Vacuum decay in quantum field theory

arXiv:hep-ph/0106091v18Jun2001VacuumdecayinquantumfieldtheoryEstebanCalzettaDepartamentodeF´ısica,UniversidaddeBuenosAires,CiudadUniversitaria,1428BuenosAires,ArgentinaAlbertRouraandEnricVerdaguer∗DepartamentdeF´ısicaFonamental,UniversitatdeBarcelona,Av.Diagonal647,08028Barcelona,SpainWestudythecontributiontovacuumdecayinfieldtheoryduetotheinteractionbetweenthelongandshort-wavelengthmodesofthefield.ThefieldmodelconsideredconsistsofascalarfieldofmassMwithacubicterminthepotential.Thedynamicsofthelong-wavelengthmodesbecomesdiffusiveinthisinteraction.ThediffusivebehaviourisdescribedbythereducedWignerfunctionthatcharacterizesthestateofthelong-wavelengthmodes.ThisfunctionisobtainedfromthewholeWignerfunctionbyintegrationofthedegreesoffreedomoftheshort-wavelengthmodes.ThedynamicalequationforthereducedWignerfunctionbecomesakindofFokker-Planckequationwhichissolvedwithsuitableboundaryconditionsenforcinganinitialmetastablevacuumstatetrappedinthepotentialwell.Asaresultafiniteactivationrateisfound,evenatzerotemperature,fortheformationoftruevacuumbubblesofsizeM−1.Thiseffectmakesasubstantialcontributiontothetotaldecayrate.I.INTRODUCTIONInthispaperwereportourpreliminaryfindingswithinalargerprogramwhichaimsatthedevelopmentofatheoryofnonequilibriumfirstorderphasetransitions,suchashaveoccurredintheEarlyUniverse(grandunifiedandelectroweaksymmetrybreaking[1])and,possibly,inthefirststagesofheavyioncollisions(chiralsymmetrybreakingandconfinement[2,3]).Forthisreason,wemustseekadescriptionofthedecayprocesswhichemphasizesthedynamicalaspects,overthestaticaspectsencodedintheeffectivepotential.Vacuumdecayinfieldtheoryisdescribedwithapotentialwhichdisplaysalocalminimum,separatedfromtheabsoluteminimumbyapotentialbarrier.Asystempreparedinthefalsevacuumstate(metastablephase)withinthepotentialwellmaydecayinessentiallytwodifferentways,namely(a)bytunnelingeffect,thatis,bygoingthroughthebarrierinaclassicallyforbiddentrajectory[4–6],orelse,(b)byactivation,thatis,byjumpingabovethebarrier[7,8].Ineithercase,thedecayprobabilityfollowsthelawP∼Aexp(−B)whichgivestheprobabilityperunittimeandunitvolumetonucleatearegionofthestablephasewithinthemetastablephase.Inthetunnelingeffect,B=S/¯h,where¯hisPlanck’sconstantandSistheactionforthetrajectorywhichgoesunderthebarrierinimaginarytime[9].Inthermalactivation,B=Vs/(kBT),wherekBisBoltzman’sconstant,Tthetemperature,andVsistheheightofthefreeenergymeasuredfromthefalsevacuum[10–12].Thus,activationdisappearsasT→0.Insystemswithfewdegreesoffreedom,theremustbeanexternalagent,typicallyathermalsource,foractivationtobepossible.Ourthesisisthatinfieldtheoriesthereisaphenomenonsimilartoactivationevenatzerotemperature.Thiscomesfromtheobservationthatinafieldtheory,whenamodedecompositionofthefieldismade,thereareonlyafewlongwavelenghtsmodeswhichareunstableanddecay.Thesenearlyhomogeneousmodesmayberegardedasanopensystem,whichinteractswiththeenvironmentprovidedbytheshorterwavelengthmodes.Itisthenpossibletodescribethequantumevolutionofthesystemintermsofaneffectivedynamics,wherebytheinteractionwiththeenvironmentresultsintheonsetofdissipationandnoise.Theultimatereasonforthepresenceofafiniteactivationrateevenatzerotemperatureisthat,foragenericfieldtheorythedynamicsofthesehomogeneousmodesisanharmonicenoughtocontainFouriercomponentswithfrequenciesabovethethresholdforexcitationoftheshortwavelengthmodes.Thisresultsinanenergytransferfromthelongwavelengthorhomogeneousmodestotheshortwavelengthorinhomogeneousmodesthroughparticlecreation.Asdemandedbytheenergybalance,andencodedinthefluctuation-dissipationtheorem,thisenergyflowiscompensatedbyastochasticforceonthehomogeneousmodes,originatedinfluctuationsoftheinhomogeneous∗AlsoatInstitutdeF´ısicad’AltesEnergies(IFAE),Barcelona,Spain.1modes.Thusthedynamicsofthehomogeneousmodesbecomesdiffussive,evenif,properlyspeaking,thereisnoexternal“environment”tothefield[13].Wewishtostressthatthisisnotonlyatheoreticalpossibility.Inthispaper,wewillshowthroughadetailedanalysisofaconcretemodelthattheactivationratemakesasubstantialcontributiontothefulldecayrateevenatzerotemperature.Intheprocess,weshalldevelopthenecessaryformalismtocomputetheactivationratetoleadingorderin¯h.ThekeyingredientwillbethedescriptionofthequantumstateofthelongwavelengthmodesofthefieldbymeansofthereducedWignerfunction.Thisfunctionhasthesameinformationthatthereduceddensitymatrixofanopenquantumsystembutissimilarinmanyaspectstoadistributionfunctioninphase-space.ThedynamicalequationforthereducedWignerfunction(masterequation)includesnoisetermsproducedbytheshortwavelengthmodestoquadraticorderintheparametercouplingtheshortandlongwavelengthmodes.Weshallderivethetunnelingratefromananalysisofthedecayofnontrivialsolutionsofthemasterequation,aftersuitableboundaryconditionshavebeenenforced.Themasterequationcontainsallrequiredinformationonvacuumdecay,includingboththe“tunneling”aspectofthehomogeneousmodeaswellas“activation”,i.e.theeffectduetothebackreactionoftheinhomogeneousmodes,butinthispaperweconcentrateintheactivationaspect.Thus,whereastheinstantonmethod[9],forinstance,seemstobebestsuit