The MSSM Electroweak Phase Transition on the Lattice

TL;DR

This study non-perturbatively investigates the MSSM electroweak phase transition at finite temperature using lattice Monte Carlo simulations of a dimensionally reduced 3d effective theory. It determines the phase diagram, critical temperatures, and order-parameter observables such as Higgs/stops expectations, latent heat, and interface tension, with careful extrapolation to infinite volume and the continuum limit. The main finding is that the transition is stronger than indicated by 2-loop perturbation theory in the considered regime (heavy Higgs and light stops), with a non-perturbative two-stage transition emerging for somewhat larger stop masses, though the second stage could be cosmologically problematic due to large interface tension and potential supercooling. These results imply that MSSM baryogenesis may be viable over a broader parameter range than perturbative analyses suggested and motivate further non-equilibrium studies and exploration of broader MSSM parameter regions.

Abstract

We study the MSSM finite temperature electroweak phase transition with lattice Monte Carlo simulations, for a large Higgs mass (m_H ~ 95 GeV) and light stop masses (m_tR ~ 150...160 GeV). We employ a 3d effective field theory approach, where the degrees of freedom appearing in the action are the SU(2) and SU(3) gauge fields, the weakly interacting Higgs doublet, and the strongly interacting stop triplet. We determine the phase diagram, the critical temperatures, the scalar field expectation values, the latent heat, the interface tension and the correlation lengths at the phase transition points. Extrapolating the results to the infinite volume and continuum limits, we find that the transition is stronger than indicated by 2-loop perturbation theory, guaranteeing that the MSSM phase transition is strong enough for baryogenesis in this regime. We also study the possibility of a two-stage phase transition, in which the stop field gets an expectation value in an intermediate phase. We find that a two-stage transition exists non-perturbatively, as well, but for somewhat smaller stop masses than in perturbation theory. Finally, the latter stage of the two-stage transition is found to be extremely strong, and thus it might not be allowed in the cosmological environment.

Figures (17)

• Figure 1: The upper bound on the 3d scalar self-coupling $x_H=\lambda_{H3}/g_{S3}^2$, for a fixed $r=0.385,x_U=0.159$ (see Sec. \ref{['sec:3d']}), and $y_H$ tuned such that we are at the transition point. This figure has been obtained with the 2-loop effective potential, requiring $v_H/T\mathop{\hbox{$>$$\sim$}} 1$. In the limit $z\to 0$, it is known from nonpert that the upper bound is $x_H=(\lambda_{H3}/g_{W3}^2) r \approx 0.04\times 0.385\approx 0.015$, independent of $y_U$. The simulation points (for the symmetric phase $\to$ broken $H$ transition) are marked with the filled circles. Since we have $x_H=0.0787$ in the simulations (see Sec. \ref{['sec:3d']}), we are always in the regime where the transition is strong enough for baryogenesis according to perturbation theory.
• Figure 2: The perturbative phase structure using the parametrization in Eq. (\ref{['prmzation']}), together with the 2-loop Landau gauge vev $v_H^L/T$ in the broken phase.
• Figure 3: The emergence of the triple point: the joint probability distribution of the SU(2) and SU(3) Higgs field lengths squared, $\tilde{H}^\dagger\tilde{H}$ and $\tilde{U}^\dagger\tilde{U}$, at $\tilde{m}_U = 67.05$ GeV, $T=84.3$ GeV, on a $12^3$, $\beta_S = 12$ lattice. Here $\left\langle {\cal O} \right\rangle \equiv \sum_x {\cal O}({\bf x}) / V$. The three peaks correspond, from left to right, to broken $\tilde{H}$, symmetric, and broken $\tilde{U}$ phases. The relative strength of the transitions is evident from the suppression of the probability density between the peaks. When the volume is increased, the suppression between the peaks grows and the peaks become sharper.
• Figure 4: One-dimensional probability distributions for the system shown in Fig. \ref{['fig:triple']}. Left: the distribution of $\tilde{H}^\dagger \tilde{H} = \sum_x \tilde{H}^\dagger\tilde{H}({\bf x})/V$. The left peak corresponds both to the symmetric phase and the broken $U$ phase, the right peak to the broken $H$ phase. Right: the probability distribution of $(V_H - V_U)$, where $V_H$ and $V_U$ are the SU(2) and SU(3) hopping terms without the Higgs field radius, Eq. (\ref{['VH']}). The three phases are clearly separated: from left to right, the peaks correspond to the broken $U$, symmetric, and broken $H$ phases.
• Figure 5: Sections of the Monte Carlo time histories of the observable $V_H$ from simulations where the multicanonical order parameter is $\tilde{H}^\dagger\tilde{H}$ (top), $(\tilde{H}^\dagger\tilde{H} - \tilde{U}^\dagger\tilde{U})$ (middle) and $V_H - V_U$ (bottom).