Equal-time two-point correlation functions in Coulomb gauge Yang-Mills theory
D. Campagnari, A. Weber, H. Reinhardt, F. Astorga, W. Schleifenbaum
aa r X i v : . [ h e p - t h ] J un Equal-time two-point correlation functionsin Coulomb gauge Yang-Mills theory
D. Campagnari a , A. Weber b ∗ , H. Reinhardt a , F. Astorga b † ,W. Schleifenbaum a a Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany b Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de HidalgoEdificio C-3, Ciudad Universitaria,A. Postal 2-82, 58040 Morelia, Michoac´an, MexicoMay 26, 2018
Abstract
We apply a functional perturbative approach to the calculation of the equal-timetwo-point correlation functions and the potential between static color charges to one-loop order in Coulomb gauge Yang-Mills theory. The functional approach proceedsthrough a solution of the Schr¨odinger equation for the vacuum wave functional toorder g and derives the equal-time correlation functions from a functional integralrepresentation via new diagrammatic rules. We show that the results coincide withthose obtained from the usual Lagrangian functional integral approach, extract the betafunction, and determine the anomalous dimensions of the equal-time gluon and ghosttwo-point functions and the static potential under the assumption of multiplicativerenormalizability to all orders. ∗ Email: [email protected] † Email: [email protected] Introduction
Finding an accurate (semi-)analytical description of the infrared sector of QCD is still oneof the most important challenges of present-day quantum field theory. In this work weconcentrate on Yang-Mills theory, QCD without dynamical quarks, since it is in this sec-tor where the peculiar properties of QCD, in particular the confining interaction betweenquarks, arise. Recently, much of the activity in this area has focused on the formulationand (approximate) solution of Yang-Mills theory in the Coulomb gauge [1]–[11], the primaryreason being that the Coulomb gauge Hamiltonian explicitly contains the color-Coulombpotential which furnishes the dominant nonperturbative contribution to the static or heavyquark potential.Semi-analytical functional approaches to the calculation of gluon and ghost propagators inthe infrared, mostly using Dyson-Schwinger equations, have been successful in Landau gaugeYang-Mills theory [12]–[14]. In the so-called ghost dominance approximation, even verysimple analytical solutions exist in the far infrared [7], [15]–[17]. Although the consistencyof these solutions is still under discussion, it is natural to inquire whether a similar approachcould be useful in the Coulomb gauge. The breaking of Lorentz covariance through theCoulomb gauge condition makes the usual Lagrangian functional integral approach quitecumbersome in this gauge, see, e.g., Ref. [18]. For this reason, semi-analytical approaches inCoulomb gauge have mostly used a Hamiltonian formulation. A set of equations similar toDyson-Schwinger equations is obtained from a variational principle using a Gaussian typeof ansatzes for the vacuum wave functional in the Schr¨odinger representation [1], [3]–[6]. Inthe ghost dominance approximation, furthermore, simple analytical solutions are availablefor the far infrared [2, 7].Nevertheless, the status of the semi-analytical and analytical solutions in the Coulombgauge is not yet entirely clear, for two reasons: first, two different solutions with an infraredscaling behavior (differing in the infrared exponents) have been found in both the analyticaland the semi-analytical approaches [2]–[7], and there is as yet no theoretical guidance towhat the physical solution should be; second, the inclusion of the Coulomb form factor (theform factor for the color-Coulomb potential, which measures the deviation of the Coulombpotential from a factorization in terms of ghost propagators) in the set of equations ofDyson-Schwinger type results problematic. In Refs. [3]–[6], the equation for the Coulombform factor has been considered subleading compared to the equations for the gluon andghost propagators and therefore treated in the tree-level approximation, while in Ref. [19]all equations have been considered to be of the same order and therefore treated on an equalfooting, with the result that solutions with infrared scaling behavior cease to exist. It shouldbe emphasized that only solutions with scaling behavior can give rise to a linearly risingCoulomb potential, and that the latest lattice calculations also show a scaling behavior forthe equal-time correlation functions in the deep infrared [8, 9]. It is not clear at present howto improve the approximation used in the variational approach in order to arrive at a uniqueand consistent solution.An interesting relation between Landau and Coulomb gauge Yang-Mills theory has beenpointed out in the ghost dominance approximation in Refs. [2, 7]: the equal-time correlation2unctions of the Hamiltonian approach in Coulomb gauge are the formal counterparts inthree dimensions of the covariant correlation functions in Landau gauge in four dimensions.Building on this analogy, a possible strategy would be to replace the variational principleby a calculation of equal-time correlation functions in the Coulomb gauge and intend toformulate Dyson-Schwinger equations for the latter. In the present work, we take a first stepin this direction: we set up a functional integral representation of the equal-time correlationfunctions (without taking a detour to the space-time correlation functions) that is the precisethree-dimensional analogue of the usual functional integral representation of the covariantcorrelation functions in the Lagrangian approach to Landau gauge Yang-Mills theory. Wealso develop a diagrammatic representation and a set of Feynman rules for the equal-timecorrelation functions. We use this formulation to calculate the equal-time gluon and ghosttwo-point correlation functions and the potential for static color charges in Coulomb gaugeperturbatively to one-loop order. We extract the one-loop beta function and determinethe asymptotic ultraviolet behavior of the equal-time two-point functions and the staticpotential. We also show that our results coincide with those obtained in a Lagrangianfunctional integral approach [20, 21] and use the latter for the renormalization of the equal-time correlation functions and the static potential.The organization of the paper is as follows: in the next section, we determine the vacuumwave functional perturbatively to order g from the solution of the Schr¨odinger equation.With the vacuum functional determined to the corresponding order, we turn to the calcu-lation of the equal-time gluon and ghost two-point correlation functions in Section 3. Wealso calculate the one-loop corrections to the static or heavy quark potential (and thus tothe Coulomb form factor) in the same section. Although for the latter calculation we needto go beyond the terms that we have calculated for the vacuum functional in Section 2,the relevant additional contributions are quite simply determined. In Section 4, we provideanother representation of the equal-time two-point functions by choosing equal times (zero)in the space-time correlation functions determined before in the Lagrangian functional inte-gral representation [20, 21] of the theory. The static potential can also be obtained from atwo-point function that arises in the Lagrangian approach. We use the alternative represen-tations of the two-point functions and the static potential to perform the renormalization ofour results. We show the nonrenormalization of the ghost-gluon vertex in the same sectionand use it to determine the beta function and the asymptotic ultraviolet behavior of thetwo-point functions. We also show that the same beta function is found from consideringthe static potential. Finally, in Section 5, we summarize our findings and comment on sev-eral possible applications. In the Appendix, we give some details on an important differencethat arises between the Lagrangian and the Hamiltonian approach when it comes to theimplementation of the Coulomb gauge. It is very simple to write down a functional integral representation of the equal-time corre-lation functions, given that they are nothing but the vacuum expectation values of products3f the field operators. In the Schr¨odinger representation of Yang-Mills theory in Coulombgauge, the equal-time n -point correlation functions in (3-)momentum space have the follow-ing representation: h A ai ( p , t = 0) A bj ( p , t = 0) · · · A fr ( p n , t = 0) i = Z D [ A ] δ ( ∇ · A ) FP ( A ) A ai ( p ) A bj ( p ) · · · A fr ( p n ) | ψ ( A ) | . (1)Here, ψ ( A ) is the true vacuum wave functional of the theory. The (absolute) square | ψ ( A ) | then plays the rˆole of the exponential of the negative Euclidean classical action in thecorresponding representation of the covariant correlation functions (in Euclidean space). FP ( A ) ≡ det[ −∇ · D ( A )], with the covariant derivative in the adjoint representation definedas D ab ( A ) = δ ab ∇ + gf abc A c , (2)is the Faddeev-Popov determinant (in 3 dimensions) which forms a part of the integrationmeasure for the scalar product of states in the Schr¨odinger representation (see Ref. [22]).Note