Probing hadron wave functions in lattice QCD · C~r,t1,t2! 5 ( ni,nf,n p,q ^hunf,p& e 2En f (p)(T/2...

PHYSICAL REVIEW D 66, 094503 ~2002!

Probing hadron wave functions in lattice QCD

C. AlexandrouDepartment of Physics, University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, Cyprus

Ph. de ForcrandInstitute fur Theoretische Physik, ETH Ho¨nggerberg, CH-8093 Zu¨rich, Switzerland

and CERN, Theory Division, CH-1211 Geneva 23, Switzerland

A. TsapalisDepartment of Physics, University of Wuppertal, Wuppertal, Germany

~Received 12 July 2002; published 22 November 2002!

Gauge-invariant equal-time correlation functions are calculated in lattice QCD within the quenched approxi-mation and with two dynamical quark species. These correlators provide information on the shape and multi-pole moments of the pion, the rho, the nucleon and theD.

DOI: 10.1103/PhysRevD.66.094503 PACS number~s!: 11.15.Ha, 12.38.Aw, 12.38.Gc, 14.70.Dj

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I. INTRODUCTION

State of the art lattice calculations of hadronic matrixements have produced very accurate spectroscopic infotion. Examples of the accuracy reached in quenched laQCD are the calculation of the masses of low lying hadro@1# and glueballs@2#. However progress in determining haron wave functions and quark distributions has not beenrapid. The initial calculations of Bethe-Salpeter amplitudwere carried out on a rather small lattice in the mid 1980s@3#for the pion and the rho. Further progress came in the e1990s when the Bethe-Salpeter amplitudes were calculon a larger lattice for the pion and the rho@4,5# as well as forthe nucleon and theD @4#. However the results were of limited interest because of their manifest dependence ongauge chosen. A different approach to explore hadrostructure was pursued by the authors of Ref.@6#; instead offixing a gauge or a path for the gluons, they consideredrelation functions of quark densities which, being expection values of local operators, are gauge invariant. This isapproach we have adopted in this work.

Our main motivation for studying density correlatorsthat they reduce, in the non-relativistic limit, to the wafunction squared and thus they provide detailed, gauinvariant information on hadron structure. The shape of hrons is one such important quantity that can be directly stied. The issue whether the nucleon is deformed fromspherical shape was raised twenty years ago@7# and is stillunsettled. Because the spectroscopic quadrupole momea spin one-half particle vanishes, in experimental studiessearches for quadrupole strength in theg* N→D transition.Spin-parity selection rules allow a magnetic dipoleM1, anelectric quadrupoleE2, or a Coulomb quadrupoleC2 am-plitude. If both the nucleon and theD are spherical then theelectric and Coulomb quadrupole amplitudes are expectebe zero. AlthoughM1 is indeed the dominant amplitudthere is mounting experimental evidence thatE2 andC2 arenonzero. The physical origin of a nonzeroE2 andC2 am-plitude is attributed to different mechanisms in the variomodels. In quark models the deformation is due to the co

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magnetic tensor force@7#. In ‘‘cloudy’’ baryon models it isdue to meson exchange currents@8#.

A recent experimental search atq250.126 GeV2 hasyielded an electric quadrupole to magnetic dipole amplituratio @9#:

REM5GE2 /GM15~22.160.262.0!%. ~1!

The larger error, an order of magnitude larger than thetistical error, is due to the model dependence in the extrtion of this ratio from the experimental data. The experimetal determination ofREM is complicated by the presence ononresonant processes coherent with the resonant excitof D(1232).

The ratioREM can be evaluated within lattice QCD without any model assumptions by computing the transition mtrix elementN to D. An early lattice calculation of this transition matrix element provided an estimate ofREM5(2368)% @10#.

Here we consider an alternative route to understandingissue of deformation, via the direct study of hadron wafunctions. We compute density-density correlators for msons and baryons and three-density correlators for baryand look for asymmtries when these are projected alongspin axis or perpendicular to it. The observables that weare described in Sec. II. In Sec. III we give the relationsthe deformation among the states of different spin projtions. Our quenched lattice results are presented in SecThey clearly give support to a deformed rho. The nucledeformation averages to zero in agreement with the factits spectroscopic quadrupole moment is zero. An analysithe intrinsic nucleon deformation requires determiningbody-fixed coordinates by diagonalization of the momentinertia tensor, which must be done configuration by configration. This is too noisy to yield a statistically significaresult. On the other hand theD has a nonzero spectroscopmoment and any deformation should be detected via protion with respect to its spin axis. In quenched QCD we detno significant deformation for theD within our presentstatistics.

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ALEXANDROU, de FORCRAND, AND TSAPALIS PHYSICAL REVIEW D66, 094503 ~2002!

In addition to providing information about deformatiobaryon wave functions, calculated for the first time ingauge invariant way for all values of the relative coordinatcan be used to study indirectly the potential among the thquarks. By performing fits to the three-density correlatowe find that a reasonableAnsatzfor the baryonic potential isprovided by the sum of two-body potentials, called theDAnsatz@11#.

