PHYSICAL REVIEW C 67, 038202 ~2003!

Charmonium mass in nuclear matter

Su Houng Lee1,2 and Che Ming Ko11Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843-3366

2Institute of Physics and Applied Physics and Department of Physics, Yonsei University, Seoul 120-749, Korea~Received 2 August 2002; published 20 March 2003!

The mass shift of charmonium states in nuclear matter is studied in the perturbative QCD approach. Theleading-order effect due to the change of gluon condensate in nuclear matter is evaluated using the leading-order QCD formula, while the higher-twist effect due to the partial restoration of chiral symmetry is estimatedusing a hadronic model. We find that while the mass ofJ/c in nuclear matter decreases only slightly, those ofc(3686) andc(3770) states are reduced appreciably. Experimental study of the mass shift of charmoniumstates in nuclear matter can thus provide valuable information on the changes of the QCD vacuum in nuclearmedium.

DOI: 10.1103/PhysRevC.67.038202 PACS number~s!: 25.75.2q, 14.40.Gx, 12.39.Hg, 12.40.Yx

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Understanding hadron mass changes in nuclear medand/or at finite temperature can provide valuable informatabout the QCD vacuum@1–3#. It is also relevant phenomenologically to the interpretation of experimental resufrom relativistic heavy ion collisions@4#, in which a hotdense matter is formed during the collisions. Previous stuhave been largely concerned with hadrons made of lquarks @3#. Only recently were there studies of the imedium masses of hadrons consisting of heavy chquarks. Using either QCD sum rules@5,6# or the quark-meson coupling model@7#, it has been found that the massD meson, which is made of a charm quark and a light quais reduced significantly in nuclear medium as a result ofdecrease of the light quark condensate. For theJ/c, whichconsists of a charm and anticharm quark pair, both the Qsum rules analysis@8# and the LO perturbative QCD calculation @9,10# show that its mass is reduced slightly in thnuclear matter mainly due to the reduction of the gluon cdensate in nuclear matter.

The change of hadron masses at finite temperature isstudied using the lattice gauge theory as its prediction ismodel dependent. Recent lattice gauge calculations at fitemperature with dynamical quarks have shown that ebelow critical temperature the interquark potential at laseparation approaches an asymptotic valueV`(T) that de-creases with increasing temperature@11#. This transitionfrom a linearly rising interquark potential in free space tosaturated one at finite temperature is due to the decreathe string tension and the formation ofQq and qQ pairs,where q denotes a light quark, when the separation oftwo heavy quarks (Q) becomes large. The decrease inV`(T)can thus be interpreted as a decrease of the open heavymeson (Qq or qQ) massmH , such as theD meson mass, afinite temperature@12,13#. Furthermore, the decrease ofmHseems to be a consequence of the reduction in the constimass of light quark as the temperature dependence ofV`(T)is similar to that of the chiral condensate^qq& @14#. Thisrelation between the massmH and the chiral order parametealso follows naturally from the heavy quark symmetry@15#.With the finite temperature interquark potential, it has beshown via the solution of corresponding Schro¨dinger equa-

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tion that the masses of charmonium states are reducefinite temperature as well@16,17#.

At finite density, lattice gauge calculations are at presnot feasible for studying the heavy quark potential or tmassmH of the open heavy quark meson. Masses of thheavy quark systems are, however, expected to changepreciably in nuclear medium. Model independent estimahave shown@18,19# that condensates of the lowest dimesional operators (as /p)G2& and ^qq& decrease, respectively, by 6 and 30 % at normal nuclear matter, which asignificant changes expected only near the critical poinfinite temperature@20#. As in the case of finite temperaturethe reduction of gluon condensate leads to the softeningthe confining part of interquark potential@21#, while the de-crease of quark condensates implies a drop of the open hquark meson mass orV` of the heavy quark potential. Bothare expected to lead to nontrivial changes in the bindenergies of the charmonium statesc(3686) andc(3770), astheir wave functions are sensitive to both the confining pand the asymptotic value of the interquark potential.

