Deconfinement in Yang-Mills theory through toroidal compactification with deformation

TL;DR

The paper develops a calculable framework to study deconfinement in Yang-Mills theory by compactifying on ${\mathbb R}^2\times {\bf S}^1_L\times {\bf S}^1_\beta$ and implementing a double-trace deformation ${\Delta S}$ that preserves center symmetry. In the large-$N$ limit with fixed $L$, the deformed theory is thermodynamically equivalent to ordinary YM, while at finite $N$ and small $L$ it enters a semi-classical regime described by a two-dimensional theory with competing electric and magnetic perturbations, enabling analytic access to the confinement–deconfinement transition. The transition is understood as a competition between electric and magnetic degrees of freedom, with a concrete estimate for the critical temperature and a mapping of the Polyakov loop onto a 2d order parameter; the results are connected to the Liao–Shuryak picture of the quark–gluon plasma. The framework yields qualitative and, in parts, quantitative insights that align with lattice results and offer avenues for extension to QCD-like theories and other gauge groups, including open problems such as fermion effects and universality classes at finite $N$.

Abstract

We introduce field theory techniques through which the deconfinement transition of four-dimensional Yang-Mills theory can be moved to a semi-classical domain where it becomes calculable using two-dimensional field theory. We achieve this through a double- trace deformation of toroidally compactified Yang-Mills theory on R2 \times S1_L \times S1_β. At large N, fixed-L, and arbitrary β, the thermodynamics of the deformed theory is equivalent to that of ordinary Yang-Mills theory at leading order in the large N expansion. At fixed-N, small L and a range of β, the deformed theory maps to a two-dimensional theory with electric and magnetic (order and disorder) perturbations, analogs of which appear in planar spin-systems and statistical physics. We show that in this regime the deconfinement transition is driven by the competition between electric and magnetic perturbations in this two-dimensional theory. This appears to support the scenario proposed by Liao and Shuryak regarding the magnetic component of the quark-gluon plasma at RHIC.

Figures (4)

• Figure 1: (Left) At $N=\infty$, the thermodynamics of the deformed and undeformed Yang-Mills theories are equivalent. (Right) Unlike pure Yang-Mills, the deformed theory has, at finite $N$, a semi-classical domain (defined by $LN\Lambda \lesssim 1$) where the confinement-deconfinement transition is analytically calculable.
• Figure 2: For $\beta > \beta_m$, $e^{i \alpha_i \cdot \bm \sigma}$ is relevant, and for $\beta < \beta_e$, $e^{i \alpha_i \cdot \bm {\widetilde{\sigma}}}$ is relevant. In $\beta_m < \beta < \beta_e$ interval, both perturbations are relevant.
• Figure 3: A more refined version of Fig.2. The putative phase transition is expected to occur at $\beta_c$ where $\Delta_e= \Delta_m=1$. For $\beta_c < \beta < \beta_e$, magnetic operators dominate, and $\beta_m < \beta < \beta_c$, electric operators dominate.
• Figure 4: Simplest possible phase diagram of $SU(N)$ deformed Yang-Mills theory on ${\bf R}^2 \times {\bf S}^1_\beta \times {\bf S}^1_L$. Above (below) the solid line, the theory is in the $({\mathbb Z}_N)_\beta$ unbroken (broken) confined (deconfined) phase. Between the solid and dashed line $T_c <T<2T_c$, and at least in the semi-classical domain, although the theory is in deconfined phase, magnetic defects are still relevant. Below the dashed line, they are irrelevant.