Thermodynamics of SU(3) Lattice Gauge Theory

TL;DR

This study delivers a nonperturbative computation of the equation of state for the pure SU(3) gauge theory on lattices with $N_ au=4,6,8$, extrapolating to the continuum to obtain $T_c/\,\sqrt{\sigma}=0.629(3)$. By combining finite-temperature thermodynamics with zero-temperature string tension scales and a nonperturbative scale setting via $a\Lambda_L=R(\beta)\lambda(\beta)$, the authors show that the pressure and energy density rise toward the ideal gas limit from below, with about a 15% deviation up to $T\approx 5T_c$. They further analyze electric/magnetic condensates and the spatial string tension to argue that the temperature-dependent running coupling remains large, $g^2(5T_c)\simeq 1.5$, and extract a stable relation for the spatial tension, $\sqrt{\sigma_s}/T=0.566(13)\,g^2(T)$. These results provide quantitative benchmarks for the thermodynamics of QCD-like plasmas and highlight persistent nonperturbative effects up to several times the critical temperature.

Abstract

The pressure and the energy density of the $SU(3)$ gauge theory are calculated on lattices with temporal extent $N_τ= 4$, 6 and 8 and spatial extent $N_σ=16$ and 32. The results are then extrapolated to the continuum limit. In the investigated temperature range up to five times $T_c$ we observe a $15\%$ deviation from the ideal gas limit. We also present new results for the critical temperature on lattices with temporal extent $N_τ= 8$ and 12. At the corresponding critical couplings the string tension is calculated on $32^4$ lattices to fix the temperature scale. An extrapolation to the continuum limit yields $T_c/\sqrtσ = 0.629(3)$. We furthermore present results on the electric and magnetic condensates as well as the temperature dependence of the spatial string tension. These observables suggest that the temperature dependent running coupling remains large even at $T\simeq 5T_c$. For the spatial string tension we find $\sqrt{σ_s}/T = 0.566(13) g^2(T)$ with $g^2(5T_c) \simeq 1.5$.

Figures (11)

• Figure 1: The Polyakov-loop susceptibility on lattices of size $32^3 \times 8$ (a) and $32^3\times 12$ (b). The solid curves are the DSM interpolations, the dashed lines show the jackknife error bands. The interpolations are based on data collected at $\beta= 6.04$, 6.06, 6.065, 6.07 and 6.08 (a) and $\beta= 6.32$, 6.33, 6.335 and 6.34 (b), respectively.
• Figure 2: The $\Delta\beta$-function, Eq. \ref{['deltabeta']}, from results of MCRG studies Akemi (squares) and from our finite temperature calculation (dots). The dashed (dashed-dotted) curve shows $\Delta\beta$ as obtained from the asymptotic form of the renormalization group equation using the effective (bare) coupling. The solid curve is derived from our ansatz, Eq. \ref{['scale']}.
• Figure 3: The difference $\Delta S$ versus $\beta- \beta_c (N_\tau, N_\sigma)$ for $N_\tau = 4$, 6 and 8.
• Figure 4: The pressure versus $T/T_c$ for $N_\tau = 4$, 6 and 8 integrating the interpolations for the action density. For $N_\tau = 4$ we show two curves, one for the temperature scale using the effective coupling scheme (dashed curve) and one for our parametrization (solid curve). For $N_\tau = 6,$ and 8 we only show the latter. The error bars indicate the uncertainties arising from the integration. The horizontal dashed line shows the ideal gas continuum value, the dashed-dotted lines the corresponding lattice values for $N_\tau =4$, 6 and 8.
• Figure 5: The difference $(\epsilon -3p)/T^4$.