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

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 ${\tilde{\Gamma }}_{4N}$. 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 ${\tilde{\Gamma }}_4$. We present three typical models with ${\tilde{\Gamma }}_4$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.

44 pages, 1 figure