Real-time pion propagation in finite-temperature Q

arXiv:hep-ph/0204226v211Oct2002Real-timepionpropagationinfinite-temperatureQCDD.T.Son1,∗andM.A.Stephanov2,3,†1InstituteforNuclearTheory,UniversityofWashington,Seattle,Washington98195-15502DepartmentofPhysics,UniversityofIllinois,Chicago,Illinois60607-70593RIKEN-BNLResearchCenter,BrookhavenNationalLaboratory,Upton,NewYork11973(Dated:April2002)AbstractWearguethatinQCDnearthechirallimit,atalltemperaturesbelowthechiralphasetransi-tion,thedispersionrelationofsoftpionscanbeexpressedentirelyintermsofthreetemperature-dependentquantities:thepionscreeningmass,apiondecayconstant,andtheaxialisospinsus-ceptibility.Thedefinitionsofthesequantitiesaregivenintermsofequal-time(static)correlationfunctions.Thus,allthreequantitiescanbedetermineddirectlybylatticemethods.TheprecisemeaningoftheGell-Mann–Oakes–Rennerrelationatfinitetemperatureisgiven.PACSnumbers:11.10.Wx,11.30.RdKeywords:Finite-temperaturefieldtheory,chiralsymmetries∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]hadronsathightemperaturesanddensitiesareofgreatinterestfrombothexperimentalandtheoreticalperspectives.Onemotivationforstudyingtemperatureef-fectsonhadronscomesfromthesuggestionthatsomefeaturesofthedileptonspectrumobservedinheavy-ioncollisionscanbeexplainedbythemodificationofmassesandwidthsofmesonsbythethermalmedium[1].Nevertheless,reliableinformationonthetemper-aturemodificationofhadronicpropertiesisstilllacking.Latticesimulations,whichrelyontheimaginary-timeformulationofquantumfieldtheory,haveseriousdifficultieswithreal-timequantities.1TheabsenceofLorentzinvarianceatfinitetemperatureimpliesthatthereisnodirectrelationshipbetweenreal-timecharacteristicsofhadrons(forexample,theso-called“polemasses,”whicharesupposedlythepositionsofpolesinpropagators)andquantitiesthatcanbeextractedfromEuclideanpropagators(e.g.,the“screeningmasses,”whichcharacterizetheexponentialfalloffofstaticEuclideancorrelators).Thus,asarule,latticemeasurementsofcorrelationfunctionsatfinitetemperaturecannotbeusedtodrawconclusionsaboutreal-timepropagationofhadrons.Theaimofthispaperistodemonstratethatpionspresentanexceptiontothisrule.Weshallarguethatitispossibletodeterminethedispersionrelationofsoftpions(moreprecisely,itsrealpart)atalltemperaturesbelowthechiralphasetransition,knowingonlyequal-time(orstatic)correlationfunctions,which,inprinciple,canbedeterminedonthelattice.ItshouldbeemphasizedthatwedonotassumethetemperatureTtobesmallcomparedtothechiralphasetransitiontemperatureTc:wemusthaveTTc,butT/Tcisallowedtobeoforder1.OurresultsarevalidinthenontrivialregimewhereneitherperturbativeQCDnorchiralperturbationtheoryarereliable.Moreover,ourmethod,withminimalmodification,canbeappliedtoanyfieldtheorywithabrokensymmetry,attemperaturesbelowsymmetryrestoration.Thatthedispersionrelationofamodecanbeexpressedfullyintermsofstaticcorrelationfunctionsisnontrivial,butbynomeansunprecedented.Werecallthesoundwaves,whosevelocityisu=(∂p/∂ǫ)1/2,wherepandǫarethepressureandtheenergydensity,respectively.Thesoundspeed,whilebeingareal-timequantity(thepoleinthecorrelatoroftheenergydensityT00),canbedeterminedsolelyfromthermodynamics.Therelationbetweenthespeedofsoundandthethermodynamicfunctionsisanexactconsequenceoftheexistenceofthehydrodynamicdescription.AlessfamiliarexampleisthevariationalFeynman-Bijlformulawhichrelatesthephononspectruminsuperfluidheliumtothestaticdensity-densitycorrelationfunction[3].Theexamplemostcloselyrelatedtoourproblem,however,isthatofspinwavesinantiferromagnets[4]:thevelocityofspinwavesatanytemperaturebelowphasetransitionisequaltotheratioofthestiffnessandthemagneticsusceptibilityinadirectionperpendiculartomagnetization.Bothquantitiescanbedefinedfromthestaticresponse1Someprogress,however,mayhavebeenmaderecently[2].2ofthesystemtoexternalfields.TheonlydifferencebetweenQCDandantiferromagnetsisthatintheformercasethesymmetryisSU(2)V×SU(2)A≃O(4),whichisspontaneouslybrokentoSU(2)V≃O(3),whileinferromagnetsO(3)isbrokendowntoO(2)[5,6].Thepaperisconstructedasfollows.InSec.IIwesummarizethefindingsofthepaper.InSec.IIIweusesimple,butnonrigorous,argumentsrelyingonaneffectiveLagrangiantounderstandtheseresults.InSec.IVtheresultsarederivedinamorerigorouswayfromasetofassumptionsaboutthereal-timecorrelationfunctions,whichcomesfromhydrodynamics.InSec.Vweshowthatourresultholdsforthesimplestfield-theoreticalmodelofascalarfieldtheorywithbrokensymmetry.IntheAppendixwegiveasimplederivationoftheknownresultaboutthedynamicalcriticalexponentz,andderivethecriticalscalingofadiffusioncoefficient.II.SUMMARYOFRESULTSWeclaimthat,inQCDwithtwolightflavors,attemperaturesbelowthechiralphasetransition,therealpartofthedispersionrelationofsufficientlysoftpionsisgivenbythefollowingequation:ω2p=u2(p2+m2).(2.1)Inthispaperthefollowingterminologyisused:uisthepionvelocity(although,strictlyspeaking,itisthepionvelocityonlywhenm=0),andmisthepionscreeningmass(weshallshowthatitisthesamescreeningmassasdefinedonthelattice).Theenergyofapionatp=0,mp=umiscalledthepionpolemass.Atfinitetemperature,themeaningofsoftpionsmayneedsomeclarifi