CP breaking in $S(3)$ flavoured Higgs model

We analyze the Higgs sector of the minimal $S(3)$-invariant extension of the Standard Model including CP violation arising from the spontaneous electroweak symmetry breaking. This extended Higgs sector includes three SU(2) doublets Higgs fields with complex vev's providing an interesting scenario to analyze the Higgs masses spectrum, trilinear Higgs self-couplings and CP violation. We present how the spontaneous electroweak symmetry breaking coming from three $S(3)$ Higgs fields gives an interesting scenario with nine physical Higgs and three Goldstone bosons when spontaneous CP violation arises from the Higgs field $S(3)$ singlet $H_S$. Furthermore, a numerical analysis of the Higgs masses and trilinear Higgs self-couplings is presented. Particularly, we find a physical solution for the scenario in which spontaneous CPB is provided by $H_S$. In this scheme, the scalar Higgs $H^0_1$ is identified, whose mass is 125 GeV and $\lambda_{H_{1}^0 H_{1}^0 H_{1}^0} \sim \lambda_{h^0 h^0 h^0}^{SM}$.