Strange-quark mass from combined e +e − and τ-decay data: test of the isospin symmetry and implications on ϵ′/ ϵ and mu, d

Abstract

I extract the strange-quark mass using a τ-like decay sum rule for the φ-meson, and some other sum rules involving its difference with the vector component of the hadronic τ-decay. As a conservative estimate, one obtains to order α s 3: m s (1 GeV) = (178±33) MeV ⇒ m s (2 GeV) = (129±24) MeV, while the positivity of the spectral function leads to the upper bound: m s(1 GeV)≤(200±28) MeV ⇒ m s(2 GeV)≤(145±20) MeV . These results are in good agreement with the existing sum rule and τ-decay results, and, in particular, with the result from the the sum rule involving the difference of the isoscalar and isovector components of the e +e −→ hadrons data. This signals small effects of the SU(2) isospin violation due to the ω– ρ mixing parameters, and questions the reliability of the existing sum rule estimates of these parameters. Combining our result with the recent data on ϵ′/ ϵ, we can estimate, within the standard model, the four-quark weak matrix elements B 6 1/2−0.54 B 3/2 8 to be about (2.8±1.3). This result may suggest a large violation of the vacuum saturation estimate similarly to the case of the four-quark condensates obtained from the sum rules analysis, and can serve as a guide for a future accurate non-perturbative extraction of such matrix elements. Combining our result with the sum rule estimate of m u + m d and with the Dashen formula for the mass ratio, one can deduce the update values: m d(2 GeV)=(6.4±1.1) MeV and m u(2 GeV)=(2.3±0.4) MeV .