2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/198671I 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 $α_s^3$: $\bar{m}_s$(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 $τ$-decay results, and, in particular, with the result from the the sum rule involving the difference of the isoscalar and isovector components of the $e^+e^-\rar$ 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\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 $\bar{m}_u(2 {\rm GeV})= (2.3\pm 0.4) {\rm MeV}$.Latex2e sources 11 pages + 1 table. Revised version written on July 99 to appear in Phys. Lett. BHigh Energy Physics - PhenomenologyHigh Energy Physics - ExperimentHigh Energy Physics - LatticeNuclear Theorytext