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An in-Medium Heavy-Quark Potential from the QQ¯ Free Energy ShuaiY.F.LiuandRalfRapp CyclotronInstituteandDepartmentofPhysicsandAstronomy,TexasA&MUniversity,CollegeStation,Texas77843-3366,USA Abstract 5 1 Weinvestigatetheproblemofextractingastaticpotentialbetweenaquarkanditsantiquarkinaquark-gluonplasma 0 2 (QGP) from lattice-QCD computationsof the singlet free energy, FQQ¯(r). We utilize the thermodynamicT-matrix formalismtocalculatethefreeenergyfromanunderlyingpotentialansatzresummedinladderapproximation.Imag- n inarypartsofboth QQ¯ potential-typeandsingle-quarkselfenergiesareincludedasestimatedfromearlierresultsof a the T-matrix approach. We find that the imaginary parts, and in particular their (low-) energydependence, induce J markeddeviationsofthe(realpartofthe)potentialfromthecalculatedfreeenergy.Whenfittinglatticeresultsofthe 0 latter,theextractedpotentialischaracterizedbysignificantlong-rangecontributionsfromremnantsoftheconfining 3 force.Webrieflydiscussconsequencesofthisfeaturefortheheavy-quarktransportcoefficientintheQGP. ] h Keywords: Heavy-quarkpotential,Bethe-SalpeterEquation,Quark-GluonPlasma p PACS:12.38.Mh,12.39.Pn,25.75.Nq - p e h 1. Introduction equation [15–19] remains the definition of the driving [ kernel or potential. Thus far, it has been bracketed by Heavy-flavor (HF) particles, containing charm or 1 using either the temperature dependent free or inter- v bottom quarks, play a central role in the analysis of nalenergies(or combinationsthereof),which, roughly 2 strongly interacting matter as produced in ultralrela- speaking,ledtoaratherweaklyorstronglycoupledsys- 9 tivistic heavy-ion collisions (URHICs) [1]. The nu- tems,respectively. 8 clear modification factor and elliptic flow of D and B 7 mesons (as well as their semileptonic decay leptons) In the present paper we address this uncertainty 0 . are key measures to establish and quantify the strong- by utilizing the thermodynamic T-matrix to derive an 1 0 coupling nature of the QCD medium [2–4], while the expression for the singlet HQ free energy, FQQ¯(r). productionpatternsofcharmoniaandbottomonia[5–8] Previous work to extract the HQ potential has been 5 1 arebelievedtoencodeinformationonitsdeconfinement donebothnonperturbativelyfromlQCDusinganeffec- : properties. Forbothobservables,agoodunderstanding tive Schro¨dingerequationand potentialfor the Wilson v of the in-mediuminteractionsof the heavy quarksand loop [20–23], and within weak-coupling schemes [18, i X quarkoniaismandatory. 24,25]. Here,wetakeadiagramaticapproachbydefin- r Lattice-QCD(lQCD)computationsallowtostudythe ingthepotentialwithinafield-theoreticscatteringequa- a properties of heavy-quark (HQ) systems in the quark- tion,whichleadstoaslightlydifferentSchro¨dinger(or gluonplasma(QGP) fromfirstprinciple. Inparticular, Dyson) equation than in previouswork. Starting from euclidean correlation functions of quarkonia, HQ sus- the finite-temperature Bethe-Salpeter equation (BSE) ceptibilitiesandstaticQQ¯ freeenergieshavebeencom- fora static quarkandantiquarkin a mixedcoordinate- puted with good precision [9, 10]. A non-perturbative frequency representation, we resum the ladders into a T-matrix approach (“Brueckner theory”) for quarko- closed form expression which naturally serves as the nia and heavy quarks in the QGP has been developed definition of the in-medium driving kernel. The per- to interpret the lQCD results [11, 12], and the calcu- tinent spectral function can then be straightforwardly lated spectral functions and transport coefficients have related to the HQ free energy. A key element in eval- beenappliedto bothhidden[13] andopenHF observ- uating the resulting expression is the energy