Non-Holomorphic $A_4$ Modular Symmetry in Type-I Seesaw: Implications for Neutrino Masses and Leptogenesis

TL;DR

This work develops a non-holomorphic, non-supersymmetric Type-I seesaw model with $A_4$ modular symmetry, where the flavor structure is entirely fixed by the complex modulus $τ$ and a small set of real parameters. It yields predictive textures for charged leptons, Dirac neutrinos, and Majorana RH neutrinos through polyharmonic Maass modular forms, linking low-energy neutrino observables to high-scale leptogenesis. Numerical scans over $τ$ and Yukawa prefactors produce neutrino masses and mixings compatible with NuFIT 6.0 while delivering CP asymmetries and Boltzmann evolution that reproduce the observed baryon asymmetry in both strong and weak washout regimes. The model anticipates an effective Majorana mass in the meV range and sub-meV $m_{ee}$ for NH, placing future neutrinoless double beta decay experiments in a position to test the proposed modular-flavor framework. Overall, it provides a tightly constrained bridge between neutrino flavor physics and cosmology, with distinctive predictions tied to the modulus parameter $τ$.

Abstract

We propose a minimal extension of the Standard Model with right-handed neutrinos, governed by a non-holomorphic $A_{4}$ modular flavor symmetry. Within this model framework, the light neutrino masses are generated via the popular type-I seesaw mechanism in which the structure of the Dirac neutrino Yukawa couplings is decided by nonholomorphic modular forms. Unlike conventional flavor models with ad hoc flavon fields, the structure of Dirac and Majorana mass matrices is entirely determined by a modulus parameter $τ$. We construct the predictive mass matrices for charged leptons, Dirac neutrinos, and right-handed Majorana neutrinos and show the compatibility with neutrino oscillation data by an appropriate choice of input model parameters. We present numerical analysis of two sets of benchmark points explaining neutrino masses while generating the correct amount of baryon asymmetry via thermal leptogenesis. We estimate numerically the values of CP-asymmetry and examine the evolution of the lepton asymmetry by studying Boltzman equations by considering both strong and washout regimes with CP-asymmetry parameter in the range $|\varepsilon_{1}| \sim 10^{-4}$--$10^{-8}$. The model predicts an effective Majorana mass in the few meV range, below current experimental bounds but within reach of next-generation $0νββ$ searches. The key feature of non-holomorphic $A_4$ modular symmetry naturally accommodates non-zero neutrino masses and mixings, minimizes the Yukawa arbitrariness, and establishes a direct connection between high-scale leptogenesis with low-energy neutrino observable parameters, thereby the model provides a testable link between neutrino flavor physics and cosmology.

Figures (13)

• Figure 1: Plots showing the correlation among Re[$Y^{(-2)}_{3,2}$] vs Re[$\tau$] that exist in the upper half plane, $\mathcal{H}$. The "$\textcolor{RedViolet}{\bullet}$" and " $\textcolor{Orchid}{\bm{\times}}$" color coded points correspond to the benchmark points BP1 ($M_D \approx \; \mathcal{O}(1\; GeV)$) and BP2 ($M_D \approx \; \mathcal{O}(10\; GeV)$) respectively as presented in Table.\ref{['tab:BP1']} and Table.\ref{['tab:BP2']}.
• Figure 2: Plots showing the correlation between Re[$\tau$] vs Im[$\tau$]. The "$\textcolor{RedViolet}{\bullet}$" and " $\textcolor{Orchid}{\bm{\times}}$" color coded points correspond to the benchmark points BP1 and BP2 respectively as presented in Table. \ref{['tab:BP1']} and Table. \ref{['tab:BP2']}.
• Figure 3: Correlation plots between $\sin^2\theta_{12}$ and $\sin^2\theta_{13}$ are shown here where the measured values of $\theta_{12}$ and $\theta_{13}$ are taken from the NuFIT6.0 for $3\sigma$ range. The "$\textcolor{RedViolet}{\bullet}$" and " $\textcolor{Orchid}{\bm{\times}}$" color coded points correspond to the benchmark points BP1 and BP2 respectively as presented in Table. \ref{['tab:BP1']} and Table. \ref{['tab:BP2']}.
• Figure 4: Plot showing correlations between between $\sin^2\theta_{12}$ and $\sin^2\theta_{23}$. We have taken the $3\sigma$ values of $\theta_{12}$ and $\theta_{23}$ from NuFIT6.0 (2024). The "$\textcolor{RedViolet}{\bullet}$" and " $\textcolor{Orchid}{\bm{\times}}$" color coded points correspond to the benchmark points BP1 and BP2 respectively as presented in Table. \ref{['tab:BP1']} and Table. \ref{['tab:BP2']}.
• Figure 5: Plots showing the correlation among the planes of $m_1$ and $m_2$ that are measured from our type-I seesaw model in $A_4$ modular symmetry framework. The "$\textcolor{RedViolet}{\bullet}$" and " $\textcolor{Orchid}{\bm{\times}}$" color coded points correspond to the benchmark points BP1 and BP2 respectively as presented in Table. \ref{['tab:BP1']} and Table. \ref{['tab:BP2']}.