Spontaneous $\mathrm{CP}$ violation and Higgs boson masses in the next-to-minimal supersymmetric model

We study the possibility of spontaneous $\mathrm{CP}$ violation in the next-to-minimal supersymmetric standard model. It is shown that the spontaneous $\mathrm{CP}$ violation is induced by the radiative effects of top quark, top squark, bottom quark, and bottom squark superfields. The available regions of parameters, which are obtained by imposing the constraints from experiments, are rather narrow. We also obtain strong constraints for light Higgs boson masses such as $m_H<~36$ GeV numerically. The sum of the masses of the two light neutral Higgs boson should set around 93 GeV and the charged Higgs boson has a somewhat higher mass, larger than 700 GeV.