Instantaneous Dynamics of QCD

We start from the observation that, in the confining phase of QCD, the instantaneous color-Coulomb potential in Coulomb gauge is confining. This suggests that, in the confining phase, the dynamics, as expressed in the set of Schwinger-Dyson equations, may be dominated by the purely instantaneous terms. We develop a calculational scheme that expresses the instantaneous dynamics in the local formulation of QCD that includes a cut-off at the Gribov horizon.


Introduction
In sections 1 through 9 we outline the properties and derivation of the instantaneous dynamics expressed in the instantaneous Schwinger-Dyson equations (ISDE). In sect. 10, we write the ISDE in diagrammatic form.

No confinement without Coulomb confinement
Gribov's insight into the mechanism of confinement [1] is substantiated by the theorem [2], for R → ∞, where the color-Coulomb potential, V coul (R), is the instantaneous part of the zero-zero component of the gluon propagator in Coulomb gauge, 1 and V wilson (R) is the gauge-invariant potential derived from a large rectangular Wilson loop. Accordingly, when V wilson (R) rises linearly, V coul (R) rises linearly or super-linearly.
Motivated by this theorem, we conjecture that, in the Coulomb gauge, the instantaneous dynamics is dominant. This will be expressed in a closed system of instantaneous Schwinger-Dyson equations (ISDE). We shall work within the framework of local quantum field theory, with a local action that encodes the cut-off at the Gribov horizon [4], and moreover that is BRST-invariant, thus preserving the geometric property of a gauge theory. This action is derived by the method of Maggiore-Schaden [5] in which BRST-symmetry is spontaneously broken [6]. A conjecture is offered in [7,8] for the identification of physical operators and the physical subspace, but this important subject will not be discussed here.
We shall derive the ISDE by extending to the local action that enforces the cut-off at the Gribov horizon the method which was previously applied to the Faddeev-Popov action [9].

Horizon function and non-local action
We shall localize the non-local Euclidean action, where S FP is the (local) Faddeev-Popov action in Coulomb gauge, H is the non-local horizon function given by and γ 1/4 is the Gribov mass. H cuts off the functional integral at the Gribov horizon, as one sees from the eigenfunction expansion, where the λ n (A) are the eigenvalues of the Faddeev-Popov operator M ≡ −∇ · D(A) in Coulomb gauge. The lowest non-trivial eigenvalue λ 1 (A) approaches 0 as the Gribov horizon is approached from within.

Horizon condition and Kugo-Ojima confinement condition
The Gribov mass γ 1/4 is not an independent parameter, but is fixed by the horizon condition It is a remarkable fact that the horizon condition and the famous Kugo-Ojima confinement condition [10,11] are the same statement This may indicate that color confinement is assured in this theory, although the precise hypotheses of the Kugo-Ojima theorem are not satisfied in this approach.
It is also a remarkable fact that the horizon condition is equivalent to the statement that the QCD vacuum is a perfect color-electric superconductor, which is the dual Meissner effect [12], where G( k) is the ghost propagator and ( k) is the dielectric constant.

Auxiliary ghosts
Just as the Faddeev-Popov determinant is localized by introducing ghosts, likewise, the horizon function H may be localized by introducing "auxiliary" ghosts [4], For reviews of this approach, see [13,14].

Cancellation of energy divergences
The Coulomb gauge is plagued by energy divergences. For example, the integrand of the ghost loop is independent of p 0 , which leads to the horrible energy divergence We get rid of such divergences by using the first-order formalism in which they cancel manifestly [3,15]. This relies on the fact that the Coulomb gauge is a unitary gauge. The first-order action is obtained by writing Here F 0i = ∂ 0 A i − D i A 0 , the Coulomb gauge condition ∂ i A i = 0 holds identically, and π i is colorelectric field. It is decomposed into its transverse and longitudinal parts, with ∂ i τ i = 0. As a result of the equality (23), given below, the c −c ghost loop cancels the λ − A 0 loop, and likewise for the pairs of auxiliary ghosts. This cancellation eliminates the unwanted energy divergences.

Local action and physical degrees of freedom in Coulomb gauge
The local action is given by 2 and B i is the color-magnetic field. Only the first term, iτ i ∂ 0 A i , contains a time derivative. Because A i is identically transverse, ∂ i A i = 0 (as is τ i ), ∂ 0 acts on the two would-be physical degrees of freedom. The remaining terms impose constraints in the local theory. The energy divergences cancel between pairs of fermi and bose ghost loops, including the first pair, i∂ i λD i A 0 − ∂ ic D i c. The last term, L 3 , mixes bose-ghost and gluon fields. In the following it will be convenient to change variables from ϕ andφ to U and V defined by The V-field mixes with the gluon field A, whereas the U-field does not. The propagators of the fields λ and A 0 are related to the ghost propagator D cc by where Γ λλ is the 2-point vertex function. The equality of the Faddeev-Popov ghost propagator and the bose-ghost propagator, assures that the corresponding fermi and bose loops yield Faddeev-Popov determinants that cancel exactly,

Instantaneous Schwinger-Dyson equations (ISDE)
The truncation scheme which we use is represented schematically in Fig. 1, with detailed diagrams given in Figs. 2 and 3. Propagators such as D λA 0 and D AV represent mixing. 2 For the action at finite temperature, see [16].  Figure 2. Free propagators that appear in the ISDE. V P is the part of the bose-ghost propagator that mixes with the gluon propagator.
In Coulomb gauge, the propagators in general decompose into an instantaneous part and a noninstantaneous part, where lim p 0 →∞ D N ( p, p 0 ) = 0. The ISDE is obtained as follows.
I. Cancel loops with two instantaneous bose-ghost propagators against similar fermi-ghost loops, to get rid of energy divergences, 3 II. Neglect loops with two non-instantaneous propagators, III. Keep loops with one instantaneous propagator and one non-instantaneous propagator, Only the equal-time part of any non-instantaneous propagator contributes to the graphs we consider, where the equal-time part of the propagator is defined by Thus, in all graphs that we consider, the non-instantaneous propagator gets replaced by its equal-time part. The diagrams corresponding to the ISDE are given in Fig. 3. Calculations with the ISDE will be reported elsewhere [17]. We have a local, renormalizable quantum field theory with the following interesting properties: • It provides a cut-off at the Gribov horizon.
• The Kugo-Ojima color confinement condition is satisfied.
• The vacuum is a perfect dielectric.
• BRST-symmetry is spontaneously broken, but perhaps only in the unphysical sector.
• In Coulomb gauge, the color-Coulomb potential rises linearly or super-linearly when the Wilson potential is linearly rising.
• In the ISDE, all loops consist of one instantaneous propagator and the equal-time part of a noninstantaneous propagator.