Renormalization of the flavor-changing neutral currents

The flavor-changing amplitudes γ$q_i$ $q_j$ and ${\mathrm{Zq}}_i$ $q_j$ are computed for off-shell particles and without approximations. The calculations are done in the ’t Hooft–Feynman gauge, and the on-shell renormalization scheme is used. The Ward-Slavnov-Taylor identities are derived and used to check the results. Putting the external quarks on shell, we found that the corresponding flavor-changing S-matrix elements are not renormalized, i.e., the sum of all counterterms vanishes. Using the renormalization scheme proposed by Sirlin the meaning of this cancellation is clarified.