# Modular $A_4$ Symmetry Models of Neutrinos and Charged Leptons

## TL;DR

This paper explores modular $A_4$ symmetry models for neutrino and charged lepton masses, introducing multiple charged lepton representations and modular weights, resulting in numerous viable models fitting experimental data.

## Contribution

It extends previous $A_4$ modular symmetry models by allowing varied charged lepton representations and weights, significantly increasing the number of viable neutrino mass models.

## Key findings

- Fourteen models fit the neutrino oscillation data for both normal and inverted ordering.
- Introduction of multiple charged lepton representations yields ten distinct Yukawa matrices.
- The analysis broadens the landscape of modular symmetry models compatible with experimental observations.

## Abstract

We present a comprehensive analysis of neutrino mass and lepton mixing in theories with $A_4$ modular symmetry, where the only flavon field is the single modulus field $\tau$, and all masses and Yukawa couplings are modular forms. Similar to previous analyses, we discuss all the simplest neutrino sectors arising from both the Weinberg operator and the type I seesaw mechanism, with lepton doublets and right-handed neutrinos assumed to be triplets of $A_4$. Unlike previous analyses, we allow right-handed charged leptons to transform as all combinations of $\mathbf{1}$, $\mathbf{1}'$ and $\mathbf{1}''$ representations of $A_4$, using the simplest different modular weights to break the degeneracy, leading to ten different charged lepton Yukawa matrices, instead of the usual one. This implies ten different Weinberg models and thirty different type I seesaw models, which we analyse in detail. We find that fourteen models for both NO and IO neutrino mass ordering can accommodate the data, as compared to one in previous analyses, providing many new possibilities.