Fermion Masses and Mixing from Double Cover and Metaplectic Cover of $A_5$ Modular Group

TL;DR

This paper explores the $A'_5$ double cover of the $A_5$ modular group, constructing modular forms and applying them to lepton and quark mass models, resulting in several phenomenologically viable models with minimal free parameters.

Contribution

It introduces the construction of level 5 modular forms, extends the modular symmetry framework with rational weights, and develops new lepton and quark mass models based on $A'_5$ and its metaplectic extension.

Findings

15 models fit lepton data with 9 parameters

9 models fit with generalized CP symmetry and 7 parameters

A neutrino mass model with specific phenomenological predictions

Abstract

We perform a comprehensive study of the homogeneous finite modular group $A'_5$ which is the double covering of $A_5$. The integral weight and level 5 modular forms have been constructed up to weight 6 and they are decomposed into the irreducible representations of $A'_5$. Then we perform a systematical analysis of the $A'_5$ modular models for lepton masses and mixing. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find out 15 models with 9 real free parameters which can accommodate the experimental data of lepton sector. After including generalized CP symmetry, 9 viable models with 7 free parameters are found out. We apply $A'_5$ modular symmetry to the quark sector, and a quark-lepton unification model is given. The framework of modular invariance is extended to include the rational weight modular forms of level 5. The ring of modular forms at level 5 can be generated by two algebraically independent weight $1/5$ modular forms denoted by $F_1(\tau)$ and $F_2(\tau)$. We give the expressions of the rational weight modular forms of level 5 up to weight $3$ and arrange them into the irreducible multiplets of finite metaplectic group $\widetilde{\Gamma}_5\cong A'_5\times Z_5$. A neutrino mass model with $\widetilde{\Gamma}_5$ modular symmetry is presented, and the phenomenological predictions of the model are analyzed numerically.