Mixing and Decay Constants of Pseudoscalar Mesons(5)

We propose a new eta-eta' mixing scheme where we start from the quark flavor basis and assume that the decay constants in that basis follow the pattern of particle state mixing. On exploiting the divergences of the axial vector currents - which embody the

powercompletely1.Inthenextsectionwewillconfrontourapproachwithexperiment.Wedeterminethemixingangleφandthebasicdecayconstantsfqandfsphenomenologicallyandlookforconsistencyandfordeviationsfromthe rstorderofSU(3)Fbreaking.

III.THEPHENOMENOLOGICALVALUESOFφ,Fq,Fs

Severalpossibilitiestoextractthevalueofthemixinganglefromexperimenthavebeendiscussedintheliterature,see,forinstance,[1,3,15,16].Wecanpro tfromthesepapersbyproperlyadaptingthemtotheqsmixingscheme.Wenotethatinthephenomenologicalanalyses[1,3,15,16]someadditionalsimplifyingassumptionshadtobemade.Thus,forinstance,OZI-suppressedcontributionsormassdependenciesofformfactorsandcouplingconstantsareignored.Westartby rstdiscussingprocesseswhichareindependent(orinsensitive)tothedecayconstantsandallowaoneparameter toftheparticlestatemixingangleφ.

ThedecayJ/ψ→Pρ:WeconsidertheratioofthedecaywidthsΓ[J/ψ→η′ρ]andΓ[J/ψ→ηρ].IntheseprocessesG-parityisnotconserved,theyproceedthroughavirtualphoton(seeFig.1a).Contributionsfromtheisospin-violatingpartofQCDaresupposedlyverysmallascanbeinferredfromthesmallnessoftheJ/ψ→φπwidthandwillbeneglected.Thecalculationofthedecaywidthsrequirestheknowledgeoftheρ–Ptransition2formfactorsatmomentumtransferq2=MJ/ψ.Onaccountofthe avorcontentoftheρmeson,thistransitionformfactoronlyprobestheηqcomponentsoftheηandη′ifOZI-suppressedcontributionsareneglected.Hence,

Fρη(q2)=cosφFρηq(q2),

Fρη′(q2)=sinφFρηq(q2).

andtherefore

Γ[J/ψ→η′ρ]

kηρ

where

222kPV=MJ/ψ[1 (MP+MV)/MJ/ψ]/2.(3.1) 3(3.2)(3.3)

Fromtheexperimentalvalue0.54±0.11forthisratioofdecaywidths[17]weobtainφ=39.9 ±2.9 .Almostthesamevalueforφhasbeenfoundinananalysisofallisospin-1J/ψ→PVdecays(includingpions)[16].Aglobal ttoallJ/ψ→PVdecaymodesonthebasisofaparticularmodel,yieldsφ=37.8 ±1.7 [16].Becauseofitsmodeldependencewewillnotusethelatterresultinevaluatingtheaverageofthemixingangle.SinceinthederivationofEq.(3.2)onlythemixingangleoftheparticlestatesenters,onemayfreelytransformfrom

sbasistotheoctet-singletoneaswasdone,forinstance,in[16].Nevertheless,thesimplerelationtheq

betweentheratioofdecaywidthsandthemixingangle,independentofthedynamics,isanadvantageoftheqsbasisusedhere.Intheoctet-singletmixingschemeonewouldhavetodealwithalinearcombinationoftwoaprioridi erentformfactors.Wewillpro tfromthisadvantagealsointhefollowing veprocesses.

Thedecaysη′→ργandρ→ηγ:Thetransitionmatrixelementscontrollingtheseprocessescanbedecomposedcovariantly[18]

( )νλσ γP(pP)|T|ρ(pρ) = egρPγ µνλσpµpρερ,Pεγ(3.4)

leadingtothefollowingexpressionsforthedecaywidths

23Γ[η′→ργ]=αgρη′γkρ,Γ[ρ→ηγ]=α

Usingthephenomenologicalparametersfoundin[7],andtranslatingthemintothequark avorbasiswithadmissionoftwomixingangles,φqandφs,de nedinanalogytoEq.(1.3),one ndsfq=1.09fπ;fs=1.38fπ;φq=39.4 andφs=38.5 .ThefactthatthetwoanglesnearlycoincidegivedirectsupporttothevalidityofEq.(1.4).1