Universal mechanism of (semi-classical) deconfinement and theta-dependence for all simple groups

TL;DR

The paper presents a universal, semi-classical mechanism for deconfinement in pure Yang–Mills theories across all simple gauge groups by exploiting a twisted partition function on ${\mathbb R^3 \times {\mathbb S^1}}$. It shows that neutral bions generate eigenvalue repulsion of the Wilson line, counteracting monopole-instanton and perturbative attraction, with the resulting phase structure mapped for ${\rm SU}(N_c)$ with $N_c\ge3$ and ${\rm G_2}$; theta-angle effects produce a multi-branched $T_c(\theta)$ with a minimum at $\theta=\pi$. The transition is first order for $N_c\ge3$ (center-symmetry changes) and for $G_2$ (discontinuous Wilson line), while the mechanism remains universal across simple groups. The approach provides a controlled bridge between weak-coupling semi-classical analysis and strong-coupling confinement, with potential extensions via resurgence and applications to related models.

Abstract

Using the twisted partition function on R^3 x S^1, we argue that the deconfinement phase transition in pure Yang-Mills theory for all simple gauge groups is continuously connected to a quantum phase transition that can be studied in a controlled way. We explicitly consider two classes of theories, gauge theories with a center symmetry, such as SU(N_c) gauge theory for arbitrary N_c, and theories without a center symmetry, such as G_2 gauge theory. The mechanism governing the phase transition is universal and valid for all simple groups. The perturbative one-loop potential as well as monopole-instantons generate attraction among the eigenvalues of the Wilson line. This is counter-acted by neutral bions --- topological excitations which generate eigenvalue repulsion for all simple groups. The transition is driven by the competition between these three effects. We study the transition in more detail for the gauge groups SU(N_c), N_c>2, and G_2. In the case of G_2, there is no change of symmetry, but the expectation value of the Wilson line exhibits a discontinuity. We also examine the effect of the theta-angle on the phase transition and critical temperature T_c(theta). The critical temperature is a multi-branched function, which has a minimum at theta=pi as a result of topological intereference.

Figures (7)

• Figure 1: Contour plots of the bion- and monopole-instanton-induced potential, as a function of the two holonomies, showing the first order phase transition for $SU(3)$ (darker shades represent smaller values of the potential). Left panel: Contour plot for $c_m < c_m^{**}<c_m^{\it cr}$ ($c_m = 2.20$, $c_m^{\it cr} = 2.446$) as a function of $b_1, b_2$. The ${\mathbb Z}_3$-symmetric (confining) minimum is at the origin. Right panel: Contour plot for $c_m^{\it cr} < c_m < c_m^{*}$ ($c_m =2.5$) as a function of $b_1, b_2$. The ${\mathbb Z}_3$-breaking global minima are clearly visible, and the ${\mathbb Z}_3$-symmetric confining minimum is meta-stable.
• Figure 2: Effective potential due to bions and monopoles for different values of $c_m$. The potential (in units of $V_{bion}^0$) is shown as a function of $b_1$ for $b_2 = 0$, corresponding to a cut along the $x$-axis in Fig. 4.
• Figure 3: The distribution of the eigenvalues of the Polyakov loop around ${\mathbb S}^1_L$ for $N_c = 4$ for different values of $c_m$, shown for $g^2 N_c = 0.1$.
• Figure 4: The distribution of the eigenvalues of the Polyakov loop around ${\mathbb S}^1_L$ for $N_c = 5$ for different values of $c_m$, shown for $g^2 N_c = 0.1$.
• Figure 5: The distribution of the eigenvalues of the Polyakov loop around ${\mathbb S}^1_L$ for $N_c = 10$ for different values of $c_m$, shown for $g^2 N_c = 0.1$.