that the fields A ai ( p ) on the left-hand side of Eq. (1) are spatially transverse, p · A a ( p ) =0. We will assume the transversality of the fields A a in all of the following formulae, whichwe could make manifest by introducing a transverse basis in momentum space. However,there is usually no need to do so explicitly.In order to write down the functional integral for the equal-time correlation functionsexplicitly, the vacuum wave functional needs to be specified. The analogy with the covarianttheory suggests to use an exponential ansatz for this wave functional, in the spirit of the e S expansion in many-body physics [23]. We consider a full Volterra expansion of the exponent: ψ ( A ) = exp (cid:18) − ∞ X k =2 k ! Z d p (2 π ) · · · d p k (2 π ) X i ,i ,...,i k X a ,a ,...,a k f a a ...a k k ; i i ...i k ( − p , . . . , − p k ) × A a i ( p ) · · · A a k i k ( p k )(2 π ) δ ( p + . . . + p k ) (cid:19) . (3)Any normalization factor can be conveniently absorbed in the functional integration measurein Eq. (1). Terms linear in A in the exponent ( k = 1) are excluded by the symmetry of thewave functional under global gauge transformations (in the absence of external color charges).Regarding notation, given that our Hamiltonian formalism is not manifestly covariant, wewill denote the contravariant spatial components of 4-vectors by (latin) subindices .We insert this ansatz for the vacuum wave functional into the Schr¨odinger equation Hψ ( A ) = E ψ ( A ) , (4)4here H is the Christ-Lee Hamiltonian for Coulomb gauge Yang-Mills theory [22], H = 12 Z d x (cid:18) − FP ( A ) δδA ai ( x ) FP ( A ) δδA ai ( x ) + B ai ( x ) B ai ( x ) (cid:19) + g Z d x d y FP ( A ) ρ a ( x ) FP ( A ) h x , a | ( −∇ · D ) − ( −∇ )( −∇ · D ) − | y , b i ρ b ( y ) . (5)Here, B ai = − ǫ ijk F ajk = (cid:16) ∇ × A a − g f abc A b × A c (cid:17) i (6)is the chromo-magnetic field, and ρ a ( x ) = ρ aq ( x ) + f abc A bj ( x ) 1 i δδA cj ( x ) (7)the color charge density, including external static charges ρ q for later use. Note that we haveextracted a factor g from the color charges in order to simplify the counting of orders of g in the rest of the paper. The notation h x , a | C | y , b i refers to the kernel of the operator C inan integral representation.In most of the following perturbative calculation, we will need the Hamiltonian only upto order g , where H = 12 Z d p (2 π ) (cid:18) − (2 π ) δδA ai ( p ) (2 π ) δδA ai ( − p ) + A ai ( − p ) p A ai ( p ) (cid:19) (8)+ 12 Z d p (2 π ) A ai ( − p ) (cid:18) N c g Z d q (2 π ) − (ˆ p · ˆ q ) ( p − q ) (cid:19) (2 π ) δδA ai ( − p ) (9)+ g Z d p (2 π ) d p (2 π ) d p (2 π ) if abc [ δ jk ( p ,l − p ,l ) + δ kl ( p ,j − p ,j ) + δ lj ( p ,k − p ,k )] × A aj ( p ) A bk ( p ) A cl ( p )(2 π ) δ ( p + p + p ) (10)+ g Z d p (2 π ) · · · d p (2 π ) (cid:2) f abe f cde ( δ ik δ jl − δ il δ jk ) + f ace f bde ( δ ij δ kl − δ il δ jk )+ f ade f bce ( δ ij δ kl − δ ik δ jl ) (cid:3) A ai ( p ) A bj ( p ) A ck ( p ) A dl ( p )(2 π ) δ ( p + p + p + p )(11)+ g Z d p (2 π ) ρ a ( − p ) 1 p ρ a ( p ) + O ( g ) . (12)The term (9) stems from the application of the functional derivative to the Faddeev-Popovdeterminant. In this term, N c stands for the number of colors, f acd f bcd = N c δ ab , and5 p ≡ p / | p | denotes a unit vector. In the absence of external charges, we get for the term (12) g Z d p (2 π ) ρ a ( − p ) 1 p ρ a ( p )= 12 Z d p (2 π ) A ai ( − p ) (cid:18) N c g Z d q (2 π ) t ij ( q )( p − q ) (cid:19) (2 π ) δδA aj ( − p ) (13) − g Z d p (2 π ) · · · d p (2 π ) (cid:18) f ace f bde δ ik δ jl ( p + p ) + f ade f bce δ il δ jk ( p + p ) (cid:19) × (2 π ) δ ( p + p + p + p ) A ai ( p ) A bj ( p )(2 π ) δδA ck ( − p ) (2 π ) δδA dl ( − p ) . (14)In the term (13) on the right-hand side, t ij ( q ) denotes the spatially transverse projector ortransverse Kronecker delta t ij ( q ) ≡ δ ij − ˆ q i ˆ q j . (15)We shall now show, explicitly up to order g , that there is a unique perturbative solutionof the Schr¨odinger equation (4) for the wave functional ψ ( A ) in Eq. (3), if we only supposethat the dominant contribution to the coefficient function f k is at least of order g k − for k ≥
2. A similar method for the determination of the vacuum wave functional has beenapplied before in Refs. [24]–[26] to a scalar theory and to Yang-Mills theory in Weyl gauge.We will consider the case without external charges to begin with, and include charges ρ q later on in the context of the static potential. To order g , the Schr¨odinger equation reads( N c − (cid:18)Z d p (2 π ) f ( p ) (cid:19) (2 π ) δ ( ) + 12 Z d p (2 π ) A ai ( − p ) h p − (cid:0) f ( p ) (cid:1) i A ai ( p ) = E , (16)where we have used that f ab ij ( p , − p ) = f ( p ) δ ij δ ab = f ( − p ) δ ij δ ab (17)(to be contracted with spatially transverse fields) as a consequence of the symmetry underthe exchange of the arguments, of spatially rotational and global gauge symmetry, and ofthe fact that f ab ij ( p , p ) is only defined for p + p = 0. Equation (16) implies that, to thecurrent order, f ( p ) = | p | , (18) E = ( N c − (cid:18)Z d p (2 π ) | p | (cid:19) (2 π ) δ ( ) . (19)Generally, the energy E cancels any field-independent terms multiplying the vacuum func-tional in the Schr¨odinger equation to any order in g . Eqs. (18) and (19) represent nothingbut the well-known solution of the free ( g = 0) theory. The choice of the sign in Eq. (18) isdictated by the normalizability of the wave functional (3) to order g . As usual, (2 π ) δ ( )is to be understood as the total volume of space.6o the next (first) order of g , the Schr¨odinger equation is not much more complicated:it reads13! Z d p (2 π ) d p (2 π ) d p (2 π ) n igf abc [ δ jk ( p ,l − p ,l ) + δ kl ( p ,j − p ,j ) + δ lj ( p ,k − p ,k )] − | p | f abc jkl ( − p , − p , − p ) o A aj ( p ) A bk ( p ) A cl ( p )(2 π ) δ ( p + p + p ) = 0 , (20)where we have already taken into account the results (18), (19) and the fact that f abc ijk ( p , p , p ) = f abc f ijk ( p , p , p ) , (21)which is a consequence of global gauge symmetry and the invariance of the vacuum wavefunctional under charge conjugation. The unique solution of Eq. (20) with the full symmetryunder the exchange of the arguments is f abc ijk ( p , p , p ) = − igf abc | p | + | p | + | p | [ δ ij ( p ,k − p ,k ) + δ jk ( p ,i − p ,i ) + δ ki ( p ,j − p ,j )] . (22)This equality, and all the following equalities with explicit spatial (Lorentz) indices, areproper equalities only after contracting with the corresponding number of transverse vectorfields A , or, equivalently, after contracting every external spatial index with a transverseprojector, for example in Eq. (22) the index i with t il ( p ).We will now consider the Schr¨odinger equation to order g . On the left-hand side,terms with four and two powers of A appear, which have to cancel separately, and an A -independent term which must equal E to this order. We begin with the term with fourpowers of A . The quartic coupling (11) in the Hamiltonian has to be cancelled by termsstemming from the second functional derivative in Eq. (8) acting on the vacuum wave func-tional, and a contribution from Eq. (14). To order g , the coefficient functions f of Eq.(18) and f of Eq. (22) contribute, as well as the function f which we will determine. Asa result, the coefficient function f in the vacuum wave functional takes the following (fullysymmetric) form to order g :( | p | + . . . + | p | ) f abcd ijkl ( p , . . . , p )= g (cid:2) f abe f cde ( δ ik δ jl − δ il δ jk ) + f ace f bde ( δ ij δ kl − δ il δ jk ) + f ade f bce ( δ ij δ kl − δ ik δ jl ) (cid:3) (23) − (cid:2) f abe ijm ( p , p , − p − p ) t mn ( p + p ) f cde kln ( p , p , p + p )+ f ace ikm ( p , p , − p − p ) t mn ( p + p ) f bde jln ( p , p , p + p )+ f ade ilm ( p , p , − p − p ) t mn ( p + p ) f bce jkn ( p , p , p + p ) (cid:3) (24) − g (cid:18) f abe f cde δ ij δ kl ( | p | − | p | )( | p | − | p | )( p + p ) + f ace f bde δ ik δ jl ( | p | − | p | )( | p | − | p | )( p + p ) f = − b − (cid:18) b b + 2 perms. (cid:19) − (cid:18) b b + 2 perms. (cid:19) Figure 1: A diagrammatic representation of Eqs. (23)–(25). Every diagram corresponds toprecisely one of the Eqs. (23)–(25), in the same order. The “2 perms.” refer to permutationsof the external legs. + f ade f bce δ il δ jk ( | p | − | p | )( | p | − | p | )( p + p ) (cid:19) . (25)This result for f is represented diagrammatically in Fig. 1. Equation (23), divided by( | p | + . . . + | p | ), is interpreted as the elementary or “bare” four-gluon vertex. The rˆoleof the factor 2 and the signs in Fig. 1 will become clear in the next section. Equation (24)and the second diagram in Fig. 1 represent the contraction of two elementary three-gluonvertices, the latter being given mathematically by Eq. (22). The contraction refers to spatialand color indices and the momenta, with opposite signs. Note that there is no “propagator”factor associated with the contraction (except for a transverse Kronecker delta), and there isa factor 1 / ( | p | + . . . + | p | ) for the external momenta which is unusual from a diagrammaticpoint of view. Finally, Eq. (25) and the last diagram in Fig. 1 describe an “elementary”Coulomb interaction between the external gluon