Strong Electroweak Phase Transitions in the Standard Model with a Singlet

TL;DR

This work analyzes how adding a real scalar singlet to the Standard Model can yield strongly first-order electroweak phase transitions through tree-level barriers, avoiding reliance on thermal cubic terms. By introducing a shift-invariant 8-parameter potential and focusing on configurations with degenerate minima, the authors develop a mean-field framework that exposes the vacuum structure and barrier conditions, including a novel flat-direction mechanism at the critical temperature. They provide a practical strategy to identify strong transitions, validate it with Z2-symmetric, supersymmetric, and light-singlet scenarios, and corroborate the mean-field picture with full one-loop finite-temperature calculations showing sizable $v_c/T_c$ in many cases. The results indicate broad regions of parameter space where strong EWPhTs are possible, with implications for electroweak baryogenesis and gravitational-wave signatures, and deliver a useful parametrization for phenomenological scalar-sector studies.

Abstract

It is well known that the electroweak phase transition (EWPhT) in extensions of the Standard Model with one real scalar singlet can be first-order for realistic values of the Higgs mass. We revisit this scenario with the most general renormalizable scalar potential systematically identifying all regions in parameter space that develop, due to tree-level dynamics, a potential barrier at the critical temperature that is strong enough to avoid sphaleron wash-out of the baryon asymmetry. Such strong EWPhTs allow for a simple mean-field approximation and an analytic treatment of the free-energy that leads to very good theoretical control and understanding of the different mechanisms that can make the transition strong. We identify a new realization of such mechanism, based on a flat direction developing at the critical temperature, which could operate in other models. Finally, we discuss in detail some special cases of the model performing a numerical calculation of the one-loop free-energy that improves over the mean-field approximation and confirms the analytical expectations.

Figures (14)

• Figure 1: The curve $h^2=D^2_h(s)$ in the $(h^2/v^2,s/w)$-plane. The different cases correspond to: $\lambda_m=0$ (upper left); $\lambda_m>0$ (upper right); $\lambda_m<0$ with $|\lambda_m|<m_{sh}^4/(m_h^2v^2)$ (lower left); $\lambda_m<0$ with $|\lambda_m|>m_{sh}^4/(m_h^2v^2)$ (lower right). The unphysical region $h^2<0$ is shaded gray. The EW breaking minimum at $h^2=v^2$ and $s=w$ is marked by a black dot.
• Figure 2: Example for the dependence of the potential on $\lambda^2$. The upper left plot shows $D^2_h(s)$ (solid blue) and $D^2_s(s)$ [along which $\partial V/\partial s=0$, see eq. (\ref{['h22']})] (solid red) for several values of $\lambda^2$: $\tilde{\lambda}^2$, $8\tilde{\lambda}^2/9$ and $\lambda_d^2$. The intersections of these two curves correspond to the stationary points of the potential. The remaining plots show the potential along $D^2_h(s)$, $D^2_s(s)$ (same color coding) and $h=0$ (dashed) at the indicated values of $\lambda^2$.
• Figure 3: Curves $D_h^2(s)$ (solid blue) and several $D_s^2(s)$ (red solid, dashed and dash-dotted) with different values of $m_s^2$, intersecting to give two potential minima (indicated by black dots) in the 4 different cases listed in eqs. (\ref{['casea']})-(\ref{['cased']}).
• Figure 4: Special scenario with $\mathbf{Z}_2$ symmetry, $\lambda_m>0$ and $\lambda^2<0$. Left, curves with $\partial V/\partial h=0$ [$D^2_h(s)$ and $h=0$, blue lines] and $\partial V/\partial s=0$ [$D^2_s(s)$ and $s=0$, red lines] intersecting in the minima at $(0,\pm w_0)$ and $(v,0)$, as indicated by the black dots. Right, corresponding potential showing the barrier between minima.
• Figure 5: Special scenario with $\mathbf{Z}_2$ symmetry, $\lambda_m>0$ and $\lambda^2=0$ showing a flat direction. Left, degenerate parabolas $D_{h,s}^2(s)$. Right, corresponding potential.