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 non-instantaneous terms?"We discovered the happy circumstance that all the non-instantaneous 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. In particular the time component of the gluon propagator is found to be purely instantaneous $D_{A_0 A_0}(t, R) = \delta(t) V(R)$, where $V(R)$ is the color-Coulomb potential. 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 $R$.


INTRODUCTION
While the quest for exotic quantum theories of gravity captivates many physicists, a much more mundane question remains unanswered: what is the qualitative mechanism for the mismatch between the UV degrees of freedom of the standard model (quarks and gluons) and the IR states we observe in the lab (baryons and mesons). In other words, an intuitive physical picture of confinement still eludes us, despite the empirical successes of the standard model in the UV. Genuinely new physics is unlikely needed; from lattice simulations, we know that non-Abelian gauge theory by itself is capable of creating gluonic flux tubes which confine quark-anti-quark pairs into mesons at low energy [1]. Yet despite our best efforts, the mathematics behind this phenomenon is unknown. The ultimate goal of science is not just to reproduce nature, but rather to understand it, and this goal is what drives the field of non-perturbative QCD.
The breakdown of perturbation theory at low energies forces one to face the non-Abelian character of Yang-Mills theory head on. Various approaches have been made over the years to use functional methods to extract information about the fully nonperturbative, dressed propagators and vertices of QCD. These quantities are crucial to understanding confinement. For example, an infrared vanishing gluon propagator violates reflection positivity and thus implies that the gluon is not an asymptotic field of the theory. Also, in Landau gauge, the divergence of the ghost dressing function at k = 0 leads to a well defined global color charge which is an important part of the Kugo-Ojima confinement scenario [2,3]. Two techniques that have greatly increased our understanding of the non-perturbative sector of QCD, constituting an infinite hierarchy of coupled equations that can be derived rigorously from the full quantum effective action, are the functional renormalization group equations (FRG) [2,[4][5][6] and the Schwinger-Dyson equations (SDE) [7][8][9][10][11][12][13][14]. A third technique, exploits a formal similarity between vacuum expectation values in the Hamiltonian formalism and correlation functions in Euclidean quantum field theory. In this approach an ansatz is made for the vacuum wave functional which confirms results found by other techniques [4,[15][16][17][18][19][20]. The advantage of the canonical approach is that with Lagrangian methods, an uncontrolled truncation must be made to complete the equations. At first glance, it seems that in the Hamiltonian approach, a truncation is still made even with a non-Gaussian ansatz: a finite order polynomial is still used for the vacuum wave functional. However, due to the gap equation found by varying the energy density, the best possible coefficients of that finite order polynomial will be found which minimizes the effect of the truncation [15]. Nonetheless, we will proceed with the approach povided by the Schwinger-Dyson equations. In contradistinction to the Hamiltonian operator method, we use a local Euclidean quantum field theory.
Coulomb gauge is a natural choice for attempting a qualitative understanding of confinement for two reasons. First, it is a unitary gauge, where Gauss's law can be resolved explicitly by the longitudinal component of the color electric field, thus only propagating physical degrees of freedom (analogous to the two polarizations of the physical photon of QED). To interpolate between UV QCD and phenomenological theories of IR QCD, tracking the physical degrees of freedom is essential. Second, the long-range nature of the color-Coulomb potential, δ(x 0 − y 0 )V Coulomb ( x − y) = A 0 (x)A 0 (y) , gives a physical picture of what does the confining. Despite being a gauge-dependent quantity, the color-Coulomb potential also gives us insight into the IR asymptotics of the gauge-invariant Wilson potential by the following argument, found in detail in [21]. Consider a quark-antiquark pair at separated points, x and y with R ≡ | x − y|. The correlator of two Wilson lines, G(R, T ), extending an amount T in the time direction is related to the