The leptonic Dirac CP-violating phase from sum rules

In the reference 3-neutrino mixing scheme with three light massive neutrinos, CP-violating effects in neutrino oscillations can be caused by the Dirac CP-violating phase delta present in the unitary neutrino mixing matrix U. Using the fact that U = Ue+U upsilon where U-e and U-upsilon are unitary matrices arising from the diagonalisation, respectively, of the charged lepton and neutrino mass matrices, we consider in a systematic way forms of U-e and U-upsilon allowing us to express delta as a function of the neutrino mixing angles present in U and the angles contained in U-upsilon. After obtaining sum rules for cos delta, we consider several forms of U-upsilon dictated by, or associated with, symmetries, such as tri- bimaximal, bimaximal, etc., for which the angles in U-upsilon are fi xed. For each of these forms and forms of U-e allowing to reproduce the measured values of the neutrino mixing angles, we construct the likelihood function for cos ffi, using the prospective uncertainties in the determination of the mixing angles. Our results show that the measurement of delta along with improvement of the precision on the neutrino mixing angles can provide unique information about the possible existence of a new fundamental symmetry in the lepton sector.