Renormalizability of the Refined Gribov-Zwanziger action in the linear covariant gauges
RRenormalizability of the Refined Gribov-Zwanziger action in the linear covariantgauges
M. A. L. Capri, ∗ D. Fiorentini, † A. D. Pereira, ‡ and S. P. Sorella § UERJ − Universidade do Estado do Rio de Janeiro,Departamento de F´ısica Te´orica, Rua S˜ao Francisco Xavier 524,20550-013, Maracan˜a, Rio de Janeiro, Brasil UFF − Universidade Federal Fluminense,Instituto de F´ısica, Campus da Praia Vermelha,Avenida General Milton Tavares de Souza s/n,24210-346, Niter´oi, RJ, Brasil
The Refined Gribov-Zwanziger framework takes into account the existence of equivalent gaugefield configurations in the gauge-fixing quantization procedure of Euclidean Yang-Mills theories.Recently, this setup was extended to the family of linear covariant gauges giving rise to a local andBRST-invariant action. In this paper, we give an algebraic proof of the renormalizability of theresulting action to all orders in perturbation theory.
CONTENTS
I. Introduction 1II. The Refined Gribov-Zwanziger action in the linear covariant gauges 6III. Renormalizability analysis: Preliminaries 7A. Conventions 7B. Introduction of external sources 8C. Extended BRST symmetry 9IV. Ward identities and algebraic characterization of the most general counterterm 11A. Parametric form of the counterterm 17B. Stability of the action Σ 20V. Conclusions 22Acknowledgements 22A. Remarks on the localization of the BRST-invariant RGZ action 22References 23
I. INTRODUCTION
A general feature of the continuum formulation of non-Abelian gauge theories is the gauge-fixing procedure. Asis widely known, this can be achieved through the so-called Faddeev-Popov method which gives consistent resultswithin the perturbative treatment of gauge theories. Nevertheless, as was shown by Gribov in [1], the Faddeev-Popovprocedure relies on some hypothesis which are not well grounded as long as one goes away from the perturbativeregime. The problem arises from the fact that a gauge-fixing condition is not enough to fix completely the gaugefreedom, allowing for gauge equivalent configurations, the so-called Gribov copies, even after imposing the gauge-fixingcondition. It was soon realized that this is not a particular problem of some specific gauge-fixing, but an intrinsicproblem related to the non-trivial geometrical structure non-Abelian gauge theories, see [2]. For a pedagogical intro-duction to the Gribov problem, we refer to [3–6]. ∗ [email protected] † [email protected] ‡ [email protected]ff.br § [email protected] a r X i v : . [ h e p - t h ] A ug In the Landau gauge, the gauge-fixing condition is expressed as ∂ µ A aµ = 0 . (1)Such a condition would be ideal if any gauge equivalent configuration A (cid:48) aµ , connected through A aµ via a gauge trans-formation, would not satisfy (1). For concreteness, one can assume that A aµ and A (cid:48) aµ are connected via an infinitesimalgauge transformation. Hence, A (cid:48) aµ = A aµ − D abµ θ b , ∂ µ A (cid:48) aµ = 0 ⇒ − ∂ µ D abµ θ b = 0 , (2)with θ a an infinitesimal gauge parameter and D abµ ≡ δ ab ∂ µ − gf abc A cµ , the covariant derivative in the adjoint repre-sentation of the gauge group . Eq.(2) reveals that the configuration A (cid:48) aµ satisfies the gauge condition (1), i.e. it isa Gribov copy of A aµ , if the operator − ∂ µ D abµ develops zero-modes. In [1], it was proven that such zero-modes existand explicit examples were constructed. As a consequence, even after the gauge-fixing procedure, a residual gaugeredundancy remains or, in other words, the gauge-fixing is not ideal. Therefore, the Faddeev-Popov procedure shouldnot be strictly applied as it stands, requiring an improvement at the non-perturbative level.Already in [1], Gribov proposed that, besides the standard gauge-fixing, a further constraint should be imposedto the path integral: the functional measure should be restricted to a region free from zero-modes of the Faddeev-Popov operator − ∂ µ D abµ . Such a region, known as the Gribov region Ω L , is defined asΩ L = (cid:8) A aµ , ∂ µ A aµ = 0 | − ∂ µ D abµ > (cid:9) , (3)where the subscript L means that one is referring to the Landau gauge. It is important to point out here that, inthis gauge, the Faddeev-Popov operator − ∂ µ D abµ , is hermitian, implying that its eigenvalues are real. This propertyallows for a meaningful requirement of the positivity of the Faddeev-Popov operator. Moreover, it has been provenin [7] that the region Ω L enjoys a set of remarkable features : i) It is bounded in all directions in field space; ii)
Thetrivial perturbative vacuum A aµ = 0 belongs to Ω L ; iii) It is convex; iv)
All gauge orbits cross Ω L at least once. From(2) and (3), it is clear that the Gribov region is free from infinitesimal Gribov copies. Unfortunately, it still containscopies generated by large gauge transformations and one should look for a region truly free from Gribov copies, knownas the fundamental modular region , see [8]. Ideally, one should restrict the path integral to the fundamental modularregion rather than the Gribov region. Nevertheless, till now, a practical operational way to restrict the functionalintegral to the fundamental modular region has not yet been achieved. We stick therefore to the Gribov region Ω L .Formally, Gribov’s proposal can written as Z = (cid:90) Ω L [ D Φ] e − ( S YM + S FP ) , (4)with S YM = 14 (cid:90) d x F aµν F aµν , S FP = (cid:90) d x (cid:0) ib a ∂ µ A aµ + ¯ c a ∂ µ D abµ c b (cid:1) , (5)where b a , ¯ c a , c a are the auxiliary Nakanishi-Lautrup field, the Faddeev-Popov antighost and ghost fields, respectively,and Φ is a shorthand notation for all the fields of the theory. The field strength F aµν is given by F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν . (6)The implementation of the restriction to the region Ω L , as expressed in eq.(4), was worked out at leading order byGribov in [1] and to all orders by Zwanziger in [9]. Although the procedures pursued are different, their equivalence We consider SU ( N ) gauge theories in four Euclidean dimensions. was established to all orders in [10]. Effectively, the restriction to Ω L is achieved through the following modificationof the original Faddeev-Popov path integral, Z = (cid:90) Ω L [ D Φ] e − ( S YM + S FP ) = (cid:90) [ D Φ] e − ( S YM + S FP + γ H ( A ) − V γ ( N − , (7)where H ( A ) = g (cid:90) d x d y f abc A bµ ( x ) (cid:2) M − (cid:3) ad ( x, y ) f dec A eµ ( y ) , (8)is the so-called horizon function , with M ab = − ∂ µ D abµ , V is the spacetime volume and N is the number of colors. Theparameter γ has mass dimension one and is known as the Gribov parameter. It is not free, but determined througha gap equation, (cid:104) H ( A ) (cid:105) = 4 V ( N − , (9)with (cid:104) . . . (cid:105) taken with the functional measure defined in (7). As is evident from (8), the horizon function is non-local,giving rise to a non-local action which, thanks to the aforementioned