Binding Energies and Melting Temperatures of Heavy Hadrons in Quark-Gluon Plasma I.M.Narodetskii, Yu.A.Simonov and A.I.Veselov ITEP,Moscow,117218Russia 1 1 Abstract. Weanalyzethestaticpotentialofaquark–antiquarkpairatT ≥Tc,whereTcisatemperatureofadeconfinement phasetransitioninQCD.Wediscussthepossibilitythatthenon-perturbativepartofthispotentialcanbestudiedthroughthe 0 modificationofthecorrelationfunctions,whichdefinethequadraticfieldcorrelatorsofthenonperturbativevaccuumfields. 2 Weusethenon–perturbative quark–antiquark potentialderivedinthiswayandthescreenedone–gluon-exchange potential n with T-dependent Debye screening mass to calculate J/y , ¡ and W bbb binding energies and melting temperatures in the a deconfinedphaseofthefull2-flavorsQCD. J Keywords: Quark-GluonPlasma,Non-PerturbativeQuarkPotential,HeavyHadrons,MeltingTemperatures 6 PACS: 12.38Lg,14.20Lq,25.75Mq 1 ] INTRODUCTION bb¯ mesonsandheavybbbbaryonsaboveT .Inarecent h c paper [5] we calculated binding energiesfor the lowest p - There is a significant change of view on physicalprop- QQ and QQQ eigenstates (Q=c,b) using both the NP p erties and underlying dynamics of quark–gluon plasma potentialof the FCM and the screenedCoulombpoten- e (QGP), produced at RHIC, see e.g. [1] and references tialwiththetemperaturedependentDebayradiuscalcu- h [ therein.Onquitegeneralgroundsitisexpected,thatfun- lated in pure SU(3) glue theory.In this talk we discuss damentalforcesbetweenquarksandgluonsgetmodified andextendtheresultsofthisanalysis,inparticular,byin- 2 at finite temperature. Instead of behaving like a gas of cludingtheeffectofDebyescreeninginthefull2-flavor v free quasiparticles – quarks and gluons, the matter cre- QCD. 0 ated in RHIC interacts much more strongly than origi- 9 8 nallyexpected.Recallthatabovethedeconfinementtem- 0 peratureTc theperturbativeone–gluon–exchangepoten- FCMATFINITETEMPERATURES . tialisexpectedtobeexponentiallyscreenedatlargedis- 2 1 tances (r 1/T) [2]. Moreover, heavy quark free en- Theapproachisbasedonthestudyofthequadraticfield 0 ergyFQQ¯(≫r,T)showslargeasymptoticvalue,FQQ¯(¥ ,T) correlators <trFmn (x)F (x,0)Fls (0)>(xisEuclidian), 1 [3].Thisvaluecanbeexplainedonlybynon-perturbative whereF (x,0)istheSchwingerparalleltransporternec- v: (NP)effects,sinceperturbativeone-gluon-exchangepo- essary to maintain gauge invariance. These correlators i tential, even with increased a (r), cannot produce sim- areexpressedintermsoftwoscalarfunctions,D(x)and X ilar effect. Therefore it is morse appropriate to describe D1(x) [4]. At T = 0, the string tension s is expressed r the(NP)propertiesoftheQCDphaseaboveT interms onlyintermsofD(x): a c of the NP part of the QCD force rather than a strongly ¥ ¥ coupledCoulombforce. s = 2 dl dn D( l 2+n 2). (1) AtT =0,thenon-perturbativequark-antiquarkpoten- Z Z tial is Vnp(r) = s r, where s is the SU(3) string ten- 0 0 p tion. This potential has been used extensively in poten- At T T