Modular symmetry at level 6 and a new route towards finite modular groups

TL;DR

This paper introduces a new way to construct finite modular groups using quotient of principal congruence subgroups, explores their properties, and applies them to models of lepton masses and mixing.

Contribution

It presents a novel construction of finite modular groups from principal congruence subgroup quotients and applies this framework to develop models of lepton masses and mixing.

Findings

Comprehensive study of b3  modular symmetry.

Development of five lepton mass and mixing models using b3  symmetry.

Construction of a benchmark model with b3(2)/b3(6) d T' group.

Abstract

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $\Gamma(N')/\Gamma(N")$, and the modular group $SL(2,\mathbb{Z})$ is extended to a principal congruence subgroup $\Gamma(N')$. The original modular invariant theory is reproduced when $N'=1$. We perform a comprehensive study of $\Gamma'_6$ modular symmetry corresponding to $N'=1$ and $N"=6$, five types of models for lepton masses and mixing with $\Gamma'_6$ modular symmetry are discussed and some example models are studied numerically. The case of $N'=2$ and $N"=6$ is considered, the finite modular group is $\Gamma(2)/\Gamma(6)\cong T'$, and a benchmark model is constructed.