properties of the Gribov region Ω L , takesinto account the existence of a huge set of Gribov copies. Remarkably, such an action can be localized through theintroduction of auxiliary fields viz. a pair of bosonic ones ( ¯ ϕ, ϕ ) abµ and a pair of anticommuting fields (¯ ω, ω ) abµ . In theLandau gauge, the local action takes the following form S GZ = S YM + S FP − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ϕ bcµ − ¯ ω acµ M ab ω bcµ + gf adl ¯ ω acµ ∂ ν (cid:0) ϕ lcµ D deν c e (cid:1)(cid:1) − γ (cid:90) d x gf abc A aµ ( ϕ + ¯ ϕ ) bcµ , (10)and is known as the Gribov-Zwanziger action. It is easy to check that, upon integration of the auxiliary fields, thenon-local action (7) is recovered. This action is renormalizable to all orders in perturbation theory [9] and providesa local framework which, thanks to the aforementioned properties of the Gribov region Ω L , takes into account theexistence of a huge set of Gribov copies in the Landau gauge.Nevertheless, It was realized in [11] that the restriction to the Gribov region leads to additional non-perturbativeinstabilities giving rise to the formation of dimension-two condensates. In particular, it was shown that the conden-sates (cid:104) A aµ A aµ (cid:105) and (cid:104) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:105) are non-vanishing already at one-loop order, besides being free of ultravioletdivergences, see also [12–14].Taking into account the existence of such condensates from the beginning, gives rise to the so-called Refined Gribov-Zwanziger (RGZ) action, which is expressed as S RGZ = S GZ + m (cid:90) d x A aµ A aµ − M (cid:90) d x (cid:0) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:1) , (11)where the mass parameters m and M are dynamically determined by their own gap equations, see [13]. The RGZaction enjoys many interesting properties. In particular, the tree-level gluon propagator reads (cid:104) A aµ ( k ) A bν ( − k ) (cid:105) = δ ab k + M ( k + m )( k + M ) + 2 g N γ (cid:18) δ µν − k µ k ν k (cid:19) , (12)from which one sees that it attains a finite value at k = 0. Such a massive/decoupling behavior is in agreementwith the most recent lattice data as well as with functional and effective methods, [15–22]. As observed in [23], in d = 2 the refinement does not occur due to the existence of infrared divergences which prevent the formation of the We underline here that the important property that all gauge orbits cross Ω L at least once, gives a well defined support to originalGribov’s proposal of restricting the domain of integration in the path integral to the region Ω L , although Ω L itself is not free fromGribov copies. condensates. As a consequence, in d = 2 the gluon propagator vanishes at zero momentum, giving rise to the so-calledscaling solution [24]. In d = 3 [26] as well as d >
4, [27], the massive/decoupling behavior persists. Such differentbehavior for d = 2 and d > A h,aµ , introduced in [47, 48].Explicitly, A h,aµ can be written as an infinite non-local series, given by A hµ = (cid:18) δ µν − ∂ µ ∂ ν ∂ (cid:19) (cid:18) A ν − ig (cid:20) ∂ ∂A, A ν (cid:21) + ig (cid:20) ∂ ∂A, ∂ ν ∂ ∂A (cid:21) + O ( A ) (cid:19) , (13)with A hµ being BRST-invariant. sA hµ = 0 , sA aµ = − D abµ ( A ) c b . (14)When written in terms of A hµ , the RGZ action in the Landau gauge can be expressed as [46]:˜ S L RGZ = S YM + S FP − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) − γ (cid:90) d x gf abc A h,aµ ( ϕ + ¯ ϕ ) bcµ + m (cid:90) d x A h,aµ A h,aµ − M (cid:90) d x (cid:0) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:1) , (15)with M ab ( A h ) = − δ ab ∂ + gf abc A h,cµ ∂ µ , with ∂ µ A h,aµ = 0 . (16)Despite the presence of the Zwanziger’s localizing fields, the action (15) is still non-local due to the presence of thenon-local field A h,aµ , see eq.(13). Notwithstanding, a localization of such an action was introduced in [49] by means ofthe use of a Stueckelberg-like field ξ a . More specifically, following [49], the non-local expression (13) is re-expressedin a local way as A hµ = h † A µ h + ig h † ∂ µ h , (17)where h = e igξ a T a , (18)with ξ being an auxiliary Stueckelberg field. In addition, one imposes the transversality condition ∂ µ A hµ = 0 . (19)When solved iteratively for the Stueckelberg field, equations (17),(19) give back the non-local expression (13), see [49]for the details.Therefore, for the local and BRST-invariant RGZ action in the Landau gauge one gets [49]: We refer to Appendix. A of [46] for the details of the construction of A h,aµ . We employ a matrix notation, as described in Appendix. A of [46]. S L RGZ = S YM + S FP − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) − γ (cid:90) d x gf abc ( A h ) aµ ( ϕ + ¯ ϕ ) bcµ + m (cid:90) d x ( A h ) aµ ( A h ) aµ − M (cid:90) d x (cid:0) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b , (20)with τ a being a Lagrange multiplier needed to imposes the transversality of A hµ , eq.(19). The fields (¯ η a , η a ) are a pairof ghosts needed to take into account the Jacobian arising from the transversality constraint (19), ∂ µ A hµ = 0. Weremind the reader to Appendix. A for the proof of the equivalence between the RGZ actions given in eqs.(20) and(11). As a consequence, the functional integral for the local and BRST-invariant RGZ action in the Landau gauge iswritten as Z = (cid:90) [ D µ ] e − S RGZ +4 V γ ( N − , (21)where [ D µ ] = [ D A ] [ D b ] [ D ¯ c ] [ D c ] [ D ¯ ϕ ] [ D ϕ ] [ D ¯ ω ] [ D ω ] [ D τ ] [ D ξ ] [ D ¯ η ] [ D η ] . (22)Explicitly, the nilpotent BRST transformations which leave the action (20) invariant are sA aµ = − D abµ c b , sc a = g f abc c b c c ,s ¯ c a = ib a , sb a = 0 ,sϕ abµ = 0 , sω abµ = 0 ,s ¯ ω abµ = 0 , s ¯ ϕ abµ = 0 ,sh ij = − igc a ( T a ) ik h kj , sA h,aµ = 0 ,sτ a = 0 , s ¯ η a = 0 ,sη a = 0 , s = 0 . (23)The BRST transformation of the Stueckelberg field ξ a can be obtained iteratively from the transformation of h ij , i.e. ( sh ij = − igc a ( T a ) ik h kj ), yielding sξ a = g ab ( ξ ) c b , (24)where g ab ( ξ ) is a power series in ξ a , namely g ab ( ξ ) = − δ ab + g f abc ξ c − g f amr f mbq ξ q ξ r + O ( g ) . (25)Having a local and BRST invariant setup which takes into account the existence of Gribov copies in the Landaugauge, it is natural to look for an extension of such a framework to different gauges. A natural generalization are theso-called linear covariant gauges, where the gauge condition reads ∂ µ A aµ = − iαb a , (26)with α a non-negative gauge parameter. Although this class of gauges preserves the linearity of the Landau gauge, itintroduces a gauge parameter α as well as a longitudinal sector for the gluon fields. In a series of papers [46, 49–55],the development of the RGZ action to linear covariant gauges was worked out. Besides being interesting by itsown as a further development of the RGZ framework, recent studies within analytic as well as numerical lattice In [49] the ghosts (¯ η, η ) a were not introduced. Although