-gluon vertex isbigger than the coupling of all other vertices it isclear that the correction will lower the gluon-to-

Ž .quark multiplicity ratio as in the case of Eq. 6 .w xThe validity of the correction 28 is therefore

assumed for the whole range of angles consid-ered. Conservatively, half of the difference ob-


tained with the lowest order prediction rsC rC and the NNLO prediction is consideredA F

as systematic uncertainty. A leading order a s

was used for the lowest order prediction and asecond order a in the other case. Considerings

that in the three-jet events mainly the gluonscale pH is varied, the resulting error estimate1

Ž .agrees with that given in Eq. 7 .11. Quantities influencing C rC : for the centralA F

result, C rC has been assumed variable in Eq.A FŽ .6 only. The stability of the result was checkedby also leaving C variable in some or all of theA

parameterizations of a and N q y.s e e

To check in how far the offset term N is con-0

stant, N has been extracted for each u -bin individ-0 1

ually fixing C to its default value. The individualA

results are consistent with the average value and notrend is observed.

Ž . Ž .Alternatively to Eq. 9 , Eq. 7 has been fitted toŽ .the data, where the OO a correction factor has beens

Ž Ž H..parameterized as 1qca p . This leads to thes 12 Ž .same fit results for C rC and x as Eq. 9 , whichA F

Ž H.implies that both corrections r p P N and1 0) ) HŽ .2 N Y qN Y Pca p as well as the val-Ž .Ž .q qq g g s 1

ues obtained for C rC agree within "1%. ItA F

should, however, be stressed that the behaviour ofthe fragmentation function requires the presence of anon-perturbative offset term.

The prediction of the multiplicity ratio given byw x Ž .35 has been tried as an alternative to Eq. 6 .Although this calculation takes recoil effects intoaccount, a non-perturbative offset term is still re-quired. The prediction differs by about 10% fromw x28 in the NNLO term. As it does not reproduce thecolour factor ratio contained in the fragmentationmodels which describe the data well, it has not beenapplied in this analysis.

Ž . Ž .Averaging the results given in Eqs. 10 and 11and adding in quadrature the systematic errors sum-marized in Table 3 gives the following final result:

CAs2.246"0.062 stat .Ž .

CF

"0.080 syst . "0.095 theo. 12Ž . Ž . Ž .

This result confirms the QCD expectation that gluonbremsstrahlung is stronger from gluons than from

quarks by the colour factor ratio C rC and is directA F

evidence for the triple-gluon coupling.This measurement yields the most precise result

obtained so far for the colour factor ratio C rC .A F

Even the best measurements from four-jet angularw xdistributions 36 suffer from the relatively small

number of four-jet events available. Furthermore,many of these measurements specify no theoreticalsystematic error as they so far rely on leading ordercalculations. It is remarkable that this measurementof C rC is performed from truly hadronic quanti-A F

ties, the charged multiplicities. Jets, i.e. partonicquantities only enter indirectly via the definition ofthe scales E) and pH .1

In order to illustrate comprehensively the contentsof the measurement of the three-jet multiplicity wecompare in Fig. 4 the multiplicity corresponding to agg and a qq final state. The qq multiplicity is takento be the multiplicity measured in eqey annihilationcorrected for the bb contribution as described above.The gg multiplicity at low scale values is taken from

w xthe CLEO measurement 10 , for which no system-atic error was specified. At higher scale, twice thedifference of the three-jet multiplicity and the qq

Ž Ž ..term the first term in Eq. 9 is interpreted as thegg multiplicity. The gg data should be extendable tohigher energies by measuring the multiplicity in ppscattering as a function of the transverse energy. Thedashed curve through the qq points is a fit of the

Ž . Ž .prediction according to Eqs. 4 or 5 . The gg lineis the perturbative expectation for back-to-back glu-

Ž .ons according to the second term of Eq. 9 . N is0Ž .taken from Eq. 13 . In principle N is a property of0

the complete three-jet event, so it is unclear if thesubtraction of the full amount of N is justified in0

order to obtain the gluon jet multiplicity. However,this only introduces a constant shift in the ‘‘ggevent’’ multiplicity, the scale dependence of thegluon jet multiplicity remains unaltered. The plotshows again that the increase of the gg multiplicitywith scale is about twice as big as in the qq case,illustrating the large gluon-to-quark colour factorratio C rC .A F

It is of interest to present also a dedicated mea-surement of the non-perturbative parameter N . In0

order to obtain this value, b-depleted events havebeen used. A fit of the three-jet event multiplicityhas then been performed with N as the only free0


Fig. 4. Comparison of the charged hadron multiplicity for an initial qq and a gg pair as function of the scale. The dashed curve is a fitŽ . Ž . Ž .according to Eqs. 4 or 5 , the full line is twice the second term of Eq. 9 . The grey band indicates the uncertainty due to the error of N .0

w xThe DELPHI gg data will be made available in the DurhamrRAL database 38 .

parameter. C rC has been set to its default value.A F

The parameterization of the eqey multiplicity ac-Ž . q ycording to Eq. 4 uses the low energy e e data as

input. The fit yields:

N s1.91"0.03 stat . "0.33 syst . 13Ž . Ž . Ž .0

The systematic error was estimated as for C rC .A F

Furthermore a normalization error due to the multi-plicity in eqey events has been added in quadrature.This error has been assumed to be given by the errorof the precise average multiplicity at the Z resonancew x37 . The actual value of N f2 corresponds to0

Ž .about one primary particle see also Section 3.1 .This is indeed a reasonable value which had already

w xbeen expected in Ref. 1 .

4. Summary

In summary, the dependence of the charged parti-cle multiplicity in quark and gluon jets on the trans-verse momentum-like scale has been investigatedand the charged hadron multiplicity in symmetric

three-jet events has been measured as a function ofthe opening angle u .1

The ratio of the variations of gluon and quark jetmultiplicities with scale agrees with the QCD expec-tation and directly reflects the higher colour chargeof gluons compared to quarks. This can also beinterpreted as direct evidence for the triple-gluoncoupling, one of the basic ingredients of QCD. It isof special importance that this evidence is due tovery soft radiated gluons and therefore complemen-tary to the measurement of the triple-gluon couplingin four-jet events at large momentum transfer.

The increase of the gluon to quark jet multiplicityratio with increasing scale is understood as being dueto a difference in the fragmentation of the leadingquark or gluon. The simultaneous description of thequark and gluon jet multiplicities with scale alsosupports the Local Parton Hadron Duality hypothesisw x33 although large non-perturbative terms for theleading quark or gluon are responsible for the ob-served relatively small gluon to quark jet multiplicityratio.

Using the novel method of measuring the evolu-tion of the multiplicity in symmetric three-jet events


with their opening angle, a precise result for thecolour factor ratio is obtained:

CAs2.246"0.062 stat .Ž .

CF

"0.080 syst . "0.095 theo.Ž . Ž .It is superior in precision to the best measurements

w xfrom four-jet events 36 . Finally it is remarkable thatthis measurement is directly performed from trulyhadronic quantities. Jets only enter indirectly via thedefinition of the energy scale of the quark-antiquarkpair and the transverse momentum scale of the gluon.These scales are calculated directly from the jetangles.

Acknowledgements

We would like to thank V.A. Khoze for hisinterest in this analysis and many enthusiastic discus-sions and explanations. We thank S. Lupia and W.

Ž .Ochs for providing us with their program for Eq. 5 .We are greatly indebted to our technical collabo-

rators, to the members of the CERN-SL Division forthe excellent performance of the LEP collider and tothe funding agencies for their support in building andoperating the DELPHI detector. We acknowledge inparticular the support of: Austrian Federal Ministryof Science and Traffics, GZ 616.364r2-IIIr2ar98;FNRS–FWO, Belgium; FINEP, CNPq, CAPES,FUJB and FAPERJ, Brazil; Czech Ministry of Indus-try and Trade, GA CR 202r96r0450 and GA AVCRA1010521; Danish Natural Research Council; Com-

Ž .mission of the European Communities DG XII ;Direction des Sciences de la Matiere, CEA, France;´Bundesministerium fur Bildung, Wissenschaft,¨Forschung und Technologie, Germany; General Sec-retariat for Research and Technology, Greece; Na-

Ž .tional Science Foundation NWO and FoundationŽ .for Research on Matter FOM , The Netherlands;

Norwegian Research Council; State Committee forScientific Research, Poland, 2P03B06015, 2P03B0-3311 and SPUBrP03r178r98; JNICT–Junta Na-cional de Investigacao Cientıfica e Tecnologica, Por-˜ ` `tugal; Vedecka grantova agentura MS SR, Slovakia,Nr. 95r5195r134; Ministry of Science and Technol-ogy of the Republic of Slovenia; CICYT, Spain,

AEN96–1661 and AEN96-1681; The Swedish Natu-ral Science Research Council; Particle Physics andAstronomy Research Council, UK; Department ofEnergy, USA, DE–FG02–94ER40817.

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11 March 1999


Strangeness enhancement at mid-rapidity in Pb–Pb collisionsat 158 A GeVrc

WA97 Collaboration

E. Andersen b, F. Antinori e,k, N. Armenise a, H. Bakke b, J. Ban g, D. Barberis f,´H. Beker e, W. Beusch e, I.J. Bloodworth d, J. Bohm m, R. Caliandro a,¨M. Campbell e, E. Cantatore e, N. Carrer k, M.G. Catanesi a, E. Chesi e,

M. Dameri f, G. Darbo f, A. Diaczek l, D. Di Bari a, S. Di Liberto n, B.C. Earl d,D. Elia a, D. Evans d, K. Fanebust b, R.A. Fini a, J.C. Fontaine i, J. Ftacnik g,´ˇ

B. Ghidini a, G. Grella o, M. Guida o, E.H.M. Heijne e, H. Helstrup c, A.K. Holme j,D. Huss i, A. Jacholkowski a, G.T. Jones d, P. Jovanovic d, A. Jusko g,

T. Kachelhoffer p, J.B. Kinson d, A. Kirk d, W. Klempt e, B.T.H. Knudsen b,K. Knudson e, I. Kralik e, V. Lenti a, R. Lietava g, R.A. Loconsole a,´

G. Løvhøiden j, M. Luptak g, V. Mack i, V. Manzari a, P. Martinengo e,´M.A. Mazzoni n, F. Meddi n, A. Michalon p, M.E. Michalon-Mentzer p,

P. Middelkamp e, M. Morando k, M.T. Muciaccia a, E. Nappi a, F. Navach a,P.I. Norman d, B. Osculati f, B. Pastircak g, F. Pellegrini k, K. Pıska m,ˇ´ ´ˇ

F. Posa a, E. Quercigh e, R.A. Ricci h, G. Romano o, G. Rosa n, L. Rossi f,e ˇ e a f ˇ e,gH. Rotscheidt , K. Safarık , S. Saladino , C. Salvo , L. Sandor ,ˇ ´

G. Segato k, M. Sene l, R. Sene l, S. Simone a, W. Snoeys e, P. Staroba m,´ ´S. Szafran l, M. Thompson d, T.F. Thorsteinsen b, G. Tomasicchio a, G.D. Torrieri d,

T.S. Tveter j, J. Urban g, M. Venables d, O. Villalobos Baillie d, T. Virgili o,´A. Volte l, M.F. Votruba d, P. Zavada m´

a Dipartimento I.A. di Fisica dell’UniÕersita e del Politecnico di Bari and Sezione INFN, Bari, Italy`b Fysisk Institutt, UniÕersitetet i Bergen, Bergen, Norway

c Høgskolen i Bergen, Bergen, Norwayd School of Physics and Astronomy, UniÕersity of Birmingham, Birmingham, UK

e CERN, European Laboratory for Particle Physics, GeneÕa, Switzerlandf Dipartimento di Fisica dell’UniÕersita and Sezione INFN, Genoa, Italy`

g Institute of Experimental Physics, SloÕak Academy of Sciences, Kosice, SloÕakiaˇh INFN, Laboratori Nazionali di Legnaro, Legnaro, Italyi GRPHE, UniÕersite de Haute Alsace, Mulhouse, France´

j Fysisk institutt, UniÕersitetet i Oslo, Oslo, Norwayk Dipartimento di Fisica dell’UniÕersita and Sezione INFN, Padua, Italy`


( )E. Andersen et al.rPhysics Letters B 449 1999 401–406402

l College de France and IN2P3, Paris, France`m Institute of Physics, Academy of Sciences of Czech Republic, Prague, Czech Republicn Dipartimento di Fisica dell’UniÕersita ‘‘La Sapienza’’ and Sezione INFN, Rome, Italy`

o Dipartimento di Scienze Fisiche ‘‘E.R. Caianiello’’ dell’UniÕersita and INFN, Salerno, Italy`p Institut de Recherches Subatomiques, IN2P3rULP, Strasbourg, France

Received 8 January 1999; revised 25 January 1999Editor: L. Montanet

Abstract

K 0, L, J , V and negative particle yields and transverse mass spectra have been measured at central rapidity in Pb–PbS

and p–Pb collisions at 158 A GeVrc. Yields are studied as a function of the number of nucleons participating in thecollision N , which is estimated with the Glauber model. From p–Pb to Pb–Pb collisions the particle yields per participantpart

increase substantially. The enhancement is more pronounced for multistrange particles, and exceeds an order of magnitudefor the V . For a number of participants, N , greater than 100, however, all yields per participant appear to be constant.part

q 1999 Published by Elsevier Science B.V. All rights reserved.

The study of relativistic heavy-ion collisions pro-vides a unique opportunity to search for a newpredicted state of matter – the quark-gluon plasmaŽ .QGP . Several experimental signatures which couldsignal the onset of the QGP phase have been pro-

Ž w x.posed. For a recent review see Ref. 1 . Recently, anumber of experimental observations that could indi-cate a phase transition to QGP have been presentedw x2–5 .

Strange particles produced in heavy-ion collisionsgive important information on the collision mecha-nism. In particular, the enhanced relative yield ofstrange and multi-strange particles in nucleus-nucleuswith respect to proton-nucleus interactions has beensuggested as one of the sensitive signatures for a

w xphase transition to a QGP state 6,7 . It is expectedthat the enhancement should be more pronounced for

w xmulti-strange than for singly strange particles 8 . Forw xa recent review of the subject, see Refs. 9,10 .

The WA97 experiment addresses strangeness pro-duction in Pb–Pb collisions and is designed to studythe yields of strange particles and antiparticles carry-ing one, two and three units of strangeness as afunction of the number of nucleons taking part in thecollision. The WA97 experimental set-up, its silicontelescope and the use of the multiplicity detectors is

w xdescribed in Ref. 4 .Recently we published data on the L, J and V

yields in Pb–Pb interactions as a function of colli-w xsion centrality and compared with yields in p–Pb 4 .

We observed a strong increase in the production atmid-rapidity for L, J and V hyperons and anti-hy-perons in Pb–Pb collisions with respect to p–Pbcollisions and this enhancement exhibited a markedhierarchy, i.e. the V enhancement is larger than thatof the J , and the J enhancement is larger than thatof the L. The present letter elaborates on these

Žfindings with improved statistics a factor of two.higher than in the previous work . The analysis of

0 Ž y.K and negative particles h is now also included.S

In the p–Pb runs we have collected data with twodifferent trigger conditions:Ž .a at least two tracks in the telescope, as required to

find V 0’s;Ž .b at least one track in the telescope.

Ž .Sample a was used for the strange particle studyŽ . yand sample b for the h study. In both cases the

effect of the trigger has been taken into account inthe calculation of the particle yields. In the Pb–Pbsample the trigger corresponded to the most central

w x40% of the total inelastic cross section, see Ref. 4 .We selected as hy those negative tracks which

pointed to the interaction vertex.The K 0 were identified by their decayS

K 0 ™pyqpqS

To ensure that K 0 is not ambiguous with L, a cutS< <for a F0.45 was made in the Armenteros-Podo-A

w xlanski plot 11 .

( )E. Andersen et al.rPhysics Letters B 449 1999 401–406 403

The L, Jy, Vy hyperons and their antiparticleswere identified by reconstructing their decays intofinal states containing only charged particles:

L ™ pqpy

y yJ ™ Lqpyπpqp

y yV ™ LqKyπpqp

The details of the analysis, i.e. the extraction of thevarious particle signals and the weighting procedures

w xare discussed in Refs. 4,12,13 .In the geometry of our experiment the feed-down

from weak decays is expected to be of minor impor-tance, and the L and J data have not been cor-rected for feed-down from cascade decays. It isestimated to be less than 5% for L and less than

y q10% for L. For both J and J the feed-downfrom V decays is less than 2%.

Ž .The mass resolution is better than 6 MeV FWHMfor all signals.

The acceptance windows for L, J and V fromPb–Pb and p–Pb collisions are shown in our previ-

w x y 0ous publication 4 . For negatives h and K theS

acceptance windows are shown in Fig. 1. The differ-ential distributions of the yield per event for each

kind of particle were fitted in their respective accep-tance windows using the expression

d2N mTas f y m exp y 1Ž . Ž .T ž /dm d y TT

where m is the transverse mass, y is the rapidityT

and as1.5. The fit was performed using the methodof maximum likelihood.

For the present analysis with limited statistics wehave assumed the rapidity distributions to be flat for< < Ž .. Ž .y y y - 0.5, i.e. in expression 1 f y scm

constant. We have investigated the systematic errorwhich this assumption could introduce in the case of

y 0p–Pb for h , K , L and L, where published dataSw x w xexist for p–Au 14 and p–S 15 collisions. We find

that using a flat rapidity distribution, instead of onew xobtained from a fit to the published data 14 , changes

the values of T by less than 2%, 5%, 5% and 10%,y 0in the case of the h , K , L and L distributions,S

respectively. The corresponding changes in the parti-Ž .cle yields, defined by Eq. 2 below, are less than

10%, 5%, 5% and 6%.For each particle species, the values for the slope

T were calculated both for the p–Pb sample and

y Ž . 0Fig. 1. Acceptance windows for h assumed to be pions and K . For Pb–Pb collisions, the symmetry of the system around midrapidityS

allows reflection of the acceptance windows around y .cm


w xPb–Pb sample. These values, given in Ref. 12 , areused in the analysis which follows.

The WA97 multiplicity detectors allow us to studyparticle yields as a function of collision centrality asmeasured by the number of participants N . To thispart

purpose the multiplicity spectrum is divided into four² :bins and the average number of participants Npart

w xfor each bin is calculated as described in Ref. 4 .For p–Pb the number of participants corresponds tothe estimated average for minimum bias collisions.The particle production yield per event, Y, in eachcentrality bin is defined by the integral

`2d Ny q0.5cm

Ys d p d y 2Ž .H HT d y d p0 y y0.5 Tcm

< <where the extrapolation to the window yyy -cm

0.5 and p )0 GeVrc is done according to expres-TŽ . w xsion 1 using the values of T given in Ref. 12 .

Fig. 2 shows particle yields per event for p–Pband Pb–Pb interactions as a function of the number

² :of participants N . The vertical error bars corre-part

spond to statistical uncertainties only, and do notinclude systematic errors from feed-down nor fromthe assumption of a flat rapidity distribution in ouracceptance window. As discussed above, these are

estimated to be small relative to the current statisticalerrors. For the hy yield in p–Pb collisions, however,a 15% systematic error has been introduced to ac-count for the uncertainties due to the single trackbackground subtraction procedure. The horizontalbars show the root-mean-square values of the num-ber of participants in the selected bins for Pb–Pbcollisions, and the range corresponding to 80% of thecross section in p–Pb.

In Fig. 2 the particles are divided into two groups.Fig. 2a shows the yields of particles with at least one

Ž ycommon valence quark with the nucleon J , L,y 0.h and of the K , which has contributions ds andS

ds. Fig. 2b refers to particles with no commonq yvalence quark with the nucleon: L, J and V q

qV . It is instructive to analyze them separately sincethe particles in the two groups are empirically knownto exhibit different production features, e.g. L andL have different rapidity spectra both in p–S and

w xS–S 14 . Fig. 3a,b shows the particle yields ex-pressed in units of the corresponding yield per p–Pb

Žinteraction i.e. each yield is rescaled so that the.value for p–Pb is set to one . The particle yields in

Ž .Pb–Pb are compared to a yield curve full linedrawn through the p–Pb points and proportional tothe number of participants, N .part

y 0 y q y qŽ . Ž . Ž .Fig. 2. Yields, defined in Eq. 2 , as a function of the number of participants for a h , K , L and J ; b L, J and V qV . NoteSq y qthat the yields for J and V qV are very similar.

( )E. Andersen et al.rPhysics Letters B 449 1999 401–406 405

Fig. 3. Yields, expressed in units of yields observed in p–PbŽ . ycollisions, as a function of the number of participants for a h ,

0 y q y qŽ .K , L and J ; b L, J and V q V . The solid lineS

represents a function through the p–Pb point proportional to the² :number of participants N . The corresponding yields perpart

Ž . Ž .participant are shown in c and d . The proton points arejuxtaposed on the horizontal scale.

All yields appear to increase with centrality fromp–Pb to Pb–Pb faster than linearly with the numberof participants. However, within our experimentalcentrality range for Pb–Pb, i.e. for N )100, wepart

observe that all particle yields per participant appearto be constant. This is illustrated in Fig. 3c and Fig.3d, where we present the particle yield per partici-

² : ² : ² :pant, Y r N , as a function of N .part part

For each particle species we then compute aglobal enhancement, E, going from p–Pb to Pb–Pbcollisions, defined as

² : ² :Y YEs , 3Ž .² : ² :ž / ž /N Npart part pyPbPbyPb

² : ² :where Y and N are averaged over the fullpart

centrality range covered by the experiment. E mea-sures the enhancement at midrapidity for the varioushadron species. The values E for each particle aredisplayed in Fig. 4. Similar enhancement values areobtained if we use the data before the extrapolationto the full yyp window. We note that the en-T

hancement E increases with the strangeness content:

y q qE V qV )E J )E LŽ . Ž . Ž .and

E Jy )E L fE K 0 .Ž . Ž . Ž .S

In summary, the strange particle yields per partic-ipant at central rapidity increase from p–Pb to Pb–Pb.The enhancement is more pronounced for multi-strange particles, and exceeds one order of magni-

w xtude in the case of V . As pointed out in Ref. 16 ,such a behaviour contradicts expectations fromhadronic rescattering models, where secondary pro-

Ž .duction of multi-strange anti baryons is hindered byhigh mass thresholds and low cross sections. Withinthe participant range N )100, corresponding topart

our Pb–Pb data, all yields are found to increaseproportionally to N , as it would be expected ifpart

Fig. 4. Strange particle enhancement versus strangeness content.


strange quarks are equilibrated in a deconfined andchirally symmetric quark gluon plasma.

Acknowledgements

We are grateful to U. Heinz, C. Lourenco and J.Rafelski for fruitful discussions.

