On Gauge Independence for Gauge Models with Soft Breaking of BRST Symmetry
aa r X i v : . [ h e p - t h ] O c t On Gauge Independence for Gauge Modelswith Soft Breaking of BRST Symmetry
Alexander Reshetnyak † , ∗ † Institute of Strength Physics and Material ScienceSiberian Branch of Russian Academy of Sciences,Akademicheskii av. 2/4, 634021 Tomsk, Russia ∗ Tomsk State Pedagogical University,Kievskaya St. 60, 634061 Tomsk, Russia
Abstract
A consistent quantum treatment of general gauge theories with an arbitrary gauge-fixingin the presence of soft breaking of the BRST symmetry in the field-antifield formalism isdeveloped. It is based on a gauged (involving a field-dependent parameter) version of finiteBRST transformations. The prescription allows one to restore the gauge-independence ofthe effective action at its extremals and therefore also that of the conventional S -matrixfor a theory with BRST-breaking terms being additively introduced into a BRST-invariantaction in order to achieve a consistency of the functional integral. We demonstrate theapplicability of this prescription within the approach of functional renormalization groupto the Yang–Mills and gravity theories. The Gribov–Zwanziger action and the refinedGribov–Zwanziger action for a many-parameter family of gauges, including the Coulomb,axial and covariant gauges, are derived perturbatively on the basis of finite gauged BRSTtransformations starting from Landau gauge. It is proved that gauge theories with softbreaking of BRST symmetry can be made consistent if the transformed BRST-breakingterms satisfy the same soft BRST symmetry breaking condition in the resulting gauge asthe untransformed ones in the initial gauge, and also without this requirement. Email: [email protected]
Keywords:
Gauge theories, field-dependent BRST transformations, Gribov–Zwanziger theory,BRST symmetry, BV quantization, Functional renormalization group
PACS:
Introduction
The contemporary progress in high-energy physics and quantum field theory is stronglyconnected with the non-pertubative features of quantum theories. The electroweak and stronginteractions are described by the Standard Model, in which Quantum Chromodynamics (QCD)is a constituent, and there are no experimental facts in conflict with QCD. While the StandardModel has been justified by the discovery of the Higgs boson, the problem of consistency in QCDis far from its solution, especially in view of the confinement phenomenon. The Lagrangian ofQCD (and generally that of the Standard Model) belongs to the class of non-Abelian gaugetheories [1], [2], [3]. It is well known that the BRST symmetry [4], being a special globalfermionic descendant of gauge invariance, plays a fundamental role in quantum field theory,since the fundamental interactions of Nature, together with gravity and perhaps some yetunknown forces, can be described in terms of gauge theories. The covariant quantization ofYang–Mills theories by means of the Faddeev–Popov procedure cannot be realized correctly,even perturbatively, for the entire spectrum of the momenta distribution due to the well-knownGribov problem [5] in the deep infra-red region for gauge fields, once a gauge condition hasbeen imposed differentially [6], since there remains an infinitely large number of discrete gaugecopies even after gauge-fixing. In order to fix the residual gauge freedom, Gribov has undertaken a detailed study of theCoulomb gauge and suggested a restriction of the domain of functional integration for gaugefields to the so-called first Gribov region, which has been effectively incorporated into thefunctional measure as the Heaviside Θ-function, thus realizing the “no-pole” condition for theghost propagator. Effectively, this restriction can be implemented, in the Landau gauge with ahermitian Faddeev–Popov operator, by a special addition introduced to the standard Faddeev–Popov action and known as the Gribov–Zwanziger functional [9], [10]. However, this additionis not gauge-invariant and is therefore non-invariant under the original BRST transformations.The idea of using the Zwanziger action in order to take account of gauge field configurationshas introduced to the path integral of the Yang–Mills theory the entire spectrum of frequencies,which has been examined in a number of papers [11] based on the breakdown of BRST symmetryin Yang–Mills theories. Notice that until now the considerations [9], [10], [11] of the Gribovhorizon in Yang–Mills theories have been carried out basically in the Landau gauge. Theanalytical proof [5] of the presence of Gribov copies in the physical spectrum has been confirmedby lattice simulations in some QCD models, such as the SU (2) gluodynamics (see, e.g., [12], [13]and references therein), which is an expected result due to the discovery of field configurations There are some other recently suggested methods of solving the Gribov problem in a consistent way: first,the procedure of imposing an algebraic (instead of a differential one) gauge on auxiliary scalar fields in a theorywhich is non-perturbatively equivalent to the Yang-Mills theory with the same gauge group [7]; second, theprocedure of averaging over the Gribov copies with a non-uniform weight in the functional integral and thereplica trick [8]. R ξ gauges for a small value of the gauge parameter ξ for anapproximation of the quantum action being quadratic in the fields; let us also notice the proposalof a new form of the horizon functional in R ξ gauges [15] in the maximal Abelian gauges [16],[17], in the Coulomb gauge [18], and on a curved Riemannian background to study the