this term does not alter the results of [49], such a term is needed in order toprove the equivalence between the local and non-local and formulations. We acknowledge U. Reinosa, J. Serreau, M. Tissier and N.Wschebor for discussions on this issue. approaches to non-perturbative Yang-Mills theories started employing the linear covariant gauges [56–61]. Hence, aninterplay between these different approaches, as performed in the Landau gauge, becomes possible, leading to a deeperunderstanding of the behavior of the correlation functions in the non-perturbative infrared region in this class of gauges.In this paper we pursue the study of the RGZ formulation initiated in [46, 49–55], by addressing the issue ofthe renormalizability properties of the local and BRST invariant RGZ action in the linear covariant gauges. In partic-ular, we prove, using the algebraic renormalization setup [62], the all orders renormalizability of the RGZ frameworkin the linear covariant gauge, a topic which was still lacking in our previous studies.The structure of the paper is the following: In Sect. II we give a brief overview of the construction of the RGZaction in the linear covariant gauges. Subsequently, in Sect. III we identify the classical complete action which willbe the starting point for the algebraic renormalization analysis. In Sect. IV we give the formal proof of the all orderrenormalizability of the RGZ action in the linear covariant gauges. Finally, we collect our conclusions. Additionalmaterial clarifying some specific technical points of this work are collected in the appendix. II. THE REFINED GRIBOV-ZWANZIGER ACTION IN THE LINEAR COVARIANT GAUGES
Within the standard Faddeev-Popov framework, the gauge-fixed Yang-Mills action in the linear covariant gaugesreads S FPLCG = S YM + (cid:90) d x (cid:16) ib a ∂ µ A aµ + α b a b a + ¯ c a ∂ µ D abµ c b (cid:17) = S YM + (cid:90) d x s (cid:18) ¯ c a ∂ µ A aµ − iα c a b a (cid:19) , (27)where the gauge parameter α is non-negative. The particular case α = 0 is the Landau gauge. Very much close to thecase of the Landau gauge, infinitesimal Gribov copies arise as long as the Faddeev-Popov operator − ∂ µ D abµ developszero-modes. Nevertheless, for non-vanishing α , such an operator is not Hermitean, a feature that jeopardizes thestandard Gribov-Zwanziger analysis for the removal of such zero-modes from the path integral measure. This problemhas been a great challenge in dealing with Gribov copies for generic values of α . A first attempt to face this issuewas to consider the operator − ∂ µ D abµ projected onto the transverse component of the gauge field, see [52–54]. In thiscase, the resulting projected operator is Hermitean and the standard procedure for the construction of a horizon-likefunction is available. This setup was worked out in [52, 53] and its renormalizability was analyzed in details in[54]. Nonetheless, it exhibits drawbacks: i) the limit α = 0 does not fully recover the standard (R)GZ action in theLandau gauge and ii) it breaks BRST symmetry softly, a feature that obscures the control of the gauge parameterindependence of the correlation functions of gauge invariant quantities. These difficulties have been overcome by theconstruction of a BRST invariant formulation of the RGZ action in the Landau gauge which, as discussed in [46],naturally leads to a BRST invariant formulation of the linear covariant gauges.Following [46], the local and BRST-invariant RGZ action in the linear covariant gauges is written as S LCGRGZ = S FPLCG − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) − γ (cid:90) d x gf abc ( A h ) aµ ( ϕ + ¯ ϕ ) bcµ + m (cid:90) d x ( A h ) aµ ( A h ) aµ − M (cid:90) d x (cid:0) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b . (28)This action is invariant under the nilpotent BRST transformations (23). As such, the following properties can beshown to hold, see [49–51]: • Correlation functions of gauge-invariant quantities are α -independent; • The mass parameters ( γ, m, M ) are independent from the gauge parameter α and, as a consequence, can enterphysical quantities; • The longitudinal part of the gluon propagator is exact and equal to the tree-level result; • The pole mass of the transverse component of the gluon propagator is independent from α . • The action (28) effectively implements the restriction of the path integral to the functional region Σ defined asΣ = (cid:110) A aµ , ∂ µ A aµ = αib a (cid:12)(cid:12)(cid:12) − ∂ µ D abµ ( A h ) > (cid:111) . (29)At the tree-level, the gluon propagator stemming from the action (28) is (cid:104) A aµ ( k ) A bν ( − k ) (cid:105) = δ ab (cid:20) k + M ( k + m )( k + M ) + 2 g N γ (cid:18) δ µν − k µ k ν k (cid:19) + αk k µ k ν k (cid:21) . (30)At this order, the transverse component of the propagator is α -independent, but one should keep in mind that aslong as higher loops are considered, α -dependent corrections might appear. Though, we underline that the polemass of the transverse component retains its independence from α to all orders [49–51]. For the longitudinal part, aspreviously mentioned, the result is exact. The tree-level propagator (30) is in good agreement with the most recentlattice data in the linear covariant gauges [60]. As in the Landau gauge, a massive/decoupling behavior is observedfor the transverse component.Owing to the aforementioned prescription for a BRST-invariant (R)GZ construction in the linear covariant gauges,it was established in [50] that, in great similarity with the Landau gauge, the formation of the refining condensateshappens in d = 3 , d = 2. Also, matter fields were introduced in [55] according to the prescriptiondeveloped in [32], giving rise to analytic expressions for the non-perturbative propagators for scalar fields in theadjoint representation of the gauge group as well as for quarks in the fundamental representation. III. RENORMALIZABILITY ANALYSIS: PRELIMINARIESA. Conventions
In order to give an algebraic proof of the renormalizability of the action (28), it is convenient, in analogy with [63],to adopt the following parametrization: A aµ → g A aµ , b a → gb a , ξ a → g ξ a , α → αg , m → m g , τ a → gτ a , (31)As a consequence, the Faddeev-Popov action in the linear covariant gauges is rewritten as S FPLCG = 14 g (cid:90) d x F aµν F aµν + (cid:90) d x (cid:16) ib a ∂ µ A aµ + α b a b a + ¯ c a ∂ µ D abµ c b (cid:17) , (32)with F aµν = ∂ µ A aν − ∂ ν A aµ + f abc A bµ A cν , D abµ = δ ab ∂ µ − f abc A cµ , (33)the redefined field strength and covariant derivative. Accordingly, the RGZ action in the linear covariant gaugesbecomes S LCGRGZ = S LCGFP − (cid:90) d x (cid:2) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ + γ f abc ( A h ) aµ ( ϕ + ¯ ϕ ) bcµ (cid:3) + m (cid:90) d x ( A h ) aµ ( A h ) aµ − M (cid:90) d x (cid:0) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b , (34)with A hµ ≡ ( A h ) aµ T a = h † A µ h + ih † ∂ µ h , h = e iξ a T a . (35)After the redefinition (31), the BRST transformations read sA aµ = − D abµ c b , sc a = 12 f abc c b c c ,s ¯ c a = ib a , sb a = 0 ,sϕ abµ = 0 , sω abµ = 0 ,s ¯ ω abµ = 0 , s ¯ ϕ abµ = 0 ,sξ a = g ab ( ξ ) c b , sA h,aµ = 0 ,sτ a = 0 , s ¯ η a = 0 ,sη a = 0 , s = 0 , (36)with g ab ( ξ ) = − δ ab + 12 f abc ξ c − f acd f cbe ξ e ξ d + O ( ξ ) . (37)Before writing the Ward identities, we need to introduced a suitable set of external sources which we describe indetails in the next subsection. B. Introduction of external sources
