Equilibrium and Non-Equilibrium Hard Thermal Loop

arXiv:hep-ph/9708363v224Aug1998EquilibriumandNon-EquilibriumHardThermalLoopResummationintheRealTimeFormalism∗MargaretE.Carrington1,HouDefu2,andMarkusH.Thoma3,4†1DepartmentofPhysics,BrandonUniversity,Brandon,Manitoba,CanadaR7A6A92Institutf¨urTheoretischePhysik,Universit¨atRegensburg,D-93040Regensburg,Germany3Institutf¨urTheoretischePhysik,Universit¨atGiessen,D-35392Giessen,Germany4EuropeanCenterforTheoreticalStudiesinNuclearPhysicsandRelatedAreas,VillaTambosi,StradadelleTabarelle286,I-38050Villazzano,Italy(February1,2008)AbstractWeinvestigatetheuseofthehardthermalloop(HTL)resummationtech-niqueinnon-equilibriumfieldtheory.WeusetheKeldyshrepresentationoftherealtimeformalism(RTF).WederivetheHTLphotonselfenergyandtheresummedphotonpropagator.Weshowthatnopinchsingularitiesappearinthenon-equilibriumHTLeffectivepropagator.Wediscussapossibleregular-izationmechanismforthesesingularitiesathigherorders.AsanexampleoftheapplicationoftheHTLresummationmethodwithintheRTFwediscussthedampingrateofahardelectron.PACSnumbers:11.10.Wx,12.38.Cy,12.38.MhTypesetusingREVTEX∗SupportedbyBMBF,GSIDarmstadt,andDFG†Heisenbergfellow1I.INTRODUCTIONPerturbationtheoryforgaugetheoriesatfinitetemperaturesuffersfrominfraredsin-gularitiesandgaugedependentresultsforphysicalquantities.Theseproblemsareavoidedbyusinganeffectiveperturbationtheory(Braaten-Pisarskimethod[1])whichisbasedontheresummationofhardthermalloop(HTL)diagramsintoeffectiveGreenfunctions.Thispowerfulmethodwasderivedwithintheimaginarytimeformalism(ITF).UsingresummedGreenfunctions,mediumeffectsoftheheatbath,suchasDebyescreening,collectiveplasmamodes,andLandaudamping,aretakenintoaccount.TheHTLresummationtechniquehasbeenappliedtoanumberofinterestingproblems,inparticulartothepredictionofsigna-turesandpropertiesofaquark-gluonplasma(QGP)expectedtobeproducedinrelativisticheavyioncollisions(forareviewsee[2]).However,theuseofthermalfieldtheoriesfordescribingaQGPinnucleus-nucleuscol-lisionsisrestrictedbythefactthatatleasttheearlystageofsuchacollisionleadstoafireball,whichisnotinequilibrium.Itisnotclearifacompletethermalandchemicalequi-libriumwillbeachievedlateron.Hence,non-equilibriumeffectsinapartongasshouldbeconsideredforpredictingsignaturesofQGPformationandforobtainingaconsistentpictureofthefireball.Thiscanbedoneinthecaseofachemicallynon-equilibratedpartongasbymeansofrateequations[3]ormoregenerallybyusingtransportmodels[4].However,theseapproachesarebasedonasemiclassicalapproximation.Inparticular,infrareddivergenceshavetoberemovedphenomenologically.ThereforeitisdesirabletoderiveaGreenfunc-tionapproachincludingmediumeffectsasinthecaseoftheHTLresummation.ForthispurposeonehastoabandontheITF,whichisrestrictedtoequilibriumsituations.Therealtimeformalism(RTF),ontheotherhand,canbeextendedtoinvestigatenon-equilibriumsystems[5,6].TheRTFinvolveschoosingacontourinthecomplexenergyplanewhichfulfillstheKubo-Martin-Schwingerboundaryconditionandcontainstherealaxis[5].Thisleadstopropagatorsandselfenergieswhicharegivenby2×2matrices.Thechoiceofthecontour2isnotunique.WewilladopttheKeldyshorclosedtimepathcontour,whichwasinventedforthenon-equilibriumcase[5].Inparticular,wewilldemonstratetheusefulnessoftheKeldyshrepresentation[7]basedonadvancedandretardedpropagatorsandselfenergiesandshowhowpotentiallydangerousterms(pinchsingularities)[8]innon-equilibriumaretreatedeasilywithinthisrepresentation.InthenextsectionwereviewtheKeldyshrepresentation.Insection3,wediscusstheequilibriumcalculation.WeconsiderQEDandgivetheresultsoftherealtimecalculation,intheHTLapproximation,ofthephotonselfenergy,theresummedphotonpropagator,andtheelectrondampingrate.Theresultsare,ofcourse,identicaltothoseoftheITF,whichdemonstratesthatalthoughtheHTLresummationschemewasderivedwithintheITF,theresultisindependentofthechoiceofcontour.Insection4,weextendtheHTLresummationtechniquetooff-equilibriumsituationsbyfollowingtheequilibriumcalculationsoutlinedinsection3.Weshowthatnopinchsingularitiesappearinthenon-equilibriumHTLeffectivepropagator.II.KELDYSHREPRESENTATIONInthissectionwereviewtheKeldyshrepresentationoftheRTF.Thebarepropagatorforbosonsreads[6]D(K)=1K2−m2+iǫ00−1K2−m2−iǫ−2πiδ(K2−m2)nB(k0)θ(−k0)+nB(k0)θ(k0)+nB(k0)nB(k0),(1)whereK=(k0,k),k=|k|,θdenotesthestepfunction,andthedistributionfunctionisgivenbynB(k0)=1/[exp(|k0|/T)−1]intheequilibriumcase.ForfermionsthebarepropagatorcanbewrittenasS(K)=(K/+m)1K2−m2+iǫ00−1K2−m2−iǫ3+2πiδ(K2−m2)nF(k0)−θ(−k0)+nF(k0)−θ(k0)+nF(k0)nF(k0),(2)wheretheFermidistributionisgivenbynF(k0)=1/[exp(|k0|)+1]inequilibrium.Thecomponentsofthesepropagatorsarenotindependent,butfulfilltherelationG11−G12−G21+G22=0,(3)whereGstandsforDorS.Byanorthogonaltransformationofthese2×2matriceswearriveatarepresentationofthepropagatorsintermsofadvancedandretardedpropagatorswhichwasfirstintroducedbyKeldysh[7].Thethreeindependentcomponentsofthisrepresentationaredefinedas[6]GR=G11−G12,GA=G11−G21,GF=G11+G22.(4)TheinvertedrelationsreadG11=12(GF+GA+GR),G12=12(GF+GA−GR),G21=12(GF−GA+GR),G22=12(GF−GA−GR).(5)Similarrelationsto(3)and(4)h