Testing CP-Violation in the Scalar Sector at Future $e^+e^-$ Colliders

We propose a {\em model-independent} method to test CP-violation in the scalar sector through measuring the inclusive cross sections of $e^+e^-\rightarrow Zh_1,Zh_2,h_1h_2$ processes with the recoil mass technique, where $h_1, h_2$ stand for the 125 GeV standard model (SM) like Higgs boson and a new lighter scalar respectively. This method effectively measures a quantity $K$ proportional to the product of the three couplings of $h_1ZZ,h_2ZZ,h_1h_2Z$ vertices. The value of $K$ encodes a part of information about CP-violation in the scalar sector. We simulate the signal and backgrounds for the processes mentioned above with $m_{2}=40\textrm{GeV}$ at the Circular Electron-Positron Collider (CEPC) with the integrated luminosity $5\textrm{ab}^{-1}$. We find that the discovery of both $Zh_2$ and $h_1h_2$ processes at $5\sigma$ level will indicate an $\mathcal{O}(10^{-2})$ $K$ value which can be measured to $16\%$ precision. The method is applied to the weakly-coupled Lee model in which CP-violation can be tested either before or after utilizing a "$p_T$ balance" cut (see \autoref{simu} for the definition). Lastly we point out that $K\neq 0$ is a sufficient but not a necessary condition for the existence of CP-violation in the scalar sector, namely $K = 0$ does not imply CP conservation in the scalar sector.