Standard model cross-over on the lattice

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 $\mathrm{SU}(2)\times \mathrm{U}(1)$ $\text{gauge}+\text{Higgs}$ 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 $\sim 5\mathrm{GeV}$. We measure the cross-over temperature from the maximum of the susceptibility of the Higgs condensate, with the result $T_c=159.5\pm 1.5\mathrm{GeV}$. Outside of the narrow cross-over region the perturbative results agree well with nonperturbative ones.

Figure 1

The parameters of the effective theory (1) as functions of the physical temperature.

Figure 2

The continuum limit of $⟨{\varphi }^{†}\varphi ⟩$ at a few selected temperature values. The statistical errors are too small to be visible at this scale.

Figure 3

The continuum result of $⟨{\varphi }^{†}\varphi ⟩$, 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 ${\chi }_{{\varphi }^{†}\varphi }$ 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 ${\chi }_{{\varphi }^{†}\varphi }$.

Figure 6

The finite volume behavior of the susceptibility ${\chi }_{{\varphi }^{†}\varphi }$, shown for the lattices of size $L/a=24$, 32 and 60 at ${\beta }_G=9$. Only the smallest volume shows appreciable deviation, which is much below the lattice spacing effects in Fig. 4.

Figure 7

The interaction measure $\mathrm{\Delta }=(e-3p)/T^4$ (top left); energy density $e$ and pressure $p$ (top right); heat capacity $C_V=de/dT$ (bottom left); speed of sound squared $c_s^2$ and the equation of state parameter $w=p/e$ (bottom right). The error bands are a combination of the statistical errors and renormalizaton scale variation $\overline{\mu }=(0.5\dots 2)\pi T$. The energy density and the pressure are affected by a systematic uncertainty of order 1%, indicated with a vertical arrow.

Figure 8

Higgs and $W^3$ screening masses, measured from ${60}^3$, ${\beta }_G=9$ and ${120}^3$, ${\beta }_G=16$ lattices.

Figure 9

The effective $\gamma -Z$ mixing as a function of the temperature. The dashed lines show the 1-loop perturbative results.