Renormalization of mixing angles and computation of the hadronic $W$ decay widths

Abstract

We provide a practical prescription for a variant of the On-Shell scheme which does not require mixing matrix counterterms at all, i.e. $δV=0$. The scheme is based on the fact that one can always choose a basis in which there are no mixing matrices (angles) and, therefore, the corresponding counterterms are superfluous. Importantly, the prescription is model- and process-independent and is formulated entirely in terms of self-energies. As an example, we compute the 1-loop hadronic $W$-boson decay widths in the Standard Model with different renormalization schemes of the quark mixing matrix found in the literature and the one found in this paper. For full consistency, the principles of this scheme are employed both for the quark mixing matrix and for the Weinberg angle.