Renormalization of the quark mass matrix

TL;DR

This paper presents a simplified and permutation-invariant formulation of the renormalization group equations for quark mixing parameters, along with approximate solutions for hierarchical or degenerate quark masses.

Contribution

It introduces a compact, rephasing-invariant framework for quark mass matrix renormalization and derives approximate solutions under specific mass hierarchies.

Findings

Renormalization group equations can be expressed in a simple, permutation-invariant form.

Approximate solutions are obtained for hierarchical and nearly degenerate quark masses.

The approach simplifies analysis of quark mixing parameter evolution.

Abstract

Using a set of rephasing-invariant variables, it is shown that the renormalization group equations for quark mixing parameters can be written in a form that is compact, in addition to having simple properties under flavor permutation. We also found approximate solutions to these equations if the quark masses are hierarchical or nearly degenerate.