References

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11 March 1999


Cumulative author index to volumes 441–449

Abbaneo, D., 445, 239; 447, 336Abbiendi, G., 443, 394; 444, 539; 447, 134,

157Abe, K., 447, 167Abe, T., 443, 394Abel, S., 444, 427Abel, S.A., 444, 119Abele, A., 446, 349Abraham, K.J., 446, 163Abramowicz, H., 443, 394Abreu, M.C., 444, 516; 449, 128Abreu, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Abriola, D., 447, 41Acciarri, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Acebal, J.L., 445, 94Achard, P., 447, 147; 448, 152Achasov, M.N., 449, 122Achterberg, E., 447, 41Ackerstaff, K., 444, 531, 539; 447, 134, 157Acosta, D., 443, 394Adam, W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Adamczyk, L., 443, 394Adams, J., 447, 240Adamus, M., 443, 394Adler, J.-O., 442, 43Adomeit, J., 446, 349Adriani, O., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Adye, T., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Adzic, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Afanasiev, S., 445, 14Affholderbach, K., 445, 239; 447, 336Agashe, K., 444, 61Aglietti, U., 441, 371Agodi, C., 442, 48Aguilar-Benitez, M., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Aguirre, R.M., 449, 161Ahlen, S., 444, 503, 569; 445, 428; 446, 368Ahn, C., 442, 109Ahn, J.K., 444, 267

Ahn, S.H., 443, 394Airapetian, A., 442, 484; 444, 531Aitala, E.M., 445, 449; 448, 303Ajaltouni, Z., 445, 239; 447, 336Ajinenko, I., 446, 75; 448, 311; 449, 364, 383Akama, K., 445, 106Akeroyd, A.G., 441, 224; 442, 335Akhmedov, E.T., 442, 152Akopov, N., 442, 484; 444, 531Akushevich, I., 442, 484; 444, 531Alamanos, N., 442, 48Alavi-Harati, A., 447, 240Alba, R., 442, 48Albrecht, Z., 448, 311; 449, 364, 383Albuquerque, I.F., 447, 240Alcaraz, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Alderweireld, T., 448, 311; 449, 364, 383Aldeweireld, T., 444, 491; 446, 62, 75Alekseev, G.D., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Alemanni, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Alemany, R., 441, 479; 444, 491; 445, 239;

446, 62, 75; 447, 336; 448, 311; 449, 364,383

ALEPH Collaboration, 445, 239; 447, 183, 336,355

Aleppo, M., 445, 239; 447, 336Alexander, G., 444, 539; 447, 134, 157Alexandre, J., 445, 351Alexopoulos, T., 447, 240Aliev, T.M., 441, 410Allaby, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Allison, J., 444, 539; 447, 134, 157Allison, W.W.M., 449, 137Allmendinger, T., 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Allport, P.P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Almehed, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Alner, G.J., 449, 137Aloisio, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152

CumulatiÕe author index to Õolumes 441–449408

Altegoer, J., 445, 439Altekamp, N., 444, 539; 447, 134, 157Altmann, M., 447, 127Alvarez, D.E., 447, 41Alviggi, M.G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Amako, K., 447, 167Amaldi, U., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Amarian, M., 442, 484; 444, 531Amato, S., 441, 479; 444, 491; 445, 449; 446,

62, 75; 448, 303, 311; 449, 364, 383Ambjørn, J., 445, 307Ambrosi, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Amelung, C., 443, 394Ammon, R., 447, 127Amsler, C., 446, 349Amzal, N., 443, 69, 82An, S.H., 443, 394Anassontzis, E.G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Anderhub, H., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Andersen, E., 449, 401Andersen, J., 446, 117Anderson, K.J., 444, 539; 447, 134, 157Anderson, S., 444, 539; 447, 134, 157Andersson, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Andreazza, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Andreev, V.P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Andringa, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Angelantonj, C., 444, 309Angelescu, T., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Angelini, C., 445, 439Angelopoulos, A., 444, 38, 43, 52Anisovich, A.V., 449, 145, 154Anjos, J.C., 445, 449; 448, 303Annand, J.R.M., 442, 43Anselmino, M., 442, 470Anselmo, F., 443, 394; 444, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Antilogus, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Antinori, F., 447, 178; 449, 401Antonelli, A., 445, 239; 447, 336Antonelli, M., 445, 239; 447, 336Antoniadis, I., 444, 284Antonioli, P., 443, 394Antoniou, N.G., 444, 583Antonov, D., 444, 208Antonuccio, F., 442, 173Anzivino, G., 446, 117

Aoi, N., 448, 180Aoki, S., 444, 267Apel, W.-D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Apostolakis, A., 444, 38, 43, 52Appel, J.A., 445, 449; 448, 303Appelshauser, H., 444, 523¨Arai, M., 449, 230Arai, Y., 447, 167Aranda, A., 443, 352Arcelli, S., 444, 539; 447, 134, 157Arcidiacono, R., 446, 117Ardouin, D., 446, 191Arefiev, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Arenhovel, H., 447, 15¨Arenton, M., 447, 240Argyres, P.C., 441, 96; 442, 180Arima, A., 445, 1Arisaka, K., 447, 240Arkhipov, V., 445, 14Armbruster, P., 444, 32Armenise, N., 449, 401Armesto, N., 442, 459Armoni, A., 449, 76Armstrong, S.R., 445, 239; 447, 336Arneodo, M., 443, 394Arnoud, Y., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Arutyunov, G.E., 441, 173Asai, M., 443, 409Asai, S., 444, 539; 447, 134, 157Asano, Y., 447, 167Aschenauer, E.C., 442, 484; 444, 531Ashby, S.F., 444, 539; 447, 134, 157Ashery, D., 445, 449; 448, 303Aslanides, E., 444, 38, 43, 52Asman, B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Astier, P., 445, 439Astruc, J., 449, 128Asztalos, S.J., 449, 6Aubert, J.J., 445, 239; 447, 336Auger, F., 442, 48Augustin, I., 446, 117Augustin, J.-E., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Augustinus, A., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Autiero, D., 445, 439Avakian, H., 442, 484; 444, 531Avakian, R., 442, 484; 444, 531Averitte, S., 447, 240Avetissian, A., 442, 484; 444, 531Avila, C., 445, 419Axen, D., 444, 539; 447, 134, 157Axenides, M., 444, 190; 447, 67Ayres, D.S., 449, 137

CumulatiÕe author index to Õolumes 441–449 409

Azcoiti, V., 444, 421Azemoon, T., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Aziz, T., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Azuelos, G., 444, 539; 447, 134, 157Azzurri, P., 445, 239; 447, 336

Bacci, C., 447, 127Bachler, J., 444, 523¨Back, T., 443, 69, 82¨Backenstoss, G., 444, 38, 43, 52Bacon, T.C., 443, 394Badaud, F., 445, 239; 447, 336Badgett, W.F., 443, 394Bagliesi, G., 445, 239; 447, 336Baglin, C., 444, 516; 449, 128Bagnaia, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bai, J.Z., 446, 356Bailey, D.C., 443, 394Bailey, D.S., 443, 394Bailey, S.J., 444, 523Bailin, D., 443, 111Baillon, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bains, B., 442, 484; 444, 531Bajc, B., 447, 313Bakas, I., 445, 69Baker, C.A., 446, 349; 449, 114, 145, 154Baker, W.F., 445, 419Bakke, H., 449, 401Bakker, B.L.G., 449, 267Baksay, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Balandras, A., 447, 147; 448, 152Balata, M., 447, 127Baldini, R., 444, 111Baldisseri, A., 445, 439Baldit, A., 444, 516; 449, 128Baldo-Ceolin, M., 445, 439Ball, A.H., 444, 539; 447, 134, 157Ball, R.C., 447, 147; 448, 152Ballesteros, H.G., 441, 330Ballocchi, G., 445, 439Bambade, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bamberger, A., 443, 394Ban, J., 449, 401´Ban, Y., 446, 356Bando, M., 444, 373Banerjee, S., 444, 503, 503, 569; 445, 428, 449;

446, 368; 447, 147; 448, 152, 303Banerjee, Sw., 444, 569; 445, 428; 446, 368;

447, 147; 448, 152Banicz, K., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Banner, M., 445, 439

Barao, F., 441, 479; 444, 491; 446, 62, 75; 448,311; 449, 364, 383

Barate, R., 445, 239; 447, 183, 336Barbagli, G., 443, 394Barberio, E., 444, 539; 447, 134, 157Barberis, D., 446, 342; 449, 401Barbero, C., 445, 249Barbiellini, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Barbier, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Barbieri, R., 445, 407Barczyk, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bardin, D.Y., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Barenboim, G., 443, 317Bargassa, P., 444, 38, 43, 52Barger, V., 442, 255Bari, G., 443, 394Barillere, R., 444, 503, 569; 445, 428; 446,`

368; 447, 147; 448, 152Bark, R., 443, 69Barker, A.R., 447, 240Barker, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Barlow, R.J., 444, 539; 447, 134, 157Barna, D., 444, 523Barnby, L.S., 444, 523Barnea, N., 446, 185Barnes, R.P., 447, 178Barnett, B.M., 446, 349Baroncelli, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Barone, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Barone, M., 447, 122Barr, G., 446, 117; 449, 137Barr, S.M., 448, 41Barreiro, F., 443, 394Barreiro, T., 445, 82Barret, O., 443, 394Barrett, W.L., 449, 137Barrow, J.D., 443, 104; 447, 246Barrow, S., 444, 531Bartalini, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bartels, J., 442, 459Barth, J., 444, 555; 445, 20Bartke, J., 444, 523Bartoldus, R., 444, 539; 447, 134, 157Barton, R.A., 444, 523Baru, S.E., 449, 122Basa, S., 445, 439Baschirotto, A., 444, 503, 569; 445, 428; 446,

368Bashindzhagyan, G.L., 443, 394Bashkirov, V., 443, 394


Basile, M., 443, 394; 444, 503, 569; 445, 428;446, 368; 447, 147; 448, 152

Baskerville, W.K., 448, 275Bass, S.A., 442, 443; 446, 191; 447, 227Bassetto, A., 443, 325Bassompierre, G., 445, 439Batalin, I., 441, 243Batalin, I.A., 446, 175Batley, J.R., 444, 539; 447, 134, 157Battaglia, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Battiston, R., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Batty, C.J., 446, 349; 449, 114, 145, 154Baubillier, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bauer, W., 444, 231Bauerdick, L.A.T., 443, 394Baulieu, L., 441, 250Baumann, S., 444, 539; 447, 134, 157Baumann, T., 444, 32Baumgarten, C., 442, 484; 444, 531Bay, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bayes, A.C., 447, 178Beane, S.R., 444, 147Beaumel, D., 448, 180Becattini, F., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Beccaria, M., 448, 129Bechtluft, J., 444, 539; 447, 134, 157Becirevic, D., 444, 401Becker, H.G., 446, 117Becker, U., 444, 503, 569; 445, 239, 428; 446,

368; 447, 147, 336; 448, 152Beckmann, M., 442, 484; 444, 531Becks, K.-H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bediaga, I., 445, 449; 448, 303Bedjidian, M., 444, 516; 449, 128Bednarek, B., 443, 394Bednyakov, V., 442, 203Begalli, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Behner, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Behnke, O., 444, 38, 43, 52Behnke, T., 444, 539; 447, 134, 157Behrend, R.E., 444, 163Behrens, U., 443, 394Behrndt, K., 442, 97Beier, H., 443, 394Beilliere, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Beker, H., 449, 401Belitsky, A.V., 442, 307Bell, K.W., 444, 539; 447, 134, 157Bella, G., 444, 539; 447, 134, 157

Bellagamba, L., 443, 394Bellantoni, L., 447, 240Bellavance, A., 447, 240Bellerive, A., 444, 539; 447, 134, 157Belli, P., 447, 127Bellia, G., 442, 48Bellotti, E., 447, 127Belokopytov, Yu., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Belostotski, S., 444, 531Belostotski, St., 442, 484Belous, K., 441, 479; 444, 491Belyaev, A., 447, 331Belz, E., 442, 484Belz, J., 447, 240Belz, J.E., 444, 531Ben-David, R., 447, 240Benakli, K., 447, 51Benayoun, M., 446, 349Benchouk, C., 445, 239; 447, 336Bencivenni, G., 445, 239; 447, 336Benczer-Koller, N., 446, 22Bender, M., 446, 117Benedetti, R., 441, 60Beneke, M., 443, 308Benelli, A., 444, 38, 43, 52Benisch, T., 444, 531Benisch, Th., 442, 484Benlliure, J., 444, 32Bennhold, C., 445, 20Benslama, K., 445, 439Bentvelsen, S., 444, 539; 447, 134, 157Benvenuti, A.C., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Berat, C., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Berdoz, A., 446, 349Berdugo, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Berdyugin, A.V., 449, 122Berezinsky, V., 444, 387; 449, 237Berg, B.A., 444, 487Berges, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Berggren, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bergman, D.R., 447, 240Bergman, O., 441, 133Bering, K., 446, 175Berkooz, M., 449, 68Berlich, R., 445, 239; 447, 336Bern, Z., 444, 273; 445, 168Bernabei, R., 447, 127Bernas, M., 444, 32Bernreuther, S., 442, 484; 444, 531Bernstein, A.M., 442, 20Bertani, M., 444, 111Berthomieu, G., 447, 127


Bertin, V., 444, 38, 43, 52Bertini, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bertini, M., 442, 398Bertolin, A., 443, 394Bertram, I., 443, 347Bertrand, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bertucci, B., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bes, D.R., 446, 93Besancon, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383BES Collaboration, 446, 356Besson, N., 445, 439Betev, B.L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bethke, S., 444, 539; 447, 134, 157Betsch, A., 446, 179Bettarini, S., 445, 239; 447, 336Betteridge, A.P., 445, 239; 447, 336Betts, S., 444, 539; 447, 134, 157Beuchert, K., 446, 349Beusch, W., 446, 342; 447, 178; 449, 401Beuselinck, R., 445, 239; 447, 336Bhadra, S., 443, 394Bhattacharya, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Białkowska, H., 444, 523Bian, J.G., 446, 356Bianchi, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bianchi, N., 442, 484; 444, 531Biasini, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bicudo, P., 442, 349Biebel, J., 448, 125Biebel, O., 444, 539; 447, 134, 157Bienlein, J.K., 443, 394Bigi, M., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Biguzzi, A., 444, 539; 447, 134, 157Biino, C., 446, 117Bijnens, J., 441, 437Biland, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bilei, G.M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bilenky, M.S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bilenky, S.M., 444, 379Bilger, R., 443, 77; 446, 179, 363Billmeier, A., 444, 523Bimonte, G., 441, 69Binetruy, P., 441, 52, 163´Bing, O., 445, 423Bini, C., 444, 111Binnie, D.M., 445, 239; 447, 336

Binon, F.G., 446, 342Bird, I., 445, 439Bird, S.D., 444, 539; 447, 134, 157Birse, M.C., 446, 300Bischoff, S., 446, 349Bizouard, M.-A., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Bizzeti, A., 446, 117Black, S.N., 445, 239; 447, 336Blaikley, H.E., 443, 394Blair, G.A., 445, 239; 447, 336Blaising, J.J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Blanc, F., 444, 38, 43, 52Blanchard, S., 444, 531Blaylock, G., 445, 449; 448, 303Bleicher, M., 442, 443; 447, 227Blick, A.M., 446, 342Blobel, V., 444, 539; 447, 134, 157Bloch, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bloch-Devaux, B., 445, 239; 447, 336Bloch, P., 444, 38, 43, 52Blom, H.M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Blondel, A., 445, 239; 447, 336Bloodworth, I.J., 444, 539; 447, 134, 157; 449,

401Blouw, J., 442, 484; 444, 531Blucher, E., 447, 240Blum, P., 446, 349; 449, 114¨Blumenfeld, B., 445, 439Blumenfeld, Y., 442, 48Blumer, H., 446, 117¨Blyth, C.O., 444, 523Blyth, S.C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bobbink, G.J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Bobinski, M., 444, 539Bobisut, F., 445, 439Boccali, T., 445, 239; 447, 336Bock, G.J., 447, 240Bock, P., 444, 539; 447, 134, 157Bock, R., 444, 503, 523, 569; 445, 428; 446,

368; 447, 147; 448, 152Bockhorst, M., 445, 20Bocquet, G., 446, 117Bode, C., 449, 137Boger, J., 447, 127Bohm, A., 444, 503, 569; 445, 428; 446, 368;¨

447, 147; 448, 152Bohm, J., 449, 401¨Bohme, J., 444, 539; 447, 134, 157¨Bohnet, I., 443, 394Bohrani, A., 449, 128Bohrer, A., 445, 239; 447, 336¨Boivin, M., 445, 423


Boix, G., 445, 239; 447, 336Bokel, C., 443, 394Boldizsar, L., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Bologna, G., 445, 239; 447, 336Bolz, M., 443, 209Bombaci, I., 447, 352Bonacorsi, D., 444, 539; 447, 134, 157Bondarev, V., 445, 14Bonesini, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bonissent, A., 445, 239; 447, 336Bonivento, W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Boonekamp, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Booth, C.N., 445, 239; 447, 336Booth, P.S.L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Borasoy, B., 447, 98Bordalo, P., 444, 516; 449, 128Border, P.M., 449, 137Borgia, B., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Borgland, A.W., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Borisov, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Borissov, A., 442, 484; 444, 531Bormann, C., 444, 523Bornheim, A., 443, 394Bortignon, P.F., 444, 1Borzemski, P., 443, 394Boscherini, D., 443, 394Bosio, C., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Bossi, F., 445, 239; 447, 336Botje, M., 443, 394Botner, O., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bottcher, H., 442, 484; 444, 531¨Botterill, D.R., 445, 239; 447, 336Boucaud, Ph., 444, 401Bouchez, J., 445, 439Boucrot, J., 445, 239; 447, 336Boudinov, E., 444, 491; 446, 62, 75; 448, 311;

449, 364, 383Bouquet, B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bourdarios, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bourilkov, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Bourquin, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Bourrely, C., 442, 479Boutemeur, M., 444, 539; 447, 134, 157Bowcock, T.J.V., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383

Bowdery, C.K., 445, 239; 447, 336Bown, C., 447, 240Bowser-Chao, D., 441, 468Boyd, S., 445, 439Boyko, I., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Boyle, O., 446, 117Bozhenok, A.V., 449, 122Bozovic, I., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bozzi, C., 445, 239; 447, 336Bozzo, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Braccini, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Brack, J., 442, 484; 444, 531Bracker, S.B., 445, 449; 448, 303Braden, H.W., 448, 195Bradley, D.A., 442, 38Brady, F.P., 444, 523Braghin, F.L., 446, 1Braibant, S., 444, 539; 447, 134, 157Brambilla, N., 442, 349Branchina, V., 445, 351Branchini, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Branco, G.C., 442, 229Brandenberger, R., 445, 323Brandt, F., 443, 147Brandt, S., 445, 239; 447, 336Branson, J.G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Bratkovskaya, E.L., 445, 265; 447, 31Brauksiepe, S., 442, 484; 444, 531Braun, B., 442, 484; 444, 531Braun, M.A., 442, 459; 444, 435Braun, V.M., 443, 308Braun, W., 444, 555; 445, 20Braune, K., 446, 349Bravina, L., 442, 443Bray, B., 444, 531Brecher, D., 442, 117Breitweg, J., 443, 394Bremer, J., 446, 117Bremner, C.A., 444, 260Brenke, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Brenner, R.A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Brient, J.-C., 445, 239; 447, 336Bright, S., 447, 240Bright-Thomas, P., 444, 539; 447, 134, 157Brigliadori, L., 444, 539; 447, 134, 157Brigljevic, V., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Brihaye, Y., 441, 77Briskin, G., 443, 394Broadhurst, D.J., 441, 345Brochu, F., 447, 147; 448, 152


Brock, I., 443, 394Brock, I.C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Brockmann, R., 444, 523Brodie, J.H., 445, 296Brodowski, W., 446, 179Brodsky, S.J., 449, 306Bronoff, S., 448, 85Brons, S., 442, 484; 444, 531Brook, N.H., 443, 394Brooks, C.B., 449, 137Broude, C., 446, 22Brown, R.M., 444, 539; 447, 134, 157Browning, F., 444, 142Bruckman, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bruckner, W., 442, 484; 444, 531¨Brugnera, R., 443, 394Bruins, E.E.W., 444, 531Brull, A., 442, 484; 444, 531¨Brummer, N., 443, 394¨Brun, R., 444, 523Brunet, J.-M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Bruni, A., 443, 394Bruni, G., 443, 394Brustein, R., 442, 74Buchbinder, E.I., 446, 216Buchbinder, I.L., 446, 216Buchel, A., 442, 180Buchholz, P., 446, 117Buchmuller, O., 445, 239; 447, 336¨Buchmuller, W., 443, 209; 445, 399; 448, 320¨Buck, P.G., 445, 239; 447, 336Budzanowski, A., 445, 20Bueno, A., 445, 439Buffini, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bugg, D.V., 449, 114, 145, 154Bugge, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Buijs, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bukin, D.A., 449, 122Bukina, E.N., 449, 93Bulten, H.J., 442, 484; 444, 531Buncic, P., 444, 523ˇ ´Bunyatov, S., 445, 439Buran, T., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Burbach, G., 445, 20Burchat, P.R., 445, 449; 448, 303Burckhart, H.J., 444, 539; 447, 134, 157Burdin, S.V., 449, 122Burgard, C., 443, 394; 444, 539Burger, J.D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Burger, W.J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152

Burgess, C.P., 447, 257Burgin, R., 444, 539¨Burgsmueller, T., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Burgwinkel, R., 444, 555; 445, 20Burnstein, R.A., 445, 449; 448, 303Burow, B.D., 443, 394Buscher, V., 445, 239; 447, 336¨Buschmann, P., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Busenitz, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Buskulic, D., 445, 239; 447, 183, 336Bussey, P.J., 443, 394Bussiere, A., 444, 516; 449, 128`Busson, P., 444, 516; 449, 128Butterworth, J.M., 443, 394Button, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Bylsma, B., 443, 394Bytsenko, A.A., 443, 121; 449, 168

Cabrera, S., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Caccia, M., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Cacciatori, S., 444, 332Cadman, R., 444, 531Cai, X.D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Caines, H.L., 444, 523Calabrese, R., 444, 111Calafiura, P., 446, 117Calderini, G., 445, 239; 447, 336Caldwell, A., 443, 394Calen, H., 446, 179´Caliandro, R., 447, 178; 449, 401Callot, O., 445, 239; 447, 336Calvetti, M., 446, 117Calvi, M., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Camacho Rozas, A.J., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 383Cameron, W., 445, 239; 447, 336Camilleri, L., 445, 439Campana, P., 445, 239; 447, 336Campanelli, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Campbell, M., 449, 401Camporesi, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Campos, A., 443, 338Canale, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cantatore, E., 449, 401Capell, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Capiluppi, P., 444, 539; 447, 134, 157Capitani, G.P., 442, 484; 444, 531


Capon, G., 445, 239; 447, 336Capua, M., 443, 394Capurro, O.A., 447, 41Cara Romeo, G., 443, 394; 444, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Carbonell, J., 447, 199Cardini, A., 445, 439Carena, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Carena, M., 441, 205Carlin, R., 443, 394Carlino, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Carlson, C.E., 441, 363Carlson, P., 444, 38, 43, 52Carman, T.S., 444, 252Carmi, I., 447, 127Carnegie, R.K., 444, 539; 447, 134, 157Carone, C.D., 441, 363; 443, 352Carosi, R., 446, 117Carr, J., 445, 239; 447, 336Carr, L.D., 444, 523Carrer, N., 449, 401Carroll, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Carroll, M., 444, 38, 43, 52Cartacci, A.M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Carter, A.A., 444, 539; 447, 134, 157Carter, J.R., 444, 539; 447, 134, 157Carter, P., 442, 484; 444, 531Carter, T., 445, 449; 448, 303Cartiglia, N., 443, 394Cartwright, S., 445, 239; 447, 336Carvalho, H.S., 445, 449; 448, 303Casado, M.P., 445, 239; 447, 336Casas, J.A., 445, 82Casaus, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Case, T., 446, 349Cashmore, R.J., 443, 394Caso, C., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Cassing, W., 447, 31Castellini, G., 443, 394; 444, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Castillo Gimenez, M.V., 441, 479; 444, 491;

446, 62, 75; 448, 311; 449, 364, 383Castor, J., 444, 516; 449, 128Cataldo, M., 448, 20Catanesi, M.G., 449, 401Catani, S., 446, 143Catford, W.N., 444, 32Cattadori, C., 447, 127Cattai, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cattaneo, M., 445, 239; 447, 336Cattaneo, P.W., 445, 439

Catterall, C.D., 443, 394Catterall, S., 442, 266Cavallari, F., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Cavallo, F.R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cavallo, N., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Cavanaugh, R., 445, 239; 447, 336Cavasinni, V., 445, 439Cawley, E., 444, 38, 43, 52Cebra, D., 444, 523Cecchi, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ceccucci, A., 446, 117Cederwall, B., 443, 69, 82Celikel, A., 443, 359Cenci, P., 446, 117Cerrada, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Cerri, C., 446, 117Cerruti, Ch., 441, 479Cerutti, F., 445, 239; 447, 336Cervera-Villanueva, A., 445, 439Cesaroni, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chabaud, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Chacko, Z., 442, 199Chaichian, M., 442, 192Chalmers, M., 445, 239; 447, 336Chambers, J.T., 445, 239; 447, 336Chambon, T., 444, 516; 449, 128Chamizo, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Chamseddine, A.H., 442, 97Chang, C.Y., 444, 539; 447, 134, 157Chang, D., 444, 142Chang, L.N., 441, 419Chang, Y.H., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Chankowski, P.H., 441, 205Chapin, D., 443, 394Chapkin, M., 444, 491; 446, 62Chappell, S.P.G., 444, 260Charalambous, S., 444, 43Charity, R.J., 446, 197Charles, E., 445, 239; 447, 336Charlot, C., 444, 516; 449, 128Charlton, D.G., 444, 539; 447, 134, 157Charpentier, Ph., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Chaturvedi, U.K., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Chaurand, B., 444, 516; 449, 128Chaussard, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Chaves, A.S., 445, 94


Chazelle, G., 445, 239; 447, 336Checchia, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Chekanov, S., 443, 394Chelkov, G.A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Chemarin, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Chemtob, M., 448, 57Chen, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chen, F.-z., 442, 223Chen, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chen, G.M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chen, G.P., 446, 356Chen, H.F., 444, 503, 569; 445, 428; 446, 356,

368; 447, 147; 448, 152Chen, H.S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chen, J.C., 446, 356Chen, Q., 444, 252Chen, S., 445, 239; 447, 336Chen, Y., 446, 356Chen, Y.B., 446, 356Chen, Y.Q., 446, 356Cheng, B.S., 446, 356Cheng, Y., 449, 194Chereau, X., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chernodub, M.N., 443, 244; 449, 267Chertok, M.B., 444, 38, 43, 52Chesi, E., 449, 401Cheu, E., 447, 240Chevrot, I., 449, 128Cheze, J.B., 446, 117Chiarella, V., 445, 239; 447, 336Chiba, M., 447, 167Chiba, T., 442, 59Chiba, Y., 447, 167Chiefari, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Chien, C.Y., 444, 503, 569; 445, 428; 446, 368Chierici, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Childress, S., 447, 240Chishtie, F., 446, 267Chiu, T.-W., 445, 371Chliapnikov, P., 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Chmeissani, M., 445, 239; 447, 336Chochula, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Chodos, A., 449, 260Choi, S.Y., 449, 207Chollet, J.C., 446, 117Chomaz, Ph., 442, 48; 447, 221