one should distinguish between electric and tial models. At T ≥Tc, s =0, but that does not mean magne≥tic ccorrelators DE(x), DH(x), DE(x), and DH(x), thattheNPpotentialdisappears.Attemptingtoguessthe 1 1 and,correspondingly,betweens E ands H.Itwasargued formofthenon-perturbativepotentialweaddresstothe in[6]andlaterconfirmedonthelattice[7]thatabovethe Field CorrelatorMethod (FCM) (see [4] and references deconfinement region DE(x) and s E vanish, while the there in). Within FCM the NP QQ¯ potential above T c colorelectric correlator DE(x) and colormagneticcorre- was suggested to occur due to the non zero correlation 1 lators DH(x) and DH(x) should stay unchangedat least functionD (x)thatisoneofthetwofunctionswhichde- 1 1 uptoT 2T.ThecorrelatorsDH(x)andDH(x)donot finethequadraticfieldcorrelatorsofthenonperturbative ∼ c 1 producestatic quark–antiquarkpotentials,theyonlyde- vaccuum fields . The most direct prediction of this ap- fine the spatial string tension s = s H and the Debye proachis the existence of boundstates of heavy cc¯and s mass mD (cid:181) √s s that grows with the temperaturein the TABLE 1. Parameters of the quark- antiquark potentials in units of GeV. dimensionallyreducedlimit[8]. Both sets of parameters correspond to theTchoeloNr–PesletacttircicQcQorpreoltaetnotriafulnatctTion≥DTEc(oxr)iginatesfrom V(¥ ,Tc)=0.505GeV 1 nf Tc M0 md(Tc) 1/T r 0 0.275 0.9 0.793 V (r,T) = dn (1 n T) l dl DE(x). (2) 2 0.165 1.08 0.545 np Z − Z 1 0 0 IntheconfinementregionthefunctionDE(x)wascalcu- 1 light quarkonia with chiral symmetry restoration to an- latedin[9]exploitingtheconnectionoffieldcorrelators otherpublication. tothegluelumpGreen’sfunction1 Recallthat,intheframeworkoftheFCM,themasses ofheavyquarkoniaaredefinedas exp( M x) DE(x) = B − 0 , (3) 1 x m2 whereB=6a sfs fM0,a sf beingthefreezingvalueofthe MQQ¯ = m QQ +m Q +E0(mQ,m Q), (7) strongcouplingconstant,s isthestingtensionatT =0, f E (m ,m )isaneigenvalueoftheHamiltonian and the parameterM has the meaningof the gluelump 0 Q Q 0 mass. Above T the analytical form of DE should stay c 1 H=H +V +V , (8) unchanged at least up to T 2T. The only change is 0 np OGE c B B(T)=x (T)B,wheret∼heT-dependentfactor where we have omitted spin-dependentand self-energy → termsproportionalto 1/m . InEq.(8)V is theone- M T T Q OGE x (T) = 1 0.36 0 − c (4) gluon-exchangepotentialwhichisexpectedtobeexpo- (cid:18) − B Tc (cid:19) nentiallyscreenedatlargedistances is determined by lattice data, see Eq. 52 of Ref. [11]. 4 a Substituting (3) into (2) and integratingover l one ob- V (r,T) = sexp( m (T)r), (9) OGE d tainsV (r,T) =V(¥ ,T) V(r,T)where −3 r − np − m (T)beingtheDebyemass.InEq.