Flavor symmetry L-e-L-mu-L-tau, atmospheric neutrino mixing, and CP violation in the lepton sector

The PMNS neutrino mixing matrix is given, in general, by the product of two unitary matrices associated with the diagonalization of the charged lepton and neutrino mass matrices. Assuming that the active flavor neutrinos possess a Majorana mass matrix which is diagonalized by a bimaximal mixing matrix, we give the allowed forms of the charged lepton mixing matrix and the corresponding implied forms of the charged lepton mass matrix. We then assume that the origin of bimaximal mixing is a weakly broken flavor symmetry corresponding to the conservation of the non--standard lepton charge $L' = L_e - L_\mu - L_\tau$. The latter does not predict, in general, the atmospheric neutrino mixing to be maximal. We study the impact of this fact on the allowed forms of the charged lepton mixing matrix and on the neutrino mixing observables, analyzing the case of $CP$--violation in detail. When compared with the case of exact bimaximal mixing, the deviations from zero $U_{e3}$ and from maximal atmospheric neutrino mixing are typically more sizable if one assumes just $L'$ conservation. In fact, $|U_{e3}|^2$ can be as small as 0.007 and atmospheric neutrino mixing can take any value inside its currently allowed range. We discuss under which conditions the atmospheric neutrino mixing angle is larger or smaller than $\pi/4$. We present also a simple see--saw realization of the implied light neutrino Majorana mass matrix and consider leptogenesis in this scenario.