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

TL;DR

This paper extends non-holomorphic modular flavor symmetry to odd weight polyharmonic Maa{ ss} forms, constructing models that successfully describe quark and lepton masses and mixings with minimal free parameters, consistent with experimental data.

Contribution

It introduces a novel framework for odd weight polyharmonic Maa{ ss} forms within modular flavor symmetry, enabling minimal-parameter models of fermion masses and mixings.

Findings

Constructed a lepton model based on b3'_3a0 with neutrino masses via type-I seesaw.

Extended the model to include quarks, describing masses and mixings with only 13 free parameters.

Found that only normal neutrino mass ordering is viable in the combined quark-lepton model.

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.