Charged Lepton Corrections to Neutrino Mixing Angles and CP Phases Revisited

We re-analyze charged lepton corrections to neutrino mixing angles and CP phases, carefully including CP phases from the charged lepton sector. We present simple analytical formulae for including the charged lepton corrections and derive compact new results for small neutrino and charged lepton mixings $\theta^{\nu}_{13}$ and $\theta^{e}_{13}$. We find a generic relation $\theta_{12} + \frac{1}{\sqrt{2}} \theta^{e}_{12} \cos(\delta - \pi) \approx \theta^{\nu}_{12}$, which relates the prediction from the neutrino sector $\theta^{\nu}_{12}$ to the charged lepton mixing $\theta^{e}_{12}$ and to the MNS neutrino oscillation phase $\delta$. We apply our formula to the examples of bimaximal or tri-bimaximal neutrino mixing. One implication is that the so-called quark-lepton complementarity relation $\theta_{12} + \theta_C = 45^\circ$ can only hold for $\delta = \pi$ and it gets modified in the presence of leptonic CP violation. On the other hand, the lepton mixing $\theta_{13}$ generated from the charged lepton correction $\theta^{e}_{12}$ is independent of CP phases and given by $\theta_{13} = \frac{1}{\sqrt{2}} \theta^{e}_{12}$. Combining these results leads to a model-independent sum rule: $\theta_{12} + \theta_{13} \cos(\delta - \pi) \approx \theta^{\nu}_{12}$ where $\theta^{\nu}_{12} = (35.26^\circ) 45^\circ$ in the case of (tri-)bimaximal neutrino mixing, for example.