Gauge dependence of the on-shell renormalized mixing matrices

It was recently pointed out that the on-shell renormalization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the method by Denner and Sack causes a gauge parameter dependence of the amplitudes. We analyze the gauge dependence of the on-shell renormalization of the mixing matrices both for fermions and scalars in general cases, at the one-loop level. We then show that this gauge dependence can be avoided by fixing the counterterms for the mixing matrices in terms of the off-diagonal wave function corrections for fermions and scalars after a rearrangement, in a similar manner to the pinch technique for gauge bosons. We finally present explicit calculation of the gauge dependence for two cases: CKM matrix in the Standard Model, and left-right mixing of scalar quarks in the minimal supersymmetric standard model.