The high temperature phase transition in SU(N) gauge theories

TL;DR

The paper computes the continuum deconfining temperature Tc in units of the string tension σ for SU(2)–SU(8) and finds the N-dependence to be Tc/√σ = 0.596(4) + 0.453(30)/N^2, indicating rapid convergence to the large-N limit. It confirms that the transition is first order for N≥3 and that the latent heat grows with N, while finite-volume corrections shrink as 1/N^2, allowing Tc to be determined on smaller volumes at large N. The authors argue that the domain-wall tension between confining and deconfined phases grows with N, linking the results to Eguchi-Kawai reduction and the Master Field concept, and they propose reduced-model strategies to study Tc efficiently in the large-N limit. These findings provide a coherent large-N picture of deconfinement and motivate further work on reduced-volume formulations and the role of Master Fields in gauge theories.

Abstract

We calculate the continuum value of the deconfining temperature in units of the string tension for SU(4), SU(6) and SU(8) gauge theories, and we recalculate its value for SU(2) and SU(3). We find that the $N$-dependence for $2 \leq N \leq 8$ is well fitted by $T_c/\sqrt{sigma} = 0.596(4) + 0.453(30)/N^2$, showing a rapid convergence to the large-N limit. We confirm our earlier result that the phase transition is first order for $N \geq 3$ and that it becomes stronger with increasing $N$. We also confirm that as $N$ increases the finite volume corrections become rapidly smaller and the phase transition becomes visible on ever smaller volumes. We interpret the latter as being due to the fact that the tension of the domain wall that separates the confining and deconfining phases increases rapidly with $N$. We speculate on the connection to Eguchi-Kawai reduction and to the idea of a Master Field.

Figures (8)

• Figure 1: Reweighted susceptibility $\chi_l$ as a function of $\beta$ for SU(3) on a $L=32$ lattice at $L_t=5$. The location of the maximum, $\beta_c$, and the value of the maximum, $\chi_l^\text{max}$, are given as well.
• Figure 2: The value of $\ln \chi_l^\text{max}$ plotted versus $\ln V$ for all our values of $N$ at $L_t=5$. The full lines are best fits for $N=3,4,6,8$ with a slope close to unity while the dashed line is the best fit for $N=2$ with a slope of $\sim 0.63$.
• Figure 3: The normalised susceptibility maxima $\chi_l^\text{max}/V$ plotted against $1/V$ for $N=2,3$ at $L_t=5$. The dashed line is a best fit with the leading scaling behaviour extracted from Fig.\ref{['fig:crit_exp_nt5_sun']} and a leading $O(1/V)$ correction.
• Figure 4: The normalised susceptibility maxima $\chi_l^\text{max}/V$ plotted against $1/V$ for $N=4,6$ at $L_t=5$. The dashed lines are the best fits with $O(1/V)$ and $O(1/V^2)$ corrections. Note the different scale for SU(6) on the $x$-axis.
• Figure 5: The normalised specific heat maxima $C_\text{max}/V$ plotted against $1/V$ for $N=3,4,6$ at $L_t=5$. The straight lines are best fits with a leading $O(1/V)$ correction. Also shown are the $V\to\infty$ limits.