Generalised CP and $A_4$ Family Symmetry

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.