Quark masses and CKM hierarchies from $S_4^\prime$ modular flavor symmetry

TL;DR

This paper develops models using $S_4^ extprime$ modular flavor symmetry to explain quark mass hierarchies and CKM mixing, emphasizing the role of the modulus $ au$ and residual symmetries, with potential explanations for coefficient hierarchies via an additional $S_3$ symmetry.

Contribution

The paper introduces specific models with $S_4^ extprime$ modular symmetry that successfully reproduce quark hierarchies and mixing patterns, highlighting the importance of residual symmetries and modular weights.

Findings

Quark hierarchies are achieved in select quark representation cases.

Models explain both small and large top-to-bottom Yukawa ratios.

Additional $S_3$ symmetry accounts for coefficient hierarchies.

Abstract

We propose models to explain the hierarchies of the quark masses and mixing by utilizing the $S_4^\prime$ modular flavor symmetry. The hierarchy is realized by the modulus $\tau$ stabilized at $\mathrm{Im}\,\tau \gg 1$, where the residual $Z_4^T$ symmetry is approximately unbroken and the Froggatt-Nielsen mechanism works. It is found that the quark hierarchies are realized only in a few cases of quark representations. We study two models with assigning the modular weights, so that the observed quark hierarchies are explained in the cases of both small and large ratios of the top to bottom Yukawa couplings. We also argue that $\mathcal{O}({0.1})$ hierarchies of the $\mathcal{O}({1})$ coefficients can be explained by imposing another $S_3$ modular symmetry.