In this paper, we evaluate the mass shift ofc(3686) andc(3770) due to changes in the gluon and quark condensin nuclear medium. The effect of the gluon condensatedetermined using the leading-order~LO! QCD formula,which was developed in Refs.@9,22# and has been used tstudy theJ/c mass in medium@10#. The effect due to thechange in quark condensates is difficult to calculate usingquark and gluon degrees of freedom as they appear as htwist effects in the operator product expansion@23,24#. How-ever, its dominant effect to a heavy quark system is to redV` as a result of the decrease ofD meson in-medium massabout 50 MeV in normal nuclear matter due to the 30reduction in the quark condensate@5–7,25#. Therefore, wecan study the effect of changing quark condensate oncharmonium states at finite density by using a hadromodel to calculate their mass shifts due to the change oDmeson in-medium mass. Combining the effects from chanin the gluon condensate and inmD , we find that bothc(3686) andc(3770) masses are reduced appreciablynormal nuclear matter density.

The mass shift of charmonium states in nuclear medican be evaluated in the perturbative QCD when the ch

©2003 The American Physical Society02-1

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BRIEF REPORTS PHYSICAL REVIEW C67, 038202 ~2003!

quark mass is large, i.e.,mc→`. In this limit, one can per-form a systematic operator product expansion~OPE! of thecharm quark-antiquark current-current correlation functbetween the heavy bound states by taking the separascale (m) to be the binding energy of the charmoniu@9,22,26#. The forward scattering matrix element of thcharm quark bound state with a nucleon then has the folling form:

T~q25mc2 !5(

n

Cn

~m!n^On&N . ~1!

Here,Cn is the Wilson coefficient evaluated with the charquark bound state wave function and^On&N is the nucleonexpectation value of local operators of dimensionn.

For heavy quark systems, there are only two independlowest dimension operators; the gluon condens@^(as /p)G2&# and the condensate of twist-2 gluon operamultiplied byas @^(as /p)GamGn

a&#. These operators can brewritten in terms of the color electric and magnetic fie^(as /p)E2& and ^(as /p)B2&. Since the Wilson coefficienfor ^(as /p)B2& vanishes in the nonrelativistic limit, the onlcontribution is thus proportional to(as /p)E2&, similar tothe usual second-order Stark effect. We shall thus calcuthe mass shift of charmonium states due to change ofgluon condensate in nuclear medium by the QCD secoorder Stark effect@10#.

The mass shift of charmonium states to leading ordedensity is obtained by multiplying the leading term in Eq.~1!by the nuclear densityrN . This gives

Dmc~e!521

9E dk2U]c~k!

]k U2 k

k2/mc1e K as

pE2L

N

•

rN

2mN.

~2!

In the above,mN and rN are the nucleon mass and thnuclear density, respectively;^(as /p)E2&N;0.5 GeV2 is thenucleon expectation value of the color electric field ande52mc2mc . In Ref. @9#, the LO mass shift formula waderived in the large charm quark mass limit. As a result,wave functionc(k) is Coulombic and the mass shift is epressed in terms of the Bohr radiusa0 and the binding en-ergy e052mc2mJ/c . This might be a good approximatiofor J/c but is not realistic for the excited charmonium staas Eq. ~2! involves the derivative of the wave functionwhich measures the dipole size of the system. We haverewritten in the above the LO formula for charmonium mashift in terms of the QCD parametersas50.84 andmc51.95, which are fixed by the energy splitting betweenJ/cand c(3686) in free space@9#. Furthermore, we take wavfunctions of the charmonium state to be Gaussian withoscillator constantb determined by their squared radii^r 2&50.472, 0.962, and 1 fm2 for J/c, c(3686), andc(3770),respectively, as obtained from the potential models@27#. Thisgives b50.52,0.39, and 0.37 GeV if we assume that thecharmonium states are in the 1S, 2S, and 1D states, respectively. Using these parameters, we find that the mass shifnormal nuclear matter density obtained from the LO QC

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formula ~2! are 28, 2100, and 2140 MeV forJ/c, c(3686), andc(3770), respectively.

