Wilson loop from a Dyson equation

The Dyson equation proposed for planar temporal Wilson loops in the context of supersymmetric gauge theories is critically analyzed thereby exhibiting its ingredients and approximations involved. We reveal its limitations and identify its range of applicability in nonsupersymmetric 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.

Figure 1

Trapezoidal temporal Wilson loop.

Figure 2

(a) Processes involved and (b),(c) processes ignored in the Dyson equation (8) illustrated in Fig. 3.

Figure 3

Graphical illustration to (a) the Dyson series for the Wilson loop and (b) the Dyson equation (8).

Figure 4

(a) Triangle and (b) rectangular shaped loops representing the Wilson loop $W(S,T=0;L)$ and $W(S,S;L)$; see text.

Figure 5

The gluon energy $\omega (k)$. Data points (MC) show the lattice results of Ref. 22. The full curve is the fit of the lattice data by Gribov’s formula (30) while the dashed curve is the result of the variational calculation [9].

Figure 6

The potential $U(r;L)$ (23) for several distances $L$.

Figure 7

Left-hand panel: The full static quark potential $V(L)$ obtained from the full propagator (32) and the perturbative potential $V_{\mathrm{pert}}(L)$ obtained from the perturbative propagator (33). Right-hand panel: The full potential minus its perturbative part.