Fermion Hierarchies in $SU(5)$ Grand Unification from $\Gamma_6^\prime$ Modular Flavor Symmetry

TL;DR

This paper proposes a model using the $ ext{Gamma}_6^ ext{'}$ modular flavor symmetry within $SU(5)$ GUT to explain fermion mass hierarchies through a Froggatt-Nielsen-like mechanism, matching observed data.

Contribution

It introduces the minimal $ ext{Gamma}_6^ ext{'}$ symmetry framework for fermion hierarchies in $SU(5)$ GUT, with specific representations and modular weights, supported by numerical coefficient fits.

Findings

Successfully reproduces fermion mass hierarchies

Identifies $ ext{Gamma}_6^ ext{'}$ as minimal symmetry for up-type quark hierarchies

Provides numerical coefficients consistent with observed data

Abstract

We construct a model in which the hierarchies of the quark and lepton masses and mixing are explained by the $\Gamma_6^\prime$ modular flavor symmetry. The hierarchies are realized by the Froggatt-Nielsen-like mechanism due to the residual $Z^T_6$ symmetry, approximately unbroken at $\tau \sim i\infty.$ We argue that the $\Gamma_6^{(\prime)}$ symmetry is the minimal possibility to realize the up-type quark mass hierarchies, since the Yukawa matrix is symmetric. We find a combination of the representations and modular weights and then show numerical values of $\mathcal{O}(1)$ coefficients for the realistic fermion hierarchies.