Predictions for the Majorana CP violation phases in the neutrino mixing matrix and neutrinoless double beta decay

We obtain predictions for the Majorana phases $\alpha_21/2$ and $\alpha_31/2$ of the $3\times 3$ unitary neutrino mixing matrix $U = U_e^\dagger \, U_\nu$, $U_e$ and $U_\nu$ being the $3\times 3$ unitary matrices resulting from the diagonalisation of the charged lepton and neutrino Majorana mass matrices, respectively. We focus on forms of $U_e$ and $U_\nu$ permitting to express $\alpha_21/2$ and $\alpha_31/2$ in terms of the Dirac phase $\delta$ and the three neutrino mixing angles of the standard parametrisation of $U$, and the angles and the two Majorana-like phases $\xi_21/2$ and $\xi_31/2$ present, in general, in $U_\nu$. The concrete forms of $U_\nu$ considered are fixed by, or associated with, symmetries (tri-bimaximal, bimaximal, etc.), so that the angles in $U_\nu$ are fixed. For each of these forms and forms of $U_e$ that allow to reproduce the measured values of the three neutrino mixing angles $\theta_12$, $\theta_23$ and $\theta_13$, we derive predictions for phase differences $(\alpha_21/2 - \xi_21/2)$, $(\alpha_31/2 - \xi_31/2)$, etc., which are completely determined by the values of the mixing angles. We show that the requirement of generalised CP invariance of the neutrino Majorana mass term implies $\xi_21 = 0$ or $\pi$ and $\xi_31 = 0$ or $\pi$. For these values of $\xi_21$ and $\xi_31$ and the best fit values of $\theta_12$, $\theta_23$ and $\theta_13$, we present predictions for the effective Majorana mass in neutrinoless double beta decay for both neutrino mass spectra with normal and inverted ordering.