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

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, $1$, $1^{\prime }$ and $1^{\prime \prime }$, 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 $M_{\nu }$ (${\mathbf{A}}_{1,2}$, ${\mathbf{B}}_{1,2,3,4}$ and $\mathbf{C}$) can be realized successfully. For the latter case, only five of them (namely ${\mathbf{A}}_{1,2}$, ${\mathbf{B}}_{3,4}$ and $\mathbf{C}$) can be achieved due to the intrinsic structure of the right-handed neutrino mass matrix $M_{\mathrm{R}}$ in our assumption for simplicity.

Comments: 23 pages, 6 tables, references added and version accepted for publication in NPB