Based on the weakly coupled spontaneous $CP$-violation two-Higgs-doublet model (called the Lee model) and the mechanism to generate the correlation between the smallness of $CP$ violation and the lightness of the scalar mass, as we proposed earlier, we predict a light $CP$ mixing scalar $\eta$ in which the pseudoscalar component is dominant. It is a natural scenario in which $m_{\eta }\sim \mathcal{O}(10\mathrm{GeV})\ll v$. It means new physics might be hidden below the electroweak scale $v$. Masses of all other scalars ($h$, $H$, $H^{\pm }$) should be around the electroweak scale $v$. Among them, the 125 GeV Higgs boson ($h$) couplings are standard-model-like, and the charged Higgs boson ($H^{\pm }$) mass should be around the heaviest neutral scalar ($H$) mass. We discussed all experimental constraints and showed that this scenario is still allowed by data. The strictest constraints come from the flavor violation experiments and the electric dipole moments of the electron and neutron. We also discussed the future tests for this scenario. It is possible to discover the extra scalars or exclude this scenario at future colliders, especially at the LHC and $e^{+}{e}^{-}$ colliders with $\mathcal{O}({\mathrm{ab}}^{-1})$ luminosity. We also pointed out that the $Z$-mediated Higgs pair production via $e^{+}{e}^{-}\to h_i{h}_j$ ($h_i$, $h_j$ stand for two of the $\eta$, $h$, $H$) would be the key observable to confirm or exclude $CP$ violation in the Higgs sector. The sensitivity to test this scenario is worth studying in greater detail.

• Received 6 February 2016
• Publisher error corrected 19 September 2016

DOI:https://doi.org/10.1103/PhysRevD.94.055008