Let us begin by introducing the following set of sources (
M, V, N, U ) abµν and express the term which contains theGribov parameter γ in (28) as (cid:90) d x γ f abc ( A h ) aµ ( ϕ + ¯ ϕ ) bcµ −→ (cid:90) d x (cid:0) M aiµ D abµ ( A h ) ϕ bi + V aiµ D abµ ( A h ) ¯ ϕ bi + N aiµ D abµ ( A h ) ω bi + U aiµ D abµ ( A h )¯ ω bi − M aiµ V aiµ + N aiµ U aiµ (cid:1) , (38)where we are employing the multi-index notation i = ( a, µ ) in the same way as done in [11, 29, 64]. We alsoemphasize that the presence of terms which are quadratic in the sources is allowed by power counting. The localsources ( M, V, N, U ) abµν enlarge the original theory (28) which is recovered by demanding that they attain a suitablephysical limit, namely M abµν (cid:12)(cid:12)(cid:12) phys = V abµν (cid:12)(cid:12)(cid:12) phys = γ δ ab δ µν ,N abµν (cid:12)(cid:12)(cid:12) phys = U abµν (cid:12)(cid:12)(cid:12) phys = 0 , (39)from which the right-hand side of eq.(38) reduces precisely to the left-hand side. Also, in order to preserve the BRSTinvariance of the theory, the sources are chosen to be BRST-singlets, i.e. sM aiµ = sV aiµ = sN aiµ = sU aiµ = 0 , (40)In addition, following the procedure of the algebraic renormalization setup [67], additional external sources coupledto the composite operators corresponding to the non-linear BRST transformations of the fields need to be introduced, i.e. S sources = (cid:90) d x (cid:20) − Ω aµ D abµ c b + 12 L a f abc c b c c + J aµ ( A h ) aµ + K a g ab ( ξ ) c b (cid:21) , (41)where the composite operator ( A h ) aµ has also been coupled to its corresponding source J aµ , see [63]. Finally, accordingto the local composite operator method (LCO) [65, 66] for evaluating the effective potential giving rise to the dimensiontwo condensates (cid:104) A haµ A haµ (cid:105) and (cid:104) ¯ ω ai ω ai − ¯ ϕ ai ϕ ai (cid:105) , two external sources J and ˜ J coupled to the corresponding operators( A haµ ( x ) A haµ ( x )) and (¯ ω ai ( x ) ω ai ( x ) − ¯ ϕ ai ( x ) ϕ ai ( x )), need be introduced, S cond = (cid:90) d x (cid:20) J ( A h ) aµ ( A h ) aµ + ˜ J (¯ ω ai ω ai − ¯ ϕ ai ϕ ai ) + θ J (cid:21) , (42)where the parameter θ appearing in the quadratic source term J of eq.(42) takes into account the UV divergencespresent in the vacuum correlation function (cid:104) (( A h ) aµ ) ( x )(( A h ) bν ) ( y ) (cid:105) when x → y . Moreover, as argued in [11], it is notnecessary to add a quadratic term in ˜ J , due to the absence of UV divergences in (cid:104) (¯ ω ai ω ai − ¯ ϕ ai ϕ ai ) x (¯ ω ai ω ai − ¯ ϕ ai ϕ ai ) y (cid:105) ,for x → y .For future use, it will be also necessary to introduce an extra term depending on external sources given by S extra = (cid:90) d x (cid:2) − Ξ aµ D abµ ( A h ) η b + X i η a ¯ ω ai + Y i η a ¯ ϕ ai + ¯ X abi η a ω bi + ¯ Y abi η a ϕ bi (cid:3) . (43)The whole new set of sources is invariant under BRST transformations, i.e. s Ω aµ = sL a = s J aµ = sK a = sJ = s ˜ J = s Ξ aµ = sX i = sY i = s ¯ X i = s ¯ Y i = 0 . (44)After the introduction of the external sources, for the complete extended classical action Σ one hasΣ = S LCGFP − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b − (cid:90) d x (cid:0) M aiµ D abµ ( A h ) ϕ bi + V aiµ D abµ ( A h ) ¯ ϕ bi − N aiµ D abµ ( A h ) ω bi + U aiµ D abµ ( A h )¯ ω bi + M aiµ V aiµ − N aiµ U aiµ (cid:1) + (cid:90) d x (cid:20) − Ω aµ D abµ c b + 12 L a f abc c b c c + J aµ ( A h ) aµ + K a g ab ( ξ ) c b (cid:21) + (cid:90) d x (cid:20) J ( A h ) aµ ( A h ) aµ + ˜ J (¯ ω ai ω ai − ¯ ϕ ai ϕ ai ) + θ J (cid:21) + (cid:90) d x (cid:2) − Ξ aµ D abµ ( A h ) η b + X i η a ¯ ω ai + Y i η a ¯ ϕ ai + ¯ X abi η a ω bi + ¯ Y abi η a ϕ bi (cid:3) , (45)with s Σ = 0 . (46) C. Extended BRST symmetry
As dicussed in [51, 67], it turns out to be convenient to extend the action of the BRST operator s on the parameter α as sα = χ , (47)where χ is a constant Grassmann parameter with ghost number 1, to be set to zero at the end of the algebraic analysis.Following [67], such a transformation plays a pivotal role in order to control the gauge parameter (in)dependence ofcorrelation functions. For the renormalizability proof we shall present, we employ (47) as well. Furthermore, it canbe shown that, besides the BRST invariance, eq.(46), the action (45) enjoys a second nilpotent exact symmetry: δ Σ = 0 , (48)where δ is given by δϕ ai = ω ai , δω ai = 0 δ ¯ ω ai = ¯ ϕ ai , δ ¯ ϕ ai = 0 δN aiµ = M aiµ , δM aiµ = 0 δV aiµ = U aiµ , δU aiµ = 0 ,δY i = X i , δX i = 0 ,δ ¯ X abi = − ¯ Y abi , δ ¯ Y abi = 0 , (49)0with δ = 0. The transformations (49) reveal a doublet structure for the localizing Zwanziger fields and sources.Moreover, by taking into account that { s, δ } = 0, one can define a single extended nilpotent operator Q defined by Q = s + δ , Q = 0 . (50)Also, for future applications, it turns out to be useful to embed the source ˜ J into a Q -doublet by means of theintroduction of the source H , transforming as Q ˜ J = H , Q H = 0 . (51)Summarizing, the full set of extended Q -transformations are given by Q A aµ = − D abµ c b , Q c a = 12 f abc c b c c , Q ¯ c a = ib a , Q b a = 0 , Q ϕ abµ = ω abµ , Q ω abµ = 0 , Q ¯ ω abµ = ¯ ϕ abµ , Q ¯ ϕ abµ = 0 , Q ξ a = g ab ( ξ ) c b , Q A h,aµ = 0 , Q τ a = 0 , Q ¯ η a = 0 , Q η a = 0 , Q α = χ , Q χ = 0 , Q N aiµ = M aiµ , Q V aiµ = U aiµ , Q U aiµ = 0 , Q Y i = X i , Q X i = 0 , Q ¯ X abi = − ¯ Y abi , Q ¯ Y abi = 0 , Q Ω aµ = 0 , Q L a = 0 , Q K a = 0 , Q J aµ = 0 , Q J = 0 , Q ˜ J = H , Q H = 0 , Q Ξ aµ = 0 . (52)The complete classical action Σ invariant under the extended transformations (52) is, explicitly,Σ = 14 g (cid:90) d x F aµν F aµν + (cid:90) d x (cid:18) ib a ∂ µ A aµ + α b a b a − i χ ¯ c a b a + ¯ c a ∂ µ D abµ c b (cid:19) − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b − (cid:90) d x (cid:0) M aiµ D abµ ( A h ) ϕ bi + V aiµ D abµ ( A h ) ¯ ϕ bi − N aiµ D abµ ( A h ) ω bi + U aiµ D abµ ( A h )¯ ω bi + M aiµ V aiµ − N aiµ U aiµ (cid:1) + (cid:90) d x (cid:20) − Ω aµ D abµ c b + 12 L a f abc c b c c + J aµ ( A h ) aµ + K a g ab ( ξ ) c b (cid:3) + (cid:90) d x (cid:20) J ( A h ) aµ ( A h ) aµ + ˜ J (¯ ω ai ω ai − ¯ ϕ ai ϕ ai ) + H ¯ ω ai ϕ ai + θ J (cid:21) + (cid:90) d x (cid:2) − Ξ aµ D abµ ( A h ) η b + X i η a ¯ ω ai + Y i η a ¯ ϕ ai + ¯ X abi η a ω bi + ¯ Y abi η a ϕ bi (cid:3) , (53)with Q Σ = 0 . (54)As already pointed out, the action Σ is an enlarged action which reduces to the original one (34) when the sourcesattain suitable physical values, summarized below1 Fields
