Generalized Scaling in Flavor Neutrino Masses

Scaling in flavor neutrino masses $M_{ij}$ ($i,j$=$e,\mu,\tau$) can be described by two angles: $\theta_{SC}$ and the atmospheric neutrino mixing angle $\theta_{23}$. For $A$=${\cos ^2}{\theta_{SC}}+{\sin ^2}{\theta_{SC}}t_{23}^4$ and B=${\cos ^2}{\theta_{SC}}-{\sin ^2}{\theta_{SC}}t_{23}^2$, where $t_{23}=\tan\theta_{23}$, our scaling ansatz dictates that $M_{i\tau}/M_{i\mu}$ = $- \kappa_it_{23}$ ($i$=$e,\mu,\tau$) with $\kappa_e$=1, $\kappa_\mu$=B/A and $\kappa_\tau$=1/B and leads to the vanishing reactor neutrino mixing angle $\theta_{13}=0$. This generalized scaling is naturally realized in seesaw textures. To obtain $\theta_{13}\neq 0$ as required by the recent experimental results, we introduce breaking terms of scaling ansatz, which are taken to keep $M_{\mu\tau}/M_{\mu\mu}$ = $- \kappa_\mu t_{23}$ intact even at $\theta_{13}\neq 0$. We derive relations that connect CP violating phases with phases of flavor neutrino masses, which are found to be numerically supported. The angle $\theta_{SC}$ is observed to be $0.91 \lesssim\sin^2\theta_{SC}\lesssim 0.93$ for the normal mass hierarchy and $\sin^2\theta_{SC}\lesssim 0.33$ for the inverted mass hierarchy. Also observed is the size of $|M_{ee}|$ to be measured in neutrinoless double beta decay, which is 0.001-0.004 eV (0.02 eV-0.05 eV) in the normal (inverted) mass hierarchy.