Comparison between quenched and full QCD ressheds light on the role of the pion cloud. It is known thquenching eliminates all or part of the pion cloud dependon the hadronic state. For the rho channel no intermedbackgoing quarks with the pion quantum numberspresent@12#, so the deformation that we observe is not duethe pion cloud. We use the SESAM configurations@13# toinvestigate unquenching effects. We find that the deformain the rho increases. We detect a deformation in theD whichremains small for the values of the dynamical quark mconsidered. The unquenched lattice results are presenteSec. V. Section VI contains our conclusions.

II. CORRELATION FUNCTIONS

The wave function of a meson is usually defined ingiven gaugeg as the equal time Bethe-Salpeter amplitude

FgBS~r !5E d3r 8^0uqf 1~r 8!Gqf 2~r 81r !uM & ~2!

whereG is a Dirac matrix with the quantum numbers of thmesonM of flavor f 1 , f 2 . Fg

BS(r ) is the minimal Fock spacestate wave function, which is an approximation to the fwave function since other multiquark components arecluded. For the Bethe-Salpeter amplitude one must eithethe gauge or connect the quarks with gluons to form gauinvariant~but path-dependent! quantities. Wave functions fobaryons are defined in an analogous way to the meson wfunctions but they involve two relative distances. E.g. tBethe-Salpeter amplitude,Fg

BS(r1 ,r2), for the proton isgiven by

(r8

^0ueabcuda~r 8,t !

3„ubT~r 81r1 ,t !Cg5dc~r 81r2 ,t !…uB&. ~3!

Summation overr 8 projects onto zero momentum. For thD1, the interpolating field that we take is

Jm~x!5eabc

A3@ua~x!„2ubT~x!Cgmdc~x!…

1da~x!„ubT~x!Cgmuc~x!…#, ~4!

whereC5g0g2 is the charge conjugation operator. We uG65(g17 ig2)/2 to create a hadron of a definite spin componentJz . Explicitly for the first term in Eq.~4! we take

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a~x!„2ubT~x!Cg3dc~x!…

2u2a~x!„2ubT~x!CG1dc~x!…#

J21/2:eabc

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a~x!„2ubT~x!Cg3dc~x!…

1u1a~x!„2ubT~x!CG2dc~x!…#

J23/2:eabc

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a~x!„2ubT~x!CG2dc~x!…# ~5!

whereqi is the i th component of the spinor. The same costruction is done for the second term of Eq.~4!. With thesecombinations we recover, in the nonrelativistic limit, thquark model wave functions of these states.

The Bethe-Salpeter amplitudes in the Coulomb and inLandau gauge were investigated in Ref.@4#. Instead of fixinga gauge, a gauge-invariant wave function, which correspoto the Bethe-Salpeter amplitude in the axial gauge, canconstructed by joining the quark and the antiquark withthin string of glue@14#:

cs~y!5^0uq~0!Gei *0ydzAy(z)q~y!uM &. ~6!

Other variations to the thin string are to smear the gluons@5#or to evolve them so that they reach their ground statetribution. It was shown in Ref.@14# that there is a muchlarger probability to find a quark-antiquark pair separatedsay, 1 fm when connected by a physical adiabatic flux tuthan when the quarks are surrounded by gluons fixed toCoulomb gauge or when they are connected by a thin stof gluons. Although in Ref.@14# only mesonic states werconsidered, generalization to baryons is straightforward, wthe gluonic strings attached to each of the three quarks ctracted at a point.

In this work we opt for the calculation of density-densicorrelators@15,16#,

C~r ,t1 ,t2!5E d3r 8^huru~r 81r ,t2!rd~r 8,t1!uh& ~7!

where the density operator is given by the normal orproduct

ru~r ,t !5:u~r ,t !g0u~r ,t !: ~8!

so that disconnected graphs are excluded.For mesons the density-density correlator is shown sc

matically in Fig. 1. Inserting a complete set of hadronstates we find

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PROBING HADRON WAVE FUNCTIONS IN LATTICE QCD PHYSICAL REVIEW D66, 094503 ~2002!

C~r ,t1 ,t2!

5 (ni ,nf ,n

(p,q

^hunf ,p&e2Enf

(p)(T/22t2)

Enf~p!

3^nf ,puruun,p1q&e2 iq•re2En(p1q)(t22t1)

En~p1q!

3^n,p1qurduni ,p&e2Eni

(p)t1

Eni~p!

^ni ,puh&. ~9!

The sum over all excitations of the initial,ni , and final,nf ,states yields the ground state hadron in the limit (T/22t2)→` and t1→`, whereT is the lattice extent in Euclideatime. Since we take periodic boundary conditions the mamum time separation isT/2. Enforcing zero momentumwould require a summation over the spatial volume onsource or sink site. This is technically not feasible sinceinvolves quark propagators from all to all spatial lattice sitInstead, the suppression of the nonzero momenta and ohigher excitations is obtained by choosing the largest psible time separation from the source and the sink. We tthe same spatial coordinate for the source and the snamely x05z. As a starting point we take, in this workdensity insertions to be always at equal times. A disadvtage of the density-density correlators is that they are subto more severe finite size effects, having typically larger stial extent (; twice! than Bethe-Salpeter amplitudes@5#.

