(S)QCD on R^3 x S^1: Screening of Polyakov loop by fundamental quarks and the demise of semi-classics

TL;DR

The paper extends the semiclassical, center-symmetric analysis of softly-broken ${ m N}=1$ SYM on ${f R}^3 imes{f S}^1$ to theories with fundamental flavors, examining how quarks modify monopole-instanton dynamics and the Polyakov loop. It shows that quarks deform monopole-instantons through quantum fluctuations but do not mediate binding, so the expected center-symmetry stabilization by neutral bions persists only when quarks are heavy; as quarks become light, the semiclassical description breaks down and a dual description is needed. In antiperiodic-fermion setups, a crossover in the Polyakov loop from approximate center symmetry to center breaking emerges with increasing $1/L$, aligning with lattice QCD expectations for massive quarks. The work also analyzes special boundary-condition cases with center symmetry preserved, derives the bion-induced potential, and argues that fundamental zeromode exchange does not contribute to neutral bion binding, thereby delineating the regime of validity for the semiclassical picture in SQCD on ${f R}^3 imes{f S}^1$.

Abstract

Recently, it was argued that the thermal deconfinement transition in pure Yang-Mills theory is continuously connected to a quantum phase transition in softly-broken N=1 SYM theory on R^3 x S^1. The transition is semiclassically calculable at small S^1 size L, occurs as the soft mass m_soft and L vary, and is driven by a competition between perturbative effects and nonperturbative topological molecules. These are correlated instanton--anti-instanton tunneling events, whose constituents are monopole-instantons "bound" by attractive long-range forces. The mechanism driving the transition is universal for all simple gauge groups, with or without a center, such as SU(N) or G_2. Here, we consider theories with fundamental quarks. We examine the role topological objects play in determining the fate of the (exact or approximate) center-symmetry in SU(2) SQCD, with or without soft-breaking terms. In theories whose large-m_soft limit is thermal nonsupersymmetric QCD with massive quarks, we find a crossover of the Polyakov loop, from approximately center-symmetric at small 1/L to maximally center-broken at larger 1/L, as seen in lattice thermal QCD with massive quarks and T=1/L. We argue that in all calculable cases, including SQCD with exact center symmetry, quarks deform instanton-monopoles by their quantum fluctuations and do not contribute to their binding. The semiclassical approximation and the molecular picture of the vacuum fail, upon decreasing the quark mass, precisely when quarks would begin mediating a long-range attractive force between monopole-instantons, calling for a dual description of the resulting strong-coupling theory.

Figures (2)

• Figure 1: The minimum of $b'$, proportional to the Polyakov loop trace, $\mathrm{tr}\, \Omega \approx {g^2 \over 4 \pi} b'$, as a function of the parameter $c= c_3' {m_{soft} \over L^2}\; \Lambda^{-3+{N_f \over 2}} M^{-{ N_f \over2}}$. The value $b'=\delta$ corresponds to an almost center-symmetric Polyakov loop and the behavior at $c>4$ indicates a transition towards collapsing eigenvalues of the Polyakov loop. The solid-blue, dashed-purple and dotted-yellow curves are for three values of parameter $c_5'=0,0.0001,0.0005$ respectively. Recall that the $c_5'=0$ case corresponds to neglecting the (suppressed by powers of $g^2$) perturbative holonomy potential, leading to a second order transition, which turns into a crossover once $c_5'\ne0$. This is the behavior seen on the lattice, see Ref. Heller:1984eq for an early study of $SU(2)$ theory with dynamical massive quarks.
• Figure 2: The graphical determination of $\delta_{min}$ for $M = L^{-1} e^{- S_0}$, for three different values of $g^2$, $N_f=1$, and all antiperiodic quarks, $\alpha=\pi$. This value of $M$ is, within leading exponential accuracy, equal to the dual photon (and superpartners) mass, see Section \ref{['bionpotentialsection']}. The three functions of $\delta$ plotted for $0<\delta<.5$ are labeled on the Figure. The relevant intersection point determines that $\delta_{min} \simeq .25$ for $g^2 \ll 1$. Clearly, the semiclassical approximation is valid, as the slopes of the lines near the intersection points are rather well separated. The plot also shows that even exponentially small values of the Dirac mass can lead to only finite deviations from center symmetry, e.g., with angular separation of the Polyakov loop eigenvalues about ${\pi \over 2}$, thus still described by a weakly-coupled Abelian theory.