Hamiltonian and the state |ψq q by where L( x, 0, T ) is a Wilson line extending from 0 to T at point x. Defining the logarithmic derivative, one can show that the Coulomb energy is obtained in the limit T → 0, and the energy of the flux tube ground state is obtained in the opposite limit, T → ∞. Since the latter is the ground state, at large R (so one can neglect the self-energy contribution), Thus the Coulomb potential must be at least linear (possibly super-linear) in order to reproduce a linearly rising Wilson potential like the one seen on the lattice. While a long-range Coulomb potential is a necessary condition for confinement, it isn't a sufficient one. Similarly to how charges screen each other to make neutral molecules despite the presence of the long-range Coulomb potential, the QCD vacuum creates quark-anti-quark pairs, confining color charge despite the presence of a long range color-Coulomb potential. Thus, even at high temperature, above the deconfinement phase transition, the long range Coulomb force is present as seen in [13]. The instantaneous character of the dynamics is of particular importance to those interested in studying the so-called quarkgluon plasma at high temperature. The common wisdom is that at high temperatures, typical momentum transfer is large, and thus, due to asymptotic freedom, quarks and gluons will behave like a weakly interacting plasma. The presence of a long range color-Coloumb potential at high temperature challenges this view, and suggests that one might expect a strongly interacting fluid, despite the approximate Stefan-Boltzmann like behavior witnessed by Karsch et al on the lattice [22]. This isn't contridictory with the renormalization group; recall that in Coulomb gauge, the physical quantity g 2 D A 0 A 0 is a renormalization-group invariant [23]. This phenomenology would be similar to N = 4 super Yang-Mills (SYM) in the planar limit as pointed out in [24]. The comparison of high temperature QCD to N = 4 SYM, a strongly-coupled integrable theory, is particularly intriguing in light of an article by Dubovsky and Gorbenko [25] which suggests that at large N, the theory of QCD flux tubes may also be integrable, evading the no-go theorem in [26] by possessing a massless pseudoscalar mode in addition to the usual goldstone modes of a string-like flux tube embedded in spacetime. If the color-Coulomb potential is indeed stronger at high temperature than at zero temperature, as the lattice calculation suggests [21], this would imply that gluons are more likely to form color singlets (ie. glueballs), rather than less, since gluon configurations not bound into their flux-tube ground state would be Boltzmann suppressed, making the flux tube description more relevant. The instantaneous character of the dynamics is crucial to accessing the physics at high temperature because it only keeps terms in correlation functions that dominate at vanishingly small temporal separation. At high temperatures, the partition function becomes vanishingly small in the Euclidean-time direction, thus yielding a dimensionally reduced theory, in addition to any instantaneous physics inherited from the higher dimensional theory. This heuristic picture is illustrated in [24] and a rigorous treatment of Gribov-Zwanger theory in Coulomb gauge at finite temperature can be found in [27].
One objective of this article is to gain a quantitative handle on the asymptotic behavior of the color-Coulomb potential. We do this by finding a self-consistent set of vertices of the full quantum effective action that satisfy the Schwinger-Dyson equations, continuing the work of [14]. More specifically, in that work, only terms in the SDEs were kept that are proportional to δ(t) in order to calculate the instantaneous part of D A 0 A 0 , a.k.a., the color-Coulomb potential. In sect. 11 of the present article, we compare the infrared critical exponents found [14] with numerical simulation in lattice gauge theory of SU(2) by Langfeld and Moyaerts [28]. The agreement is striking. There is also reasonably good agreement with Burgio, Quandt and Reinhardt [29] for SU (2), and with Nakagawa et al [30] for SU (3). This led us to question why the agreement was so good, in view of the neglect of the non-instantaneous terms. We have discovered that the non-instantaneous terms vanish because of the invariance under time-dependent gauge transformations g(t). These form the remnant gauge symmetry group of gauge transformations that are not fixed by the Coulomb gauge condition ∂ i A i = 0.