In the case of baryons three density insertions are neeand the correlator is given by

C~r1 ,r2 ,t !5E d3r 8^hurd~r 8,t !ru~r 81r1 ,t !

3ru~r 81r2 ,t !uh& ~10!

where we have taken all the density insertions to be at etimes. This involves two relative distances. Like Eq.~7!, itcan be computed efficiently by fast Fourier transform~FFT!.

The relevant diagrams are shown in the upper part of F2 for the nucleon or theD1, for the general case of densitinsertions at three unequal times. In addition to one deninsertion for eachu-quark line, one can have two densiinsertions on the sameu-quark line. Evaluation of this second diagram requires the quark propagatorG(r2 ,r1) for allpartial distances of the two arguments. This is beyond

FIG. 1. Density-density correlator for a meson.t1 , t2 , T/22t1

and T/22t2 are taken large enough to isolate the mesonic grostate.

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present resources, and this diagram is not included. To chthat the first diagram that we calculate provides by itselreasonable description of the baryon wave function, we acompute the baryon wave function with two density instions for theu and d quarks as shown in the lower part oFig. 2. This may be viewed as the square of the one-partwave function obtained from the full wave function by intgrating over one relative coordinate. If the contributionthe second diagram is small then we expect that

E d3r 2C~r1 ,r2 ,t !;E d3r 8^hurd~r 8,t !

3ru~r 81r1 ,t !uh& ~11!

will be satisfied, even when the left-hand side~LHS! of theequation is calculated using only the diagram with one dsity insertion on each quark line. This comparison willperformed in Sec. IV B.

III. RELATIONS AMONG THE DEFORMATIONSIN DIFFERENT CHANNELS

The interpolating field for a rho meson is taken toJm(x)5d(x)gmu(x). The physical states of spin projection0 and61 for the rho are obtained using interpolating fielJ0(x)5d(x)g3u(x) and J65d(x)@(g17 ig2)/2#u(x) re-spectively. Because of rotational invariance the correlato

Css~r !5(x

^Js~z!rd~x!ru~x1r !Js†~0!& ~12!

d

FIG. 2. The upper two diagrams show the three-density crelator for a baryon as defined in Eq.~10!. As in Fig. 1 the densityoperators are inserted far enough from the source and the sinthat the baryonic ground state of interest is isolated. The lodiagram shows the equivalent two-density correlator after integtion of one of the relative distances.

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satisfies the relation C11(r•e1)5C22(r•e2)5C33(r•e3)whereej is a unit vector along thej axis. This means that fothe 61 channels we have (C221C11)/25C111C22. Dueto parity symmetryC225C11 which is satisfied after ensemble averaging and leads to the relations

C115C225C111C22. ~13!

If we denote byT(r ) the tranverse and byL(r ) the lon-gitudinal projection ofC33(r ) with respect to the spin axisthe deformationa(r ) is defined byL(r )/T(r )511a(r ).The deformation in the61 channels is then 2T(r )/(T(r )1L(r ));12a(r )/2 for smalla i.e. if the spin-0 state of therho is elongated~prolate! along thez axis then the spin61states will be ‘‘flat’’ ~oblate! by approximately half thisamount. This observation is consistent with the data shoin Fig. 3.

Nonrelativistically one can use the Wigner-Eckart therem to obtain a similar result. The deformation is obtainedmeasuring the quadrupole moment defined by

FIG. 3. Density-density correlator for the rho in a definite spstateJz50 ~top! andJz511 ~bottom!, for quark separations transverse~line with Z label! and longitudinal~line with X andY labels!with respect to the spin axis. The hopping parameter isk50.154.

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5^JMu2r 2Y20~u,f!uJM&

5~2 !J2MS J 2 J

2M 0 M D ^Ju2r 2P2„cos~u!…uJ& ~14!

for a stateuJM& whereJ is the spin of the state,M the spin

projection along thez axis and (2M 0 MJ 2 J) is the 3-j symbol.

Therefore if we know the deformation for one spin projetion we can relate it to the rest. Evaluating the 3-j symbol forthe quantum numbers of the rho we find that the deformafor M50 is twice and opposite in sign to that forM561.We will thus only show results for the spin-0 state, whiexhibits the largest deformation.

The interpolating fields for projecting to the physicalD1

spin states were given in Eq.~4!. Just like for the deforma-tion of the rho in different spin projections, similar relationcan be obtained for the physical components of theD1. Theinterpolating fields for positive spin projections 3/2 and 1are no longer related to the negative ones since they invdifferent spinor components. However the cross terms oftype g1g2 contribute an order of magnitude less to the crelator and if we neglect these terms we find for the densdensity correlator of the63/2 state thatC3/2;C23/2;12a(r )/2,1 wherea(r ) is the deformation if we takeg3 inEq. ~4!.

