Half-integral weight modular forms and application to neutrino mass models

TL;DR

This paper extends the modular invariance framework to include half-integral weight modular forms, constructing explicit examples and applying them to neutrino mass models with phenomenological analysis.

Contribution

It introduces a consistent theory of half-integral weight modular forms for congruence subgroups and applies this to develop novel neutrino mass models with modular symmetry.

Findings

Constructed explicit half-integral weight modular forms up to weight 6.

Decomposed forms into irreducible representations of the metaplectic group.

Presented and analyzed three neutrino mass models with phenomenological predictions.

Abstract

We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral weight modular forms for congruence subgroup $\Gamma(4N)$ and show that they can be decomposed into the irreducible multiplets of finite metaplectic group $\widetilde{\Gamma}_{4N}$. We construct concrete expressions of the half-integral/integral modular forms for $\Gamma(4)$ up to weight 6 and arrange them into the irreducible representations of $\widetilde{\Gamma}_4$. We present three typical models with $\widetilde{\Gamma}_4$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.