discussion on the deviation from the tri-bi maximal mixing is in parallel. The $S_4$ symmetry model building is reviewed in [@Bazzocchi:2012st]. In Appendix A, we summarised representations and tensor products of $S_4$ group.

For quark sector, the CKM matrix is close to the identity. Deviation from the identity has a hierarchy structure parametrized by some powers of the Cabibbo angle, $\lambda=\sin\theta_C$. On the other hand, the PMNS matrix, mixing matrix in the lepton sector has large mixing angles. Even the smallest mixing angle, $\theta_{13}$ is in the order of $\lambda$. To explain this, it is natural to assume that $u-$ and $d-$ quark sectors have almost the same structure under the discrete flavor symmetry whereas the charged lepton and the right-handed neutrino sectors do not. This picture can be realised by introducing appropriate ‘flavons’ charged under discrete symmetry group and more symmetries can be introduced to forbid useless couplings. Here, we consider the symmetry group $S_4 \times Z_4 \times {\rm U(1)}_L$, where U(1)$_L$ represents a lepton number, which may be discretized. In this paper, we consider superpotential for See-Saw mechanism with flavons $\Phi$ and $\chi$, [$$\begin{split}W= -\l_{1ij} \bar{E}_i \Phi L_j H_d + \l_{2ij} N_i L_j H_u+\frac12 \l_{3ij} X N_i \chi 