Neutrino mixing and CP violation from Dirac-Majorana mixing

We consider a minimal condition that predicts the 1-3 lepton mixing angle $θ_{13} \simeq θ_{\rm C}/\sqrt{2}$ with $θ_{\rm C}$ the Cabibbo angle, and give the improved prediction of $θ_{13}$. In the case of normal mass ordering, the theoretical value of $θ_{13}$ is predicted as $θ_{13}=8.6^\circ$, which is in good agreement with the current global best fit. In the case of inverted mass ordering, the theoretical value is predicted as $θ_{13}=9.7^\circ$, which is far from the current global best fit. We also study the leptonic CP violation. We show that any values of the leptonic Dirac CP phase $δ_{\rm CP}$ can be obtained by fine tuning. Without fine tuning, $|\sinδ_{\rm CP}|$ should be very small, typically of ${\cal O}(λ^2)$ where $λ\equiv \sinθ_{\rm C}$. Furthermore, a model-independent measure of CP violation is proposed, which is applicable to any flavor models.

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