ExplicitCPviolation in the general two-doublet model, with real CKM matrix

Taking the most general two-doublet model with explicit $\mathrm{CP}$ violation in the Higgs sector but no phase in the Cabibbo-Kobayashi-Maskawa matrix, we analyze the $K^0$ $\mathrm{CP}$ observables with pure new physics, the $\mathrm{CP}$ violation coming from the Higgs sector only. It is found that there is a direct correlation between the strength of FCNC couplings and the Higgs boson masses, and the lightest Higgs boson is nearly pure CP-even, but its small CP-odd composition gives a large enough contribution to ${\epsilon }_K.$