The Wilson loop from a Dyson equation

TL;DR

This paper critically analyzes a Dyson equation for Wilson loops in gauge theories, revealing its limitations, applicability range, and demonstrating its use in deriving a linearly rising potential in both continuum and lattice approaches.

Contribution

It identifies the specific conditions under which the Dyson equation for Wilson loops is valid and solves it using gluon propagators to obtain the potential.

Findings

The Dyson equation applies only to strongly asymmetric Wilson loops.

The Wilson potential can be extracted only at intermediate distances.

The equation yields a linearly rising color potential in both continuum and lattice calculations.

Abstract

The Dyson equation proposed for planar temporal Wilson loops in the context of supersymmetric gauge theories is critically analysed thereby exhibiting its ingredients and approximations involved. We reveal its limitations and identify its range of applicability in non-supersymmetric gauge theories. In particular, we show that this equation is applicable only to strongly asymmetric planar Wilson loops (consisting of a long and a short pair of loop segments) and as a consequence the Wilsonian potential can be extracted only up to intermediate distances. By this equation the Wilson loop is exclusively determined by the gluon propagator. We solve the Dyson equation in Coulomb gauge for the temporal Wilson loop with the instantaneous part of the gluon propagator and for the spatial Wilson loop with the static gluon propagator obtained in the Hamiltonian approach to continuum Yang-Mills theory and on the lattice. In both cases we find a linearly rising color potential.