---------------- ---------------- ---------- ------- --------- -------- ----------------
    $A_{4}$    1, $1^{\prime}, 1^{\prime \prime}$                                  1,$1''$,$1'$                                        1                3             3         1        3        1            1
    $Z_{4}$                    1                                                        1                                              1                1             1         1        1        i            -1
   $SU(2)_L$                   2                                                        1                                              2                2             1         1        1        1            2
   $U(1)_Y$              $\frac{1}{2}$                                                  1                                        $-\frac{1}{2}$   $-\frac{1}{2}$      0         0        0        0      $-\frac{1}{2}$

  : \[tab:a\] Field content and transformation properties under $A_4 \times Z_4 $ symmetry of the model.

The renormalisable Yukawa Lagrangian for leptons can be written as $$\begin{aligned}
\mathcal{L}_Y & \supset Y_e \overline{L_e} H e_R +Y_{\mu} \overline{L_{\mu}} H \mu_R + Y_{\tau} \overline{L_{\tau}} H \tau_R + (Y_{\nu 1} \overline{L_e}+Y_{\nu 2} \overline{L_{\mu}}+Y_{\nu 3} \overline{L_{\tau}}) \tilde{H}_T N_R \nonumber \\
& + (Y_{\xi}\xi+Y_N \phi_N) N_R N_R +{\rm h.c.}\end{aligned}$$ where $\tilde{H}_T = \tau_2 H^*_T$. Due to the additional $Z_4$ symmetry, the fields $\psi, \eta$ do not have any renormalisable coupling with SM leptons or right handed neutrinos. We assume $\psi$ be vector like so that a bare mass term $M_{\psi} \bar{\psi} \psi$ is allowed in the Lagrangian. As mentioned earlier, gravity is not supposed to obey such global symmetries and hence we can write down Planck suppressed terms which explicitly break both $A_4$ and $Z_4$. The dimension five terms involving one of the $A_4$ triplet flavon 