Lightness of Higgs Boson and Spontaneous CP Violation in Lee Model

We propose a mechanism in which the lightness of Higgs boson and the smallness of CP-violation are correlated based on the Lee model, namely the spontaneous CP-violation two-Higgs-doublet-model. In this model, the mass of the lightest Higgs boson $m_h$ as well as the quantities $K$ and $J$ are $\propto t_{\beta}s_{\xi}$ in the limit $t_{\beta}s_{\xi}\rightarrow0$ (see text for definitions of $t_\beta$ and $\xi$), namely the CP conservation limit. Here $K$ and $J$ are the measures for CP-violation effects in scalar and Yukawa sectors respectively. It is a new way to understand why the Higgs boson discovered at the LHC is light. We investigated the important constraints from both high energy LHC data and numerous low energy experiments, especially the measurements of EDMs of electron and neutron as well as the quantities of B-meson and kaon. Confronting all data, we found that this model is still viable. It should be emphasized that there is no standard-model limit for this scenario, thus it is always testable for future experiments. In order to pin down Lee model, it is important to discover the extra neutral and charged Higgs bosons and measure their CP properties and the flavor-changing decays. At the LHC with $\sqrt{s}=14\textrm{TeV}$, this scenario is favored if there is significant suppression in the $b\bar{b}$ decay channel or any vector boson fusion (VBF), V+H production channels. On the contrary, it will be disfavored if the signal strengths are standard-model-like more and more. It can be easily excluded at $(3\sim5)\sigma$ level with several $\textrm{fb}^{-1}$ at future $e^+e^-$ colliders, via the accurately measuring the Higgs boson production cross sections.