Although the higher twist effects on the charmoniumasses are expected to be nontrivial, the result forJ/c isconsistent with those from other nonperturbative QCD sties, such as the QCD sum rules@8,24# and the effective po-tential model@28,29#, which are all based on the dipole interactions between quarks in the charmonium and thosthe nuclear matter. To go beyond the leading order inOPE, we need to calculate the contributions from highermensional operators in Eq.~1!, which include light quarkoperators. Explicit calculations from QCD sum rules forJ/cup to dimension 6 operators@24# show that the effect due tochange in the condensates of light quark operators at dimsion 6, which include qGqqGq& and ^qDGq& @23,24#, isunimportant for the mass shift ofJ/c. However, such a cal-culation cannot be easily generalized to the excited charnium statesc(3686) andc(3770), where the sum rules dnot exist even in the vacuum. On the other hand, the higtwist effects due to the light quark operators can be estimaby considering the coupling of the charmonium to theDDstates as in the potential model for charmonium states@27#.Therefore, instead of summing up the nonconvergent conbutions from the change in the light quark condensates inOPE of Eq.~1!, we estimate its contribution by evaluatinthe charmed meson one-loop effect on the mass of a cmonium with in-mediumD meson mass predicted from thQCD sum rules@5,6# or the quark-meson coupling model@7#.

Following the studies in Ref.@30# on r-p interactions andin Ref. @31# on f-K interactions, we use the following Lagrangian for interacting charmoniumc andD meson:

L51

2~ uDmDu22mD

2 uDu2!21

4FmnFmn1

1

2mc

2cmcm, ~3!

whereFmn5]mcn2]ncm is the charmonium field strengthDm5]m2 i2gcDDcm , andD5(D0,D1).

The coupling constantgcDD can be determined using th3P0 model@32#. In this model, the coupling constant is proportional to the overlap integral between the relative quwave functions of charmonium and the two outgoicharmed mesons as well as to a coupling parameterg, whichcharacterizes the probability of producing a light quaantiquark pair in the3P0 state. The result can be read ofrom Refs.@33,34# and is given by

gcDD2 ~q!5g2p3/2

mc3

bD3

f c~q2,r !e2q2/2bD2 (112r 2), ~4!

whereq is the three-momentum ofD mesons in thec restframe andr 5b/bD with bD (b) being the oscillator con-stant forD meson (c) wave function. For theb ’s, the samevalues will be used as in the LO QCD calculation. The vues forg andbD are taken to be 0.35 and 0.31 GeV, respetively, to reproduce both the decay width ofc(3770) toDDand the partial decay width ofc(4040) toDD, DD* , andD* D* @33,35#. The functionf c(q2,r ) denotes

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BRIEF REPORTS PHYSICAL REVIEW C67, 038202 ~2003!

26r 3~11r 2!2

~112r 2!5

for J/c,

25~312r 2!2~123r 2!2

3~112r 2!7

3S 122r 2~11r 2!

~112r 2!~312r 2!~3r 221!

q2

bD2 D 2

for c(3686), and

295r 7

3~112r 2!7 S 12~11r 2!

5~112r 2!

q2

bD2 D 2

for c(3770).Because of its momentum dependence,gcDD(q) takes

into account the form factor at thecDD vertex and allowsalso the coupling of the charmonium to off-shellD mesons.For c(3770), it can decay toDD in free space, and its onshell coupling constant isgc(3770)DD@q5(mc

2/42mD2 )1/2#

515.4. The coupling constants atq50 are 15.3, 18.7, and16.8 for J/c, c(3686), and c(3770), respectively. Thevalue for J/c coupling toD mesons is slightly larger thathat estimated by the vector meson dominance model@36,37#and by the QCD sum rules@38#. As theD meson momentumincreases, its coupling constant toJ/c has a simple exponential fall off due to the 1S quark wave function of theJ/c. Incontrast, the coupling constants of excited charmonistatesc(3686) andc(3770) toD mesons fall off exponentially with the D meson momentum but vanish at certainq2

as a result of the nodes in the 2S or 1D wave functions ofthe excited charmonium states@33,35#.

Similar to the method introduced in Ref.@31#, we haveused the above Lagrangian to evaluate the self-energycharmonium due to theD meson loop. After performing theenergy integral in the rest frame of thec, i.e., k5(mc,0),the invariant part of the polarizationP(k) then has the fol-lowing form:

P~k!51

6p2PE dq2gcDD

2 ~q2!F q

AmD*21q2

3S 4q2

mc224mD*

224q213D 2~mD* 5mD!G , ~5!

wheremD* is the in-mediumD meson mass andP denotesthat only the principle value of the integral is evaluated. Tsubtracted term in the above equation is a renormalizaconstant which is determined by requiringP(k25mc

2)50whenmD* 5mD . This ensures that theD meson loop does nocontribute to the real part of the charmonium self-energyfree space. The mass shift of the charmonium at finite dsity is then given byDmc5P(k25mc

2).

