We study the Higgs sector of the next-to-minimal supersymmetric standard model (NMSSM) with explicit CP violation at the one-loop level, where the radiative corrections due to the quarks and squarks of the third generation are taken into account. We
expect that, within a reasonable region of the parameter space of the present model, at least one of five neutral Higgs bosons may be produced at the future $e^+ e^-$ International Linear Collider (ILC) with $sqrt{s} = 500$ GeV, with cross section larger than 12 fb, 15 fb, and 1.5 fb, respectively, via the Higgs-strahlung process, the $WW$ fusion process, and the $ZZ$ fusion process. We find that the effect of the CP phase in the present model yields significant influences upon the production cross sections of the five neutral Higgs bosons. We also study the decay modes of the five neutral Higgs bosons to find that their decay widths are similarly affected by the CP phase. Some of the decay modes in the present model behave differently from those of the Standard Model.