A relation among all the spin projections of theD1 can beobtained in the nonrelativistic limit by applying the WigneEckart theorem Eq.~14!. We find that the deformation for theM563/2 states is equal and opposite to that for theM561/2. Among the physical states we will therefore shoresults only for the13/2. Since we expect the deformatioto be maximal when ag3 is used in Eq.~4! for the interpo-lating field, we will also look for deformation in this channein addition to the13/2 physical state. In the unquenchedata where we observe a small deformation these relatbetween the various amplitudes, as far as the relative sare concerned, are indeed satisfied.

IV. QUENCHED LATTICE RESULTS

A. Density-density correlation functions

We have analyzed 220 quenched configurations ab56.0 for a lattice of size 163332 obtained from the NERSCarchive@17#, using the Wilson Dirac operator with hoppinparameterk50.15,0.153,0.154 and 0.155. The ratio of tpion mass to the rho mass at these values ofk is0.88,0.84,0.78 and 0.70 respectively. Using the relat2amq51/k21/kc , with the critical valuekc50.1571, weobtain for the naive quark massmq values of about 300, 170130 and 90 MeV respectively, where we useda21

1The small difference betweenC3/2 andC23/2 is due to our limitedstatistics.

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51.94 GeV (a50.103 fm) from the string tension@18# toset the scale. Alternatively, the scale could be set fromrho mass in the chiral limit. This approach yieldsa21

52.3 GeV (a50.087 fm), also on increasingly larger latices @19,20#, with a systematic error of about 10% cominfrom the choice of fitting range and chiral extrapolation asatz, which is about twice as large as the statistical one@19#.In our discussion of quenched data we will use the value oadetermined from the string tension. However, to comparequenched with the unquenched results we will use the vaextracted from the rho mass in the chiral limit since thdetermination is applicable both in the quenched and inunquenched theory.

We fix the source and the sink for maximum separationt i5a andt f517a, wherea is the lattice spacing. The densitinsertions are taken in the middle of the time interval i.e.t59a. To check that the time intervalut2t i u5ut2t f u58a issufficient, we have performed an analysis on 27 configutions atk50.153, varying the timet where the density operators are inserted.

The results for the rho correlator are shown in Fig. 4~a!for four different insertion times atk50.153. As can beseen, density insertions att58a and 9a give the same cor-relator, reassuring us that the time separation is large enoto isolate the ground state of the rho. In the other channthe results are similar, with larger statistical noise for tbaryons. In Fig. 4~b! we show in addition the correlator fothe spin-0 state of the rho, for quark separations alongspin axis~z! and perpendicular to it (x). As expected, thedeformation is the same when measured from density intions att58a and t59a, since the ground state is isolateby then. More surprisingly, thisz2x asymmetry is almosunchanged when measured att54a, even though thez andxprofiles change appreciably. These findings indicate thatdeformation which we observe in more detail below is

FIG. 4. ~a! The density-density correlator for the rho, measurfrom density insertions at time 9a, 8a, 6a, and 4a. To avoidcluttering, the data as a function ofr are averaged over bins of sizDr 50.07a. ~b! The density-density correlator for the rho, measurfrom density insertions at time 9a,8a and 4a, for quark separationsalong thez and x axes. In both cases, 27 configurations are usand statistical error bars are omitted for clarity.

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robust, physical property of the rho meson in its ground stas well as its low-lying excited states.

In Fig. 5 we collect the correlators for the pion, the rhthe nucleon and theD1 for the four different quark masse(k values! considered. We confirm an observation made

FIG. 6. ~a! Density-density correlators,C(r ), for the pion, therho, the nucleon and theD1 at k50.154 vsur u. ~b! Same as~a! butwith two dynamical quarks atk50.157. The dynamical results wilbe discussed in Sec. V. We used the rho mass to set the scale inthe quenched and the unquenched theory. Errors bars are omfor clarity.

,

FIG. 5. Density-density correlators,C(r ), for ~a! the pion,~b!the rho,~c! the nucleon and~d! theD1 versusur u at k50.15, 0.153,0.154, and 0.155. Errors bars are omitted for clarity.

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earlier studies@3,4# that the wave functions are not very sesitive to the bare quark mass. This behavior is as expectethe bag model, but is inconsistent with nonrelativistic quamodels where a much stronger mass dependence isdicted. Insensitivity on the bare mass can be understood fthe consideration that the quarks are dressed and their etive mass is therefore not very much affected by changthe bare mass. From Fig. 5 we also see that the rho wfunction depends more on the quark mass than the othree. In particular the nucleon andD1 wave functionshardly change as we go fromk50.154 tok50.155 whichcorresponds to reducing the naive quark mass frommq;130 MeV tomq;90 MeV.

A direct comparison of the sizes of the four hadrons cbe made in Fig. 6~a! where we plot the density-density corelators fork50.154. The pion has the smallest size, aproximately half that of theD, whereas the rho has a sizcomparable to the nucleon and theD.

Any asymmetry with respect to the spin axisz is best seenby comparing the correlatorC(r ) for r5(x,y,0) and (x,0,z),i.e. in a plane perpendicular to the spin axis and a pl

FIG. 7. Contour plot of the pion correlator,C(r ), at k50.153, whenr lies in thexz plane~solid lines! or in thexy plane~dashed lines! of size 162.

