Anatomy of the Electroweak Phase Transition for Dark Sector induced Baryogenesis

We investigate the electroweak phase transition patterns for a recently proposed baryogenesis model with CP violation originated in the dark sector. The model includes a complex scalar singlet-Higgs boson portal, a $U(1)_l$ gauge lepton symmetry with a $Z^\prime$ gauge boson portal and a fermionic dark matter particle. We find a novel thermal history of the scalar sector, featuring a $Z_2$ breaking singlet vacuum in the early Universe driven by a dark Yukawa coupling, that induces a one-step strongly first order electroweak phase transition. We explore the parameter space that generates the observed matter-antimatter asymmetry and dark matter relic abundance, while being consistent with constraints from electric dipole moment and collider searches, and dark matter direct detection bounds. The complex singlet can be produced via the Higgs portal and decays into Standard Model particles after traveling a certain distance. We explore the reach for long-lived singlet scalars at the {\unit[13]{TeV}} Large Hadron Collider with $\mathcal{L}=\unit[139]{fb^{-1}}$ and show its impact on the parameter space of the model. Setting aside currently unresolved theoretical uncertainties, we estimate the gravitational wave signatures detectable at future observatories.