depen- ables[14]inURHICs. Themaintheoreticaluncertainty dence of the spectral function, in particular of imagi- in this and other approachesbased on the Schro¨dinger narypartsinducedbybothone-andtwo-bodycorrela- PreprintsubmittedtoElsevier February2,2015 tions,i.e.,single-quarkself-energiesandQQ¯ potential- The task in the following is to calculate G˜>(−iτ,r) in typecontributions.Theformerhavebeencalculatedbe- Eq. (2) within a BSE (or potential model) in the Mat- forewithintheselfconsistentT-matrixformalism[26], subara formalism. In a “mixed” frequency-coordinate whileforthelatterwe takeguidancefromperturbative representationitisgivenby calculations[18,24, 25]. Therealpartofthepotential 1 willbetakenfromafield-theoreticansatzforascreened T˜(zλ,vn,vm|r)= K(vn−vm,r)− βXK(vn−vk,r) Cornell potential [27], where four parameters charac- k terizeitsstrengthandscreeningproperties. Thisset-up ×G(2)(z −v ,v |r)T˜(z ,v ,v |r) (4) 0 λ k k λ k m willthenbedeployedtoconductafittolQCDdatafor whereK(v −v ,r)istheinteractionkerneland thestaticcolor-singletfreeenergy. n m G(2)(z −v ,v |r)≡G˜(1))(z −v )G˜(1)(v ) (5) 0 λ n n λ n n 2. StaticQQ¯ FreeEnergyinLadderApproximation the uncorrelatedtwo-particle propagator. The external frequency z is conserved because of imaginary-time Our objective in this section is to relate the static λ translationinvariance.Thefulltwo-particleGreenfunc- correlation function of (infinitely massive) quark and tiontakestheform[32,33] antiquark to the HQ free energy, F (r) [18, 28–30], QQ¯ and evaluate it in ladder approximation (BSE or T- G˜(z ,r)=G˜(2)(z ) matrix scheme). Toward this end we first formally λ 0 λ 1 carry out the ladder-diagram resummation of the BSE + G˜(2)(z −v ,v )T˜(z ,v ,v|r)G˜(2)(z −v,v), intheimaginary-time(Matsubara)formalism,andthen β2 Xk,l 0 λ k k λ k l 0 λ l l reduce the BSE to the T-matrix (potential approxima- (6) tion). Theone-particlepropagatorintheinfinite-masslimit where we defined G˜(02)(zλ) ≡ −1β kG˜(02)(zλ − vk,vk), isgivenby[18] and likewise for G˜(z ,r). The discPrete energy depen- λ denciesandsummationsrenderEq.(4) ofmatrixtype, G(1)(cid:0)vn,r′(cid:1)=Z (d23πp)′3eip′·r′ivn−εp′ −1Σ(1)(vn,p′) Tanfmorm=alKsnomlu+tioPnk[vKiaGm(02a)]tnrkixTkimn,vewrshiiocnh,iTsnmam=enabkl[e1t−o ≈ d3p′ eip′·r′ 1 ≡δ r′ G˜(1)(v ) , KG(02)]−nk1Kkm,which,however,isnotverypracticPal. Z (2π)3 iv −M−Σ(1)(v ) n To proceed,we adoptthe apotentialapproximation, n n (cid:0) (cid:1) (1) i.e., neglecting energy transfers in the interaction kernel so that K(v − v ,r) → V(z ,r); we keep a pos- n k λ where the δ-function implies that the particle never sible dependence on the external energy parameter z , λ moves; Σ(1) denotes the single-particle selfenergy and in analogyto what has been done, e.g., in the descrip- M is its (heavy) mass; vn = (2n+1)π/β is a discrete tionofelectromagneticplasmas[34]wherea“dynami- fermionic Matsubara frequency (n integer, T = 1/β: calscreening”caninducesuchadependence.Withthis temperature),adoptingtheconventionsofRef.[31]with approximation, the Matsubara summations in Eq. (4) ~ = 1. The two-particleGreen functioninheritsthe δ- decoupleandonecanresumthegeometricseriestofind functionstructure[18]1: T˜(z ,v ,v |r)= λ n m G>(−iτ,r1,r2|r′1,r′2) V(z ,r)− 1 V(z ,r)G˜(2)(z −v ,v )T˜(z ,v ,v |r) ≡δ r1−r′1 δ r2−r′2 G˜>(−iτ,r), (2) λ βXk λ 0 λ k k λ k m (cid:0) (cid:1) (cid:0) (cid:1) V(z ,r) nwohteartieorn=is|ru1s−edr2t|o.HdeerneoatendthienrthedeufcoelldowGirnegenthfeu“ntciltdioen” = 1−V(z ,λr)G˜(2)(z ) ≡T˜(zλ|r) (7) λ 0 λ withoutthespatialδ-functions. Thestatic QQ¯ freeen- Thecorrespondingfulltwo-particleGreenfunctioncan ergy, FQQ¯, can be defined in terms of the QQ¯ Green alsobewrittenincompactformas functionas[18] 1 1 G˜(z ,r)= . (8) FQQ¯(r)=−βln(cid:16)G˜>(−iβ,r)(cid:17) . (3) λ G˜(02)(zλ) −1−V(zλ,r) h i Theaboveexpressonisakeyformulaofourderivation, 1Colorandspinindicesaresuppressed. asitcanserveasthedefinitionoftheinteractionkernel 2 inthe T-matrixformalismItslightlydiffersfromwhat tion,Eq.