Chorowicz, V., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Chrisman, D., 444, 539; 447, 134, 157Christiansen, H.R., 441, 185; 445, 8Chudoba, J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Chumney, P., 442, 484; 444, 531Chung, K.S., 444, 267Chung, M.S., 444, 267Chwastowski, J., 443, 394Ciafaloni, P., 446, 278Ciborowski, J., 443, 394Ciesielski, F., 447, 199Cieslik, K., 449, 364Cifarelli, L., 443, 394; 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Cindolo, F., 443, 394; 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Ciocca, C., 444, 539; 447, 134, 157Cirilli, M., 446, 117Cirio, R., 443, 394Cisbani, E., 442, 484; 444, 531Ciuchini, M., 441, 371Ciulli, V., 445, 239; 447, 336Civinini, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Civitarese, O., 446, 93Cizewski, J.A., 446, 22Clare, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Clare, R., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Clark, R.M., 449, 6Clarke, P.E.L., 444, 539; 447, 134, 157Clavelli, L., 446, 153Clay, E., 444, 539; 447, 134, 157Clement, G., 449, 12´Clement, H., 443, 77; 446, 179, 363Clifft, R.W., 445, 239; 447, 336Cline, J.M., 448, 321Close, F.E., 446, 342Cloth, P., 443, 394Coarasa, J.A., 442, 326Cobb, J.H., 449, 137Coboken, K., 443, 394¨Cocks, J.F.C., 443, 69, 82Coffin, J.-P., 446, 191Cogan, J., 446, 117Cohen, I., 444, 539; 447, 134, 157Coignet, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Colaleo, A., 445, 239; 447, 336Colangelo, G., 441, 437Colas, P., 445, 239; 447, 336Coldewey, C., 443, 394Cole, J.E., 443, 394Coleman, R., 447, 240Coles, J., 445, 239; 447, 336


Colijn, A.P., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Colino, N., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Collaboration, Z.E.U.S., 443, 394Collazuol, G., 445, 439Collins, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Colo, G., 444, 1`Colomer, M., 441, 479; 444, 491Colonna, M., 442, 48Colrain, P., 445, 239; 447, 336Combley, F., 445, 239; 447, 336Comelli, D., 446, 278Conboy, J.E., 444, 539; 447, 134, 157Conforto, G., 445, 439; 447, 122Coniglione, R., 442, 48Conta, C., 445, 439Contalbrigo, M., 445, 439Contardo, D., 444, 516; 449, 128Contin, A., 443, 394Contogouris, A.P., 442, 374Contreras, J.G., 446, 158Contri, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cooke, O.C., 444, 539; 447, 134, 157Cooper, F., 449, 260Cooper, G.E., 444, 523Cooper-Sarkar, A.M., 443, 394Copeland, E.J., 443, 97Coppola, N., 443, 394Copty, N.K., 445, 449; 448, 303Coquereaux, R., 443, 221Corbo, G., 441, 371`Corcella, G., 442, 417Corcoran, M.D., 447, 240Corden, M., 445, 239; 447, 336Cormack, C., 443, 394Corradi, M., 443, 394Corriveau, F., 443, 394Cortes, J.L., 444, 451Ćorti, G., 447, 240Cortina, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cortina-Gil, D., 444, 32Cosme, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cossutti, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Costa, M., 443, 394Costantini, F., 446, 117Costantini, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Cotorobai, F., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Cottingham, W.N., 443, 394Cotton, R.J., 449, 137Cougo-Pinto, M.V., 446, 170

Courant, H., 449, 137Court, G.R., 442, 484; 444, 531Courtat, P., 445, 423Cousins, R., 445, 439Couyoumtzelis, C., 444, 539; 447, 134, 157Cowan, G., 445, 239; 447, 336Coward, D., 446, 117Cowell, J.-H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cowin, R.L., 444, 260Cowsik, R., 449, 219Cox, B., 447, 240Coxe, R.L., 444, 539; 447, 134, 157Coyle, P., 445, 239; 447, 336CPLEAR Collaboration, 444, 38, 43, 52Cramer, J.G., 444, 523Cramer, O., 446, 349Craps, B., 445, 150Crawford, G., 445, 239; 447, 336Crawford, G.I., 442, 43Crawley, H.B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Creanza, D., 445, 239; 447, 336Crede, V., 446, 349Ćremaldi, L.M., 445, 449; 448, 303Cremonesi, O., 447, 127Crennell, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Crepe, S., 446, 117; 448, 311; 449, 364, 383´ Ćrespo, J.M., 445, 239; 447, 336Cribier, M., 447, 127Crisler, M.B., 447, 240Cristinziani, M., 444, 523Crittenden, J., 443, 394Croni, M., 446, 363¨Crosetti, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Cross, R., 443, 394Crowe, K.M., 446, 349Cruz, J., 449, 30Crystal Barrel Collaboration, 446, 349Csato, P., 444, 523Csikor, F., 441, 354Csilling, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Csizmadia, P., 443, 21Csorgo, T., 443, 21¨ ˝Cudell, J.R., 448, 281Cuevas Maestro, J., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Cuffiani, M., 444, 539; 447, 134, 157Cui, X.Z., 446, 356Cullen, D.M., 443, 69, 82Cundy, D., 446, 117Curtis, L., 445, 239; 447, 336Czarnecki, A., 449, 354Czellar, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383


Dado, S., 444, 539; 447, 134, 157Dagan, S., 443, 394D’Agostini, G., 443, 394Dai, T.S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Daigo, M., 447, 167Dal Corso, F., 443, 394D’Alessandro, R., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Dallavalle, G.M., 444, 539; 447, 134, 157Dalpiaz, P., 446, 117Dalpiaz, P.F., 442, 484; 444, 531Dameri, M., 449, 401Damgaard, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Damgaard, P.H., 445, 366; 446, 175D’Angelo, S., 447, 127Daniels, D., 445, 439Danielsen, K.M., 446, 342Danielsson, M., 444, 38, 43, 52Dann, J.H., 445, 239; 447, 336Darbo, G., 449, 401Dardo, M., 443, 394Darling, C., 445, 449; 448, 303Das, S.R., 445, 142Dasgupta, A., 445, 279Dasgupta, I., 447, 284Da Silva, W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Daskalakis, G., 445, 239; 447, 336Daugas, J.M., 444, 32Davenport, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Davidson, S., 445, 191Davier, M., 445, 239; 447, 336Davies, J.P., 447, 178Davis, A.-C., 446, 238Davis, E.D., 447, 209Davis, R., 444, 539; 447, 134, 157De Angelis, A., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383De Asmundis, R., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152De Beer, M., 446, 117De Boer, W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383De Brabandere, S., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Debruyne, D., 441, 1Debu, P., 446, 117Decamp, D., 445, 239; 447, 183, 336De Campos, F., 447, 331De Carlos, B., 445, 82De Clercq, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383De Conti, C., 444, 14De Falco, A., 444, 111Deffayet, C., 441, 52, 163

Deffner, R., 443, 394Degaudenzi, H., 445, 439Degener, T., 446, 349Deghorain, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Degl’Innocenti, S., 441, 291; 444, 387Deglon, P., 447, 147; 448, 152Degre, A., 444, 503, 569; 445, 428; 446, 368;´

447, 147; 448, 152Deiters, K., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Dejardin, M., 444, 38, 43, 52De Jong, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152De Jong, S., 444, 539; 447, 134, 157De la Cruz, B., 444, 503, 569; 445, 428; 446,

368; 447, 147, 178; 448, 152Delbourgo, R., 446, 332Del Duca, V., 445, 168De Leo, R., 442, 484Delepine, D., 442, 229´Deleplanque-Stephens, M.-A., 449, 6Delfino, M., 445, 239; 447, 336Delheij, P.P.J., 444, 531Delius, G.W., 444, 217Della Ricca, G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Della Volpe, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Dell’Orso, R., 445, 239; 447, 336De Lotto, B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Del Peso, J., 443, 394DELPHI Collaboration, 441, 479; 444, 491;

446, 62, 75; 448, 311; 449, 364, 383Delpierre, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Del Pozo, L.A., 444, 539Del Prete, T., 445, 439Del Zoppo, A., 442, 48Demaria, N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383De Martino, A., 447, 292De Mello Neto, J.R.T., 445, 449; 448, 303De Min, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383De Miranda, J.M., 445, 449; 448, 303Dempsey, J.F., 446, 197Demuth, D.M., 449, 137Denes, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Deng, C., 441, 285Denig, A., 443, 77Denisenko, K., 445, 449; 448, 303DeNotaristefani, F., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152De Oliviera, F., 444, 32De Palma, M., 445, 239; 447, 336


De Pasquale, S., 443, 394De Paula, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Deppe, O., 443, 394De Rafael, E., 443, 255De Roeck, A., 444, 539; 447, 134, 157Derre, J., 444, 38, 43, 52Derrick, M., 443, 394Dervan, P., 447, 134, 157De Salvo, A., 447, 147; 448, 152DeSalvo, R., 445, 419De Sanctis, E., 442, 484; 444, 531De Santo, A., 445, 439Desch, K., 444, 539; 447, 134, 157Deschamps, O., 445, 239; 447, 336De Schepper, D., 442, 484; 444, 531Descroix, E., 444, 516; 449, 128Deshpande, A., 443, 394Desler, K., 443, 394Dessagne, S., 445, 239; 447, 336DESY-Munster Collaboration, 446, 209¨Devaux, A., 444, 516; 449, 128Devenish, R.C.E., 443, 394Devitsin, E., 442, 484; 444, 531De Vivie de Regie, J.-B., 445, 239; 447, 336´De Wit, B., 443, 153De Witt Huberts, P.K.A., 442, 484; 444, 531De Wolf, E., 443, 394Dey, J., 443, 293; 447, 352Dey, M., 443, 293; 447, 352Dhamotharan, S., 445, 239; 447, 336Dhawan, S., 443, 394Diaczek, A., 449, 401Diakonos, F.K., 444, 583Dias de Deus, J., 442, 395Dıaz, M.A., 441, 224´Di Bari, D., 447, 178; 449, 401Dibon, H., 446, 117Di Cecio, G., 441, 319Di Ciaccio, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Di Diodato, A., 441, 479Diehl, M., 449, 306Diemoz, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Dienes, B., 444, 539; 447, 134, 157Dieperink, A.E.L., 441, 17; 446, 15Dietl, H., 445, 239; 447, 336Dignan, T., 445, 439Di Gregorio, D.E., 447, 41Dijkstra, H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Di Lella, L., 445, 439Di Liberto, S., 449, 401Dillon, G., 448, 107Dillon, G.K., 444, 260Di Lodovico, F., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152

Dimopoulos, K., 446, 238Dimopoulos, S., 441, 96Dimova, T.V., 449, 122Dine, M., 444, 103Di Nezza, P., 442, 484; 444, 531Ding, H.L., 446, 356Dinh Dang, N., 445, 1Dinius, J., 446, 197Dionisi, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Di Salvo, E., 441, 447Dissertori, G., 445, 239; 447, 336Dittmaier, S., 441, 383Dittmar, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Dixit, M.S., 444, 539; 447, 134, 157Dixon, L., 444, 273Djaoshvili, N., 446, 349Doble, N., 446, 117Do Couto e Silva, E., 445, 439Dolbeau, J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Dolgopolov, A.V., 446, 342Dolgoshein, B.A., 443, 394Dolgov, A.D., 442, 82Dolinsky, S.I., 449, 122D’Oliveira, A.B., 445, 449; 448, 303Dominguez, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Dominguez, C.A., 448, 93Dong, L.Y., 446, 356Donnachie, A., 448, 281Donnelly, I.J., 445, 439Donskov, S.V., 446, 342Dorey, N., 442, 145Dorey, P., 448, 249Doria, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Dornan, P.J., 445, 239; 447, 336Doroba, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Doser, M., 446, 349Dos Reis, A.C., 445, 449; 448, 303Dosselli, U., 443, 394Dostrovsky, I., 447, 127Dova, M.T., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Doyle, A.T., 443, 394Draayer, J.P., 442, 7Dracos, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Dragon, N., 446, 314Drapier, O., 444, 516; 449, 128Drees, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Drees, M., 447, 116Drevermann, H., 445, 239; 447, 336Drews, G., 443, 394


Dris, M., 441, 479; 444, 491; 446, 62, 75; 448,311; 449, 364, 383

Drummond, I.T., 442, 279; 447, 298Druzhinin, V.P., 449, 122Du, Z.Z., 446, 356Dubbert, J., 444, 539; 447, 134, 157Dubovik, V.M., 449, 93Dubrovin, M.S., 449, 122Duchesneau, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Duchovni, E., 444, 539; 447, 134, 157Duckeck, G., 444, 539; 447, 134, 157Duclos, J., 446, 117Dudas, E., 441, 163Duerdoth, I.P., 444, 539; 447, 134, 157Duflot, L., 445, 239; 447, 336Dufournand, D., 447, 147; 448, 152Duinker, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Dulinski, Z., 443, 394´Dumarchez, J., 445, 439Dumitru, A., 446, 326Dunn, J., 444, 523Dunnweber, W., 446, 197, 349¨Duperrin, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Duran, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Durand, J.-D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Duren, M., 442, 484; 444, 531¨Durkin, L.S., 443, 394Dvoredsky, A., 442, 484; 444, 531Dyring, J., 446, 179

E-811 Collaboration, 445, 419Ealet, A., 444, 38, 43, 52; 445, 239; 447, 336Earl, B.C., 446, 342; 449, 401Eartly, D.P., 445, 419Easo, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Eatough, D., 444, 539; 447, 134, 157Ebersberger, C., 446, 117Ebert, D., 444, 208Ebert, K.H., 447, 127Eboli, O.J.P., 447, 116Echols, R., 444, 103; 449, 60Eckardt, V., 444, 523Ecker, G., 441, 437Eckert, M., 443, 394Eckhardt, F., 444, 523Edgecock, T.R., 445, 239; 447, 336Edmonds, J.K., 443, 394Ehara, M., 445, 14Ehmanns, A., 446, 349Ehret, R., 441, 479

Eigen, G., 441, 479; 444, 491; 446, 62, 75; 448,311; 449, 364, 383

Eisenberg, Y., 443, 394Eisenhardt, S., 443, 394Ekelof, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ekspong, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ekstrom, C., 446, 179¨Elbakian, G., 442, 484; 444, 531Eleftheriadis, C., 444, 38, 43, 52Elia, D., 447, 178; 449, 401Elias, V., 446, 267Elitzur, S., 449, 180Ellert, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ellis, G., 445, 239; 447, 336Ellis, J., 444, 367Ellis, M., 445, 439Ellis, P.J., 443, 63El Mamouni, H., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Elmer, P., 445, 239; 447, 336Elsing, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ely, J., 442, 484; 444, 531Emerson, J., 444, 531Emmanuel-Costa, D., 442, 229Engel, J.-P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Engelen, J., 443, 394Engelhardt, D., 446, 349Engler, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152En’yo, H., 444, 267Epperson, D., 443, 394Eppling, F.J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Erhan, S., 445, 455Erhardt, A., 446, 363Ermolov, P.F., 443, 394Erne, F.C., 444, 503, 569; 445, 428; 446, 368;´

447, 147; 448, 152Ernst, C., 442, 443; 447, 227Ernst, J., 444, 555; 445, 20Erwin, A.R., 447, 240Erzen, B., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Escribano, R., 444, 397Eskreys, A., 443, 394Espagnon, B., 444, 516; 449, 128Espirito Santo, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Estabrooks, P.G., 444, 539; 447, 134, 157Etzion, E., 444, 539; 447, 134, 157Evans, D., 446, 342; 447, 178; 449, 401


Evans, H.G., 444, 539Evans, N., 449, 281Extermann, P., 444, 503, 569; 446, 368; 447,

147; 448, 152Eyal, G., 441, 191

Fabbri, A., 449, 30Fabbri, F., 444, 539; 447, 134, 157Fabbro, B., 445, 239; 447, 336Fabre, M., 444, 503, 569; 446, 368; 447, 147;

448, 152Fabre, P.E.M., 445, 428Faccini, R., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Faddeev, L., 449, 214Faessler, A., 442, 203; 443, 7Faessler, M.A., 446, 349Fagerstroem, C.-P., 443, 394Faıf, G., 445, 239; 447, 336¨Fajfer, S., 447, 313Falagan, M.A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Falciano, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Faldt, G., 445, 423¨Falk, E., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Falk, T., 444, 367, 427Fallon, P., 449, 6Falvard, A., 445, 239; 447, 336Fanebust, K., 449, 401Fanourakis, G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Fantechi, R., 446, 117; 447, 336Fanti, M., 444, 539; 447, 134, 157Fanti, V., 446, 117Fantoni, A., 442, 484; 444, 531Fantoni, S., 446, 99Faraggi, A.E., 445, 77, 357Faravel, L., 444, 38, 43, 52Farchioni, F., 443, 214Fargeix, J., 444, 516; 449, 128Farina, C., 446, 170Fassouliotis, D., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Faust, A.A., 444, 539; 447, 134, 157Favara, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Fay, J., 444, 503, 569; 445, 428; 446, 368; 447,

147; 448, 152Fayard, L., 446, 117Fayot, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Fazio, T., 445, 439Fechtchenko, A., 442, 484; 444, 531Fedin, O., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Feindt, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383

Feinerman, O., 449, 180Feinstein, A., 441, 40Felcini, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Feldman, G.J., 445, 439Feldmann, Th., 449, 339Fenyuk, A., 441, 479; 444, 491; 446, 62Ferdi, C., 445, 239; 447, 336Ferenc, D., 449, 347Ferguson, D.P.S., 445, 239; 447, 336Ferguson, M.I., 444, 523Ferguson, T., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Fermilab E791 Collaboration, 445, 449; 448,

303Fernandez, A., 445, 449; 448, 303Fernandez-Bosman, M., 445, 239; 447, 336Fernandez, E., 445, 239; 447, 336Fernandez, J.P., 443, 394´Fernandez, L.A., 441, 330´Fernandez Niello, J.O., 447, 41´Fernando, S., 445, 52Ferrante, I., 445, 239; 447, 336Ferrari, F., 444, 167Ferrari, N., 447, 127Ferrari, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ferrari, R., 445, 439Ferreira, R., 444, 516; 449, 128Ferrer, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ferrer, F., 446, 111Ferrer, M.L., 444, 111Ferrer-Ribas, E., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Ferrere, D., 445, 439`Ferrero, M.I., 443, 394Ferroni, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ferstl, M., 442, 484; 444, 531Fesefeldt, H., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Fetscher, W., 444, 38, 43, 52Feverati, G., 444, 442Fiandrini, E., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Fichet, S., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Fick, D., 442, 484; 444, 531Fidecaro, M., 444, 38, 43, 52Fiedler, F., 444, 539; 447, 134, 157Fiedler, K., 442, 484; 444, 531Field, J.H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Field, S., 447, 240Fields, T.H., 449, 137Fierro, M., 444, 539; 447, 134, 157Figiel, J., 443, 394Figueiredo, J.M., 445, 94


Fil’chenkov, M.L., 441, 34Filges, D., 443, 394Filipcic, A., 444, 38, 43, 52ˇ ˇFilipov, G., 445, 14Filippone, B.W., 442, 484; 444, 531Filthaut, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Finch, A.J., 445, 239; 447, 336Fini, R., 447, 178Fini, R.A., 449, 401Finocchiaro, P., 442, 48Fioravanti, D., 447, 277Fiorentini, G., 441, 291; 444, 387Fiorini, E., 447, 127Firestone, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Fischer, G., 446, 117Fischer, H., 442, 484; 444, 531Fischer, H.G., 444, 523Fischer, P.-A., 441, 479; 444, 491; 446, 62, 75Fisher, P.H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Fisk, I., 444, 503, 569; 445, 428; 446, 368; 447,

147; 448, 152Fissum, K.G., 442, 43Flagmeyer, U., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Flaminio, V., 445, 439Fleck, I., 444, 539; 447, 134, 157Fleuret, F., 444, 516; 449, 128Flierl, D., 444, 523Flohr, M.A.I., 444, 179Floratos, E., 444, 190Floratos, E.G., 445, 69; 447, 67Focardi, E., 445, 239; 447, 336Fodor, Z., 441, 354; 444, 523Foeth, H., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Fohl, K., 443, 77; 446, 363¨Foka, P., 444, 523Fokitis, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Folman, R., 444, 539; 447, 134, 157Fontaine, J.C., 449, 401Fontanelli, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Foot, R., 443, 185Force, P., 444, 516; 449, 128Forconi, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ford, R., 447, 240Forgacs, P., 441, 275´Formica, A., 446, 117Fort, H., 444, 174Forte, S., 448, 295Fortune, H.T., 444, 531Forty, R.W., 445, 239; 447, 336

Foss, J., 445, 239; 447, 336Foster, B., 443, 394Foster, F., 445, 239; 447, 336Fouchez, D., 445, 239; 447, 336Foudas, C., 443, 394Fox, B., 442, 484; 444, 531Fox, G.F., 445, 449; 448, 303Fox, H., 446, 117Frabetti, P.L., 446, 117Frabetti, S., 442, 484; 444, 531Francis, D., 444, 38, 43, 52Franco, E., 441, 371Franek, B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Frank, M., 445, 239; 447, 336Frankel, S., 441, 425Fransson, K., 446, 179Franz, J., 442, 484; 444, 531Frascaria, N., 442, 48Fraternali, M., 445, 439Frati, W., 441, 425Fredj, L., 444, 503, 516, 569; 445, 428; 446,

368; 447, 147; 448, 152Freedman, S.J., 449, 6Freer, M., 444, 260French, B.R., 446, 342Frere, J.-M., 444, 397`Freudenreich, K., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Freund, P., 444, 523Fricke, U., 443, 394Fries, R.J., 443, 40Friese, V., 444, 523Frishman, Y., 449, 76Frisken, W.R., 443, 394Fritsch, T., 447, 127Frodesen, A.G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Froggatt, C.D., 446, 256Frolov, S.A., 441, 173Fruhwirth, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 383Frullani, S., 442, 484; 444, 531Fry, J., 444, 38, 43, 52Ftacnik, J., 449, 401´ˇFuchs, J., 441, 141; 447, 266Fuchs, M., 444, 523Fujikawa, B.K., 449, 6Fukawa, M., 447, 167Fukuda, N., 448, 180Fukuda, T., 444, 267Fukui, S., 445, 14Fukushima, Y., 443, 409; 447, 167Fulda-Quenzer, F., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Funahashi, H., 444, 267Funk, M.-A., 442, 484; 444, 531


Funk, W., 446, 117Furetta, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Furtjes, A., 444, 539; 447, 134, 157¨Furuno, K., 442, 53Fusayasu, T., 443, 394Fuster, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Futamase, T., 447, 46Futyan, D.I., 444, 539; 447, 134, 157

Gabathuler, E., 444, 38, 43, 52Gaberdiel, M.R., 441, 133Gabler, F., 444, 523Gacougnolle, R., 445, 423Gadaj, T., 443, 394Gaff, S., 446, 197Gagnon, P., 444, 539; 445, 449; 447, 134, 157;

448, 303Gago, J., 444, 516; 449, 128Gagunashvili, N.D., 442, 484; 444, 531Gaillard, J-M., 445, 439Gal, J., 444, 523Galaktionov, Yu., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Galan-Gonzalez, V., 449, 288´ ´Galante, A., 444, 421Galea, R., 443, 394Galeao, A.P., 444, 14˜Galindo-Uribarri, A., 443, 89Gallagher, H.R., 449, 137GALLEX Collaboration, 447, 127Gallo, E., 443, 394Galloni, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Galumian, P., 444, 531; 445, 439Gamba, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gamblin, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gamboa, J., 444, 451Gamet, R., 444, 38, 43, 52Gandelman, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Gangler, E., 445, 439Ganguli, S.N., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Ganis, G., 445, 239; 447, 336Ganz, R., 444, 523Gao, C.S., 446, 356Gao, H., 442, 484; 444, 531Gao, M.L., 446, 356Gao, S.Q., 446, 356Gao, Y., 445, 239; 447, 336Gaponenko, I.A., 449, 122Garattini, R., 446, 135Garber, Y., 442, 484; 444, 531¨Garcia-Abia, P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152

Garcıa, A.O., 443, 221´Garcia, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Garcıa, G., 443, 394´Garcia-Garcia, C., 449, 137Garcia, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 383Garcia, J.A., 441, 198Gardestig, A., 445, 423˚Garfagnini, A., 443, 394Garibaldi, F., 442, 484; 444, 531Garrido, L., 445, 239Garrido, Ll., 447, 336Gary, J.W., 444, 539; 447, 134, 157Gascon, J., 444, 539; 447, 134, 157Gascon-Shotkin, S.M., 444, 539; 447, 134, 157Gaspar, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gaspar, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gasparini, U., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gataullin, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Gatignon, L., 446, 117Gau, S.S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Gauzzi, P., 444, 111Gavillet, Ph., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gavrilov, G., 442, 484; 444, 531Gay, P., 445, 239; 447, 336Gaycken, G., 444, 539; 447, 134, 157Gazdzicki, M., 444, 523´Gazis, E.N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Geich-Gimbel, C., 444, 539; 447, 134, 157Geiger, P., 442, 484; 444, 531Geiser, A., 444, 358; 445, 439Geiss, J., 447, 31Geissel, H., 444, 32Geist, W., 444, 523Gelao, G., 445, 239; 447, 336Gelbke, C.K., 446, 197Gele, D., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Gendner, N., 443, 394Genovese, M., 442, 398Gentile, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Georgiopoulos, C., 445, 239; 447, 336Geppert, D., 445, 439Gerber, H.-J., 444, 38, 43, 52Gerber, J.-P., 441, 479Gerdyukov, L., 446, 62, 75; 448, 311; 449, 364,

