A modular $A_4$ symmetry realization of two-zero textures of the Majorana neutrino mass matrix

TL;DR

This paper presents a modular $A_4$ symmetry framework to realize two-zero textures in the Majorana neutrino mass matrix without flavons, successfully covering all viable textures in the Weinberg operator case and most in the seesaw case.

Contribution

It introduces a modular $A_4$ invariant model that achieves two-zero textures in neutrino mass matrices without flavons, expanding the understanding of neutrino mass structures.

Findings

All seven two-zero textures are realizable with the Weinberg operator.

Five of seven textures are achievable with the type-I seesaw mechanism.

The model simplifies the realization of neutrino mass textures without additional flavon fields.

Abstract

We show how to realize two-zero textures of the Majorana neutrino mass matrix $M_\nu$ based on modular $A_4$ invariant models without flavons. In these models, all matter fields are assigned to three inequivalent singlets, ${\bf 1}$, ${\bf 1^\prime}$ and ${\bf 1^{\prime\prime}}$, of the finite modular group $\Gamma_3 \simeq A_4$. Considering tensor products of the $A_4$ group, it is easy to make the charged lepton mass matrix $M_\ell$ diagonal. Since not all modular forms of a specific weight and level 3 can be arranged into three inequivalent singlets of $A_4$ simultaneously, we can always make some entries in $M_\nu$ vanish by properly assigning the representations and modular weights for the matter fields. We consider two cases where neutrino masses originate from the Weinberg operator and the type-I seesaw mechanism, respectively. For the former case, all seven viable two-zero textures of $M_\nu$ (${\bf A_{1,2}}$, ${\bf B_{1,2,3,4}}$ and ${\bf C}$) can be realized successfully. For the latter case, only five of them (namely ${\bf A_{1,2}}$, ${\bf B_{3,4}}$ and ${\bf C}$) can be achieved due to the intrinsic structure of the right-handed neutrino mass matrix $M_{\rm R}$ in our assumption for simplicity.