$A_4$ and CP symmetry and a model with maximal CP violation

Understand

We study a second CP symmetry compatible with the $A_4$ flavor group, which interchanges the representations $\mathbf{1}'$ and $\mathbf{1}"$.

• We analyze the lepton mixing patterns arising from the $A_4$ and CP symmetry broken to residual subgroups $Z_3$ and $Z_2\times CP$ in the charged lepton and neutrino sectors respectively.
• One phenomenologically viable mixing pattern is found, and it predicts maximal atmospheric mixing angle as well as maximal Dirac CP phase, trivial Majorana phase and the correlation $\sin^2\theta_{12}\cos^2\theta_{13}=1/3$.
• We construct a concrete model based on the $A_4$ and CP symmetry, the above interesting mixing pattern is achieved, the observed charged lepton mass hierarchy is reproduced, and the reactor mixing angle $\theta_{13}$ is of the correct order.