influenceof the curvature tensor on changing the size of the Gribov region [19].There is a large freedom in the choice of admissible gauges used to obtain a correct pathintegral in Yang–Mills theories with account of the Gribov problem; it is also well known that theGreen functions are gauge-dependent; however, this dependence has such a specific characterthat it should be cancelled in physical combinations such as the S-matrix. Contemporaryproofs of the gauge-independence of the S-matrix in Yang–Mills theories are based on theBRST symmetry, see, e.g., [20], and also apply to more general gauge theories [21]. Therearises an immediate problem of consistency for a gauge theory in case the BRST invariance ofthe resulting quantum action (such as the Gribov–Zwanziger action) turns out to be broken.The study of this problem has been initiated by [22], [23]. These studies investigated bothYang–Mills and general gauge theories, such as supergravity, superstrings with open algebras,and higher-spin fields as reducible gauge theories, see, e.g., [24], with an introduction of so-called soft breaking of the BRST symmetry (under the natural assumption of the existence ofthe Gribov horizon and Gribov–Zwanziger functional for any theory with a non-Abelian gaugealgebra) and achieved their results on a basis of the field-antifield method [25], [26]. Namely, in[22], [23], it was shown (with some peculiar features studied in [27]) that the gauge-independenceof the effective action for a gauge theory with soft BRST symmetry breaking on the mass shellrequires the fulfillment of a quite strong condition for the BRST symmetry breaking term, andtherefore we come to the conclusion [22]: “It is argued that gauge theories with a soft breakingof BRST symmetry are inconsistent.” The same statement has been shown to take place in theGribov–Zwanziger theory with the R ξ -gauge.As a next step in solving the problem of determining the horizon functional for the Yang–Mills theory in gauges beyond the Landau gauge, there is the recent concept of so-called finitefield-dependent BRST transformations [28], earlier used in the infinitesimal form [25], [26]in order to establish the gauge-independence of the vacuum functional, but now explicitlyconstructed to relate the Faddeev–Popov action in the Landau gauge and in the covariant R ξ -gauge. The concept of field-dependent BRST transformations, first suggested [29] in a finite(however, different) form (see also [30], [31]) permits one to obtain perturbatively an explicitform of the Gribov horizon functional in the R ξ -gauge [32], starting from its form in the Landaugauge. This provides a different perspective of the problem of gauge-dependence for the Gribov–Zwanziger theory and allows one to revisit this problem more generally in a gauge theory withsoft BRST breaking symmetry. 2n this work, we present a study of gauge-dependence in general gauge theories with softbreaking of the BRST breaking and develop, for this purpose, a concept of gauged (equivalently,field-antifield-dependent) BRST transformations. We present an explicit calculation of theJacobian of the corresponding change of field-antifield variables in the partition function, todetermine and solve a non-linear equation for an unknown field-antifield-dependent odd-valuedparameter Λ. We establish a coincidence of the vacuum functional without a BRST broken termin a gauge determined by a gauge Fermion Ψ with the vacuum functional in a different gaugedetermined by Ψ + ∆Ψ. On this basis, we examine the properties of the average effective actionwithin the approach of the functional renormalization group to the Yang–Mills and gravitytheories. We also suggest the Gribov–Zwanziger horizon functional for a many-parameter familyof linear gauges, including the Coulomb, the axial, and the R ξ gauges, used in non-HermitianFaddeev–Popov operators.The paper is organized as follows. In Section 2, we introduce the concept of finite gauged(field-dependent) BRST symmetry transformations and investigate the related change of vari-ables in the functional integral for general gauge theories in the field-antifield formalism. InSection 3, we use the field-dependent BRST transformations to formulate the study of gauge-dependence for the generating functionals of Green’s functions for a general gauge theory withBRST-broken terms in arbitrary gauges (using a suitable regularization scheme); we also for-mulate the main result of this study. An application of the general results to the functionalrenormalization group approach to the Yang–Mills and gravity theories is considered in Sec-tion 4. In Section 5, we examine different choices for gauged BRST transformations in order tofind the form of the Gribov–Zwanziger action and of the refined Gribov–Zwanziger action in amany-parameter family of gauges including the Coulomb, axial, Landau and covariant gauges,starting from the Landau gauge. Finally, in Section 6 we discuss some issues and perspectivesrelated to the suggested procedure. In Appendix A, we analyze the existence of a solution to anon-linear functional equation for an unknown field-antifield-dependent odd-valued parameterΛ, which establishes the coincidence of the vacuum functionals in different gauges.We use the condensed notation of DeWitt [33] and our previous notation [22], [23], [15].Derivatives with respect to sources and antifields are taken from the left, while those withrespect to fields are taken from the right. Left derivatives with respect to fields are labelled bythe subscript “ l ”. The Grassmann parity of a quantity A is denoted by ε ( A ). In this section, we recall the basic notions and properties of the field-antifield formalism forgeneral gauge theories. We also introduce gauged (field-dependent) BRST transformations andcalculate the Jacobian for the change of variables determined by these transformations.3 .1 Overview of Field-antifield Formalism
