Fermion mixing renormalization and gauge invariance

We study the renormalization of the fermion mixing matrix in the Standard Model and derive the constraints that must be satisfied to respect gauge invariance to all orders. We demonstrate that the prescription based on the on-shell renormalization conditions is not consistent with the Ward-Takahashi Identities and leads to gauge dependent physical amplitudes. A simple scheme is proposed that satisfies all theoretical requirements and is very convenient for practical calculations. Despite a few interesting papers have been devoted to the fermion mixing renormalization [1, 2, 3, 4], the subject has so far escaped much attention. This is mostly due to the fact that, as a result of GIM cancelations, the radiative corrections related to the renormalization of the CKM matrix can be made very small, O(GF m 2 q), where mq is the mass of a light quark. They are therefore of little practical importance in the context of the Standard Model (SM). Still, the subject has some conceptual interest in its own, not to mention the relevance of mixing in many extension of the SM. In this letter, we reconsider it from a different point of view, which allows us to point out some inconsistency in previous analyses and to propose an alternative solution. For definiteness, we concentrate on the case of the CKM matrix renormalization in the SM. A convenient framework to study this issue is provided by the Ward Takahashi Identities (WTI) of the theory with background fields [5, 6, 7]. Indeed, as the diagonalization of the fermion mass matrix is achieved by field redefinitions that do not commute with the gauge transformations, the CKM elements appear explicitly in the WTI, unlike masses and gauge couplings. This will give us a strong constraint. At the functional level, the WTI which represent the σ generator of SU(2)L are implemented by the local functional operator W+ acting on the effective action Γ (see [7]). For our purposes, the relevant part of W+ is the one which contains the quark fields: W + = ∑