The Standard Model cross-over on the lattice

TL;DR

This work addresses the finite-temperature electroweak cross-over in the Standard Model at the physical Higgs mass by simulating a 3D SU(2)×U(1) gauge–Higgs theory on the lattice, including hypercharge and performing continuum extrapolations across multiple lattice spacings and large volumes. The authors measure the Higgs condensate and its susceptibility to locate the pseudocritical temperature and compute fundamental thermodynamic quantities and screening masses, establishing a narrow cross-over window of about $5$ GeV centered near $T_c\approx 159.6$ GeV. By combining non-perturbative lattice results with perturbative inputs, they obtain the equation of state, energy density, pressure, and related observables, finding good agreement with perturbation theory away from the cross-over and highlighting the small but well-defined non-perturbative effects near $T_c$. The inclusion of the hypercharge field and a continuum extrapolation provide more reliable results than prior studies, enhancing the reliability of predictions for early-Universe electroweak thermodynamics and sphaleron-related processes.

Abstract

With the physical Higgs mass the Standard Model symmetry restoration phase transition is a smooth cross-over. We study the thermodynamics of the cross-over using numerical lattice Monte Carlo simulations of an effective SU(2) X U(1) gauge + Higgs theory, significantly improving on previously published results. We measure the Higgs field expectation value, thermodynamic quantities like pressure, energy density, speed of sound and heat capacity, and screening masses associated with the Higgs and Z fields. While the cross-over is smooth, it is very well defined with a width of only approximately 5 GeV. We measure the cross-over temperature from the maximum of the susceptibility of the Higgs condensate, with the result $T_c = 159.5 \pm 1.5$ GeV. Outside of the narrow cross-over region the perturbative results agree well with non-perturbative ones.

Figures (9)

• Figure 1: The parameters of the effective theory (\ref{['contaction']}) as functions of the physical temperature.
• Figure 2: The continuum limit of $\langle \phi^\dagger\phi\rangle$ at a few selected temperature values. The statistical errors are too small to be visible at this scale.
• Figure 3: The continuum result of $\langle\phi^\dagger\phi\rangle$, compared with the perturbative broken and symmetric phase results. The shaded bands are estimations of unknown higher order corrections to perturbative results. The solid continuous line is an interpolation to the data.
• Figure 4: Above: susceptibility $\chi_{\phi^\dagger\phi}$ shown at $\beta_G=6$, $9$ and $16$, together with the interpolating functions. The continuum limit is shown with a heavy line. Below: As above, zoomed-in to the shaded band near the cross-over region.
• Figure 5: Continuum extrapolation of the maximum location of $\chi_{\phi^\dagger\phi}$.