FIG. 8. Same as Fig. 7 but for the rho forJz50. We haveincluded a circle to guide the eye.

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containing it. The resulting contour plots are shown fork50.154 in Figs. 7 and 8 for the pion and the rho and in Fi9 and 10 for the nucleon and theD1. The cigar shape isclearly visible in the case of the rho, whereas the pion athe nucleon look spherical~up to small lattice distortions! asexpected since their spectroscopic quadrupole is zero.

A small asymmetry appears for theD1, but it is not sta-tistically significant. This is true for all theD1 spin projec-tions. In Fig. 10 we select the unphysical state obtained usinterpolating fieldJ3 because it should show maximal defomation. In the case of theD, quenching removes only part othe pion cloud, contrary to the case of the rho where itremoved completely. Still, our negative finding does not rout a deformation of theD induced by pions. It may be thathe asymmetry only shows up when the quark mass is furdecreased so that the pion has a mass close to its phyvalue, although we do not observe a statistically significincrease in the deformation as we go from quenched quof naive quark massmq;300 MeV to mq;90 MeV. Itcould also be that the asymmetry is enhanced and onlycomes visible in full QCD, with a complete pion cloud madof light enough pions. This issue is addressed in the nsection.

FIG. 9. Same as Fig. 7 but for the nucleon.

FIG. 10. Same as Fig. 8 but for theD1. The interpolating fieldis J3.

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A more quantitative determination of meson deformatiocan be obtained by computing the second moments ofquark separationr along the three axes. Figure 11 shows tsecond moments along the spin axis,^z2&, and the average othe moments along the two transverse axes,^(x21y2)/2&,plotted versus the pion mass squared. Here we use themass to convert to physical units in order to be able to copare with the corresponding unequenched results. The spstate of the rho shows an elongation along the spin awhich increases as the quark mass is decreased. As alrmentioned, since this is a quenched calculation, this demation is not due to the pion cloud. Therefore the situatmay change if dynamical quarks are included. This is studin Sec. V.

From the second moments we can obtain the chargemean square~rms! radius of the mesons defined in the quamodel by

^r ch2 &5(

qeq^~rq2R!2&

5

(q

eqE d3r ~r /2!2C~r !

E d3rC~r !

~15!

whereR is the coordinate of the center of mass andeq is theelectric charge of the quarks. In the chiral~quenched! limitwe estimate for the pionAr p

2 ;0.35 fm using the rho mass tset the scale and 0.42 fm using the string tension@18# to becompared with the experimental value of 0.53 fm@21#. Forthe rho, using the rho mass and the string tension we

FIG. 11. ^z2& and^(x21y2)/2& versus the pion mass squaredphysical units. The lines are linear fits to the quenched data.pion ~dash-dotted line! is spherical. It is smaller than the rho, whotransverse size with respect to the spin axis~solid line! is smallerthan its longitudinal size~dashed line!. The rho~in the spin-0 pro-jection! is shown to be cigar shaped, particularly in the chiral limThe circles show full QCD results, discussed in Sec. V.

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respectivelyAr r2;0.37 fm and 0.44 fm. The ratioAr r

2/r p2

;1.06 from the lattice data is somewhat small comparedthe value 1.15, obtained from the experimental value 0.53for the pion@21# and 0.61 fm for the rho@22# calculated in aDyson-Schwinger equation approach usingf p593 MeV.This discrepancy between the experimental and lattice resmay be due to two reasons:~1! we are using the quenchetheory where the pion cloud is eliminated and~2! we are farfrom the chiral limit and making a linear extrapolation inmp

2

may be problematic, especially since chiral loops givelogarithmically divergent contribution to the pion and protocharge radii@23#. Lighter quark masses will be required tcheck the chiral extrapolation. In Sec. V we will examine teffects of unquenching.

A similar comparison of z2& with ^(x21y2)/2& for theD1, using as interpolating fieldJ3 defined in Eq.~4!, gives^z2&.^(x21y2)/2& ~the reverse is true for the states wispin projection63/2), giving suspicion of a deformationHowever, statistical errors are larger than the signal, sowe cannot claim to see any significant deformation.

B. Three-density correlation functions

We have analyzed 30 configurations atk50.15 andk50.154. Since we now have to consider two relative dtances the calculation of the three-density correlationsmore demanding even using FFT. As explained in Sec. IIonly compute the diagram with density insertions on diffeent quark lines. However, we can check the quality of tapproximation by integrating our three-density correlatiover one relative distance, and comparing the result withtwo-density correlation studied in the last subsection. Ifthree-density correlation contains complete information, bexpressions should agree as per Eq.~11!.

This comparison is performed in Fig. 12 for the nucleofor two quark masses. Although the three-density correlatas expected, is subject to larger finite-size effects visiblelarge quark separation, both methods give virtually identi

e

FIG. 12. Single particle density for the nucleon~a! for k50.15 and~b! for k50.154.

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results, indicating that we captured the dominant contrition to the three-density correlation. A similar conclusiholds for theD.

In any case, the usefulness of three-density correlatoto supplement two-density correlators and expose moretailed structure: single-particle observables, such as the qrupole moment, can be extracted directly from the twdensity correlators.

