Neutrino mixing and CP violation from Dirac-Majorana mixing

Abstract

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