lines.With this result in hand, we can go on to consider the terms quadratic in A in theSchr¨odinger equation to order g . The relevant contributions originate from Eqs. (8), (9),(13), and (14), and involve the functions f and f . We obtain the following equation forthe coefficient function f to order g : (cid:0) f ( p ) (cid:1) δ ab δ ij = (cid:18) p − N c g | p | Z d q (2 π ) − (ˆ p · ˆ q ) ( p − q ) (cid:19) δ ab δ ij + 12 Z d q (2 π ) f abcc ijkl ( − p , p , − q , q ) t kl ( q ) − N c g δ ab Z d q (2 π ) | p | − | q | ( p − q ) t ij ( q ) . (26)The explicit expression for f to order g is f ( p ) = | p | − N c g Z d q (2 π ) − (ˆ p · ˆ q ) ( p − q ) + N c g | p | Z d q (2 π ) | p | + 2 | q | (27) − N c g | p | Z d q (2 π ) (cid:0) δ ik p l + δ kl q i − δ li p k (cid:1) t km ( p − q ) t ln ( q )2 | p | + 2 | q |× (cid:0) δ jm p n + δ mn q j − δ nj p m (cid:1) t ij ( p )( | p | + | q | + | p − q | ) (28) − N c g | p | Z d q (2 π ) p · ˆ q ) | p | + 2 | q | ( | p | − | q | ) ( p − q ) (29)8 f = (cid:0) bc (cid:1) − − b b − b − b b − b b − b b rs + Figure 2: The diagrams corresponding to Eqs. (27)–(30). The bare propagator, the inverse2 | p | of which appears in Eq. (27), is marked with an open circle for later use. The firsttwo one-loop diagrams correspond to the integrals in Eq. (27), in the same order. Thefollowing diagrams represent Eqs. (28)–(30), respectively. See the text for a motivation ofthe “crossed” gluon propagator notation in the last loop diagram. − N c g | p | Z d q (2 π ) (cid:0) p · ˆ q ) (cid:1) | p | − | q | ( p − q ) , (30)where we have used the contraction of an arbitrary tensor T ij ( p )12 T ij ( p ) t ij ( p ) = T t ( p ) (31)in order to extract the transverse part. We have presented the diagrams corresponding toEqs. (27)–(30) in Fig. 2. The first loop integral in Eqs. (26) and (27) results from Eq. (9) andis represented in Fig. 2 as a ghost loop because it stems from the Faddeev-Popov determinant.The following three loop diagrams in Fig. 2 are obtained by contracting two external legs inthe diagrams of Fig. 1, see Eq. (26). The last loop integral in Eq. (26), or the integral (30),on the other hand, originates from the terms (13) and (14) in the Hamiltonian. Lackinga better notation, we distinguish this contribution from the contraction of Eq. (25), theprevious diagram, by marking the gluon propagator with a cross (because there is no term | q | in the denominator that would indicate the presence of an internal gluon propagator — infact, the diagram may be interpreted to contain a ΠΠ-correlator, where Π is the momentumconjugate to A ).We have thus completed the determination of the (exponent of the) perturbative vacuumwave functional to order g . The result is given in Eqs. (22), (23)–(25), and (27)–(30), tobe substituted in Eq. (3). We can also extract the perturbative vacuum energy to the sameorder from the A -independent terms in the Schr¨odinger equation with the result E = ( N c − (cid:18)Z d p (2 π ) f ( p ) (cid:19) (2 π ) δ ( ) (32)[cf. Eq. (16)], where Eqs. (27)–(30) have to be substituted for f ( p ). The explicit expressionis not relevant for our purposes. We shall come back to the vacuum energy later in thecontext of the static potential in the presence of external charges. It should also be clear bynow how to take the determination of the perturbative vacuum functional and the vacuumenergy systematically to higher orders. 9 Equal-time two-point correlation functions
For the calculation of the equal-time correlation functions, we need to include the Faddeev-Popov determinant in the measure of the functional integral, see Eq. (1). For our diagram-matic procedure, it is very convenient to introduce ghost fields and write FP ( A ) = Z D [ c, ¯ c ] exp (cid:18) − Z d x ¯ c a ( x )[ −∇ · D ab ( A )] c b ( x ) (cid:19) . (33)In our conventions, we have explicitly Z d x ¯ c a ( x )[ −∇ · D ab ( A )] c b ( x ) = Z d p (2 π ) ¯ c a ( − p ) p c a ( p ) (34)+ g Z d p (2 π ) d p (2 π ) d p (2 π ) if abc p ,j ¯ c a ( p ) c b ( p ) A cj ( p )(2 π ) δ ( p + p + p ) . (35)Note that p ,j under the integral in Eq. (35) can be replaced by − p ,j due to the transversalityof A .We now have a representation of the equal-time correlation functions as a functionalintegral over the transverse components of A , the ghost and the antighost fields, see Eq. (1).The integration measure, which would be the exponential of the negative of the Euclideanaction in the usual four-dimensional formulation (in Euclidean space), is now given by theexponential in Eq. (33) and the square of Eq. (3). Note that the vacuum functional is real(at least to order g ) because the coefficient functions fulfill the reality condition (cid:0) f a ...a k k ; i ...i k ( − p , . . . , − p k ) (cid:1) ∗ = f a ...a k k ; i ...i k ( p , . . . , p k ) . (36)We shall use the analogy of this representation with the familiar functional integral repre-sentation of the covariant correlation functions in the usual four-dimensional formulation forthe perturbative determination of the equal-time correlation functions (1). The correspond-ing Feynman rules are easily identified: the (static) gluon propagator is the inverse of 2 | p | ,cf. Eq. (18) (the factor of two is due to the square of the wave functional in the measure), theother contributions − (cid:0) f ( p ) − | p | (cid:1) and the other coefficient functions − f ( p , p , p ) and − f ( p , . . . , p ) determine the two-, three-, and four-gluon vertices. Furthermore, from Eq.(34) we identify the free ghost propagator 1 / p and from Eq. (35) the ghost-gluon vertex.For the case of φ theory in (1 + 1) dimensions, the calculation of equal-time correlationfunctions from a representation analogous to (1) has been discussed in Ref. [27].We consider the gluon equal-time two-point function h A ai ( p ) A bj ( p ) i (with t = 0 inthe arguments of the gluon fields to be understood) first. One of the contributions to betaken into account is the ghost loop, constructed from two ghost-gluon vertices (35) andtwo ghost propagators [see Eq. (34)], and furthermore two static gluon propagators fromEq. (18) for the external lines. As it turns out, this contribution is exactly cancelled bythe other contribution with the same graph “topology” which arises from contracting one ofthe two-gluon vertices, (minus twice) the first integral in Eq. (27), with two external gluon10 b bc bcbcbc + b b bc bc = 0 b bc bcbc + b bc bc = E (cid:18) b bc bcbc (cid:19) b b bc bcbcbc + b b bc bcbc + b b bc bc = E (cid:18) b b bc bcbcbc (cid:19) b b bc bcbc + b b bc bc + b b bc bc rs + = E (cid:18) b b bc bcbc (cid:19) + b b bc bc rs + Figure 3: Diagrammatic representation of the various contributions to the gluonic equal-timetwo-point function, see Eqs. (37)–(41). The propagators marked with open circles are takenfrom Eqs. (18) and (34), respectively, while the “direct” contractions without open circlesrefer to the contractions that appear in the course of the determination of the vacuum wavefunctional, see Figs. 1 and 2, so that the corresponding parts of the diagrams translate into(minus two times) the mathematical expressions (24)–(25) [divided by ( | p | + . . . + | p | )]and (27)–(30). The notation “ E ( · )” for the sum of all diagrams with the same topology isexplained in the text, following Eq. (44).propagators. Both contributions are represented diagrammatically in the first line of Fig.3. The cancellation of the ghost loop contribution from the perturbative gluon two-pointfunction (to one-loop order) is interesting given that this contribution plays a major role inthe nonperturbative approaches to the infrared behavior of the equal-time gluon two-pointfunction [2]–[7]. The cancellation of ghost loops was found to be a general feature in theLagrangian functional integral approach [18, 20]. An alternative way to see the cancellationin our present approach is to write the Faddeev-Popov determinant as FP ( A ) = exp (cid:2) tr ln (cid:0) − ∇ · D ( A ) (cid:1)(cid:3) . (37)The coefficient of the term quadratic in A in tr ln (cid:0) − ∇ · D ( A ) (cid:1) precisely equals twice thefirst integral in Eq. (27) and hence cancels out in the exponent.Next we turn to the tadpole contribution which is obtained from the elementary four-gluon vertex extracted from Eq. (23) appropriately contracted with three static gluon prop-agators. Again, there is a second contribution with the same “topology” given by the two-gluon vertex from the last integral in Eq. (27) contracted with two external propagators,cf. the second line in Fig. 3. The sum of these two contributions to the gluon equal-time11wo-point function is (with p ≡ p ) − N c g (2 | p | ) Z d q (2 π ) | p | + 2 | q | | q | − N c g (2 | p | ) Z d q (2 π ) | p | + 2 | q | = − N c g (2 | p | ) Z d q (2 π ) | q | , (38)to be multiplied with δ ab t ij ( p ).The most complicated contribution to the two-point function comes from diagrams withthe gluon loop topology (with two three-gluon vertices). There are three different diagramsof this type, represented in the third line of Fig. 3, the first from contracting two three-gluon vertices extracted from Eq. (22) with four static gluon propagators (two internal andtwo external), the second from contracting the part of the four-gluon vertex given by Eq.