  We perform a nonperturbative lattice study of the electroweak phase
transition in the real singlet scalar extension of the Standard Model.We
consider both the heavy and light singlet-like scalar regimes at non-zero
singlet-doublet mixing angle. After reviewing features of the lattice method
relevant for phase transition studies, we analyze the dependence of phase
transition thermodynamics on phenomenologically relevant parameters. In the
heavy singlet-like scalar regime, we find that the transition is crossover for
small doublet-singlet mixing angles, despite the presence of an energy barrier
in the tree-level potential. The transition becomes first order for
sufficiently large mixing angles. We find two-loop perturbation theory to agree
closely with the lattice results for all thermodynamical quantities considered
here (critical temperature, order parameter discontinuity, latent heat) when
the transition is strongly first order. For the light singlet-like scalar
regime relevant to exotic Higgs decays, we update previous one-loop
perturbative results using the two-loop loop dimensionally reduced effective
field theory and assess the nature of the transition with lattice simulations
at set of benchmark parameter points. For fixed singlet-like scalar mass the
transition becomes crossover when the magnitude of the Higgs-singlet portal
coupling is small. We perform our simulations in the high-temperature effective
theory, which we briefly review, and present analytic expressions for the
relevant lattice-continuum relations.
