First-order electroweak phase transition in a complex singlet model with $\mathbb{Z}_3$ symmetry

We consider an extension of the Standard Model with a complex singlet scalar, where a global $U(1)$ symmetry is explicitly broken to $\mathbb{Z}_3$ symmetry. We study the two-step electroweak phase transition in the model and find that it can be of first-order if the heavy scalar mass falls in the range of $1-2$~TeV and the mixing angle $\left | \theta \right |\gtrsim 0.2$ ($11.5^{\circ}$). The Higgs signal strength measurements at the LHC, on the other hand, restrict the mixing angle $\left | \theta \right |\lesssim 0.4$ ($23^{\circ}$). Future colliders including high-luminosity LHC can probe the remaining parameter space of first-order phase transition in this scenario. After the $U(1)$ symmetry breaking, the pseudo-Goldstone boson becomes a dark matter candidate due to a hidden $\mathbb{Z}_2$ symmetry of the model. We find that the pseudo-Goldstone boson can make up a small fraction of the observed dark matter and escape from the constraints of current direct detection. We also show that the stochastic gravitational wave signals from the phase transition are potentially discoverable with future space-based interferometers.