What is the Criterion for a Strong First Order Electroweak Phase Transition in Singlet Models?

TL;DR

The paper probes whether the singlet extension of the Standard Model alters the conventional criterion for a strong first-order electroweak phase transition, by computing the sphaleron energy at the critical temperature $T_c$ in the SM+S model. It finds that the commonly used $\Omega_c/T_c$ criterion, based on the distance in field space, does not reliably predict strong EWPT in the presence of a singlet, because the sphaleron energy does not scale with $\Omega(T)$ and the singlet contributes only weakly to the sphaleron energy. The authors show that the traditional $v_c/T_c$ measure remains the meaningful predictor for a strong transition over a broad parameter range, even allowing Higgs masses above the SM bound. Practically, when the singlet couples only to the Higgs, one can treat the problem via an effective two-Higgs-doublet description by replacing $S$ with its vev, an approach with implications for NMSSM analyses.

Abstract

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 Omega (Tc)/Tc>1, where Omega =(v^2+(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 vc/Tc>1 is more meaningful, and it is satisfied for a large part of the parameter space for physically allowed Higgs masses.

Figures (5)

• Figure 1: A, B, C and D represent the profiles of the functions f, L and R for the sets of parameters A, B, C and D in table \ref{['tab1']} respectively. The continuous lines represent the profiles at zero temperature and the dashed ones represent the profiles of the functions at finite temperature.
• Figure 2: For the points above the dash-dotted lines in (a) and (b), the electroweak phase transition is strongly first order according to ( \ref{['con1']}) and (\ref{['con2']}), respectively. In (a), the continuous curve represents $\mathit{E}_{\mathit{SP}}\mathit{(T}_{\mathit{c}}\mathit{\ )/T}_{\mathit{c}}$ as a function of the Higgs mass for the case of the Standard Model.
• Figure 3: The solid line denotes the ratio $\upsilon \left( T\right) /\upsilon$, the dashed one denotes $\Omega \left( T\right) /\Omega$; and the dot-dashed one denotes $E_{Sp}(T)/E_{Sp}(0)$. All the plots end at the critical temperature.
• Figure 4: $E_{Sp}(T_{c})/T_{c}$ vs $\Omega _{c}/T_{c}$ for 3000 randomly chosen sets of parameters.
• Figure 5: $E_{Sp}(T_{c})/T_{c}$ vs $\upsilon _{c}/T_{c}$ for 3000 randomly chosen sets of parameters.