LOCAL ON-SHELL FADDEEV-POPOV ACTION IN COULOMB GAUGE
The Faddeev-Popov quantization of Yang-Mills theory in Coulomb gauge is defined in phase-space formalism by the Lagrangian density, [23]. The connection A a µ as well as the Nakanishi-Lautrup and Faddev-Popov ghost fields b a , c a andc a are all fields in the adjoint representation of the global SU(N) color group. Color components are represented by Latin superscripts. To streamline notation we adopt the convention that X · Y ≡ ∑ a X a Y a and (X × Y ) a ≡ ∑ bc g f abc X b Y c , where f abc are the su(N) structure constants and g is the gauge coupling. In this notation the gauge-covariant derivative in the adjoint representation is D µ X = ∂ µ X + A µ × X. If one integrates out the canonically conjugate color-electric field π i , one gets the Coulomb-gauge Faddeev-Popov Lagrangian density in the second-order formalism, Next, we integrate out the b-field, so the gauge condition is satisfied on-shell, and A is purely transverse, We separate the transverse and longitudinal parts of π, where ∂ i τ i = 0. The Faddeev-Popov action with the on-shell gauge condition is given by The time derivative appears only in the first term,

TIME-DEPENDENT GAUGE TRANSFORMATIONS AND THEIR CONSEQUENCE FOR PROPAGATORS
The gauge condition ∂ i A i = 0 does not fix time-dependent gauge transformations g(t). Moreover the action S is invariant under such gauge transformations, where the fields transform according to and Φ α = t a Φ a α and A 0 = t a A a 0 . The t a are a basis of the Lie algebra of the gauge structure group, [t a ,t b ] = f abc t c , and Φ a α = (A a i , τ a i , λ a , c a ,c a ) represents all fundamental fields besides A a 0 . Under these transformations, F µν and π transform gauge covariantly, g F µν = g −1 F µν g, and g π i = g −1 π i g. (In general it will be understood that g = g(t).) A symmetry of the action implies that expectation values are invariant under the same symmetry transformation, This symmetry is generally ignored in analytic calculations, because it is broken in usual approximation schemes. For example, it is not a symmetry of the tree-level theory. 1 However it is a powerful symmetry. 1 Indeed the tree-level Lagrangian density in Coulomb gauge contains a time derivative in the term 1 2 (∂ 0 A i ) 2 , and only in this term. Under the infinitesimal time-dependent gauge transformation δA a i = f abc A b i × ω c (t), this term breaks the symmetry, There is no other term in the tree-level Lagrangian with a time derivative to cancel this. Statement: Let φ a 1 (x) and φ a 2 (y) be two fields that transform covariantly under time-dependent gauge transformations. Then their propagator has a δ-function singularity in time The proof is immediate. The infinitesimal form of the time-dependent gauge transformation, Eq. (3.2), is where i = 1, 2, and invariance under infinitesimal time-dependent gauge transformations, Eq. (3.4), reads Global gauge invariance, that is, for g = const, implies that φ a This holds for all ω(t). The general solution to this condition, which is a well defined distribution, is Eq. (3.5), as asserted. The proof holds for other non-trivial representations such as the fundamental representation. It also extends immediately to the lattice. Propagators whose time-dependence is given by δ(x 0 − y 0 ) will be called "instantaneous."

PROPAGATORS IN COULOMB GAUGE
The scalar fields A 0 and λ and the ghost pair c andc appear at most quadratically in the action, Eq. (2.1), with fixed A i and τ i . To calculate the propagators of these fields, one may integrate out the fields A 0 and λ or c andc by Gaussian integration, and one obtains the well-known formulas is the d-dimensional Faddeev-Popov operator that depends only on the transverse dynamical field A i , K is the operator with kernel and are the potentials produced by the color charge density ρ ≡ gτ i ×A i of the dynamical gluons (and of quarks, if quarks are present). The Faddeev-Popov operator is hermitian,

SCHWINGER-DYSON EQUATIONS
We wish to explore the hypothesis that there exists an asymptotic infrared limit of the DSE which is dominated by loops containing an instantaneous propagator. Details of the derivation of the SD equations are given in [14]. (There is a slight change of notation. The substitutions from [14] to the present article are φ → λ, The time derivative appears in the action Eq. (2.5) only once, in the canonical term iτ · ∂ 0 A i , so the fields τ i and A i propagate in time, and there is no instantaneous term, with factor δ( The only propagators with the instantaneous factor δ(x 0 − y 0 ) occur in the equations (4.1) for the scalar propagators. In the DSE there are some loops that contain at least one factor of δ(x 0 − y 0 ), and some loops that contain none. The DSE holds separately for each of these sets, and we shall retain only those loops that contain at least one factor of δ(x 0 − y 0 ). (It will turn out happily that this gives us a closed system of equations.) Because