Renormalization group improved $m_s$ and $\vert V_{us}\vert$ determination from hadronic $\tau$ decays

We determine the strange quark mass ($m_s$) and quark mixing element $\vert V_{us}\vert$, and their joint determination from the Cabibbo suppressed hadronic $\tau$ decays in various perturbative schemes. Compared to the previous analysis based on the optimal renormalization or the renormalization group summed perturbation theory (RGSPT) scheme, we have improved this analysis by replacing the theoretical longitudinal contributions with phenomenological parametrization, and the RGSPT coefficients are used for the dimension-4 Adler functions. The improved analysis results in the extraction of $m_s(2\,\mathrm{GeV})=98\pm19\,\mathrm{MeV}$ and $\vert V_{us}\vert=0.2191\pm0.0043$ from the RGSPT scheme.