Modular invariant flavor model of $\rm A_4$ and hierarchical structures at nearby fixed points

Abstract

In the modular invariant flavor model of $\rm A_4$, we study the hierarchical structure of lepton/quark flavors at nearby fixed points of $\tau=i$ and $\tau=\omega$ of the modulus, which are in the fundamental domain of $\rm PSL(2,\mathbb{Z})$. These fixed points correspond to the residual symmetries $\mathbb{Z}_2^{S}=\{I, S \}$ and $\mathbb{Z}_3^{S}=\{ I, ST, (ST)^2 \}$ of $\rm A_4$, where $S$ and $T$ are generators of the $A_4$ group. The infinite $\tau= i \infty$ also preserves the residual symmetry of the subgroup $\mathbb{Z}^T_3=\{ I,T,T^2 \}$ of $\rm A_4$. We study typical two-type mass matrices for charged leptons and quarks in terms of modular forms of weights $2$, $4$ and $6$ while the neutrino mass matrix with the modular forms of weight $4$ through the Weinberg operator. Linear modular forms are obtained approximately by performing Taylor expansion of modular forms around fixed points. By using them, the flavor structure of the lepton and quark mass matrices are examined at nearby fixed points. The hierarchical structure of these mass matrices is clearly shown in the diagonal base of $S$, $T$ and $ST$. The observed PMNS and CKM mixing matrices can be reproduced at nearby fixed points in some cases of mass matrices. By scanning model parameters numerically at nearby fixed points, our discussion are confirmed for both the normal hierarchy and inverted one of neutrino masses. Predictions are given for the sum of neutrino masses and the CP violating Dirac phase of leptons at each nearby fixed point.