The $m_{\scriptscriptstyle W}-m_{\scriptscriptstyle Z}$ interdependence in the Standard Model: a new scrutiny

The $m_{\scriptscriptstyle W}-m_{\scriptscriptstyle Z}$ interdependence in the Standard Model is studied at $O(\alpha^2) $ in the $\overline{MS}$ scheme. The relevant radiative parameters, $\Delta \hat{\alpha} (\mu),\, \Delta \hat{r}_{W},\, \hat{\rho}$ are computed at the full two-loop level augmented by higher-order QCD contributions and by resummation of reducible contributions. We obtain $m_{\scriptscriptstyle W} = 80.357 \pm 0.009 \pm 0.003$ GeV where the errors refer to the parametric and theoretical uncertainties, respectively. A comparison with the known result in the On-Shell scheme gives a difference of $ \approx 6$ MeV. As a byproduct of our calculation we also obtain the $\overline{MS}$ electromagnetic coupling and the Weinberg angle at the top mass scale, $\hat{\alpha} (M_t) = (127.73)^{-1} \pm 0.0000003$ and $\sin^{2}\!\hat{\theta}_{W} (M_t) = 0.23462 \pm 0.00012$.