Non-holomorphic modular flavor symmetry and odd weight polyharmonic Maa{\ss} form

Abstract

We extend the framework of non-holomorphic modular flavor symmetry to include the odd weight polyharmonic Maa{\ss} forms. The integer weight polyharmonic Maa{\ss} forms of level $N$ can be arranged into multipltets of the homogeneous finite modular group $\Gamma'_N$. We propose to construct the integer weight, including weight one, non-holomorphic polyharmonic Maa{\ss} forms from the non-holomorphic Eisenstein series. The previous results of even weight polyharmonic Maa{\ss} forms are reproduced. We apply this formalism to address the flavor structure of the standard model. An example lepton model based on the modular group $\Gamma'_3\cong T'$ is constructed, where neutrino masses are generated via type-I seesaw mechanism with two right-handed neutrinos. This model can accommodate the experimental data for both normal and inverted neutrino mass orderings. We further extend this model to include quarks, so that the masses and mixing parameters of both quark and lepton sectors can be successfully described in terms of only thirteen real free parameters. It is the modular invariant model with the smallest number of free parameters so far, only normal ordering neutrino mass is viable after including quarks, and the correlations among the input parameters and flavor observables are analyzed.