A b c ¯ c ξ ¯ ϕ ϕ ¯ ω ω α χ τ η ¯ η Dimension 1 2 0 2 0 1 1 1 1 0 0 2 0 2 c -ghost number 0 0 1 − − η -ghost number 0 0 0 0 0 0 0 0 0 0 0 0 1 − U ( f )-charge 0 0 0 0 0 − − L K J J M N U V ˜ J H Ξ X Y ¯ X ¯ Y Dimension 3 4 4 2 3 2 2 2 2 2 2 3 3 3 3 3 c -ghost number − − − − − η -ghost number 0 0 0 0 0 0 0 0 0 0 0 − − − − − U ( f )-charge 0 0 0 0 0 − − − − χ (cid:12)(cid:12)(cid:12) phys = Ω aµ (cid:12)(cid:12)(cid:12) phys = L a (cid:12)(cid:12)(cid:12) phys = K a (cid:12)(cid:12)(cid:12) phys = J aµ (cid:12)(cid:12)(cid:12) phys = H (cid:12)(cid:12)(cid:12) phys = 0 , Ξ aµ (cid:12)(cid:12)(cid:12) phys = X i (cid:12)(cid:12)(cid:12) phys = Y i (cid:12)(cid:12)(cid:12) phys = ¯ X abi (cid:12)(cid:12)(cid:12) phys = ¯ Y abi (cid:12)(cid:12)(cid:12) phys = 0 ,J (cid:12)(cid:12)(cid:12) phys = m , ˜ J (cid:12)(cid:12)(cid:12) phys = M ,M abµν (cid:12)(cid:12)(cid:12) phys = V abµν (cid:12)(cid:12)(cid:12) phys = γ δ ab δ µν , N abµν (cid:12)(cid:12)(cid:12) phys = U abµν (cid:12)(cid:12)(cid:12) phys = 0 , (55)so that Σ (cid:12)(cid:12)(cid:12) phys = S LCGRGZ . (56)Since the action S LCGRGZ can be seen as a particular case of the more general extended action Σ, the renormalizabilityof Σ will imply that of S LCGRGZ .Let us also notice that, thanks to the parametrization (31), it is simple to check that g ∂ Σ ∂g = − g (cid:90) d x F aµν F aµν , (57)a feature that will be exploited in the proof of the renormalizability. For the benefit of the reader, the quantumnumbers of fields, sources and parameters of the theory are collected in tables I and II. IV. WARD IDENTITIES AND ALGEBRAIC CHARACTERIZATION OF THE MOST GENERALCOUNTERTERM
The classical extended action Σ defined by eq.(53) enjoys a rich set of symmetries characterized by the followingWard identities, • Slavnov-Taylor identity S Q (Σ) = 0 , (58)2with S Q (Σ) = (cid:90) d x (cid:18) δ Σ δA aµ δ Σ δ Ω aµ + δ Σ δc a δ Σ δL a + δ Σ δξ a δ Σ δK a + ib a δ Σ δ ¯ c a + ω ai δ Σ δϕ ai + ¯ ϕ ai δ Σ δ ¯ ω ai + M aiµ δ Σ δN aiµ + U aiµ δ Σ δV aiµ + H δ Σ δ ˜ J + X i δ Σ δY i − ¯ Y abi δ Σ δ ¯ X abi (cid:19) + χ ∂ Σ ∂α . (59) • Anti-ghost equation δ Σ δ ¯ c a + ∂ µ δ Σ δ Ω aµ = i χb a . (60) • Gauge-fixing condition δ Σ δb a = i∂ µ A aµ + αb a − i χ ¯ c a . (61) • Equation of motion of the Lagrange multiplier τ a δ Σ δτ a − ∂ µ δ Σ δ J aµ = 0 . (62) • Global U ( f ) symmetry U ij Σ = 0 , (63)with U ij = (cid:90) d x (cid:32) ϕ ai δδϕ aj − ¯ ϕ aj δδ ¯ ϕ ai + ω ai δδω aj − ¯ ω aj δδ ¯ ω ai − M ajµ δδM aiµ + V aiµ δδV ajµ − N aj δδN ai + U ai δδU aj + X i δδX j + Y i δδY j − ¯ X abj δδ ¯ X abi − ¯ Y abj δδ ¯ Y abi (cid:19) . (64) • Linearly broken constraints δ Σ δ ¯ ϕ ai + ∂ µ δ Σ δM aiµ + f abc V biµ δ Σ δ J cµ = − ˜ Jϕ ai + Y i η a , (65) δ Σ δϕ ai + ∂ µ δ Σ δV aiµ − f abc ¯ ϕ bi δ Σ δτ c + f abc M biµ δ Σ δ J cµ = − ˜ J ¯ ϕ ai − H ¯ ω ai + ¯ Y bai η b , (66) δ Σ δ ¯ ω ai + ∂ µ δ Σ δN aiµ − f abc U biµ δ Σ δ J cµ = ˜ Jω ai − Hϕ ai − X i η a , (67) δ Σ δω ai + ∂ µ δ Σ δU aiµ − f abc ¯ ω bi δ Σ δτ c + f abc N biµ δ Σ δ J cµ = − ˜ J ¯ ω ai − ¯ X bai η a . (68)3 • c -ghost number and η -ghost number Ward identities (cid:90) d x (cid:18) c a δ Σ δc a − ¯ c a δ Σ δ ¯ c a + ω ai δ Σ δω ai − ¯ ω ai δ Σ δ ¯ ω ai − Ω aµ δ Σ δ Ω aµ − L a δ Σ δL a − K a δ Σ δK a + U ai δ Σ δU ai − N aiµ δ Σ δN aiµ − ˜ J ∂ Σ δ ˜ J + X i δ Σ δX i − ¯ X abi δ Σ δ ¯ X abi (cid:19) + χ ∂ Σ ∂χ = 0 , (69) (cid:90) d x (cid:18) η a δ Σ δη a − ¯ η a δ Σ δ ¯ η a − Ξ aµ δ Σ δ Ξ aµ − X i δ Σ δX i − Y i δ Σ δY i − ¯ X abi δ Σ δ ¯ X abi − ¯ Y abi δ Σ δ ¯ Y abi (cid:19) = 0 . (70) • Exact R ij symmetry R ij Σ = 0 , (71)with R ij = (cid:90) d x (cid:32) ϕ ai δδω aj − ¯ ω aj δδ ¯ ϕ ai + V aiµ δδU ajµ − N ajµ δδM aiµ + ¯ X abj δδ ¯ Y abi + Y i δδX j (cid:33) . (72) • Local ¯ η equation δ Σ δ ¯ η a + ∂ µ δ Σ δ Ξ aµ = 0 . (73) • Integrated linearly broken η equation (cid:90) d x (cid:18) δ Σ δη a + f abc ¯ η b δ Σ δτ c − f abc Ξ bµ δ Σ δ J cµ (cid:19) = (cid:90) d x (cid:0) − ¯ Y abi ϕ bi + ¯ X abi ω bi + X ¯ ω ai − Y i ¯ ϕ ai (cid:1) . (74) • Identities that mix the Zwanziger ghosts with the new ghosts W i (1) (Σ) = (cid:90) d x (cid:18) ¯ ω ai δ Σ δ ¯ η a + η a δ Σ δω ai + N aiµ δ Σ δ Ξ aµ + ˜ J δ Σ δX i (cid:19) = 0 , (75) W i (2) (Σ) = (cid:90) d x (cid:18) ¯ ϕ ai δ Σ δ ¯ η a − η a δ Σ δϕ ai + M aiµ δ Σ δ Ξ aµ − ˜ J δ Σ δY i + H δ Σ δX i (cid:19) = 0 , (76) W i (3) (Σ) = (cid:90) d x (cid:18) ϕ ai δ Σ δ ¯ η a − η a δ Σ δ ¯ ϕ ai − f abc δ Σ δ ¯ Y abi δ Σ δτ c − V aiµ δ Σ δ Ξ aµ + ˜ J δ Σ δ ¯ Y aai (cid:19) = 0 , (77) W i (4) (Σ) = (cid:90) d x (cid:18) ω ai δ Σ δ ¯ η a − η a δ Σ δ ¯ ω ai + f abc δ Σ δ ¯ X abi δ Σ δτ c + U aiµ δ Σ δ Ξ aµ + ˜ J δ Σ δ ¯ X aai + H δ Σ δ ¯ Y aai (cid:19) = 0 . (78)In order to characterize the most general invariant counterterm, which can be freely added to all orders in perturbationtheory, we follow the setup of the algebraic renormalization [62] and perturb the classical action Σ by adding anintegrated local quantity in the fields and sources, Σ CT , with dimension bounded by four and vanishing c and η -ghostnumber. We demand thus that the perturbed action, (Σ + (cid:15) Σ CT ), where (cid:15) is an expansion parameter, fulfills, to thefirst order in (cid:15) , the same Ward identities obeyed by the classical action, i.e. S Q (Σ + (cid:15) Σ CT ) = O ( (cid:15) ) , (cid:18) δδ ¯ c a + ∂ µ δδ Ω aµ (cid:19) (Σ + (cid:15) Σ CT ) − i χb a = O ( (cid:15) ) ,δδb a (Σ + (cid:15) Σ CT ) = i∂ µ A aµ + αb a − i χ ¯ c a + O ( (cid:15) ) , (cid:18) δδτ a − ∂ µ δδ J aµ (cid:19) (Σ + (cid:15) Σ CT ) = O ( (cid:15) ) ,U ij (Σ + (cid:15) Σ CT ) = O ( (cid:15) ) , (cid:18) δδ ¯ ϕ ai + ∂ µ δδM aiµ + f abc V biµ δδ J cµ (cid:19) (Σ + (cid:15) Σ CT ) = − ˜ Jϕ ai + Y i η a + O ( (cid:15) ) , (cid:18) δδϕ ai + ∂ µ δδV aiµ − f abc ¯ ϕ bi δδτ c + f abc M biµ δδ J cµ (cid:19) (Σ + (cid:15) Σ CT ) = − ˜ J ¯ ϕ ai − H ¯ ω ai + ¯ Y bai η b + O ( (cid:15) ) (cid:18) δδ ¯ ω ai + ∂ µ δδN aiµ − f abc U biµ δδ J cµ (cid:19) (Σ + (cid:15) Σ CT ) = ˜ Jω ai − Hϕ ai − X i η a + O ( (cid:15) ) , (cid:18) δδω ai + ∂ µ δδU aiµ − f abc ¯ ω bi δδτ c + f abc N biµ δδ J cµ (cid:19) (Σ + (cid:15) Σ CT ) = − ˜ J ¯ ω ai − ¯ X bai η b + O ( (cid:15) ) , R ij (Σ + (cid:15) Σ CT ) = O ( (cid:15) ) , (cid:18) δδ ¯ η a + ∂ µ δδ Ξ aµ (cid:19) (Σ + (cid:15) Σ CT ) = O ( (cid:15) ) , (cid:90) d x (cid:18) δδη a + f abc ¯ η b δδτ c − f abc Ξ bµ δδ J cµ (cid:19) (Σ + (cid:15) Σ CT ) = (cid:90) d x (cid:0) − ¯ Y abi ϕ bi + ¯ X abi ω bi + X ¯ ω ai − Y i ¯ ϕ ai (cid:1) + O ( (cid:15) ) ,W i (1 , , , (Σ + (cid:15) Σ CT ) = O ( (cid:15) ) . (79)As a consequence of the first condition of eqs.