Light Quark Masses and Quarkonium Decays

After discussing the intrinsic ambiguity in determining the light quark mass ratio mu/md, we reexamine the recent proposal that this ambiguity can be resolved by applying the QCD multipole expansion for the heavy quarkonium decays. It is observed that, due to instanton effects, some matrix elements which have been ignored in previous works can give a significant contribution to the decay amplitudes, which results in a large uncertainty in the value ofmu/md deduced from quarkonium phenomenology. This uncertainty can be resolved only by a QCD calculation of some second order coefficients in the chiral expansion of the decay amplitudes. It has been observed by a number of authors [1−5] that second order corrections in chiral perturbation theory can significantly affect the estimate of the light quark masses. In particular, it was pointed out that the determination of mu/md suffers from a large uncertainty due to the instanton-induced mass renormalization [1, 3]: M → M̄(ω) ≡ M + ωMI (1) where the real matrix M = diag(mu, md, ms) denotes the light quark masses in the QCD lagrangian, MI ≡ 1 4πf (detM )(M ) = 1 4πf (mdms, mums, mumd) is the instanton-induced second order mass with the pion decay constant f ≃ 93 MeV, and ω is a dimensionless parameter of order unity. Most of the previous analyses on M do not distinguish M from M̄ , and thus the corresponding results can be interpreted as those on M̄ for an arbitrary value of ω, which leads to a large uncertainty in the extracted value of mu/md. Recently it was argued that the above mentioned difficulty can be overcome by noting that the instanton-induced mass MI is distinguished from the bare mass M through its θ-dependence where θ denotes the CP violating QCD vacuum angle [6]. If one keeps the θ-dependence explicitly, MI always appears with the phase e iθ due to the winding number of instantons. Note that M and eMI have the same transformation under the QCD chiral symmetry SU(3)L × SU(3)R × U(1)A under which M → eLMR, θ → θ + 3α, (2) where L ∈ SU(3)L, R ∈ SU(3)R, and α generates the anomalous U(1)A rotation. Then one may be able to measure M directly, not M̄ involving an arbitrary unknown parameter ω, by probing the θ-dependence of the QCD dynamics. Among quantities that probe the θ-dependence, the matrix elements 〈