As the initial point of our study, we consider a theory of gauge fields, A i , i = 1 , , . . . , n ,with ε ( A i ) = ε i , determined by a classical action, S = S ( A ), invariant under infinitesimalgauge transformations δA i = R iα ( A ) ξ α for α = 1 , , . . . , m , implying the Noether identities S ,i ( A ) R iα ( A ) = 0 , < m < n. (2.1)Here, the gauge transformations are parameterized by m arbitrary (usually supposed to besmall) functions, ξ α , of the space-time coordinates, with ε ( ξ α ) = ε α , whereas S ,i ≡ δS /δA i ,while R iα ( A ) are the generators of the gauge transformations, with ε ( R iα ) = ε i + ε α .The generators may be dependent in the case rank k R iα k S ,i =0 = m < m , implying thepresence of zero eigenvectors, Z α α ( A ), α = 1 , ..., m , for the generators on the mass shell S ,i = 0, thus determining a reducible gauge theory, so that in the case rank k Z α α k S ,i =0 < m the eigenvectors should be dependent as well. Thus, an L -th-stage reducible gauge theory ofthe fields A i is determined by the relations Z α s − α s ( A ) Z α s α s +1 ( A ) = S ,i ( A ) K iα s − α s +1 ( A ) , for α s +1 = 1 , ..., m s +1 , s = 0 , ..., L − , (2.2) and rank k Z α s − α s k S ,i =0 < m s , rank k Z α L − α L k S ,i =0 = m L , (2.3) where Z α − α ≡ R iα , ε ( Z α s α s +1 ) = ε α s + ε α s +1 , ε ( K iα s − α s +1 ) = ε i + ε α s − + ε α s +1 . (2.4)The total configuration space M of all the fields { Φ A } in the BV method depends on theirreducible [25] or reducible [26] nature of a given classical gauge theory. In the case of an L -thstage reducible theory, M is parameterized by the fieldsΦ ≡ { Φ A } = { A i , C α s , C α s s ′ , B α s s ′ } , s = 0 , ..., L, s ′ = 0 , ..., s, (2.5)with ε ( C α s , C α s s ′ , B α s s ′ ) = ( ε α s + s + 1 , ε α s + s + 1 , ε α s + s ), ε (Φ A ) = ε A and the following ghostnumber distribution: gh (cid:0) A i , C α s , C α s s ′ , B α s s ′ (cid:1) = (cid:0) , s + 1 , s ′ − s − , s ′ − s (cid:1) , which obeys an additive composition law when calculated on monomials. Here, the respectiveclassical, minimal-ghost, antighost, extra-ghost and Nakanishi–Lautrup fields are explicitlyindicated in the BV method. For L = 0, the gauge theory is irreducible, with C α , C α ≡ C α , B α ≡ B α being the ghost, antighost and Nakanishi–Lautrup fields.The BV method demands the introduction of an odd cotangent bundle Π T ∗ M ≡ T ∗ (0 , M ,usually known as the field-antifield space (for a more involved geometry, based on the field-antifield formalism, see, e.g., Refs. [34], [35], [36], [37], [38], [39], [40]), where each field Φ A in M has a corresponding antifield Φ ∗ ≡ Φ ∗ A , { Φ ∗ A } = { A ∗ i , C ∗ α s , C ∗ s ′ α s , B ∗ s ′ α s } , with ( ε, gh )(Φ ∗ A ) = ( ε A +1 , − − gh (Φ A )) . (2.6)4n the total field-antifield space { Φ A , Φ ∗ A } , one defines a bosonic functional, ¯ S = ¯ S (Φ , Φ ∗ ),being a special extension of the classical action to Π T ∗ M with the boundary condition ofvanishing antifields Φ ∗ A and Planck constant, ¯ S (Φ , (cid:12)(cid:12) ~ =0 = S , with gh ( S ) = 0, encoding thegauge algebra functions and satisfying a quantum master equation (within the class of gauge-invariant regularizations, with ∆ ¯ S ∼ δ (0) = 0 for a local ¯ S ) in two equivalent forms:∆ exp (cid:26) i ~ ¯ S (cid:27) = 0 ⇐⇒ ( ¯ S, ¯ S ) = i ~ ∆ ¯ S. (2.7)These equations are written in terms of a natural (in Π T ∗ M ) odd Poisson bracket, ( • , • ),(known as the antibracket) and a nilpotent odd Laplacian, ∆,( • , • ) = δ • δ Φ A δ • δ Φ ∗ A − δ r • δ Φ ∗ A δ l • δ Φ A , ∆ = ( − ε A δ l δ Φ A δδ Φ ∗ A . (2.8)We assume that formal manipulations with ∆ are supported by a suitable regularization scheme.This is a nontrivial requirement, since ∆ is not well-defined on local functionals, because forany local functional F one finds that ∆ F ∼ δ (0). The standard way to solve this problem is touse a regularization similar to the dimensional one [42], when δ (0) = 0. In this paper, just asin [23], we consider a more general class of regularizations.The quantum action is constructed as a special representative from the set of solutions tothe master equation (2.7) and is described by the transformationexp (cid:26) i ~ S X (cid:27) = exp {− [∆ , X ] } exp (cid:26) i ~ ¯ S (cid:27) , for ε ( X ) = 1 , gh ( X ) = − , (2.9)with the supercommutator [ , ] and some functional X = X (Φ , Φ ∗ ), whose form controls thechoice of a Lagrangian surface in Π T ∗ M , on which the restriction of the Hessian for S X shouldbe non-degenerate. Choosing X = Ψ(Φ) as the gauge fermion (e.g., Ψ(Φ) = C α χ α ( A, B ) forirreducible theories with an admissible gauge χ α ( A, B ) = 0), one makes the quantum action S Ψ non-degenerate in the configuration space M ,exp (cid:26) i ~ S Ψ (cid:27) = exp (cid:26) δ Ψ δ Φ A δδ Φ ∗ A (cid:27) exp (cid:26) i ~ ¯ S (cid:27) ⇐⇒ S Ψ (Φ , Φ ∗ ) = ¯ S (cid:0) Φ , Φ ∗ + δ Ψ δ Φ (cid:1) . (2.10)By construction, the action S Ψ satisfies the master equation (2.7) due to the supercommu-tativity of the operators exp {− [∆ , Ψ] } and ∆, namely,[∆ , exp {− [∆ , Ψ] } ] = 0 = ⇒ ∆ exp (cid:26) i ~ S Ψ (cid:27) = 0 , (2.11)and is used to construct the path integral and the generating functionals of Green’s functionsin the field-antifield formalism [25], [26]. The generating functionals