What is the criterion for a strong first order electroweak phase transition in singlet models?

It is widely believed that the existence of singlet scalars in some standard model extensions can easily make the electroweak phase transition strongly first order, which is needed for the electroweak baryogenesis scenario. In this paper, we will examine the strength of the electroweak phase transition in the simplest extension of the standard model with a real singlet using the sphaleron energy at the critical temperature. We find that the phase transition is stronger by adding a singlet; and also that the criterion for a strong phase transition Ω(Tc)/Tc1, where Ω=(υ²+(x-x₀)²)1/2 and x (x₀) is the singlet vev in the broken (symmetric) phase, is not valid for models containing singlets, even though often used in the literature. The usual condition υc/Tc1 is more meaningful, and it is satisfied for a large part of the parameter space for physically allowed Higgs masses.