Fermion Hierarchies in $SU(5)$ Grand Unification from ${\Gamma }_6^{\prime }$ Modular Flavor Symmetry

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 $Z_6^T$ 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.

Comments: 18 pages