of the usual, Z = Z ( J, Φ ∗ ), There is another proposal [41] to define an odd Laplacian, thus solving the problem of δ (0). W = W ( J, Φ ∗ ), Green functions extended by external [those which do not enterthe integration measure in (2.12)] antifields in the BV formalism [25], [26] can be presented asexp (cid:26) i ~ W (cid:27) = Z = Z D Φ exp n i ~ (cid:0) S Ψ (Φ , Φ ∗ ) + J A Φ A (cid:1)o , (2.12)with sources J A ( ε ( J A ) = ε A ), whereas the effective action Γ = Γ(Φ , Φ ∗ ) is determined by theLegendre transformation of W with respect to J A ,Γ(Φ , Φ ∗ ) = W ( J, Φ ∗ ) − J A Φ A , with Φ A = δWδJ A , δ Γ δ Φ A = − J A . (2.13)The standard properties of the above generating functionals are inherited from the gauge in-variance of the classical action, transformed into the BRST invariance, being an invarianceunder global N = 1 supersymmetry transformations in the extended configuration space M , δ µ Φ A = (Φ A , S Ψ ) µ, δ µ Φ ∗ A = 0 , (2.14)with a constant anticommuting parameter µ .First, the integrand in Eq. (2.12) for Z Ψ ≡ Z (0 , Φ ∗ ) is invariant with respect to the trans-formations (2.14).Second, the vacuum functional Z Ψ is independent with respect to a variation of the gaugecondition, Ψ → Ψ + δ Ψ, if one makes in Z Ψ+ δ Ψ the change of variablesΦ A → Φ ′ A = Φ A + (Φ A , S Ψ ) µ (Φ) , with Φ ′∗ A = Φ ∗ A , (2.15)referred to as field-dependent (i.e., gauged) BRST transformations, now with an arbitraryanticommuting µ (Φ), µ (Φ) = 0, being, however, infinitesimal, µ (Φ) = i ~ δ Ψ. Indeed, in thiscase we have Z Ψ+ δ Ψ = Z Ψ + o ( δ Ψ).The next consequence of the transformations (2.14), based on the equivalence theorem [43],is the presence of the Ward identities for
Z, W,
Γ, namely, J A δZδ Φ ∗ A = 0 , J A δWδ Φ ∗ A = 0 , (Γ , Γ) = 0 . (2.16)Finally, the study of gauge dependence for the generating functionals of Green’s functions Z, W,
Γ leads to the following variations [21, 22, 23] under the change of the gauge conditionΨ → Ψ + δ Ψ: δZ ( J, Φ ∗ ) = i ~ J A δδ Φ ∗ A δ Ψ (cid:0) ~ i δδJ (cid:1) Z ( J, Φ ∗ ) , (2.17) δW ( J, Φ ∗ ) = J A δδ Φ ∗ A δ Ψ (cid:0) δWδJ + ~ i δδJ (cid:1) (2.18) δ Γ(Φ , Φ ∗ ) = − (Γ , h δ Ψ i ) for h δ Ψ i = δ Ψ( b Φ) · , b Φ A = Φ A + i ~ (Γ ′′ − ) AB δ l δ Φ B , (2.19) Despite the term “gauged”, the parameter µ (Φ) should be considered as an odd-valued functional, i.e., notas an arbitrary space-time function, such as the gauge parameter ξ α . ′′ − ) being reciprocal to the Hessian (Γ ′′ ), with the elements(Γ ′′ ) AB = δ l δ Φ A (cid:16) δ Γ δ Φ B (cid:17) : (Γ ′′ − ) AC (Γ ′′ ) CB = δ AB . (2.20)The above local representation for δ Γ can be rewritten with the use of differential conse-quences of the Legendre transformation (2.13) in a non-local form: δ Γ = δ Γ δ Φ A h − δδ Φ ∗ A + ( − ε B ( ε A +1) (Γ ′′ − ) BC (cid:16) δ l δ Φ C δ Γ δ Φ ∗ A (cid:17) δ l δ Φ B i h δ Ψ i . (2.21)Indeed, in order to derive (2.17) we make the change of variables (2.15) with µ (Φ) = − i ~ δ Ψ inthe functional Z ( J, Φ ∗ ) ≡ Z Ψ ( J, Φ ∗ ), constructed with respect to the action S Ψ , then, extractingthe functional Z Ψ+ δ Ψ ( J, Φ ∗ ), we obtain, with accuracy up to the first order in δ Ψ, Z Ψ ( J, Φ ∗ ) = Z D Φ ′ exp n i ~ (cid:0) S Ψ (Φ ′ , Φ ′∗ ) + J A Φ ′ A (cid:1)o = Z D Φ exp n i ~ (cid:0) S Ψ+ δ Ψ (Φ , Φ ∗ ) + J A Φ A (cid:1)o(cid:16) − (cid:16) i ~ (cid:17) J A δS Ψ δ Φ ∗ A δ Ψ(Φ) (cid:17) = Z Ψ+ δ Ψ ( J, Φ ∗ ) − i ~ J A δδ Φ ∗ A δ Ψ (cid:0) ~ i δδJ (cid:1) Z Ψ ( J, Φ ∗ ) , (2.22)where the final line has been derived using the differentiation of the functional integral withrespect to the sources J and external antifields Φ ∗ .From the final variations of Z, W,
Γ, we can see, on the extremals, that for
Z, W with J A = 0 and equivalently for Γ with δ Γ δ Φ A = 0, the corresponding variations given by Eqs.(2.17), (2.18) and (2.21) are vanishing. This result for Z, W is identical with that for thevacuum functionals Z Ψ , W Ψ = ~ i ln Z Ψ . The same results are valid for renormalizable generatingfunctionals Z R , W R , Γ R with an appropriate gauge-invariant regularization respecting the Wardidentities (2.16) and their differential consequences.Due to Gribov’s [5] and, in general, Singer’s [6] results, we notice that the above-listedBV quantization rules correctly describe physics within the functional integral technique in theperturbative way only for Abelian gauge theories in any gauges and non-Abelian gauge theoriesin a connected domain of the configuration space where the Faddeev–Popov operator (forcontinuous gauges with space-time derivatives) for a theory in question has positive eigenvalues. Because of the crucial importance of gauged BRST transformations (2.15), we now considerthem in detail, assuming that, in general, an infinitesimal value of the odd-valued parameter µ can be changed to a finite nilpotent one, Λ(Φ , Φ ∗ ), Λ = 0, being dependent, as a functional,7n the entire set of fields Φ A and antifields Φ ∗ A (however, not on the space-time coordinates ina manifest form) as follows:Φ ′ A = Φ A + (Φ A , S Ψ )Λ(Φ , Φ ∗ ) = ⇒ δ Φ A = S A Ψ Λ(Φ , Φ ∗ ) , for S A Ψ ≡ δS Ψ δ Φ ∗ A . (2.23)The corresponding extended (due to the antifields) Slavnov variation , s e F (Φ , Φ ∗ ) = δFδ Φ A S A Ψ , ofan arbitrary functional F (Φ , Φ ∗ ) generally fails to be nilpotent, s e F (Φ , Φ ∗ ) = δFδ Φ A S A Ψ ,B S B Ψ = δFδ Φ A (cid:0) S Ψ , B S AB Ψ − i ~ ∆ S A Ψ (cid:1) ( − ε A = 0 , (2.24)even for a local action functional, when ∆ S A Ψ ∼ δ (0). In spite of the result (2.24), i.e., that( s e ) = 0, the observation that for any constant odd scalar parameters Λ , Λ with gh(Λ ) =gh(Λ ) there exists a real number a such that Λ = a Λ implies that the right transformations Gg (Λ) = (1 + s e G Λ) acting on any functional G = G (Φ , Φ ∗ ) form an Abelian one-parametricsupergroup, since g (Λ ) g (Λ ) = g (Λ + Λ ) for any odd Λ i , i = 1 ,