383Gerland, L., 447, 227Gerschel, C., 444, 516; 449, 128Gerzon, S., 448, 303


Gevaert, A., 444, 397Geweniger, C., 445, 239; 447, 336Gharib, T., 444, 231Gharibyan, V., 442, 484; 444, 531Gheordanescu, N., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Gherghetta, T., 446, 28Ghete, V.M., 445, 239; 447, 336Ghez, P., 445, 239; 447, 183, 336Ghidini, B., 447, 178; 449, 401Ghilencea, D., 442, 165Ghodbane, N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Giacconi, P., 441, 257Giacomelli, G., 444, 539; 447, 134, 157Giacomelli, P., 444, 539; 447, 134, 157Giagu, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Gialas, I., 443, 394Giannini, G., 445, 239; 447, 336Gianoli, A., 446, 117Gianotti, F., 445, 239; 447, 336Giarritta, P., 446, 349Giassi, A., 445, 239; 447, 336Gibbons, G.W., 443, 138Gibbs, W.R., 444, 252Gibin, D., 445, 439Gibson, B.F., 444, 252Gibson, M., 446, 256Gibson, V., 444, 539; 447, 134, 157Gibson, W.R., 444, 539; 447, 134, 157Giehl, I., 445, 239; 447, 336Gignoux, C., 447, 199Gil, I., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Gillibert, A., 442, 48Gilmore, J., 443, 394Gingrich, D.M., 444, 539; 447, 134, 157Ginsburg, C.M., 443, 394Giordjian, V., 442, 484Girone, M., 445, 239; 447, 336Girtler, P., 445, 239; 447, 336Giudice, G.F., 446, 28Giudici, S., 446, 117Giunti, C., 444, 379Giusti, P., 443, 394Giveon, A., 449, 180Gladilin, L.K., 443, 394Gładysz, E., 444, 523Glander, K.H., 444, 555; 445, 20Glashow, S.L., 445, 412Glasman, C., 443, 394Glege, F., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Glenzinski, D., 444, 539; 447, 134, 157Glockle, W., 447, 216¨Gluck, M., 443, 298¨Gninenko, S., 445, 439

Go, A., 444, 38, 43, 52Gobbo, B., 445, 239; 447, 336Gobel, C., 445, 449; 448, 303Godley, A., 445, 439Goebel, F., 443, 394Goers, S., 444, 555; 445, 20Gokieli, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Golak, J., 447, 216Goldberg, H., 444, 68Goldberg, J., 444, 539; 447, 134, 157Goldfarb, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Goldstein, J., 444, 503, 569; 445, 428; 446, 368Golendoukhin, A., 444, 531Golendukhin, A., 442, 484Golob, B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Golubev, V.B., 449, 122Golubkov, Yu.A., 443, 394Gomez-Cadenas, J-J., 445, 439Gomez-Ceballos, G., 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gomez Nicola, A., 449, 288´Gomis, J., 443, 147Goncalves, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gong, Z.F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Gonidec, A., 446, 117Gonorazky, S., 449, 187Gonzalez-Arroyo, A., 442, 273´Gonzalez-Rey, F., 448, 37Gonzalez, S., 445, 239; 447, 336´Gonzalez Caballero, I., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Gonzalez Felipe, R., 442, 229´Goodman, M.C., 449, 137Goodsir, S., 445, 239; 447, 336Gopal, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gorini, B., 446, 117Gorn, L., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Gorn, W., 444, 539; 447, 134, 157Gorodetzky, P., 444, 516; 449, 128Gorska, M., 444, 32´Gorski, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Gosdzinsky, P., 441, 265Goßling, C., 445, 439¨Gosset, J., 445, 439Goto, Y., 444, 267Gottlicher, P., 443, 394¨Gouanere, M., 445, 439`Gougas, A., 444, 503, 569; 445, 428; 446, 368Gould, C.R., 447, 209Gounder, K., 445, 449; 448, 303


Gourio, D., 446, 191Gouz, Yu., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Govi, G., 446, 117Goy, C., 445, 239; 447, 183, 336Grabosch, H.J., 443, 394Gracco, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Graciani, R., 443, 394Grafstrom, P., 446, 117¨Graham, G., 447, 240Graham, J., 447, 240Graham, K., 447, 134, 157Grahl, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Gran, R., 449, 137Grandi, C., 444, 539; 447, 134, 157Granier-De-Cassagnac, R., 446, 117Grant, A., 445, 439Grater, J., 443, 77¨Gratta, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Grauges, E., 445, 239; 447, 336`Graw, G., 442, 484; 444, 531Grawe, H., 444, 32Gray, J.P., 444, 103; 449, 60Graziani, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Graziani, G., 445, 439; 446, 117Grazzini, M., 446, 143Grebeniouk, O., 442, 484; 444, 531Grebeniuk, M.A., 442, 125Grebieszkow, J., 444, 523Green, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Green, M.G., 445, 239; 447, 336Green, P.W., 442, 484; 444, 531Greene, J.P., 449, 6Greene, P.B., 448, 6Greenhalgh, B., 444, 260Greeniaus, L.G., 442, 484; 444, 531Greening, T.C., 445, 239; 447, 336Greenlees, P.T., 443, 69, 82Gregorio, A., 445, 239; 447, 336Greiner, C., 446, 191; 447, 31Greiner, W., 442, 443; 447, 227; 448, 290Grella, G., 449, 401Grifols, J.A., 446, 111Griguolo, L., 443, 325Grimani, C., 447, 122Grimm, H.-J., 444, 491; 446, 62, 75; 448, 311;

449, 364, 383Gris, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Grispos, G., 442, 374Grivaz, J.-F., 445, 239; 447, 336Gronau, M., 449, 321Groot Nibbelink, S., 442, 185

Grosdidier, G., 448, 311; 449, 364, 383Große-Knetter, J., 443, 394Gross, E., 444, 539; 447, 134, 157Grosshauser, C., 442, 484; 444, 531Grossiord, J.Y., 444, 516; 449, 128Grothe, M., 443, 394Grozin, A.G., 445, 165Gruenewald, M.W., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Grunhaus, J., 444, 539; 447, 134, 157Grupen, C., 445, 239; 447, 336Gruwe, M., 444, 539; 447, 134, 157´Grzelak, G., 443, 394Grzelak, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Grzywacz, R., 444, 32Gu, J.H., 446, 356Gu, S.D., 446, 356Gu, W.X., 446, 356Gu, Y.F., 446, 356; 449, 361Guadagnini, E., 441, 60Guasch, J., 442, 326Gubarev, F.V., 443, 244Guglielmi, A., 445, 439Guichard, A., 444, 516; 449, 128Guicheney, C., 445, 239; 447, 336Guida, M., 449, 401Guidal, M., 442, 484Guidal, M.G., 444, 531Guillaud, J.P., 444, 516; 449, 128Gulminelli, F., 447, 221Gunion, J.F., 444, 136Gunther, J., 444, 523¨Gunther, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Guo, Y.N., 446, 356Gupta, V.K., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Gurtu, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Guse, B., 445, 20Guss, C., 445, 419Gustafsson, L., 446, 179Gutay, L.J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Gute, A., 442, 484; 444, 531Gutperle, M., 445, 296Guy, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Gyulassy, M., 442, 1; 443, 45Gyurjyan, V., 444, 531, 531Gyurusi, J., 441, 275¨ ¨

Haas, D., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Haas, J.P., 444, 531Haas, K.-M., 445, 20Haas, T., 443, 394


Haba, J., 447, 167Hadad, M., 442, 74Haddock, R.P., 446, 349; 449, 114, 145, 154Haeberli, W., 442, 484; 444, 531Hagan, K., 447, 240Hagelberg, R., 447, 336Haggstrom, S., 446, 179¨ ¨Hagler, P., 448, 99¨Hagner, C., 445, 439Hahn, F., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Hahn, R.L., 447, 127Hahn, S., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Haidenbauer, J., 444, 25Haider, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Hain, W., 443, 394Halkiadakis, E., 447, 240Hall, L.J., 445, 407Hall-Wilton, R., 443, 394Halley, A.W., 445, 239; 447, 336Hallgren, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Halling, A.M., 445, 449; 448, 303Ham, S.W., 441, 215Hamacher, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Hamasaki, H., 447, 167Hamatsu, R., 443, 394Hamidian, H., 448, 234Hammon, N., 448, 290Hampel, W., 447, 127Han, S.W., 446, 356Han, Y., 446, 356Hanagaki, K., 447, 240Handt, J., 447, 127Hanhart, C., 444, 25Hanke, P., 445, 239; 447, 336Hanna, D.S., 443, 394Hannappel, J., 444, 555; 445, 20Hansen, J., 448, 311; 449, 364, 383Hansen, J.-O., 442, 484; 444, 531Hansen, J.B., 445, 239; 447, 336Hansen, J.D., 445, 239; 447, 336Hansen, J.R., 445, 239; 447, 336Hansen, K., 442, 43Hansen, P.H., 445, 239; 447, 336Hanson, G.G., 444, 539; 447, 134, 157Hansper, G., 445, 239; 447, 336Hansroul, M., 444, 539; 447, 134, 157Hansson, T.H., 448, 168Hapke, M., 444, 539; 447, 134, 157Harder, K., 444, 539; 447, 134, 157Harder, M.K., 443, 82Harel, A., 447, 134, 157Hargrove, C.K., 444, 539; 447, 134, 157Harnew, N., 443, 394

Haroutunian, R., 444, 516; 449, 128Harris, F.J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Harris, J.W., 444, 523Harris, M.G., 448, 185Hart, A., 441, 319Hart, J.C., 443, 394Hartmann, B., 441, 77; 444, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Hartmann, C., 444, 539; 447, 134, 157Hartmann, F.X., 447, 127Hartmann, H., 443, 394Hartmann, J., 443, 394Hartner, G.F., 443, 394Harvey, J., 445, 239; 447, 336Hasan, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Hasch, D., 442, 484; 444, 531Hasegawa, S., 445, 14Hasegawa, T., 445, 14Haselden, A., 444, 38, 43, 52Hasell, D., 443, 394Hashimoto, M., 441, 389Hass, M., 446, 22Hata, K., 442, 53Hattori, T., 445, 106Hatzifotiadou, D., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Hauschild, M., 444, 539; 447, 134, 157Hausser, O., 444, 531¨Hawkes, C.M., 444, 539; 447, 134, 157Hawkings, R., 444, 539; 447, 134, 157Hay, B., 446, 117Hayakawa, T., 442, 53Hayes, M.E., 443, 394Hayes, O.J., 445, 239; 447, 336Haymaker, R.W., 441, 319Hayman, P.J., 444, 38, 43, 52Hazumi, M., 447, 240He, J., 446, 356; 449, 194He, J.T., 446, 356He, K.L., 446, 356He, M., 446, 356He, X.-G., 444, 75; 445, 344Heaphy, E.A., 443, 394Heath, G.P., 443, 394Heath, H.F., 443, 394Hebbeker, T., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Hebbel, K., 443, 394Hedberg, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Hegyi, S., 444, 523Heidt, D., 447, 127Heijne, E.H.M., 449, 401Heinloth, K., 443, 394; 445, 20Heinsius, F.H., 442, 484; 444, 531; 446, 349Heinz, L., 443, 394


Heinz, U., 449, 347Heinzelmann, M., 446, 349Heising, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Heitger, J., 441, 354Helariutta, K., 443, 69, 82Heller, U.M., 445, 366Hellstrom, M., 444, 32¨Helstrup, H., 447, 178; 449, 401Hemingway, R.J., 444, 539; 447, 134, 157Hemmi, Y., 443, 409; 447, 167Henderson, R., 444, 531Henkel, T., 444, 523Henkes, T., 444, 531Henoch, M., 442, 484; 444, 531Henrard, P., 445, 239; 447, 336Henrich, E., 447, 127Henry-Couannier, F., 444, 38, 43, 52Hepp, V., 445, 239; 447, 336Herbstrith, A., 446, 349HERMES Collaboration, 442, 484; 444, 531Hernandez, J.J., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Hernandez, J.M., 443, 394´Hernando, J., 445, 439Herndon, M., 444, 539; 447, 134, 157Herquet, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Herr, H., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Herrera, G., 445, 449; 448, 303Herrera-Siklody, P., 442, 359´Herten, G., 444, 539; 447, 134, 157Hertenberger, R., 442, 484; 444, 531Herve, A., 444, 503, 569; 445, 428; 446, 368;´

447, 147; 448, 152Herz, M., 446, 349Hessey, N.P., 446, 349Hessing, T.L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Heuer, R.D., 444, 539; 447, 134, 157Heusch, C., 443, 394Heuser, J.-M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Heusse, Ph., 445, 239; 447, 336Heusser, G., 447, 127Hibou, F., 445, 423Hidaka, S., 447, 240Hidas, P., 444, 503, 569; 445, 428; 446, 349,

368; 447, 147; 448, 152Higashi, A., 444, 267Higon, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Higuchi, M., 447, 167Hikami, K., 443, 233Hildreth, M.D., 444, 539; 447, 134, 157Hilger, E., 443, 394Hill, J.C., 444, 539; 447, 134, 157

Hill, L.A., 444, 523Hillier, S.J., 444, 539Hino, T., 446, 342Hip, I., 443, 214Hirai, M., 448, 180Hiro-Oka, H., 449, 230Hirose, T., 443, 394; 447, 167Hirschfelder, J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Hisano, J., 445, 316Ho, J., 445, 27Hobson, P.R., 444, 539; 447, 134, 157Hoch, M., 447, 134, 157Hochman, D., 443, 394Hocker, A., 444, 539; 447, 134, 157, 336Hodd, C., 449, 114, 145, 154Hofer, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Hoffman, K., 447, 134, 157Hoffmann, C., 445, 239; 447, 336Hoistad, B., 446, 179¨Holden, J., 446, 22Holder, M., 446, 117Holland, K., 443, 338Hollander, R.W., 444, 38, 43, 52Holler, Y., 442, 484; 444, 531Hollik, W., 442, 326Holm, U., 443, 394Holme, A.K., 447, 178; 449, 401Holmgren, S.-O., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Holt, P.J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Holt, R.J., 442, 484; 444, 531Holthuizen, D., 441, 479; 444, 491; 446, 62, 75Holtzhaußen, C., 446, 349; 449, 114Holzner, G., 447, 147; 448, 152Homer, R.J., 444, 539; 447, 134, 157Homma, K., 443, 394Homma, Y., 447, 167Hong, D.K., 445, 36Hong, S.J., 443, 394Hong Tuan, R., 445, 100Honma, A.K., 444, 539; 447, 134, 157Honscheid, K., 445, 20Hoorani, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Hoorelbeke, S., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Hoprich, W., 442, 484; 444, 531Horgan, R.R., 447, 298Horikawa, N., 445, 14Horowitz, C.J., 443, 58Horvath, D., 444, 539; 447, 134, 157´Hossain, K.R., 444, 539; 447, 134, 157Hou, S.R., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Hou, W.-S., 445, 344


Houlden, M., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Howard, R., 444, 539; 447, 134, 157Howe, P.S., 444, 341Howell, C.R., 444, 252Howell, G., 443, 394Hoyer, P., 449, 306Hristov, P., 446, 117Hrubec, J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Hsiung, Y.B., 447, 240Hsu, S.D.H., 449, 281Hu, G., 444, 503, 569; 445, 428; 446, 368Hu, G.Y., 446, 356Hu, H., 445, 239; 447, 336Hu, H.M., 446, 356Hu, J.L., 446, 356Hu, Q.H., 446, 356Hu, T., 446, 356Hu, X.Q., 446, 356Huang, C.-G., 441, 285Huang, C.-S., 442, 209Huang, H.-W., 441, 396Huang, M.J., 446, 197Huang, X., 445, 239; 447, 336Huang, Y.Z., 446, 356Hubbard, D., 445, 439Huber, D., 447, 216¨Huehn, T., 445, 239; 447, 336Huet, K., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Hufner, J., 445, 223¨Hughes, G., 445, 239; 447, 336Hughes, G.J., 446, 62, 75; 448, 311; 449, 364,

383Hughes, V.W., 443, 394Huitu, K., 445, 394; 446, 285; 448, 234Hultqvist, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Hummler, H., 444, 523¨Huntemeyer, P., 444, 539; 447, 134, 157¨Hurst, P., 445, 439Hurvits, G., 445, 449; 448, 303Huss, D., 449, 401Hussein, A., 444, 252Huttmann, K., 445, 239; 446, 349; 447, 336¨Hyett, N., 445, 439Hyun, S., 441, 116

Iacobucci, G., 443, 394Iacopini, E., 445, 439; 446, 117Iannotti, L., 443, 394Iaselli, G., 445, 239; 447, 336Iashvili, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ibanez, L.E., 447, 257´˜Iconomidou-Fayard, L., 446, 117Ideguchi, E., 448, 180

Ieiri, M., 444, 267Iga, Y., 443, 394Igo, G., 444, 523Igo-Kemenes, P., 444, 539; 447, 134, 157Ihssen, H., 442, 484; 444, 531Iijima, T., 444, 267Iinuma, M., 444, 267Ikegami, Y., 443, 409Ilgenfritz, E.-M., 443, 244Imai, K., 444, 267Imai, N., 448, 180Imbo, T.D., 441, 468Imrie, D.C., 444, 539; 447, 134, 157Inaba, S., 446, 342Inuzuka, M., 443, 394Inyakin, A.V., 446, 342Iodice, M., 442, 484; 444, 531Ireland, D.G., 442, 43Irmscher, D., 444, 523Isaksson, L., 442, 43Isgur, N., 448, 111Ishida, T., 446, 342Ishihara, M., 448, 180Ishihara, N., 447, 167Ishii, K., 444, 539; 447, 134, 157Ishii, T., 443, 394Ishiyama, H., 442, 53Isupov, A., 445, 14Itakura, K., 442, 217Ito, K., 441, 155; 449, 48Itow, Y., 444, 267Ivanchenko, V.N., 449, 122Ivanov, D.Yu., 442, 453Ivanov, E., 445, 60Ivanov, M.A., 442, 435; 448, 143Ivashchuk, V.D., 442, 125Iwasa, N., 444, 32Iwasaki, H., 448, 180Iwata, T., 445, 14Iwata, Y., 447, 167Izotov, A., 442, 484; 444, 531

Jacholkowska, A., 445, 239; 447, 336Jacholkowski, A., 446, 342; 447, 178; 449, 401Jack, I., 443, 177Jackson, H.E., 442, 484; 444, 531Jackson, J.N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Jacob, F.R., 444, 539; 447, 134, 157Jacobs, P., 444, 523Jacobsen, T., 446, 342Jacobsson, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Jadach, S., 449, 97Jaffe, D.E., 445, 239; 447, 336Jahnen, T., 445, 20Jaimungal, S., 441, 147Jakob, H.-P., 443, 394


Jakobs, K., 445, 239; 447, 336Jakovac, A., 446, 203´Jalocha, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383James, C., 445, 449; 448, 303Jaminon, M., 443, 33Jamnik, D., 446, 349Janas, Z., 444, 32Janik, R., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Janik, R.A., 442, 300; 446, 9Janot, P., 445, 239; 447, 336Janssens, R.V.F., 446, 22Jarlskog, C., 448, 311; 449, 240Jarlskog, Ch., 441, 479; 444, 491; 446, 62, 75;

449, 364, 383Jarlskog, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Jarry, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Jawahery, A., 444, 539; 447, 134, 157Jean-Marie, B., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Jeitler, M., 446, 117Jejer, V., 447, 240Jelen, K., 443, 394´Jennings, J., 447, 240Jensen, D.A., 447, 240Jeoung, H.Y., 443, 394Jeremie, H., 444, 539; 447, 134, 157Jezequel, S., 445, 239; 447, 336Jgoun, A., 442, 484; 444, 531Jiang, C.H., 446, 356Jimack, M., 444, 539; 447, 134, 157Jin, B.N., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Jin, C., 448, 119Jin, H.Q., 443, 89Jin, S., 445, 239; 447, 336Jin, Y., 446, 356Jing, Z., 443, 394Jinghua, F., 444, 563Joepen, N., 444, 555Joffe–Minor, T., 449, 137Johanson, J., 446, 179Johansson, A., 446, 179Johansson, E.K., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Johansson, T., 446, 179Johnson, K.F., 443, 394Johnson, P.T., 447, 240Jon-And, K., 444, 38, 43, 52Jones, C., 444, 531Jones, C.R., 444, 539; 447, 134, 157Jones, D.R.T., 443, 177Jones, G.T., 446, 342; 449, 401Jones, L.W., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152

Jones, P., 443, 82Jones, P.G., 444, 523Jones, P.M., 443, 69Jones, R.W.L., 445, 239; 447, 336Jones, T.W., 443, 394Jonsson, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Jonsson, T., 449, 253Jopen, N., 445, 20¨Joram, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Jorjadze, G., 448, 203Josa-Mutuberria, I., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Joseph, C., 445, 439Jost, B., 445, 239; 447, 336Jostlein, H., 445, 419Jouan, D., 444, 516; 449, 128Jousset, J., 445, 239; 447, 336Jovanovic, P., 444, 539; 447, 134, 157; 449,

401Joyce, M., 448, 321Juget, F., 445, 439Juillot, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Julin, R., 443, 69, 82Jumatsu, T., 442, 53Junghans, A.R., 444, 32Jungst, H., 445, 20¨Jungst, H.G., 444, 555¨Junk, T.R., 444, 539; 447, 134, 157Jusko, A., 449, 401Juste, A., 445, 239; 447, 336Juutinen, S., 443, 69, 82

Kachelhoffer, T., 449, 401Kadija, K., 444, 523Kado, M., 445, 239; 447, 336Kafka, T., 449, 137Kageya, T., 445, 14Kainulainen, K., 448, 321Kaiser, R., 442, 484; 444, 531Kalinowsky, H., 444, 555; 445, 20; 446, 349Kallosh, R., 443, 143Kalter, A., 446, 117Kamada, H., 447, 216Kammel, P., 446, 349Kammle, B., 446, 349¨Kamrad, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kananov, S., 443, 394Kang, G., 445, 27Kang, K., 442, 249Kang, S.K., 442, 249Kankaanpaa, H., 443, 69, 82¨¨Kantar, M., 443, 359Kanzaki, J., 447, 167Kaplan, D.B., 449, 1


Kappes, A., 443, 394Kapusta, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kapusta, J.I., 443, 63Kapustinsky, J.S., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Karafasoulis, K., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Karat, E., 445, 337Karch, A., 441, 235Karlen, D., 444, 539; 447, 134, 157Karny, M., 444, 32Karsch, F., 442, 291Karshon, U., 443, 394Kartvelishvili, V., 444, 539; 447, 134, 157Kasahara, S.M.S., 449, 137Kasemann, M., 443, 394Kashirin, V., 445, 14Kasper, P.A., 445, 449; 448, 303Kasser, A., 448, 152Kato, M., 442, 53Katsanevas, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Katsoufis, E.C., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Katz, U.F., 443, 394Kaur, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kaw, P.K., 446, 104Kawagoe, K., 444, 539; 447, 134, 157Kawamoto, T., 444, 539; 447, 134, 157Kawamura, Y., 446, 228Kawano, M., 445, 14Kayal, P.I., 444, 539; 447, 134, 157Kazakov, D.I., 449, 201Kcira, D., 443, 394Ke, Z.J., 446, 356Keeler, R.K., 444, 539; 447, 134, 157Keenan, A., 443, 69, 82Kehagias, A., 444, 190; 445, 69KEK-PS E224 Collaboration, 444, 267Kekelidze, V., 446, 117Kellogg, R.G., 444, 539; 447, 134, 157Kelly, M.S., 445, 239; 447, 336Kennedy, B.W., 444, 539; 447, 134, 157Keranen, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kerger, R., 443, 394Kernan, P.J., 445, 412Kersevan, B.P., 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Kesseler, G., 446, 117Kessler, R., 447, 240Kettle, P.-R., 444, 38, 43, 52Kettunen, H., 443, 69, 82Keung, W.-Y., 444, 142Khakzad, M., 443, 394Khan, R.A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152

Khaustov, G.V., 446, 342Khein, L.A., 443, 394Khokhlov, Yu., 441, 479Khomenko, B.A., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Khovanski, N.N., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Khoze, V.V., 442, 145Khrenov, A., 445, 14Khriplovich, I.B., 444, 98Kienzle-Focacci, M.N., 444, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Kiiskinen, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kiko, J., 447, 127Kikuchi, R., 447, 167Kikukawa, Y., 448, 265Kilian, K., 446, 179Kim, B.R., 441, 215Kim, C.L., 443, 394Kim, C.S., 441, 410Kim, C.W., 444, 204Kim, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kim, D.H., 444, 503, 569; 445, 428; 446, 368;

447, 134, 147, 157; 448, 152Kim, D.W., 447, 336Kim, H.-C., 449, 299Kim, H.Y., 445, 239; 447, 336Kim, J.E., 442, 249; 447, 110Kim, J.K., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kim, J.Y., 443, 394Kim, N.J., 441, 83Kim, S.C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kinashi, T., 446, 342King, B., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383King, B.A., 441, 468King, S.F., 442, 68; 445, 191King, S.L., 443, 69, 82Kinney, E., 442, 484; 444, 531Kinnison, W.W., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Kinson, J.B., 446, 342; 447, 178; 449, 401Kinvig, A., 446, 62, 75; 448, 311; 449, 364,

383Kirch, U., 444, 555; 445, 20Kirchner, R., 446, 209Kirk, A., 446, 342; 447, 178; 449, 401Kirkby, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kirkby, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kirkby, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kirsanov, M., 445, 439Kirsch, M., 444, 531