The spatialu- andd-quark distributions in the nucleon cabe investigated by fixing the relative position of the othtwo quarks. We note that, since we are using degenerauandd quarks, only the electric charge differentiates the pton from the neutron. In Fig. 13 we show the spatial disbutions of theu andd quark in the neutron, when the relativdistance between the other two quarks is fixed to zero.both quark masses considered, thed-quark spatial distribu-tion is slightly broader than that of theu quark. Since thetotal charge of the twod quarks is -2/3 and that of theuquark 12/3, the broaderd-quark spatial distribution indi-cates that the charge root mean square radius of the neis negative. For theD1 the two distributions are the samThe charge radius squared can be evaluated using

^r ch2 &5

E d3r1E d3r2(q51

3

eqrq2~r1 ,r2!C~r1 ,r2!

E d3r1E d3r2C~r1 ,r2!

~16!

whererq2(r1 ,r2) is the distance of each quark to the center

mass, in terms of the relative distancesr1 andr2. Estimatingthis integral by a discrete lattice sum we obtain for the procharge rms (r p /a)252063.5 or Ar p

2;0.4560.04 fm at

k50.15 and (r p /a)252765.5 orAr p2;0.5260.06 fm atk

50.154. If we extrapolate these two values linearly inmp2 to

the chiral limit we find;0.59(4) fm, compared to the ex

FIG. 13. u- andd-quark spatial distributions in the neutron fok50.15 andk50.154. The errors bars for ther du50 case arecomparable to those for ther dd50 case and are omitted for clarity

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-

ise-d-

-

r

--

or

ron

f

n

perimental value 0.81 fm. Again we have useda21

51.94 GeV as determined from the string tension to convto physical units. Using the nucleon mass in the chiral limto set the scale givesa2151.88(7) GeV, very close to thevalue obtained from the string tension. If instead we wouuse the rho mass to set the scale, thena21'2.3 GeV whichgives for the proton, in the chiral limit, a smaller value equto Ar p

2;0.50(4) fm. It has been pointed out that chiral locan affect the chiral extrapolation of the radii and increatheir value@23#. To check whether the chiral logs will produce larger values closer to the experimental result, ational k values closer to the chiral limit will be needed. Thneutron charge radius squarer n

2 comes out negative for bothvalues ofk as expected. We obtainr n

2/r p2;20.2560.08 and

20.2960.12 atk50.15 andk50.154 respectively. Thesvalues are consistent with the experimental value o20.146 albeit with large errors. Computing as a checksame quantity for theD1, we obtain zero within error barsin agreement with our earlier observation that thed- andu-quark spatial distributions in theD1 are the same.

Instead of fixing the relative distance between the twouquarks in the proton orD1 to zero, we can fix it to variousnonzero values. In Fig. 14 we fix theu2u separation,r uu~between 0 and 8a along a principal lattice axis!, and showthe distribution of distancesr cm between thed quark and theu2u center of mass. Detailed information about nuclestructure is contained in this figure.

In particular, let us compare configurations (uu)2dwhere the twou quarks at at the same place i.e.r uu50, andu2d2u where thed quark lies in the middle of the twouquarks, i.e.r cm50 where r cm is the distance from thed

FIG. 14. d-quark spatial distribution with respect to the centermass of the twou quarks, for differentu2u separations~a! in thenucleon spin11/2 projection and~b! in the D1 spin 13/2 projec-tion (k50.15). The straight lines mark the value of the wave funtion whenr uu50 and theu2d quark separation is 0.2, 0.4, and 0fm, to be compared withr cm50 and r uu50.2, 0.4, and 0.8 fm,respectively.

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quark to the center of mass of the twou quarks. As we willsee in the next paragraph, effective models~both Y and DAnsatze! predict equality of the wave functions among thetwo configurations provided they have the same total [email protected]. r cm in (uu)2d equal to r uu in u2d2u]. Inequalityreveals a finer structure than captured by effective modelis clear from Fig. 14 that the top horizontal line@(uu)2dwith r cm50.2 fm] falls below the1 sign atr cm50 (u2d2u with r uu50.2 fm), indicating a relative suppressionthe (uu)2d configuration, i.e. a mutualu2u repulsion. Thesame effect is visible for a larger configuration of size 0.4~second horizontal line!, but goes away or even gets invertefor yet larger configurations. Somewhat surprisingly, thisu2u repulsion at close range seems weaker in theD1 wherethe twou quarks are in the same spin state@Fig. 14~b!# thanin the nucleon where they can be in opposite spin [email protected]~a!#. This example, where a quantitative refinement cobe obtained straightforwardly by considering lighter quaon a larger lattice, serves to illustrate the wealth of informtion contained in three-density correlators.

