Effective theories for real-time correlations in h

arXiv:hep-ph/9709449v225Sep1997UW/PT-97-22Effectivetheoriesforreal-timecorrelationsinhotplasmasPeterArnoldDepartmentofPhysics,UniversityofVirginia,Charlottesville,VA22901LaurenceG.YaffeDepartmentofPhysics,UniversityofWashington,Seattle,Washington98195(September1997)AbstractWediscussthesequenceofeffectivetheoriesneededtounderstandthequali-tative,andquantitative,behaviorofreal-timecorrelatorshA(t)A(0)iinultra-relativisticplasmas.WeanalyzeindetailthecasewhereAisagauge-invariantconservedcurrent.Thiscaseisofinterestbecauseitincludesacorrelationrecentlymeasuredinlatticesimulationsofclassical,hot,SU(2)-Higgsgaugetheory.Wefindthatsimpleperturbationtheory,freekinetictheory,lin-earizedkinetictheory,andhydrodynamicsareallneededtounderstandthecorrelationfordifferentrangesoftime.Weemphasizehowcorrelationsgener-icallyhavepower-lawdecaysatverylargetimesduetonon-linearcouplingstolong-livedhydrodynamicmodes.I.INTRODUCTIONThereisarichvarietyofspatialandtemporalscalesinhot,weaklycoupled,ultra-relativisticgaugetheories,aselectionofwhichareshowninTableI.Dependingontheproblemofinterest,awidevarietyofmethodscanbeappropriatefordescribingthephysicsondifferentscales:naiveperturbationtheory,hardthermalloops,dimensionalreduction,kinetictheory,Vlasovequations,Langevinequations,hydrodynamics,andmore.Forstaticpropertiesofplasmas,thetowerofeffectivetheoriesrelevanttodifferentdistancescalesiswellunderstood[1].TheseeffectivetheoriesareallEuclideanquantumfieldtheories,in3or4dimensions,andmaybesystematicallyconstructedusingstandardrenormalizationgrouptechniques.Forreal-timeproperties,includingtheanalysisofnon-equilibriumresponse,construct-inganappropriatesequenceofeffectivetheorieswhichcorrectlydisentanglesdynamicsondifferentdistance(ortime)scalesismuchlessstraightforward.Someofthedifficultiesareillustratedbythelongconfusionoveroneofthebasictimescalesofhotnon-Abelianplasmas:thetimescalefornon-perturbativelylargefluctuationsingaugefields(suchasthoseresponsibleforbaryonnumberviolationinelectroweaktheory)[2].Toelucidatethe1real-timedynamicsofhotgaugetheories,oneshouldunderstandthetowerofeffectivethe-oriesappropriatefordifferenttimescalesand,atleastinsimpleexamples,calculatetheappropriatematchingbetweensuchtheories.typicalparticlewavelengthT−1typicalparticleseparationT−1Debyelength(gT)−1plasmafrequency(gT)−1non-perturbativemagneticlengthscale(g2T)−1(non-Abeliantheories)meanfreepath:any-anglescattering(g2Tln)−1meanfreepath:large-anglescattering(g4Tln)−1temporalscalefortopologicaltransitions(g4T)−1(non-Abeliantheories)TABLEI.Ordersofmagnitudeofvariousdistanceandtimescalesinhot,ultrarelativistic,gaugetheoryforweakcoupling.Here,andhenceforth,“ln”standsforln(g−2),exceptforthecaseofthemeanfreepathforany-anglescatteringinanAbeliangaugetheory,whereitisagenuinelogarithmicinfrareddivergence.(Naturally,c≡¯h≡1.)Thecurrentinterestinhotelectroweakbaryonnumberviolationhasmotivatedsubstan-tialeffortsinnumericalsimulationsofhot,classical,latticegaugetheory[3–6].Forcertaintimeanddistancescales,ithasbeenarguedthatclassicalsimulationscanadequatelyre-producethebehavioroftheunderlyingquantumplasmaofinterest(ormoregenerally,thatcertainclassicallatticeresultscanbetranslatedintocorrespondingresultsforthecontin-uumquantumtheory[7].)Hence,thesesimulationsareasourceof“data”forcomparisonwiththeoreticalpicturesofhotgaugetheories.Inparticular,therehavebeenrecentmea-surementsbyTangandSmit[6]oftime-dependentcorrelationshA(t)A(0)iforseveralgaugeinvariantoperatorsA.Aninterestingtheoreticalgoalisthentounderstandhowbesttodescribethephysicsof,andcalculatethebehaviorof,suchtime-dependentcorrelatorsfordifferentrangesoftime.Inordertokeepthediscussionfocussed,weshallspecializetotheparticularclassofcorrelationswheretheoperatorAisthespatialpartjofaconservedgauge-invariantcurrentjμ(suchastheelectricchargecurrentinQED,orfermionnumbercurrentinQCD).Asweshallexplainbelow,oneofthecorrelationsmeasuredbyTangandSmitfallsintothisclass.Weshallalsofocusonthespatialaverageofsuchcurrents,A(t)=j(t)≡V−1Zd3xj(x,t),(1.1)asdidTangandSmit.However,theprinciplesandtechniquesunderlyingourdiscussionarenotlimitedtothischosenclassofexamples.Asourinterestisinthedynamicsofhotgaugetheories,weshallassumethatthecurrentjμinvolvesparticleswhichfeelagaugeforcewithcouplingg.Wealsoassumethatthereisnochemicalpotential,sothatthetotalchargeRd3xj0haszeroexpectation.Fig.1qualitativelydepictsourbasicresultsforthetimedependenceofthecurrent-currentcorrelator,andthephysicalpicturesbestsuitedtounderstandingthedynamicsatdifferenttimescales.Thepurposeofthispaperwillbetoexplainthisgraph.Wehaveassumedthatthegaugesymmetry,andthesymmetrycorrespondingtothecurrent,are23/234Tg(lnlinearizedkinetictheoryperturbationtheoryfreekinetic0hydrodynamicskinetictheory)14Tgln1T4Tg1exponentialdecaytimeRej(t)j(0)Vpower-lawdecayT3T32t-3/2FIG.1.Aqualitativepictureoftherealpartofthecurrent-currentcorrelationVhj(t)j(0)iinahotplasma,forthecaseofbosonicchargecarriers.Thelabelsontheaxesareonlymeanttodenoteordersofmagnitude,exceptthatthe“2T3”markisinfacttwicethe“T3”markinthesmall-cou