We study the scalar Higgs sector of the next-to-minimal supersymmetric standard model with an extra U(1), which has two Higgs doublets and a Higgs singlet, in the light leptophobic $Z$ scenario where the extra neutral gauge boson $Z$ does not couple to charged leptons. In this model, we find that the sum of the squared coupling coefficients of the three neutral scalar Higgs bosons to $ZZ$, normalized by the corresponding SM coupling coefficient is noticeably smaller than unity, due to the effect of the extra U(1), for a reasonable parameter space of the model, whereas it is unity in the next-to-minimal supersymmetric standard model. Thus, these two models may be distinguished if the coupling coefficients of neutral scalar Higgs bosons to $ZZ$ are measured at the future International Linear Collider by producing them via the Higgs-strahlung, $ZZ$ fusion, and $WW$ fusion processes.
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