2, due to the fact thatΛ · Λ = a Λ = 0.The usual Slavnov variation sF (Φ) acting on a functional in the configuration space F (Φ) = F (Φ ,
0) and determined at the classical level (in the tree approximation) for S Ψ = P k ≥ S ( k )Ψ isnot nilpotent, compared to first-rank gauge theories, including the Yang–Mills theory [1], sF (Φ) = δFδ Φ A S (0) A Ψ (Φ , , sF (Φ) = s e F (Φ , Φ ∗ ) (cid:12)(cid:12) Φ ∗ =0 , (2.25) s F (Φ) = δFδ Φ A S (0) A Ψ ,B S (0) B Ψ (cid:12)(cid:12) Φ ∗ =0 = δFδ Φ A S (0)Ψ , B S (0) AB Ψ ( − ε A (cid:12)(cid:12) Φ ∗ =0 = 0 . (2.26)This is explained by an open algebra, described here by the terms S (0) AB Ψ | Φ ∗ =0 , which emergesin the general Lie bracket for the generators of gauge transformations, R iα ( A ), R iα , j ( A ) R jβ ( A ) − R iβ , j ( A ) R jα ( A ) = − R iγ ( A ) F γ α β ( A ) + S ,j ( A ) M ijα β ( A ) , (2.27)in the form of the coefficients M ijα β ( A ) at the extremals, which, together with the functions F γ α β ( A ), satisfy the properties of generalized antisymmetry[ F γ α β , M ijα β ] = − ( − ε α ε β [ F γ β α , M ijβ α ] , M ijα β = − ( − ε i ε j M jiα β . Let us now calculate the Jacobian of the change of variables generated by the finite gaugedBRST transformations (2.23), namely,Sdet (cid:13)(cid:13)(cid:13)(cid:13) δ Φ ′ A δ Φ B (cid:13)(cid:13)(cid:13)(cid:13) = exp (cid:26) Str ln (cid:18) δ AB + δ ( S A Ψ Λ) δ Φ B (cid:19)(cid:27) = exp (cid:26) − Str X n =1 ( − n n (cid:18) δ ( S A Ψ Λ) δ Φ B (cid:19) n (cid:27) , (2.28) The fact that the odd operator s e is not nilpotent implies that one cannot restore a finite BRST flow(transformations) in Π T ∗ M following the Frobenius theorem, because the odd-valued vector field s e (Φ , Φ ∗ ) = ←− δδ Φ A ( s e Φ A ) does not have to be nilpotent. (cid:18) δ ( S A Ψ Λ) δ Φ B (cid:19) n = δ ( S A Ψ Λ) δ Φ B δ ( S B Ψ Λ) δ Φ B · · · δ ( S B n − Ψ Λ) δ Φ A ( − ε A . (2.29)Explicitly, the supermatrix k ( S A Ψ Λ) , B k in (2.29) can be presented as the sum of two terms:( S A Ψ Λ) , B = S A Ψ , B Λ( − ε B + S A Ψ Λ , B ≡ P AB + Q AB , for Λ B ≡ Λ , B , (2.30)such that only the first supermatrix is nilpotent, S A Ψ , B Λ S B Ψ , C Λ = P AB P BC = 0, and furthermore P AB Q BC · · · Q C k − C k P C k D Q DD · · · Q D l − D l = 0 , (2.31)for any natural numbers k, l .Using the property of supercommutativity for arbitrary even supermatrices F, G under thesymbol “Str”, Str(
F G ) = Str( GF ), we obtain from Eqs. (2.29), (2.30) the representationStr (cid:18) P AB + Q AB (cid:19) n = n X k = n − C kn Str (cid:0) ( P n − k ) AC ( Q k ) CB (cid:1) = n Str (cid:0) P AC ( Q n − ) CB (cid:1) + Str ( Q n ) AB , (2.32)with the number of combinations being C kn = n ! k !( n − k )! .Consequently, we have − ∞ X n =1 ( − n n Str (cid:18) δ ( S A Ψ Λ) δ Φ B (cid:19) n = ∞ X n =1 ( − n n (Λ C S C Ψ ) n − ∞ X n =2 ( − n (Λ C S C Ψ ) n − Λ A S A Ψ , B S B Ψ Λ + S A Ψ , A Λ= ∞ X n =1 ( − n n ( s e Λ) n − ∞ X n =2 ( − n ( s e Λ) n − Λ A (cid:0) s e S A Ψ (cid:1) Λ + (cid:0) ∆ S Ψ (cid:1) Λ= − ln (cid:0) s e Λ (cid:1) − ∞ X n =1 ( − n − ( s e Λ) n − Λ A (cid:0) s e S A Ψ (cid:1) Λ + (cid:0) ∆ S Ψ (cid:1) Λ= − ln (cid:0) s e Λ (cid:1) − (cid:0) s e Λ (cid:1) − Λ A (cid:0) s e S A Ψ (cid:1) Λ + (cid:0) ∆ S Ψ (cid:1) Λ . (2.33)As a result, we obtain the Jacobian for general gauged (field-dependent) BRST transformations:Sdet (cid:13)(cid:13)(cid:13)(cid:13) δ Φ ′ A δ Φ B (cid:13)(cid:13)(cid:13)(cid:13) = (cid:0) s e Λ (cid:1) − exp n − (cid:0) s e Λ (cid:1) − Λ A (cid:0) s e S A Ψ (cid:1) Λ + (cid:0) ∆ S Ψ (cid:1) Λ o = (cid:0) s e Λ (cid:1) − n − (cid:0) s e Λ (cid:1) − Λ A (cid:0) s e S A Ψ (cid:1) Λ + (cid:0) ∆ S Ψ (cid:1) Λ o = (cid:0) s e Λ (cid:1) − n ←− s e Λ on (cid:0) ∆ S Ψ (cid:1) Λ o , (2.34)where G ←− s e ≡ s e G , for any G = G (Φ , Φ ∗ ) and account has been taken of the identityexp {− a Λ } = 1 − a Λ, in view of Λ = 0. Let us emphasize that the superdeterminant (2.28) for vanishing antifields Φ ∗ was calculated also in [45]; s e coincides with s , (cid:16) ∆ S Ψ = 0 , S A Ψ , B S B Ψ = s = 0 (cid:17) = ⇒ Sdet (cid:13)(cid:13)(cid:13)(cid:13) δ Φ ′ A δ Φ B (cid:13)(cid:13)(cid:13)(cid:13) = (cid:0) s Λ (cid:1) − , (2.35)and which has also been taken into account as regards the independence of the generators S A Ψ (Φ) of BRST transformations on the antifields Φ ∗ A , whereas the odd-valued functional Λshould now be regarded as a field-dependent one, Λ = Λ(Φ). The representation (2.35) is nowvalid for a gauge theory of rank 1 with a closed algebra, and with the additional requirementfor the generators S A Ψ to be divergentless: ∆ S Ψ = S A Ψ , A = 0.For the functional integral G (Φ ∗ ) = Z D Φ exp n i ~ Q (Φ , Φ ∗ ) o , the change of variables (2.23) leads to the representation G (Φ ∗ ) = Z D Φ ′ exp n i ~ Q (Φ ′ , Φ ∗ ) o = Z D Φ exp n i ~ (cid:16) Q (Φ , Φ ∗ )+ s e Q Λ(Φ , Φ ∗ ) − i ~ (cid:0) ∆ S Ψ (cid:1) Λ − i ~ ln h(cid:0) s e Λ (cid:1) − (cid:0) ←− s e Λ (cid:1)i (cid:17)o , (2.36)which is different from the similar result [28] for the Yang–Mills theory due to the terms (cid:0) ←− s e Λ (cid:1) in the final line and the presence of i ~ (cid:0) ∆ S Ψ (cid:1) Λ.A repeated application of gauged BRST transformations with the same gauged parameterΛ is not nilpotent due to the equality δ Λ ( δ Λ F (Φ , Φ ∗ )) = δ Λ ( s e F (Φ , Φ ∗ )Λ) = − s e F (Φ , Φ ∗ )Λ + s e F (Φ , Φ ∗ ) s e (Λ)Λ , (2.37)and due to the vanishing commutator [ δ µ , δ µ ] F = 2( s e F ) µ µ = 0 of global BRST transfor-mations with constant parameters µ , µ , such that µ = a · µ , as shown by Eq. (2.24) for s e F = 0 and the subsequent relations. Of course, for a constant Λ (i.e., Λ = µ ) the nilpotency in however, any explicit calculation is absent. At the same time, the first version of the present work appearedas arXiv:1312.2092v1[hep-th] earlier than the above paper [45], which appeared as arXiv:1312.2802v1[hep-th].Note that in the first version of [45] this superdeterminant was calculated in (3.10) incorrectly. A correctcalculation algorithm, including a representation of the superdeterminant as a series in (2.33), was given inthe 3rd line of (2.33) in arXiv:1312.2092v1[hep-th]. A small error in the representation of the second series inthe 3rd line, which appeared starting from the 4th line, was corrected in arXiv:1312.2092v3[hep-th]. Turningto [45], we have to pay attention to the fact that the introduction of “finite supersymmetric two-parametrictransformations” linear in the anticommuting parameters ξ a , a = 1 , ξ a )transformations, instead of (2.3); see [64] for details. (cid:13)(cid:13)(cid:13)(cid:13) δ Φ ′ A δ Φ B (cid:13)(cid:13)(cid:13)(cid:13) = exp (cid:8) ∆ S Ψ µ (cid:9) . Let us now examine the generating functionals Z Ψ ( J, Φ ∗ ), Z Ψ+∆Ψ ( J, Φ ∗ ) in Eq. (2.22) forthe same gauge theory, however, given by different (not necessarily related to each other bysmall variations) gauges described by the gauge fermions Ψ(Φ), [Ψ + ∆Ψ](Φ), which differ bya Grassmann-odd functional, ∆Ψ(Φ), subject to the conditions ( ε, gh )∆Ψ(Φ) = (1 , − S Ψ , we obtain the generating functional Z Ψ ( J, Φ ∗ ), namely, Z Ψ ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:16) S Ψ + i ~ ln (cid:0) s e Λ (cid:1) + i ~ (cid:0) s e Λ (cid:1) − Λ A (cid:0) s e S A Ψ (cid:1) Λ + J A (Φ A + δ Φ A ) (cid:17)o . (2.38)In its turn, Z Ψ+∆Ψ ( J, Φ ∗ ) corresponding to a finite change of the gauge fermion takes the form Z Ψ+∆Ψ ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:16) S Ψ (Φ , Φ ∗ ) + s e (cid:0) ∆Ψ(Φ) (cid:1) + X n ≥ n ! ∆Ψ A · · · ∆Ψ A n S A n ...A Ψ (Φ , Φ ∗ ) + J A Φ A (cid:17)o . (2.39)Consider a functional equation for an unknown odd-valued functional, Λ, following the require-ment of coincidence of the above representations (2.38) and (2.39) for J A = 0: Z Ψ+∆Ψ (0 , Φ ∗ ) = Z Ψ (0 , Φ ∗ ) i ~ n ln (cid:0) s e Λ (cid:1) + (cid:0) s e Λ (cid:1) − Λ A (cid:0) s e S A Ψ (cid:1) Λ o = X n ≥ n ! ∆Ψ A · · · ∆Ψ A n S A n ...A Ψ (Φ , Φ ∗ ) ⇐⇒ − i ~ ln n(cid:0) s e Λ (cid:1) − (cid:0) ←− s e Λ (cid:1)o = (cid:16) exp n − [∆ , ∆Ψ] o − (cid:17) S Ψ . (2.40)Having in mind the fact that for an infinitesimal ∆Ψ = δ Ψ, with accuracy up to the first orderin δ Ψ from Eq. (2.40), we have a linearized (with respect to Λ and Λ A ) and easily solvedequation, i ~ s e Λ = s e δ Ψ(Φ) = ⇒ Λ = − i ~ δ Ψ and Λ = Λ(Φ) . (2.41)This fact has been used to verify the gauge-independence property Z Ψ = Z Ψ+ δ Ψ for the vacuumfunctional Z Ψ in Section 2.1.Therefore, we hope that the highly non-linear equation (2.40), which provides a compensa-tion for a finite change of the gauge Fermion in Z Ψ by means of the Jacobian for the change11f variables generated by the gauged BRST transformations (2.23), also has a solution, whichshould be of the form Λ (cid:0) Φ , Φ ∗ | ∆Ψ (cid:1) = Λ (cid:0) ∆Ψ (cid:1) . (2.42)We analyze a justification of this representation in Appendix A.Using the above result, we argue that that for any finite change of the gauge ∆Ψ thereexists a gauged (field-dependent) BRST transformation (2.15) with an odd-valued functionalΛ (cid:0) ∆Ψ (cid:1) in (2.42) such that, due to the equivalence theorem [43, 44], there is a coincidence ofthe two representations (2.38) and (2.39), which is also valid for the vacuum functional: Z Ψ+∆Ψ (0 , Φ ∗ ) = Z Ψ (0 , Φ ∗ ) . (2.43)This is the main result of this section, which we use in the study of the gauge-(in)dependenceproblem for a theory with BRST symmetry breaking terms. Let us turn to the problem of gauge dependence for a gauge theory determined by Eqs.(2.1), (2.2), (2.4) with a quantum action S Ψ (Φ , Φ ∗ ) additively extended along the lines of ourprevious study [22], [23] by a soft BRST breaking term M (Φ , Φ ∗ ) defined in a gauge Ψ(Φ) upto an action S (Φ , Φ ∗ ) determining the generating functional of Green’s functions, Z M ( J, Φ ∗ ), S = S Ψ + M, Z M ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:0) S (Φ , Φ ∗ ) + J A Φ A (cid:1)o , (3.1)with the boundary condition Z M ( J, Φ ∗ ) (cid:12)(cid:12) M =0 = Z ( J, Φ ∗ ) . We remind that, at the classical level, since we assume the bosonic functional M (Φ , Φ ∗ ) to havea regular decomposition in powers of ~ , M (Φ , Φ ∗ ) = P n ≥ ~ n M n (Φ , Φ ∗ ), the condition of a softbreaking of BRST symmetry implies( M , M ) = 0 and m , i R iα = 0 , for m (Φ) = M (Φ , Φ ∗ ) (cid:12)(cid:12) Φ ∗ =0 , (3.2)whereas in the case of a regularization more general than dimensional-like ones, the totalgenerating equation for M (Φ , Φ ∗ ) reads [23]∆ (cid:26) − i ~ M (cid:27) = 0 ⇐⇒ ( M, M ) = − i ~ ∆ M. (3.3) One may examine a more general BRST symmetry breaking functional, not satisfying Eq. (3.3) or Eq.(3.2), without changing the results for the dependence of the effective action (as will be seen later); however,we will follow the study of [22], [23], because the solution of these equations restricts the rank condition for theHessian of M to be no greater than dim M , and to be such that the functional integral in (3.1) is well-defined.