Having, in the nonrelativistic limit, the wave function ofbaryon it is interesting to ask whether we can deducepotential which would yield this wave function. The relevapotential will of course depend on the two relative distanbetween the quarks. The issue is whether we can furreduce the degrees of freedom to effectively write the cfining potential in terms of one distance only. Two such pposals exist in the literature, which make different predtions for the linear rise of the baryonic potential.~i! Theso-calledY Ansatzcan be derived by a strong coupling argment @24#: the baryon potential grows in proportion to thminimal length of gluonic string necessary to join togeththe three static quarks. The three strings join at the Stepoint. ~ii ! The so-calledD Ansatzis derived from a centevortex picture of confinement@11#. The baryonic potential issimply a sum of two-bodyqq potentials and grows like(s/2)(r 121r 231r 31), wheres is the qq string tension andr i j the qi2qj distance. The difference between these tAnsatzedepends on the geometry of the three quark systFor instance it vanishes when the three quarks are aligsince then the Steiner point which minimizes the lengthflux tube joining the three quarks coincides with the positof the middle quark. Unfortunately this difference never eceeds;15%, so that very accurate results at large quseparations;1 fermi are needed to ascertain which modelany, is correct. In a recent lattice calculation of the baryopotential it was shown that theD Ansatzis favored for dis-tances up to 0.8 fm@25#. Although this finding is in agreement with the results of@26#, a different analysis of similarlattice results has led others@27# to the conclusion that theYAnsatz is preferred. This issue is currently under furthstudy both on the lattice@28# and phenomenologically@29#.

To examine whether our wave function results favor oof these twoAnsatze, we plot them as a function ofr Y @Fig.15~a!# and of r D @Fig. 15~b!#, wherer Y is the minimal totallength of the flux strings andr D51/2(r 121r 231r 31). Thescatter of the data is visibly smaller when plotted as a fution of r D , indicating thatr D is a better effective variable

09450

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etser---

rer

o.

d,f

-kfc

r

e

-

than r Y . Furthermore, consider the solution to the nonretivistic Schrodinger equation with potentialVY}r Y or VD

}r D . It is an Airy function which asympotically decays aexp(2cr3/2). Fits of the wave function of the nucleon to thasymptotic form are shown in Fig. 15. The fit using ther D asthe relevant distance yields ax2/DOF50.4 whereas usingr Yone getsx2/DOF51.0. Therefore bothAnsatze provide asurprisingly good description of the wave function, withpreference for theD Ansatz.

As a more stringent test, we fix the relative distancur1u5ur2u between thed and the twou quarks, and study thewave function as a function of theudu angle, cos(u)5r1•r2 /ur1uur2u, that the relative distances make. Both for theDandY Ansa¨tzethe wave function should only depend on thangle but with a different functional dependence. In Fig.we show the wave function for the nucleon versus the cosof the angle. Again theD Ansatzprovides a better descriptioof the data.

V. DENSITY-DENSITY CORRELATORS WITH TWODYNAMICAL QUARKS

As we noted in the previous section, if the pion cloudresponsible for hadron deformation, then the quenchedunquenched results may differ significantly. To investiga

FIG. 15. Nucleon wave function~a! versus half theD distanceand ~b! versus theY distance.

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the importance of dynamical quarks, we used the SES@13# configurations with two dynamical degenerate quaspecies atb55.6 on a lattice of size 163332. The latticespacing determined from the rho mass in the chiral limita2152.3 GeV @13# which is the same as for the quenchtheory atb56.0 and therefore the physical volume is tsame as in our quenched calculation. We use this determtion of the lattice spacing, which is applicable in both tquenched and the unquenched theory, to compare reamong both. We have analyzed 150 configurations ak50.156 and 200 atk50.157. The ratio of the pion mass trho mass is 0.83 atk50.156 and 0.76 atk50.157. Thesevalues are close to the quenched mass ratios measuredk50.153~0.84! andk50.154~0.78! respectively, allowing usto make pairwise quenched-unquenched comparisons.

In Fig. 6~b! we plot the two-density correlators for thfour hadrons for light sea quarks (k50.157). Comparingwith the quenched results (k50.154), one sees that the piosize remains the same, but that the rho, nucleon andD1 sizesincrease. TheD1 is now clearly larger than the rho and thrho correlator decays more slowly than the nucleon. Tslower decay is also visible in the quenched case fok50.154 and 0.155 but not for the smaller values ofk. Theinvariance of the pion size can also be seen in Fig. 11 whshows the second moments. This result is consistentcalculations in the Dyson-Schwinger framework which cocluded that the pion cloud, consisting of physical pions, ctributes only 15% to the root mean square radius of the p@30#. In our simulations the effect is expected to be evsmaller since the pion mass is larger than 600 MeV. Onother hand, Fig. 11 shows that unquenching increases thesize and therefore the ratio of the rho charge radius topion’s approaches the experimental value. This ratio is

FIG. 16. Theta dependence of the nucleon wave function wthe two relative distances are fixed to 0.660.001 fm. The filledsquares are the central values of the wave function. The solid linthe fit using theD Ansatz for the angular dependence and tdashed line is the corresponding fit using theY Ansatz. Note that forcos(u),20.5 theY Ansatzcoincides with theD Ansatz.

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s

a-

lts

t

is

hth--nnehoe

x-

pected to be more reliable than the individual moments silattice artifacts partly cancel out.

From Fig. 17 we compare the quenched and unquencasymmetry in the rho for comparable ratiosmp /mr . Themain observation is that the asymmetry in the rho growsfull QCD. This is clearly seen in the~a!-~b! comparison, i.e.for the heavier dynamical quarks. For the lighter quarks~c!-~d!, this growth in the asymmetry is still there but the effeseems less pronounced. On the other hand, finite-size efare more important. Disentangling the two would reqularger lattices.

The same analysis for theD1 gives no definite resultsregarding the deformation. Whereas in all channels weobserve a deformation obeying the sign relations obtaineSec. II with the Wigner-Eckart theorem, the deformationmains small with large statistical errors. However it is inteesting to look at a three-dimensional contour plot for theD1

in the13/2 spin state and compare it to the rho 0-spin staIn Fig. 18 we show these contour plots for the heavierquarks that we analyzed. The elongation in the rho is cleavisible whereas theD1 appears to be squeezed. Note thsqueezing in this channel means that the unphysical chausing an interpolating field aJ3 in Eq. ~4! is elongated. Thestatistical uncertainties are larger for the lighter sea quark

n

is

FIG. 17. Asymmetry for the rho wave function~a! in thequenched approximation atk50.153 and~c! at k50.154 and~b!for two dynamical quarks atk50.156 and~d! k50.157 .

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FIG. 18. Three-dimensional contour plot of the correlator~black!: upper for the rho state with 0 spin projection~cigar shape! and lowerfor theD1 state with13/2 ~slightly oblate! spin projection for two dynamical quarks atk50.156. Values of the correlator~0.5 for the rho,0.8 for theD1) were chosen to show large distances but avoid finite-size effects. We have included for comparison the contour of~gray!.

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the trend for the deformation remains the same. The oveconclusion is that unquenching seems to increase the dmation for ther and theD, which would imply that the pioncloud contributes to the deformation of these hadrons. Hever more statistics, a bigger lattice and lighter quark maswill be needed to consolidate this observation.

VI. CONCLUSIONS

Two-density correlators for mesons and baryonsshown to contain rich information on their structure. For tbaryons these correlators reduce to the one-particle densthe nonrelativistic limit.

The overall quark mass dependence of the correlatorsthe pion, the rho, the nucleon and theD1 is found to berather weak. The rho correlator shows the strongest dedence on the quark mass, and the nucleon and theD theweakest. For the quark masses that we have studied inwork unquenching has the strongest effect on hadron sfor the rho and theD, and the weakest for the pion and thnucleon. In the quenched approximation we find that thestate with 0-spin projection is prolate, whereas for theD1 nostatistically significant deformation is seen. Since for thethere are no virtual pions from backward moving quarksthe quenched approximation, the rho deformation is anamic property of quarks forming a vector particle. Addisea quarks while keeping the pion to rho mass ratio conswe observed an increase in the rho deformation, and a soblate deformation of theD1 13/2 spin state. The pioncloud may be responsible for this effect. As already poinout, lighter sea quark masses together with a larger latticwell contain the hadrons will be needed in order to stufurther the role of the pion cloud.

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dtoy

Three-density correlators for baryons, although compuneglecting insertions on the same quark line, are shownreproduce the two-density correlators when integrated oone relative coordinate. Therefore one expects the contrtion of the neglected diagram with two insertions on tsame quark line to be small. With the three-density corretors baryonic structure can be explored in more detail. Inneutron we clearly detect a broaderd-quark spatial distribu-tion compared to that of theu quark. This accounts for thenegative charge square radius of the neutron observedperimentally. By comparing theu- and d- spatial distribu-tions in the proton and in theD1 we observe that there ispreference for the twou quarks to be at 180° rather thanthe same place. More statistics are needed to consolidattrend observed here.

Information on the baryonic potential can be extractfrom fits to the three-density correlators of the nucleon.performing a fit to the radial and angular dependence wethat the baryonic wave functions are better described bconfining potential which is the sum of two-body potentiaknown as theD Ansatz, at least for relative distances o;1 fm that we can probe in this work.

Note added in proof.We would like to point out that thequenched results were obtained using point sources. Intrast, the unquenched results were obtained using ‘‘Wuptal’’ ~gauge-invariantly smeared! sources, which create a different superposition of hadronic eigenstates. This differehas no consequence when these hadronic states are allto propagate sufficiently far in Euclidean time to isolate tground state. It should be kept in mind however when exaining our lightest quark results, which are most sensitiveexcited state contamination.

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ACKNOWLEDGMENTS

We thank C. N. Papanicolas for encouraging us to lointo the issue of deformation and for discussions. T

D

ys

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SU(3)163332 quenched lattice configurations were otained from the Gauge Connection archive@17#. We thankthe SESAM Collaboration for giving us access to their dnamical lattice configurations.

ys.

. D

a,

062

A

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Probing hadron wave functions in lattice QCD · C~r,t1,t2! 5 ( ni,nf,n p,q ^hunf,p& e 2En f (p)(T/2...

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Transcript of Probing hadron wave functions in lattice QCD · C~r,t1,t2! 5 ( ni,nf,n p,q ^hunf,p& e 2En f (p)(T/2...