N_j,\end{split}$$]{} where $i,j=1,2,3$ are the generation indices and $X$ is a SUSY breaking spurion. For this, $S_4$, $Z_4$, U(1)$_L$ and U(1) R-symmetry quantum numbers are given in Table \[table:charges\].

The charged lepton Yukawa couplings can be constructed from $\bar{E}\Phi L H_d$, the neutrino Dirac Yukawa coupling can be constructed from $NL$, and the Majorana mass of the heavy neutrinos can be constructed from $X N\chi N$. On the other hand, $\Phi^2$, $\chi^2$, and $\Phi\chi$ cannot couple to the combinations $\bar{E} L H_d$, $NL$, and $XNN$ to make singlets. Note that U(1)$_R$ is introduced to forbid unwanted coupling $ N \chi N$, which makes $B_N$ in a matrix form, not a constant.

The discrete symmetry quantum number can be extended to the quark sector, such as $Q: ({\bf 3},1,1, 1, 1)$, $\bar{U} : ({\bf 2+1},2,0, 0)$, and $\bar{D} : ({\bf 2+1},2,0, 0)$ under $S_4\times Z_4\times {\rm U(1)}_L \times$U(1)$_R$. The flavons $\Phi : ({\bf 3+3^\prime}, 1,0)$ make the singlet combinations ${\bar U}\Phi Q H_u+{\bar D}\Phi Q H_d$ and Yukawa couplings $Y_U$ and $Y_D$ have the same form as the charged lepton Yukawa coupling. They are diagonalized by the same unitary matrix so CKM matrix is the identity in the leading order. If 