Kirsten, T., 447, 127Kisiel, J., 446, 349Kisielewska, D., 443, 394Kiss, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kisselev, A., 442, 484; 444, 531Kisslinger, L.S., 445, 271Kitamura, S., 443, 394Kitching, P., 442, 484; 444, 531Kittel, W., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kivel, N., 443, 308Kjaer, N.J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Klanner, R., 443, 394Klapp, O., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Klein, F., 444, 555; 445, 20Klein, F.-J., 444, 555Klein, F.J., 445, 20Klein, H., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Kleinknecht, K., 445, 239; 446, 117; 447, 336Klempt, E., 444, 555; 445, 20; 446, 349Klempt, W., 446, 342; 449, 401Klier, A., 444, 539; 447, 134, 157Klimek, K., 443, 394Klimentov, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Klimov, O., 445, 439Kluberg, L., 444, 516; 449, 128Kluge, E.E., 445, 239; 447, 336Kluge, W., 443, 77Kluit, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Kluth, S., 444, 539; 447, 134, 157Knaepen, B., 441, 198Knecht, M., 443, 255Kneringer, E., 445, 239; 447, 336Knoblauch, D., 441, 479Knowles, I.G., 446, 117Knudsen, B.T.H., 449, 401Knudson, K., 447, 178; 449, 401Ko, C.M., 444, 237; 445, 265Ko, P., 442, 249Ko, U., 443, 394Kobakhidze, A.B., 448, 243Kobayashi, H., 442, 484; 444, 531Kobayashi, T., 441, 235; 442, 192; 444, 539;

447, 134, 157Kobel, M., 444, 539; 447, 134, 157Kobrak, H.G.E., 447, 240Koch, H., 446, 349Koch, N., 442, 484; 444, 531Koch, U., 446, 117Koch, W., 443, 394Koetke, D.S., 444, 539; 447, 134, 157Koffeman, E., 443, 394

Kofman, L., 448, 6Kogan, I.I., 442, 136Kokkas, P., 444, 38, 43, 52Kokkinias, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kokkonen, J., 445, 439Kokott, T.P., 444, 539; 447, 134, 157Kolesnikov, V., 445, 14Kolesnikov, V.I., 444, 523Kolo, C., 446, 349Kolosov, V., 446, 342Kolrep, M., 444, 539; 447, 134, 157Kolstein, M., 442, 484; 444, 531Kolster, H., 442, 484; 444, 531Komamiya, S., 444, 539; 447, 134, 157Komatsubara, T., 442, 53Konashenok, A., 444, 523Kondashov, A.A., 446, 342Kondo, T., 447, 167Konig, A.C., 444, 503, 569; 445, 428; 446,¨

368; 447, 147; 448, 152Konigsmann, K., 442, 484; 444, 531¨Konopelchenko, B.G., 444, 299Konopliannikov, A., 441, 479Konstantinidis, N., 445, 239; 447, 336Kooijman, P., 443, 394Koop, T., 443, 394Kopeliovich, B., 446, 321; 447, 308Kopeliovich, B.Z., 445, 223Kopke, L., 446, 117¨Kopp, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Koratzinos, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Korchin, A.Yu., 441, 17Korhonen, T.T., 447, 167Korner, J.G., 442, 435; 448, 143¨Korol, A.A., 449, 122Korolko, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Korostelev, M.S., 449, 122Korotkov, V., 442, 484; 444, 531Korotkova, N.A., 443, 394Korpa, C.L., 446, 15Korsch, W., 442, 484; 444, 531Korthals Altes, C.P., 448, 85Korzhavina, I.A., 443, 394Koshuba, S.V., 449, 122Kosmas, T.S., 443, 7Kossakowski, R., 444, 516; 449, 128Kostioukhine, V., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Kostov, I.K., 444, 196Kostrewa, D., 445, 20Kotanski, A., 443, 394´Kotikov, A.V., 441, 345Kourkoumelis, C., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383


Koutsenko, V., 444, 503, 569; 445, 428; 446,368; 447, 147; 448, 152

Kouznetsov, O., 441, 479; 444, 491; 446, 62,75; 448, 311; 449, 364, 383

Kovacs, T.G., 443, 239´Kovalenko, S., 442, 203Kovzelev, A., 445, 439Kowal, A.M., 443, 394Kowalewski, R.V., 444, 539; 447, 134, 157Kowalski, H., 443, 394Kowalski, M., 444, 523Kowalski, T., 443, 394Kozela, A., 444, 555; 445, 20Kozlov, V., 442, 484; 444, 531Kraemer, R.W., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Krakauer, D., 443, 394Kralik, I., 449, 401´Kramer, L.H., 442, 484; 444, 531Kramer, M., 441, 383¨Krammer, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kraniotis, G.V., 443, 111Krasnitz, A., 445, 366Krasnoperov, A., 445, 439Krause, B., 444, 531Krauss, L.M., 445, 412Krcmar, M., 442, 38ˇKrecak, Z., 442, 38ˇKrehl, O., 444, 25Krenz, W., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kress, T., 444, 539; 447, 134, 157Kreuger, R., 444, 38, 43, 52Kreuter, C., 441, 479; 444, 491; 446, 62, 75Kreyerhoff, G., 441, 215Krieger, P., 444, 539; 447, 134, 157Krishnaswami, G.S., 441, 429Krivokhijine, V.G., 442, 484; 444, 531Kriznic, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Krmpotic, F., 444, 14; 445, 249´Krocker, M., 445, 239; 447, 336¨Kroll, P., 449, 339Krstic, J., 441, 479; 444, 491; 446, 62, 75; 449,

364, 383Krstic, P., 448, 311Krumstein, Z., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Krzywicki, A., 448, 257KTeV Collaboration, 447, 240Ktorides, C.N., 444, 583Kubinec, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kubischta, W., 446, 117Kubo, J., 441, 235

Kubo, T., 448, 180Kucewicz, W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 383Kuckes, M., 444, 531¨Kuhl, T., 444, 539; 447, 134, 157Kuhn, C., 446, 191Kuhn, D., 445, 239; 447, 336Kuiroukidis, A., 443, 131Kullander, S., 446, 179Kumagai, H., 448, 180Kumbartzki, G., 446, 22Kummell, F., 442, 484; 444, 531¨Kun, S.Yu., 448, 163Kunde, G.J., 446, 197Kunin, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Kunz, J., 441, 77Kunze, M., 446, 349Kupsc, A., 446, 179´´Kuraev, E.A., 442, 453Kurashige, H., 443, 409; 447, 167Kurilla, U., 446, 349Kurisuno, M., 442, 484Kurosawa, K., 445, 316Kurowska, J., 446, 62, 75; 448, 311; 449, 364,

383Kurvinen, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Kurz, G., 446, 179Kusaka, K., 448, 180Kuusiniemi, P., 443, 69, 82Kuze, M., 443, 394Kuzenko, S.M., 446, 216Kuzmin, V.A., 443, 394Kuznetsov, A.V., 446, 378Kuznetsov, V.E., 445, 439Kwan, S., 445, 449; 448, 303Kyae, B., 447, 110Kyberd, P., 444, 539; 447, 134, 157Kyle, G., 442, 484; 444, 531Kyriakis, A., 445, 239; 447, 336

Labarga, L., 443, 394Lacaprara, S., 445, 439Lacentre, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lachnit, W., 442, 484; 444, 531Ladron de Guevara, P., 444, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152LaDue, J., 447, 240Ladygin, V., 445, 14Lafferty, G.D., 444, 539; 447, 134, 157LaFosse, D., 443, 89Lahanas, M., 444, 583Lai, A., 446, 117Lai, Y.F., 446, 356


Lakata, M., 446, 349Laktineh, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lamberti, L., 443, 394Lampe, B., 446, 163Lamsa, J.W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lancon, E., 445, 239; 447, 336Landaud, G., 444, 516; 449, 128Landi, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Landolfi, G., 444, 299Landshoff, P.V., 448, 281Landsman, H., 447, 134, 157Landua, R., 446, 349Lane, D.W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lane, J.B., 443, 394Lang, C.B., 443, 214Lang, P.F., 446, 356Langefeld, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Langs, D.C., 445, 449; 448, 303Lanske, D., 444, 539; 447, 134, 157Lanza, A., 445, 439Lapin, V., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Lapoint, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152La Rotonda, L., 445, 439Lasiuk, B., 444, 523Lassalle, J.C., 447, 178Lassila-Perini, K., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Lath, A., 447, 240Laubenstein, M., 447, 127Lauber, J., 444, 539; 447, 134, 157Laucelli Meana, M., 447, 59Laugier, J.-P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lauhakangas, R., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Laurelli, P., 445, 239; 447, 336Laurenti, G., 443, 394Laurikainen, P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Lauritsen, T., 446, 22Lautenschlager, S.R., 444, 539; 447, 134, 157Laveder, M., 445, 439Lavorato, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Lavoura, L., 442, 390Lawall, R., 444, 555; 445, 20Lawson, I., 444, 539; 447, 134, 157Layter, J.G., 444, 539; 447, 134, 157Lazarides, G., 441, 46Lazic, D., 444, 539; 447, 134, 157Lazzeroni, C., 445, 439

Lazzizzera, I., 444, 167L3 Collaboration, 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Leader, E., 445, 232Lebeau, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lebedev, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Lebedev, O., 441, 419Le Bornec, Y., 445, 423Lebrun, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lecomte, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lecoq, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Le Coultre, P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Leder, G., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Le Diberder, F., 447, 336Lednev, A.A., 446, 342Lednicky, R., 446, 191Ledovskoy, A., 447, 240Ledroit, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lee, A.M., 444, 539; 447, 134, 157Lee, C., 449, 223Lee, C.-H., 448, 168Lee, D., 444, 474; 447, 98Lee, D.-S., 449, 274Lee, H.J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lee, H.W., 441, 83Lee, I.-Y., 449, 6Lee, I.Y., 446, 22Lee, J.H., 443, 394Lee, J.M., 444, 267Lee, J.S., 447, 110Lee, K., 445, 387Lee, S.B., 443, 394Lee, S.W., 443, 394Lee, T., 447, 83Lee, U.W., 444, 204Leenhardt, S., 444, 32Lees, J.-P., 445, 239; 447, 183, 336Leeson, W., 449, 137Le Faou, J.H., 442, 48Lefebure, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lefrancois, J., 445, 239; 447, 336Le Gac, R., 444, 38, 43, 52Le Goff, J.M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Lehraus, I., 445, 239; 447, 336Lehto, M., 445, 239; 447, 336Leimgruber, F., 444, 38, 43, 52Leino, M., 443, 69, 82


Leinonen, L., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Leisos, A., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Leiste, R., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Leitner, R., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Lellouch, D., 444, 539; 447, 134, 157Lemaire, M.-C., 445, 239; 447, 336Lemmon, R., 443, 69, 82Lemmon, R.C., 446, 197Lemonne, J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 383Lenti, M., 446, 117Lenti, V., 446, 342; 447, 178; 449, 401Lenzen, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Leonardi, E., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Leontaris, G.K., 447, 67Lepeltier, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Leroy, J.P., 444, 401Leroy, O., 445, 239; 447, 336Lesiak, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Leslie, J., 445, 449; 448, 303Letessier-Selvon, A., 445, 439Lethuillier, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Letts, J., 444, 539; 447, 134, 157Leung, C.N., 443, 185Levai, P., 442, 1; 444, 523Levi, G., 443, 394Levinson, L., 444, 539; 447, 134, 157Levman, G.M., 443, 394Levtchenko, P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Levy, A., 443, 394Levy, J-M., 445, 439Lewitowicz, M., 444, 32Li, C., 444, 503, 569; 445, 428; 446, 368; 447,

147; 448, 152Li, C.G., 446, 356Li, C.S., 444, 224Li, D., 446, 356Li, G., 443, 58Li, H.B., 446, 356Li, J., 446, 356Li, P.Q., 446, 356Li, R.B., 446, 356Li, W., 446, 356Li, W.G., 446, 356Li, X.H., 446, 356; 449, 361Li, X.N., 446, 356Li, Z., 445, 271Lianshou, L., 444, 563

Liao, Y., 441, 383Libanov, M.V., 442, 63Libby, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Lichtenstadt, J., 448, 303Lidsey, J.E., 443, 97Liebisch, R., 444, 539; 447, 134, 157Lien, J.A., 447, 178Lietava, R., 449, 401Ligabue, F., 445, 239; 447, 336Liko, D., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Lim, H., 443, 394Lim, I.T., 443, 394Lima, C.L., 448, 1Limentani, S., 443, 394Lin, C.H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lin, J., 445, 239; 447, 336Lin, W.T., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lin, Z., 444, 245Linde, F.L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lindemann, L., 443, 394; 445, 20Ling, T.Y., 443, 394Link, J., 444, 555; 445, 20Linssen, L., 445, 439Lipkin, H.J., 445, 403Lipniacka, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lippi, I., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Lising, L.J., 449, 6Lissia, M., 441, 291List, B., 444, 539; 447, 134, 157Lista, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Litchfield, P.J., 449, 137Litke, A.M., 445, 239; 447, 336Littlewood, C., 444, 539; 447, 134, 157Litvinenko, A., 445, 14Liu, D., 444, 539; 446, 332; 447, 134, 157Liu, F., 444, 523Liu, H.M., 446, 356Liu, J., 446, 356Liu, R., 441, 473Liu, R.G., 446, 356Liu, W., 443, 394Liu, Y., 446, 356Liu, Z.A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ljubicic, A., 442, 38; 445, 439ˇ ´Lloyd, A.W., 444, 539; 447, 134, 157Lloyd, S.L., 444, 539; 447, 134, 157Locci, E., 445, 239; 447, 336Loconsole, R.A., 447, 178; 449, 401Loebinger, F.K., 444, 539; 447, 134, 157


Loerstad, B., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Lohmann, W., 444, 503, 569; 445, 428; 446,368; 447, 147; 448, 152

Lohr, B., 443, 394¨Lohrmann, E., 443, 394Loinaz, W., 445, 178Lokajicek, M., 441, 479; 446, 62Loken, J.G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Long, G.D., 444, 539; 447, 134, 157Long, J., 445, 439Long, K.R., 443, 394Longley, N.P., 449, 137Longo, E., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Loomis, C., 445, 239; 447, 336Lopes, J.H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lopez-Duran Viani, A., 443, 394Lopez-Fernandez, R., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Lopez, J.M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lorenzon, W., 442, 484; 444, 531Losty, M.J., 444, 539; 447, 134, 157Loukachine, K., 442, 48Loukas, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lourenco, C., 444, 516; 449, 128Love, A., 443, 111Løvhøiden, G., 449, 401Lu, C.-D., 445, 394¨Lu, D.H., 441, 27; 443, 26Lu, F., 446, 356Lu, J.G., 446, 356Lu, J.X., 443, 167Lu, W., 444, 503, 569; 445, 428; 446, 368; 447,

147; 448, 152Lu, Y.S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Lubelsmeyer, K., 444, 503, 569; 445, 428; 446,¨

368; 447, 147; 448, 152Lubicz, V., 444, 401Lubrano, P., 446, 117Luci, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Luckey, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Luckmann, S., 446, 209Lucotte, A., 445, 239; 447, 183, 336Ludwig, J., 444, 539; 447, 134, 157Luitz, S., 446, 117Lukacs, B., 443, 21´Lukina, O.Yu., 443, 394Lukyanov, S.M., 448, 180Luminari, L., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152

Lundberg, B., 445, 449; 448, 303Lundin, M., 442, 43Lung, A., 444, 531Lunin, O., 442, 173Luo, X.L., 446, 356Lupi, A., 445, 439Luppi, E., 444, 111Luptak, M., 449, 401´Luquin, L., 449, 128Lusiani, A., 445, 239; 447, 336Lustermann, W., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Lutjens, G., 445, 239; 447, 336¨Lutz, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383L’Yi, W.S., 445, 134; 448, 218Lynch, J.G., 445, 239; 447, 336Lynch, W.G., 446, 197Lyons, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Lyth, D.H., 448, 191Lyuboshitz, V.L., 446, 191Lyubovitskij, V.E., 442, 435; 448, 143

Ma, B.-Q., 441, 461Ma, E., 442, 238; 444, 391Ma, E.C., 446, 356Ma, J.M., 446, 356Ma, K.J., 443, 394Ma, W.G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Maccarrone, G., 443, 394Macchiavelli, A.O., 446, 22; 449, 6Macchiolo, A., 444, 539; 447, 134, 157Macdonald, N., 443, 394Machefert, F., 445, 239; 447, 336Machleidt, R., 445, 259Mack, V., 449, 401MacLeod, R.W., 449, 6MacNaughton, J., 444, 491; 446, 62; 448, 311;

449, 364, 383Macpherson, A., 444, 539; 447, 134, 157Mader, W., 444, 539; 447, 134, 157Maedan, S., 442, 217Maggi, G., 445, 239; 447, 336Maggi, M., 445, 239; 447, 336Magierski, P., 443, 69Magill, S., 443, 394Magnea, L., 448, 295Magnin, J., 445, 8Magueijo, J., 443, 104; 447, 246Mahanthappa, K.T., 441, 178Mahapatra, D.P., 449, 109Mahon, J.R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Maio, A., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Maiolino, C., 442, 48


Maity, M., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Majumdar, P., 445, 129Majumder, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Makeenko, Y.M., 445, 307Makino, S., 444, 267Makins, N.C.R., 442, 484; 444, 531Malakhov, A., 445, 14Malakhov, A.I., 444, 523Malek, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Maley, P., 445, 239; 447, 336Malgeri, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Malinin, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Maljukov, S., 446, 342Mallik, U., 443, 394Malmgren, T.G.M., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Maltoni, F., 441, 257Malychev, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mamier, G., 447, 336Mana, C., 444, 503, 569; 445, 428; 446, 368;˜

447, 147; 448, 152Manaenkov, S.I., 442, 484; 444, 531Mandic, I., 444, 38, 43, 52´Mandl, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mandry, R., 444, 516; 449, 128Manduci, L., 446, 197Mangeol, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Mann, W.A., 449, 137Mannelli, I., 446, 117Mannelli, M., 444, 539; 447, 134, 157Manner, W., 445, 239; 447, 336¨Mannert, C., 445, 239; 447, 336Mannocchi, G., 445, 239; 447, 336Manns, J., 445, 20Manola-Poggioli, E., 445, 439Mansouri, F., 445, 52Manthos, N., 444, 38, 43, 52Manvelyan, R., 444, 86Manzari, V., 447, 178; 449, 401Mao, H.S., 446, 356Mao, Z.P., 446, 356Marcellini, S., 444, 539; 447, 134, 157March-Russell, J., 441, 96Marchesini, P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Marchetto, F., 446, 117Marchionni, A., 445, 439Marciniewski, P., 446, 179

Marco, J., 441, 479; 444, 491; 446, 62, 75; 448,311; 449, 364, 383

Marco, R., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Marechal, B., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Marel, G., 444, 38, 43, 52Margetis, S., 444, 523Margoni, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Margotti, A., 443, 394Marian, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Mariano, A., 445, 249Marin, A., 444, 503, 569; 445, 428; 446, 368Marin, J.-C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Marinelli, N., 445, 239; 447, 336Marini, G., 443, 394Mariotti, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Markert, C., 444, 523Markopoulos, C., 444, 539; 447, 134, 157Markou, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Markou, C., 445, 239; 447, 336Markun, P., 443, 394Markytan, M., 446, 117Marnelius, R., 441, 243Marras, D., 446, 117Marrocchesi, P.S., 445, 239; 447, 336Marshak, M.L., 449, 137Marshakov, A., 448, 195Mart, T., 445, 20Martell, E.C., 441, 468Martelli, F., 445, 439Martens, F.K., 444, 531Martens, K., 442, 484Martı, G.V., 447, 41´Marti i Garcia, S., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Martin, A.D., 443, 301Martin, A.J., 444, 539; 447, 134, 157Martin, E.B., 445, 239; 447, 336Martin, F., 445, 239; 447, 336Martin, J.F., 443, 394Martin, J.M., 445, 423Martin, J.P., 444, 503, 539, 569; 445, 428; 446,

368; 447, 134, 147, 157; 448, 152Martin, J.W., 442, 484; 444, 531Martın-Mayor, V., 441, 330´Martin, O., 448, 99Martin, V.J., 446, 117Martinelli, G., 441, 371; 444, 401Martinengo, P., 446, 342; 449, 401Martinez, G., 444, 539; 447, 134, 157


Martınez, M., 443, 394; 445, 239; 447, 336´Martinez-Rivero, C., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Martinez-Vidal, F., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Martini, M., 446, 117Martins, C.J.A.P., 445, 43Marukyan, H., 442, 484Marzano, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Marzulli, V., 446, 117Masaike, A., 444, 267Maselli, S., 443, 394Mashimo, T., 444, 539; 447, 134, 157Masik, J., 448, 311; 449, 364, 383Masip, M., 444, 352Masoli, F., 442, 484; 444, 531Masoni, A., 444, 111Massam, T., 443, 394Massaro, G.G.G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Mastroberardino, A., 443, 394Mastroyiannopoulos, N., 441, 479; 444, 491;

446, 62, 75; 448, 311; 449, 364, 383Matchev, K.T., 445, 331Mateos, A., 442, 484; 444, 531Mateos, D., 443, 147Matheys, J.P., 446, 117Mathur, N., 444, 7Mato, P., 445, 239; 447, 336Matone, M., 445, 77, 357Matono, Y., 443, 409Matorras, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Matsuda, E.K., 447, 167Matsuda, M., 449, 240Matsuda, T., 445, 14Matsuda, Y., 444, 267Matsui, T., 447, 167Matsushita, T., 443, 394Matsuura, K., 442, 53Matsuyama, Y., 444, 267Matsuzaki, M., 445, 254Matteuzzi, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Matthay, H., 446, 349¨Matthiae, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mattig, P., 444, 539; 447, 134, 157¨Mattingly, M.C.K., 443, 394Mattingly, S., 443, 394Mattis, M.P., 442, 145Mayer, R.H., 446, 22MayTal-Beck, S., 445, 449; 448, 303Mazik, J., 441, 479; 444, 491; 446, 62, 75Mazumdar, K., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Mazur, P.O., 444, 284Mazzoni, M.A., 449, 401

Mazzucato, E., 446, 117Mazzucato, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mazzucato, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383McAllister, S., 442, 43McAndrew, M., 442, 484; 444, 531McCance, G.J., 443, 394McCrady, R., 446, 349Mc Cubbin, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383McCubbin, N.A., 443, 394McDonald, W.J., 444, 539; 447, 134, 157McFall, J.D., 443, 394McGeorge, J.C., 442, 43McGovern, J.A., 446, 300McIlhany, K., 442, 484; 444, 531Mc Kay, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383McKellar, B.H.J., 444, 75McKenna, J., 444, 539; 447, 134, 157McKeown, R.D., 442, 484; 444, 531Mckigney, E.A., 444, 539; 447, 134, 157McMahon, T.J., 444, 539; 447, 134, 157McManus, A.P., 447, 240McNabb, D.P., 446, 22McNamara III, P.A., 445, 239; 447, 336McNeil, M.A., 445, 239; 447, 336McNeil, R.R., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Mc Nulty, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mc Pherson, G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383McPherson, R.A., 444, 539; 447, 134, 157Meadows, B., 445, 449Meadows, B.T., 448, 303Mechain, X., 445, 439´Medcalf, T., 445, 239; 447, 336Meddi, F., 449, 401Mehen, T., 445, 378Meier, J., 446, 349Meier, R., 443, 77; 446, 363Meijers, F., 444, 539; 447, 134, 157Meißner, F., 442, 484Meißner, U.-G., 447, 1Meissner, F., 444, 531Mele, S., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Melikhov, D., 442, 381; 446, 336Melikyan, A., 444, 86Melkumov, G.L., 444, 523Mellado, B., 443, 394Menary, S., 443, 394Menden, F., 442, 484; 444, 531Mendiburu, J-P., 445, 439Meng, X.C., 446, 356Menichetti, E., 446, 117Menke, S., 444, 539; 447, 134, 157


Menze, D., 444, 555; 445, 20Mercer, D., 444, 531Merebashvili, Z., 442, 374Merino, G., 445, 239; 447, 336Merkel, H., 445, 20Merkel, R., 445, 20Merle, E., 445, 239; 447, 183, 336Merola, L., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Merola, S.M.L., 444, 503Meroni, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Merritt, F.S., 444, 539; 447, 134, 157Mertens, G., 444, 252Mes, H., 444, 539; 447, 134, 157Meschini, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Mescia, F., 444, 401Messi, R., 444, 111Messineo, A., 445, 239; 447, 336Mestvirishvili, A., 446, 117Metz, A., 442, 484; 444, 531Metzger, W.J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Meyer, A., 443, 394Meyer, C.A., 446, 349Meyer, J., 444, 539; 447, 134, 157Meyer, J.-P., 445, 439Meyer-Larsen, A., 443, 394Meyer, W.T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Meyners, N., 442, 484; 444, 531Mezzetto, M., 445, 439Mi, Y., 448, 152Miagkov, A., 441, 479; 446, 75; 448, 311; 449,

364, 383Michaels, R., 448, 275Michalon, A., 449, 401Michalon-Mentzer, M.E., 449, 401Michel, B., 445, 239; 447, 336Michelini, A., 444, 539; 447, 134, 157Michetti, A., 446, 117Middelkamp, P., 449, 401Migani, D., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Migliore, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Migneco, E., 442, 48Mihalcea, D., 445, 449; 448, 303Mihara, S., 444, 267, 539; 447, 134, 157Mihul, A., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Mihul, D.M.A., 444, 503Mikelsons, P., 447, 240Mikenberg, G., 444, 539; 447, 134, 157Mikhailov, A.A., 449, 237Mikheev, N.V., 446, 378Mikloukho, O., 442, 484; 444, 531