(7)m arethebare d Q V(¥ ,T)= B(T) 1 T 1 exp M0 , (5) quark masses, and einbeins m Q are treated as c-number M02 (cid:20) −M0(cid:18) − (cid:18)− T (cid:19)(cid:19)(cid:21) variationalparameters2.Withsuchsimplifyingassump- tions the spinless Hamiltonian takes an apparentlynon- and relativistic form, with einbein fields playing the role of 1/T theconstituentmassesofthequarks.Inwhatfollowwe V(r,T)= B(T) (1 n T)e−√n 2+r2M0dn . (6) takemc=1.4GeV,mb=4.8GeV.Asintheconfinement M0 Z − region, the constituent masses m Q only slightly exceed 0 the bare quark masses m that reflect smallness of the Q One observes the characteristic feature of the static po- kineticenergiesofheavyquarks.Thedissociationpoints tential V (r,T) produced by the correlator DE(x): the aredefinedasthosetemperaturevaluesforwhichtheen- np 1 potential gives rise to the constant termV(¥ ,T) in the ergygapbetweenV(¥ ,T)andE disappears. 0 QQ¯ interaction at large distances, which can be viewed uponasthesumofselfenergiesofQandQ¯. QUARK-ANTIQUARKSTATES WecannowexploittherelativisticHamiltoniantech- nics[13]successfullyappliedformesons,baryons,glue- The non-perturbative quark-antiquark potential is de- balls and hybrids in the confinement phase. This tech- fined by the two parameters B and M0. In what fol- nic does not take into account chiral degrees of free- lows we take s = 0.18 GeV2 and a f = 0.6, so that f s dom and is applicable when spin-dependentinteraction B = 0.648M , and vary the gluelump mass M around 0 0 canbetreatedasperturbation.Thereforebelowwecon- the central value M = 1 GeV in order to maintain 0 sider only heavy quarkonia and heavy baryons, leaving the asymptotic valueV(¥ ,T )=0.505GeV. This value c 1 Recallthatgluelumpsareactuallyboundstatesofthegluonfieldin 2 TheeigenvaluesE0(mQ,m Q)arefoundasfunctionsofthebarequark astaticcolor-octetsourcethathavebeenstudiedfirstinLatticeQCD massesmQandeinbeinsm Q,andarefinallyminimizedwithrespectto [10]. them Q.OncemQisfixed,thequarkoniaspectrumisdescribed. TABLE2. J/y statesabovethedeconfinementregion.Allthequantitiesexcept forT/Tcandr0aregiveninunitsofGeV,thedimensionofr0isGeV−1. T/Tc md V(¥ ,T) m c E0(T) V(¥ ,T) r0 Mcc − 1 0.545 0.505 1.462 -0.026 6.89 3.281 1.2 0.609 0.433 1.438 -0.0098 11.45 3.334 1.3 0.640 0.398 1.426 -0.0046 16.86 3.164 1.4 0.671 0.365 1.415 -0.0013 25.51 3.164 TABLE3. ¡ statesabovethedeconfinementregion.Thenotationsarethesame asinTable2. T/Tc md V(¥ ,T) m b E0(T)−V(¥ ,T) r0 Mbb 1 0.545 0.504 4.948 -0.345 1.17 9.768 1.6 0.733 0.302 4.954 -0.182 1.54 9.725 2.0 0.853 0.187 4.937 -0.102 2.01 9.688 2.8 1.082 0 4.837 -0.008 7.88 9.592 TABLE 4. Dissociation temperatures 1S bottomoniium undergos very little modification till (in units of Tc) for cc, bb, and W bbb T 2T. The melting temperatures for the J/y and ¡ states.W ccc isunboundbothfornf =0 ∼ c areshowninTable4. andnf =2. J/y ¡ W bbb nf =0 1.29 2.57 1.8 QQQBARYONS nf =2 1.48 2.96 2.35 The three quark potential is given by V = QQQ 12 (cid:229) i<jVQQ¯(rij,T), where 12 is the color factor andVQQ¯ agreeswith lattice estimate forthe free quark-antiquark is the sum of the perturbative and NP quark-antiquark energy 3. The strong coupling constant was taken a = potential.WesolvethethreequarkSchrödingerequation