Strange-quark mass from combined e+e- and tau-decay data:test of the isospin symmetry and implications on epsilon'/\epsilon and m_{u,d}

TL;DR

This paper determines the strange-quark mass using sum rules from e+e- and tau-decay data, testing isospin symmetry and exploring implications for CP violation and quark mass ratios, with results consistent with previous estimates.

Contribution

It introduces a novel combined sum rule approach for extracting the strange-quark mass and assesses isospin symmetry violations with implications for CP violation parameters.

Findings

Strange-quark mass at 2 GeV: 129 ± 24 MeV

Small isospin violation effects observed

Estimate of four-quark weak matrix elements

Abstract

I extract the strange-quark mass using a $\tau$-like decay sum rule for the $\phi$-meson, and some other sum rules involving its difference with the vector component of the hadronic $\tau$-decay. As a conservative estimate, one obtains to order $\alpha_s^3$: $\bar{m}_s$(1 GeV) = $(178\pm 33)$ MeV $\lrar \~\bar{m}_s$(2 GeV) = $(129\pm 24)$ MeV, while the positivity of the spectral function leads to the upper bound: $\bar{m}_s(1 {\rm GeV})\leq (200\pm 28) {MeV} \Longrightarrow~\bar{m}_s(2~{\rm GeV})\leq (145\pm 20) {MeV}$. These results are in good agreement with the existing sum rule and $\tau$-decay results, and, in particular, with the result from the the sum rule involving the difference of the isoscalar and isovector components of the $e^+e^-\rar$ hadrons data. This signals small effects of the SU(2) isospin violation due to the $\omega$-$\rho$ mixing parameters, and questions the reliability of the existing sum rule estimates of these parameters. Combining our result with the recent data on $\epsilon'/\epsilon$, 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\pm 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: $\bar{m}_d(2 {\rm GeV})= (6.4\pm 1.1) {\rm MeV}$ and $\bar{m}_u(2 {\rm GeV})= (2.3\pm 0.4) {\rm MeV}$.