Non-holomorphic modular flavor symmetry

TL;DR

This paper develops a formalism for non-holomorphic modular flavor symmetry using polyharmonic Maa{ }ss forms, extending modular invariance and successfully modeling lepton masses and neutrino oscillations.

Contribution

It introduces a novel approach to modular flavor symmetry with non-holomorphic forms and demonstrates its compatibility with CP symmetry in lepton sector models.

Findings

Lepton masses and neutrino data fit well within the models.

Predictions for CP violation phases are provided.

The formalism extends the modular invariance approach to negative weights.

Abstract

The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level $N$ polyharmonic Maa{\ss} forms satisfying the Laplacian condition. We find that the integer (even) weight polyharmonic Maa{\ss} forms of level $N$ can be decomposed into multiplets of the finite modular group $\Gamma'_N$ ($\Gamma_N$). The original modular invariance approach is extended by the presence of negative weight polyharmonic Maa{\ss} forms. The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry. We present three example models for lepton sector based on the $\Gamma_3\cong A_4$ modular symmetry, the charged lepton masses and the neutrino oscillation data can be accommodated very well, and the predictions for the leptonic CP violation phases and the effective Majorana neutrino mass are studied.