the fourier transform of an instantaneous propagator is independent of k 0 , we obtain the instantaneous parts by making the substitutions The SDE is represented graphically in Fig. 1. However most terms vanish. We discard those, and keep the remaining terms. The tree-level terms are retained. The renormalization term (penguin diagram) is canceled by a mass counter-term. Consider the other one-loop graph which is the product of two propagators. The possibilities are: both propagators are instantaneous, or one is instantaneous and the other is not, or neither is. If they are both instantaneous, such as there is a terrible divergence, characteristic of the Coulomb gauge. Fortunately these terms cancel, as we shall see shortly. If one propagator is instantaneous, such as V (x − y)δ(x 0 − y 0 ), and the other, D N (x − y), is non-instantaneous, the product is instantaneous, , and gives an instantaneous contribution to Γ. The instantaneous one-loop graphs are represented in Fig. 2 (where the dressed 3-vertices have been replaced by the tree-level 3-vertices, as will be discussed shortly). If both propagators in the loop are non-instantaneous, the result is neglected, because the product does not have a factor of δ(x 0 − y 0 ), and is not instantaneous. Now consider the two two-loop graphs in Fig.1. Both of these graphs contain a tree-level 4-vertex which originates from the quartic (A i × A j ) 2 term in the action Eq. (2.5). 2 Three propagators emerge from the treelevel 4-vertex. Each of these propagators starts from the vector field A i so none of them is instantaneous. It follows that their product is not instantaneous, and their contribution may be neglected. So far our calculations are exact. We now make our only truncation: replace the remaining dressed 3-vertex (in the graph in Fig. 1) by the corresponding tree-level vertex. The result is given in Fig. 2 and in the following equations. This truncation has been explored in depth, and is found to be robust numerically in both Coulomb and Landau gauge [2, 5-7, 10, 12, 20, 31-33]. This results from two properties of the ghost-ghost-gluon vertex in Coulomb gauge [23]: (1) The external ghost momenta factor out of the corresponding Feynman integrals. This depresses the degree of convergence of the integrals, so (2) the vertex does not require renormalization Z 1 = 1. These properties severely restrict the allowed form of the complete vertex, and investigation did not reveal a new acceptable solution of the SD equation [14]. The same properties hold in the Landau gauge [34].
The resulting equations are represented graphically in Fig. 2, and, analytically by where the indices run over all degrees of freedom. As discussed above, terms on the right-hand side such as D AA (p)D AA (k − p) (and quark loops if any), where both factors are non-instantaneous, do not contribute to these SD equations.
As follows from Eq. (5.2), we set D λλ = 0 on the right hand side of the SD equations where it appears, namely in the first term of the second line of Eq. (5.4). The remaining two terms in the second line of Eq. (5.4) appear to suffer from terrible divergences. In position space each is the product of two instantaneous propagators. For example the second term in that line is δ 2 (x 0 − y 0 )D λA 0 (x − y)D λA 0 (x − y). The second and third terms contain the divergent integral d p 0 = ∞, which is the momentum-space manifestation of the divergent factor δ 2 (x 0 − y 0 ). These are the famous energy divergences of the Coulomb gauge which cancel between the second and third term [35][36][37][38],  [36].) Each of the remaining terms in the SDE is the product of an instantaneous propagator and an equal-time propagator, which together give a finite instantaneous contribution.
With these results the DSE simplifies to Consider the loop integral for Γ AA (k). The only appearance of p 0 in the integrand occurs in D ττ (p 0 , p), so the loop integral over p 0 takes the form where the right-hand side is the equal-time propagator. Indeed it is a special case of the fourier transform, at t = 0, D ET ττ (p) = D ττ (0, p). The remaining integration d d p is an integral over the space dimension d. The same is true for all the loop integrals.
We now show that is a solution of Eq. (5.15). Indeed, suppose this is true. It gives which implies It follows that the integral in Eq. (5.15) vanishes, which gives Γ τA (k 0 , k) = −k 0 so Eq. (5.15) is satisfied. (There may also be a non-perturbative solution which cannot be expressed as a power series in g.) The SD equations now read Γ τA (k) = −k 0 (6.8) Upon close inspection of equation Eq. (6.4), something may seem amiss. The k 0 dependence in the propagators implies that these propagators are non-instantaneous, which violates the symmetry discussed in section 3. A brief calculation with a powerlaw ansatz is provided in Appendix C to show that in the infrared limit, this symmetry is restored, exhibiting a remarkable self consistency of the approach.

