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Two-Point Functions of Coulomb Gauge Yang-Mills Theory

The functional approach to Coulomb gauge Yang-Mills theory is considered within the standard, second order, formalism. The Dyson-Schwinger equations and Slavnov-Taylor identities concerning the two-point functions are derived explicitly and one-loop perturbative results are presented.


INTRODUCTION
Coulomb gauge Yang-Mills theory (and by extension Quantum Chromodynamics) is a fascinating, yet frustrating endeavor. On the one hand, Coulomb gauge offers great potential for understanding such issues as confinement [1,2]; on the other, the intrinsic noncovariance of the formalism makes any perturbative calculation formidably complicated. Many approaches to solving (or providing reliable approximations to solving) the problems in Coulomb gauge have been forwarded. Recent among these are the Hamiltonian approach of Ref. [3], based on the original work of Christ and Lee [4]. A lattice version of the Coulomb gauge action also exists [5], which has led to numerical studies, for example Refs. [6]. Functional methods based on the Lagrangian formalism have also been considered, especially within the first order (phase space) formalism [1,7] and most recently, one-loop perturbative results for both the ultraviolet divergent and finite parts of the various two-point functions have been obtained [8]. Similar results were previously obtained for the gluon propagator functions under a different formalism (using the chromoelectric field directly as a degree of freedom and without ghosts) and using different methods to evaluate the integrals [9].
In this paper, we consider the (standard, second order) functional approach to Coulomb gauge Yang-Mills theory. We derive the Dyson-Schwinger equations and Slavnov-Taylor identities for the two-point functions that arise in the construction and using the techniques of [8] we present results for the one-loop perturbative dressing functions.
The paper is organized as follows. In the next section, the functional formalism used is described. Section 3 concerns the decomposition of the functions used. The (nonperturbative) Dyson-Schwinger equations and Slavnov-Taylor identities relating the various Green's functions are derived in Section 4. In Section 5, the one-loop perturbative results are obtained. Finally, there is a summary and outlook.