12s a consequence of Eqs. (2.11), (3.3), the total action now satisfies ( S, S ) − i ~ ∆ S = ( S, M ) , (3.4)so that, in the classical limit for S = S + O ( ~ ), Eq. (3.4) implies the equation ( S , S ) = ( S , M ) . (3.5)The properties of the generating functionals of the usual, Z M ( J, Φ ∗ ), connected, W M ( J, Φ ∗ ),( W M = ~ i ln Z M ) and vertex, Γ M (Φ , Φ ∗ ), Green functions, introduced via the Legendre trans-formation of W M ( J, Φ ∗ ) with respect to the sources J A ,Γ M (Φ , Φ ∗ ) = W M ( J, Φ ∗ ) − J A Φ A , Φ A = δW M ( J, Φ ∗ ) δJ A , (3.6)have been studied in [22], [23]. These properties include the Ward identities and the calcu-lation of variations of all the generating functionals under a variation of the gauge condition(Grassmann-odd functional), Ψ(Φ) → Ψ(Φ) + δ Ψ(Φ). The properties were derived on the basisof functional averaging of the master equations (2.11) for S Ψ in a dimensional-like regulariza-tion, as applied to the local functional S [22], and in more general regularizations [23].These properties can only be obtained by means of global BRST and field-dependent(gauged) BRST transformations. There follow the Ward identities for Z M ( J, Φ ∗ ), after thechange of variables (2.14) in the integrand of (3.1), with account taken of Eqs. (2.11), (3.3), Z M ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:16) S (Φ , Φ ′∗ ) + δSδ Φ A δS Ψ δ Φ ∗ A µ − i ~ ∆ S Ψ µ + J A (cid:2) Φ A + δS Ψ δ Φ ∗ A µ i(cid:17)o = Z M ( J, Φ ∗ ) + i ~ Z D Φ (cid:0) J A + M A (cid:1) δS Ψ δ Φ ∗ A exp n i ~ (cid:16) S (Φ , Φ ′∗ ) + J A Φ A (cid:17)o µ, = ⇒ (cid:16) J A + M A (cid:0) ~ i δδJ , Φ ∗ (cid:1)(cid:17) (cid:18) ~ i δδ Φ ∗ A − M A ∗ (cid:0) ~ i δδJ , Φ ∗ (cid:1)(cid:19) Z M ( J, Φ ∗ ) = 0 , (3.7)where the notation M A (cid:0) ~ i δδJ , Φ ∗ (cid:1) ≡ δM (Φ , Φ ∗ ) δ Φ A (cid:12)(cid:12)(cid:12) Φ → ~ i δδJ and M A ∗ (cid:0) ~ i δδJ , Φ ∗ (cid:1) ≡ δM (Φ , Φ ∗ ) δ Φ ∗ A (cid:12)(cid:12)(cid:12) Φ → ~ i δδJ (3.8)has been used. In case M = 0, identity (3.7) is reduced to the usual Ward identity (2.16) for Z ( J, Φ ∗ ), as well as to the Ward identities for W M ( J, Φ ∗ ), Γ M (Φ , Φ ∗ ), which follow from (3.7), (cid:16) J A + M A (cid:0) δW M δJ + ~ i δδJ , Φ ∗ (cid:1)(cid:17) (cid:18) δW M ( J, Φ ∗ ) δ Φ ∗ A − M A ∗ (cid:0) δW M δJ + ~ i δδJ , Φ ∗ (cid:1)(cid:19) = 0 , (3.9) (Γ M , Γ M ) = δ Γ M δ Φ A c M A ∗ + c M A δ Γ M δ Φ ∗ A − c M A c M A ∗ . (3.10)Here, we have used a notation introduced in [22]: c M A ≡ δM (Φ , Φ ∗ ) δ Φ A (cid:12)(cid:12)(cid:12) Φ → b Φ and c M A ∗ ≡ δM (Φ , Φ ∗ ) δ Φ ∗ A (cid:12)(cid:12)(cid:12) Φ → b Φ , (3.11)13ith account taken for the conventions (2.19), (2.20), adapted to the case of broken BRSTsymmetry, i.e., according to the change Γ → Γ M . For completeness, note that the functionalΓ M satisfies the functional integro-differential equationexp n i ~ Γ M (Φ , Φ ∗ ) o = Z dϕ exp n i ~ h S Ψ (Φ + ~ ϕ, Φ ∗ ) + M (Φ + ~ ϕ, Φ ∗ ) − δ Γ M (Φ , Φ ∗ ) δ Φ ~ ϕ io , (3.12)determining the loop expansion Γ M = P n ≥ ~ n Γ nM . Thus, the tree-level (zero-loop) and one-loop approximations of (3.12) correspond toΓ M (Φ , Φ ∗ ) = S Ψ0 (Φ , Φ ∗ ) + M (Φ , Φ ∗ ) , (3.13)Γ M (Φ , Φ ∗ ) = S Ψ1 (Φ , Φ ∗ ) + M (Φ , Φ ∗ ) − i2 ln Sdet (cid:13)(cid:13)(cid:13) ( S ′′ ) AB (Φ , Φ ∗ ) (cid:13)(cid:13)(cid:13) , (3.14)so that the tree-level part of the Ward identity (3.10) for Γ M , (Γ M , Γ M ) = δS Ψ0 δ Φ A M A ∗ + M A δS Ψ0 δ Φ ∗ A + M A M A ∗ is fulfilled identically, due to the tree-level approximation to the generating equations (2.11) for S Ψ0 and (3.2) for M .In order to study the gauge-dependence problem, we examine, first of all, the representationfor Z M ( J, Φ ∗ ) within the gauge determined by the gauge functional, Ψ + ∆Ψ, similar to Eq.(2.39), but without the use of field-dependent BRST transformations: Z M ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:16) S Ψ (Φ , Φ ∗ ) + M (Φ , Φ ∗ ) + s e (cid:0) ∆Ψ(Φ) (cid:1) + X n ≥ n ! ∆Ψ A · · · ∆Ψ A n S A n ...A Ψ (Φ , Φ ∗ ) + ∆ M (Φ , Φ ∗ ) + J A Φ A (cid:17)o , (3.15)where account has been taken for the fact that the functional M = M Ψ (Φ , Φ ∗ ) should have thefollowing representation in the above gauge, because of the relation of gauged BRST transfor-mations with functional Λ (cid:0) Φ , Φ ∗ | ∆Ψ (cid:1) (2.42) which should compensate a finite change of thegauge ∆Ψ in Z Ψ (0 , Φ ∗ ): M Ψ+∆Ψ (Φ , Φ ∗ ) = M Ψ (Φ , Φ ∗ ) + ∆ M (Φ , Φ ∗ ) . (3.16)It should be noted that M Ψ+∆Ψ does not have the form of a gauge-invariant action, S Ψ+∆Ψ ,as regards the dependence on the variation ∆ M , despite the fact that an introduction of theadditive term ∆Ψ by means of the transformation (2.9) applied to the action, S Ψ ,exp (cid:26) i ~ S Ψ+∆Ψ (cid:27) = exp (cid:8) − [∆ , ∆Ψ] (cid:9) exp (cid:26) i ~ S Ψ (cid:27) = exp (cid:26) i ~ S Ψ (Φ , Φ ∗ + δ ∆Ψ δ Φ ) (cid:27) , (3.17)14s a transformation which turns a solution of the soft BRST symmetry breaking equation (3.3)for M Ψ into another solution, however, not having the form M Ψ+∆Ψ . This takes place, since inthe case of the functional M Ψ , being BRST-non-invariant, the gauge condition is not determinedvia a shift of the antifields: M Ψ+∆Ψ = M Ψ (cid:0) Φ , Φ ∗ + δ ∆Ψ δ Φ ) (cid:1) . (3.18)As we turn to Eq. (3.15), let us present the finite change ∆ Z M ( J, Φ ∗ ) = Z M +∆ M, Ψ+∆Ψ ( J, Φ ∗ ) − Z M, Ψ ( J, Φ ∗ ) in an equivalent form:∆ Z M ( J, Φ ∗ ) = Z D Φ h exp n i ~ (cid:16) s e (cid:0) ∆Ψ(Φ) (cid:1) + X n ≥ n ! ∆Ψ A · · · ∆Ψ A n S A n ...A Ψ (Φ , Φ ∗ )+∆ M (Φ , Φ ∗ ) (cid:17)o − i exp n i ~ (cid:0) S (Φ , Φ ∗ ) + J A Φ A (cid:1)o = Z D Φ h exp n i ~ ∆ M (Φ , Φ ∗ ) o exp n δ ∆Ψ(Φ) δ Φ A (cid:16) δδ Φ ∗ A − i ~ M A ∗ (Φ , Φ ∗ ) (cid:17)o − i × exp n i ~ (cid:0) S (Φ , Φ ∗ ) + J A Φ A (cid:1)o , (3.19)with allowance for the identity h exp n δ ∆Ψ δ Φ A δS Ψ δ Φ ∗ A o − i exp n i ~ S o = h exp n δ ∆Ψ δ Φ A (cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17)o − i exp n i ~ S o . (3.20)Considering the general term n δ ∆Ψ δ Φ A (cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17)o n , for n ≥
