SUSY SU(5) $\times S_{4}$ GUT Flavor Model for Fermion Masses and Mixings with Adjoint, Large $θ^{PMNS}_{13}$

Abstract

We propose an $S_{4}$ flavor model based on supersymmetric (SUSY) SU(5) GUT. The first and third generations of \textbf{10} dimensional representations in SU(5) are all assigned to be $1_{1}$ of $S_{4}$. The second generation of \textbf{10} is to be $1_{2}$ of $S_{4}$. Right-handed neutrinos of singlet \textbf{1} and three generations of $\bar{\textbf{5}}$ 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 $\langleΦ^ν\rangle \sim λ_{c}\langleΦ^{e}\rangle$. 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 $m_{e}=\frac{8}{27}m_{d}$ in which the novel Clebsch-Gordan (CG) factor arises from the adjoint field $H_{24}$. The GUT relation leads to a sizable mixing angle $θ^{e}_{12} \sim θ_{c}$ 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 $θ^{PMNS}_{13}\simeqθ_{c}/\sqrt{2}$, in excellent agreement with experimental results. A Dirac CP violation phase $ϕ_{12}\simeq\pmπ/2$ is required to make the deviation from $θ^ν_{12}$ small. We also present some phenomenological numerical results predicted by the model.