# Seesaw mirroring between light and heavy Majorana neutrinos with the help of the $S^{}_3$ reflection symmetry

## TL;DR

This paper explores how $S_3$ reflection symmetry constrains neutrino mass matrices in the seesaw mechanism, revealing a mirroring relationship between light and heavy neutrinos and implications for flavored leptogenesis.

## Contribution

It classifies neutrino mass matrices under $S_3$ reflection symmetry, uncovering structural similarities and a seesaw mirroring relationship between light and heavy Majorana neutrinos.

## Key findings

- 22 categories of neutrino mass matrices identified
- Structural equalities between $M_\nu$ and $M_{\rm R}$ found
- Flavored leptogenesis is viable in specific categories

## Abstract

In the canonical seesaw mechanism we require the relevant neutrino mass terms to be invariant under the $S^{}_3$ charge-conjugation transformations of left- and right-handed neutrino fields. Then both the Dirac mass matrix $M^{}_{\rm D}$ and the right-handed neutrino mass matrix $M^{}_{\rm R}$ are well constrained, so is the effective light Majorana neutrino mass matrix $M^{}_\nu$ via the seesaw formula. We find that these mass matrices can be classified into 22 categories, among which some textures respect the well-known $\mu$-$\tau$ permutation or reflection symmetry and flavor democracy. It is also found that there exist remarkable structural equalities or similarities between $M^{}_\nu$ and $M^{}_{\rm R}$, reflecting a seesaw mirroring relationship between light and heavy Majorana neutrinos. We calculate the corresponding light neutrino masses and flavor mixing parameters as well as the CP-violating asymmetries in decays of the lightest heavy Majorana neutrino, and show that only the flavored leptogenesis mechanism is possible to work for three categories of $M^{}_{\rm D}$ and $M^{}_{\rm R}$ in the $S^{}_3$ reflection symmetry limit.