s 0.354.TheparametersofthepotentialarelistedinTable by the hyperspherical harmonics method. The wave 1,whereforthereferencewealsoindicatethevaluesof functioninthehypercentralapproximationiswrittenas theSDomebeyedemtaaislssmofd(oTucr)[c1a2lc]u.lation for the full nf = 2 Y (R,T) = √1p 3 u(RR5,/T2), (10) QCD can be inferredfrom Tables 2, 3 5. At T =T we c wherethehyperradius obtaintheweaklyboundccstate. Themeltingtempera- m tTuarbeleis4∼.T1h.e3cThcafromronnfiu=m0maasnsdes1.li4e8iTncthfoerinntfer=val2,3.s1ee- R2 = 3Q r122+r223+r321 (11) (cid:0) (cid:1) 3.3GeV.Notethatatthemeltingpointr(J/y ) ¥ that is invariant under quark permutations. Averaging the → isconsistentwithnearly-freedynamics. three–quark potential over the six-dimensional sphere As expected, the ¡ state is much more bound and one obtains the one-dimensional Schrödinger equation remainsintactuptothelargertemperatures,T 2.3Tc. forthereducedfunctionu(R,T) ∼ This is in agreementwith the lattice study of Ref. [15]. d2u(R,T) 15 3 ThemassesoftheL=0bottomoniumlieintheinterval +2 E V(R,T) u(R,T)=0, (12) 9.6–9.8GeV,about0.2–0.3GeVhigherthan9.460GeV, dR2 (cid:20) 0−8R2 −2 (cid:21) themassof¡ (1S)atT =0.AtT =Tc thebbseparation whereV(R,T)=VOGE(R,T)+Vnp(R,T)and r is 0.25 fm that is compatible with r = 0.28 fm at T0= 0 (at the melting point r0 → ¥ ).0Note that the VOGE(R,T)=−136pa s Zp /2exp(−mRˆd(T)Rˆ) sin2(2q ), 0 (13) 3 However,thedifferenceintheparameterM0causesthesmalldiffer- ae4ntTcinehyeoefafcVfce(oc¥tuan,Tstco)offmothrpeTarrue>ndnwTinci.tghaths(erc)aisnetohfeaCcoounlsotmanbtpaost=en0ti.a3l5pbroodthucfeosr Vnp(R,T)=V(¥ ,T)−4xp (MT0)B× theenergiesandwavefunctions[14]. p /2 5 Thecorresponding results forthepuregluodynamics (nf = 0)are RˆK (Rˆ) T e Rˆ(1+Rˆ) sin2(2q )dq , (14) giveninRef.[14] Z (cid:18) 1 −M0 − (cid:19) 0 TABLE 5. W state above the deconfinement region. The interquark distances bbb q<ri2j >=q<Rmb2>. T/Tc md V(¥ ,T) m b E0(T) V(¥ ,T) √<R2> Mbbb − 1 0.545 0.757 4.962 -0.327 3.44 14.837 1.4 0.672 0.548 4.926 -0.185 4.22 14.768 2.0 0.853 0.281 4.919 -0.192 7.56 14.641 2.3 0.940 0.166 4.830 -0.0034 18.50 14.563 2.4 0.969 0.131 4.812 +0.0021 32.40 14.533 V(¥ ,T)beinggivenbyEq.(5),andRˆ=2M Rsinq /m . thegroundstateofJ/y survivesuptoT 1.3 1.5T, 0 Q c In Eq. (14) we use the approximate expression for the andthereisnoboundW stateatT T∼.Ont−heother ccc c non-perturbativeQQ¯ potential(6) hand,thebbandbbbstatessurviveup≥tohighertemper- ature,T 2.6 3.0T andT 1.8 2.4T forn =0,2, c c f B(T) T respectiv∼ely. T−he results sug∼gest t−hat the systems are V(r,T) xK (x) exp( x)(1+x) , ≈ M2 (cid:18) 1 − M − (cid:19) stronglyinteractingaboveT . 0 0 c (15) wherex=M randK (x)istheMcDonnaldfunction, 0 1 ThetemperaturedependentmassofthecolorlessQQQ ACKNOWLEDGMENTS statesis ThisworkwassupportedinpartbyRFBRgrants08-02- 3m2 3 M = Q + m +E (m ,m ), (16) 00657,08-02-00677,and09-02-00629. 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