Dual electroweak phase transition in the two-Higgs-doublet model with the $S_3$ discrete symmetry

In this work, dual electroweak phase transition (EWPT) consisting of two phases, is carefully studied in the two-Higgs-doublet model with the $S_3$ discrete symmetry. The role of $S_3$ here is to further separate the stages of the electroweak phase transition, compared to that of the original two-Higgs-doublet model (2HDM). The strength of the electroweak phase transition $(S)$ in the model under consideration is large enough for the first-order EWPT, specifically $1 < S < 2.8$. The ratio between the two vacuum expectation values (VEVs), $\tan\beta = v_2/v_1$, is proven to have no effects on the strength of the phase transition. This ratio only affects the mass domain that causes the first-order phase transition. Furthermore, in this paper we will show clearly that when studying the EWPT in models of more than one scalar field that generates masses, one needs to analyze the problem of phase transition under multiple stages. In other words, the effect of the first stage of symmetry breaking to the second one, is to simplify by suggestion that vacuum expectation value of the Higgs boson responsible for the initial stage is proportional to that of the field for the next stage.