(24) with one internal and two external gluon propagators, and the third by contractingthe two-gluon vertex from Eq. (28) with two static gluon propagators. The sum of thesecontributions is2 N c g (2 | p | ) Z d q (2 π ) | p | + 2 | p − q | ( | p | + | q | + | p − q | ) | q | | p − q |× (cid:0) δ km p n + δ mn q k − δ nk p m (cid:1) t mr ( p − q ) t ns ( q ) (cid:0) δ lr p s + δ rs q l − δ sl p r (cid:1) t kl ( p ) (39)(again, to be multiplied with δ ab t ij ( p ), and p ≡ p ). The tensor structure in this expressionis invariant under the transformation q → p − q , a fact we can use to replace 2 | p | + 2 | p − q | in the numerator with 2 | p | + | q | + | p − q | . The tensor structure itself can be simplified byperforming the contractions explicitly. A straightforward, but somewhat tedious calculationgives (cid:0) δ km p n + δ mn q k − δ nk p m (cid:1) t mr ( p − q ) t ns ( q ) (cid:0) δ lr p s + δ rs q l − δ sl p r (cid:1) t kl ( p )= (cid:16) − (ˆ p · ˆ q ) (cid:17) (cid:18) p + 2 q + p q + ( p · q ) ( p − q ) (cid:19) . (40)Finally, we turn to the contributions that involve the (non-abelian) Coulomb potential.There are, again, three such terms, represented in the last line of Fig. 3, the first from thefour-gluon vertex derived from Eq. (25) contracted with three static gluon propagators, andthe other two using the two-gluon vertices corresponding to the two integrals (29) and (30)contracted with two gluon propagators each. The sum of these terms is2 N c g (2 | p | ) Z d q (2 π ) p · ˆ q ) | q | ( | p | − | q | ) ( p − q ) + 2 N c g (2 | p | ) Z d q (2 π ) (cid:0) p · ˆ q ) (cid:1) | p | − | q | ( p − q ) = 2 N c g (2 | p | ) Z d q (2 π ) p · ˆ q ) | q | | p | − | q | ( p − q ) , (41)12o be multiplied with δ ab t ij ( p ) as before, and p ≡ p . On the left-hand side of Eq. (41),we have added up the contributions from the contraction of the four-gluon vertex and thetwo-gluon vertex in Eq. (29) to give the first loop integral. The left-hand side of Eq. (41) isrepresented diagrammatically as the right-hand side of the last line in Fig. 3.Putting it all together, the result for the equal-time gluon two-point function is, to order g , h A ai ( p ) A bj ( p ) i = (cid:20) | p | − N c g (2 | p | ) Z d q (2 π ) | q | (42)+ 2 N c g (2 | p | ) Z d q (2 π ) (cid:0) − (ˆ p · ˆ q ) (cid:1) (2 | p | + | q | + | p − q | )( | p | + | q | + | p − q | ) | q | | p − q |× (cid:18) p + 2 q + p q + ( p · q ) ( p − q ) (cid:19) (43)+ 2 N c g (2 | p | ) Z d q (2 π ) p · ˆ q ) | q | p − q ( p − q ) (cid:21) δ ab t ij ( p )(2 π ) δ ( p + p ) . (44)Looking at the diagrams on the left-hand sides of the first three lines in Fig. 3, it is clear thatthere is precisely one diagram among those of the same topology that is contructed exclusivelyfrom the elementary vertices (22), (23), and (35) and the “bare” propagators taken from Eqs.(18) and (34). We will call this kind of diagram an “F-diagram”. Interestingly, the sum ofall diagrams with the same topology which we will refer to as an “E-diagram”, can beconstructed from the corresponding F-diagram by a formal operation that we call the “E-operator”. It is denoted as “ E ( · )” in Fig. 3 and will be illustrated considering the exampleof the third line in Fig. 3: starting from the mathematical expression for the correspondingF-diagram, one multiplies the integrand (of the integral over loop momentum) with the sumof all | k | where k runs over the momenta of all propagators in the diagram, and divides bythe corresponding sum restricted to the momenta of the external propagators. Indeed, theresult of this operation is given by Eq. (39) in the ( q → p − q )-symmetric form, see theremark after Eq. (39).The same rule applies to the second line in Fig. 3, or Eq. (38), only that the propagatorthat starts and ends at the same vertex has to be counted twice in the sum over (internaland external) | k | . Similarly, the E-operator can be used to sum the first two diagrams onthe left-hand side of the last line in Fig. 3. We have to consider the Coulomb interaction asan elementary vertex given by (25) to this end, and count the gluon propagator that startsand ends at this vertex twice in the sum over | k | just as in the case of the other elementaryfour-gluon vertex. The same rule for the generation of the E-diagrams (given the elementaryvertices and propagators) has been shown to hold up to two-loop order for the equal-timetwo-point function and to one-loop order for the four-point function in the context of a scalar φ theory [28] (a detailed account will be given elsewhere). Note that two contributions, thesecond diagram in the first line of Fig. 3 and the third diagram in the last line in the samefigure corresponding to the first loop integral in Eq. (27) and to Eq. (30), respectively, donot fit into this general scheme. 13 c ¯ c i = bc + b b bc bcbcbc Figure 4: Diagrammatic representation of Eq. (45).The calculation of the equal-time ghost two-point function from the graphical rules ismuch simpler. One obtains directly h c a ( p )¯ c b ( p ) i = (cid:18) p + N c g p Z d q (2 π ) − (ˆ p · ˆ q ) ( p − q ) | q | (cid:19) δ ab (2 π ) δ ( p + p ) , (45)corresponding to the diagrams in Fig. 4. Note that one of the factors 1 / p for the externalghost propagators cancels against the momentum dependence of the ghost-gluon vertices.We shall close this section with a calculation of the static (heavy quark) potential, theenergy for a configuration of static external color charges ρ q ( x ). To this end, we introducecharges ρ q ( x ) into the Hamiltonian, see Eqs. (7) and (12). Compared to the Coulomb term(13)–(14) which was calculated in the absence of external charges, there are two new termsof order g : g Z d p (2 π ) ρ aq ( − p ) 1 p ρ aq ( p ) , (46)which is A -independent and hence only contributes to the vacuum energy but leaves thevacuum wave functional unchanged, and g Z d p (2 π ) d p (2 π ) d p (2 π ) f abc p (2 π ) δ ( p + p + p ) ρ aq ( p ) A bj ( p ) 1 i (2 π ) δδA cj ( − p ) . (47)The latter term, when applied to the vacuum wave functional, generates the following (prop-erly symmetrized) expression to order g , − g Z d p (2 π ) d p (2 π ) d p (2 π ) if abc | p | − | p | p ρ aq ( p ) A bj ( p ) A cj ( p )(2 π ) δ ( p + p + p ) , (48)which implies that the vacuum wave functional to order g has to be modified in order tofulfill the Schr¨odinger equation with this new term.A term cancelling the expression (48) in the Schr¨odinger equation can only result fromthe second derivative term in the Hamiltonian [Eq. (8)]. It is then simple to see that wehave to add the expression − g Z d p (2 π ) d p (2 π ) d p (2 π ) if abc p | p | − | p || p | + | p | ρ aq ( p ) A bj ( p ) A cj ( p )(2 π ) δ ( p + p + p ) (49)to the negative of the exponent of the vacuum wave functional in order to satisfy theSchr¨odinger equation to order g . This term describes the back-reaction of the vacuum14o the presence of the external charges to order g . Observe that due to the presence ofthe external charges ρ q ( p ), the coefficient function of A ai ( p ) A bj ( p ) in the vacuum wavefunctional ceases to be of the form f ( p ) δ ab δ ij (2 π ) δ ( p + p ). Furthermore, contrary tothe terms found before, the contribution (49) is imaginary. The rest of the vacuum wavefunctional determined in the previous section remains without change.We now have to calculate the vacuum energy in the presence of the external charges.To order g , the result is the former one, Eq. (32), without any contribution from thenew term (49), plus Eq. (46) which is the part of the energy that depends on the externalcharges and hence defines the potential to this order. Of course, this is just the well-knownCoulomb potential of electrodynamics. What we are really interested in are the first quantumcorrections to this “bare” potential, which are of order g .In general, the vacuum energy is given by E = g Z d p (2 π ) ρ aq ( − p ) 1 p ρ aq ( p ) − Z d p (2 π ) (2 π ) δδA ai ( p ) (2 π ) δδA ai ( − p ) ψ ( A ) (cid:12)(cid:12)(cid:12)(cid:12) A =0 , (50)which reduces to Eq. (32) in the absence of external charges. A contribution of order g can hence only originate from the terms in the vacuum wave functional that are quadraticin A . As long as we are only interested in the potential between static sources, we canconcentrate on terms that contain precisely two powers of ρ q . We then start by identifyingall the contributions to the Schr¨odinger equation of order g that contain two powers of A and two powers of ρ q . One of these contributions results from expanding the Coulomb kernel h x , a | ( −∇ · D ) − ( −∇ )( −∇ · D ) − | y , b i (51)in Eq. (5) to second order in A for ρ = ρ q . The result is the term − g Z d p (2 π ) · · · d p (2 π ) (cid:18) f ace f bde p ,i p ,j p p ( p + p ) + f ade f bce p ,j p ,i p p ( p + p ) (cid:19) × ρ aq ( p ) ρ bq ( p ) A ci ( p ) A dj ( p )(2 π ) δ ( p + . . . + p ) (52)on the left-hand side of the Schr¨odinger equation.Another contribution of the same type arises from the second functional derivative in Eq.