Mass Matrices and Their Renormalization

TL;DR

This paper derives renormalization group equations for quark mass matrices using rephasing invariant parameters, showing how hierarchies develop with energy scale and highlighting the need for new models to match experimental observations.

Contribution

It explicitly formulates RG equations for quark mass matrices in rephasing invariant terms, revealing their evolution and the necessity for novel models to explain observed hierarchies.

Findings

Mass matrices evolve rapidly with energy scale.

Hierarchies develop as energy decreases.

New models are needed to match experimental hierarchies.

Abstract

We obtain explicitly the renormalization group equations for the quark mass matrices in terms of a set of rephasing invariant parameters. For a range of assumed high energy values for the mass ratios and mixing parameters, they are found to evolve rapidly and develop hierarchies as the energy scale decreases. To achieve the experimentally observed high degree of hierarchy, however, the introduction of new models with specific properties becomes necessary.