Nonperturbative Vacuum Effect in the Quantum Field

arXiv:hep-ph/0105094v211May2001NonperturbativeVacuumEffectintheQuantumFieldTheoryofMesonMixingChueng-RyongJiandYuriyMishchenkoDepartmentofPhysics,NorthCarolinaStateUniversityRaleigh,NorthCarolina27695-8202USAReplacingtheperturbativevacuumbythenonperturbativevacuum,weextendarecentdevelopmentofaquantumfieldtheoreticframeworkforscalarandpseudoscalarmesonmixing.TheunitaryinequivalenceoftheFockspaceofbase(unmixed)eigenstatesandthephysicalmixedeigenstatesisinves-tigatedandtheflavorvacuumstatestructureisexplicitlyfound.Thisisexploitedtodevelopformulasfortwoflavorbosonoscillationsinsystemsofarbitrarybosonoccupationnumber.Weapplytheseformulastoanalyzethemixingofηwithη′andcommentontheothermeson-mixingsystems.Inaddition,weconsiderthemixingofbosoncoherentstates,whichmayhavefutureapplicationsintheconstructionofmesonlasers.1I.INTRODUCTIONThestudyofmixingtransformationsplaysanimportantpartinparticlephysicsphe-nomenology.[1]TheStandardModelincorporatesthemixingoffermionfieldsthroughtheKobayashi-Maskawa[2]mixingof3quarkflavors,ageneralizationoftheoriginalCabibbo[3]mixingmatrixbetweenthedandsquarks.Inaddition,neutrinomixingandoscillationsarethelikelyresolutionofthefamoussolarneutrinopuzzle[4].Inthebosonsector,themixingofK0withK0viaweakcurrentsprovidedthefirstevidenceofCPviolation[5]andtheB0¯B0mixingplaysanimportantroleindeterminingthepreciseprofileofaCKM[2,3]unitarytriangle[6]inWolfensteinparameterspace[7].Theηη′mixingintheSU(3)flavorgroupalsoprovidesauniqueopportunityfortestingQCDandtheconstituentquarkmodel.Furthermore,theparticlemixingrelationsforboththefermionandbosoncasearebelievedtoberelatedtothecondensatestructureofthevacuum.Thenon-trivialnatureofthevacuumisexpectedtoholdtheanswertomanyofthemostsalientquestionsregardingconfinementandthesymmetrybreakingmechanism.Theimportanceofthefermionmixingtransformationshasrecentlypromptedafun-damentalexaminationofthemfromaquantumfieldtheoreticperspective[8].Asimilaranalysisinthebosonicsectorhasalsobeenundertaken[9].However,morerecentanalysis[10]onthefermionmixingindicatedthatthepreviousresult[8]basedontheperturbativevacuumisonlytheapproximationwithrespecttotheexactonebasedonthenonpertur-bative(flavor)vacuum.Inthiswork,weshowthatthesameistrueforthebosonicsector.Uponthecompletionofourwork,wenoticethatthesameconclusionwasalsodrawninarecentliterature[11].Inourwork,however,theorthogonalitybetweenmassandflavorvacuaisshowninastraightforwardalgebraicmethodratherthansolvingadifferentialequationfortheinnerproductoftwovacuaaspresentedin[11].Asevidencedinthepreviouslitera-tures[8,10–12]themethodofusingadifferentialequationtoprovetheunitaryinequivalencebetweenthetwoFockspaceshasbeenknownforsometimeandouralgebraicmethodisanewdevelopmentinthisrespect.Moreover,weanalyzethestructureofthenonperturba-tiveflavorvacuuminagreatdetailcontrastingtothefermioncase.Thedetailsofflavorvacuum,itsperturbativeexpansioninthemixingangleandalsosomeclarifyingremarksontheGreenfunctionmethodandthearbitrarymassparametrizationaresummarizedintheaccompanyingAppendices.WebegininSectionIIwithinvestigationofthevacuumstructureusingtherelation2betweenthebaseeigenstateandthephysicalmixed-eigenstatefields.Wederivetherepre-sentationforPontecorvomixingtransformationforbosoncaseandexplicitlycalculatetheflavorvacuumstatestructureinthequantumfieldtheory(QFT).WetheninvestigatetheunitaryinequivalenceofthetwoFockspaces-oneisthespaceofmasseigenstatesandtheotheristhespaceofflavoreigenstates.InSectionIII,theladderoperatorsareconstructedinthemixedbasis.Theseareusedtoderivetimedependentoscillationformulasfor1-bosonstates,n-bosonstates,andbosoncoherentstates.Consequencesfromthereplacementoftheperturbativevacuumbytheexactnonperturbative(flavor)vacuumaredemonstrated.SectionIVisdevotedtostudyspecificcasesinourformalism,suchastheηη′system.Weshowthenumericaldifferencesbetweenthetworesults:onefromtheperturbativevacuumandtheotherfromthenonperturbativevacuum.ConclusionsanddiscussionsfollowinSectionV.InAppendixA,wepresentaderivationofanexplicitexpressionfortheflavorvacuumoperatingtheladderoperatorsofparticleandantiparticletothevacuumofmasseigenstates.InAppendixB,wediscusstheregionofvalidityforaperturbativeexpansionoftheflavorvacuum.InAppendixC,wemakesomeclarifyingremarksontheGreenfunctionmethodandthearbitrarymassparametrizationdiscussedinrecentliteratures[10,11,13].II.THEMIXINGRELATIONANDVACUUMSTRUCTUREWestartouranalysisbyconsideringthePontecorvomixingrelationship[14]fortwofields:φα=cosθϕ1+sinθϕ2φβ=−sinθϕ1+cosθϕ2,(2.1)whereϕ1,2arethefreefieldswithdefinitemassesm1,2andφα,βaretheinteractingfieldswithdefiniteflavorsα,β,respectively.TheabovementionedrelationshipnaturallyarisesbyconsideringthemixingproblemforthetwoquantumfieldswiththelagrangianoftheformL=L0,α+L0,β−λ(φ†αφβ+φ†βφα),(2.2)whereL0,α(β)arethefreeflavor-fieldlagrangians(i.e.L0,α(β)=12(∂φ†α(β)∂φα(β)−m2α(β)φ†α(β)φα(β)))andλisthecouplingconstantresponsibleformixing.Itisstraightforwardtoshowthattheabovelagrangiancanbeimmediatelydiagonalizedbythetransformation3givenbyEq.(2.1)withanappropriatechoiceofmixingangleθ[10].Theparametersofdiagon