(8) acting (twice) on the term (49), which gives the contribution g Z d p (2 π ) · · · d p (2 π ) p p (cid:18) f ace f bde t ij ( p + p ) | p + p | − | p || p + p | + | p | | p + p | − | p || p + p | + | p | + f ade f bce t ij ( p + p ) | p + p | − | p || p + p | + | p | | p + p | − | p || p + p | + | p | (cid:19) × ρ aq ( p ) ρ bq ( p ) A ci ( p ) A dj ( p )(2 π ) δ ( p + . . . + p ) (53)15 (cid:18) b b + 1 perm. (cid:19) − (cid:18) b b + 1 perm. (cid:19) Figure 5: Diagrammatic representation of the contributions (55) (multiplied by 2) to thevacuum wave functional.to the Schr¨odinger equation. The last contribution of the same type comes from the “mixed”term where the operator (47) acts upon the expression (49) in the wave functional. The resultis − g Z d p (2 π ) · · · d p (2 π ) p p (cid:20) f ace f bde t ij ( p + p ) (cid:18) | p + p | − | p || p + p | + | p | + | p + p | − | p || p + p | + | p | (cid:19) + f ade f bce t ij ( p + p ) (cid:18) | p + p | − | p || p + p | + | p | + | p + p | − | p || p + p | + | p | (cid:19)(cid:21) × ρ aq ( p ) ρ bq ( p ) A ci ( p ) A dj ( p )(2 π ) δ ( p + . . . + p ) , (54)to be included in the Schr¨odinger equation. It can be shown that no other contributionsquadratic in A and in ρ q exist to order g .In analogy to the determination of the expression (49) from Eq. (48), the three contri-butions (52)–(54) to the Schr¨odinger equation are taken care of by including the followingexpression in the negative exponent of the vacuum wave functional [in addition to (22),(23)–(25), (27)–(30), and (49)] − g Z d p (2 π ) · · · d p (2 π ) p p ( | p | + | p | ) (cid:26) f ace f bde (cid:20) p ,i p ,j ( p + p ) − t ij ( p + p ) × (cid:18) | p + p | − | p || p + p | + | p | | p + p | − | p || p + p | + | p | − | p + p | − | p || p + p | + | p | − | p + p | − | p || p + p | + | p | (cid:19)(cid:21) + f ade f bce (cid:20) p ,j p ,i ( p + p ) − t ij ( p + p ) (cid:18) | p + p | − | p || p + p | + | p | | p + p | − | p || p + p | + | p | − | p + p | − | p || p + p | + | p |− | p + p | − | p || p + p | + | p | (cid:19)(cid:21)(cid:27) ρ aq ( p ) ρ bq ( p ) A ci ( p ) A dj ( p )(2 π ) δ ( p + . . . + p ) . (55)This result (multiplied by 2) is represented diagrammatically in Fig. 5, where we have denotedthe “Coulomb propagator” 1 / p as a double line. The first diagram (and its permutation)corresponds to the expression (52), while the second diagram (plus its permutation) corre-sponds to the sum of the expressions (53) and (54). From Eq. (50), we find the contribution16o the vacuum energy g Z d p (2 π ) ρ aq ( − p ) 1( p ) (cid:26) N c g Z d q (2 π ) | q | (cid:20) p − ( p · ˆ q ) ( p − q ) + t ij ( q ) t ij ( p − q ) (cid:18) | p − q | − | q || p − q | + | q | (cid:19) − | p − q | − | q || p − q | + | q | ! + 3 p − ( p · ˆ q ) ( p + q ) + t ij ( q ) t ij ( p + q ) (cid:18) | p + q | − | q || p + q | + | q | (cid:19) − | p + q | − | q || p + q | + | q | ! ρ aq ( p ) . (56)This latter expression can be simplified by shifting q → q − p in the last two terms (in theround bracket). Together with Eq. (46), we find for the part of the vacuum energy that isquadratic in the external static charge ρ q , E ( ρ q , = g Z d p (2 π ) ρ aq ( − p ) V ( p ) ρ aq ( p ) , (57)the following result for the static potential to order g V ( p ) = 1 p + N c g p Z d q (2 π ) − (ˆ p · ˆ q ) | q | ( p − q ) (58) − N c g ( p ) Z d q (2 π ) (cid:0) ( p − q ) · q (cid:1) ( p − q ) q ! ( | p − q | − | q | ) | p − q | | q | ( | p − q | + | q | ) . (59)Note that to the proper color Coulomb potential (see, e.g., Ref. [31]), only the antiscreeningterm, the integral in Eq. (58), contributes, while the full static potential also contains thescreening contribution (59). The semi-analytical variational approaches [3, 4, 5, 6, 19] haveonly considered the proper color Coulomb potential so far.We can associate diagrams with the different contributions in Eqs. (58)–(59) in a naturalway. The vertex that joins two Coulomb (double) lines and one gluon line corresponds to thesame mathematical expression as the ghost-gluon vertex since both objects originate fromthe Faddeev-Popov operator ( −∇ · D ) (or its inverse). On the other hand, the vertex withtwo gluon lines and one Coulomb line translates to the expression igf abc | p | − | p || p | + | p | δ jk , (60)where the gluon lines carry the momenta p and p , (spatial) Lorentz indices j and k , andcolor indices b and c [cf. Eq. (49)]. Note that the “elementary” Coulomb interaction (25)is different from the contraction of two such vertices with a Coulomb propagator. Withthese conventions, we can represent the static potential as in Fig. 6. The E-operator in Fig.6 exclusively refers to the internal gluon propagators and thus amounts to multiplying theintegrand with | p − q | + | q | . 17 = + 3 b b bc + 2 E (cid:18) b b bcbc (cid:19) Figure 6: A diagrammatic interpretation of the static potential to order g .We hence have succeeded in calculating the equal-time gluon and ghost two-point func-tions and the static potential to one-loop order in our functional perturbative approach, withthe results (42)–(45) and (58)–(59). The same results can be obtained from a straightforwardapplication of Rayleigh-Schr¨odinger perturbation theory [29]. Compared to these latter cal-culations, we have here developed a functional integral and diagrammatical approach that ispotentially advantageous in higher-order perturbative calculations. We have also describeda set of simplified diagrammatic rules (the “E-operator”) for the determination of equal-time correlation functions that is expected to carry over to higher perturbative orders and ishoped to eventually lead to nonperturbative equations for the equal-time correlation func-tions analogous to Dyson-Schwinger equations. Naive power counting shows that the results of the preceding section, Eqs. (42)–(45) and(58)–(59), are ultraviolet (UV) divergent and need to be renormalized. However, some of thedenominators occuring in the loop integrals are of a different type from those that usuallyappear in covariant perturbation theory, and efficient techniques for the handling of theseterms have yet to be developed. These remarks apply in particular to Eqs. (43) and (59).The equal-time correlation functions we have been calculating are a special or limitingcase of the usual space-time correlation functions, whence we naturally obtain the represen-tation h A ai ( p , t = 0) A bj ( p , t = 0) i = Z ∞−∞ dp , π dp , π h A ai ( p , p , ) A bj ( p , p , ) i (61)(with the space-time correlation functions written in Euclidean space-time). Since the reg-ularization and renormalization program has been developed for the space-time correlationfunctions, this representation is quite useful for our purposes. In the case of Coulomb gaugeYang-Mills theory, however, covariance is explicitly broken through the gauge condition, andthe calculation of the space-time correlation functions in the usual Lagrangian functionalintegral approach represents a difficulty by itself. Techniques have been developed to over-come these difficulties and applied in Ref. [20] to the calculation of the two-point correlationfunctions to one-loop order.Before properly considering the renormalization of our results, we will verify that thesecoincide on a formal level with the expressions obtained via Eq. (61), taking for the space-timecorrelation functions the formulas derived in the Lagrangian functional integral approach in18ef. [20]. In our notation, h A ai ( p ) A bj ( p ) i = p − N c g (cid:0) p (cid:1) Z d q (2 π ) q (62)+ N c g (cid:0) p (cid:1) Z d q (2 π ) (cid:0) δ km p ,n + δ mn q k − δ nk p ,m (cid:1) t mr ( p − q ) t ns ( q ) q × (cid:0) δ lr p ,s + δ rs q l − δ sl p ,r (cid:1) t kl ( p )( p − q ) (63)+ N c g (cid:0) p (cid:1) Z d q (2 π ) t kl ( p ) t kl ( q ) ( p , − q ) q ( p − q ) ! δ ab t ij ( p )(2 π ) δ ( p + p ) , (64)with p ≡ p + p , in Euclidean space-time. Formal integration of p , and p , as in Eq. (61)and of the component q of the loop momentum, most easily using the residue theorem, leadsto our equal-time correlation function (42)–(44). In Eq. (62), we have included the tadpolediagram in order that the correspondence with the equal-time gluon two-point function (42)–(44) be term by term. The tadpole diagram was not considered explicitly in Refs. [20, 21]because it vanishes in dimensional regularization.The case of the ghost two-point function is even simpler, because it is instantaneousalready in the Lagrangian approach: from Ref. [20], h c a ( p )¯ c b ( p ) i = p + N c g (cid:0) p (cid:1) Z d q (2 π ) p ,i p ,j t ij ( q ) q ( p − q ) ! δ ab (2 π ) δ ( p + p ) , (65)and the fact that the dependence on p , and p , is exclusively through the delta functionfor energy conservation implies that h c a ( p , t )¯ c b ( p , t ) i contains the factor δ ( t − t ). Inte-grating over p , and p , or putting t and t to zero then results in a factor δ (0). In thiscase, in order to reproduce Eq. (45), we integrate either over p , or over p , , which justeliminates the delta function for energy conservation [more symmetrically, one may integrateover ( p , + p , ) instead]. Performing the integral over q then converts Eq. (65) to Eq. (45).Although it is certainly not surprising that the Lagrangian functional integral approachof Refs. [18, 20, 21] gives the same results for the equal-time correlation functions as ourHamiltonian approach, it is