Predictions for the Majorana CP Violation Phases in the Neutrino Mixing Matrix and Neutrinoless Double Beta Decay

Abstract

We obtain predictions for the Majorana phases ${\alpha }_{21}/2$ and ${\alpha }_{31}/2$ of the $3\times 3$ unitary neutrino mixing matrix $U=U_e^{†}{U}_{\nu }$, $U_e$ 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.

Comments: 59 pages, includes 8 figures and 6 tables; a discussion of the effects of the uncertainties in the values of neutrino mixing angles on the predictions for the Majorana phases and four new related figures (two in the main text and two in a new appendix) added