Real and imaginary-time $Qbar{Q}$ correlators in a

arXiv:0712.4394v1[nucl-th]28Dec2007Realandimaginary-timeQQcorrelatorsinathermalmediumA.Beraudo1,J.P.Blaizot1andC.Ratti1,21ECT*,stradadelleTabarelle286,I-38050Villazzano(Trento),Italy;2DepartmentofPhysicsandAstronomy,SUNYStony-Brook,NY11794.AbstractWeinvestigatethebehaviorofapairofheavyfermions,denotedbyQand¯Q,inahot/densemedium.AlthoughwehaveinmindthesituationwhereQand¯Qdenoteheavyquarks,ourtreatmentwillbelimitedtosimplifiedmodels,whichbearonlysomegeneralsimilaritieswithQCD.Westudyinparticularthelimitingcasewherethemassoftheheavyfermionsisinfinite.Thenanumberofresultscanbederivedexactly:aSchr¨odingerequationcanbeestablishedforthecorrelatoroftheheavyquarks;theinteractioneffectsexponentiate,leadingtoasimpleinstantaneouseffectivepotentialforthisSchr¨odingerequation.WeconsidersimplemodelsforthemediuminwhichtheQ¯Qpairpropagates.Inthecasewherethemediumisaplasmaofphotonsandlightchargedfermions,animaginarypartdevelopsinthiseffectivepotential.Wediscussthephysicalinterpretationofthisimaginarypartintermsofthecollisionsbetweentheheavyparticlesandthelightfermionsofthemedium;thesamecollisionsalsodeterminethedampingrateoftheheavyfermions.Finallywestudytheconnectionbetweenthereal-timepropagatoroftheheavyfermionpairanditsEuclideancounterpart,andshowthattherealpartofthepotentialenteringtheSchr¨odingerequationforthereal-timepropagatoristhefreeenergycalculatedintheimaginary-timeformalism.Keywords:Quarkonium,FinitetemperatureQCD,QuarkGluonPlasma,Mesoncorrelators,Polyakovloopcorrelators,Heavyquarkfree-energies,MesonSpectralFunctionsPACS:11.10.St,11.10.Wx,12.38.Gc,12.38.Mh,21.65.Qr,25.75.Cj,25.75.Nq1IntroductionThepropagationofheavyquarkboundstates,suchastheJ/Ψ,inahotanddensemediumhasbeenthesubjectofmanystudies,sincetheoriginalproposalbyMatsuiandSatz[1]suggestingthatthesuppressedproductionofPreprintsubmittedtoElsevierScience2February2008suchboundstatesinultra-relativisticheavy-ioncollisionscouldbeusedasasignalofdeconfinement.Theinitialsuggestion[1]assumedthatthedominanteffectoftheinteractionwiththemediumisthescreeningoftheQ¯Qinteractionpotential,leadingtoadisappearanceoftheboundstateswhenthescreeningissufficientlystrong.Thescreenedpotentialcanbeobtainedfromequilibriumcalculationsoftheforcebetweentwoinfinitelyheavycolorcharges,aquantitywhichcan,inprinciple,beextractedfromthelatticedataforthecorrelatoroftwoPolyakovlines.SuchacorrelatorisinfactrelatedtothechangeoffreeenergyoccurringwhenaheavyQQpairisaddedintoahotQCDmedium[2],andtherelationbetweenthisfree-energyandtheeffectivepotentialtobeinsertedinaScr¨odingerequation(orinaT-matrixcalculation,likein[3,4])remainsanopenquestion.Thus,theeffectivepotentialhasbeenidentifiedalternativelywiththeQQfreeenergy[5–8],theinternalenergy[9,10]orevenwithalinearcombinationofboth[11–13],leadingtodifferentdissociationtemperaturesfortheJ/Ψ,dependingonthechoicemade.Inanotherlineofinvestigationoneconsidersthespectralfunctionsofac¯candb¯bmesonsinahotenvironment[14–18].Suchobjectscontaininprinciplethefullphysicalinformationonthepropertiesofthemesonicexcitationsinamedium,andarealsodirectlyrelatedtoobservablequantities,likethedileptonproductionrateinthevectorchannel.Unfortunatelyspectralfunctionsarenotdirectlymeasuredinalatticesimulation,buthavetobereconstructedfromEuclideancorrelators,andthisreconstructionisplaguedwithlargeuncer-tainties.Togoaroundsuchdifficulties,onemayattempttocomparedirectlythespectralfunctionsandthecorrespondingimaginary-timecorrelatorsthatareobtainedfromlatticecalculationstothosethatcanbecalculatedfromaSchr¨odingerequationwithascreenedpotential.Suchstudieswerecarriedoutrecently[4,13,19–22].Morerecently,stillanotherapproachwasdeveloped,inwhichthefocusisonthereal-timepropagatorofaheavy-quarkpair.InRefs.[23–27],itwasarguedthatsuchapropagatorobeysaSchr¨odingerequationwithaneffectivepotentialwhichwasderivedfirstperturbatively,andthenestimatedinclassicallatticegaugesimulations.Aninterestingobservationmadein[23]isthatthiseffectivepotentialmaycontainanimaginarypartthatmaystronglyaffectthecalculatedspectralfunction.Inthispaperweshalldiscussanumberofissuestriggeredbythelatterstud-ies.Weshallexamineunderwhichconditions,thein-mediumQQ-propagatorobeysaSchr¨odingerequation.Todoso,weshallconsiderthegeneralequationofmotionsatisfiedbytheheavyfermionpaircorrelator.Thisispartofaninfi-nitehierarchyofcoupledequationsforthen-pointfunctions.However,inthelimitwherethemassoftheheavyfermionisinfinite,theequationfortheQQcorrelatordecouplesandreducestoaclosed,Schr¨odingerequation.WeshallalsoestablishalinkbeweentheeffectivepotentialenteringthisSchr¨odinger2equationandtheheavy-quarkfree-energythatcanbeextractedfromlatticedata;inotherwords,weshallinvestigatetherelationsbetweenthereal-timeandimaginary-timecorrelators,andunderstandtheoriginoftheimaginarypartfoundin[23–27].Thediscussionwillbegeneralandpedagogical:ourpurposeistoillustratesimpleandgeneralphysicaleffects,ratherthantoobtainquantitativeresults.Weshallfocusonaspecificcorrelator,whichisdefinedinSec.2,wherewealsorecallbasicpropertiesofrealandimaginarytimecorrelatorsandtheirrelationthrou