Right-handed neutrinos and the CDF II anomaly

We point out that right-handed neutrinos can resolve the tension between the latest CDF II measurement of the $W$-boson mass, $M_W$, and the standard model. Integrating out the new states yields a single $d=6$ operator, which induces a deviation from unitarity in the PMNS matrix. This alters the extraction of the Fermi constant from muon decay and increases the prediction for $M_W$, in line with the CDF II result. Non-unitarity of the PMNS matrix would also affect beta, meson, and tau decays. We find that the CDF II value for $M_W$ can be explained without conflicting with lepton flavour universality constraints or the invisible decay width of the $Z$. However, the so-called Cabibbo angle anomaly is worsened if right-handed neutrinos are the origin of the $d=6$ operator. The situation improves if the operator coefficient is left unconstrained, implying additional sources of new physics, but a common explanation of both anomalies is in tension with universality bounds.