Spontaneous CP-violation in the Simplest Little Higgs Model and its Future Tests at $e^+e^-$ Colliders: the Scalar Sector

We proposed the possibility of spontaneous CP-violation in the simplest little Higgs model. The CP-symmetry is spontaneously broken after the pseudoscalar field requires its vacuum expected value. The scalars with different CP-charge mix with each other hence they become CP-mixing state. The existence of CP-violation in scalar sector can be confirmed through measuring a nonzero $K$ (see the text). Facing all constraints, especially the electric dipole moment of electron, this model is still alive. We discussed the tests on this model at future $e^+e^-$ colliders. For the recent allowed regions, $5\sigma$ discovery for CP-violation in the scalar sector is possible when the extra scalar is lighter than about $70~\textrm{GeV}$ at CEPC with $5~\textrm{ab}^{-1}$ luminosity. If the extra scalar is heavier, more simulation studies are needed. At the end of the discussion section, we also briefly discussed the tests on this model at LHC for the case with a heavy extra scalar.