# Phase broken $\mu-\tau$ symmetry and the neutrino mass hierarchy

## TL;DR

This paper explores how phase deviations from exact $- au$ symmetry in the neutrino mass matrix can explain neutrino mixing angles and mass hierarchy, providing testable predictions and constraints on neutrino parameters.

## Contribution

It introduces a phase-breaking mechanism of $- au$ symmetry in neutrino mass matrices, linking phase deviations to observable neutrino mixing and mass hierarchy predictions.

## Key findings

- Allows non-zero $ heta_{13}$ through phase deviations
- Predicts a relationship between $|V_{\mu3}|$ and $|V_{e3}|$
- Provides bounds on the lightest neutrino mass and $m_{ee}$

## Abstract

Inspired by the neutrino oscillations data, we consider the exact $\mu-\tau$ symmetry, implemented at the level of the neutrino mass matrix, as a good initial framework around which to study and describe neutrino phenomenology. Working in the diagonal basis for the charged leptons, we deviate from $\mu-\tau$ symmetry by just modifying the phases of the neutrino mass matrix elements. This deviation is enough to allow for a non-vanishing neutrino mixing entry $|V_{e3}|$ (i.e. $\theta_{13}$) but it also gives a very stringent (and eventually falsifiable) prediction for the atmospheric neutrino mixing element $|V_{\mu3}|$ as a function of $|V_{e3}|$. The breaking by phases is characterized by a single phase and is shown to lead to interesting lower bounds on the allowed mass of the lightest neutrino depending on the ordering of neutrino masses (normal or inverted) and on the value of the Dirac ${\cal CP}$ violating phase $\delta_{CP}$. The allowed parameter space for the effective Majorana neutrino mass $m_{ee}$ is also shown to be non-trivially constrained.