(8);wewritethepertinentspectralfunctionas has been adopted in previous approaches [18, 20, 23, 1 25],ascanbeseenfrominspectingtheresultingDyson σ¯V(E,r)=−πG˜I(E+2M+iǫ|r)≈ equationforthemesonspectralfunction. Nextwe usethe analyticityoftheGreenfunctionto 1 − VI(E,r)+Σ(I2)(E) h i , obtainthe correlationfunctionin imaginary-timevia a π E−V (E,r)−Σ(2)(E) 2+ V (E,r)+Σ(2)(E) 2 R R I I spectralrepresentation, h i h i (13) ∞ eE′(β−τ) whereweuseΣ(2) todenotethetwo-particleselfenergy G˜>(−iτ,r)= dE′σ E′,r , (9) thatarisesfromthe uncorrelatedsingle-particleselfen- Z V eβE′ −1 (cid:0) (cid:1) ergies[26,32]. We nowmakeconcreteansa¨tzeforthe −∞ potential and selfenergies in terms of a field-theoretic withthespectralfunctioncorrespondingtoEq.(8), modelforthein-mediumCornellpotential[27]andpre- vious results from the T-matrix formalism [26]. The 1 σ (E,r)=− G˜ (E+iǫ|r) . (10) formerleadstothefollowingrealpartsofpotentialand V π I uncorrelatedselfenergy, We use the subscript “V” to indicate that the spectral 4 e−mDr e−msr V (E,r)=− α −σ (14) functioniscalculatedinpotentialapproximation. Here R 3 s r m s and in the following, the subscripts ”R” and “I” are 4 1 Σ(2)(E)=− α m +σ , (15) short-handnotationsforthe real andimaginarypartof R 3 s D m s a given quantity. Since we work in the static limit corresponding to the r-dependent and r-independent (M → ∞), we redefine the spectral function relative terms of the screened Cornell potential, respectively to the QQ¯ threshold by a shift E = E′ − 2M, i.e., (additional contributions from the uncorrelated 2- σ¯ (E,r) ≡ σ (E + 2M,r), which further simplifies V V particle propagator are small [26] and therefore ne- Eq.(9)to glected in the present exploratory calculation). Their strengthsarecharacterizedbythestrongcoupling,α = ∞ s G˜>(−iτ,r) g2/(4π), and string tension σ, along with correspond- G¯>(−iτ,r)≡ = dEσ¯ (E,r)e−τE (11) e−τ(2M) Z V ing screening masses mD and ms, respectively. As in −∞ ourpreviouswork[12],weutilizethelatterasindepen- dentfit parameters, where, for simplicity, we choose a CombiningEqs.(3),(10)and(11),wearriveatthefinal linear ansa¨tze, m = c gT with two constant coef- expressionforF (r)withintheT-matrixapproach, D,s D,s QQ¯ ficients, c (which turn outto be close of orderone). D,s The imaginary part of the uncorrelated 2-particle self- ∞ 1 energy, Σ(2), follows from the folding integral of the FQQ¯ =−βln−Z∞ dEσ¯V(E,r)e−βE (12) stiivneglien--pmareItdiciulemprsoelpfaegnaetrogrisesw. hIicnhpcrionnctiapilne,ththeisreqsupaenc-- tity follows automatically from a selfconsistent calcu- In the following section, in the spirit of a variational lation,oncethepotentialisspecified,aswascarriedout approach,we will fit a physically motivatedansatz for in Ref. [26]. For our exploratory study in the present thepotentialtolQCDdataofthein-mediumfreeenergy. work,wedonotenforceselfconsistencybutinsteadem- ploytheenergydependenceofΣ(1)asfoundinRef.[26] I (seeFig.3inthere)tocomputeΣ(2). Itturnsoutthatthe 3. PotentialExtractionfromtheFreeEnergy I lattercanbewellrepresentedbyGaussianform, Thetwo-particleGreenfunctioninmediumreceives (E− Σ(R2))2 4 Σ(2)(E)= Dexp− 2  α T . (16) ackroeirnsnitnreiglb,ufVrtoi,omannsidnffdrrioovmmidutthhaeelisn(imntegerladeci-utpimoarn-tsmicoloefdesiaeficlefhden)peairrngttiiecerlsae,cwΣti(io1t)hn, WithC ≃I1.6wecanreprodu2c(eCtThe)2ener3gysdependence the heatbathwhich figureintoG(0). Both are, in prin- ofthesingle-particlewidthofRef.