WIMP and SIMP Dark Matter from the Spontaneous Breaking of a Global Group

We propose and study a scalar extension of the Standard Model which respects a $\mathbb{Z}_3$ symmetry remnant of the spontaneous breaking of a global $U(1)_\text{DM}$ symmetry. Consequently, this model has a natural dark matter candidate and a Goldstone boson in the physical spectrum. In addition, the Higgs boson properties are changed with respect to the Standard Model due to the mixing with a new particle. We explore regions in the parameter space taking into account bounds from the measured Higgs properties, dark matter direct detection as well as measurements of the effective number of neutrino species before recombination. The dark matter relic density is determined by three classes of processes: the usual self-annihilation, semi-annihilation and purely dark matter $3 \to 2$ processes. The latter has been subject of recent interest leading to the so-called `Strongly Interacting Massive Particle' (SIMP) scenario. We show under which conditions our model can lead to a concrete realization of such scenario and study the possibility that the dark matter self-interactions could address the small scale structure problems. In particular, we find that in order for the SIMP scenario to work, the dark matter mass must be in the range $7-115$ MeV, with the global symmetry energy breaking scale in the TeV range.