REDUCTION TO THREE UNKNOWNS
The propagators and inverse propagators, D rs Γ st = δ rt , of the scalar fields are related by where we have used Γ A 0 A 0 (k) = 0. This gives D λλ (k) = 0, in accordance with Eq. (5.2). Correspondingly for the dynamical propagators, we have Eq. (6.4), which gives for the equal-time propagators Note that D τA (k 0 , k) is odd in k 0 which gives D ET τA (k) = dk 0 D τA (k 0 , k) = 0, as claimed. From the last equation we have the simple identity, which determines D ET ττ . There remain only three independent unknown functions D ET AA (k), D A 0 A 0 (k) and D A 0 λ (k). We also have from Eq. (7.3), The last two equations give We now substitute the right hand side of the SDE for Γ AA (k) and Γ ττ (k), Eq. (6.6) and Eq. (6.7), into the last equation, which gives where the last two equations come from Eq. (6.10) and Eq. (6.11), and we have used Altogether there are three equations for the three propagators D ET AA (k), D cc (k) and D A 0 A 0 (k). These three quantities are invariant under the remnant gauge symmetry g(t). Suppose the three equations are solved, so these three quantities are known. Then one can recover a 4th quantity, D ET ττ (k), from 4D ET ττ (k)D ET AA (k) = 1. These 4 quantities are all that appear on the right hand side of eqs. (6.6) through (6.11), from which one can recover all Γ αβ and hence all propagators D βγ .

GAUGE CONDITION ON THE LATTICE AND IN THE CONTINUUM
Beside imposing the Coulomb gauge condition, ∂ i A i = 0, we must also address the non-perturbative issue of Gribov copies [39][40][41][42].
A gauge choice that is accessible to numerical simulation is implemented by minimizing (the lattice analog of) the spatial Hilbert norm, with respect to gauge transformations g(x), where A µ = 1 2 iτ b A b µ and g A µ = g −1 A µ g + g −1 ∂ µ g. At a global or local minimum, the gauge condition ∂ i A i = 0 is satisfied, and all eigenvalues of the Faddeev-Popov operator M(A) are non-negative λ n (gA) ≥ 0. The set of continuum configurations that satisfy these conditions is designated by Ω and is called the "(first) Gribov region." It is a convex region in configuration space (A-space) that is bounded in every direction. Its boundary, ∂Ω, is called the "Gribov horizon." At large volume V , Ω is specified by H(gA) ≤ (N 2 − 1)dV , where the "horizon function," H(gA), is defined in Eq. (A.2) [43]. The actual lattice simulation with which we shall compare was gauge-fixed by finding one local minimum of the minimizing functional for each gauge orbit. 3 The set Λ of absolute minima of the minimizing functional provides a complete gauge fixing. It would be nice if we could perform the (functional) integral over Λ, but we cannot, because we do not have an explicit description of Λ in the physical limit of large volume V , as we do for Ω. 4 In this situation we make the approximation which consists in integrating over Ω instead of Λ. This approximation introduces a certain "gauge-fixing error," and the total error of the present calculation is the compound of this gauge-fixing error with the error introduced by the truncation of terms in the SDE.
In the limit of large volume V , the functional integral over the Gribov region Ω gets concentrated on its surface ∂Ω, 5 and the cut-off at the Gribov horizon is replaced by insertion of the factor δ[(N 2 − 1)dV − H], which enforces the "horizon condition." 6 In Appendix A, it is shown that the horizon condition H = (N 2 − 1)dV , and the maximum-b condition, lim |k|→0 b(k) = ∞, are equivalent, where b(k) ≡ k 2 D cc (k) is the ghost dressing function. 7 The maximum-b condition states that the ghost propagator is of longer range than the electrostatic potential, which is the same as requiring that the ghost propagator D cc (k) be more singular than 1 k 2 at k = 0, or equivalently that the inverse ghost propagator Γc c (k) = D −1 cc (k) vanishes more rapidly than k 2 . The last condition is imposed by subtracting the term of order k 2 on the right hand side of the SD equation for Γc c , so it reads There is an overall coefficient k 2 , and the integrand vanishes at k = 0, so the right hand side vanishes faster than k 2 . It is not obvious whether the last integral is positive for all k, as it should be if M(A) is a positive matrix, so it is a nice check that when it is evaluated below, I(α, γ), given in Eq. (10.7), it is in fact positive.

