$Z_3$ Scalar Singlet Dark Matter

We consider the minimal scalar singlet dark matter stabilised by a $Z_3$ symmetry. Due to the cubic term in the scalar potential, semi-annihilations, besides annihilations, contribute to the dark matter relic density. Unlike in the $Z_2$ case, the dark matter spin independent direct detection cross section is no more linked to the annihilation cross section. We study the extrema of the potential and show that a too large cubic term would break the $Z_3$ symmetry spontaneously, implying a lower bound on the direct detection cross section, and allowing the whole parameter space to be tested by XENON1T. In a small region of the parameter space the model can avoid the instability of the standard model vacuum up to the unification scale. If the semi-annihilations are large, however, new physics will be needed at TeV scale because the model becomes non-perturbative. The singlet dark matter mass cannot be lower than 53.8 GeV due to the constraint from Higgs boson decay into dark matter.