Flavor mixing, gauge invariance, and wave-function renormalization

Abstract

We clarify some aspects of the LSZ formalism and wave-function renormalization for unstable particles in the presence of electroweak interactions when mixing and $\mathrm{CP}$ violation are considered. We also analyze the renormalization of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix which is closely related to wave-function renormalization. We critically review earlier attempts to define a set of ``on-shell'' wave-function renormalization constants. With the aid of extensive use of the Nielsen identities complemented by explicit calculations we corroborate that the counterterm for the CKM mixing matrix must be explicitly gauge independent and demonstrate that the commonly used prescription for the wave-function renormalization constants leads to gauge parameter dependent amplitudes, even if the CKM counterterm is gauge invariant as required. We show that a proper LSZ-compliant prescription leads to gauge independent amplitudes. The resulting wave-function renormalization constants necessarily possess absorptive parts, but we verify that they comply with the expected requirements concerning $\mathrm{CP}$ and $\mathrm{CPT}.$ The results obtained using this prescription are different (even at the level of the modulus squared of the amplitude) from the ones neglecting the absorptive parts in the case of top decay. The difference is numerically relevant.