Model-independent determination of $\overline{m}_s$ from $τ$-like inclusive decays in $e^+e^-$ and implications for the $χSB$-parameters

Abstract

We determine the strange quark running mass of the $\overline{MS}$-scheme by $simulating$ $τ$-$like$ inclusive processes for the $old$ Das-Mathur-Okubo sum rule relating the $e^+e^-$ into $I=0$ and $I=1$ hadrons total cross-sections data. We obtain to three-loop accuracy: $\overline{m}_s$(1 GeV)=($196.7\pm 29.1$) MeV. By combining this result with the pseudoscalar sum rule estimate of $(\overline{m}_d +\overline{m}_u)$ and the standard current algebra values of the light quark mass ratios, we deduce the $average$: $\overline{m}_d$(1 GeV)=$(10.3\pm 1.5)$ MeV, $\overline{m}_u$(1 GeV)=$(5.0\pm 1.5)$ MeV and $\frac{1}{2}\la \bar uu +\bar dd \ra$ (1 GeV) $ \simeq -[(228.6\pm 9.1) \mbox{MeV}]^3$. Using also our value of $m_s$, we update the estimates of the $K'(1.46)$ and $K^*_0(1.43)$ decay constants and of the (pseudo)scalar two-point correlator subtraction constants. Then, we deduce a deviation of about 34\% from kaon PCAC and the ratio of the {\it normal-ordered} condensates: $\la \bar ss \ra /\la \bar uu \ra = 0.68^{+0.15}_{-0.29}$, which confirm previous findings from QCD spectral sum rules. Finally, using the recent value of the $\overline{m}_b$ from the $Υ$-sum rules, we deduce the scale independent quark-mass ratio: $m_b/m_s=34\pm 4$.