Dynamics near the critical point the hot renormali

arXiv:hep-ph/0110012v231Jan2002Dynamicsnearthecriticalpoint:thehotrenormalizationgroupinquantumfieldtheoryD.Boyanovsky1,∗andH.J.deVega2,†1DepartmentofPhysicsandAstronomy,UniversityofPittsburgh,Pittsburgh,Pennsylvania15260,USA2LPTHE,Universit´ePierreetMarieCurie(ParisVI)etDenisDiderot(ParisVII),Tour16,1er.´etage,4,PlaceJussieu,75252Paris,Cedex05,France(Dated:February1,2008)Theperturbativeapproachtothedescriptionoflongwavelengthexcitationsathightemperaturebreaksdownnearthecriticalpointofasecondorderphasetransition.Westudythedynamicsoftheseexcitationsinarelativisticscalarfieldtheoryatandnearthecriticalpointviaarenormaliza-tiongroupapproachathightemperatureandanǫexpansionind=5−ǫspace-timedimensions.Thelongwavelengthphysicsisdeterminedbyanon-trivialfixedpointoftherenormalizationgroup.Atthecriticalpointwefindthatthedispersionrelationandwidthofquasiparticlesofmomentumpisωp∼pzandΓp∼(z−1)ωprespectively,thegroupvelocityofquasiparticlesvg∼pz−1vanishesinthelongwavelengthlimitatthecriticalpoint.AwayfromthecriticalpointforT&Tcwefindωp∼ξ−z1+(pξ)2z12andΓp∼(z−1)ωp(pξ)2z1+(pξ)2zwithξthefinitetemperaturecorrelationlengthξ∝|T−Tc|−ν.Thenewdynamicalexponentzresultsfromanisotropicrenormalizationinthespa-tialandtimedirections.ForatheorywithO(N)symmetrywefindz=1+ǫN+2(N+8)2+O(ǫ2).Thisdynamicalcriticalexponentdescribesanewuniversalityclassfordynamicalcriticalphenomenainquantumfieldtheory.Criticalslowingdown,i.e,avanishingwidthinthelong-wavelengthlimit,andthevalidityofthequasiparticlepictureemergenaturallyfromthisanalysis.PACSnumbers:11.10.Gh,11.10.Wx,64.60.Ak,05.10.Cc,11.10.-zI.INTRODUCTIONTheexperimentalpossibilityofstudyingthephasetransitionsofQCDviaultrarelativisticheavyioncollisionswiththecurrenteffortatRHICandtheforthcomingprogramatLHCmotivatesatheoreticalefforttounderstandthedynamicalaspectsofphasetransitionsathightemperature.QCDisconjecturedtofeaturetwophasetransitions,theconfinement-deconfinement(orhadronization)andthechiralphasetransitions.Detailedlatticestudies[1]seemtopredictthatbothtransitionsoccurataboutthesametemperatureTc∼170Mev.Whilelatticegaugetheoriesfurnishanon-perturbativetooltostudythethermodynamicequilibriumaspectsofthetransitionthedynamicalaspectscannotbeaccessedwiththisapproach.Inacondensedmatterexperimentthetemperatureistypicallyacontrolparameteranditcanbevariedsufficientlyslowsoastoensurethataphasetransitionoccursinlocalthermodynamicequilibrium.Inanultrarelativisticheavyioncollisionthecurrenttheoreticalunderstandingsuggeststhatathermalizedquark-gluonplasmamaybeformedatatimescaleoforder1fm/cwithatemperaturelargerthanthecritical.Thisquark-gluonplasmathenexpandshydrodynamicallyandcoolsalmostadiabatically,thetemperaturefallingoffasapoweroftimeT(t)∼Ti(ti/t)1/3untilthetransitiontemperatureisreachedatatimescale∼10−50fm/cdependingontheinitialtemperature[2].Whetherthephasetransitionoccursinlocalthermodynamicequilibriumornotdependsontheratioofthecoolingtimescaletcool∼T(t)/˙T(t)totherelaxationorthermalizationtimescaleofafluctuationofagivenwavelengthp−1,trel(p).Iftcool≫trelthenthefluctuationrelaxesintimescalesmuchshorterthanthatofthetemperaturevariationandreacheslocalthermodynamicequilibrium.If,ontheotherhand,tcool≪trelthefluctuationdoesnothavetimetorelaxtolocalthermodynamicequilibriumandfreezesout.Forthesefluctuationsthephasetransitionoccursveryfastandoutofequilibrium.Thusanimportantdynamicalaspectistounderstandtherelaxationtimescalesforfluctuations.Alargebodyoftheoretical,experimentalandnumericalworkincondensedmatterphysicsrevealthatwhiletyp-icallyshortwavelength(p≫T)fluctuationsreachlocalthermalequilibrium,nearacriticalpointlongwavelengthfluctuationsrelaxveryslowly,andundergocriticalslowingdown[3,4].Aphenomenologicaldescriptionofthedynam-icsnearaphasetransitiontypicallyhingesonthetime-dependentLandau-Ginzburgequationwhichisgeneralized∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]2toincludeconservationlaws[3,4].Inthesimplestcaseofanon-conservedorderparameter,suchasinascalarfieldtheorywithdiscrete(Ising-like)symmetry,thetimedependentLandauGinzburgequationispurelydissipative.Whilephenomenological,thisapproachhasprovedverysuccesfulinavarietyofexperimentalsituationsandislikelytoprovideasuitabledescriptionofthedynamicsformacroscopic,coarse-grainedsystemssuchasbinarymixtures,etc.[3].ThephenomenologicalapproachbasedonthetimedependentLandau-Ginzburgequationswhicharefirstorderintimederivativesseemtoprovideasuitabledescriptionofcoarse-grainedmacroscopicdynamicsinnon-relativisticsystems.Howeveritisclearthatsuchapproachisnotjustifiedinarelativisticquantumfieldtheory,sincetheunderlyingequationsofmotionaresecondorderandtimereversalinvariant.InparticularinthecaseofpurelydissipativetimedependentLandauGinsburg(phenomenological)description[3,4],frequencyandmomentaenterwithdifferentpowersinthepropagators,andatthemeanfield(ortree)level,thisresultsinadynamicalscalingexponentz=2.Thissituationmustbecontrastedtothatofarelativisticquantumfieldtheorywhereattreelevel(meanfield)frequenciesandmomentaenterwiththesamepowerinthepropagatorleadingtoa