(79), i.e. S Q (Σ + (cid:15) Σ CT ) = O ( (cid:15) ), one gets B Q Σ CT = 0 , (80)where B Q is the so-called linearized nilpotent Slavnov-Taylor operator, B Q = (cid:90) d x (cid:18) δ Σ δA aµ δδ Ω aµ + δ Σ δ Ω aµ δδA aµ + δ Σ δc a δδL a + δ Σ δL a δδc a + δ Σ δξ a δδK a + δ Σ δK a δδξ a + ib a δδ ¯ c a + ω ai δδϕ ai + ¯ ϕ ai δδ ¯ ω ai + M aiµ δδN aiµ + U aiµ δδV aiµ + H δδ ˜ J + X i δδY i − ¯ Y abi δδ ¯ X abi (cid:19) + χ ∂∂α , (81)with B Q B Q = 0 . (82)Equation (80) tells us that the invariant counterterm Σ CT belongs to the cohomolgy of B Q in the space of the integratedlocal polynomials in the fields and sources with c and η -ghost number zero and bounded by dimension four. Owingto the general results on the cohomology of Yang-Mills theories, see [62], the general solution of (80) can be writtenas Σ CT = ∆ + B Q ∆ ( − , (83)with ∆ and ∆ ( − integrated local polynomials in the fields and sources of dimension bounded by four and ghostnumber zero and minus one, respectively, and B Q ∆ = 0, with ∆ (cid:54) = B Q ( . . . ). At this point one sees the usefulness ofthe extended operator Q . The auxiliary fields and sources introduced due to the restriction of the functional measureto the Gribov region are doublets with respect to Q , implying that they belong to the exact part of the cohomology5of B Q [62], i.e. they appear only in ∆ ( − . Keeping this fact in mind, the most general structure allowed for ∆ canbe written as∆ = (cid:90) d x (cid:20) c g F aµν F aµν + c ( ∂ µ ( A h ) aµ )( ∂ ν ( A h ) aν ) + c ( ∂ µ ( A h ) aν )( ∂ µ ( A h ) aν ) + c f abc ( A h ) aµ ( A h ) bν ∂ µ ( A h ) cν + λ abcd ( A h ) aµ ( A h ) bµ ( A h ) cν ( A h ) dν + ˆ J aµ O aµ ( A, ξ ) + J O ( A, ξ ) + c ( ∂ µ ¯ η a + Ξ aµ )( ∂ µ η a )+ f abc ( ∂ µ ¯ η a + Ξ aµ ) P bµ ( A, ξ ) η c + c θ J (cid:21) , (84)where ( c , c , . . . , c , λ ) are arbitrary dimensionless coefficients, while O aµ ( A, ξ ), O ( A, ξ ) and P bµ ( A, ξ ) are local expres-sions in A aµ and ξ a with ghost number zero and dimension one and two, respectively.In equation (84) we have already taken into account the fact that, due to the Ward identity (62), the variables( τ a , J aµ ) can enter the counterterm only through the combinationˆ J aµ = J aµ − ∂ µ τ a . (85)Furthermore, from (83), one gets B Q O aµ ( A, ξ ) = Q O aµ ( A, ξ ) = s O aµ ( A, ξ ) = 0 , B Q O ( A, ξ ) = Q O ( A, ξ ) = s O ( A, ξ ) = 0 , B Q P aµ ( A, ξ ) = QP aµ ( A, ξ ) = s P aµ ( A, ξ ) = 0 , (86)implying the BRST-invariance of O aµ ( A, ξ ), O ( A, ξ ) and P aµ ( A, ξ ). In [63], the general solution of eqs.(86) was workedout , yielding O aµ ( A, ξ ) = b ( A h ) aµ , (87)and O ( A, ξ ) = b A h ) aµ ( A h ) aµ , (88) P aµ ( A, ξ ) = b ( A h ) aµ , (89)with ( b , b , b ) free dimensionless parameters. As a consequence, the most general expression for ∆ after the impositionof (86) is expressed as∆ = (cid:90) d x (cid:20) c g F aµν F aµν + c ( ∂ µ ( A h ) aµ )( ∂ ν ( A h ) aν ) + c ( ∂ µ ( A h ) aν )( ∂ µ ( A h ) aν ) + c f abc ( A h ) aµ ( A h ) bν ∂ µ ( A h ) cν + λ abcd ( A h ) aµ ( A h ) bµ ( A h ) cν ( A h ) dν + b ˆ J aµ ( A h ) aµ + b J A h ) aµ ( A h ) aµ + c ( ∂ µ ¯ η a + Ξ aµ )( ∂ µ η a )+ b f abc ( ∂ µ ¯ η a + Ξ aµ )( A h ) bν η c + c θ J (cid:21) . (90)We should remark that, since the parameters ( α, χ ) were introduced as a Q -doublet, they do not enter in the non-trivial part of the cohomology of Q . As a consequence, these parameters do not appear in ∆. Although the explicit solution for P aµ ( A, ξ ) was not shown in [63], the reasoning is exactly the same as for O aµ ( A, ξ ) and O ( A, ξ ). i.e. ∆ ( − , one notices that if we set J = ˜ J = M = N = V = U = H = χ = K = J = Ξ = X = Y = ¯ X = ¯ Y = 0 (91)in (53), the resulting action is Σ LCG = 14 g (cid:90) d x F aµν F aµν + (cid:90) d x (cid:16) ib a ∂ µ A aµ + α b a b a + ¯ c a ∂ µ D abµ c b (cid:17) − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b , (92)which is the Yang-Mills gauge-fixed action in the linear covariant gauges with the addition of the following terms: − (cid:90) d x (cid:0) ¯ ϕ acµ M ab ( A h ) ϕ bcµ − ¯ ω acµ M ab ( A h ) ω bcµ (cid:1) + (cid:90) d x τ a ∂ µ ( A h ) aµ − (cid:90) d x ¯ η a M ab ( A h ) η b . (93)However, upon integration over ( ¯ ϕ, ϕ, ¯ ω, ω ) and ( τ, ¯ η, η ), the terms (93) give rise to a unity. Therefore, correlationfunctions of the original fields of the Faddeev-Popov quantization, i.e. ( A, ¯ c, c, b ), are the same as those computedwith the standard Yang-Mills action in the linear covariant gauges (27). From this observation, it follows that, in thelimit (91), the counterterm (90) should reduce to the standard one in the Faddeev-Popov action in linear covariantgauges, see also [63]. This gives c = c = c = 0 , c = b , λ abcd = 0 , (94)yielding ∆ = (cid:90) d x (cid:20) c g F aµν F aµν + b ˆ J aµ ( A h ) aµ + b J A h ) aµ ( A h ) aµ + c ( ∂ µ ¯ η a + Ξ aµ ) D abµ ( A h ) η b + c θ J (cid:21) . (95)Finally, the Ward identity (74) imposes the following constraint c = − b . (96)Let us proceed thus with the characterization of the trivial part of the cohomology of B Q , i.e. ∆ ( − . Keeping in mindthat ∆ ( − must have dimension bounded by four, be a local expression in the fields and sources and ghost numberminus one, it follows that the most general term allowed by the constraint (79) is∆ ( − = (cid:90) d x (cid:2) f ab ( ξ, α )(Ω aµ + ∂ µ ¯ c a ) A bµ + f ab ( ξ, α ) c a L b + K a f ab ( ξ, α ) ξ b + − b (cid:0) V aiµ N aiµ + V aiµ D abµ ( A h )¯ ω bi + N aiµ D abµ ( A h ) ϕ bi + ( ∂ µ ¯ ω ai ) D abµ ( A h ) ϕ bi (cid:1)(cid:3) , (97)with f ab ( ξ, α ), f ab ( ξ, α ) and f ab ( ξ, α ) arbitrary functions of ξ a and α . Invoking again the limit (91), one is able toconclude that f ab ( ξ, α ) = δ ab d , f ab ( ξ, α ) = δ ab d , (98)where ( d , d ) are free parameters which might be α -dependent. Acting with B Q on ∆ ( − one obtains Modulo the standard external BRST sources (Ω , L ) terms. B Q ∆ ( − = (cid:90) d x (cid:26) d (cid:18) δ Σ δA aµ + i∂ µ b a (cid:19) A aµ − d (Ω aµ + ∂ µ ¯ c a ) δ Σ δ Ω aµ + d (cid:18) δ Σ δL a L a + δ Σ δc a c a (cid:19) + δ Σ δξ a f ab ( ξ ) ξ b − K b δ Σ δK a (cid:18) ∂f bc ∂ξ a ξ c + f ba ( ξ ) (cid:19) + b f abc ( A h ) cµ (cid:0) U aiµ ¯ ω bi + V aiµ ¯ ϕ bi + M aiµ ϕ bi − N aiµ ω bi − ω ai ∂ µ ¯ ω bi − ϕ ai ∂ µ ¯ ϕ bi (cid:1) − b (cid:2) U aiµ N aiµ + V aiµ M aiµ + U aiµ ∂ µ ¯ ω ai + V aiµ ∂ µ ¯ ϕ ai + M aiµ ∂ µ ϕ ai − N aiµ ∂ µ ω ai + ( ∂ µ ¯ ϕ ai ) ∂ µ ϕ ai − ( ∂ µ ¯ ω ai ) ∂ µ ω ai (cid:3) + χ ∂d ∂α L a c a + χK a ∂f ab ∂α ξ b + χ ∂d ∂α (Ω aµ + ∂ µ ¯ c a ) A aµ (cid:27) . (99)Therefore, for Σ CT , we getΣ CT = ∆ + B Q ∆ ( − = (cid:90) d x (cid:20) c g F aµν F aµν + b ˆ J aµ ( A h ) aµ + b J A h ) aµ ( A h ) aµ − b ( ∂ µ ¯ η a + Ξ aµ ) D abµ ( A h ) η b + c J (cid:21) + (cid:90) d x (cid:26) d (cid:18) δ Σ δA aµ + i∂ µ b a (cid:19) A aµ − d (Ω aµ + ∂ µ ¯ c a ) δ Σ δ Ω aµ + d (cid:18) δ Σ δL a L a + δ Σ δc a c a (cid:19) + δ Σ δξ a f ab ( ξ ) ξ b − K b δ Σ δK a (cid:18) ∂f bc ∂ξ a ξ c + f ba ( ξ ) (cid:19) + b f abc ( A h ) cµ (cid:0) U aiµ ¯ ω bi + V aiµ ¯ ϕ bi + M aiµ ϕ bi − N aiµ ω bi − ω ai ∂ µ ¯ ω bi − ϕ ai ∂ µ ¯ ϕ bi (cid:1) − b (cid:2) U aiµ N aiµ + V aiµ M aiµ + U aiµ ∂ µ ¯ ω ai + V aiµ ∂ µ ¯ ϕ ai + M aiµ ∂ µ ϕ ai − N aiµ ∂ µ ω ai + ( ∂ µ ¯ ϕ ai ) ∂ µ ϕ ai − ( ∂ µ ¯ ω ai ) ∂ µ ω ai (cid:3) + χ ∂d ∂α L a c a + χK a ∂f ab ∂α ξ b + χ ∂d ∂α (Ω aµ + ∂ µ ¯ c a ) A aµ (cid:27) . (100)Having determined the most general invariant local counterterm compatible with the symmetries of the theory givenby (100), one should check the stability of the theory, namely, that (100) can be reabsorbed in the original action (53)through a redefinition of the fields, parameters and sources. Before showing this, it is important to express (100) inthe parametric form, see [63], a task which will be done in the following subsection. A. Parametric form of the counterterm