FUNCTIONAL FORMALISM
Let us begin by considering Coulomb gauge Yang-Mills theory. We use the framework of functional methods to derive the basic equations that will later give rise to the Dyson-Schwinger equations, Slavnov-Taylor identities, Feynman rules etc. Throughout this work, we will use the notation and conventions established in [7,8]. We work in Minkowski space (until the perturbative integrals are to be explicitly evaluated) with metric g µν = diag(1, − 1).
The Yang-Mills action is defined as where the (antisymmetric) field strength tensor F is given in terms of the gauge field A a µ : In the above, the f abc are the structure constants of the SU (N c ) group whose generators obey T a , T b = ıf abc T c . The Yang-Mills action is invariant under a local SU (N c ) gauge transform characterized by the parameter θ a x : 3) The field strength tensor can be expressed in terms of the chromoelectric and chromomagnetic fields (σ = A 0 ) such that S Y M = (E 2 − B 2 )/2. The electric and magnetic terms in the action do not mix under the gauge transform which for the gauge fields is written Given an infinitesimal transform U x = 1 − ıθ a x T a , the variation of the gauge field is where the covariant derivative in the adjoint representation is given bŷ Consider the functional integral where Φ denotes the collection of all fields. Since the action is invariant under gauge transformations, Z is divergent by virtue of the integration over the gauge group. To overcome this problem we use the Faddeev-Popov technique and introduce a gauge-fixing term along with an associated ghost term [10]. Using a Lagrange multiplier field to implement the gauge-fixing, in Coulomb gauge ( ∇· A = 0) we can then write The new term in the action is invariant under the standard BRS transform whereby the infinitesimal gauge parameter θ a is factorized into two Grassmann-valued components θ a = c a δλ where δλ is the infinitesimal variation (not to be confused with the colored Lagrange multiplier field λ a ). The BRS transform of the new fields reads It is at this point that this work diverges from Ref. [7] in that we remain here within the standard (second order) formalism. By including source terms to Z, we construct the generating functional, Z[J]: It is convenient to introduce a compact notation for the sources and fields and we denote a generic field Φ α with source J α such that the index α stands for all attributes of the field in question (including its type) such that we can write where summation over all discrete indices and integration over all continuous arguments is implicitly understood.
Expanding the various terms we have explicitly (2.14) The field equations of motion are derived from the observation that the integral of a total derivative vanishes, up to boundary terms. The boundary terms vanish, although this is not trivial in the light of the Gribov problem [2] (the reader is directed to Ref. [7] and references therein for a discussion of this topic). Writing S = S Y M + S f p , we have that The explicit form of the field equations of motion is given in Appendix A.
In addition to the field equations of motion, there exist identities derived by considering the BRS invariance of the action (these eventually form the Slavnov-Taylor identities). The BRS transform is continuous and we can regard it as a change of variables in the functional integral. Given that the Jacobian of such a change of variables is trivial and that the action is invariant, we have that (2.17) We define the classical fields to be The generating functional of proper Green's functions is the effective action, Γ, which is a function of the classical fields and is defined through a Legendre transform of W : We introduce a bracket notation for derivatives of W with respect to sources and of Γ with respect to classical fields (no confusion arises since the two sets of derivatives are never mixed): It is now possible to present the field equations of motion in terms of proper functions (the Dyson-Schwinger equations are functional derivatives of these equations). Using the results listed in Appendix A we have: It is also useful to express the λ equation of motion in terms of connected functions: The identity stemming from the BRS invariance is also best expressed in terms of both connected and proper functions and reads: where we have used the common trick of using the ghost equation of motion in order to reexpress one of the interaction terms transversely, with the transverse projector in configuration space being . This manipulation will be useful when we consider the Slavnov-Taylor identities for the two-point functions later on.
At this stage it is useful to explore some consequences of the above equations that lead to exact statements about the Green's functions. Introducing our conventions and notation for the Fourier transform, we have for a general two-point function (connected or proper) which obeys translational invariance: 4 . Starting with Eq. (2.23), we have that the only non-zero functional derivative is and all other proper Green's functions involving derivatives with respect to the λ-field vanish (even in the presence of sources). In terms of connected Green's functions, Eq. (2.23) becomes Eq. (2.25) and the only non-zero functional derivative is Because Eq. (2.25) involves the contraction of a vector quantity, the information is less restricted than previously. However, we can write down the following (true once sources have been set to zero such that the tensor structure is determined): where t ji ( k) = δ ji − k j k i / k 2 is the transverse projector in momentum space. These relations encode the transverse nature of the vector gluon field. Turning to Eq. (2.26), we recognize that if we functionally differentiate with respect to ıη d y , again with respect to ıξ e z and set sources to zero, we get that <ıξ e z ıξ d y >= 0.
(2.32) In effect, the auxiliary Lagrange multiplier field λ drops out of the formalism to be replaced by the transversality conditions, as it is supposed to.

FEYNMAN RULES AND DECOMPOSITIONS
Let us now discuss the Feynman rules and general decompositions of Green's functions that will be relevant to this work. The Feynman rules for the propagators can be derived from the field equations of motion (written in Appendix A) by neglecting the interaction terms and functionally differentiating. Denoting the tree-level quantities with a superscript (0), the corresponding equations read: The tree-level ghost propagator is then and we identify the momentum space propagator as The rest of the propagators follow a similar pattern and their momentum space forms (without the common color factor δ ab ) are given in Table I. Note that it is understood that the denominator factors involving both temporal and spatial components implicitly carry the relevant Feynman prescription, i.e., 1 The ghost proper two-point function in momentum space is and the rest are presented (without color factors) in Table I. It is immediately apparent that the gluon polarization is not transverse in contrast to Landau gauge. The tree-level vertices are determined by taking the various interaction terms of Eqs. (2.21-2.24) and functionally differentiating. Since, in this study, we are interested only in the eventual one-loop perturbative results we omit the tree-level four-point functions (Γ 4A and Γ AAσσ ). Defining all momenta as incoming, we have: (3.7) In addition to the tree-level expressions for the various two-point functions (connected and proper) it is necessary to consider their general nonperturbative structures. These structures are determined by considering the properties of the fields under the discrete transforms of time-reversal and parity (the noncovariant analogue of Lorentz invariance arguments for covariant gauges). Using the same techniques as in Ref. [7] we can easily write down the results in momentum space. For the ghost, we have and the rest are presented in Table II. With the exception of the ghost, all dressing functions are scalar functions of two independent variables, k 2 0 and k 2 . The ghost dressing functions are functions of k 2 only for exactly the same reasons as in the first order formalism [7]. At tree-level, all dressing functions are unity.
The dressing functions for the propagators and two-point proper functions are related via the Legendre transform. The connection follows from δıJ β δıJ α = δ αβ = −ı δ δıJ α <ıΦ β >= δΦ γ δıJ α <ıΦ γ ıΦ β >=<ıJ α ıJ γ ><ıΦ γ ıΦ β > . (3.9) (Recall here that there is an implicit summation over all discrete indices and integration over continuous variables labeled by γ.) Considering all the possibilities in turn, we find that Actually, whilst we have included D σλ up to this point, since there is no vertex involving the λ-field this propagator will not directly play any role in the formalism. However, indirectly it does turn out to have a meaning as will be shown in the next section.

