Mixing and Decay Constants of Pseudoscalar Mesons(9)

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

and

θc= z 2Mηc(4.6)

witha2asgiveninEq.(2.15).UsingthephenomenologicalparametervaluesquotedinTableI,we ndthefollowing2=m2numericalresults:z=0.35±0.03;θc= 1.0 ±0.1 .Sincez2a2=0.03GeV2wehaveMηcctoaverygoodc′approximation.Thecharmdecayconstantsoftheηandtheηtakethevalues

cfη= (2.4±0.2)MeVcfη′= (6.3±0.6)MeV(4.7)

Theirvaluesareinroughagreementwiththeresultspresentedin[13,27,28]butindramaticcon ictwiththevaluescquotedin[10,11].fη′lieswellwithintheboundestimatedin[7].Ouranalysissupportstheconclusionsdrawnin[13]thatthecharmcontentoftheη′isnotthesolutionfortheabnormallylargeB→Kη′decaywidth,theexplanationofwhichremainsanopenproblem.

Usingtheabovevaluesforthemixinganglesφ,θyandθc,we ndforthequarkcontentofthephysicalmesons

|η =0.77|ηq 0.63|ηs 0.006|ηc0 |η′ =0.63|ηq +0.77|ηs 0.016|ηc0 |ηc =0.015|ηq +0.008|ηs +|ηc0 .(4.8)

Thecharmadmixturestotheηandη′aresomewhatsmallerthanestimatedin[1]butslightlylargerthanquotedin

[28].Apossibletestfortheηc0contentisprovidedbytheradiativeJ/ψdecays.ForthedecaysJ/ψ→ηγ,η′γweusedalreadytheactionofthegluonsasdescribedbythematrixelementsoftheanomaly(notethatθ8=θy).Sincetheη′hastheηc0contentθccosθ8whiletheηc0contentofηcispracticallyone,weexpect

3 αs 3′ 0|Γ[J/ψ→ηγ]√=(4.9)kηckηc

Theexperimentalnumberforthisratio,0.33±0.10[17],givesusanother–admittedlylessreliable–determinationofthecharmadmixtureinη′.Theresult|θccosθ8|=0.014±0.002isingoodagreementwiththenumbercontainedinEq.(4.8).

V.SUMMARY

Inthedescriptionofη-η′mixingthereare veparametersinvolved,themixingangleoftheparticlestatesandfourdecayconstants.Motivatedbytheobservationofnearlyidealmixinginvectorandtensorparticleswetakeasourbasisthestatesaccordingtotheirquark avorcompositions.Ourcentralassumptionisthenthatinthisparticularbasisthemixingofthedecayconstantsfollowsthatofstatemixing.Thisnewmixingschemeisveryrestrictive.It xesthestructureofthemassmatrixandpredictsthemixingangleandthefourdecayconstantsupto rstorderin avorsymmetrybreaking:

i)Thefourdecayconstantsareimmediatelyreducedtotwoconstantsfqandfsandasingleangleφthatisidenticaltothestatemixingangleanddescribesthedeviationfromidealmixing.

ii)Thedivergencesoftheaxialvectorcurrentsprovideuswithamassmatrixquadraticintheparticlemasseswitho diagonalelementsentirelydeterminedbytheanomaly.Theoldproblemofquadraticversuslinearmassmatriceshasfounditsanswer.

iii)ingthisresultthemixingangleφcanbecalculatedfromfq/fs.iv)SU(3)Frelations xfqandfsintermsoffπandfKto rstorderin avorsymmetrybreaking,and xthose22partsofthemassmatrixwhichcontainthecurrentquarkmassesintermsofMπandMK.Thedecayconstantsobtainedthiswayobeytherequirementsofchiralperturbationtheorywhichareknownuptoorder1/Nccorrections.v)Withtheseingredientsandbyusingtheknownmassesofthephysicalstatesthemassmatrixisover-determined.Althoughsizeablecorrectionstothe avorsymmetryresultscouldhavebeenexpected,theresultingparameter-freede-terminationofthemixingangleandthedecayconstantsisinreasonableagreementwithapreviousphenomenologicalanalysiswithunconstrainedparameters.

vi)Weperformedanewanalysisanddeterminedphenomenologicallyφandfqandfsfromseveralindependentexperiments.Allresultswereconsistentwitheachother.Thus,theweightedaveragevalueforthemixingangleis