Double Cover of Modular $S_4$ for Flavour Model Building

TL;DR

This paper develops a formalism for the double cover of the modular group S_4, constructs modular forms using theta constants, and applies these to build viable lepton flavour models with CP symmetry.

Contribution

It introduces the $S'_4$ double cover formalism, constructs modular forms explicitly, and demonstrates their application in lepton flavour model building.

Findings

Constructed lowest-weight modular forms using theta constants.

Derived $S'_4$ multiplication rules and Clebsch-Gordan coefficients.

Built phenomenologically viable lepton flavour models.

Abstract

We develop the formalism of the finite modular group $\Gamma'_4 \equiv S'_4$, a double cover of the modular permutation group $\Gamma_4 \simeq S_4$, for theories of flavour. The integer weight $k>0$ of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight ($k=1$) modular forms in terms of two Jacobi theta constants, denoted as $\varepsilon(\tau)$ and $\theta(\tau)$, $\tau$ being the modulus. We show that these forms furnish a 3D representation of $S'_4$ not present for $S_4$. Having derived the $S'_4$ multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to $k=10$. These are expressed as polynomials in $\varepsilon$ and $\theta$, bypassing the need to search for non-linear constraints. We further show that within $S'_4$ there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.