Pseudo-Dirac neutrinos from flavour dependent CP symmetry

Discrete residual symmetries and flavour dependent CP symmetries consistent with them have been used to constrain neutrino mixing angles and CP violating phases. We discuss here role of such CP symmetries in obtaining a pseudo-Dirac neutrino which can provide a pair of neutrinos responsible for the solar splitting. It is shown that if (a) $3\times 3$ Majorana neutrino matrix $M_{\nu }$ is invariant under a discrete $Z_2\times Z_2$ symmetry generated by $S_{1,2}$, (b) CP symmetry $X$ transform $M_{\nu }$ as $X^T{M}_{\nu }X=M_{\nu }^{\ast }$, and (c) $X$ and $S_{1,2}$ obey consistency conditions $X{S}_{1,2}^{\ast }{X}^{†}=S_{2,1}$, then two of the neutrino masses are degenerate independent of specific forms of $X$, $S_1$ and $S_2$. Explicit examples of this result are discussed in the context of $\Delta \left(6n^2\right)$ groups which can also be used to constrain neutrino mixing matrix $U$. Degeneracy in two of the masses does not allow complete determination of $U$ but it can also be fixed once the perturbations are introduced. We consider explicit perturbations which break $Z_2\times Z_2$ symmetries but respect CP. These are shown to remove the degeneracy and provide a predictive description of neutrino spectrum. In particular, a correlation $\sin 2{\theta }_{23}\sin {\delta }_{CP}=\pm Im\left[p\right]$ is obtained between the atmospheric mixing angle ${\theta }_{23}$ and the CP violating phase ${\delta }_{CP}$ in terms of a group theoretically determined phase factor $p$. Experimentally interesting case ${\theta }_{23}=\frac{\pi }{4}$, ${\delta }_{CP}=\pm \frac{\pi }{2}$ emerges for groups which predict purely imaginary $p$. We present detailed predictions of the allowed ranges of neutrino mixing angles, phases and the lightest neutrino mass for three of the lowest $\Delta \left(6n^2\right)$ groups with $n=2,4,6$.