On Wilson Criterion

TL;DR

This paper investigates the continuum limits of U(1) gauge theory with the Villain action on a lattice, revealing that the Wilson criterion for confinement is ambiguous and depends on the chosen scaling, with implications for the convergence to Euclidean electrodynamics.

Contribution

It demonstrates the dependence of continuum limits on scaling choices and clarifies the conditions under which correlation functions converge to those of Euclidean electrodynamics.

Findings

Correlation functions converge under special scaling only.

Wilson criterion for confinement is ambiguous.

Large loop perimeter asymptotics depend on loop smearing density.

Abstract

U(1) gauge theory with the Villain action on a cubic lattice approximation of three- and four-dimensional torus is considered. The naturally chosen correlation functions converge to the correlation functions of the R-gauge electrodynamics on three- and four-dimensional torus as the lattice spacing approaches zero only for the special scaling. This special scaling depends on a choice of a correlation function system. Another scalings give the degenerate continuum limits. The Wilson criterion for the confinement is ambiguous. The asymptotics of the smeared Wilson loop integral for the large loop perimeters is defined by the density of the loop smearing over a torus which is transversal to the loop plane. When the initial torus radius tends to infinity the correlation functions converge to the correlation functions of the R-gauge Euclidean electrodynamics.