Wilson Renormalization Group formulation of Real T

arXiv:hep-ph/9601375v226Apr1996CERN-TH/96-23SHEP96-05WilsonRenormalizationGroupformulationofRealTimethermalfieldtheoriesM.D’AttanasioDepartmentofPhysics,UniversityofSouthampton,UnitedKingdomandINFNGruppocollegatodiParma,ItalyM.PietroniTheoryDivision,CERNCH-1211Geneva23,SwitzerlandAbstractWeapplyRenormalizationGrouptechniquestotheRealTimeformulationofthermalfieldtheory.DuetotheseparationbetweentheT=0andtheT6=0partsofthepropagatorinthisformalism,onecanderiveexactevolutionequationsfortheGreenfunctionsdescribingtheeffectofintegratingoutthermalfluctuationsofincreasingwavelengths,theinitialconditionsbeingtherenormalizedGreenfunctionsoftheT=0theory.Asafirstapplication,westudythephasetransitionfortherealscalartheory,computingtheorderofthetransition,thecriticaltemperature,andcriticalexponents,indifferentapproximationstotheevolutionequationsforthescalarpotential.CERN-TH/96-23February19961IntroductionThedynamicsofasecondorder,orweaklyfirstorder,phasetransitionisgovernedbylongwavelengthfluctuationsoftheorderparameter.Thesefluctuations,whoselengthscaleismuchlargerthantheinversetemperatureofthesystem,areessentiallyclassical,sincetheprobabilityofaquantumfluctuationoversuchscalesishighlysuppressed.Forthisreason,thedetailsofthemicroscopictheoryarenotrelevanttothedescriptionofthecriticalbehaviourofthesystem,whichcanbesuccessfullystudiedbyusingclassicalmodels,suchastheIsingmodelforferromagnetsortheGinsburg-Landautheoryforsuperconductors.Inthesemodels,afreeenergycanbedefinedasafunctionalofamacroscopicorderparameter,whichisallowedtovaryonlyonlargelengthscales.Theeffectoftheshortwave-length(quantumandthermal)fluctuationsisincorporatedbytheparametersappearinginthefreeenergy(masses,couplingconstants,etc.),whichareusuallytreatedasphenomeno-logicalparametersbutshouldinprinciplebecomputable,startingfromtheunderlyingtheory.TheWilsonRenormalizationGroup(RG)[1]providesthenaturalframeworkinwhichthisprocedurecanbesystematicallycarriedout.Themainideaistostartfromamicro-scopictheory,theparametersofwhicharesupposedtobeknown,andthenprogressivelyintegrateoutthehighfrequencymodesoftheorderparameterdowntosomeinfra-redcutoffΛ.Inthiswayoneobtainsacoarse-grainedorderparameterandthecorrespondingeffectiveaction,whichcanbeusedastherelevanttooltodescribethesystem.Inthispaper,wewillapplythisideatoquantumfieldtheoryatfinitetemperature.Ourapproachwillbethefollowing.First,wewillassumethatwe“know”thezero-temperaturerenormalizedquantumfieldtheory,whichmeansthatthereexistssomereli-ableapproximationmethod(perturbationtheory,latticesimulations,etc.)tocomputethezero-temperaturerenormalizedGreenfunctions.Inthisway,therenormalizationconstantscanbefixedbytheexperimentalmeasurementsandalltheparametersofthetheoryareknown.InaRGlanguage,weassumethatallthequantumfluctuationshavealreadybeenintegratedout.Second,wewillintegrateoutthermalmodesonly,forfrequencieshigherthanthein-fraredcutoffΛ.Fornon-vanishingvaluesforΛ,wewillobtaincoarse-grainedorderpa-rameterandfreeenergy,whicharetheappropriateobjectstostudythedynamicsoflongwavelengththermalfluctuations.InthelimitΛ→0,wewillobtainthefinitetemperaturequantumfieldtheoryinthermalequilibrium.Inthisapproach,sinceallthequantumfluctuationsareintegratedoutfromthebegin-ning,itwillbepossibletorelatethethermalfieldtheoryandthephysical(renormalized)quantumfieldtheoryatzerotemperatureinatransparentway.Theidealframeworktoperformthisprogramofcoarse-grainingofthermalfluctuationsistheRealTime(RT)formulationofthermalfieldtheories[2],inwhichthethermalpartinthefreepropagatorsiswellseparatedfromthezero-temperaturequantisticone.Asiswellknown,thepricetopayforthisisadoublingofthenumberofdegreesoffreedom.These“ghost”fieldsarenecessarytocanceltheso-calledpinchsingularities,whichareduetothe1factthatthethermalfluctuationsareon-shell.BecauseofthesetechnicalcomplicationsMatsubara’sImaginaryTime(IT)approach[3]ismorepopularintheliterature.Inparticular,thecoarse-grainingprocedurecanbepreciselyformulatedintheso-calledClosed-Time-Path(CTP)formalism[4](seerefs.[5,6]foraclearpresentation).Indeed,thedescriptionofasysteminwhichonlytheshortwavelengthfluctuationsareinthermalequilibriumcanbeachievedbymodifyingthedensitymatrixwithrespecttothethermalone,andtheCTPwasdesignedjusttodescribesystemswithagenericdensitymatrix.Anyway,thereadernotfamiliarwiththeCTPformalismshouldnotworrytoomuchaboutit.AswewilldiscussinAppendixA,themodificationofthedensitymatrixthatwewillconsiderisequivalenttoworkinginthemorefamiliarRTformulationofNiemiandSe-menoff[2,7]withamodifieddistributionfunction,givenbytheBose-Einsteindistributionfunction(ortheFermi-Dirac,forfermions)multipliedbythecutofffunction.Asafirststageofthisprogram,inthispaperweillustrateourmethodbyconsideringthewell-studiedself-interactingrealscalartheory.Asiswellknown,thismodelbelongstotheuniversalityclassoftheIsingmodel,andthenithasasecondorderphasetransition.However,perturbationtheoryfailstoreproducethisresultevenaftertheresummationofdaisyandsuper-daisydiagrams[8],unlessthegapequationsaresolvedtoO(λ2),λbeingthequarticcouplingconstant[9].Wederivethe“exact”WilsonRG