Zero modes of fermions with general mass matrix

TL;DR

This paper investigates fermionic zero modes in a model with general neutrino mass matrices, revealing conditions under which zero modes exist or vanish, and linking these to topological properties of the background string configurations.

Contribution

It provides a comprehensive analysis of zero energy solutions of the Dirac equation with general mass matrices, extending understanding of topological sectors in neutrino mass models.

Findings

Neutrino and electron zero modes exist if either Majorana mass vanishes.

Large harmonic mean of Majorana masses suppresses neutrino zero modes.

Presence of a single zero mode implies a topologically distinct sector.

Abstract

We analyze zero energy solutions of the Dirac equation in the background of a string-like configuration in an extension of the standard model which accommodates the most general fermionic mass matrix for neutrinos. If either the left- or the right-handed Majorana mass vanishes, neutrino and electron zero modes are found to exist. If the harmonic mean of the neutrino Majorana masses is large compared to the Dirac mass, we can prove that normalizable neutrino zero modes cease to exist. This leads to an odd number of fermionic zero modes -- namely, only the electron zero mode -- which we argue implies that the bosonic background is in a topologically distinct sector from the vacuum. This is confirmed by noting that the bosonic sector of the model involves the breaking of a global U(1) symmetry and hence possesses topological global U(1) strings.