Renormalization of the energy-momentum tensor in n

arXiv:hep-th/0403068v213Mar2004Renormalizationoftheenergy-momentumtensorinnoncommutativecomplexscalarfieldtheoryS.Belluccia,I.L.Buchbinderb,V.A.Krykhtinc∗aINFN,LaboratoriNazionalidiFrascati,P.O.Box13,I-00044Frascati,ItalybDepartmentofTheoreticalPhysics,TomskStatePedagogicalUniversity,Tomsk634041,RussiacLaboratoryofMathematicalPhysicsandDepartmentofTheoreticalandExperimentalPhysics,TomskPolytechnicUniversity,Tomsk634050,RussiaAbstractWestudytherenormalizationofdimensionfourcompositeoperatorsandtheenergy-momentumtensorinnoncommutativecomplexscalarfieldtheory.Theproperoperatorbasisisdefinedanditisprovedthatthebarecompositeopera-torsareexpressedviarenormalizedoneswiththehelpofanappropriatemixingmatrixwhichiscalculatedintheone-loopapproximation.Thenumberandformoftheoperatorsinthebasisandthestructureofthemixingmatrixessentiallydif-ferfromthoseinthecorrespondingcommutativetheoryandinnoncommutativerealscalarfieldtheory.Weshowthattheenergy-momentumtensorinthenon-commutativecomplexscalarfieldtheoryisdefineduptosixarbitraryconstants.Thecanonicallydefinedenergy-momentumtensorisnotfiniteandmustbereplacedbythe”improved”one,inordertoprovidefiniteness.Suitable”improving”termsarefound.Renormalizationofdimensionfourcompositeoperatorsatzeromomen-tumtransferisalsostudied.Itisshownthatthemixingmatricesaredifferentforthecasesofarbitraryandzeromomentumtransfer.Theenergy-momentumvector,unliketheenergy-momentumtensor,isdefinedunambigouslyanddoesnotrequire”improving”,inordertobeconservedandfinite,atleastintheone-loopapproximation.∗e-mail:[e-mail],[e-mail],[e-mail]1IntroductionThestudyofnoncommutativefieldtheorieshasattractedmuchattentionlately,duetotheirprofoundlinkswiththestringtheory[1]andremarkablepropertiesinclassicalandquantumdomains(seee.g.thereviews[9–11]).Thereexisttwogeneralaspectsofrenormalizationprocedureinanyfieldtheory.Firstly,therenormalizationofGreenfunctionsoreffectiveactionandsecondly,therenor-malizationofcompositeoperators(seee.g.[2]foradiscussionofthisprobleminthecommutativetheories).TheproblemofrenormalizationofGreen’sfunctions(i.e.fields,masses,andcouplingconstants)wasstudiedformanynoncommutativefieldtheoriestodifferentapproximationorders(seee.g.[3–8]andthereviews[9–11]).Thepresentpaperisdevotedtotheproblemofrenormalizationofcompositeoperatorsandtheenergy-momentumtensorinnoncommutativecomplexscalarfieldtheory.Theanalogousprobleminnoncommutativerealfieldtheorywasconsideredin[12].Aswewillsee,therenormalizationofcompositeoperatorsinnoncommutativecomplexscalarfieldtheoryessentiallydiffersfromthatinnoncommutativerealfieldtheory.1Anoncommutativefieldtheoryisusuallyconstructedfromthecorrespondingcommu-tativetheory,byreplacingthepointwiseproductofthefieldswiththestaronef·g→(f⋆g)(x)=exp(i2θμν∂uμ∂vν)f(x+u)g(x+v)u=v=06=(g⋆f)(x),(1)wheretheconstantsθμνarenoncommutativityparameterswithdimensionofalengthsquared.Thestarproduct(1)isnoncommutative,so,incontrasttothecommutativefieldtheories,thereisaproblemoforderingofthefieldsintheLagrangianofnoncommutativetheory.Inthenoncommutativerealscalarfieldtheorytherewasonlyonekindoffieldandthisproblemwasabsent.Thereforeinthatcasebothcommutativeandnoncommutativetheorieshadonecouplingconstant.Inthecaseofnoncommutativecomplexscalarfieldtheorywehavetwokindsoffieldsandtheproblemoffieldsorderingarises.Thereforewehavetotakeintoaccountallpossiblewaysoffieldordering.TheactionofthetheoryistheLagrangianintegratedoverthewholespace-time.However,whenweintegratethestarproduct(1)overthewholespace-time,wecanprove,integratingbyparts,thefollowingconsequences:Zd4xf⋆g=Zd4xf·g=Zd4xg⋆f,(2)Zd4xf1⋆f2⋆···⋆fN=Zd4xf2⋆···⋆fN⋆f1.(3)Eq.(2)leadsustoconcludethatthefreepartofanactioninnoncommutativetheoryisthesameasinthecorrespondingcommutativemodel,andfromeq.(3)weseethatinteractiontermswhichdifferintheLagrangianbyacyclicpermutationarethesameintheaction.Forexample,inthetheoryofnoncommutativecomlexscalarfieldtheorywhichweshallstudy,therearetwodifferentinteractionterms[7,11]andtheactionmaybewrittenasS=Zd4x∂μφ∗⋆∂μφ−m2φ∗⋆φ−λa4!φ∗⋆φ⋆φ∗⋆φ−λb4!φ∗⋆φ∗⋆φ⋆φ.(4)1Problemofconstructingtheclassicalenergy-momentumtensorinnoncommutativefieldtheoriesisdiscussedin[13–18].2Heretherearetwopossibilitiesoforderingtheoperatorsintheinteractionterms,there-forewecanintroduceingeneraltwoindependentcouplingconstants,incontrasttothecommutativecase,wherethereisonlyoneinteractionandonlycouplingconstant.Thisdistinguishesthecaseofnoncommutativecomplexscalarfieldtheoryfromtherealoneandmakestheconsiderationoftherenormalizationofcompositeoperatorsinthistheoryalsointeresting.Inthepresentpaperwestudythisproblemandcompareitwithanalogousproblemsbothinnoncommutativerealscalarfieldtheoryandincommutativecomplexscalarfieldtheory.Thepaperisorganizedasfollows.Inthenextsectionwederivetheclassicalenergy-momentumtensorofthenoncommutativecomplexscalarfieldtheorywhichfollowsfromtheNoether’stheoremanddiscusssomepointsconcerningitsderivationinthenoncom-mutativecase.Insection3wepresentthegeneralrenormalizationstructureofdimensionfourcompositeoperatorsandtheninsection4w