DYSON-SCHWINGER EQUATIONS AND SLAVNOV-TAYLOR IDENTITIES
With the observation that δ δıΦ β <ıJ γ ıJ α >= − <ıJ γ ıJ ε ><ıΦ ε ıΦ β ıΦ δ ><ıJ δ ıJ α > (4.1) [stemming from the Legendre transform and following from Eq. (3.9)], the derivation of the Dyson-Schwinger equations becomes relatively straightforward. Starting with Eq. (2.21), omitting the terms that will not contribute at one-loop perturbatively and recognizing the tree-level vertices in configuration space, we have that Taking the functional derivative with respect to ıA f lw , using Eq. (4.1), setting sources to zero and Fourier transforming to momentum space (each step is straightforward so we omit the details for clarity) we get the Dyson-Schwinger equation for the gluon polarization: Turning now to Eq. (2.22), we have where again, terms that do not contribute at the one-loop perturbative level are omitted. There are two functional derivatives of interest, those with respect to ıσ f w and ıA f lw , which give rise to the following two Dyson-Schwinger equations: Next we consider the ghost equation, Eq. (2.24), which can be written The ghost Dyson-Schwinger equation is subsequently In addition to the Dyson-Schwinger equations, the Green's functions are constrained by Slavnov-Taylor identities. These are the functional derivatives of Eq. (2.27). Since Eq. (2.27) is Grassmann-valued, we must first functionally differentiate with respect to ıc d y . We are not interested (here) in further ghost correlations, so we can then set ghost sources to zero. Also, there is no further information to be gained by considering the Lagrange multiplier field λ a , and we set its source to zero also. Equation (2.27) then becomes (4.9) Taking the functional derivatives of this with respect to ıσ e z or ıA e kz and setting all remaining sources to zero gives rise to the following two equations: Now, using Eq. (4.1), we have that (4.14) Since the ghost Green's functions are independent of the ghost line's energy scale [7], after ω 0 has been integrated out, there is no external energy scale andΣ σ;cc (k) =Σ σ;cc ( k). we can see automatically that in momentum space,Σ Aj;cc (k) ∼ k j and that the transverse projector that acts on it in Eqs. (4.10) and (4.11) will kill the term. We thus have <ıσ e z ıA a ix > <ıc a x ıc d y > , (4.18) <ıA e kz ıA a ix > <ıc a x ıc d y > , (4.19) which in terms of the momentum space dressing functions gives The Slavnov-Taylor identities for the two-point functions above are rather revealing. They are the Coulomb gauge equivalent of the standard covariant gauge result that the longitudinal part of the gluon polarization remains bare [11]. We notice that they relate the temporal, longitudinal and ghost degrees of freedom in a manner reminiscent of the quartet mechanism in the Kugo-Ojima confinement criterion [12]. Also, they represent Gauß' law as applied to the Green's functions. Equation (4.9) suggests that proper functions involving the temporal σ-field can be systematically eliminated and replaced by functions involving the vector A and ghost fields although whether this is desirable remains to be seen.
We can now return to the general decompositions of the two-point functions. We see that as a consequence of either of the two Slavnov-Taylor identities above, Eqs. (4.20) or (4.21), Eq. (3.10) reduces to D σλ = D c , reassuring us that at least the formalism is consistent. We also see that there are only three independent two-point dressing functions, whereas (accounting for the tensor structure of the gluon polarization) we have five Dyson-Schwinger equations. We will investigate this perturbatively in the next section.

