Generalised CP and $A_4$ Family Symmetry

Abstract

We perform a comprehensive study of family symmetry models based on $A_4$ combined with the generalised CP symmetry $H_{\mathrm{CP}}$. We investigate the lepton mixing parameters which can be obtained from the original symmetry $A_4⋊H_{\mathrm{CP}}$ breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely $G_{\mathrm{CP}}^{\nu }\cong Z_2^S\times H_{\mathrm{CP}}^{\nu }$ in the neutrino sector and $G_{\mathrm{CP}}^l\cong Z_3^T⋊H_{\mathrm{CP}}^l$ 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 $A_4⋊H_{\mathrm{CP}}$ 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.

38 pages, 3 figures

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