We study the textures of SM fermion mass matrices and their mixings in a supersymmetric adjoint SU(5) Grand Unified Theory with modular S4 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 S4. 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 τ . 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 𝒴d𝒴μ/𝒴e𝒴s = 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.

Abstract:We propose an S 4 flavor model based on supersymmetric (SUSY) SU(5) GUT. The first and third generations of 10 dimensional representations in SU(5) are all assigned to be 1 1 of S 4 . The second generation of 10 is to be 1 2 of S 4 . Right-handed neutrinos of singlet 1 and three generations of5 are all assigned to be 3 1 of S 4 . The VEVs of two sets of flavon fields are allowed a moderate hierarchy, that is Φ ν ∼ λ c Φ e . Tri-Bimaximal (TBM) mixing can be produced at both leading order (LO) and next to next to leading order (NNLO) in neutrino sector. All the masses of up-type quarks are obtained at LO. We also get the bottom-tau unification m τ = m b and the popular Georgi-Jarlskog relation m µ = 3m s as well as a new mass relation m e = 8 27 m d in which the novel Clebsch-Gordan (CG) factor arises from the adjoint field H 24 . The GUT relation leads to a sizable mixing angle θ e 12 ∼ θ c and the correct quark mixing matrix V CKM can also be realised in the model. The resulting CKM-like mixing matrix of charged leptons modifies the vanishing θ ν 13 in TBM mixing to a large θ PMNS 13 θ c / √ 2, in excellent agreement with experimental results. A Dirac CP violation phase φ 12 ±π/2 is required to make the deviation from θ ν 12 small. We also present some phenomenological numerical results predicted by the model.

Assuming a diagonal Majorana neutrino mass matrix, we investigate the neutrino Yukawa textures which lead to a non-zero reactor mixing angle θ 13 . The neutrino effective coupling matrix κ ef f is pre-diagonalized by a constant mixing pattern V ν with a vanishing θ ν 13 . The resulting pre-diagonal symmetrical matrix κ is set to be four texture zeros with two types of off-diagonal elements nonzero, which is κ 13 and κ 23 , respectively. With the expectation of simple textures we thoroughly classify the linear combinations, α i , β i and γ i of Yukawa elements λ ij in a same row, according to the values vanishing or not. Each set of the classifications can lead to a Yukawa texture which may have implications for the discrete flavor model buildings. We also present a model based on A 4 according to one set of the constraints on the three combinations with a specific choice of a coefficient in Yukawa texture. *