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

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\mathrm{.}Atmuchweakercoupling,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 ${\mathrm{\Lambda }}_L$ is $\sqrt{\sigma }\approx 5{\mathrm{\Lambda }}_L$.