Pseudo-Goldstone Dark Matter Model with CP violation

Understand

We consider an explicitly CP-violating model with two Higgs doublets and one complex singlet scalar.

• The singlet $S$ is charged under a global $\rm U(1)$ symmetry which is softly broken by a mass term $\mu^2 S^2+\hc$.
• Imaginary part of $S$ is a stable dark matter candidate which at the tree level, in the limit of zero momentum transfer, decouples from nucleons naturally satisfying all existing direct detection limits on dark matter scattering cross-section.
• It is explicitly shown that within this framework in the alignment limit CP-violation is still present in contrast to a popular version of a 2-Higgs doublet model with softly broken $Z_2$ symmetry.