  We consider a simple Higgs portal model in beyond the standard model
scenario: an extra real gauge singlet scalar that couples to the Higgs. We
calculate the higher-loop corrections to the cross section of the
Higgsstrahlung process $e^+e^- \rightarrow Z h$, along with the tri-Higgs
coupling and the wave-function renormalization. We find noticeable contribution
to the total Higgsstrahlung cross section, especially coming from two-loop
diagrams. We also find that the correction to the tri-Higgs coupling is complex
when this extra scalar is lighter than half of the center of mass energy. It
indicates a new source of CP violation. In the region of this extra scalar
being lighter than a couple of hundreds of GeVs, we argue that the higher-loop
calculation is a more reliable approach than the effective field theory
calculation.
