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 H_{1,2} and a gauge singlet field N. Within such a framework, the Cabibbo-Kobayashi-Maskawa matrix is real and all CP-violating phenomena can be described by two phases phi_D and phi_N which arise from the complex vacuum expectation values <H_{1,2}^0> and <N> of the neutral Higgs doublet and singlet fields respectively. 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 which is still consistent with present experimental data. We investigate, in particular, its dependence on the CP-violating phase phi_N as well as the admixture of the gauge singlet field, and 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-\bar{K}^0 mixing, we find that the upper bound on the lightest Higgs-boson mass becomes stronger. A comment is made on the electric dipole moments of electron and neutron which may provide a key test of the consistency of the NMSSM with spontaneous CP violation.