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 $\mathrm{\ensuremath{\Gamma}}(4N)$ and show that they can be decomposed into the irreducible multiplets of finite metaplectic group ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{4N}$. We construct concrete expressions of the half-integral/integral modular forms for $\mathrm{\ensuremath{\Gamma}}(4)$ up to weight 6 and arrange them into the irreducible multiplets of ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{4}$. We present three typical models with ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Gamma}}}}_{4}$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.