Resummation of Perturbation Series in Non-Equilibr

arXiv:hep-ph/9407249v17Jul1994CERN-TH.7336/94RESUMMATIONOFPERTURBATIONSERIESINNON-EQUILIBRIUMSCALARFIELDTHEORYT.Altherr1TheoryDivision,CERN,CH-1211Geneva23,SwitzerlandAbstractThegeneralbehaviourofperturbationseriesinnon-equilibriumscalarfieldtheoryisanalysedinsomedetail,withaparticularemphasisonthe“patho-logicalterms”,generatedbymultipleproductsofδ-functions.Usinganintu-itiveregularizationmethod,itisshownthatthesetermsgivelargecontribu-tionsatallorders,evenwhenconsideringsmalldeviationsfromequilibrium.Fortunately,thesetermscanalsoberesummedandIgivethegeneralexpres-sionsfortheresummedpropagatorsinnon-equilibriumscalarfieldtheory,regardlessofthesizeofdeviationsfromequilibrium.SubmittedtoPhysicsLettersBCERN-TH.7336/94June941OnleaveofabsencefromL.A.P.P.,BP110,F-74941Annecy-le-VieuxCedex,France.1IntroductionInapreviouswork[1],Seibertandthepresentauthorshowedthatthestandardtheoreticalframeworkofnon-equilibriumquantumfieldtheoryisplaguedwithseriousdifficultieswhenperturbationseriesisconsidered.Theproblemoriginatesfromthenon-cancellationofthe“pathologicalterms”[2]thatareassociatedtomultipleproductsofδ-functions.In[1],ithasbeenprovedthatthesetermsdonotcancelunlesstheparticledistributionsarethoseforasysteminthermalandchemicalequilibrium.Thispaperisdevotedtothediscussionofapossiblesolutiontothisproblem.Inparticular,thebehaviouroftheperturbationseries,thatisintermsofthecouplingconstant,isscrutinizedinthelightofthepresenceofthesepathologies.Firstneededisaregularizationscheme,inordertodealwiththemathematicallyill-definedpathologies.Sincetheproblemoriginatesfromtheinfinitesimaliǫprescriptioninthefreepropagator,arathernaturalwaytoperformthisregularizationistointroduceafinitewidth,ordamping,inthepropagators.Throughoutthepaper,the2×2Keldyshmatrixstructure[3]isusedasatoy-modelofanon-equilibriumquantumfieldsystem,D11(K)D12(K)D21(K)D22(K)=ΔR(K)(1+n(k0))+ΔA(K)n(k0)(1+n(k0))(ΔR(K)+ΔA(K))n(k0)(ΔR(K)+ΔA(K))ΔR(K)n(k0)+ΔA(K)(1+n(k0)),(1)withtheRetarded/AdvancedpropagatorsdefinedasΔR(A)(K)=+(−)iK2−m2+(−)iγk0.(2)1ThewidthγisanarbitraryfinitefunctionofK.Thedistributionn(k0)isdefinedsothatn(k0)=−1−n(−k0)=n(k),(3)wheren(k)representsthenon-equilibriumdistributionofthequanta,namelyTrhρa†(k)a(p)i=n(k)δ3(k−p),(4)whereρisanarbitrarydensityoperator.ItisassumedthatthesystemevolvessufficientlyslowlyintimeandspacesothattheFouriertransform,forshortspace-timeseparations,hasthesimpleexpressionshownineq.(1).Byintroducingthisfinitewidthinthebarepropagators,theperturba-tionseriesnowbecomesmathematicallywell-defined:pathologiesassociatedwithmultipleproductsofδ-functionsareregularized.Inprinciple,thiswidth,whichisintroducedheuristically“byhand”,canbecalculatedbyusingper-turbationtheory.Ishalldiscussthispointinmoredetaillateron.Thequestionisnow:Howdothepathologicaltermschangethebehaviouroftheperturbationseriesintermsofthecouplingconstantg?2g2φ4theoryFromnowon,Iconsiderthemasslesscase.Thesimplestthingtocomputein(g2/4!)φ4theoryisthetadpolediagram(inn=4−2ǫdimensions)ReΣ=12g2ZdnK(2π)n(1+2n(k0))γk0K4+γ2k20.(5)Asusual,theultravioletsingularitiesarepresentinthevacuumpartonly,providedthedistributionn(k0)dropssufficientlyfastwhenk0→∞,asin2theequilibriumcase.Forthematterpart(denotedbyδΣinthefollowing),Iobtain,afteraCauchyintegration:Re(δΣ)=g28π2Z∞0dk0k0n(k0)s1+iγk0+s1−iγk0!,(6)whereIhaveassumed,forthesakeofsimplificationinthenotation,thatγdependsonlyonk0.Onerecoversthewell-knownthermalmass,δm2=(g2/24)T2,whenγ→0andwhenn(k0)istheBose–Einsteindistribution[4].Atoneloop,theimaginarypartofΣvanishes,asinthecaseofzerowidth.Hence,eq.(6)providesthecompletenon-equilibriumgeneralizationoftheoneloopthermalmassatequilibrium,regardlessoftheparticlenumberdistributionn(k0).Inthecaseofasmalldampingonehasδm2=g24π2Z∞0dk0k0n(k0).(7)Aninspectionofthetwoloopdiagramsdoesnotrevealanyseriousin-fraredproblem.Forthefirsttopology,showninfig.1a,termsthatareassoci-atedwithpotentialpathologies(δ2(K2)terms)cancelinthelimitofvanishingγ[1].Therefore,providedn(k0)isnomoresingularthantheBose–Einsteindistribution,theinfraredproblemsarenotworstthanatequilibrium.Forinstance,forasystemthatisclosetoequilibrium,thetwoloopdiagramscontributeatO(g3T2)tothethermalmass[4].Thesameistrueforthesecondtopology,showninfig.1b,wherenopathologiesarepresent.Thisdiagramhasanimaginarypart,whichalsogivesthewidth:γ(K)=i2k0(Σ12(K)−Σ21(K)).(8)3Forsmalldeviationsfromequilibrium,γisjusttheusualequilibriumdamp-ingrateing2φ4theory,thatisγhard=O(g4T),(9)timeseventuallysomeln1/gthatarisesfromsomelogarithmicinfraredsin-gularities(Idonotdiscusstheseminorcorrectionshere).Thesubscripthardreferstothecaseofahardexternalmomentum(K∼T).Thesoft(K∼gT)dampingrateiseasilycalculableandisgivenbyγsoft=132√6π3g3ln1gT,(10)toleadingorderinln1/g.Asmotivatedfromthebeginning,thedampings,eithersoftorhard,eventuallyprovideawaytoregularizethepathologicalterms.However,thereremainstoperformtheresummationofthedampingeffectsthatleadtothepropagator(2)(whichisonlyausefulassumptionforthetimebeing).Potentialproblemsariseatthenextorder,thatisatthethreelooporder,whichmakestheanal