Fixed points of the renormalisation group running of quark and fermion mixing matrices in the Standard Model and beyond

TL;DR

This paper investigates the fixed points of fermion mixing matrices under renormalisation group evolution in the Standard Model and beyond, identifying their properties and stability.

Contribution

It identifies six fixed points at 1-loop order and argues these remain fixed points at all orders, linking them to geometric properties of the mixing matrix space.

Findings

Six fixed points found at 1-loop in the massless case.

Fixed points are associated with specific geometric properties.

Number of fixed points increases factorially with sterile neutrinos.

Abstract

The renormalisation group running of fermion mixing matrices in the Standard model and beyond is studied. For the massless 1-loop running with three generations six fixed points are found. Their associated anomalous dimension matrices are calculated and the nature of each fixed point, whether attractive, repulsive or mixed, is determined. An argument is given that the fixed points found at 1-loop must remain fixed points to all orders in perturbation theory and even non-perturbatively, as they are associated with certain differential geometric properties of vector fields on the space of mixing matrices. With $N_g$ dark or sterile neutrinos there are at least $N_g!$ fixed points of the fermion mixing matrix.