Modular flavor symmetry and vector-valued modular forms

TL;DR

This paper extends the concept of modular flavor symmetry by incorporating vector-valued modular forms, enabling the construction of new finite modular groups and lepton mass models with simplified invariant structures.

Contribution

It introduces a general framework using vector-valued modular forms for finite modular groups, expanding the toolkit for flavor symmetry models in particle physics.

Findings

Constructed two new lepton mass models based on finite groups

Listed finite modular groups up to order 72

Provided a differential equation method for modular multiplet construction

Abstract

We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including $\Gamma_N$ and $\Gamma'_N$ as the flavor symmetry. The theory of vector-valued modular forms provide a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-valued modular forms. The general finite modular groups are listed up to order 72. We apply the formalism to construct two new lepton mass models based on the finite modular groups $A_4\times Z_2$ and $GL(2,3)$.