Having characterized the most general local invariant counterterm compatible with the Ward identities (79), letus proceed to prove that it can be re-absorbed in the original action Σ by means of a suitable redefinitions of fields,sources and parameters. In this case, it turns out to be useful to cast the counterterm in the parametric form, see[63]. To this end, we express the counterterm (100) as Σ CT = (cid:88) n =1 Σ CT n (101) In the following, we have implemented the following redefinitions ( c , c , d , d ) → ( a , b , a , a ). Also, since we have already exploitedthe dependence from α of the counterterm, we can set χ = 0 from now on. CT1 = a g (cid:90) d x F aµν F aµν , Σ CT2 = b (cid:90) d x J aµ A h,aµ , Σ CT3 = b (cid:90) d x (cid:0) τ a ∂ µ A h,aµ − ( ∂ µ ¯ η a + Ξ aµ ) D abµ ( A h ) η b (cid:1) , Σ CT4 = (cid:90) d x (cid:18) b J A h,aµ A h,aµ + b θ J (cid:19) , Σ CT5 = a (cid:90) d x (cid:0) − ib a ∂ µ A aµ (cid:1) , Σ CT6 = a (cid:90) d x ¯ c a ∂ µ δ Σ δ Ω aµ , Σ CT7 = (cid:90) d x (cid:18) a A aµ δ Σ δA aµ − a Ω aµ δ Σ δ Ω aµ + a L a δ Σ δL a + a c a δ Σ δc a + f ab ( ξ ) ξ b δ Σ δξ a − K a δ Σ δK a ∂f bc ∂ξ a ξ c − K b δ Σ δK a f ba ( ξ ) (cid:19) , Σ CT8 = b (cid:90) d x (cid:0) ¯ ϕ ai M ab ( A h ) ϕ bi − ¯ ω ai M ab ( A h ) ω bi + U aiµ D abµ ( A h )¯ ω bi + V ai D abµ ( A h ) ¯ ϕ bi + M aiµ D abµ ( A h ) ϕ bi − N aiµ D abµ ( A h ) ω bi + U aiµ N aiµ + V aiµ M aiµ (cid:1) . (102)One can employ eq.(57) to write Σ CT1 = − a g ∂ Σ ∂g . (103)Also, one should notice that δ Σ δ J aµ = A h,aµ ,δ Σ δτ a = ∂ µ A h,aµ ,δ Σ δ ¯ η a = ∂ µ D abµ ( A h ) η b ,δ Σ δη a = − D abµ ( A h ) ∂ µ ¯ η b − ¯ Y abi ϕ bi + ¯ X abi ω bi + X i ¯ ω ai − Y i ¯ ϕ bi − D abµ ( A h )Ξ bµ ,δ Σ δX i = η a ¯ ω ai , δ Σ δY i = η a ¯ ϕ ai , δ Σ δ ¯ X abi = η a ω bi , δ Σ δ ¯ Y abi = η a ϕ bi ,δ Σ δJ = 12 A h,aµ A h,aµ + θJ ,θ ∂ Σ ∂θ = (cid:90) d x θ J , (104)which implyΣ CT2 = b (cid:90) d x J aµ δ Σ δ J aµ , Σ CT3 = b (cid:90) d x (cid:20) τ a δ Σ δτ a + 12 (cid:18) ¯ η a δ Σ δ ¯ η a + η a δ Σ δη a + Ξ aµ δ Σ δ Ξ aµ − X i δ Σ δX i − Y i δ Σ δY i − ¯ X abi δ Σ δ ¯ X abi − ¯ Y abi δ Σ δ ¯ Y abi (cid:19)(cid:21) , Σ CT4 = b (cid:90) d x J δ Σ δJ + ( b − b ) θ ∂ Σ ∂θ . (105)Concerning the term Σ CT5 , one recognizes that it can be expressed in parametric form by taking into account that9 δ Σ δb a = i∂ µ A aµ + αb a , ∂ Σ ∂α = 12 b a b a . (106)Hence, Σ CT5 = − a (cid:90) d x b a δ Σ δb a + 2 a α ∂ Σ ∂α . (107)The term Σ CT6 can be expressed in parametric form by using the fact that δ Σ δ ¯ c a = − ∂ µ δ Σ δ Ω aµ , (108)which yields Σ CT6 = − a (cid:90) d x ¯ c a δ Σ δ ¯ c a . (109)In order to write Σ CT8 in parametric form, one should employ the relations (cid:90) d x ¯ ϕ ai δ Σ δ ¯ ϕ ai = (cid:90) d x (cid:104) ¯ ϕ ai ∂ ϕ ai − f abc A h,cµ ϕ ai ∂ µ ¯ ϕ bi − V aiµ ∂ µ ¯ ϕ ai + f abc A h,cµ V aiµ ¯ ϕ bi − ˜ J ¯ ϕ ai ϕ ai + Y i η a ¯ ϕ ai (cid:105) , (cid:90) d x ¯ ω ai δ Σ δ ¯ ω ai = (cid:90) d x (cid:104) − ¯ ω ai ∂ ω ai + f abc ¯ ω ai ∂ µ ( A h,cµ ω bi ) − ¯ ω ai ∂ µ U aiµ − f bac ¯ ω ai U biµ A h,cµ + ˜ J ¯ ω ai ω ai + H ¯ ω ai ϕ ai + X i η a ¯ ω ai (cid:3) , (cid:90) d x ϕ ai δ Σ δϕ ai = (cid:90) d x (cid:104) ϕ ai ∂ ¯ ϕ ai + f abc ϕ bi ( ∂ µ ¯ ϕ ai ) A h,cµ + ϕ ai ∂ µ M aiµ + f abc ϕ bi M aiµ A h,cµ − ˜ J ¯ ϕ ai ϕ ai + H ¯ ω ai ϕ ai + ¯ Y abi η a ϕ bi (cid:3) , (cid:90) d x ω ai δ Σ δω ai = (cid:90) d x (cid:104) ω ai ∂ ¯ ω ai + f abc ω bi ( ∂ µ ¯ ω ai ) A h,cµ + ω ai ∂ µ N aiµ + f abc ω bi N aiµ A h,cµ − ˜ Jω ai ¯ ω ai + ¯ X abi η a ω bi (cid:105) , (cid:90) d x M aiµ δ Σ δM aiµ = (cid:90) d x (cid:2) − M aiµ D abµ ( A h ) ϕ bi − M aiµ V aiµ (cid:3) , (cid:90) d x V aiµ δ Σ δV aiµ = (cid:90) d x (cid:2) − V aiµ D abµ ( A h ) ¯ ϕ bi − V aiµ M aiµ (cid:3) , (cid:90) d x N aiµ δ Σ δN aiµ = (cid:90) d x (cid:2) N aiµ D abµ ( A h ) ω bi + N aiµ U aiµ (cid:3) , (cid:90) d x U aiµ δ Σ δU aiµ = (cid:90) d x (cid:2) − U aiµ D abµ ( A h )¯ ω bi − U aiµ N aiµ (cid:3) , (110)and δ Σ δ ˜ J = ¯ ω ai ω ai − ¯ ϕ ai ϕ ai , δ Σ δH = ¯ ω ai ϕ ai . (111)This entails thatΣ CT8 = − b (cid:90) d x (cid:20) ¯ ϕ ai δ Σ δ ¯ ϕ ai + ϕ ai δ Σ δϕ ai + ¯ ω ai δ Σ δ ¯ ω ai + ω ai δ Σ δω ai + M aiµ δ Σ δM aiµ + V aiµ δ Σ δV aiµ + N aiµ δ Σ δN aiµ + U ai δ Σ δU aiµ + 2 ˜ J δ Σ δ ˜ J + 2 H δ Σ δH − X i δ Σ δX i − Y i δ Σ δY i − ¯ X abi δ Σ δ ¯ X abi − ¯ Y abi δ Σ δ ¯ Y abi (cid:21) . (112)0Consequently, the expression of the counterterm Σ CT in parametric form isΣ CT = − a g ∂ Σ ∂g + b (cid:90) d x J aµ δ Σ δ J aµ + b (cid:90) d x (cid:20) τ a δ Σ δτ a + 12 (cid:18) ¯ η a δ Σ δ ¯ η a + η a δ Σ δη a + Ξ aµ δ Σ δ Ξ aµ (cid:19)(cid:21) + b (cid:90) d x J δ Σ δJ + ( b − b ) θ ∂ Σ ∂θ − a (cid:90) d x b a δ Σ δb a + 2 a α ∂ Σ ∂α − a (cid:90) d x ¯ c a δ Σ δ ¯ c a + (cid:90) d x (cid:20) a A aµ δ Σ δA aµ − a Ω aµ δ Σ δ Ω aµ + a L a δ Σ δL a + a c a δ Σ δc a + f ab ( ξ ) ξ b δ Σ δξ a − K a δ Σ δK a (cid:18) ∂f bc ∂ξ a ξ c + f ba ( ξ ) (cid:19)(cid:21) − b (cid:90) d x (cid:20) ¯ ϕ ai δ Σ δ ¯ ϕ ai + ϕ ai δ Σ δϕ ai + ¯ ω ai δ Σ δ ¯ ω ai + ω ai δ Σ δω ai + M aiµ δ Σ δM aiµ + V aiµ δ Σ δV aiµ + N aiµ δ Σ δN aiµ + U ai δ Σ δU aiµ + 2 ˜ J δ Σ δ ˜ J + 2 H δ Σ δH (cid:21) , (113)which can be immediately rewritten as Σ CT = R Σ , (114)where R stands for the operator R = − a g ∂∂g + b (cid:90) d x J aµ δδ J aµ + b (cid:90) d x (cid:20) τ a δδτ a + 12 (cid:18) ¯ η a δδ ¯ η a + η a δδη a + Ξ aµ δδ Ξ aµ (cid:19)(cid:21) + b (cid:90) d x J δδJ + ( b − b ) θ ∂∂θ − a (cid:90) d x b a δδb a + 2 a α ∂∂α − a (cid:90) d x ¯ c a δδ ¯ c a + (cid:90) d x (cid:20) a A aµ δδA aµ − a Ω aµ δδ Ω aµ + a L a δδL a + a c a δδc a + f ab ( ξ ) ξ b δδξ a − K a (cid:18) ∂f bc ∂ξ a ξ c + f ba ( ξ ) (cid:19) δδK a (cid:21) − b (cid:90) d x (cid:20) ¯ ϕ ai δδ ¯ ϕ ai + ϕ ai δδϕ ai + ¯ ω ai δδ ¯ ω ai + ω ai δδω ai + M aiµ δδM aiµ + V aiµ δδV aiµ + N aiµ δδN aiµ + U ai δδU aiµ + 2 ˜ J δδ ˜ J + 2 H δδH (cid:21) . (115)Expression (114) turns out to be quite useful for the analysis of the satbility of the starting action Σ, as addressed inthe next subsection. B. Stability of the action Σ In order to end the algebraic proof of the renormalizability of (53), one should prove that the counterterm (114)can be re-absorbed into the initial action Σ through a suitable redefinition of fields, sources and parameters. Thedetermination of those redefinitions is made very easy once one knows the counterterm written in its parametricform, as in (113). If the counterterm (113) can be re-abosrbed in the starting action, then, to the first order in theparameter expansion (cid:15) , the following relation should hold [62], i.e.
