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

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 αs3: ms(1 GeV) = (178±33) MeV ⇒ ms(2 GeV) = (129±24) MeV, while the positivity of the spectral function leads to the upper bound: ms(1 GeV)≤(200±28) MeV ⇒ ms(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 B61/2−0.54B3/28 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 mu+md and with the Dashen formula for the mass ratio, one can deduce the update values: md(2 GeV)=(6.4±1.1) MeV and mu(2 GeV)=(2.3±0.4) MeV.