Mixing and Decay Constants of Pseudoscalar Mesons(3)

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

Transformingthenon-strangeandstrange

axial-vector

currentsintooctetandsingletcurrents,onecanalsoconnectthedecayconstantsde nedinEq.(1.1)tofqandfs

81 fηfη=U(φ)FU (θideal)(2.4)81fη′fη′

withtheresult

f8=

f1=

andthus 2fs/fq),2fq/fs)

√(2.5)tan(θ1 θ8)=

q-s

4π .GG(2.7)

itsdual;midenotethecurrentquarkmasses.Thevacuum–mesonGdenotesthegluon eldstrengthtensorandG

transitionmatrixelementsoftheaxialvectorcurrentdivergencesaregivenbytheproductofthesquareofthemeson2mass,MP,andtheappropriatedecayconstant.Forinstance,

s2s 0| µJµ5|η =Mηfη.(2.8)

Themassfactors,whichnecessarilyappearquadraticallyhere,canbeconsideredastheelementsoftheparticlemassmatrix 2 M0ηM2=.(2.9)20Mη′

iWiththehelpofEq.(2.3)thematrixelementsof µJµ5(i=q,s)canthenbeidenti edasthoseofthematrixproduct

M2U(φ)F.Transformingittothequark avorbasisandsolvingforthemassmatrix

2M2qs=U(φ)MU(φ),(2.10)

oneeasily nds

M2qs= m2qq+

2√fq 0|4παsfs |ηs GG 0|1αsm2ss+4π |ηs GG(2.11)

whereweusetheabbreviations

√m2qq=fq¯iγ5d|ηq ; 0|muu¯iγ5u+mddm2ss=2

2 0|αs 0|αsfs

.(2.13)