Horizontal symmetries of leptons with a massless neutrino

TL;DR

This paper explores how residual symmetries in neutrino and charged lepton mass matrices within finite discrete groups can lead to specific leptonic mixing patterns, especially with a massless neutrino, providing models that approximate realistic mixing angles.

Contribution

It analyzes the structure of residual symmetries in finite groups and demonstrates how they can produce realistic leptonic mixing with a massless neutrino, including explicit examples and models.

Findings

Groups $\Sigma(3N^3)$ reproduce specific columns of $U_{PMNS}$

Inverted hierarchy predicts $ heta_{13}=0$, $ heta_{23}=rac{ p}{4}$ with certain groups

An explicit $S_4(2)$ model yields viable mixing angles with a massless neutrino

Abstract

Residual symmetry $G_\nu$ of neutrino mass matrix with a massless neutrino and embedding of $G_\nu$ and the residual symmetry $G_l$ of the charged lepton mass matrix into finite discrete groups $G$ is discussed. Massless neutrino results if $G_\nu$ and hence $G$ are subgroups of $U(3)$ rather than of $SU(3)$. Structure of the resulting leptonic mixing matrix $U_{PMNS}$ is discussed in three specific examples based on groups (a) $\Sigma(3N^3)$, (b) $\Sigma(2N^2)$ and (c) $S_4(2) \equiv A_4\rtimes Z_4$. $\Sigma(3N^3)$ groups are able to reproduce either the second or the third column of $U_{PMNS}$ correctly. $\Sigma(2N^2)$ groups lead to prediction $\theta_{13}=0$, $\theta_{23}=\frac{\pi}{4}$ for the reactor and atmospheric mixing angles respectively if neutrino mass hierarchy is inverted. Solar angle remains undetermined in this case. This also gets determined when $G=S_4(2)$ which can give bi-maximal mixing for inverted hierarchy. Examples (b) and (c) provide a good zeroth order approximation to realistic leptonic mixing with a massless neutrino. We also present an example of the specific model based on $S_4(2)$ symmetry in which a massless neutrino and viable leptonic mixing angles are obtained.