Mikulec, I., 446, 117Mikuz, M., 444, 38, 43, 52ˇMilburn, R.H., 445, 449; 448, 303; 449, 137Milcent, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Milewski, J., 443, 394Milite, M., 443, 394Miller, C.A., 442, 484; 444, 531Miller, D.B., 443, 394Miller, D.J., 444, 539; 447, 134, 157Miller, J., 444, 38, 43, 52Miller, M.A., 442, 484; 444, 531Miller, W.H., 449, 137Milner, R., 442, 484; 444, 531Minakata, H., 449, 260Minard, M.-N., 445, 239; 447, 183, 336Minashvili, I., 446, 342Minic, D., 442, 102Minten, A., 445, 239; 447, 336Miquel, R., 445, 239; 447, 336Mir, Ll.M., 447, 336Mir, L.M., 445, 239Mir, R., 444, 539; 447, 134, 157Mirabelli, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Mirabito, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mironov, A., 448, 195Mishra, G.C., 449, 109Mishra, S.R., 445, 439Mitaroff, W.A., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Mitra, P., 441, 89Mitsyn, V., 442, 484; 444, 531Miu, G., 449, 313Miyake, K., 447, 167Miyamura, O., 443, 331Mizoguchi, S., 441, 123Mizukoshi, J.K., 447, 116Mjoernmark, U., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Mkrtchyan, R., 444, 86Mnich, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Moa, T., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Moch, M., 448, 311; 449, 364, 383Mochida, S., 447, 240Mock, A., 444, 523Moeller, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Moenig, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mohanty, B., 449, 109Mohanty, S., 445, 185Mohapatra, R.N., 441, 299; 442, 199Mohr, W., 444, 539; 447, 134, 157Molnar, J., 444, 523´


Molnar, P., 444, 569; 445, 428; 446, 368; 447,147; 448, 152

Molodtsov, S.V., 443, 387Monaco, V., 443, 394Mondardini, M.R., 445, 419Monderen, D., 444, 397Moneta, L., 445, 239Monge, M.R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Monig, K., 443, 394¨Monnier, E., 447, 240Montanari, A., 444, 539; 447, 134, 157Montanet, F., 444, 38, 43, 52Montanet, L., 449, 114Monteil, S., 445, 239; 447, 336Monteiro, T., 443, 394Monteleoni, B., 444, 569; 445, 428; 446, 368;

447, 147; 448, 152Monteleoni, P.M.B., 444, 503Montero, A., 442, 273Montret, J-C., 445, 239; 447, 336Montvay, I., 446, 209Moore, C.F., 444, 252Moore, K.N., 446, 117Moore, R., 444, 503, 569; 445, 428; 446, 368Moore, R.W., 446, 117Moorhead, G.F., 445, 439Morandin, M., 443, 394Morando, M., 449, 401Morawitz, P., 445, 239; 447, 336Moreau, G., 448, 57Moreau, X., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Moreno, J.M., 445, 82Morettini, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mori, S., 447, 167Mori, T., 444, 539; 447, 134, 157Moriconi, M., 447, 292Morishima, T., 447, 46Moroi, T., 447, 75Morosov, B., 446, 179Morozov, A., 448, 195Morpurgo, G., 448, 107Morris, C., 444, 252Morton, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mortsell, A., 446, 179¨Moßbauer, R., 447, 127¨Mosel, U., 447, 31Moser, H.-G., 445, 239; 447, 336Mossuz, L., 445, 439Most, A., 442, 484; 444, 531Mota, A.L., 445, 94Motsch, F., 445, 239; 447, 336Mottola, E., 444, 284Moukhai, E.A., 447, 8Moulik, T., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152

Mount, R., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Mourgues, S., 449, 128Moutoussi, A., 445, 239Moutoussi, L.M.A., 447, 336Mozzetti, R., 444, 531Mualem, L., 449, 137Muanza, G.S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Muccifora, V., 442, 484; 444, 531Muciaccia, M.T., 449, 401Mueller, A.C., 444, 32Mueller, U., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Muenich, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Muheim, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Muijs, A.J.M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Muikku, M., 443, 69, 82Mukhopadhyay, N.C., 444, 7Mukhopadhyaya, B., 443, 191Mulders, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Mulet-Marquis, C., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Muller, A., 444, 38, 43, 52Muller-Kirsten, H.J.W., 444, 86; 445, 287¨Mullins, S.M., 443, 89Multamaki, T., 445, 199¨Munday, D.J., 446, 117Munoz Sudupe, A., 441, 330˜Murakami, K., 443, 409Muresan, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Murgia, F., 442, 470Murray, W.J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Murray, W.N., 443, 394Murtas, F., 445, 239; 447, 336Murtas, G.P., 445, 239; 447, 336Muryn, B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Musa, L., 446, 117Musgrave, B., 443, 394Musto, R., 441, 69Muther, H., 445, 259¨Myatt, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Myklebust, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Myung, Y.S., 441, 83

Nachtman, J.M., 445, 239; 447, 336NA38 Collaboration, 444, 516, 523; 446, 117;

449, 128Nagai, K., 444, 539; 447, 134, 157Nagaitsev, A., 442, 484; 444, 531


Nagano, K., 443, 394Nagashima, Y., 447, 167Nagoshi, C., 444, 267Nahn, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Nakada, T., 444, 38, 43, 52Nakagawa, T., 446, 342Nakagawa, Y., 447, 167Nakamura, I., 444, 539; 447, 134, 157Nakamura, T., 447, 167; 448, 180Nakamura, T.T., 443, 409Nakano, I., 447, 167Nakaya, T., 447, 240Nakayama, H., 445, 14Nam, S.W., 443, 394Nandi, B.K., 449, 109Nania, R., 443, 394Napier, A., 445, 449; 448, 303; 449, 137Napolitano, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Nappi, A., 446, 117Nappi, E., 442, 484; 449, 401Naraghi, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Nardi, M., 442, 14Naryshkin, Y., 442, 484; 444, 531Nash, J., 445, 239; 447, 336Nason, P., 447, 327Nassalski, J., 446, 117Natale, A.A., 442, 369Nathan, A.M., 442, 484; 444, 531Nauenberg, M., 447, 23Nauenberg, U., 447, 240Nauta, B.J., 444, 463Navach, F., 447, 178; 449, 401Navarra, F.S., 443, 285Navarria, F.L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Navarro-Salas, J., 449, 30Navas, S., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Nawrocki, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Nayak, G.C., 442, 427Nayak, T.K., 449, 109Neal, H.A., 444, 539; 447, 134, 157Nedelec, P., 445, 439´ ´Needham, M.D., 446, 117Nefedov, Yu., 445, 439Negri, P., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Negus, P., 445, 239; 447, 336Neise, L., 447, 227Nellen, B., 444, 539; 447, 134, 157Nelson, J.M., 444, 523Nelson, K.S., 447, 240Nemecek, S., 446, 75; 448, 311; 449, 383Nemes, M.C., 445, 94

Nersessian, A., 445, 123Nessi-Tedaldi, F., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Neubert, M., 441, 403Neuerburg, W., 444, 555; 445, 20Neufeld, N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Neuhofer, G., 446, 117Neumeister, N., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Neunreither, F., 442, 484; 444, 531Newman, H., 444, 569; 445, 428; 446, 368;

447, 147; 448, 152Ng, J.N., 441, 419Ng, K.-W., 449, 274Nguyen, A., 445, 449; 448, 303Nguyen, H., 447, 240Nguyen-Mau, C., 445, 439Nicolaidou, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Niczyporuk, M., 444, 531Nie, J., 446, 356Nie, S., 449, 89Nief, J.-Y., 445, 239; 447, 183, 336Nielsen, B.S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Nielsen, H.B., 446, 256Nielsen, J., 445, 239; 447, 336Nielsen, M., 443, 285Niemi, A.J., 449, 214Niessen, H.N.T., 444, 503Niessen, T., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Nigro, A., 443, 394Nikiforov, A., 445, 14Nikitin, N., 442, 381Nikolaenko, V., 441, 479Nikolaev, N.N., 442, 398Nikolenko, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Nikonov, V.A., 449, 145, 154Nilsson, B., 442, 43Nilsson, B.S., 445, 239; 447, 336Nippe, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Nisati, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Nishimura, M., 446, 37Nishimura, T., 443, 394Nisius, R., 444, 539; 447, 134, 157Nogga, A., 447, 216Nogueira, F.S., 441, 339Nojiri, S., 443, 121; 444, 92; 449, 39, 173NOMAD Collaboration, 445, 439Nomokonov, V., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Nomura, I., 444, 267Nomura, T., 443, 409


Nomura, Y., 445, 316Nordmann, D., 445, 439Norman, K.L., 446, 342Norman, P.I., 449, 401Normand, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Norrbin, E., 442, 407Norton, A., 446, 117Norton, P.R., 445, 239; 447, 336Notani, M., 448, 180Notz, D., 443, 394Novaes, S.F., 447, 331Novikov, I.D., 442, 82Nowak, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Nowak, M.A., 442, 300; 446, 9Nowak, R.J., 443, 394Nowak, W.-D., 442, 484; 443, 379; 444, 531Nowakowski, M., 446, 111Noyes, V.A., 443, 394Nunez, C., 441, 185´˜Nupieri, M., 444, 531Nurnberger, H.-A., 445, 239; 447, 336¨Nussinov, S., 441, 299Nuzzo, S., 445, 239; 447, 336Nygren, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Nylander, P., 443, 394

Oakes, R.J., 447, 313Oakham, F.G., 444, 539; 447, 134, 157Oblakowska-Mucha, A., 441, 479Obraztsov, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Obregon, O., 449, 173Obst, A., 444, 252Ocariz, J., 446, 117Ochs, A., 443, 394Oda, I., 444, 127Odaka, S., 447, 167Ødegard, S., 443, 69˚O’Dell, V., 447, 240Odintsov, S.D., 443, 121; 444, 92; 449, 39, 173Odoom, D., 449, 114Odorici, F., 444, 539; 447, 134, 157Odyniec, G., 444, 523Oelert, W., 446, 179Oelwein, P., 444, 531Oevers, M., 442, 291Ogami, H., 444, 531Ogawa, H., 448, 180Ogawa, K., 447, 167Ogren, H.O., 444, 539; 447, 134, 157Oh, B.Y., 443, 394Oh, K., 442, 109Oh, P., 444, 469Oh, S., 441, 178

Oh, S.K., 441, 215Oh, Y.D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ohama, T., 447, 167Ohlsson-Malek, F., 444, 516; 449, 128Ohsugi, T., 447, 167Ohta, N., 441, 123; 445, 287Ohyama, H., 447, 167Okabe, K., 447, 167Okada, N., 449, 230Okamoto, A., 447, 167Okrasinski, J.R., 443, 394´Oldenburg, M., 444, 523Oleari, C., 447, 327Olive, K.A., 444, 367Oliver, W.P., 449, 137Olshevski, A.G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383O’Neale, S.W., 444, 539; 447, 134, 157O’Neill, T.G., 442, 484; 444, 531Ono, A., 447, 167Onofre, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383OPAL Collaboration, 444, 539; 447, 134, 157Openshaw, R., 444, 531Orava, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Orazi, G., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Orear, J., 445, 419Oreglia, M.J., 444, 539; 447, 134, 157Orejudos, W., 445, 239; 447, 336Orestano, D., 445, 439Organtini, G., 444, 569; 445, 428; 446, 368;

447, 147; 448, 152Orito, S., 444, 539; 447, 134, 157Orr, R.S., 443, 394Osada, E., 445, 14Osculati, B., 449, 401Osetrin, K.E., 449, 173O’Shaughnessy, K., 445, 449; 448, 303O’Shea, V., 445, 239; 447, 336Osterberg, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ostonen, G.O.R., 444, 503Ostonen, R., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Ostrick, M., 444, 555Ouared, R., 446, 349Ould-Saada, F., 446, 349Ouraou, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ouyang, J., 442, 484; 444, 531Ouyang, Q., 445, 239; 447, 336Owen, B., 444, 531Owen, B.R., 442, 484Oz, Y., 444, 318


Pac, M.Y., 443, 394Pacheco, A., 445, 239; 447, 336Pacheco, A.J., 447, 41Pachos, J., 444, 469Padhi, S., 443, 394Paganetti, M., 444, 555; 445, 20Paganoni, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Page, R.D., 443, 69, 82Pagels, B., 444, 38, 43, 52Paiano, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Pain, R., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Paiva, R., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Pajares, C., 442, 395; 444, 435Pakhtusova, E.V., 449, 122Palacios, J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Paladini, B., 448, 76Palestini, S., 446, 117Palinkas, J., 444, 539; 447, 134, 157´ ´Palka, H., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Palla, F., 445, 239; 447, 336Palla, G., 444, 523Pallavicini, M., 441, 447Pallin, D., 445, 239; 447, 336Palmonari, F., 443, 394Palomares, C., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Pan, F., 442, 7Pan, Y.B., 445, 239; 447, 336Panagiotakopoulos, C., 446, 224Panagiotou, A.D., 444, 523Pandoulas, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Pang, M., 447, 240Panzer-Steindel, B., 446, 117Paoletti, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Paolucci, P., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Paoluzi, L., 444, 111; 447, 127Papadopoulos, D.B., 443, 131Papadopoulos, G., 443, 159Papadopoulos, I., 444, 38, 43, 52Papadopoulou, T.D., 441, 479Papadopoulou, Th.D., 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Papageorgiou, K., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Papavassiliou, V., 442, 484; 444, 531Pape, L., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Papillon, S., 449, 128

Papoyan, V.V., 444, 293Papp, G., 442, 300; 446, 9Parikh, J.C., 446, 104Parikh, M.K., 449, 24Park, C., 444, 156Park, H.K., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Park, I., 448, 37Park, I.C., 445, 239; 447, 336Park, I.H., 443, 394; 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Park, I.S., 444, 267Park, P.P.H.K., 444, 503Park, Q.-H., 449, 223Park, S.K., 443, 394Parker, M.A., 446, 117Parkes, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Parodi, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Parrini, G., 445, 239; 447, 336Parsons, H.L.C., 446, 117Parsons, J.A., 443, 394Parzefall, U., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Pascale, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Paschos, E.A., 443, 201Pascual, A., 445, 239; 447, 336Pasqualucci, E., 444, 111Pasquinucci, A., 444, 318Passalacqua, L., 445, 239; 447, 336Passaleva, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Passeri, A., 441, 479; 446, 62, 75; 448, 311;

449, 364, 383Passon, O., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Pasti, P., 447, 251Pastircak, B., 449, 401ˇ´Pastore, F., 445, 439Pasyuk, E., 444, 252Pasztor, G., 444, 539; 447, 134, 157´Pate, S.F., 442, 484; 444, 531Pater, J.R., 444, 539; 447, 134, 157Patkos, A., 446, 272´Patrascioiu, A., 445, 160Patricelli, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Patrick, G.N., 444, 539; 447, 134, 157Patt, J., 444, 539; 447, 134, 157Patzold, J., 443, 77; 446, 179, 363¨Paul, E., 443, 394; 444, 555; 445, 20Paul, T., 444, 569; 445, 428; 446, 368; 447,

147; 448, 152Pauluzzi, M., 444, 569; 445, 428; 446, 368;

447, 147; 448, 152


Pauluzzi, T.P.M., 444, 503Paus, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pauss, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pavel, N., 443, 394Pavlopoulos, P., 444, 38, 43, 52Pawlak, J.M., 443, 394Pawlak, R., 443, 394Pawlowski, M., 444, 293Payre, P., 445, 239; 447, 336Peach, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Peak, L.S., 445, 439Pearce, G.F., 449, 137Pearce, P.A., 444, 163Pedace, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Peeters, K., 443, 153Pegoraro, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Pei, Y.J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Peigne, S., 449, 306´Peigneux, J.P., 446, 342Pelfer, P., 443, 394Pellegrini, F., 449, 401Pellegrino, A., 443, 394Pelucchi, F., 443, 394Pene, O., 446, 336´Peng, K.C., 445, 449; 448, 303Penin, A.A., 443, 264; 447, 354Penionzhkevich, Yu.E., 448, 180Pennacchio, E., 445, 439Pennanen, J., 447, 167Pensotti, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pepe-Altarelli, M., 445, 239; 447, 336Pepe, M., 446, 117Peralta, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Perelstein, M., 444, 273Perera, L.P., 445, 449; 448, 303Perez-Ochoa, R., 444, 539; 447, 134, 157Perez, P., 445, 239; 447, 336Perez-Victoria, M., 442, 315´Peris, S., 443, 255Pernicka, E., 447, 127Pernicka, M., 441, 479; 444, 491; 446, 62, 75,

117; 448, 311; 449, 364Peroni, C., 443, 394Perret-Gallix, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Perret, P., 445, 239; 447, 336Perrodo, P., 445, 239; 447, 183, 336Perrotta, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Perroud, J-P., 445, 439

Pervushin, V.N., 444, 293Pesci, A., 443, 394Pessard, H., 445, 439Petcov, S.T., 444, 584Peter, P., 441, 52Peters, K., 446, 349Petersen, B., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Peterson, E.A., 449, 137Petkou, A.C., 446, 306Petkova, V.B., 444, 163Petrak, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Petreczky, P., 442, 291Petridis, A., 444, 523Petridou, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Petrolini, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Petrucci, F., 446, 117Petrucci, M.C., 443, 394Petti, R., 445, 439Petyt, D.A., 449, 137Petzold, S., 444, 539; 447, 134, 157Pevsner, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Peyaud, B., 446, 117Pfeifenschneider, P., 444, 539; 447, 134, 157Pfeiffer, M., 443, 394Pfutzner, M., 444, 32¨Pham, T.N., 447, 313Phillips, H.T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Piana, G., 441, 479Piattelli, P., 442, 48Piccioni, D., 443, 394Piccolo, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pick, B., 446, 349Piechocki, W., 448, 203Pierazzini, G., 446, 117Pierce, D.M., 445, 331Pieri, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pierre, F., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Pietra, C., 446, 349Pietrzyk, B., 445, 239; 447, 183, 336Pilcher, J.E., 444, 539; 447, 134, 157Pilipenko, Yu., 445, 14Pillin, M., 448, 227Pimenta, M., 444, 491; 446, 62, 75; 448, 311;

449, 364, 383Pinciuc, C., 443, 394Pinder, C.N., 446, 349; 449, 114, 145Pinfold, J., 444, 539; 447, 134, 157Pinsky, S., 442, 173Piotrzkowski, K., 443, 394


Piotto, E., 441, 479; 444, 491; 446, 62, 75; 448,311; 449, 364, 383

Piper, A., 444, 523Pirjol, D., 449, 321Piroue, P.A., 444, 503, 569; 445, 428; 446,´

368; 447, 147; 448, 152Pıska, K., 449, 401´ˇPistolesi, E., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pitt, M., 444, 531Pivovarov, A.A., 443, 264; 447, 354Pizzi, J.R., 444, 516; 449, 128Placci, A., 445, 439Plane, D.E., 444, 539; 447, 134, 157Plefka, J., 443, 153Pliszka, J., 444, 136Plotzke, R., 444, 555; 445, 20¨Plouin, F., 445, 423Plumacher, M., 443, 209¨Pluquet, A., 445, 439Plyaskin, V., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Podlyski, F., 445, 239; 447, 336Podobnik, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Poelz, G., 443, 394Poffenberger, P., 444, 539; 447, 134, 157Pohl, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Pojidaev, V., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Pokorski, S., 441, 205Pol, M.E., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Polarski, D., 446, 53Polenz, S., 443, 394Polesello, G., 445, 439Policarpo, A., 444, 43Polikarpov, M.I., 449, 267Polini, A., 443, 394Polivka, G., 444, 38, 43, 52Poljsak, M., 444, 411ˇPollmann, D., 445, 439Pollock, M.D., 448, 13Polls, A., 445, 259Polok, G., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Polok, J., 444, 539; 447, 134, 157Polonyi, J., 445, 351Polovnikov, S.A., 446, 342Pol’shin, S.A., 449, 56Polunin, A.A., 449, 122Polyakov, V.A., 446, 342Poolman, H.R., 444, 531Popescu, R., 446, 197Popov, B., 445, 439Porcu, M., 446, 117Pordes, R., 447, 240

Poropat, P., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Porter, R.J., 444, 523Posa, F., 449, 401Poskanzer, A.M., 444, 523Posocco, M., 443, 394; 444, 111Postema, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Potashov, S., 442, 484; 444, 531Pothier, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Potrebenikov, I., 446, 117Potterveld, D.H., 442, 484; 444, 531Poulsen, C., 445, 439Poves, A., 443, 1Povh, B., 446, 321Pozdniakov, V., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Prange, G., 445, 239; 447, 336Prasad, V., 447, 240Pratt, S., 444, 231Prelovsek, S., 447, 313ˇPrice, L.E., 449, 137Prindle, D.J., 444, 523Prinias, A., 443, 394Privitera, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Prodanov, E.M., 445, 112Produit, N., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Prokofiev, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Proskuryakov, A.S., 443, 394Pruss, S.M., 445, 419Przybycien, M., 443, 394; 444, 539; 447, 134,´

157Przybycien, M.B., 443, 394´Przysiezniak, H., 445, 239; 447, 336Puente Penalba, J., 447, 59˜Puga, J., 443, 394Puhlhofer, F., 444, 523¨Pukhaeva, N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Pullia, A., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Puolamaki, K., 446, 285; 448, 234¨Purohit, M.V., 445, 449; 448, 303Putz, J., 445, 239; 447, 336Putzer, A., 445, 239; 447, 336Pyata, E.E., 449, 122

Qi, N., 445, 239Qi, N.D., 446, 356Qi, X.R., 446, 356Qian, C.D., 446, 356Qiao, C., 447, 240Qiu, J.F., 446, 356Qu, Y.H., 446, 356


Quadt, A., 443, 394Quandt, M., 446, 290Quarati, P., 441, 291Quartieri, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Quast, G., 445, 239; 447, 336Que, Y.K., 446, 356Quercigh, E., 447, 178; 449, 401Quevedo, F., 447, 257Quinn, B., 445, 449; 447, 240; 448, 303

Raach, H., 443, 394Racca, C., 444, 516; 449, 128Rademacher, R., 447, 284Radeztsky, S., 445, 449; 448, 303Radojicic, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Radyushkin, A.V., 449, 81Rae, W.D.M., 444, 260Raeven, B., 445, 239Rafatian, A., 445, 449; 448, 303Ragazzi, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ragusa, F., 445, 239; 447, 336Rahal-Callot, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Rahkila, P., 443, 69Rahmani, H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rahmfeld, J., 443, 143Raine, C., 445, 239; 447, 336Raja, N., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Rajeev, S.G., 441, 429Rakness, G., 442, 484; 444, 531Rakoczy, D., 441, 479; 444, 491; 446, 75; 448,

311; 449, 364, 383Ramberg, E.J., 447, 240Rames, J., 444, 491Ramond, P., 441, 163Ramos, E., 445, 123Ramos, S., 444, 516; 449, 128Rancoita, P.G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Rander, J., 445, 239; 447, 336Ranieri, A., 445, 239; 447, 336Ranjard, F., 445, 239; 447, 336Raso, G., 445, 239; 447, 336Raso, M., 443, 394Ratajczak, M., 446, 349Rathouit, P., 445, 439Ratoff, P.N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rattaggi, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Rau, W., 447, 127Rauch, W., 444, 523Ravanini, F., 444, 442

Raven, G., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Ravindran, V., 445, 206, 214Ray, R.E., 447, 240Ray, S., 447, 352Razis, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Read, A.L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Reali, A., 442, 484; 444, 531Reay, N.W., 445, 449; 448, 303Rebecchi, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Redaelli, N.G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Redondo, I., 443, 394Redwine, R., 442, 484; 444, 531Reeder, D.D., 443, 394Regan, P.H., 444, 32Regenfus, C., 446, 349Regler, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Reich, J.C., 449, 6Reid, D., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Reid, J.G., 444, 523Reidy, J.J., 445, 449; 448, 303Reinhardt, H., 446, 290Reinhardt, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rejmund, M., 444, 32Rembser, C., 444, 539; 447, 134, 157Ren, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Renard, F.M., 448, 129Renardy, J.-F., 445, 239; 447, 336Renfordt, R., 444, 523Renk, B., 445, 239; 446, 117; 447, 336Renken, R., 442, 266Rensch, B., 445, 239; 447, 336Renton, P.B., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Renzoni, G., 445, 439Reolon, A.R., 442, 484; 444, 531Repond, J., 443, 394Resag, S., 446, 349Rescigno, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Resvanis, L.K., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Retamosa, J., 443, 1Retyk, W., 444, 523Reucroft, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Reviol, W., 443, 89Rey, S.-J., 449, 68Reya, E., 443, 298Reznikov, S., 445, 14


