the leading order contributions. Such corrections will propagate to the light neutrino mass matrix through the type I seesaw formula.

-   there can be corrections to the right handed neutrino mass matrix of the type $ (Y'_{\xi}\xi^2+Y'_N \phi^2_N) N_R N_R/M_{\rm Pl}$ which can change the structure of $M_R$ from the $\mu-\tau$ symmetric leading order form mentioned before. Once again this correction will propagate to the light neutrino mass matrix via type I seesaw formula.

All these corrections can be simultaneously or separately sufficient to generate correct neutrino mixing. For representative purpose, we will consider corrections to the Dirac neutrino mass matrix only and that too via Higgs doublet $H$. Addition of other corrections will make the calculations complicated without any new insights. A more general analysis is beyond the scope of this work and left for future studies.

Now, following equation (\[plH\]), the correction to the Dirac neutrino mass matrix now can be written as, $$M^1_D=\frac{u v_1}{M_{\rm Pl}}\left(\begin{array}{ccc}
(Y'_{3})_{11} & (Y'_{3})_{12} & (Y'_{3})_{13} \\
(Y'_{3})_{21} & (Y'_{3})_{22} &(Y'_{3})_{23} \\
(Y'_{3})_{31} &(Y'_{3})_{32} & (Y'_{3})_{33}
\end{array}\right), 
\label{mdirac1}$$ In the present construction, following $A_4$ multiplication rules given in the appendix, we find that $(Y'_{3})_{11}=(Y'_{3})_{23}=(Y'_{3})_{32}$ , $(Y'_{3})_{12}=(Y'_{3})_{21}=(Y'_{3})_{33}$ and $(Y'_{3})_{13}=(Y'_{3})_{22}= (Y'_{3})_{31}$ . With such correction 