also not trivial. Concerning the gauge fixing procedure, thereis the following important difference between the two approaches: in the Lagrangian formu-lation, the Weyl gauge A ≡ ∇ · A ≡ A -field rather than setting A to zero yields an expression in the exponent of the measurefor the functional integral that resembles the Christ-Lee Hamiltonian [18]. The derivation ofthe Hamilton operator (5) by Christ and Lee [22], on the other hand, relies on the existenceof a gauge transformation that makes any gauge field A µ satisfy both the Coulomb and Weylgauge conditions. We discuss the possibility of simultaneously implementing the Weyl andCoulomb gauges in the Hamiltonian and the Lagrangian approaches in the Appendix.19lthough not defined as an equal-time correlation function in our approach, it turns outthat the static potential (58)–(59) is related to the space-time two-point function h A a ( p ) A b ( p ) i in the Lagrangian functional integral approach, as was first pointed out by Zwanziger [31].The formal expression for the space-time correlation function is [20] h A a ( p ) A b ( p ) i = p + N c g (cid:0) p (cid:1) Z d q (2 π ) p ,i p ,j t ij ( q ) q ( p − q ) (66)+ N c g (cid:0) p (cid:1) Z d q (2 π ) q p , p · ( p − q ) q ( p − q ) t ij ( p − q ) t ij ( q ) ! δ ab (2 π ) δ ( p + p ) . (67)Integrating over either p , or p , and over the energy component q of the loop momentum,we obtain the antiscreening contribution (58) to the static potential from Eq. (66) becausethe latter is already instantaneous. In order to find Eq. (59) starting from Eq. (67), wehave to put the respective other energy component, p , or p , , to zero in addition [this isnot necessary in the cases of Eqs. (65) and (66), because there the result of integrating overone of the energy components is independent of the other]. For h A a ( p , t ) A b ( p , t ) i , thisprocedure amounts to integrating over the relative time t − t , which is, in fact, intuitivelyquite appealing for a non-instantaneous contribution to the potential between static sources.We shall now use the representation (61) of the equal-time gluon two-point function andthe corresponding representations of the ghost two-point function and the static potential forthe renormalization of these equal-time correlation functions. To this end, we make use ofthe explicit expressions obtained for Eqs. (62)–(67) in Ref. [20] in dimensional regularization.Thus, by integrating the result for (62)–(64) according to Eq. (61), we obtain for the equal-time correlation function h A ai ( p ) A bj ( p ) i = (cid:20) | p | + N c g (4 π ) | p | (cid:18) ǫ − ln p µ + C A (cid:19)(cid:21) δ ab t ij ( p )(2 π ) δ ( p + p ) (68)in the limit ǫ →
0, where d = 3 − ǫ is the dimension of space and µ an arbitrary mass scale.The value of the constant C A is not relevant to our purposes, but being an integral over anexplicitly known function of p , / p , we have carefully checked that it is finite.From the explicit expressions for Eqs. (65)–(67) in dimensional regularization [20], wefind directly h c a ( p )¯ c b ( p ) i = (cid:20) p + N c g (4 π ) p (cid:18) ǫ − ln p µ + C c (cid:19)(cid:21) δ ab (2 π ) δ ( p + p ) , (69) V ( p ) = 1 p + N c g (4 π ) p (cid:18) ǫ − ln p µ + C V (cid:19) . (70)This procedure to regularize the equal-time correlation functions finds further supportin the cases where the equal-time functions in the form (42)–(45) and (58)–(59) can beevaluated directly in dimensional regularization (in d = 3 − ǫ dimensions). For Eqs. (44)20nd (45), identical results are obtained in both ways [29] [also trivially for Eq. (42) and theloop integral in Eq. (58) which is just three times the one of Eq. (45)].The results (68) and (69) for the equal-time two-point correlation functions can be renor-malized in analogy to the procedures developed for covariant theories: we introduce renor-malized correlation functions (or correlation functions of the renormalized fields) h A aR,i ( p ) A bR,j ( p ) i = 1 Z A h A ai ( p ) A bj ( p ) i , h c aR ( p )¯ c bR ( p ) i = 1 Z c h c a ( p )¯ c b ( p ) i . (71)The simplest choice of the normalization conditions is h A aR,i ( p ) A bR,j ( p ) i (cid:12)(cid:12) p = κ = 12 | p | δ ab t ij ( p )(2 π ) δ ( p + p ) , h c aR ( p )¯ c bR ( p ) i (cid:12)(cid:12) p = κ = 1 p δ ab (2 π ) δ ( p + p ) , (72)at the renormalization scale κ . With these normalization conditions and the results (68),(69), we obtain Z A ( κ ) = 1 + N c g (4 π ) (cid:18) ǫ − ln κ µ + C A (cid:19) ,Z c ( κ ) = 1 + N c g (4 π ) (cid:18) ǫ − ln κ µ + C c (cid:19) , (73)to order g .The expression (70) for the static potential needs to be renormalized, too. This is mostnaturally achieved by a renormalization of the coupling constant as was first suggested inRef. [32, 33], through g V ( p ) (cid:12)(cid:12) p = κ = ¯ g R ( κ ) p , (74)see Eq. (57). Hence, from Eq. (70),¯ g R ( κ ) = g (cid:20) N c g (4 π ) (cid:18) ǫ − ln κ µ + C V (cid:19)(cid:21) , (75)which implies for the corresponding beta function to one-loop order, κ ∂∂κ ¯ g R ( κ ) = ¯ β (4 π ) ¯ g R ( κ ) , (76)that ¯ β = − N c . (77)21 = − b − b bb bcbc bc − b bb bcbc bc Figure 7: The proper ghost-gluon vertex to one-loop order.This is the well-known result from covariant perturbation theory (for Yang-Mills theory incovariant gauges), and has also been found in Ref. [20].For the rest of this section, we will pursue a more conventional way of renormalizingthe coupling constant (which, however, leads to the same result). To this end, we considerthe equal-time ghost-gluon three-point correlation function h c a ( p )¯ c b ( p ) A ci ( p ) i to order g (one loop). The calculation of this correlation function is performed in analogy with thedetermination of the equal-time two-point correlation functions in Section 3, using the resultfor the vacuum wave functional obtained in Section 2. In this particularly simple case (andto the order considered), the external “propagators” (equal-time two-point functions) canbe factorized to define the equal-time proper three-point vertex Γ abci ( p , p , p ) as h c a ( p )¯ c b ( p ) A ci ( p ) i = − Z d p (2 π ) d p (2 π ) d p (2 π ) h c a ( p )¯ c d ( − p ) i× Γ defj ( p , p , p ) h c e ( − p )¯ c b ( p ) i h A fj ( − p ) A ci ( p ) i . (78)The explicit perturbative result isΓ abcj ( p , p , p ) = − igf abc p ,k − N c g Z d q (2 π ) (cid:2) p · p − ( p · ˆ q )( p · ˆ q ) (cid:3) ( p ,k − q k )2 | q | ( p − q ) ( p + q ) (79)+ 2 N c g Z d q (2 π ) p ,l p ,n t lm ( p − q ) t nr ( p + q ) q | p − q | | p + q |× δ km ( p ,r − p ,r − q r ) − δ mr ( p ,k − p ,k − q k ) − δ rk ( p ,m − p ,m + q m ) | q | + | p − q | + | p + q | ! (80) × t jk ( p )(2 π ) δ ( p + p + p ) . It is represented diagrammatically in Fig. 7. Note that due to the transversality of the gauge,two powers of the external momenta can be factorized from the loop integrals [cf. Eq. (45)]and, as a result, the integrals are UV finite. This phenomenon is well-known in anothertransverse gauge, the Landau gauge [34, 35]. For future use, we note that by very lengthy22lgebra the tensor structure in Eq. (80) can be simplified as follows: p ,l p ,n t lm ( p − q ) t nr ( p + q ) h δ km ( p ,r − p ,r − q r ) − δ mr ( p ,k − p ,k − q k ) − δ rk ( p ,m − p ,m + q m ) i t jk ( p )(2 π ) δ ( p + p + p )= 2 q p ,k + ( p · p ) q k − ( p − q ) · ( p + q )( p − q ) ( p + q ) n [ q · ( p − q )][ q · ( p + q )] p ,k + [ p · ( p − q )][ p · ( p + q )] q k o! t jk ( p )(2 π ) δ ( p + p + p ) . (81)We define the renormalized coupling constant in analogy with the covariant case asΓ abcR,j ( p , p , p ) (cid:12)(cid:12)(cid:12) p = p = p = κ ≡ Z c ( κ ) Z / A ( κ ) Γ abcj ( p , p , p ) (cid:12)(cid:12)(cid:12) p = p = p = κ = − ig R ( κ ) f abc p ,k t jk ( p )(2 π ) δ ( p + p + p ) (82)at the symmetric point. As a consequence, using Eq. (73) and the UV finiteness of the loopintegrals (79)–(80), g R ( κ ) = g (cid:20) N c g (4 π ) (cid:18) ǫ − ln κ µ + C (cid:19)(cid:21) , (83)with a finite constant C given by (11 / C = (4 / C c + (1 / C A + C v , where C v is obtainedfrom the finite loop integrals in Eqs. (79)–(80).For the beta function defined in analogy with Eq. (76) we obtain from Eq. (83) β = − N c , (84)which coincides with the one obtained in Eq. (77) before with the renormalized couplingconstant defined through the static potential. We should mention that we could have ex-tracted the beta function directly from the cutoff dependence of the vacuum wave functional,as it has actually been done in Ref. [26] for Yang-Mills theory in Weyl gauge. Here, how-ever, our intention was to closely follow the procedure applied in the Lagrangian covariantformulation.The integration of the renormalization group equation (76) gives the well-known (one-loop) result g R ( κ ) = (4 π ) N c ln κ Λ QCD ! (85)[and the same for ¯ g R ( κ ) (75)]. It must be noted that for renormalization group improve-ments like Eq. (85) to be sensible we have to suppose that the three-dimensional formulation23resented here is multiplicatively renormalizable to all orders in the same way as the usualformulation of a renormalizable covariant quantum field theory, which is not known at present(even the renormalizability of the Lagrangian functional integral approach to Coulomb gaugeYang-Mills