SUSY SU(5) $\times S_4$ GUT Flavor Model for Fermion Masses and Mixings with Adjoint, Large ${\theta }_{13}^{PMNS}$

We propose an $S_4$ flavor model based on supersymmetric (SUSY) SU(5) GUT. The first and third generations of $\mathbf{\text{10}}$ dimensional representations in SU(5) are all assigned to be $1_1$ of $S_4$. The second generation of $\mathbf{\text{10}}$ is to be $1_2$ of $S_4$. Right-handed neutrinos of singlet $\mathbf{\text{1}}$ and three generations of $\overline{\mathbf{\text{5}}}$ 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 $\langle\Phi^{\nu}\rangle \sim \lambda_{c}\langle\Phi^{e}\rangle$. 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 $m_{\tau }=m_b$ and the popular Georgi-Jarlskog relation $m_{\mu }=3m_s$ 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 ${\theta }_{12}^e\sim {\theta }_c$ 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 ${\theta }_{13}^{\nu }$ in TBM mixing to a large ${\theta }_{13}^{PMNS}\simeq {\theta }_c\mathrm{/}\sqrt{2}$, in excellent agreement with experimental results. A Dirac CP violation phase ${\varphi }_{12}\simeq \pm \pi \mathrm{/}2$ is required to make the deviation from ${\theta }_{12}^{\nu }$ small. We also present some phenomenological numerical results predicted by the model.