A. First SD equation for critical exponents
We now assume that the propagators approach an asymptotic limit at small k = |k| which is a power law, with critical exponents defined in Table I. We substitute this power-law Ansatz into Eq. (7.7), For this to yield a bona fide solution, the loop integral must converge. There is a singularity due to the color-Coulomb potential, 1/|p − k| δ . However the factor k γ p γ − p γ k γ vanishes at p = k like (p − k) 2 , so the integral converges at p = k provided δ < d + 2. There is a singularity at p = 0 due to the terms p γ and p −γ , so the integral does not converge at p = 0 unless d > |γ|. The loop integral must also converge at high p to assure that the infrared dynamics decouples from the other degrees of freedom. Suppose γ is positive γ > 0. In this case the highest power of p in the last integrand comes from the power p γ in the second term in the parenthesis, and the integral will not converge at high p unless d + γ < δ. Now suppose instead that γ is negative, γ < 0. In this case the highest power of p comes from the first term in parenthesis 1/p γ , and the loop integral will not converge unless d − γ < δ, and we have established, By power-counting one sees that the right-hand side of Eq. (9.1) is proportional to k d−δ . The inequalites just obtained imply that in the infrared asymptotic limit, k → 0, the right-hand side is dominant over each term on the left-hand side. Indeed it dominates the first term in this limit provided δ − d > γ − 2, that is, if δ > d + γ − 2, which holds by virtue of Eq. (9.2). Likewise it dominates the second term on the left provided δ − d > −γ, that is, if δ > d − γ, which is also true. Therefore in the infrared asymptotic limit only the right-hand side survives, and the first SDE reads, 3) The inequality δ > d, just derived, is none other than the condition in space dimension d, for the color-Coulomb potential to be confining, lim r→∞ V C (r) = ∞, for we have 8 Thus Eq. (9.3) is a sufficient condition for the color-Coulomb potential to be confining.

B. Second SD equation for critical exponents
Insertion of the power laws into (8.2) yields By counting powers of k and p on the left and right hand sides, we obtain α = 2 + d − γ − α, which gives the "sum rule" Moreover the right hand side has a coefficient k 2 , and an integrand that vanishes with k, so the right hand side vanishes with k more rapidly than k 2 . We conclude that α > 2, so the ghost propagator is more singular than the free propagator. 9 This was imposed by the horizon condition. We require that the loop integral (9.5) converges at high p. This yields the inequality d < γ + α + 2. Indeed the subtraction term cancels the leading term in 1/p at high p, and the next power is killed by angular integration, so the subtraction term increases the power of 1/p by 2. We substitute into the last inequality and obtain α < 4, and thus 2 < α < 4. (9.8)

C. Third SD equation for critical exponents
Upon insertion of the power Ansatz into Eq. (7.8) we obtain The integral converges provided d − γ − δ < 0, which agrees with the confinement bound, Eq. (9.2). In this case the tree-level term is negligible compared to the loop term in the infrared asymptotic limit, and this ISD simplifies to where Likewise from Eq. (9.10) we obtain One has The integral I(α, γ) was evaluated in (A.17) of [48], with I(α) = I G (α G ) and α = 2 + 2α G , with the result 10 [I(α, γ) is positive because α + γ − d = 2 − α < 0, provided that α < 4, which holds by Eq.  A third relation between the critical exponents is provided by Eq. (9.3). An obvious solution to that equation is provided by We have searched diligently for another solution, but we have not found any. This solution and the sum rule (9.7) then give and we finally obtain the following condition, expressed with a new parameter, θ ≡ δ − d − 1, where We have used Γ(x)Γ(1 − x) = π/ sin(πx). The change of variable from δ to θ is convenient because E(θ, d) is even in θ, For a given space dimension d, let θ(d) be a solution to Eq. (10.12), then the critical exponent of the color-Coulomb potential δ is recovered from (10.14) The function E(θ, d) is finite and positive for θ in the interval −1 < θ < 1 (which corresponds to d < δ < d + 2), and is divergent at the end-points, θ = ±1, where cos(πθ/2), which is in the denominator, vanishes, cos(πθ/2) = 0. Since E(θ, d) is even in θ, if θ(d) is a solution to Eq. (10.12), then so is −θ(d), and the solutions to (10.12) form two branches θ + (d) and θ − (d) = −θ + (d), as shown in Fig. 3. We are interested in integer space dimensions d = 2 and d = 3. However it is helpful to take d to be a continuous variable in the interval 2 < d < 3. The function F(d) diverges in the limit d → 2 which tells us that θ(2) = ±1, for we have just seen that E(θ, d) is divergent at θ = ±1. 11 These values of θ correspond to δ(d = 2) = d + 1 ± 1 = 3 ± 1, so for d = 2, there are two solutions δ − (2) = 2 and δ + (2) = 4. One sees in Fig. 4 that, as d increases from d = 2, the two branches, θ + (d) and θ − (d) approach each other monotonically, and at a critical dimension, d c = 2.9677... , (10.15) they merge at θ = 0, which corresponds to δ(θ = 0) = d c + 1 [14].