Ribeiro, E., 442, 349Ricci, B., 441, 291; 444, 387Ricci, R.A., 449, 401Rich, J., 447, 127Richard, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Richter, A., 443, 1Rick, H., 444, 539; 447, 134, 157Rickenbach, R., 444, 38, 43, 52Ridky, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Riedinger, L.L., 443, 89Riemann, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Riles, K., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Rinaudo, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Riotto, A., 442, 68; 445, 323; 446, 28Riska, D.O., 444, 21Ristinen, R., 442, 484; 444, 531Rith, K., 442, 484; 444, 531Ritter, H.G., 444, 523Ritz, F., 447, 15Ritz, S., 443, 394Riu, I., 445, 239; 447, 336Riveline, M., 443, 394Rizzo, G., 445, 239; 447, 336Roberts, B.L., 444, 38, 43, 52Roberts, R.G., 443, 301Robertson, A.N., 445, 239; 447, 336Robertson, S., 444, 539; 447, 134, 157Robins, S.A., 444, 539; 447, 134, 157Robohm, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Robutti, E., 441, 447Roda, C., 445, 439Rodin, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Rodin, V.A., 447, 8Rodning, N., 444, 539; 447, 134, 157Rodrigues da Silva, P.S., 442, 369Roe, B.P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Roethel, W., 446, 349Rohde, M., 443, 394Rohne, E., 445, 239; 447, 336Rohne, O., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rohrich, D., 444, 523¨Roland, C.R.G., 444, 523Rolandi, L., 445, 239; 447, 336Roldan, J., 443, 394´Roloff, H., 442, 484; 444, 531Romana, A., 444, 516; 449, 128Romano, G., 449, 401Romanovsky, V., 446, 342

Romeo, A., 441, 265Romero, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Romero, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ronceux, B., 449, 128Ronchese, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rondio, E., 446, 117Roney, J.M., 444, 539; 447, 134, 157Rong, G., 446, 356Ronningen, R., 446, 197Ronzhin, A., 447, 240Roodman, A., 447, 240Roose, F., 445, 150Roper, C.D., 444, 252Roper, G., 444, 531¨Rosa, G., 449, 401Roscoe, K., 444, 539; 447, 134, 157Rosenberg, E.I., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Rosenfeld, R., 447, 331Rosier-Lees, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Rosinsky, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rosner, J.L., 441, 403Rosowsky, A., 445, 239; 447, 336Ross, G.G., 442, 165Rosselet, Ph., 448, 152Rossi, A.M., 444, 539; 447, 134, 157Rossi, L., 449, 401Rossi, P., 442, 484; 444, 531Roth, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Rothberg, J., 445, 239; 447, 336Rotscheidt, H., 446, 342; 449, 401Roudeau, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rouge, A., 445, 239; 447, 336´Rousseau, D., 445, 239; 447, 336Rovelli, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Rowe, M.A., 449, 6Rowley, J.K., 447, 127Roy, D.P., 444, 391; 445, 185Roy, M.S., 443, 293Roy, S., 443, 167, 191Roynette, J.C., 442, 48Royon, Ch., 446, 62, 75; 448, 311; 449, 364,

383Rozen, Y., 444, 539; 447, 134, 157Rozowsky, J.S., 444, 273Rubakov, V.A., 442, 63Rubbia, A., 445, 439Ruber, R.J.M.Y., 446, 179Rubin, H.A., 445, 449; 448, 303


Rubinstein, R., 445, 419Rubio, J.A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ruckl, R., 448, 320Ruddick, K., 449, 137Rudnitsky, S., 442, 484; 444, 531Rudolph, G., 445, 239; 447, 336Rudolph, H., 444, 523Ruf, T., 444, 38, 43, 52Ruggieri, F., 445, 239; 447, 336Ruh, M., 442, 484; 444, 531Ruhlmann-Kleider, V., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Ruijter, H., 442, 43Ruiz, A., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Ruiz Arriola, E., 443, 33Rukoyatkin, P., 445, 14Rulikowska-Zarebska, E., 443, 394Rumpf, M., 445, 239; 447, 336Rumyantsev, O.A., 443, 51Rumyantsev, V., 446, 342Runge, K., 444, 539; 447, 134, 157Runolfsson, O., 444, 539; 447, 134, 157Ruschmeier, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Rusetsky, A.G., 442, 435Ruske, O., 443, 394Ruspa, M., 443, 394Russakovich, N., 446, 342Rust, D.R., 444, 539; 447, 134, 157Ryan, J.J., 443, 394Rybicki, A., 444, 523Rychenkova, P., 443, 138Ryckbosch, D., 442, 484; 444, 531Ryckebusch, J., 441, 1Rykaczewski, H., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Rykaczewski, K., 444, 32Ryskin, M.G., 446, 48

Saadi, Y., 445, 239; 447, 336Saarikko, H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sabetfakhri, A., 443, 394Sabra, W.A., 442, 97Sacchi, R., 443, 394Sachs, K., 444, 539; 447, 134, 157Sacquin, Y., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sadamoto, M., 447, 240Sadovsky, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sadrozinski, H.F.-W., 443, 394Saeki, T., 444, 539; 447, 134, 157Safarık, K., 447, 178; 449, 401ˇSagawa, H., 444, 1Sahr, O., 444, 539; 447, 134, 157Saito, K., 441, 27; 443, 26; 447, 233

Saito, N., 444, 267Sajot, G., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Sakamoto, H., 443, 409; 447, 167Sakar, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Sakemi, Y., 442, 484; 444, 531Sakuda, M., 447, 167Sakurai, H., 448, 180Saladino, S., 449, 401Salehi, H., 443, 394Salgado, P., 448, 20Salicio, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Salinas, F., 444, 252Salnikov, A.A., 449, 122Salt, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Salvatore, F., 445, 439Salvo, C., 449, 401Samanta, B.C., 447, 352Samoylenko, V.D., 446, 342Sampson, S., 443, 394Sampsonidis, D., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Sanchez, E., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Sanchez, F., 445, 239; 447, 336Sanchez, M., 449, 137Sandell, A., 442, 43Sander, H.-G., 445, 239; 447, 336Sanders, D.A., 445, 449; 448, 303Sanders, M.P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Sandhas, W., 442, 43Sandor, L., 449, 401´Sandoval, A., 444, 523Sang, W.M., 444, 539; 447, 134, 157Sanguinetti, G., 445, 239; 447, 336Sann, H., 444, 523Sann, M., 447, 127Sannino, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Santha, A.K.S., 445, 449; 448, 303Santoni, C., 444, 43Santonocito, D., 442, 48Santoro, A.F.S., 445, 449; 448, 303Santos, F.C., 446, 170SAPHIR Collaboration, 444, 555; 445, 20Sapienza, P., 442, 48Saradzhev, F.M., 442, 259Sarakinos, M.E., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Sarantites, D.G., 443, 89Sarantsev, A.V., 449, 145, 154Sarantsev, V.V., 449, 145, 154Sarkar, U., 442, 243; 444, 391; 445, 185Sarkisyan, E.K.G., 444, 539; 447, 134, 157Sartorelli, G., 443, 394


Sasaki, S., 443, 331; 449, 230Sasaki, Y., 442, 53Sasao, N., 443, 409Sato, M., 447, 167Sato, N., 447, 167Sato, T., 441, 105Satteson, M., 446, 22Satuła, W., 443, 89Saturnini, P., 444, 516; 449, 128Satz, H., 442, 14Saull, P.R.B., 443, 394Savage, M.J., 449, 1Savcı, M., 441, 410Savelius, A., 443, 69, 82Savin, A.A., 443, 394Savin, I., 442, 484; 444, 531Savoy, C.A., 444, 119Savrie, M., 446, 117´Savvidy, G.K., 449, 253Sawyer, R.F., 448, 174Saxon, D.H., 443, 394Sbarra, C., 444, 539; 447, 134, 157Scadron, M.D., 446, 332Scarlett, C., 442, 484; 444, 531Scarpaci, J.A., 442, 48Scarr, J.M., 447, 336Schael, S., 445, 239; 447, 336Schafer, A., 443, 40; 448, 99¨Schafer, C., 444, 503, 569; 445, 428; 446, 368;¨

447, 147; 448, 152Schafer, E., 444, 523¨Schafer, M., 444, 38, 43, 52¨Schafke, A., 446, 290¨Schahmaneche, K., 445, 439Schaile, A.D., 444, 539; 447, 134, 157Schaile, O., 444, 539; 447, 134, 157Schaller, L.A., 444, 38, 43, 52Schalm, K., 446, 247; 448, 37Schanne, S., 446, 117Schapler, D., 443, 77Schaposnik, F.A., 441, 185; 449, 187Scharf, F., 444, 539; 447, 134, 157Scharff-Hansen, P., 444, 539; 447, 134, 157Schatzman, E., 447, 127Schegelsky, V., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Scheidt, J., 446, 117Schenk, U., 445, 20Schepkin, M., 443, 77Schepkin, M.G., 446, 179Schieck, J., 444, 539; 447, 134, 157Schietinger, T., 444, 38, 43, 52Schilcher, K., 448, 93Schildknecht, D., 449, 328Schiller, A., 442, 453; 443, 244Schinzel, D., 446, 117Schioppa, M., 443, 394Schlatter, D., 445, 239; 447, 336

Schlein, P.E., 445, 455Schlenstedt, S., 443, 394Schmeling, S., 445, 239; 447, 336Schmidke, W.B., 443, 394Schmidt, B., 445, 439Schmidt, C.R., 445, 168Schmidt, F., 442, 484; 444, 531Schmidt, I., 441, 461; 444, 451Schmidt, J., 446, 117Schmidt-Kaerst, S., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Schmidt, K.E., 446, 99Schmidt, P., 446, 349Schmischke, D., 444, 523Schmitt, B., 444, 539; 447, 134, 157Schmitt, H., 442, 484; 444, 531Schmitt, M., 445, 239; 447, 336Schmitt, S., 444, 539; 447, 134, 157Schmitz, D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Schmitz, N., 444, 523Schneekloth, U., 443, 394Schneider, H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Schneider, O., 445, 239; 447, 336Schnell, G., 442, 484; 444, 531Schneps, J., 449, 137Schnetzer, S., 447, 240Schnurbusch, H., 443, 394Scholmann, J., 445, 20Scholten, O., 441, 17Scholz, N., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Schonfelder, S., 444, 523¨Schonharting, V., 446, 117¨Schoning, A., 444, 539; 447, 134, 157¨Schopper, A., 444, 38, 43, 52Schopper, H., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Schotanus, D.J., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Schramm, S., 446, 191Schroder, B., 442, 43¨Schroder, M., 444, 539; 447, 134, 157¨Schroder, Y., 447, 321¨Schroers, W., 449, 299Schub, M.H., 449, 137Schue, I., 446, 117´Schulday, I., 444, 555; 445, 20Schuler, G.A., 449, 328Schuler, K.P., 442, 484; 444, 531¨Schumacher, M., 444, 539; 445, 20; 447, 134,

157Schutz, P., 445, 20¨Schvellinger, M., 449, 161Schwartz, A.J., 445, 449; 448, 303Schwarzer, O., 443, 394Schweigert, C., 441, 141; 447, 266


Schwemling, Ph., 441, 479; 444, 491; 446, 62,75; 448, 311; 449, 364, 383

Schwenke, J., 444, 503, 569; 446, 368; 447,147; 448, 152

Schwering, G., 444, 503, 569; 446, 368; 447,147; 448, 152

Schwering, J.S.G., 445, 428Schwetz, M., 449, 281Schwick, C., 444, 539; 447, 134, 157Schwickerath, U., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Schwieger, J., 443, 7Schwille, W.J., 444, 555; 445, 20Schwind, A., 442, 484; 444, 531Schyns, M.A.E., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Sciaba, A., 445, 239; 447, 336`Sciacca, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Sciarrino, D., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Sciulli, F., 443, 394Scoccola, N.N., 444, 21; 446, 93Scopetta, S., 442, 28Scott, I., 449, 114, 145, 154Scott, I.J., 445, 239; 447, 336Scott, J., 443, 394Scott, W.G., 444, 539; 447, 134, 157Scuri, F., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Seager, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sedgbeer, J.K., 443, 394; 445, 239; 447, 336Sedykh, Y., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Segar, A.M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Segato, G., 449, 401Seibert, J., 442, 484; 444, 531Seibert, R., 446, 349Seiden, A., 443, 394Seidlein, R., 449, 137Seiler, E., 445, 160Sekimoto, M., 444, 267Sekulin, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Selonke, F., 443, 394Selvaggi, G., 445, 239; 447, 336Semenoff, G.W., 445, 307Semenov, A., 445, 14; 446, 342Semenov, A.Yu., 444, 523Semenova, I., 445, 14Sen, S., 445, 112Sene, M., 446, 342; 447, 178; 449, 401´Sene, R., 446, 342; 447, 178; 449, 401´SenGupta, S., 445, 129; 446, 104Senyo, K., 447, 240Serbo, V.G., 442, 453

Serednyakov, S.I., 449, 122Serin, L., 445, 239; 447, 336Serrano, M., 445, 439Serreau, J., 448, 257Servoli, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Seth, K.K., 441, 479Settles, R., 445, 239; 447, 336Seuster, R., 444, 539; 447, 134, 157Sevior, M.E., 445, 439Sevrin, A., 443, 153Sexton, J.C., 448, 76Seyboth, P., 444, 523Seyerlein, J., 444, 523Seymour, M.H., 442, 417Seywerd, H., 445, 239; 447, 336Sguazzoni, G., 445, 239; 447, 336Shabelski, Yu.M., 446, 48Shafi, Q., 448, 46Shagin, P.M., 446, 342Shah, T.P., 443, 394Shanahan, P., 447, 240Shang, S.-Q., 449, 6Shao, Y.Y., 446, 356Shary, V.V., 449, 122Shatunov, Yu.M., 449, 122Shawhan, P.S., 447, 240Shcheglova, L.M., 443, 394Sheaff, M., 445, 449; 448, 303Shears, T.G., 444, 539; 447, 134, 157Sheikh, J.A., 443, 16Shellard, E.P.S., 445, 43Shellard, R.C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Shen, B.C., 444, 539; 447, 134, 157Shen, B.W., 446, 356Shen, D.L., 446, 356Shen, H., 446, 356Shen, X.Y., 446, 356Sheng, H.Y., 446, 356Shepherd-Themistocleous, C.H., 444, 539; 447,

134, 157Sher, M., 449, 89Sheridan, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sherwood, P., 444, 539; 447, 134, 157Shevchenko, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Shi, H.Z., 446, 356Shibata, T.-A., 442, 484; 444, 531Shibatani, K., 442, 484Shim, J.S., 449, 207Shimizu, H., 446, 342Shin, T., 442, 484; 444, 531Shin, Y.M., 444, 267Shioden, M., 447, 167Shirai, J., 447, 167Shirkov, D.V., 442, 344


Shiromizu, T., 443, 127Shivarov, N., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Shizuma, T., 442, 53Shmatikov, M., 446, 43Shoutko, V., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Shovkovy, I.A., 441, 313Shukla, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Shukla, S., 445, 419Shumilov, E., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Shutov, V., 442, 484; 444, 531Shuvaev, A.G., 446, 48Shvedov, O.Yu., 443, 373Shvorob, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Sideris, D., 443, 394Sidorov, A.V., 445, 232Sidorov, V.A., 449, 122Sidwell, R.A., 445, 449; 448, 303Siebel, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Siedenburg, T., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Sievers, M., 443, 394Sikler, F., 444, 523Silagadze, Z.K., 449, 122Sillou, D., 445, 439Silva, G., 449, 187Silva-Marcos, J.I., 443, 276Silva, S., 444, 516; 449, 128Silva Neto, M.B., 441, 339Silvestre, R., 441, 479; 444, 491Silvestrini, L., 441, 371Silvestris, L., 445, 239; 447, 336Sim, K.S., 444, 267Simani, C., 442, 484Simani, M.C., 444, 531Simard, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Simmons, D., 443, 394Simmons, E.H., 443, 347Simon, A., 442, 484; 444, 531Simon, J., 443, 147´Simone, S., 449, 401Simonetto, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Simopoulou, E., 445, 239; 447, 336Simpson, J., 443, 69, 82Sims, D.A., 442, 43Simula, S., 442, 381Sin, S.-J., 444, 156Sinclair, L.E., 443, 394Singer, P., 445, 394Singer, S.M., 444, 260Singh, H., 444, 327Singovsky, A.V., 446, 342

Sinram, K., 442, 484; 444, 531Siroli, G.P., 444, 539; 447, 134, 157Sisakian, A.N., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sittler, A., 444, 539; 447, 134, 157Sjostrand, T., 442, 407; 449, 313¨Skaali, T.B., 441, 479; 444, 491; 446, 62, 75Skadhauge, S., 449, 240Skillicorn, I.O., 443, 394Skrinsky, A.N., 449, 122Skrzypczak, E., 444, 523Skrzypek, M., 449, 354Skuja, A., 444, 539; 447, 134, 157Slater, W., 447, 240Slaughter, A.J., 445, 449; 448, 303Slaus, I., 444, 252Slavich, P., 442, 484; 444, 531Smadja, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Smalska, B., 443, 394Smend, F., 445, 20Smirichinski, V.I., 444, 293Smirnov, N., 444, 491; 446, 62Smirnova, O., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Smith, A.M., 444, 539; 447, 134, 157Smith, B., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Smith, B.H., 443, 89Smith, D., 445, 239Smith, G.R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Smith, W.H., 443, 394Smolik, L., 445, 239; 447, 336Smyrski, J., 444, 555; 445, 20Smythe, W.R., 444, 531Snellings, R., 444, 523Snigirev, A.M., 443, 387Snoeys, W., 449, 401Snow, G.A., 444, 539; 447, 134, 157Sobie, R., 444, 539; 447, 134, 157Sobol, A., 446, 342Sobotka, L.G., 446, 197Soff, S., 446, 191; 447, 227Soffer, J., 441, 461; 442, 479Sofianos, S.A., 442, 43Sokatchev, E., 444, 341Sokoloff, M.D., 445, 449; 448, 303Sokolov, A., 446, 75; 448, 311; 449, 364, 383Sola, J., 442, 326`Solano, A., 443, 394Solano, J., 445, 449; 448, 303Soldati, R., 441, 257Soldner-Rembold, S., 444, 539; 447, 134, 157¨Soler, F.J.P., 445, 439Solodukhin, S.N., 448, 209Solomey, N., 447, 240Solomin, A.N., 443, 394Solovianov, O., 446, 75; 448, 311; 449, 383


Soloviev, O.A., 442, 136Solovjev, A., 446, 342Solovtsov, I.L., 442, 344Somalwar, S.V., 447, 240Sommer, J., 445, 239; 447, 336Son, D., 443, 394; 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Sonderegger, P., 444, 516; 449, 128Song, H.S., 449, 207Song, W.Y., 449, 207Song, X.F., 446, 356Sonnenschein, J., 449, 76Sonoda, H., 446, 58Sopczak, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sorokin, D., 447, 251Sosnowski, R., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sowinski, J., 444, 531Sozzi, G., 445, 439Sozzi, M., 446, 117Spagnolo, P., 445, 239; 447, 336Spagnolo, S., 447, 134, 157; 448, 129Spanderen, K., 446, 209Spanier, S., 446, 349Spassov, T., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Spector, D., 442, 159Spengos, M., 442, 484; 444, 531Speth, J., 444, 25Speth, W., 445, 20Spieles, C., 442, 443; 446, 191, 326; 447, 227Spillantini, P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Spinetti, M., 444, 111Spira, M., 441, 383Spiriti, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Spiro, M., 447, 127Sponholz, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Springer, R.P., 449, 1Sproston, M., 444, 539; 447, 134, 157Squarcia, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Squier, G.T.A., 444, 523Sreekantan, B.V., 449, 219Srivastava, P.P., 448, 68St-Laurent, M., 443, 394Stahl, A., 444, 539; 447, 134, 157Staiano, A., 443, 394Stairs, D.G., 443, 394Stamenov, D.B., 445, 232Stampfer, D., 441, 479; 444, 491; 446, 62, 75;

449, 364, 383Stanco, L., 443, 394Stanek, R., 443, 394

Stanescu, C., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Stanic, S., 441, 479; 444, 491; 446, 62, 75; 448,311; 449, 364, 383

Stanishkov, M., 447, 277Stanton, N.R., 445, 449; 448, 303Stapnes, S., 441, 479; 444, 491; 446, 62, 75Staroba, P., 449, 401Stassinaki, M., 446, 342Stassinakis, A., 449, 137Stech, B., 449, 339Steele, D., 445, 439Steele, T.G., 446, 267Stefanis, N.G., 449, 299Stefanski, R.J., 445, 449; 448, 303Steffens, E., 442, 484; 444, 531Steffens, F.M., 447, 233Steininger, M., 445, 439Stenger, J., 442, 484; 444, 531Stenson, K., 445, 449; 448, 303Stenzel, H., 445, 239; 447, 336Stepaniak, J., 446, 179Stephan, F., 445, 239; 447, 336Stephens, F.S., 449, 6Stephens, K., 444, 539; 447, 134, 157Stephenson Jr., G.J., 444, 75Sterbenz, S., 444, 252Stern, A., 441, 69Steuer, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Steuerer, J., 444, 539; 447, 134, 157Stevenson, K., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Stewart, I.W., 445, 378Stewart, J., 442, 484; 444, 531Stickland, D.P., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Stiegler, U., 445, 439Stielglitz, L., 447, 127Stifutkin, A., 443, 394Stipcevic, M., 442, 38; 445, 439ˇ ´Stirling, W.J., 443, 301Stocchi, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Stock, F., 444, 531Stock, H., 446, 349¨Stock, R., 444, 523Stocker, H., 442, 443; 446, 191; 447, 227; 448,¨

290Stoenner, R.W., 447, 127Stoesslein, U., 444, 531Stolarczyk, T., 445, 439Stoll, K., 444, 539; 447, 134, 157Stone, A., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Stone, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152


Stone, R.L., 447, 240Stonjek, S., 443, 394Stoßlein, U., 442, 484¨Stoyanov, B., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Straessner, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Straßburger, C., 446, 349Straub, P.B., 443, 394Strauss, J., 441, 479; 446, 62, 75; 448, 311;

449, 364, 383Strickland, E., 443, 394Strobele, H., 444, 523¨Strohbusch, U., 446, 349Strohmer, R., 444, 539; 447, 134, 157¨Stroili, R., 443, 394Strom, D., 444, 539; 447, 134, 157Strong, J.A., 445, 239; 447, 336Stroot, J.P., 446, 342Strub, R., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Struck, Chr., 444, 523Strumia, A., 445, 407Stugu, B., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Sudhakar, K., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Suehiro, M., 443, 409Suffert, M., 446, 349Sugonyaev, V.P., 446, 342Suh, J.S., 446, 349Sukhanov, A., 446, 179Sultanov, G., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Sultansoy, S., 443, 359Sumiyoshi, T., 447, 167Summerer, K., 444, 32¨Summers, D.J., 445, 449; 448, 303Sun, F., 446, 356Sun, H.S., 446, 356Sun, L.Z., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Sun, Y., 446, 356Sun, Y.Z., 446, 356Suomijarvi, T., 442, 48¨Suranyi, P., 447, 284Surguladze, L.R., 446, 153Surrow, B., 443, 394; 444, 539; 447, 134, 157;

449, 328Susa, T., 444, 523Susinno, G., 443, 394Susinno, G.F., 444, 503, 569; 445, 428; 446,

368Susukita, R., 444, 267Suszycki, L., 443, 394Suter, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Sutter, M., 442, 484; 444, 531Sutton, M.R., 443, 394

Suzuki, I., 443, 394; 447, 240Svaiter, N.F., 441, 339Swain, J.D., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Swallow, E.C., 447, 240Swanson, R.A., 447, 240Szafran, S., 449, 401Szczekowski, M., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Szentpetery, I., 444, 523Szep, Zs., 446, 272´Szeptycka, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Sziklai, J., 444, 523Szillasi, Z., 444, 503, 569; 446, 368; 447, 147;

448, 152Szleper, M., 446, 117

Tabarelli, T., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Tachibana, M., 442, 217Tadic, D., 445, 249´Taegar, S.A., 447, 240Takabayashi, N., 445, 14Takach, S., 445, 449; 448, 303Takacs, G., 444, 442´Takada, Y., 447, 167Takamatsu, K., 446, 342Takasaki, F., 447, 167Takashima, R., 444, 267Takeuchi, Y., 443, 409Takeutchi, F., 444, 267Takita, M., 447, 167Talbot, S.D., 444, 539; 447, 134, 157Talby, M., 445, 239; 447, 336Tallini, H., 442, 484; 444, 531Tamura, N., 447, 167Tamvakis, K., 446, 224Tanabe, K., 445, 1Tanaka, R., 445, 239; 447, 336Tanaka, S., 444, 539; 447, 134, 157Tang, H.-B., 443, 63Tang, S.Q., 446, 356Tang, X.W., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Tanigawa, T., 445, 254Taniguchi, T., 443, 409Tanii, Y., 446, 37Tanimoto, M., 449, 240Tao, C., 447, 127Tapper, A.D., 443, 394Tapper, R.J., 443, 394Taras, P., 444, 539; 447, 134, 157Tarasov, Yu.A., 441, 453Tareb-Reyes, M., 445, 439Tarem, S., 444, 539; 447, 134, 157Taroian, S., 442, 484; 444, 531Tarrago, X., 444, 516; 449, 128Tassi, E., 443, 394


