Minimal lepton models with non-holomorphic modular $A_{4}$ symmetry

TL;DR

The paper develops and tests a comprehensive set of minimalist lepton models based on the non-holomorphic $A_{4}$ modular symmetry at level $3$, where Yukawa couplings are polyharmonic Maaß forms and neutrino masses arise from the Weinberg operator. It systematically classifies all viable assignments of lepton representations and modular weights, yielding 1820 potential models, and then evaluates their compatibility with current lepton-sector data via a global $ ext{χ}^{2}$ analysis against NuFIT 6.0. Without gCP symmetry, 147 NO and 6 IO models are viable; with gCP, these numbers shrink to 47 NO and 5 IO, demonstrating substantial predictive power. A representative model is analyzed in detail, illustrating tight correlations among input parameters, mixing angles, CP phases, and neutrino masses, and showing how future JUNO, DUNE, Hyper-K, and $0 uetaeta$ experiments can decisively test this class of theories. Overall, the work provides a flavon-free, highly predictive framework for lepton flavor that connects modular symmetry, Maaß forms, and neutrino phenomenology in a testable way.

Abstract

We present a comprehensive bottom-up analysis of lepton mass and mixing based on the non-holomorphic $A_{4}$ modular symmetry. Neutrinos are assumed to be Majorana particles and the light neutrino masses are generated through the Weinberg operator. In this framework, we construct all phenomenologically viable models with minimal number of free parameters, where the Yukawa couplings are expressed in terms of polyharmonic Maaß forms of weights $\pm4$, $\pm2$ and $0$ at level $N=3$. Without imposing generalized CP (gCP) symmetry, we identify 147 (6) viable models with seven real free parameters that successfully reproduce the current experimental data of lepton sector for the normal (inverted) mass ordering. When gCP symmetry consistent with $A_{4}$ modular symmetry is included, the number of free parameters is reduced by one, yielding 47 (5) phenomenologically viable models in the normal (inverted) mass ordering. Finally, we present detailed numerical analyses of a representative model for both mass orderings to illustrate these results.

Figures (7)

• Figure 1: The best fit values of modulus $\tau$ for 147 (47) and 6 (5) viable models in the case without (with) gCP symmetry for NO and IO neutrino mass spectra, respectively.
• Figure 2: The results of the best fit values of the minimum value of $\chi^2$, the three lepton mixing angles and three CP-violating phases for viable models without (147 models) and with (47 models) gCP symmetry in the NO case. The red dashed lines in the first four panels represent the best fit values, and the light blue bounds represent the $1\sigma$ and $3\sigma$ ranges from NuFIT 6.0 with Super-Kamiokande atmospheric data Esteban:2020cvm. The lighter green band in the panel of $\sin^{2}\theta_{12}$ is the prospective $3\sigma$ range after 6 years of JUNO running JUNO:2022mxj. The lighter green regions in the panels of $\sin^{2}\theta_{23}$ and $\delta_{CP}$ are the resolution in degrees after 15 years of DUNE running DUNE:2020ypp for true values of them corresponding to their best fit values given by NuFIT 6.0.
• Figure 3: The results of the best fit values of the minimum value of $\chi^2$, the three lepton mixing angles and three CP-violating phases for all viable models without and with gCP symmetry in the IO case.
• Figure 4: The best fit values of the minimum value of $\chi^2$, the effective mass $m_{\beta\beta}$ in $0\nu\beta\beta$-decay, the kinematical mass $m_{\beta}$ in beta decay and the three neutrino mass sum $\sum_{i=1}^{3} m_{i}$. In the panel of the neutrino mass sum $\sum_{i=1}^{3} m_{i}$, the vertical bands indicate the current most stringent limit $\sum_{i=1}^{3} m_{i}<120\,\text{meV}$ from the Planck $+$ lensing $+$ BAO Planck:2018vyg, the next-generation experiments sensitivity ranges $\sum_{i=1}^{3} m_{i}<(44-76)\,\text{meV}$ of Euclid+CMB-S4+LiteBIRD Euclid:2024imf, and the red dashed line represents the limitation of the NO case ($\sum_{i=1}^{3} m_{i}\geq 57.75\,\text{meV}$). In the panel of the kinematical mass $m_{\beta}$ in beta decay, the gray region represents Project 8 future bound ($m_{\beta}<0.04\,\text{meV}$) Project8:2022wqh. In the panel of the effective Majorana mass $m_{\beta\beta}$, the vertical bands indicate the latest result $m_{\beta\beta}<(28-122)\,\text{meV}$ of KamLAND-Zen KamLAND-Zen:2024eml, and the next-generation experiments sensitivity ranges $m_{\beta\beta}<(9-21)\,\text{meV}$ from LEGEND-1000 LEGEND:2021bnm and $m_{\beta\beta}<(4.7-20.3)\,\text{meV}$ from nEXO nEXO:2021ujk.
• Figure 5: The best fit values of the minimum value of $\chi^2$, the effective mass $m_{\beta\beta}$, the kinematical mass $m_{\beta}$ and the mass sum $\sum_{i=1}^{3} m_{i}$.