# New $A_4$ lepton flavor model from $S_4$ modular symmetry

## TL;DR

This paper develops a new lepton flavor model based on $A_4$ symmetry derived from the $S_4$ modular group, successfully explaining neutrino masses and mixing parameters with specific predictions.

## Contribution

It introduces a novel $A_4$ flavor model from $S_4$ modular symmetry, detailing the derivation of modular forms and neutrino mass matrix construction.

## Key findings

- Viable neutrino mass matrices for NH and IH.
- Predictions for $ heta_{23}$ and $ ext{CP}$ phase depending on neutrino mass sum.
- Decomposition of $S_4$ modular forms into $A_4$ representations.

## Abstract

We study a flavor model with $A_4$ symmetry which originates from $S_4$ modular group. In $S_4$ symmetry, $Z_2$ subgroup can be anomalous, and then $S_4$ can be violated to $A_4$. Starting with a $S_4$ symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only $A_4$ symmetry when $Z_2$ in $S_4$ is anomalous. We obtain modular forms of two singlets and a triplet representations of $A_4$ by decomposing $S_4$ modular forms into $A_4$ representations. We propose a new $A_4$ flavor model of leptons by using those $A_4$ modular forms. We succeed in constructing a viable neutrino mass matrix through the Weinberg operator for both normal hierarchy (NH) and inverted hierarchy (IH) of neutrino masses. Our predictions of the CP violating Dirac phase $\delta_{CP}$ and the mixing $\sin^2\theta_{23}$ depend on the sum of neutrino masses for NH.