Modular $S_4$ and $A_4$ Symmetries and Their Fixed Points: New Predictive Examples of Lepton Mixing

TL;DR

The paper develops a modular-flavor framework where a finite modular group $\Gamma_N$ acts on the modulus $\tau$, and Yukawa couplings are modular forms whose values fix residual symmetries at special points. Focusing on level $N=4$ ($S_4$) and level $N=3$ ($A_4$), it derives the fixed points in the fundamental domain and their conjugacy classes, then analyzes the resulting triplet modular forms at these points up to weight 6. Applying these alignments to lepton mixing in tri-direct modular models with two and three right-handed neutrinos, the authors obtain three phenomenologically viable two-RHN cases, including a Littlest Modular Seesaw with $n=1+\sqrt{6}$, and construct several $N=3$/$A_4$ examples. The results demonstrate that modular symmetry with residual fixed-point structures can yield highly predictive lepton mixing patterns (notably TM1) and concrete mass spectra, without flavon fields, underscoring the practical impact of modular invariance in flavor physics.

Abstract

In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup $Γ_N$ of the modular symmetry, broken by the vacuum expectation value (VEV) of a modulus field $τ$. If the VEV of the modulus $τ$ takes some special value, a residual subgroup of $Γ_N$ would be preserved. We derive the fixed points $τ_S=i$, $τ_{ST}=(-1+i\sqrt{3})/2$, $τ_{TS}=(1+i\sqrt{3})/2$, $τ_T=i\infty$ in the fundamental domain which are invariant under the modular transformations indicated. We then generalise these fixed points to $τ_f=γτ_S$, $γτ_{ST}$, $γτ_{TS}$ and $γτ_{T}$ in the upper half complex plane, and show that it is sufficient to consider $γ\inΓ_{N}$. Focussing on level $N=4$, corresponding to the flavour group $S_4$, we consider all the resulting triplet modular forms at these fixed points up to weight 6. We then apply the results to lepton mixing, with different residual subgroups in the charged lepton sector and each of the right-handed neutrinos sectors. In the minimal case of two right-handed neutrinos, we find three phenomenologically viable cases in which the light neutrino mass matrix only depends on three free parameters, and the lepton mixing takes the trimaximal TM1 pattern for two examples. One of these cases corresponds to a new Littlest Modular Seesaw based on CSD$(n)$ with $n=1+\sqrt{6}\approx 3.45$, intermediate between CSD$(3)$ and CSD$(4)$. Finally, we generalize the results to examples with three right-handed neutrinos, also considering the level $N=3$ case, corresponding to $A_4$ flavour symmetry.

Figures (2)

• Figure 1: The fixed points of the modular group, it is impossible to display all of them because there are infinite fixed points. The red region and yellow region are the fundamental domains of $\overline{\Gamma}$ and $\overline{\Gamma}(4)$ respectively. The fixed points are displayed in solid (hollow) circles and diamonds in (outside) the fundamental domain of $\overline{\Gamma}(4)$.
• Figure 2: The contour plots of $\sin^2\theta_{12}$, $\sin^2\theta_{13}$, $\sin^{2}\theta_{23}$ and $m^2_{2}/m^2_{3}$ in the $\eta/\pi-r$ plane for cases $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$. The cyan, red, green and blue areas denote the $3\sigma$ regions of $\sin^{2}\theta_{23}$, $\sin^{2}\theta_{13}$ and $m_{2}^{2}/m_{3}^{2}$ respectively. The solid lines denote the 3 sigma upper bounds, the thin lines denote the 3 sigma lower bounds and the dashed lines refer to their best fit values, as adopted from NuFIT 4.1 Esteban:2018azc.