With the problem as stated, there is no solution for space dimension above the critical value d > d c . However 2.9677 is tantalizingly close to 3, and it may be that there is a solution for d = 3 (a) if the gauge-fixing error noted above were corrected, (b) if the truncation error were corrected, or (c) if perhaps the true asymptotic behavior isn't a pure power-law, but rather dominated by a power with multiplicative log corrections. We shall suppose one of these possibilities is in effect, and we consider that it is an approximation to replace the physical value d = 3 by d = d c , which is a difference of about 1%. For d = 3 and θ = 0, we have Instead of an equality, there is a difference of about 3%. The color-Coulomb potential,  [49] which asserts that the color-Coulomb potential V coul (R) is bounded below by the gauge-invariant Wilson potential V W (R) that is linear at large R V coul (R) ≥ V W (R) = σR, (10.18) whereas the lower branch does not. (Both branches accord with the exact bound δ > 2α − 2 = d [47].) The most natural choice between the two branches is to take the upper branch to be the physical solution because it satisfies the last inequality. In d = 2 space dimensions, the calculation reported here yields, δ(d = 2) = 4, which corresponds to a color-Coulomb potential that rises like r 2 at large r. This is an unexpectedly steep rise. One might speculate that the physical solution is a superposition or mixture of the two branches, so that the physical solution corresponds to a value of θ that lies between the two branches. If so, then θ must also satisfy θ ≥ 0, corresponding to δ ≥ d + 1, to be consistent with the last inequality, and because linear rise corresponds to δ = d + 1. Finally we note that linear rise is energetically favorable compared to superlinear. A lattice calculation in two space dimensions (if it is not already in the literature) would throw some light on this matter.
In any case the comparison we shall make with lattice gauge theory is with critical dimension d c = 2.9677... , which corresponds uniquely to linear rise, δ = d + 1.

A. Comparison with Lattice Gauge Theory
Lattice calculations have been reported for SU (2) [28,29] and SU(3) [30]. In Table II we compare our results with Langfeld and Moyaerts [28]. In the first column of Table II are the critical exponents, α, γ, and δ, of the propagators of the ghost, the spatial gluon, and the temporal gluon respectively, that are defined in Table I. In the second column, the values of these exponents found in the present article are expressed in terms of the dimension of space d. In the third column, the critical exponents defined in the present article are expressed, for the reader's convenience, in terms of the parameters defined by [28]. (δ LM is the infrared exponent designated δ in [28].) The 4th column gives the numerical values of the critical exponents for the critical dimension found above, d c = 2.9677... . 12 . The final column is result of the numerical simulation [28]. These authors do not give a numerical value for the infrared exponent of the transverse equal-time gluon propagator, γ, but state "At small momentum, the propagator becomes roughly momentum-independent and seems to approach a constant in the IR limit |p| → 0." This is consistent with our result, which gives for this infrared exponent, γ = 0. If it is not accidental, the agreement between the 4th and the 5th column is remarkable for the accuracy of both the lattice simulation and the SD equation.

B. Features of Gluodynamics in the Asymptotic Infrared Limit
We summarize the basic features of gluodynamics in the asymptotic infrared limit, under the assumption that the agreement between the SDE and the lattice gauge calculation is not accidental.
