Determination of Majorana-Type Phases from the Time Evolution of Lepton Numbers

Abstract We have investigated an approach for determining Majorana-type phases using the time evolution of lepton family numbers. We show how the second-order time derivative of the expectation values for the lepton family numbers depends on the sum of the Majorana-type phases. Furthermore, others have connected the Majorana-type phases to the orientation of unitary triangles for the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, and the usual Majorana phases $\alpha _{21}$ and $\alpha _{31}$. Theoretically, this allows for the extraction of the orientation of the unitary triangles and the Majorana phases from lepton family numbers. We study three example situations: first, how to extract the Majorana-type phases and the lightest neutrino mass for three massive neutrinos, and when a neutrino is massless, i.e. $m_{1,3}=0$; second, the determination of the Majorana phase and the lightest neutrino mass for a two-generation toy model; and third, simplified realizations of the type I seesaw model with two gauge singlet neutrinos and two families of lepton doublets. We calculate how the Majorana phases and Majorana-type phases are related to CP violation for leptogenesis at high energies. At first, the effective Majorana mass matrix is parametrized with real and positive diagonal elements at low energies. In this basis, the phases of the off-diagonal elements are related to the CP-violating phases in the PMNS matrix. We explicitly show the relationship between the single Majorana phase and the phase of the effective Majorana mass matrix for the toy model with two generations of active neutrinos. Then, for the model with two gauge singlet neutrinos and two families of lepton doublets, we study the effective Majorana mass matrix generated by the seesaw model. In that model, we can show how the Majorana phase at low energy is related to the two CP-violating phases of the seesaw matrix. This relationship between the phases depends on the lepton number asymmetries of the heavy Majorana neutrino decays for the toy models.