Noncompact Gauge-Invariant Simulations of U(1), SU(2), and SU(3)

TL;DR

This paper introduces a new gauge-invariant, noncompact Monte Carlo simulation method for U(1), SU(2), and SU(3) gauge theories, successfully reproducing known results and demonstrating confinement and perturbative behavior.

Contribution

The paper presents a novel noncompact, gauge-invariant Monte Carlo approach applicable to multiple gauge theories, with results matching exact and perturbative predictions.

Findings

Creutz ratios match exact U(1) results for β ≥ 0.5

SU(2) and SU(3) show confinement at specific β values

Results align with perturbation theory at weak coupling

Abstract

We have applied a new gauge-invariant, noncompact, Monte Carlo method to simulate the $U(1)$, $SU(2)$, and $SU(3)$ gauge theories on $8^4$ and $12^4$ lattices. The Creutz ratios of the Wilson loops agree with the exact results for $U(1)$ for $\beta \ge 0.5$ apart from a renormalization of the charge. The $SU(2)$ and $SU(3)$ Creutz ratios robustly display quark confinement at $\beta = 0.5$ and $\beta = 2$, respectively. At much weaker coupling, the $SU(2)$ and $SU(3)$ Creutz ratios agree with perturbation theory after a renormalization of the coupling constant. For $SU(3)$ the scaling window is near $ \beta = 2 $, and the relation between the string tension $\sigma $ and our lattice QCD parameter $ \Lambda_L $ is $\sqrt{\sigma} \approx 5 \Lambda_L$.