Generalised CP and trimaximal TM1 lepton mixing in S4S4 family symmetry

We construct two flavor models based on S4S4 family symmetry and generalised CP symmetry. In both models, the S4S4 family symmetry is broken down to the Z2SU subgroup in the neutrino sector, as a consequence, the trimaximal TM1 lepton mixing is produced. Depending on the free parameters in the flavon potential, the Dirac CP is predicted to be either conserved or maximally broken, and the Majorana CP phases are trivial. The two models differ in the neutrino sector. The flavon fields are involved in the Dirac mass terms at leading order in the first model, and the neutrino mass matrix contains three real parameters such that the absolute neutrino masses are fixed. Nevertheless, the flavon fields enter into the Majorana mass terms at leading order in the second model. The leading order lepton mixing is of the tri-bimaximal form which is broken down to TM1 by the next to leading order contributions.