[26],beingverysim- 2 ciple, complex quantities and are contained in our T- ilarforbothinternalandfreeenergiesasunderlyingpo- matrixexpressionforthe fulltwo-particleGreen func- tential. The strength parameter D, however, which we 3 definerelativetotheperturbativevalue, 4α T,markedly internalenergies,U = F +TS withS = −dF/dT,also 3 s dependsontheunderlyingpotential,varyingbyafactor agree with the lattice data. The underlying two-body of2-3betweenfreeandinternalenergies;asmentioned potential turns out to lie significantly above F , but QQ¯ above,wehereuseitasafitparameter. below UQQ¯, at intermediate and long distances. How- Finally,wehavetospecifytheimaginarypartofthe ever,thepotentialdiffersfrombothfreeandinternalen- potential; its magnitude and r dependence has been ergyinthatitsderivative,characterizingtheinter-quark determined previously in the weak coupling limit as force,islargerthanforboth FQQ¯ andUQQ¯, inaregion Vpert = 4α TΦ(m r) [18, 24, 25], where Φ(x) is a around r ≃ 1fm. This is a remnant of the confining I 3 s D smoothlyvaryingfunctionwhichvanishesfor x=0and stringterminthehotmedium,causedbytherelatively approaches1forlarge x. However,inourdefinitionof smallscreeningmass,ms. Eventhoughthisforceisnot thepotential,wehaveidentifiedthenon-zeropartatin- solarge,ithasaratherlongrange,encompassingarela- finite distance with the uncorrelated selfenergy. Upon tivelylargevolume.Intheheavy-lightquarksector,this subtractingthispartandagainallowingforanonpertur- enablesheavyquarkstoeffectivelycoupletomoreparti- bativeenhancementwiththesamestengthparameterD clesintheheatbath,relativetoashort-rangeforce.Our and energy dependenceas in Eq. (16) we arrive at the preliminarycalculationsoftheHQdiffusioncoefficient form indeed indicate that the latter is comparable to earlier calculationsusingtheinternalenergy[12],whichhasa (E− Σ(R2))2 4 strongerforcebutoverashorterrange.Inthissense,the V (E,r)= Dexp− 2  α T[Φ(Bm r)−1]. potentialobtainedinthepresentstudymaystillimplya I  2(CT)2 3 s D (17) stroTnhgelyabcoovuepelexdmQapGlPe.ofafitisnotunique,mostlydue Toaccommodatethepossibilitythattheimaginarypart touncertaintiesintheimaginaryparts,whichinthefu- of the string interaction can have a different r depen- turewillhavetobecalculatedinaselfconsistentscheme dencethangivenbyΦ(albeitwiththesameasymptotic as laid outin Ref. [26].2 However, severalqualitative, limits),werescaleditsargumentbyaconstantB(which model-independentobservationscan bemade. First, if turnsouttobeclosetoone).Wenotethatnontrivialen- the imaginary parts, (V + Σ(2)) (E), are taken to zero, I ergydependenciesinbothimaginaryparts,VI andΣ(I2), thespectralfunctionσV becomesaδfunctioninenergy are crucial to ensure convergenceof the integration in andthepotentialinthedenominatorisforcedtobecome Eq.(12). In this sense, the potential is required to go thefreeenergy,thusrecoveringtheweak-couplinganal- beyondthestaticapproximation. ysis of Ref. [18]. This underlines the key role of the As an example of our method we perform a fit to imaginarypartsofbothpotentialandsingle-quarkself- a set of free energies computed in lattice QCD with energy in raising the potential above the free energy, Nf = 2 + 1 dynamical flavors [35], for temperatures i.e., rendering a stronger force. The E-depedence of T=1.2, 1.7, 2.2 Tc, with Tc=196MeV [35]. The fol- (V+Σ(2))I(E)(the“widthofthewidth”),encodedinthe lowingparametervalueswereused: TheCoulombDe- widthparameterC,alsoplaysanimportantrole,which bye mass is mD = 1.22gT, close to the expectedHTL reiteratestheneedformicroscopiccalculations,suchas value,whilethestringinteractionisscreenedlesswith theT-matrixapproach.Thedependenceontherealpart ms =0.8gT;forthestringtensionitselfaconstantvalue is more robust and “model independent” in the sense of σ = 1.58GeV/fm turns out to be sufficient, which that it can be largely