• 1. The dynamics occurs in a single time slice. More precisely, the ghost and temporal gluon propagators, D cc and D A 0 A 0 are both instantaneous, that is, proportional to δ(x 0 − y 0 ), and the spatial gluon propagator is taken at equal time, x; t, y). This is due to the fact that in the Coulomb gauge, Gauss's law, D i π i = ρ quark , is a constraint that is satisfied as an equation of motion.
• 2. In the asymptotic infrared limit, these propagators are fit by power laws with critical exponents whose values are given in the table.
(a) Compared to the tree-level propagator 1/|k| 2 , the ghost propagator is moderately long range. (b) The color-Coulomb propagator is long-range, corresponding to a linear rise in r, or close to it. (c) The infrared limit of the equal-time spatial gluon propagator has critical exponent 0 or close to 0. 13 • 3. The horizon condition H(gA) = (N 2 − 1)dV , and the divergence of the ghost dressing function, lim |k→0| k 2 D cc (k) = ∞, are identical gauge conditions. This is shown in Apppendix A, and applied in sect. 8 where the gauge condition is imposed by subtracting the k 2 term in the SDE.
• 4. There is a shadow cast on these considerations because we have found no solution to the SDE at space dimension d = 3, but only close to it, at d = d c = 2.9677... . We must figure out what mechanism, if any, acts so there is a solution at d = 3. A small effect in the right direction would be sufficient. This could be provided by a dressed vertex replacing a tree-level vertex.
• 5. In Appendix B, the contribution of gluon propagators to the Wilson loop W = N −1 TrP exp( igt b A b µ dx µ ) is calculated. It is found that the spatial gluon propagator D A i A j does not contribute at all. Only the instantaneous temporal gluon propagator contributes, which moreover exponentiates, Eq. (B.2). Consequently the calculation of the contribution of the gluon propagator to the path-ordered exponential is particularly simple in Coulomb gauge as compared to Lorentzcovariant gauges. This may be true for other expectation values.
We observe that when the horizon condition, Eq. (A.1), is satisfied, the term of order k 2 is precisely killed, so We call this the maximum-b condition. The converse is also true: if the maximum-b condition is satisfied, then the horizon condition holds. We conclude that, in the Ω-theory, the horizon condition and the maximum b-condition b(0) = ∞, are equivalent. Thus it is justified to subtract the k 2 term in the equation for the inverse ghost propagator G −1 (k). but the temporal propagator D A 0 A 0 (x−y) = δ(x 0 −y 0 )V (x − y) has the factor δ(x 0 −y 0 ), whereas the spatial propagator D A i A j (x− y) is finite at x 0 − y 0 = 0, D A i A j (0, x − y) = D ET A i A j (x − y). Consequently, when the line integral dx µ D A µ A ν (x 0 − y 0 , x − y) at fixed y, crosses the time-slice x 0 = y 0 , it receives a finite contribution if the gluon propagator is temporal (µ = 0), but it receives no contribution when the gluon propagator is spatial (µ = 1, 2, 3). 15 Thus the spatial gluon propagators do not contribute to the Wilson loop, as asserted. The remaining temporal propagators form the horizontal rungs of ladder diagrams, as illustrated in Fig.5. The path-ordering makes the two ends of the lowest rung adjacent to each other so, for the lowest rung, the Lie algebra gives t a δ ab t b = C F , where C F is the Casimir in the fundamental representation. Since it is proportional to the identity matrix, it may be removed from the path ordering. The same is then true for the next lowest rung etc., so path ordering gives a factor of C n/2 F . The combinatorics are then such that the propagators exponentiate exactly, 16 and we obtain W = exp − 1 2 g 2 C F dx µ δ µ0 dy ν δ ν0 δ(x 0 − y 0 )V (x − y) (B.4) from which Eq. (B.2) follows, as asserted. The important point is that the spatial gluon propagators have dropped out, and only the color-Coulomb potential contributes to the force on the Wilson loop, in this approximation where we have neglected all connected subgraphs with three or more legs. In Lorentz-covariant gauges, such as the Landau gauge, the gluon propagator is not instantaneous, and whereas in Coulomb gauge the propagators form the horizontal rungs of a ladder and the path ordering is easily evaluated, as we have just seen, in a Lorentz-covariant gauge, the would-be rungs run every which way, and one does not know how to disentangle the path ordering. For this problem, calculation in the Coulomb gauge is simpler than in a Lorentz-covariant gauge.