Adjoint SU(5) GUT model with Modular $S_4$ Symmetry

Abstract

We study the textures of SM fermion mass matrices and their mixings in a supersymmetric adjoint SU(5) Grand Unified Theory with modular $S_4$ being the horizontal symmetry. The Yukawa entries of both quarks and leptons are expressed by modular forms with lower weights. Neutrino sector has an adjoint SU(5) representation 24 as matter superfield, which is a triplet of $S_4$. The effective light neutrino masses is generated through Type-III and Type-I seesaw mechanism. The only common complex parameter in both charged fermion and neutrino sectors is modulus $\tau$. Down-type quarks and charged leptons have the same joint effective operators with adjoint scalar in them, and their mass discrepancy in the same generation depends on Clebsch-Gordan factor. Especially for the first two generations the respective Clebsch-Gordan factors made the double Yukawa ratio $y_dy_{\mu}/y_ey_s=12$, in excellent agreement with the experimental result. We reproduce proper CKM mixing parameters and all nine Yukawa eigenvalues of quarks and charged leptons. Neutrino masses and MNS parameters are also produced properly with normal ordering is preferred.