String Tension Scaling in High-Temperature Confined SU(N) Gauge Theories

TL;DR

This work analyzes SU($N$) gauge theories with adjoint fermions under periodic boundary conditions to realize confinement at high temperature via a $Z(N)$-symmetric Polyakov-loop minimum. It derives the one-loop Polyakov-loop potential and shows that, for sufficiently light adjoint fermions, the high-temperature confining phase is favored, enabling perturbative evaluation of timelike string tensions $\sigma_k^{(t)}$; these tensions exhibit an inverted hierarchy relative to standard Casimir or sine-law scaling, with a massless limit giving $(\sigma_k^{(t)}/T)^2 = rac{(2N_f-1) g^2 T^2}{3N}igl[3\, ext{csc}^2( frac{\pi k}{N})-1igr]$. The magnetic sector is described semiclassically by a $U(1)^{N-1}$ dual sine-Gordon model, yielding a bound on spatial tensions $\sigma_k^{(s)}$ that implies a square-root-Casimir scaling $\sigma_k^{(s)}/\sigma_1^{(s)} \, aise0.3ex extasciitilde\le\}\, \\sqrt{ rac{k(N-k)}{N-1} }$, distinct from conventional scaling laws. Together these results illuminate a coherent high-temperature confinement mechanism tied to monopole dynamics, predict measurable differences from low-temperature confinement, and propose lattice tests to distinguish timelike versus spatial tension scaling.

Abstract

SU(N) gauge theories, extended with adjoint fermions having periodic boundary conditions, are confining at high temperature for sufficiently light fermion mass $m$. In the high temperature confining region, the one-loop effective potential for Polyakov loops has a Z(N)-symmetric confining minimum. String tensions associated with Polyakov loops are calculable in perturbation theory, and display a novel scaling behavior in which higher representations have smaller string tensions than the fundamental representation. In the magnetic sector, the Polyakov loop plays a role similar to a Higgs field, leading to an apparent breaking of SU(N) to $U(1)^{N-1}$. This is turn yields a dual effective theory where magnetic monopoles give rise to string tensions for spatial Wilson loops. The spatial string tensions are calculable semiclassically from kink solutions of the dual system. We show that the spatial string tensions $σ^{(s)}_k$ associated with each $N$-ality $k$ obey a variant of Casimir scaling $σ^{(s)}_k /σ^{(s)}_1 \leq \sqrt{k(N-k)/(N-1)} $. Although lattice simulations indicate that the high temperature confining region is smoothly connected to the confining region of low-temperature pure SU(N) gauge theory, the electric and magnetic string tension scaling laws are different and readily distinguishable.

Figures (1)

• Figure 1: $k=1,2,3$ paths in the $Tr_F P$ complex plane for $SU(6)$. The $k=1$ path is along the boundary of allowed values of $Tr_F P$