Inverse Seesaw Model in Non-holomorphic Modular $A_4$ Flavor Symmetry

TL;DR

This work develops a non-holomorphic modular $A_4$ flavor framework to realize neutrino masses via an inverse seesaw at accessible scales. It constructs three benchmark models with distinct representations and modular weights, then performs a detailed numerical scan of the modulus $τ$ within the fundamental domain under a generalized CP constraint, fitting neutrino oscillation data and charged-lepton masses. The analysis yields region-by-region predictions for the absolute neutrino mass scale, Dirac and Majorana CP phases, and $m_{β}$ and $m_{ββ}$, with clear signatures for upcoming $0νββ$ and precision oscillation experiments. The results demonstrate the viability and testability of non-holomorphic modular flavor approaches, while also underscoring the need to address modulus stabilization in a UV-complete theory.

Abstract

This paper investigates an inverse seesaw model of neutrino masses based on non-holomorphic modular $A_4$ symmetry, extending the framework of modular-invariant flavor models beyond the conventional holomorphic paradigm. After the general theoretical framework is established, three concrete model realizations distinguished by their $A_4$ representation assignments and modular weight configurations for the matter fields are analyzed. Focusing on these three specific realizations, a comprehensive analysis of neutrino phenomenology is performed. By constraining the modulus parameter $τ$ to the fundamental domain and systematically scanning the parameter space, regions compatible with current neutrino oscillation data are identified. The numerical results provide predictions for currently unmeasured quantities, including the absolute neutrino mass scale, Dirac CP-violating phase, and Majorana phases. These predictions establish specific, testable signatures for upcoming neutrino experiments, particularly in neutrinoless double beta decay and precision oscillation measurements. The framework offers a well-defined target for future experimental verification or exclusion, while demonstrating the phenomenological viability of non-holomorphic modular symmetry approaches to flavor structure.

Figures (16)

• Figure 1: Distribution of viable $\tau$ values for model 1 (NO) in the complex plane. The three distinct regions (A, B, and C) correspond to different solutions that reproduce neutrino oscillation data within $3\sigma$. The color gradient represents the $\chi^2$ values, with brighter regions indicating better agreement with experimental data.
• Figure 2: Parameter correlations for model 1 NO in the three $\tau$ regions A, B, and C. The top row panels A(1)-A(6) correspond to region A, the middle row B(1)-B(6) to region B, and the bottom row C(1)-C(4) to region C. The panels display two-dimensional correlations of neutrino parameters including sum of neutrino masses $\sum m_i$, mixing angles $\sin^2\theta_{23}$ and $\sin^2\theta_{13}$, CP-violating phases $\delta_{\rm CP}$, $\eta_1$, and $\eta_2$, absolute neutrino mass $m_1$, the effective Majorana mass $m_{\beta\beta}$, and the modulus $\tau$. The color gradient represents the $\chi^2$ value.
• Figure 3: Prediction of model 1 for the effective Majorana neutrino mass $m_{\beta\beta}$ as a function of the lightest neutrino mass $m_{1}$ under NO. Regions A, B (some overlap with A), and C denote distinct phenomenological regimes. The gray dashed area indicates parameter space favored by NO. Experimental constraints include KamLAND-Zen exclusion limit (brown band, $m_{\beta\beta} < 0.028$–$0.122~\mathrm{eV}$KamLAND-Zen:2024eml), target sensitivities of next-generation $0\nu\beta\beta$ experiments such as LEGEND-1000 (green band, $0.009$–$0.021~\mathrm{eV}$LEGEND:2021bnm) and nEXO (blue band, $0.0047$–$0.0203~\mathrm{eV}$nEXO:2021ujk), as well as cosmological exclusion of $m_{1} \gtrsim 0.037~\mathrm{eV}$ (vertical gray band GAMBITCosmologyWorkgroup:2020rmf).
• Figure 4: Distribution of the modular parameter $\tau$ in model 1 for IO case. There exist three distinct viable regions A, B, and C that reproduce observed oscillation parameters. Regions A and B occupy the high-imaginary regime ($\text{Im}(\tau) > 2$), while region C appears at intermediate values.
• Figure 5: Parameter correlations for model 1 under IO case, organized by three regions A, B, and C. Top three panels exhibit the following: (a) $m_\beta$ vs. $\sum m_i$, (b) $\sum m_i$ vs. $\text{Im}(\tau)$, (c) $m_{\beta\beta}$ vs. Lightest mass $m_3$. Lower panels show region-specific correlations for regions A (A(1)-A(3)), B (B(1)-B(3)), and C (C(1)-C(3)), with columns representing CP phases and $\theta_{23}$ octant dependence relationships respectively.