theory has not yet been shown). Equation (85) and the developments to followare therefore to some degree speculative, but it seemed of some interest to us to explore theconsequences of the natural assumption of multiplicative renormalizability.With these qualifications, we go on to use a standard renormalization group argumentto extract the asymptotic UV behavior of the equal-time two-point correlation functions.To this end, we differentiate Eq. (71) with respect to κ using the κ -independence of the“bare” two-point functions. It is then seen that the κ -dependence of the renormalized two-point functions is determined by the anomalous dimensions ( κ ∂ ln Z A,c /∂κ ). Evaluatingthe latter from Eq. (73) and replacing g in the results with g R ( κ ), we obtain the desiredrenormalization group equations for the equal-time two-point functions, explicitly κ ∂∂κ h A aR,i ( p ) A bR,j ( p ) i = N c g R ( κ )(4 π ) h A aR,i ( p ) A bR,j ( p ) i ,κ ∂∂κ h c aR ( p )¯ c bR ( p ) i = 43 N c g R ( κ )(4 π ) h c aR ( p )¯ c bR ( p ) i . (86)In these equations, we substitute from Eq. (85) for g R ( κ ) and integrate. Using the normal-ization conditions (72) for the determination of the integration constants, one obtains themomentum dependence of the equal-time two-point functions: h A aR,i ( p ) A bR,j ( p ) i = 12 | p | ln κ Λ QCD ! ln p Λ QCD ! / δ ab t ij ( p )(2 π ) δ ( p + p ) , h c aR ( p )¯ c bR ( p ) i = 1 p ln κ Λ QCD ! ln p Λ QCD ! / δ ab (2 π ) δ ( p + p ) . (87)The momentum dependence of the “bare” two-point functions, obtained from Eq. (87) simplyby multiplying with the corresponding wave function renormalization constants Z A,c , is ob-viously the same. By solving the renormalization group equations for Z A and Z c that involvethe anomalous dimensions, it may be shown explicitly that the bare two-point functions are κ -independent, as they must be.For the static potential, on the other hand, we immediately obtain from Eqs. (74) and2485) [for ¯ g R ( κ )] the renormalization group improved result g V ( p ) = (4 π ) N c p ln p Λ QCD ! . (88)Note that this one-loop formula constitutes a very direct expression of asymptotic freedom.The result (87) for the momentum dependence of the equal-time two-point functions hasalso been obtained in Ref. [36] from a Dyson-Schwinger equation for the equal-time ghostcorrelator, where the gauge-invariant one-loop running (85) of the renormalized coupling con-stant is used as an input. We briefly discuss that derivation here, adapted to the conventionsof the present paper.The renormalized equal-time two-point functions are parameterized as h A aR,i ( p ) A bR,j ( p ) i = 12 ω ( p ) δ ab t ij ( p )(2 π ) δ ( p + p ) (89)and h c aR ( p )¯ c bR ( p ) i = d ( p ) p δ ab (2 π ) δ ( p + p ) , (90)and normalized according to the conditions (72). The renormalized coupling constant isdefined as before in Eq. (82). Then the Dyson-Schwinger equation for the equal-time ghosttwo-point function reads [3]–[6] d − ( p ) = Z c − N c g R ( κ ) Z d q (2 π ) − (ˆ p · ˆ q ) ω ( q ) d (cid:0) ( p − q ) (cid:1) ( p − q ) . (91)Here we have approximated the full ghost-gluon vertex appearing in the exact equationby the tree-level vertex, as it is appropriate in order to obtain the (renormalization-groupimproved) one-loop expressions.In order to solve Eq. (91), we make the following, properly normalized, ansatzes for thetwo-point functions in the ultraviolet limit p ≫ Λ QCD , | p | ω ( p ) = ln κ Λ QCD ! ln p Λ QCD ! γ , d ( p ) = ln κ Λ QCD ! ln p Λ QCD ! δ , (92)with the exponents γ and δ to be determined. The integral in Eq. (91) can then be calculatedin the limit p ≫ Λ QCD and the Dyson-Schwinger equation yields the relation [36]ln − δ κ Λ QCD ! ln δ p Λ QCD ! = N c g R ( κ ) 1(4 π ) δ ln γ + δ κ Λ QCD ! ln − γ − δ p Λ QCD ! , (93)25rom which we infer the sum rule γ + 2 δ = 1 (94)for the exponents as well as the identity g R ( κ ) 1(4 π ) δ N c ln κ Λ QCD ! = 1 (95)for the coefficients. Consistency of the latter relation with the well-known perturbative result(85) yields the exponents γ = 311 , δ = 411 , (96)where we have used the sum rule (94) again. We have thus regained the result of Eq. (87). In this work, we have accomplished a systematic perturbative solution of the Yang-MillsSchr¨odinger equation in Coulomb gauge for the vacuum wave functional following the e S method in many-body physics. This resulted in a functional integral representation for thecalculation of equal-time correlation functions. We have derived a diagrammatical represen-tation of these functions, order by order in perturbation theory, where the vertices in thediagrams are determined from the perturbative calculation of the vacuum wave functional.The number of the vertices, which by themselves have a perturbative expansion, grows withthe perturbative order. We have determined the equal-time gluon and ghost two-point cor-relation functions and the potential between static color charges to one-loop order in thisway.The results coincide with those of a straightforward calculation in Rayleigh-Schr¨odingerperturbation theory [29], and also with the values for equal times of the two-point space-time correlation functions from a Lagrangian functional integral representation [20]. Wehave emphasized that the latter coincidence is not trivial since the gauge fixing proceduresin the Hamiltonian and the Lagrangian approach are profoundly different. We have also usedthe results of the Lagrangian approach to renormalize the equal-time two-point correlationfunctions and the static potential.With the help of the nonrenormalization of the ghost-gluon vertex which we also show,or, alternatively, from the static potential, we have extracted the running of the correspond-ingly defined renormalized coupling constant. The result for the beta function is the onealso found in covariant and other gauges, β = − (11 / N c to one-loop order. We have usedstandard renormalization group arguments to determine the asymptotic ultraviolet behav-ior of the equal-time two-point functions and the static potential under the assumption of26ultiplicative renormalizability to all orders, with the result that h A ai ( p ) A bj ( p ) i ∝ (cid:0) ln( p / Λ QCD ) (cid:1) − / | p | δ ab t ij ( p )(2 π ) δ ( p + p ) , h c a ( p )¯ c b ( p ) i ∝ (cid:0) ln( p / Λ QCD ) (cid:1) − / p δ ab (2 π ) δ ( p + p ) ,g V ( p ) ∝ (cid:0) ln( p / Λ QCD ) (cid:1) − p (97)to one-loop order in the perturbative (asymptotically free) regime.It is clear from the presence of an infinite number of vertices in the functional integralrepresentation of the equal-time correlation functions (to infinite perturbative order) that thecorresponding Dyson-Schwinger equations contain an infinite number of terms, a very seriousproblem for the determination of an appropriate approximation scheme for nonperturbativesolutions. The existence of simplified diagrammatic rules for the calculation of equal-timecorrelation functions via the E-operator, to be appropriately extended to all perturbativeorders, seems to point toward the possibility of formulating similar nonperturbative equationswith a finite number of terms. It would indeed be very interesting to repeat the type ofinfrared analysis applied before to Yang-Mills theory in the Landau gauge [7], [12]–[17] andto a variational ansatz in the Coulomb gauge [1]–[7] for such a set of equations.The perturbative expression for the vacuum wave functional determined in this paper,in particular the coefficient functions f and f in Eqs. (22)–(25), can be used to motivateimproved ansatzes for the vacuum functional going beyond the Gaussian form in a variationalapproach as employed in Refs. [1], [3]–[6]. An appropriate extension of a Gaussian ansatzwould make it possible, e.g., to reproduce the correct beta function and the anomalousdimensions of the equal-time two-point functions at least to one-loop order in this approach.This idea is currently being pursued. We note in this context that the relation (26) betweenthe coefficient functions f and f following from the Schr¨odinger equation can easily bepromoted to an exact gap equation, i.e., valid to any perturbative order, by replacing | p | and | q | on the right-hand side with f ( p ) and f ( q ), respectively: (cid:0) f ( p ) (cid:1) δ ab δ ij = (cid:18) p − f ( p ) N c g Z d q (2 π ) − (ˆ p · ˆ q ) ( p − q ) (cid:19) δ ab δ ij + 12 Z d q (2 π ) f abcc ijkl ( − p , p , − q , q ) t kl ( q ) − N c g δ ab Z d q (2 π ) f ( p ) − f ( q )( p − q ) t ij ( q ) . (98)Similarly, the explicit perturbative expression for the vacuum wave functional in thepresence of static external color charges, see Eqs. (49) and (55), is expected to providevaluable information in the quest for a detailed understanding of the (quenched) interactionbetween quarks. Different ansatzes for such a wave functional in the nonperturbative regime27ave been considered in Refs. [36, 37], while relevant results from lattice calculations can befound in Refs. [38, 39].Finally, the results (97) for the ultraviolet behavior of the two-point functions are relevantto the corresponding results of a numerical evaluation on space-time lattices. A recentnumerical calculation of the equal-time ghost two-point function [9] gives a value of 0 . / ≈ .
