# Broken $S^{}_{3{\rm L}} \times S^{}_{3{\rm R}}$ Flavor Symmetry and Leptonic CP Violation

## TL;DR

This paper proposes a scenario where breaking an $S_3$ flavor symmetry in the leptonic sector determines the neutrino mixing matrix, predicts a hierarchical neutrino mass spectrum, and links leptonic CP violation to cosmological matter-antimatter asymmetry.

## Contribution

It introduces a simple explicit breaking of $S_3$ flavor symmetry that relates neutrino mixing to mass ratios and connects low- and high-energy CP violation phenomena.

## Key findings

- Neutrino mass spectrum is hierarchical with normal ordering.
- Only the first octant of $ heta_{23}$ is allowed.
- Predicted Dirac CP phase $
egative22^ ext{o}$ deviates from maximal CP violation.

## Abstract

In the framework of canonical seesaw model, we present a simple but viable scenario to explicitly break an $S^{}_{3{\rm L}} \times S^{}_{3{\rm R}}$ flavor symmetry in the leptonic sector. It turns out that the leptonic flavor mixing matrix is completely determined by the mass ratios of charged leptons (i.e., $m^{}_e/m^{}_\mu$ and $m^{}_\mu/m^{}_\tau$) and those of light neutrinos (i.e., $m^{}_1/m^{}_2$ and $m^{}_2/m^{}_3$). The latest global-fit results of three neutrino mixing angles $\{\theta^{}_{12}, \theta^{}_{13}, \theta^{}_{23}\}$ and two neutrino mass-squared differences $\{\Delta m^2_{21}, \Delta m^2_{31}\}$ at the $3\sigma$ level are used to constrain the parameter space of $\{m^{}_1/m^{}_2, m^{}_2/m^{}_3\}$. The predictions for the mass spectrum and flavor mixing are highlighted: (1) The neutrino mass spectrum shows a hierarchical pattern and a normal ordering, e.g., $m^{}_1 \approx 2.2~{\rm meV}$, $m^{}_2 \approx 8.8~{\rm meV}$ and $m^{}_3 \approx 52.7~{\rm meV}$; (2) Only the first octant of $\theta^{}_{23}$ is allowed, namely, $41.8^\circ \lesssim \theta^{}_{23} \lesssim 43.3^\circ$; (3) The Dirac CP-violating phase $\delta \approx -22^\circ$ deviates significantly from the maximal value $-90^\circ$. All these predictions are ready to be tested in the ongoing and forthcoming neutrino oscillation experiments. Moreover, we demonstrate that the cosmological matter-antimatter asymmetry can be explained via resonant leptogenesis, including the individual lepton-flavor effects. In our scenario, the leptonic CP violation at low- and high-energy scales are closely connected.