Dirac leptonic angle matrix versus Majorana leptonic angle matrix and their renormalization group running behaviors

Abstract

Enlightened by the idea of the 3 times 3 CKM angle matrix proposed recently by Harrison et al., we introduce the Dirac angle matrix Phi and the Majorana angle matrix Psi in the lepton sector for Dirac and Majorana neutrinos respectively. We show that in presence of the CP violation, the angle matrix Phi or Psi is entirely equivalent to the complex MNS matrix V itself, but has the advantage of being real, phase rephasing invariant, directly associated to the leptonic unitarity triangles (UTs) and do not depend on any particular parametrization of V. In this paper, we further analyzed how the angle matrices evolve with the energy scale. The one-loop Renormalization Group Equations (RGEs) of Phi, Psi and some other rephasing invariant parameters are derived and the numerical analysis is performed to compare between the case of Dirac and Majorana neutrinos. Different neutrino mass spectra are taken into account in our calculation. We find that apparently different from the case of Dirac neutrinos, for Majorana neutrinos the RG-evolutions of Phi, Psi and the Jarlskog strongly depend on the Majorana-type CP-violating parameters and are quite sensitive to the sign of Delta m^{2}_{31}. They may receive significant radiative corrections in the MSSM if three neutrino masses are nearly degenerate.