Spontaneous CP violation in the next-to-minimal supersymmetric standard model revisited

We re-examine spontaneous CP violation at the tree level in the context of the next-to-minimal supersymmetric standard model (NMSSM) with two Higgs doublets and a gauge singlet field. We analyse the most general Higgs potential without a discrete Z(3) symmetry, and derive an upper bound on the mass of the lightest neutral Higgs boson consistent with present experimental data. We investigate, in particular, its dependence on the admixture and CP-violating phase of the gauge singlet field, as well as on tan beta. To assess the viability of the spontaneous CP violation scenario, we estimate epsilon (K) by applying the mass insertion approximation. We find that a non-trivial flavour structure in the soft-breaking A terms is required to account for the observed CP violation in the neutral kaon sector. Furthermore, combining the minimisation conditions for spontaneous CP violation with the constraints coming from K-0-(K) over bar (0) mixing, we find that the upper bound on the lightest Higgs-boson mass becomes stronger. We also point out that the electric dipole moments of electron and neutron are a serious challenge for SUSY models with spontaneous CP violation.