The Electroweak Phase Transition: A Non-Perturbative Analysis

TL;DR

This work performs a non-perturbative lattice study of the 3d SU(2)+Higgs model to illuminate the electroweak phase transition in high-temperature 4d theories. By exploiting exact constant-physics curves, it achieves controlled continuum limits and determines the critical temperature, latent heat, and interface tension, including a quantified 3-loop correction to the effective potential. The study also analyzes correlation lengths, metastability, and effective theories for large Higgs masses, and provides a detailed map from 3d lattice observables to 4d physical theories (Standard Model and extensions). The findings reveal substantial deviations from perturbation theory near Tc, establish non-perturbative benchmarks for cosmological implications, and set the stage for incorporating U(1) effects and sphaleron-rate calculations in future work.

Abstract

We study on the lattice the 3d SU(2)+Higgs model, which is an effective theory of a large class of 4d high temperature gauge theories. Using the exact constant physics curve, continuum ($V\to\infty, a\to 0$) results for the properties of the phase transition (critical temperature, latent heat, interface tension) are given. The 3-loop correction to the effective potential of the scalar field is determined. The masses of scalar and vector excitations are determined and found to be larger in the symmetric than in the broken phase. The vector mass is considerably larger than the scalar one, which suggests a further simplification to a scalar effective theory at large Higgs masses. The use of consistent 1-loop relations between 3d parameters and 4d physics permits one to convert the 3d simulation results to quantitatively accurate numbers for different physical theories, such as the Standard Model -- excluding possible nonperturbative effects of the U(1) subgroup -- for Higgs masses up to about 70 GeV. The applications of our results to cosmology are discussed.

Figures (28)

• Figure 1: The autocorrelation function of the observable $L= V^\dagger(\hbox{\bf x}) U_i(x)V(\hbox{\bf x}+i)$ calculated from an $m_H^*=60$ GeV, $V=14^3$ lattice. (a) Heat bath/Metropolis, (b) Heat bath/Metropolis with global $R$-update, (c) $4\times$ (overrelaxation with even-odd traversal) + $1\times$ (heat bath + global update), and (d) $4\times$ (wavefront overrelaxation) + $1\times$ (heat bath + global update). In all the cases, one update means going once through all the lattice points.
• Figure 2: Data for $\langle\phi^\dagger\phi(T^*)\rangle/T^*$ as a function of $1/\beta_G$ for $m_H^*=80$, $T^*=110$ GeV computed from measured values of $\langle R^2\rangle$ using eq. (\ref{['rl2']}) with $\mu=T^*$, $g_3^2=0.44015T^*$ (parametrisation in eq. (\ref{['g3eff']})). The perturbative values corresponding to $\beta=0,20,40,50$, calculated with the CW-method at the scale $\mu=2.37 m_T$ and then run to $\mu=T^*$ with eq. (\ref{['scaledep']}), are shown on the vertical axis.
• Figure 3: As fig. \ref{['beta1']}, but for $T^*=145$ GeV and $T^*=165$ GeV.
• Figure 4: The values of $\langle\phi^\dagger\phi(T^*)\rangle/T^*$ in the $\hbar$ expansion as a function of temperature for $m_H^*=80$ GeV from eq. (\ref{['sp_for_pdp']}). The 3-loop curve contains the known part thereof, with $\beta(h)=0$. It is seen that the $\hbar$-calculation becomes increasingly unreliable as one approaches the critical temperature.
• Figure 5: The probability distribution of the average Higgs length squared $R^2$ for $m_H^*=60$ and $\beta_G=5$, 8, 12 and 20.