Topological Structure in the SU(2) Vacuum

TL;DR

This paper introduces a lattice-based framework to study SU(2) topology by combining a fixed-point-inspired action, an inverse-blocking smoothing procedure, and an algebraic topological charge operator. The approach preserves long-distance physics (e.g., the string tension) while isolating topological objects, revealing an instanton-dominated action after smoothing with average size ~0.2 fm and density ~2 fm^{-4}, and showing strong clustering rather than a dilute gas. The results challenge the notion that instantons alone drive confinement, instead suggesting that confinement arises from infrared structures beyond individual instantons. The methodology provides a controlled path to quantify topological content and can be extended to include fermions and more refined FP actions, enabling deeper insight into nonperturbative QCD dynamics.

Abstract

We study the topological content of the vacuum of SU(2) pure gauge theory using lattice simulations. We use a smoothing process based on the renormalization group equation which removes short distance fluctuations but preserves long distance structure. The action of the smoothed configurations is dominated by instantons, but they still show an area law for Wilson loops with a string tension equal to the string tension on the original configurations. Yet it appears that instantons are not directly responsible for confinement. The average radius of an instanton is about 0.2 fm, at a density of about 2 fm^(-4). This is a much smaller average size than other lattice studies have indicated. The instantons appear not to be randomly distributed in space, but are clustered.

Figures (11)

• Figure 1: Action vs. instanton radius for the SU(2) action of Table 1.
• Figure 2: Action vs. instanton radius for the SU(3) action of Table 2.
• Figure 3: Topological charge vs. instanton radius from our algebraic charge operator, measured on a set of smooth instanton configurations. The step function is the geometric charge on the same configurations. The long-short dashed line shows the topological charge measured using the naive definition.
• Figure 4: The evolution of the charge densitry profile of an ideal I-A pair both having a radius of $\rho=1.5$ with their centers 2.5 lattice spacings apart. These snapshots were taken at smoothing step 1,2,6 and 9, always on the inverse blocked lattice.
• Figure 5: Action vs. cycle number for a typical lattice.