particle which decays only to $\psi$ due to the Planck scale effects on $Z_2$ symmetry breaking.

Lepton masses and mixings {#sec2}
=========================

Up to leading order, owing to the $A_4$ symmetry, one can write the charged lepton mass matrix in the diagonal form as $$M_{\ell}=\left(\begin{array}{ccc}
Y_e& 0&0\\
0& Y_{\mu} & 0 \\
0 & 0 & Y_{\tau}
\end{array}\right)v_1
\label{mclepton}$$ where $v_1$ is the vacuum expectation value (VEV) of the neutral component of the Higgs doublet $H$. We use the $A_4$ product rules given in appendix \[appen1\] in order to evaluate the forms of different mass matrices at renormalisable level. Considering the vacuum alignment of $A_4$ triplet scalar field $H_T = (v_2, 0, 0)$, the Dirac neutrino mass matrix at leading order can be written as, $$M_D^0=\left(\begin{array}{ccc}
Y_{\nu 1}v_2& 0&0\\
0& 0 &Y_{\nu 2}v_2 \\
0 &Y_{\nu 3}v_2& 0
\end{array}\right), 
\label{mdirac}$$ where, without any loss of generality, one can consider $Y_{\nu 1}=Y_{\nu 2}=Y_{\nu 3}$.

Considering triplet flavon vacuum alignment as $\langle \phi_N \rangle = u(1,1,1)$ and $\langle \xi \rangle = u_{\xi}$ the right handed neutrino mass matrix at leading order is given by $$M_{R}= \left(\begin{array}{ccc}
a+\frac{2 b}{3} &-\frac{b}{3} &-\frac{b}{3} \\
-\frac{b}{3} &\frac{2b}{3}  & a-\frac{b}{3}  \\
-\frac{b}{3} &a-\frac{b}{3} &\frac{2b}{3}
\end{array}\right),
\label{mright}$$ where 