Casimir scaling of domain wall tensions in the deconfined phase of D=3+1 SU(N) gauge theories

TL;DR

The paper investigates spatial 't Hooft k-string tensions in the deconfined phase of SU($N$) gauge theories on the lattice for $N=2,3,4,6$, demonstrating Casimir scaling with $k(N-k)$ at high temperature and showing this dependence persists in the near-$T_c$ regime despite strong nonperturbative corrections. It reveals wetting phenomena near $T_c$, including perfect wetting for $k=N/2$, and shows that the wall tension $ ilde{\sigma}_k$ decreases rapidly as $T$ approaches $T_c$, suggesting a possible 't Hooft string condensation temperature below $T_c$ and a speculative dual picture to Hagedorn-type confinement transitions. The results indicate that Casimir scaling of domain walls is a universal feature with rich nonperturbative structure near the confinement transition, and they point to further refinements, continuum extrapolations, and eigenvalue-based analyses to sharpen the understanding of the deconfined phase and its dualities.

Abstract

We perform lattice calculations of the spatial 't Hooft k-string tensions in the deconfined phase of SU(N) gauge theories for N=2,3,4,6. These equal (up to a factor of T) the surface tensions of the domain walls between the corresponding (Euclidean) deconfined phases. For T much larger than Tc our results match on to the known perturbative result, which exhibits Casimir Scaling, being proportional to k(N-k). At lower T the coupling becomes stronger and, not surprisingly, our calculations show large deviations from the perturbative T-dependence. Despite this we find that the behaviour proportional to k(N-k) persists very accurately down to temperatures very close to Tc. Thus the Casimir Scaling of the 't Hooft tension appears to be a `universal' feature that is more general than its appearance in the low order high-T perturbative calculation. We observe the `wetting' of these k-walls at T around Tc and the (almost inevitable) `perfect wetting' of the k=N/2 domain wall. Our calculations show that as T tends to Tc the magnitude of the spatial `t Hooft string tension decreases rapidly. This suggests the existence of a (would-be) 't Hooft string condensation transition at some temperature which is close to but below Tc. We speculate on the `dual' relationship between this and the (would-be) confining string condensation at the Hagedorn temperature that is close to but above Tc.

Figures (14)

• Figure 1: Action per unit area of the $k=1$ domain wall in SU(4) with $aT=0.25$. Monte Carlo values, $+$, compared with perturbation theory based on verious couplings: $g^2(a)$, solid line, $g^2_I(a)$, long dashed line, $g^2_{SF}(T)$, short dashed line. The dotted line is the interpolation in eqn(\ref{['eqn_Ffit']}).
• Figure 2: The running coupling $g^2(T)$ obtained from the lattice bare coupling (solid line), the mean field improved bare coupling (dashed line), and the Schrodinger functional coupling (short dashed line). All for SU(4) and $aT=0.25$.
• Figure 3: Surface tension in units of $T$, using the interpolation shown in Fig. \ref{['fig_wallPT']}. For comparison we show the 2-loop perturbative result using the mean field improved coupling. All for the $k=1$ wall in SU(4).
• Figure 4: The values in the complex plane taken by the Polyakov loop for $x_3\in [0,L-1]$, for the $k=1$ wall in SU(4) with $aT=0.25$ and $T\simeq 1.88 T_c$. Dashed line is one-loop perturbation theory.
• Figure 5: The values in the complex plane taken by the Polyakov loop for $x_3\in [0,L-1]$, for the $k=1$ wall in SU(4) with $aT=0.25$ and $T\simeq 1.02 T_c$. Dashed line is one-loop perturbation theory.