The $m_W-m_Z$ interdependence in the Standard Model: a new scrutiny

The $m_W-m_Z$ 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 $m_W=80.357\pm 0.009\pm 0.003$ 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, $\widehat{\alpha }(M_t)=(127.73{)}^{-1}\pm 0.0000003$ and ${\sin }^2\text{\hspace{0.17em}⁣}{\widehat{\theta }}_W(M_t)=0.23462\pm 0.00012$.