We perform a comprehensive study of family symmetry models based on $A_4$ combined with the generalised CP symmetry $H_{ \rm{ CP}}$. We investigate the lepton mixing parameters which can be obtained from the original symmetry $A_4\rtimes H_{ \rm{ CP}}$ breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely $G^{ \nu}_{ \rm{ CP}}\cong Z^{ S}_2\times H^{ \nu}_{ \rm{ CP}}$ in the neutrino sector and $G^{ l}_{ \rm{ CP}}\cong Z^{ T}_3\rtimes H^{ l}_{ \rm{ CP}}$ in the charged lepton sector, leading to the prediction of no CP violation, namely $\delta_{ CP}$ and the Majorana phases $\alpha_{ 21}$ and $\alpha_{ 31}$ are all equal to either zero or $\pi$. We then propose an effective supersymmetric model based on the symmetry $A_4\rtimes H_{ \rm{ CP}}$ in which trimaximal lepton mixing is predicted together with either zero CP violation or $\delta_{ CP}\simeq\pm \pi/2$ with non-trivial Majorana phases. An ultraviolet completion of the effective model yields a neutrino mass matrix which depends on only three real parameters. As a result of this, all three CP phases and the absolute neutrino mass scale are determined, the atmospheric mixing angle is maximal, and the Dirac CP can either be preserved with $\delta_{ CP}=0,\pi$ or maximally broken with $\delta_{ CP}=\pm \pi /2$ and sharp predictions for the Majorana phases and neutrinoless double beta decay.
Categories: hep-ph