# Multiple modular symmetries as the origin of flavour

## TL;DR

This paper introduces a formalism for multiple modular symmetries in particle physics, applying it to neutrino and lepton sectors, resulting in a model that explains lepton mixing patterns without flavons.

## Contribution

It develops a novel formalism for multiple moduli with associated modular symmetries and demonstrates its application to explain lepton mixing patterns.

## Key findings

- Achieves trimaximal TM1 lepton mixing consistent with data
- Uses multiple moduli and residual symmetries to break the original symmetry
- Provides a flavon-free explanation for lepton flavor structure

## Abstract

We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries $S_4^A$, $S_4^B$ and $S_4^C$, associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bi-triplet scalars to the diagonal $S_4$ subgroup. The low energy effective theory involves the three independent moduli fields $\tau_A$, $\tau_B$ and $\tau_C$, which preserve the residual modular subgroups $Z_3^A$, $Z_2^B$ and $Z_3^C$, in their respective sectors, leading to trimaximal TM$_1$ lepton mixing, consistent with current data, without flavons.