Residual $Z_2$ symmetries and leptonic mixing patterns from finite discrete subgroups of $U(3)$

Abstract: We study embedding of non-commuting $Z_2$ and $Z_m$, $m\geq 3$ symmetries in discrete subgroups (DSG) of $U(3)$ and analytically work out the mixing patterns implied by the assumption that $Z_2$ and $Z_m$ describe the residual symmetries of the neutrino and the charged lepton mass matrices respectively. Both $Z_2$ and $Z_m$ are assumed to be subgroups of a larger discrete symmetry group $G_f$ possessing three dimensional faithful irreducible representation. The residual symmetries predict the magnitude of a column of the leptonic mixing matrix $U_{\rm PMNS}$ which are studied here assuming $G_f$ as the DSG of $SU(3)$ designated as type C and D and large number of DSG of $U(3)$ which are not in $SU(3)$. These include the known group series $\Sigma(3n^3)$, $T_n(m)$, $\Delta(3n^2,m)$, $\Delta(6n^2,m)$ and $\Delta'(6n^2,j,k)$. It is shown that the predictions for a column of $|U_{\rm PMNS}|$ in these group series and the C and D types of groups are all contained in the predictions of the $\Delta(6N^2)$ groups for some integer $N$. The $\Delta(6N^2)$ groups therefore represent a sufficient set of $G_f$ to obtain predictions of the residual symmetries $Z_2$ and $Z_m$.