Schwinger-Dyson equations in Coulomb gauge consistent with numerical simulation

In the present work, we undertake a study of the Schwinger-Dyson equation (SDE) in the Euclidean formulation of local quantum gauge field theory, with Coulomb gauge condition ${\partial }_i{A}_i=0$. We continue a previous study which kept only instantaneous terms in the SDE that are proportional to $\delta (t)$ in order to calculate the instantaneous part of the time component of the gluon propagator $D_{A_0{A}_0}(t,R)$. We compare the results of that study with a numerical simulation of lattice gauge theory and find that the infrared critical exponents and related quantities agree to within 1% to 3%. This raises the question, “Why is the agreement so good, despite the systematic neglect of noninstantaneous terms?” We discovered the happy circumstance that all the noninstantaneous terms are in fact zero. They are forbidden by the symmetry of the local action in Coulomb gauge under time-dependent gauge transformations $g(t)$. This remnant gauge symmetry is not fixed by the Coulomb gauge condition. The numerical result of the present calculation is the same as in the previous study; the novelty is that we now demonstrate that all the non-instantaneous terms in the SDE vanish. We derive some elementary properties of propagators which are a consequence of the remnant gauge symmetry. Our results support the simple physical scenario in which confinement is the result of a linearly rising color-Coulomb potential, $V(R)\sim \sigma R$ at large $R$. We also show that the horizon condition $⟨H(gA)⟩=(N^2-1)dV$, and the divergence of the ghost dressing function at $\mathbf{k}=0$, $\underset{|\mathbf{k}\to 0|}{\lim }{\mathbf{k}}^2{D}_{c\overline{c}}(\mathbf{k})=\infty$, are identical gauge conditions.

Figure 1

Graphic representation of the SD equation. The undressed line and vertices represent tree-level quantities. The circles, triangles and square represent dressed propagators, dressed 3-vertices, and dressed 4-vertices respectively.

Figure 2

Diagrammatic representation of the SDE. The shaded circle represents an instantaneous propagator and the empty circle an equal-time propagator. The instantaneous propagators are straight lines and the equal-time propagators are wavy.

Figure 3

A near miss at space dimension $d=3$. As $d$ decreases below $d_c$, the curved line intersects the straight horizontal line twice.

Figure 4

Plot of the two solutions ${\theta }_{\pm }(d)$. The critical exponent of the color-Coulomb potential is given by $\delta =d+1+{\theta }_{\pm }(d)$.

Figure 5

Temporal gluon propagators form the rungs of Wilson loop, ladder diagrams.