36. As for the staticpotential, numerical results are only available for the instantaneous antiscreening part, givento one-loop order by Eq. (58). The asymptotic behavior of this so-called color Coulombpotential has been determined in Ref. [40] to be g V C ( p ) ∝ (cid:0) ln( p / Λ QCD ) (cid:1) − p (99)to one-loop order, the same as for the full static potential (except for an overall factor 12 / h A ai ( p ) A bj ( p ) i ∝ | p | − η | p | δ ab t ij ( p )(2 π ) δ ( p + p ) (100)with η = 0 . a s /a t → ∞ (with the spatial and temporal lattice spacings a s and a t ).Given that the question of multiplicative renormalizability of the gluonic two-point cor-relation function (at equal and at different times) plays an important rˆole in the controversyabout the scaling violations [8, 9], it would certainly be interesting to confirm multiplicativerenormalizability in our approach to the two-loop level. The extension of the calculations pre-sented here to two loops appears relatively straightforward, albeit lengthy. Let us mention,in this context, that the static potential has never been worked out explicitly in Coulombgauge at the two-loop level. In our approach, the main problem with the two-loop calcula-tions is expected to be the correct renormalization of the expressions. Corresponding resultsfor the space-time correlation functions from a Lagrangian functional integral approach arenot known to this level, so one cannot proceed in analogy with Section 4. The calculationswith the Lagrangian functional integral method are complicated, at the two-loop level, bythe appearance of Christ-Lee-Schwinger terms [22] (see also Refs. [49, 50]) which are re-quired in order to produce results corresponding to the Hamiltonian (5) with its specificoperator ordering (in particular, the insertion of powers of the Faddeev-Popov determinant).28s an alternative to the procedure of Section 4, one may consider the use of a simple three-dimensional ultraviolet momentum cutoff. Care has to be taken, however, since such a cutoffcan break the covariance of the theory (see Ref. [51] for an example in Yukawa theory), inwhich case appropriate noncovariant counterterms have to be included. Acknowledgments
It is a pleasure to thank Adam Szczepaniak and Peter Watson for many valuable discussionson the Coulomb gauge. A.W. is grateful to the Institute for Theoretical Physics at the Uni-versity of T¨ubingen for the warm hospitality extended to him during a two-months stay in thesummer of 2008. Support by the Deutscher Akademischer Austauschdienst (DAAD), Cona-cyt grant 46513-F, CIC-UMSNH, Deutsche Forschungsgemeinschaft (DFG) under contractRe 856/6-3, and Cusanuswerk–Bisch¨ofliche Studienf¨orderung is gratefully acknowledged.
A Gauge transformations in Lagrangian and Hamilto-nian formalisms
While the Lagrangian approach to Yang–Mills theory offers some convenient features (suchas manifestation of Lorentz invariance), the more cumbersome Hamiltonian approach yieldsequations of motion invariant under a larger set of gauge transformations. In what fol-lows, we discuss gauge invariance starting from the classical Lagrangian and Hamiltonianfunctions, respectively, prior to quantization. We will employ standard covariant notationin this appendix; in particular, spatial subindices refer to the covariant components of thecorresponding 4-vector or tensor.The Lagrangian function of the gauge sector, L = − Z d x F aµν ( x ) F µνa ( x ) , (101)is invariant under gauge transformations of the gauge field A µ ( x ) ≡ A aµ ( x ) T a , A µ ( x ) → U ( x ) A µ ( x ) U † ( x ) + 1 g U ( x ) ∂ µ U † ( x ) , (102)where U ∈ SU ( N ) and [ T a , T b ] = f abc T c .The Weyl gauge, A a ( x ) = 0, can be found by choosing the time-ordered exponential U † ( x ) = T exp (cid:18) − g Z t dt ′ A ( x , t ′ ) (cid:19) . (103)To remain in the Weyl gauge, the transformation (103) may be followed by time-independenttransformations U ( x ) only. We can therefore fix the Coulomb gauge, ∂ i A ia ( x ) = 0, at oneinstant of time but it is impossible to fix both gauges simultaneously for all times.29n the Hamiltonian formalism, on the other hand, gauge transformations are generated by(first-class) constraints in configuration space [52]. To see that, supplement the Hamiltonianfunction H = 12 Z d x (cid:0) Π a ( x ) + B a ( x ) (cid:1) − Z d x A a ( x ) ˆ D abi ( x )Π ib ( x ) (104)by the constraints φ a ( x ) = Π a ( x ) ≈ , φ a ( x ) = ˆ D abi ( x )Π ib ( x ) ≈ { λ ak ( x ) } , H E = H + X k =1 , Z d x λ ak ( x ) φ ak ( x ) . (106)We defined Π aµ ( x ) = F aµ ( x ) and ˆ D abi ( x ) = δ ab ∂ i − gf abc A ci ( x ). The extended Hamiltonian H E in Eq. (106) is equivalent to the original Hamiltonian H since the constraints { φ ak ( x ) } vanish weakly (in the Dirac sense [52]). The infinitesimal time evolution of the gauge field A aµ ( x , t ) from t to t = t + δt , generated by H E through the Poisson brackets, A aµ ( x , t ) = A aµ ( x , t ) + δt { A aµ ( x , t ) , H } + δt X k =1 , Z d y λ bk ( y ) { A aµ ( x , t ) , φ bk ( y ) } , (107)gives for two different sets of Lagrange multiplier functions { λ ′ bk ( x ) } and { λ ′′ bk ( x ) } two differ-ent results A ′ aµ and A ′′ aµ , respectively. These differ to O ( δt ) by A ′′ aµ ( x , t ) − A ′ aµ ( x , t ) = δt X k =1 , Z d y (cid:0) λ ′′ bk ( y ) − λ ′ bk ( y ) (cid:1) { A aµ ( x , t ) , φ bk ( y ) } (108)and are physically equivalent. Thus, the function G = X k =1 , Z d y τ ak ( y ) φ ak ( y ) (109)generates infinitesimal gauge transformations in the (extended) Hamiltonian formalism witharbitrary functions τ a ( x ) and τ a ( x ). Computing the Poisson brackets in Eq. (108) yields A a ( x ) → A a ( x ) + τ a ( x ) (110) A ai ( x ) → A ai ( x ) − ˆ D abi ( x ) τ b ( x ) (111)The difference to the gauge transformations (102) in the Lagrangian formalism is that thetime component and the spatial components of the gauge field transform independently.The two functions τ a ( x ) and τ a ( x ) allow for a larger set of gauge transformations than thesingle function U ( x ) in the Lagrangian formalism. The simultaneous fixing of Weyl andCoulomb gauges, which is impossible in the Lagrangian formalism, can be accomplished inthe Hamiltonian formalism by appropriately choosing τ a ( x ) and τ a ( x ) (see Ref. [53] for theabelian case). Subsequently, the non-abelian gauge-fixed theory can be canonically quantizedwith projection on the physical Hilbert space [22], or with Dirac brackets [36] enforcing allconstraints strongly. Both quantization prescriptions produce the Hamiltonian operatorgiven by Eq. (5). 30 eferenceseferences