Tatar, R., 442, 109Tateo, R., 448, 249Tatichvili, G., 446, 117Tatischeff, B., 445, 423Tatsumi, D., 447, 167Tatsumi, T., 441, 9Taureg, H., 446, 117Taurok, A., 446, 117Tauscher, L., 444, 38, 43, 52, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Tavartkiladze, Z., 448, 46Taxil, P., 441, 376Taylor, G., 445, 239; 447, 336Taylor, G.N., 445, 439Taylor, L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Tchikilev, O., 449, 364Tchlatchidze, G., 446, 342Tegenfeldt, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Teixeira-Dias, P., 445, 239; 447, 336Tejessy, W., 445, 239; 447, 336Tempesta, P., 445, 239; 447, 336Tenchini, R., 445, 239; 447, 336Teranishi, T., 448, 180Tereshchenko, S., 445, 439Terkulov, A., 442, 484; 444, 531Terranova, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Terron, J., 443, 394´Tesarek, R.J., 447, 240Teschendorff, A., 443, 159Testoni, J.E., 447, 41Teubert, F., 445, 239; 447, 336Teuscher, R., 444, 539; 447, 134, 157Theis, U., 446, 314Thibault, C., 444, 38, 43, 52Thiergen, M., 444, 539; 447, 134, 157Thiessen, D.M., 444, 531Thoma, U., 444, 555; 446, 349Thomas, A.W., 441, 27; 443, 26; 447, 233Thomas, E., 444, 531Thomas, J., 441, 479; 444, 491; 446, 62, 75;

447, 134, 157; 448, 311; 449, 364, 383Thompson, A.S., 445, 239; 447, 336Thompson, D.J., 443, 201Thompson, J.C., 445, 239; 447, 336Thompson, L.F., 445, 239; 447, 336Thompson, M., 449, 401Thomson, E., 445, 239; 447, 336Thomson, G.B., 447, 240Thomson, M.A., 444, 539; 447, 134, 157Thorne, K., 445, 449; 448, 303Thorne, R.S., 443, 301Thorsteinsen, T.F., 449, 401Thron, J.L., 449, 137Thulasidas, M., 445, 239; 447, 336Tiator, L., 444, 555

Tiecke, H., 443, 394Tilquin, A., 441, 479; 444, 491; 445, 239; 446,

62, 75; 447, 336Timmermans, C., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Timmermans, J., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Timoteo, V.S., 448, 1´Ting, S.C.C., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Ting, S.M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Tinti, N., 446, 62, 75; 448, 311; 449, 364, 383Tinyakov, P.G., 442, 63Tipton, B., 442, 484; 444, 531Tischhauser, M., 446, 349¨Tittel, K., 445, 239; 447, 336Tkabladze, A., 443, 379Tkatchev, A., 446, 117Tkatchev, L.G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Tlusty, P., 444, 267´Toale, P.A., 447, 240Tobimatsu, K., 447, 167Todorov, T., 441, 479Todorova, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Toet, D.Z., 441, 479; 444, 491; 446, 62, 75Tokushuku, K., 443, 394Tomalin, I.R., 445, 239; 447, 336Tomaradze, A., 441, 479; 444, 491; 446, 62;

448, 311; 449, 364Tomasicchio, G., 449, 401Tomboulis, E.T., 443, 239Tome, B., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Tonazzo, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Tong, D., 448, 33Tong, G.L., 446, 356Tonin, M., 447, 251Tonwar, S.C., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Toothacker, W.S., 443, 394Tormanen, S., 443, 69¨ ¨Tornow, W., 444, 252Toropin, A., 445, 439Torrence, E., 444, 539; 447, 134, 157Torrente-Lujan, E., 441, 305Torrieri, G.D., 449, 401Tort, A., 446, 170Tortora, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Toth, J., 444, 503, 569; 445, 428; 446, 368;´

447, 147; 448, 152Touchard, A-M., 445, 439Touchard, F., 444, 38, 43, 52Touramanis, C., 444, 38, 43, 52


Tournefier, E., 445, 239; 447, 336Tovey, S.N., 445, 439Towers, S., 444, 539; 447, 134, 157Toy, M., 444, 523Trabelsi, A., 445, 239; 447, 336Tracas, N.D., 447, 67Traini, M., 442, 28Trainor, T.A., 444, 523Tran, H.N., 445, 20Tran, M-T., 445, 439Tran, M.Q., 444, 555; 445, 20Transtromer, G., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Trautmann, W., 446, 197Treille, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Trentalange, S., 444, 523Tricomi, A., 445, 239; 447, 336Trigger, I., 444, 539; 447, 134, 157Trinchero, R., 443, 221Tripathi, A., 447, 240Tripathi, A.K., 445, 449; 448, 303Tristram, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Trivedi, S.P., 445, 142Trochimczuk, M., 446, 62, 75; 448, 311; 449,

364, 383Trocsanyi, Z., 444, 539; 447, 134, 157´ ´Troitsky, S.V., 449, 17Troncon, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Trucks, M., 445, 117Trudel, A., 444, 531Tsabar, D., 449, 180Tsang, M.B., 446, 197Tschirhart, R., 447, 240Tseng, B., 446, 125Tsesmelis, E., 445, 439Tsirou, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Tsuboyama, T., 447, 167Tsur, E., 444, 539; 447, 134Tsuru, T., 446, 342Tsurugai, T., 443, 394Tsushima, K., 441, 27; 443, 26; 447, 233Tuchming, B., 445, 239; 447, 336Tully, C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Tung, K.L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Tuning, N., 443, 394Turcot, A.S., 444, 539; 447, 134Turcot, E.T.A.S., 447, 157Turkot, F., 445, 419Turlay, R., 446, 117Turluer, M.-L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Turner-Watson, M.F., 444, 539; 447, 134, 157Turowiecki, A., 446, 179

Tveter, T.S., 449, 401Tyapkin, I.A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Tymieniecka, T., 443, 394Tytgat, M., 442, 484; 444, 531Tzamarias, S., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383

Uchida, Y., 444, 503, 569; 445, 428; 446, 368;447, 147; 448, 152

Uchiyama, K., 442, 53Ueberschaer, B., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Ueda, I., 447, 134, 157Uehara, S., 447, 167Ulbricht, J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ullaland, O., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Ullrich, T., 444, 523Ulrichs, J., 445, 439Uman, I., 446, 349Umemori, K., 443, 394Unal, G., 446, 117Unno, Y., 447, 167Urban, J., 449, 401´Urciuoli, G.M., 442, 484; 444, 531Urin, M.H., 443, 51; 447, 8Uros, V., 445, 439Uusitalo, J., 443, 69, 82Uvarov, V., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383

Vacavant, V., 445, 439Vaiciulis, A., 443, 394Vaidya, R., 442, 243Vairo, A., 442, 349Valassi, A., 445, 239; 447, 336Valdata-Nappi, M., 445, 439Valent, G., 445, 60Valente, E., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Valenti, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Vallage, B., 445, 239; 446, 117; 447, 336Vallazza, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Valle, J.W.F., 441, 224Valuev, V., 445, 439Van Apeldoorn, G.W., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Van Baal, P., 448, 26Vance, S.E., 443, 45Van Dalen, J.A., 447, 147; 448, 152Van Dam, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Van den Brand, J.F.J., 442, 484; 444, 531Van der Steenhoven, G., 442, 484; 444, 531Vander Velde, C., 444, 491; 446, 62, 75


Van de Vyver, R., 442, 484; 444, 531Van Dierendonck, D., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Van Doninck, W.K., 444, 491; 446, 62, 75;

449, 364Vandoren, S., 442, 145Van Eijk, C.W.E., 444, 38, 43, 52Van Eldik, J., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Van Gemmeren, P., 445, 239; 447, 336Van Gulik, R., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Van Hoek, W.C., 444, 503, 569; 445, 428; 446,

368Van Holten, J.W., 442, 185Vanhove, P., 444, 196Van Hunen, J.J., 442, 484; 444, 531Van Isacker, P., 443, 16, 82Van Kooten, R., 444, 539; 447, 134, 157Van Lysebetten, A., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Van Mil, A.J.W., 444, 503, 569; 445, 428; 446,

368Van Neck, D., 441, 17Van Neerven, W.L., 445, 206, 214Vannerem, P., 444, 539; 447, 134, 157Van Nespen, W., 441, 1Van Nieuwenhuizen, P., 446, 247Vannini, C., 445, 239; 447, 336Vannucci, F., 445, 439Van Pee, H., 444, 555; 445, 20Van Rhee, T., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Van Sighem, A., 443, 394Van Vulpen, I., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Van Weert, Ch.G., 444, 463Vanzo, L., 449, 168Varela, J., 444, 516; 449, 128Varvell, K.E., 445, 439Vassilevskaya, L.A., 446, 378Vassiliadis, G., 447, 178Vassiliev, V., 449, 137Vassiliou, M., 444, 523Vassilopoulos, N., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Vattolo, D., 446, 117Vayaki, A., 445, 239; 447, 336Vaz, C., 442, 90Vazeille, F., 444, 516Vazquez-Mozo, M.A., 441, 40´V. Dombrowski, S., 446, 349Vegni, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Veillet, J.-J., 445, 239; 447, 336Velasco, M., 446, 117Velikzhanin, Yu.S., 449, 122

Veltri, M., 445, 439Venables, M., 446, 342; 447, 178; 449, 401Vento, V., 442, 28Ventura, L., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Venturi, A., 445, 239; 447, 336Venus, W., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383VENUS Collaboration, 447, 167Verbeure, F., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Vercesi, V., 445, 439Verdini, P.G., 445, 239; 447, 336Veres, G., 444, 523Verkerke, W., 443, 394Verkindt, D., 445, 439Verlato, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Veropoulos, G., 442, 374Vertogradov, L.S., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Verzegnassi, C., 448, 129Verzi, V., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Verzocchi, M., 444, 539; 447, 134, 157Veselov, A.I., 449, 267Vesztergombi, G., 444, 503, 523, 569; 445,

428; 446, 368; 447, 147; 448, 152Vetlitsky, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Vetterli, M.C., 442, 484; 444, 531V. Feilitzsch, F., 447, 127Videau, H., 445, 239; 447, 336Videau, I., 445, 239; 447, 336Vieira, J-M., 445, 439Viertel, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Vignaud, D., 447, 127Vilanova, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Vilja, I., 445, 199Villa, S., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Villalobos Baillie, O., 446, 342; 447, 178; 449,

401Villaume, G., 449, 137Vincter, M., 442, 484; 444, 531Vinogradova, T., 445, 439Virey, J.-M., 441, 376Virgili, T., 449, 401Vissani, F., 443, 191Visser, J., 442, 484; 444, 531Vitale, L., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Vitale, P., 441, 69Vivargent, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152


Vlachos, N.D., 441, 46; 446, 306Vlachos, S., 444, 38, 43, 52, 503, 569; 445,

428; 446, 368; 447, 147; 448, 152Vlasov, E., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Vo, M-K., 445, 439Voci, C., 443, 394; 444, 111Vodopyanov, A.S., 441, 479; 444, 491; 446,

62, 75; 448, 311; 449, 364, 383Vogel, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Vogl, W., 445, 20Vogt, H., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Vogt, M., 447, 336Volcker, C., 446, 349¨Volk, E., 442, 484; 444, 531Volkov, S., 445, 439Vollmer, C., 444, 491; 446, 62, 75; 448, 311;

449, 364, 383Voloshin, M.B., 449, 17Volte, A., 447, 178; 449, 401Von der Mey, M., 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Von Krogh, J., 444, 539; 447, 134, 157Von Neumann-Cosel, P., 443, 1Von Torne, E., 444, 539; 447, 134, 157¨Von Wimmersperg-Toeller, J.H., 445, 239; 447,

336Vorobiev, I., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Vorobyov, A.A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Vorvolakos, A., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Voss, H., 444, 539; 447, 134, 157Vossebeld, J., 443, 394Vossnack, O., 446, 117Votano, L., 443, 394Votruba, M.F., 446, 342; 447, 178; 449, 401Voulgaris, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Vranic, D., 444, 523´Vrba, V., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Vreeswijk, M., 445, 239; 447, 336

Waananen, A., 445, 239; 447, 336¨¨ ¨Wachsmuth, H., 445, 239; 447, 336Wackerle, F., 444, 539; 447, 134, 157¨WA102 Collaboration, 446, 342; 447, 178; 449,

401Wadhwa, M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Wagner, A., 444, 539; 447, 134, 157Wagner, C.E.M., 441, 205Wagner, G.J., 443, 77; 446, 179, 363Wah, Y.W., 447, 240Wahl, H., 446, 117

Wahlen, H., 441, 479; 444, 491; 446, 62, 75;448, 311; 449, 364, 383

Wakai, A., 445, 14Wakely, S., 449, 137Walck, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Walczak, R., 443, 394Walker, R., 443, 394Wall, D., 449, 137Wallis-Plachner, S., 446, 349Wallraff, W., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Walsh, J., 445, 239; 447, 336Walter, R.L., 444, 252Walther, D., 446, 349Wander, W., 442, 484; 444, 531Wands, D., 443, 97Wang, F., 444, 523; 446, 356Wang, J.C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Wang, L.S., 446, 356Wang, L.Z., 446, 356Wang, M., 446, 356Wang, P., 446, 356Wang, P.L., 446, 356Wang, S., 441, 473Wang, S.M., 443, 394; 446, 356Wang, T., 445, 239; 447, 336Wang, T.J., 446, 356Wang, X.-N., 443, 45; 444, 245Wang, X.L., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Wang, X.N., 444, 237Wang, Y.Y., 446, 356Wang, Z., 441, 473Wang, Z.M., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Wanninger, S., 447, 127¨Ward, B.F.L., 449, 97Ward, C.P., 444, 539; 447, 134, 157Ward, D.R., 444, 539; 447, 134, 157Ward, J.J., 445, 239; 447, 336Warner, D.D., 443, 16Warner, S., 442, 266Waroquier, M., 441, 17Was, Z., 449, 97Wasserbaech, S., 445, 239; 447, 336Wasserman, E.G., 449, 6Watanabe, S., 445, 449; 448, 303Watanabe, T., 447, 167Watanabe, Y.X., 448, 180Watase, Y., 447, 167Waters, D.S., 443, 394Watkins, P.M., 444, 539; 447, 134, 157Watson, A.T., 444, 539; 447, 134, 157Watson, D.L., 444, 260Watson, N.K., 444, 539; 447, 134, 157Watts, G., 448, 249Waugh, R., 443, 394


Weathers, L., 446, 197Webber, B.R., 444, 81Weber, A., 443, 394; 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Weber, F., 445, 439Weber, H., 442, 443Weber, P., 444, 38, 43, 52Wedemeyer, R., 445, 20Weerasundara, D.D., 444, 523Wehnes, F., 444, 555; 445, 20Wei, C.L., 446, 356Weibe, S., 444, 267Weigel, H., 447, 1Weiler, T.J., 442, 255Weirich, F., 447, 127Weiser, C., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Weiss-Babai, R., 445, 449; 448, 303Weisse, T., 445, 439Weissman, L., 446, 22Welch, T.P., 444, 531Wells, J.D., 443, 196; 445, 178Wells, P.S., 444, 539; 447, 134, 157Weneser, J., 447, 127Wenig, S., 444, 523Werlen, M., 445, 439Wermes, N., 444, 539; 447, 134, 157Werner, S., 445, 239; 447, 336West, N., 449, 137West, P.C., 444, 341Westphal, D., 443, 394Westphalen, J., 446, 209Wheater, J.F., 448, 185Whisnant, K., 442, 255White, D., 446, 197White, H.B., 447, 240White, J.S., 444, 539; 447, 134, 157White, T.O., 446, 117Whiteley, C.R., 444, 252Whitmore, J., 447, 240Whitmore, J.J., 443, 394Whitten, C., 444, 523Whitton, M., 444, 252Wichmann, R., 443, 394Wick, K., 443, 394Wicke, D., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Wickens, J.H., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Wieber, H., 443, 394Wiedemann, U.A., 449, 347Wiedenmann, W., 445, 239; 447, 336Wiedner, U., 446, 349Wiegers, B., 444, 555; 445, 20Wieland, F.W., 444, 555; 445, 20Wielgosz, U.M., 449, 137Wiener, J., 445, 449; 448, 303Wienold, T., 444, 523Wiese, U.-J., 443, 338

Wigger, O., 444, 38, 43, 52Wiggers, L., 443, 394Wilczek, F., 449, 24Wildschek, T., 443, 394Wilhelm, R., 446, 117Wilhelmi, Z., 446, 179Wilkin, C., 445, 423Wilkinson, G.R., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Williams, A.G., 441, 27Williams, D.C., 443, 394Williams, M.D., 445, 239; 447, 336Williams, M.I., 445, 239; 447, 336Williams, R.W., 445, 239; 447, 336Williamson, S.E., 442, 484; 444, 531Willis, N., 445, 423Wilson, F., 445, 439Wilson, G.W., 444, 539; 447, 134, 157Wilson, J.A., 444, 539; 447, 134, 157Wilson, J.N., 443, 89Wing, M., 443, 394Wingerter, I., 446, 117Winharting, A., 446, 117Winstein, B., 447, 240Winston, R., 447, 240Winter, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Winton, L.J., 445, 439Wirrer, G., 446, 117Wirstam, J., 448, 168Wise, M.B., 449, 1Wise, T., 442, 484; 444, 531Wißkirchen, J., 444, 555; 445, 20Wislicki, W., 446, 117Witała, H., 447, 216Witchey, N., 445, 449; 448, 303Witek, M., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Witten, L., 442, 90Wittgen, M., 446, 117Wittmack, K., 446, 349Wodarczyk, M., 443, 394Wojcik, M., 447, 127Wolf, A., 445, 20Wolf, G., 441, 479; 443, 394; 444, 491; 445,

239; 446, 62, 75; 447, 336; 448, 311; 449,364, 383

Wolfle, S., 443, 394¨Wolin, E., 445, 449; 448, 303Woller, K., 442, 484; 444, 531Wollmer, U., 443, 394Wolter, M., 444, 38, 43, 52Wood, L., 444, 523Wotton, S.A., 446, 117Wright, A.E., 445, 239; 447, 336Wroblewski, A.K., 443, 394´Wronka, S., 446, 117Wu, G.-H., 447, 83Wu, J.-Y., 447, 240


Wu, S.L., 445, 239; 447, 336Wu, S.X., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Wu, X., 445, 239; 447, 336Wu, Y.G., 446, 356Wu, Z.C., 445, 274Wunsch, M., 445, 239; 447, 336Wurzinger, R., 445, 423Wyatt, T.R., 444, 539; 447, 134, 157Wyler, D., 448, 320Wynhoff, S., 444, 503, 569; 445, 428; 446,

368; 447, 147; 448, 152Wyss, R., 443, 69

Xi, D.M., 446, 356Xia, X.M., 446, 356Xie, P.P., 446, 356Xie, Y., 445, 239; 446, 356; 447, 336Xie, Y.H., 446, 356Xing, Z., 443, 365Xiong, C.-S., 441, 155Xu, G.F., 446, 356Xu, J., 444, 503, 569; 445, 428; 446, 368Xu, N., 444, 523Xu, R., 445, 239; 447, 336Xu, Z.Z., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Xue, S., 445, 239; 447, 336Xue, S.T., 446, 356

Yabsley, B.D., 445, 439Yabuki, F., 447, 167Yamada, A., 448, 265Yamada, S., 443, 394Yamada, Y., 447, 167Yamagata, T., 447, 167Yamanaka, T., 447, 240Yamashita, S., 444, 267, 539; 447, 134, 157Yamashita, T., 443, 394Yamauchi, K., 443, 394Yamazaki, Y., 443, 394Yan, J., 446, 356Yan, Q.-S., 442, 209Yan, W.G., 446, 356Yanagida, T., 445, 399Yang, B.Z., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Yang, C.G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Yang, C.M., 446, 356Yang, C.Y., 446, 356Yang, H.J., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Yang, J., 446, 356Yang, M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Yang, S.-K., 441, 155Yang, S.M., 445, 449; 448, 303

Yang, X.F., 446, 356Yasu, Y., 446, 342Yasuda, O., 443, 185Yasuhira, M., 441, 9Yates, T.A., 444, 523Ye, J.B., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ye, M.H., 446, 356Ye, S.W., 446, 356Ye, Y.X., 446, 356Yeh, S.C., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Yekutieli, G., 444, 539; 447, 134, 157Yi, D., 445, 449; 448, 303Yi, J., 441, 479; 444, 491; 446, 62, 75; 448,

311; 449, 364, 383Yokkaichi, S., 444, 267Yoneda, K., 448, 180Yoneyama, S., 442, 484; 444, 531Yonezawa, Y., 447, 167Yonnet, J., 445, 423Yoshida, A., 448, 180Yoshida, H., 447, 167Yoshida, K., 444, 267Yoshida, M., 444, 267Yoshida, R., 443, 394Yoshida, S., 445, 449; 448, 303Yoshida, T., 444, 267Yoshioka, K., 444, 373You, J.M., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Youngman, C., 443, 394Yu, C.S., 446, 356Yu, C.X., 446, 356Yu, G.W., 446, 356Yu, Y.H., 446, 356Yu, Z.Q., 446, 356Yuan, C.Z., 446, 356Yuan, Y., 446, 356Yuanfang, W., 444, 563Yusa, K., 447, 167Yushchenko, O., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383

Zabierowski, J., 446, 179Zaccone, H., 445, 439Zacek, V., 444, 539; 447, 134, 157Zachariadou, K., 445, 239; 447, 336Zahed, I., 442, 300; 446, 9; 448, 168Zaitsev, A., 441, 479; 444, 491; 446, 62Zajac, J., 443, 394Zakharov, B.G., 442, 398Zakharov, Y., 447, 127Zakrzewski, J.A., 443, 394Zalewska, A., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Zalewski, P., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383


Zalite, A., 444, 503; 446, 368Zalite, An., 444, 569; 445, 428; 447, 147; 448,

152Zalite, Yu., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zaliznyak, R., 445, 449; 448, 303Zamiralov, V.S., 449, 93Zamora Garcia, Y., 443, 394Zanelli, J., 444, 451Zanon, D., 444, 332Zanotti, L., 447, 127Zapfe-Duren, K., 444, 531¨Zarnecki, A.F., 443, 394Zavada, P., 449, 401´Zavrtanik, D., 441, 479; 444, 38, 43, 52, 491;

446, 62, 75; 448, 311; 449, 364, 383Zawiejski, L., 443, 394Zeitnitz, C., 445, 239; 447, 336Zemp, P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zeng, Y., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zer-Zion, D., 444, 539; 447, 134, 157Zerbini, S., 449, 168Zernov, A., 446, 179Zerwas, D., 445, 239; 447, 336Zerwas, P.M., 441, 383Zetsche, F., 443, 394Zeuner, W., 443, 394Zevgolatakos, E., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Zghiche, A., 445, 423Zhang, B., 444, 237Zhang, B.Y., 446, 356Zhang, C., 445, 449; 448, 303Zhang, C.C., 446, 356Zhang, D., 446, 356Zhang, D.-X., 445, 394; 446, 285; 448, 234Zhang, D.H., 446, 356Zhang, H.L., 446, 356Zhang, J., 445, 239; 446, 356; 447, 336Zhang, J.W., 446, 356Zhang, L., 445, 239; 447, 336Zhang, L.S., 446, 356Zhang, Q.J., 446, 356Zhang, R.-J., 447, 89Zhang, S.Q., 446, 356Zhang, X.F., 444, 237Zhang, X.Y., 446, 356Zhang, Y.Y., 446, 356Zhang, Z.P., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zhao, D.X., 446, 356Zhao, H.W., 446, 356Zhao, J., 446, 356Zhao, J.W., 446, 356Zhao, M., 446, 356Zhao, W., 445, 239; 447, 336

Zhao, W.R., 446, 356Zhao, Z.G., 446, 356Zheng, J.P., 446, 356Zheng, L.S., 446, 356Zheng, Z.P., 446, 356Zhou, B., 444, 503, 569; 445, 428; 446, 368Zhou, B.-R., 444, 455Zhou, B.Q., 446, 356Zhou, G.P., 446, 356Zhou, H.S., 446, 356Zhou, J.-G., 445, 287Zhou, L., 446, 356Zhu, G.Y., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zhu, K.J., 446, 356Zhu, Q., 443, 394Zhu, Q.M., 446, 356Zhu, R.Y., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zhu, S.H., 444, 224Zhu, X.-Z., 444, 523Zhu, Y.C., 446, 356Zhu, Y.S., 446, 356Zhuang, B.A., 446, 356Zichichi, A., 443, 394; 444, 503, 569; 445, 428;

446, 368; 447, 147; 448, 152Ziegler, F., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Ziegler, T., 445, 239; 447, 336Zilizi, G., 444, 503, 569; 445, 428; 446, 368;

447, 147; 448, 152Zimanyi, J., 444, 523Zimin, N.I., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Zimmerman, D., 444, 38, 43, 52Zimmerman, E.D., 447, 240Zinchenko, A., 446, 117Zinovjev, G.M., 443, 387Ziolkowski, M., 446, 117Zioutas, K., 443, 201Zito, G., 445, 239; 447, 336Złomanczuk, J., 446, 179´Zobernig, G., 445, 239; 447, 336Zohrabian, H., 442, 484; 444, 531Zolin, L., 445, 14Zotkin, S.A., 443, 394Zou, B.S., 449, 114, 145, 154Zoupanos, G., 441, 235Zuber, J.-B., 444, 163Zuber, K., 445, 439Zucchelli, G.C., 441, 479; 444, 491; 446, 62,

75; 448, 311; 449, 364, 383Zuccon, P., 445, 439Zumerle, G., 441, 479; 444, 491; 446, 62, 75;

448, 311; 449, 364, 383Zurmuhle, R., 444, 531¨Zybert, R., 444, 523

An effective field theory calculation of the parity violating...

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