Σ[Φ ] = Σ[Φ] + (cid:15) Σ CT [Φ] + O ( (cid:15) ) , (116)where Φ stands for all fields, sources and parameters of the theory. From (114), one concludes thatΣ[Φ ] = Σ[Φ] + (cid:15) R Σ[Φ] + O ( (cid:15) ) , (117)and due to the form of R , it is easy to see that Φ = (1 + (cid:15) R )Φ . (118)The fields, sources and parameters are redefined thus according to1 A = Z / A A , b = Z / b b , c = Z / c c , ¯ c = Z / c ¯ c ,ξ a = Z abξ ( ξ ) ξ b , τ = Z / τ τ , η = Z / η η , ¯ η = Z / η ¯ η , ¯ ϕ = Z / ϕ ¯ ϕ , ϕ = Z / ϕ ϕ , ¯ ω = Z / ω ¯ ω , ω = Z / ω ω , Ω = Z Ω Ω , L = Z L L , K a = Z abK ( ξ ) K b , J = Z J J ,J = Z J J , ˜ J = Z ˜ J ˜ J , H = Z H H , g = Z g g ,α = Z α α , θ = Z θ θ , M = Z M M , V = Z V V ,N = Z N N , U = Z U U , Ξ = Z Ξ Ξ , X = Z X X ,Y = Z Y Y , ¯ X = Z ¯ X ¯ X , ¯ Y = Z ¯ Y ¯ Y . (119)with Z / A = 1 + (cid:15)a , Z / c = 1 + (cid:15)a , Z g = 1 − (cid:15) a , Z / τ = 1 + (cid:15)b , Z J = 1 + (cid:15)b Z θ = 1 + (cid:15) ( b − b ) , Z abξ ( ξ ) = δ ab + (cid:15)f ab ( ξ ) , Z abK ( ξ ) = δ ab − (cid:15) (cid:18) f ba ( ξ ) + ∂f bc ∂ξ a ξ c (cid:19) . (120)For the other fields, sources and parameters, the following relations hold Z / A = Z − = Z − / c = Z − / b = Z / α ,Z / τ = Z ¯ η = Z η = Z = Z J ,Z / c = Z L , Z X = Z Y = Z ¯ X = Z ¯ Y = 1 , (121)while for those fields and sources introduced to implement the Gribov horizon one has Z − / τ = Z / ϕ = Z / ϕ = Z / ω = Z / ω = Z M = Z V = Z N = Z U = Z / J = Z / H . (122)We see thus that, under an appropriate redefinition of fields, sources and parameters as described in eq.(120), (121)and (122), the most general local invariant counterterm compatible with the Ward identities can be re-absorbed in theclassical action (53). Hence, by the algebraic renormalization framework [62], the theory is renormalizable at all ordersin perturbation theory. It is important to emphasize that, due to the fact that the Stueckelberg field is dimensionless,the associated renormalization factors, ( Z abξ ( ξ ) , Z abK ( ξ )) are nonlinear in ξ a , a feature typical of dimensionless fields,see also [63].An interesting reamark is that the renormalization factor of the Gribov parameter γ is not an independent quantityof the theory, being expressed in terms of other renormalization factors. In fact, taking the physical limit of thesources, see (39) and (55), it turns out that Z γ = Z − / J = Z / J Z − / τ = 1 − (cid:15) b . (123)In addition, we also have Z J Z / ϕ Z / ϕ = 1 , Z J Z / ω Z / ω = 1 , (124)which express the nonrenormalization properties of the vertices ( A h φ ¯ φ ) and ( A h ω ¯ ω ), already noticed in [54] wherethe study of the renormalizability of the Refined Gribov-Zwanziger action in the linear covariant gauges in theapproximation A h ≈ A T , with A T the transverse component of the gauge field, was studied.In conclusion, the action (53) takes into account the existence of infinitesimal Gribov copies in the linear covari-ant gauges in a local, BRST invariant and renormalizable way.2 V. CONCLUSIONS
In the Landau gauge, dealing with Gribov copies has brought non-trivial infrared effects which might be relatedto the confinement of gluons, see [5, 68–70], a fact that has raised the investigation of such effects in other gauges.In particular, the issue of the Gribov copies in the linear covariant gauges has been object of intense research in thelast years, see [46, 49–55]. In particular, a local and BRST invariant action which takes into account the existenceof Gribov copies in the linear covariant gauges was proposed in [49], within the Gribov-Zwanziger framework. In thepresent work, we have been pursuing the study of the action constructed in [49], by proving its renormalizability to allorders in perturbation theory. This provides a consistent framework in order to perform loop computations, a subjectwhich is under current investigation. The present work can be naturally extended to the study of the renormalizabilityof the Refined-Gribov-Zwanziger action in the presence of non-perturbative matter coupling, as devised in [32]. Thecases of the the maximal Abelian and Curci-Ferrari gauges, see [71, 72], can be addressed as well.
ACKNOWLEDGEMENTS
The authors are thankful to U. Reinosa, J. Serreau, M. Tissier and N. Wschebor for discussions. The ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq-Brazil) and The Coordena¸c˜ao de Aperfei¸coamento dePessoal de N´ıvel Superior (CAPES) are acknowledged for support.
Appendix A: Remarks on the localization of the BRST-invariant RGZ action
In this appendix, we explicitly show that the BRST invariant local formulation of the Refined Gribov-Zwanzigeraction (20) in terms of A hµ in the Landau gauge is equivalent to the original construction presented in [11]. We beginwith the BRST invariant Refined Gribov-Zwanziger action in the Landau gauge expressed as S L RGZ = S YM + (cid:90) d x (cid:0) ib a ∂ µ A aµ + ¯ c a ∂ µ D abµ c b (cid:1) + (cid:90) d x (cid:2) ϕ acµ ∂ ν D abν ( A h ) ϕ bcµ − ¯ ω acµ ∂ ν D abν ( A h ) ω bcµ − γ f abc A h,aµ ( ϕ + ¯ ϕ ) bcµ (cid:3) + m (cid:90) d x A h,aµ A h,aµ − M (cid:90) d x (cid:0) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:1) (cid:90) d x τ a ∂ µ A h,aµ + (cid:90) d x ¯ η a ∂ µ D abµ ( A h ) η b . (A1)The partition function is written as Z = (cid:90) [ D Φ] e − S L RGZ , (A2)where Φ = { A, b, ¯ c, c, ¯ ϕ, ϕ, ¯ ω, ω, ξ, τ, ¯ η, η } . Integrating out the fields ( b, τ, ¯ η, η ) one obtains Z = (cid:90) (cid:104) D ˜Φ (cid:105) δ (cid:0) ∂ µ A aµ (cid:1) δ (cid:0) ∂ µ A h,aµ (cid:1) det (cid:0) − ∂ µ D abµ ( A h ) (cid:1) e − (cid:82) d x ( ... ) , (A3)with ( . . . ) a shorthand notation for the remaining terms in the action (A1) and ˜Φ = { A, ¯ c, c, ¯ ϕ, ϕ, ¯ ω, ω, ξ } . In order todeal with the delta function δ (cid:0) ∂ µ A h,aµ (cid:1) imposing the transversality condition ∂ µ A h,aµ = 0, we make use of δ ( f ( x )) = δ ( x − x ) | f (cid:48) ( x ) | , (A4)with f ( x ) = 0. Of course, this relation holds if f (cid:48) ( x ) exists and is non-vanishing. It is possible to construct aniterative solution for ∂ µ A h,aµ = 0, as described in Appendix A of [46]. Such a solution ξ is expressed as ξ = 1 ∂ ∂ µ A µ + ig∂ (cid:20) ∂ µ A µ , ∂ ν A ν ∂ (cid:21) + ig∂ (cid:20) A µ , ∂ µ ∂ ν A ν ∂ (cid:21) + ig ∂ (cid:20) ∂ µ A µ ∂ , ∂ ν A ν (cid:21) + O ( A ) , (A5)where we have employed the matrix notation of Appendix A of [46]. The important feature of (A5) is that all termsalways contain the divergence of the gauge field, i.e. ∂ µ A aµ .3Hence, the analogue of (A4) is δ ( ∂ µ A h,aµ ) = δ ( ξ − ξ )det (cid:0) − ∂ µ D abµ ( A h ) (cid:1) , (A6)where, due to the restriction of the domain of integration in the functional integral to the Gribov region, we havetaken into account that det (cid:0) − ∂ µ D abµ ( A h ) (cid:1) >
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