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In Fig. 1, we show the mass shifts of charmonium staas functions ofmD* . It is seen that the mass shift ofc(3770)is negative for small negative shift ofD meson mass bubecomes positive when theD meson mass drop is large. Icontrast, the mass shift ofc(3686) is negative for all negative mass shifts of theD meson. This difference can be understood from Eq.~5!, where the integral is a convolution othe form factor gcDD

2 (q2) with the terms in the squarebracket, which are singular whenq25mc

2/42mD*2 and q2

5mc2/42mD

2 . The integrand thus changes signs whenetheD meson momentumq passes through these singularitiand finally becomes negative whenq2 is larger than any ofthe singularities, which correspond to the energies ofvirtual intermediateD meson states. As in second-order pturbation theory, the contribution is attractive when the eergy of the intermediate state is larger than the charmonmass. However, the form factor decreases exponentially wq2 and can even be zero, the large negative contributionpected for a constant form factor is suppressed, leadingto an increase of thec(3770) mass whenmD2mD* >10MeV. On the other hand, the singularity of the integrandEq. ~5! in the case ofc(3683) occurs only when 2mD* fallsbelow its mass and therefore has only a small positive ctribution whenq2 is very small, leading to a reduction of itmass for anyD meson mass shift. ForJ/c, we find that itsmass only increases slightly with droppingD meson massand depends weakly onmD* . For mD2mD* 550 MeV, whichis the expected mass shift ofD meson at normal nucleamatter density, the mass shift ofJ/c is about 3 MeV. Thisresult is consistent with that from the QCD sum rules@24#and is also expected from the potential model@27#, where theJ/c wave function has only a smallDD component. We notethat the density dependence of the mass shifts of charnium states, particularly the excited ones, is nonlinear ifuse a linearly density-dependentD in-medium mesonmD*5mD250 r/r0 MeV in the denominator of Eq.~5!. On the

1.79 1.81 1.83 1.85 1.87 1.89mD

* (GeV)40

20

0

20

40

∆mΨ (

MeV

)

J/ΨΨ(3686)Ψ(3770)

-

-

FIG. 1. Mass shifts of charmonium statesJ/c ~solid curve!,c(3686) ~long dashed curve!, andc(3770) ~short dashed curve! asfunctions ofD meson in-medium massmD* . The circle indicates theexpected mass shifts at normal nuclear matter density.

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BRIEF REPORTS PHYSICAL REVIEW C67, 038202 ~2003!

other hand, the mass shift obtained from the LO QCD fmula in Eq.~2! depends linearly on the nuclear density.

Adding the mass shift from theD meson loop effect to theresult from the LO QCD calculation, we find that massescharmonium states are changed by the following amounnormal nuclear matter density:

DmJ/c52813 MeV,

Dmc(3686)52100230 MeV,

Dmc(3770)52140115 MeV, ~6!

where the first number represents the shift from the LO Qwhile the second number is from theD meson loop. Theabove results thus show that masses of excited charmonstates are reduced significantly in nuclear matter, largelyto the nontrivial decrease of the in-medium gluon condensin the LO QCD formula for their masses.

The mass shifts of bothc(3686) andc(3770) in nuclearmedium are large enough to be observed in experimentsvolving p2A annihilation as proposed in the future accele

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tor facility at the German Heavy Ion Accelerator Cent~GSI! @39#. In these experiments,c(3770) andc(3686) pro-duced inside a heavy nucleus will be studied via the dilepspectrum emitted from their decays. While the lifetimeJ/c has been shown to remain constant in nuclear mathose of c(3686) andc(3770) are reduced to less tha5 fm/c due to increases in their width in nuclear matter@33#.Therefore, an appereciable fraction of produced excited cmonium states in these experiments are expected to dinside the nucleus@40#, leading to an observable shift of thpeak positions in the dilepton spectrum. The observationsuch shifts in the masses of excited charmonium statethese experiments would give us valuable information onnontrivial changes of the QCD vacuum in nuclear mediuand on the origin of masses in QCD.

This paper is based on work supported by the NatioScience Foundation under Grant No. PHY-0098805 andWelch Foundation under Grant No. A-1358. S.H.L was asupported by the Brain Korea 21 project of Korean Minisof Education and by KOSEF under Grant No. 1999-2-11005-5.

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