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Generalized Scaling in Flavor Neutrino Masses

Scaling in flavor neutrino masses $M_{ij}$ ($i,j$=$e,μ,τ$) can be described by two angles: $θ_{SC}$ and the atmospheric neutrino mixing angle $θ_{23}$. For $A$=${\cos ^2}{θ_{SC}}+{\sin ^2}{θ_{SC}}t_{23}^4$ and B=${\cos ^2}{θ_{SC}}-{\sin ^2}{θ_{SC}}t_{23}^2$, where $t_{23}=\tanθ_{23}$, our scaling ansatz dictates that $M_{iτ}/M_{iμ}$ = $- κ_it_{23}$ ($i$=$e,μ,τ$) with $κ_e$=1, $κ_μ$=B/A and $κ_τ$=1/B and leads to the vanishing reactor neutrino mixing angle $θ_{13}=0$. This generalized scaling is naturally realized in seesaw textures. To obtain $θ_{13}\neq 0$ as required by the recent experimental results, we introduce breaking terms of scaling ansatz, which are taken to keep $M_{μτ}/M_{μμ}$ = $- κ_μt_{23}$ intact even at $θ_{13}\neq 0$. We derive relations that connect CP violating phases with phases of flavor neutrino masses, which are found to be numerically supported. The angle $θ_{SC}$ is observed to be $0.91 \lesssim\sin^2θ_{SC}\lesssim 0.93$ for the normal mass hierarchy and $\sin^2θ_{SC}\lesssim 0.33$ 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.