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 do ublets and a gauge singlet field. We analyse the most general Higgs potenti al without a discrete Z(3) symmetry, and derive an upper bound on the mass of the lightest neutral Higgs boson consistent with present experimental da ta. We investigate, in particular, its dependence on the admixture and CP-v iolating phase of the gauge singlet field, as well as on tan beta. To asses s the viability of the spontaneous CP violation scenario, we estimate epsil on (K) by applying the mass insertion approximation. We find that a non-tri vial flavour structure in the soft-breaking A terms is required to account for the observed CP violation in the neutral kaon sector. Furthermore, comb ining the minimisation conditions for spontaneous CP violation with the con straints coming from K-0-(K) over bar (0) mixing, we find that the upper bo und on the lightest Higgs-boson mass becomes stronger. We also point out th at the electric dipole moments of electron and neutron are a serious challe nge for SUSY models with spontaneous CP violation.