Simple On-Shell Renormalization Framework for the Cabibbo-Kobayashi-Maskawa Matrix

TL;DR

This paper introduces a clear on-shell renormalization method for the CKM matrix at one-loop level, ensuring gauge independence, unitarity, and finite results even in degenerate mass limits.

Contribution

It develops a novel procedure to separate mixing corrections into gauge-independent and gauge-dependent parts, and provides explicit, gauge-invariant CKM counterterms.

Findings

The framework guarantees gauge independence and finiteness of one-loop corrections.

It maintains unitarity of the CKM matrix after renormalization.

The method ensures non-singular amplitudes in mass-degenerate limits.

Abstract

We present an explicit on-shell framework to renormalize the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix at the one-loop level. It is based on a novel procedure to separate the external-leg mixing corrections into gauge-independent self-mass (sm) and gauge-dependent wave-function renormalization contributions, and to adjust non-diagonal mass counterterm matrices to cancel all the divergent sm contributions, and also their finite parts subject to constraints imposed by the hermiticity of the mass matrices. It is also shown that the proof of gauge independence and finiteness of the remaining one-loop corrections to W -> q_i + anti-q_j reduces to that in the unmixed, single-generation case. Diagonalization of the complete mass matrices leads then to an explicit expression for the CKM counterterm matrix, which is gauge independent, preserves unitarity, and leads to renormalized amplitudes that are non-singular in the limit in which any two fermions become mass degenerate.