Renormalization group effects for a rank degenerate Yukawa matrix and the fate of the massless neutrino

TL;DR

This paper investigates how renormalization group effects influence the rank and masslessness of neutrinos in Type-I seesaw models with two or three right-handed neutrinos, showing the massless state persists under certain conditions.

Contribution

It analyzes the impact of renormalization group effects on the rank degeneracy of the neutrino mass matrix in Type-I seesaw models with different numbers of right-handed neutrinos.

Findings

The massless neutrino state remains under renormalization effects in the 3-2 model.

In the 3-3 model, the rank of the neutrino mass matrix depends on kernel solutions of RG equations.

For simplest kernel solutions, the neutrino mass matrix remains rank degenerate.

Abstract

The Type-I seesaw model is a common extension to the Standard Model that describes neutrino masses. The Type-I seesaw introduces heavy right-handed neutrinos with Majorana mass that transform as Standard Model electroweak gauge singlets. We initially study a case with two right-handed neutrinos called the 3-2 model. At an energy scale above the right-handed neutrinos, the effective neutrino mass matrix is rank degenerate implying the lightest neutrino is massless. After considering renormalization effects below the two right-handed neutrinos, the effective neutrino mass matrix remains rank degenerate. Next, we study a model with three right-handed neutrinos called the 3-3 model. Above the energy scale of the three right-handed neutrinos, we construct the effective neutrino mass matrix to be rank degenerate. After solving for the renormalization effects to energies below the three right-handed neutrinos, we find the rank of the effective neutrino mass matrix depends on the kernel solutions of the renormalization group equations. We prove for the simplest kernel solutions the effective neutrino mass matrix remains rank degenerate.