determinedby the small distance is larger than in vacuum but significantly smaller than behavior of free energy where Φ(x) forces the imagi- in previousfits to the in-mediumfree energy[12, 27]. naryparttobecomesmall. Letusalsocommentonthe For the strong coupling we utilize a weak temperature magnitude of the string tension exceeding its vacuum running,αs = 0.288−0.05(1.2TTc −1),decreasingfrom valuebyabout60%. Thisappearstobearathergeneric 0.288at1.2Tcto0.238at2.4Tc. Forthedimensionless featurewithinthescreenedCornellpotentialansatz. In coefficients we have B = 0.73, which delays the ap- previousworksutilizingthisansatz[12,27],wherethe proachofΦtoitsasymptoticvalueofone,andC=2.07 strength of the string term was additionallyallowed to in Eq. (16), which producesslightly longer tails in the develop a temperature dependence, fits to the free en- energy dependence of the HQ selfenergies than in the ergypreferredstringtensionsofuptoafactor3-4times microscopiccalculationsofRef.[26]. The resulting fit to the lQCD free energies is dis- 2Theimaginarypartsimplicitinourfitarecomparabletowhatwas playedinFig.1,showinggoodagreement.Reproducing foundbeforewiththeinternalenergyaspotential,whichindicatesthat thetemperaturedependenceessentiallyensuresthatthe theyareintheexpectedballparkoftheselfconsistentsolution. 4 Figure1:Ourfit(bluelines)tothecolor-singletQQ¯freeenergiescomputedinlatticeQCD(blackdots)[35]at3differenttemperatures,T=1.2,1.7, 2.2Tcintheleft,middleandrightpanel,respectively. Theredlinesaretherealpartsoftheunderlyingpotential,whilethegreenlinesrepresent thecorrepsondinginternalenergies. thevacuum,albeitwithamuchstrongerscreeningthan (inwhichcaseitwouldbeclosetothefreeenergy),and in ourpresentpotentialfit. Asa result, the in-medium exhibits a rather long-range remnant of the confining free energy never exceeds the vacuum potential (as in force surviving well above T . The associated imagi- c ourpresentwork).Similarfeaturesarealsofoundinap- narypartsarelarge,notinconsistentwithwhatwasob- plicationsoftheDyson-Schwingerapproachtoquarko- tainedearlierusingtheinternalenergyasapotentialap- niaatfiniteT [36]. Apossiblemicroscopicmechanism proximation. First estimates indicate thatthe pertinent for an enhanced string tension could be the release of HQ diffusion coefficient is around D (2πT) ≃ 3 − 5, s magneticmonopoleinthenear-T region,assuggested, suggestiveforastrongcouplingregime. c e.g.,inRef.[37]. Several further investigations are in order to scruti- nizeourinitialestimateswithintheT-matrixapproach. First,thecalculationneedstobecarriedoutselfconsis- 4. ConclusionsandOutlook tently,bycomputingtheheavy-quarkselfenergiesfrom theunderlyingT-matricesandreinsertingtheformerin Utilizing a thermodynamic T-matrix approach, we thelatter[26]. Second,thefreeenergiesshouldbeup- have derived an expression for the static heavy-quark dated with the most recentlattice-QCD results. Third, free energy that directly relates to the underlying in- quarkonium correlators, HQ suscpetibilities and trans- teraction kernel, resummed in ladder approximation. portcoefficientsshouldbecomputedandsystematically The such defined kernel serves as our definition of an compared to lattice data. The role of possible three- in-medium potential, V (r), in the context of a dia- QQ¯ bodycorrelationsmayneedtobeaddressed. Ifthethe- grammaticmany-bodyapproach. Key elementsof this oretical framework passes these tests, applications to framework are a non-trivial energy dependence of the heavy-quarkandquarkoniumphenomenologyarewar- pertinent QQ¯ spectral function, induced by the imagi- ranted. nary parts of both one- and two-body type, i.e., in the single-quark selfenergies and the potential. 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