Non-holomorphic modular flavor symmetry and odd weight polyharmonic Maaß form

TL;DR

This work extends the non-holomorphic modular flavor symmetry to include odd-weight polyharmonic Maaß forms, enabling a unified, modular-flavor description of quarks and leptons. By constructing integral-weight forms from non-holomorphic Eisenstein series, the authors develop explicit level-$N$ multiplets for $N=3,4,5$ and apply them to construct predictive lepton and quark models based on the finite group $\Gamma'_3\cong T'$. The lepton sector with a type-I seesaw and two right-handed neutrinos achieves good fits for both mass orderings, and a quark sector with a minimal parameter set reproduces masses and CKM parameters; a joint quark–lepton model sharing the modulus $\tau$ yields a highly constrained, 13- or 14-parameter framework with NO favored and testable flavor correlations. Overall, the paper demonstrates that odd-weight polyharmonic Maaß forms provide a powerful, predictive alternative to holomorphic modular models, with potential implications for GUTs and future flavor experiments.

Abstract

We extend the framework of non-holomorphic modular flavor symmetry to include the odd weight polyharmonic Maaß forms. The integer weight polyharmonic Maaß forms of level $N$ can be arranged into multipltets of the homogeneous finite modular group $Γ'_N$. We propose to construct the integer weight, including weight one, non-holomorphic polyharmonic Maaß forms from the non-holomorphic Eisenstein series. The previous results of even weight polyharmonic Maaß 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 $Γ'_3\cong T'$ 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.

Figures (12)

• Figure 1: Allowed regions in the $\tau$ plane for the lepton model with gCP in section \ref{['subsec:lepton_model-seesaw']}. Different color shadings correspond to the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels.
• Figure 2: Allowed regions for the lepton input parameters from the lepton-only (blue) and combined (red) analyses. Different color shadings correspond to the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels.
• Figure 3: Allowed regions for the lepton observables from the lepton-only (blue) and combined (red) analyses. Different color shadings correspond to the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels.
• Figure 4: Allowed regions for the lepton input parameters in case of IO neutrino masses spectrum. Different color shadings correspond to the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels.
• Figure 5: Allowed regions for the lepton observables in case of IO neutrino masses spectrum. Different color shadings correspond to the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels.