Flavor symmetry $L_e-L_{\mu }-L_{\tau }$, atmospheric neutrino mixing, and $CP$ violation in the lepton sector

The Pontecorvo-Maki-Nakagawa-Sakata 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 nonstandard lepton charge $L^{\prime }=L_e-L_{\mu }-L_{\tau }$. 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 $L^{\prime }$ 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 seesaw realization of the implied light neutrino Majorana mass matrix and consider leptogenesis in this scenario.

Figure 1

Scatter plots of some of the $\lambda$ parameters, $\sin {\theta }_{23}^{\nu }$ and the neutrino mixing observables for the $3\sigma$ allowed ranges of values of the neutrino mixing parameters given in Eq. (3). All ${\lambda }_{ij}$ are small and conservation of $CP$ is assumed.

Figure 2

Scatter plots of ${tan}^2{\theta }_{\odot }$ against $|U_{e3}{|}^2$ for two different limits on ${sin}^{2}2{\theta }_{\mathrm{atm}}$ and for the cases ${\theta }_{23}^{\nu }=\pi /4$ and free ${\theta }_{23}^{\nu }$. A CKM-like hierarchy of the ${\lambda }_{ij}$ is assumed and it is required that the neutrino mixing parameters ${\tan }^2{\theta }_{\odot }$ and $|U_{e3}|$ have values in their respective $3\sigma$ allowed intervals, given in Eq. (3).

Figure 3

Scatter plots of ${tan}^2{\theta }_{\odot }$ against $|U_{e3}{|}^2$, $|U_{e3}{|}^2$ against ${tan}^2{\theta }_{\mathrm{atm}}$, ${tan}^2{\theta }_{\odot }$ against $J_{CP}$ and ${tan}^2{\theta }_{\mathrm{atm}}$ against ${tan}^2{\theta }_{23}$. We choose ${\lambda }_{12}\equiv \lambda =0.24$, ${\lambda }_{13}={\lambda }^3$ and ${\lambda }_{23}={\lambda }^2$ (except for the lower right plot, in which also ${\lambda }_{23}=\lambda$ is used). The neutrino mixing parameters are required to lie within their respective $3\sigma$ allowed ranges of values, given in Eq. (3).