ONE-LOOP PERTURBATION THEORY
Let us now consider the one-loop perturbative form of the two-point dressing functions that are derived from the Dyson-Schwinger equations. So far, all quantities are expressed in Minkowski space. The perturbative integrals must however be evaluated in Euclidean space. The analytic continuation to Euclidean space (k 0 → ık 4 ) is straightforward given the Feynman prescription for denominator factors. Henceforth, all dressing functions will be written in Euclidean space and are functions of k 2 4 and k 2 . The Euclidean four momentum squared is k 2 = k 2 4 + k 2 . We write the perturbative expansion of the two-point dressing functions as follows: The loop integrals will be dimensionally regularized with the (Euclidean space) integration measurē . The coupling acquires a dimension: where µ is the square of some non-vanishing mass scale squared. This factor is included in Γ (1) αβ such that the new coupling and Γ (1) are dimensionless. By inserting the appropriate tree-level factors into the Dyson-Schwinger equations, extracting the color and tensor algebra we get the following integral expressions for the various two-point proper dressing functions: At this stage, we are in a position to check the two Slavnov-Taylor identities for the two-point functions. The first of these, Eq. (4.20), reads at one-loop: Inserting the integral expressions above and eliminating overall constants, the left-hand side reads By expanding the transverse projectors and scalar products, it is relatively trivial to show that this does indeed vanish. The second identity, Eq. (4.21), reads These integrals cancel explicitly, though it should be remarked that this cancellation is more obvious in the first order formalism [8]. Secondly, with respect to the temporal variable x, all the results above are strictly finite for Euclidean and spacelike Minkowski momenta -any singularities occur for z = x/y = −1 (the light-cone) with branch cuts extending in the timelike direction. This means that the analytic continuation between Euclidean and Minkowski space can be justified. Thirdly, the coefficient of the ε-pole for D σσ and the combination D AA D 2 c is 11N c /3(4π) 2 which is minus the value of the first coefficient of the β-function. This confirms that g 2 D σσ [13] and g 2 D AA D 2 c (the Coulomb gauge analogue of the Landau gauge nonperturbative running coupling) are renormalization group invariants at this order in perturbation theory. Fourthly, the results above for D AA , D σσ and D c are identical to those calculated within the first order formalism [8].

SUMMARY AND OUTLOOK
The two-point functions (connected and proper) of Coulomb gauge Yang-Mills theory have been considered within the standard, second order formalism. Functional methods have been used to derive the relevant Dyson-Schwinger equations and Slavnov-Taylor identities. One-loop perturbative results have been presented and the Slavnov-Taylor identities that concern them verified.
Suffice it to say that it is tautological for the situation in Coulomb gauge to be somewhat different from covariant gauges such as Landau gauge. The proper A-A two-point function is explicitly not transverse, nor does its longitudinal component remain bare beyond tree-level. This longitudinal component can however be written in terms of the temporal gluon and ghost two-point functions via the Slavnov-Taylor identities. Indeed, the Slavnov-Taylor identities show that there are only three independent two-point dressing functions: the (transverse) spatial gluon propagator dressing function (D AA ), the temporal gluon propagator dressing function (D σσ ) and the ghost propagator dressing function (D c ). With the exception of the ghost dressing function, all are noncovariantly expressed in terms of two variables: k 2 4 (or k 2 0 in Minkowski space) and k 2 . Perturbatively it is seen that the analytic continuation between Euclidean and Minkowski space (and vice versa) is valid and that the Slavnov-Taylor identities hold.
There are many further questions to be addressed. The perturbative structure of the vertex functions, the addition of the quark sector and the construction of physical scattering matrix elements from noncovariant components are all important next steps. The issue of noncovariant renormalization prescriptions must also be understood. The connection of the functional formalism with other approaches such as the Hamiltonian formalism [3] and lattice calculations must also be established. Clearly, there is a lot of work yet to be done.