1, inside the decomposition (3.19)and integrating by parts in the path integral, we obtain Z D Φ exp n i ~ ∆ M o δ ∆Ψ δ Φ A n ∆Ψ B (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)o n − (cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17) exp n i ~ (cid:0) S + J A Φ A (cid:1)o = i ~ Z D Φ exp n i ~ ∆ M o(cid:20) ∆ M A n ∆Ψ B (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)o n − (cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17) − n n − X k =1 k − Y l =1 ∆Ψ B l (cid:16) δδ Φ ∗ Bl − i ~ M B l ∗ (cid:17) ∆Ψ B k M B k ∗ A n − Y l = k +1 ∆Ψ B l (cid:16) δδ Φ ∗ Bl − i ~ M B l ∗ (cid:17)o(cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17) + n ∆Ψ B (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)o n − (cid:16) J A + M A (cid:17)(cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17)(cid:21) ∆Ψ exp n i ~ (cid:0) S + J A Φ A (cid:1)o , (3.21)where account has been taken of the generating equations (2.11) for S Ψ and (3.3) for M , aswell as the notation M B k ∗ A ≡ δδ Φ A ( M B k ∗ ) and the following properties of the functional ∆Ψ: (cid:0) ∆Ψ (cid:1) ≡ , and ∆Ψ AB (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)(cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17) ≡ . Z M ( J, Φ ∗ ) can therefore be finally presented as∆ Z M = Z D Φ (cid:20) exp n i ~ ∆ M o X n ≥ n ! n ∆Ψ B (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)o n − (cid:21) exp n i ~ (cid:0) S + J A Φ A (cid:1)o = i ~ exp n i ~ ∆ M ( ~ i δδJ , Φ ∗ ) o(cid:20) ∆ M A X n ≥ n ! n ∆Ψ B (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)o n − − X n ≥ n ! n n − X k =1 k − Y l =1 ∆Ψ B l (cid:16) δδ Φ ∗ Bl − i ~ M B l ∗ (cid:17) ∆Ψ B k M B k ∗ A n − Y l = k +1 ∆Ψ B l (cid:16) δδ Φ ∗ Bl − i ~ M B l ∗ (cid:17)o + X n ≥ n ! n ∆Ψ B (cid:16) δδ Φ ∗ B − i ~ M B ∗ (cid:17)o n − (cid:16) J A + M A (cid:17)(cid:21)(cid:16) δδ Φ ∗ A − i ~ M A ∗ (cid:17) ∆Ψ( ~ i δδJ ) Z M + (cid:20) exp n i ~ ∆ M ( ~ i δδJ , Φ ∗ ) o − (cid:21) Z M , (3.22)where the arguments ~ i δδJ are implied to be substituted instead of the fields, Φ, in ∆ M A , M B ∗ , M A , M B k ∗ A , ∆Ψ B in the last equality, in accordance with the conventions (3.8).The general result (3.22) for the variation of Z M in the approximation linear in powers ofthe variations ∆Ψ , ∆ M reads as follows:∆ Z M ( J, Φ ∗ ) = i ~ h(cid:0) J A + M A ( ~ i δδJ , Φ ∗ ) (cid:1)(cid:16) δδ Φ ∗ A − i ~ M A ∗ ( ~ i δδJ , Φ ∗ ) (cid:17) ∆Ψ( ~ i δδJ )+∆ M ( ~ i δδJ , Φ ∗ ) i Z M ( J, Φ ∗ ) , (3.23)and coincides with the result for δZ M first obtained in [23].In the particular case of the absence of BRST symmetry breaking terms, i.e., when M = 0,we derive from (3.22) a new representation for a finite variation of the functional Z ( J, Φ ∗ ) undera finite variation of the gauge condition,∆ Z ( J, Φ ∗ ) = i ~ "X n ≥ n + 1)! (cid:16) ∆Ψ B ( ~ i δδJ ) δδ Φ ∗ B (cid:17) n J A δδ Φ ∗ A ∆Ψ( ~ i δδJ ) Z ( J, Φ ∗ ) , (3.24)which is reduced, in the case of a small variation, ∆Ψ = δ Ψ, to the form (2.17), well-known inthe BV formalism, with the notation ∆ Z = δZ .To complete the research of gauge-dependence in the theory with broken BRST symmetry,let us calculate the variations of W M ( J, Φ ∗ ) and Γ M (Φ , Φ ∗ ) under a finite change of the gaugecondition, taken into account for the relations ∆ W M = ~ i Z − M ∆ Z M and ∆ W M = ∆Γ M . First of16ll, ∆ W M reads∆ W M = exp n i ~ ∆ M (cid:0) ~ i δδJ + δW M δJ , Φ ∗ (cid:1)o(cid:20) ∆ M A X n ≥ n ! n ∆Ψ B (cid:16) δδ Φ ∗ B + i ~ δW M δ Φ ∗ B − i ~ M B ∗ (cid:17)o n − × (cid:16) δδ Φ ∗ A + i ~ δW M δ Φ ∗ A − i ~ M A ∗ (cid:17) − X n ≥ n ! n n − X k =1 k − Y l =1 ∆Ψ B l (cid:16) δδ Φ ∗ Bl + i ~ δW M δ Φ ∗ Bl − i ~ M B l ∗ (cid:17) ∆Ψ B k M B k ∗ A × n − Y l = k +1 ∆Ψ B l (cid:16) δδ Φ ∗ Bl + i ~ δW M δ Φ ∗ Bl − i ~ M B l ∗ (cid:17)o(cid:16) δδ Φ ∗ A + i ~ δW M δ Φ ∗ A − i ~ M A ∗ (cid:17) + X n ≥ n ! n ∆Ψ B (cid:16) δδ Φ ∗ B + i ~ δW M δ Φ ∗ B − i ~ M B ∗ (cid:17)o n − (cid:16) J A + M A (cid:17) δδ Φ ∗ A (cid:21) ∆Ψ( ~ i δδJ + δW M δJ )+ X n ≥ n ! (cid:18) i ~ (cid:19) n − ∆ M ( ~ i δδJ + δW M δJ , Φ ∗ ) , (3.25)where use has been made of the Ward identity (3.9) and substitution of the arguments, (cid:0) ~ i δδJ + δW M δJ (cid:1) , instead of Φ in ∆ M A , M B ∗ , M A , ∆Ψ B should be made. Again, without BRST symmetrybreaking terms for M = 0, we can obtain from Eq. (3.25) a new representation for a finitevariation for the generating functional W ( J, Φ ∗ ), namely,∆ W = "X n ≥ n + 1)! (cid:16) ∆Ψ B (cid:0) δWδJ + ~ i δδJ (cid:1)h i ~ δWδ Φ ∗ B + δδ Φ ∗ B i(cid:17) n J A δδ Φ ∗ A ∆Ψ (cid:0) δWδJ + ~ i δδJ (cid:1) . (3.26)For the first order in powers of ∆Ψ and ∆ M , the variation (3.25) for ∆ W M ( J, Φ ∗ ) has the form∆ W M = (cid:16) J A + M A (cid:0) δW M δJ + ~ i δδJ , Φ ∗ (cid:1)(cid:17) δδ Φ ∗ A ∆Ψ (cid:0) δW M δJ + ~ i δδJ (cid:1) + ∆ M (cid:0) δW M δJ + ~ i δδJ , Φ ∗ (cid:1) , (3.27)identical (after the change ∆ → δ ) with the one obtained in [22], [23].Second, on order to derive a finite form of the gauge variation for the effective action, wecan use the calculations of [23]. Namely, the change of variables ( J A , Φ ∗ A ) → (Φ A , Φ ∗ A ) from theLegendre transformation (3.6) implies δδ Φ ∗ (cid:12)(cid:12)(cid:12) J = δδ Φ ∗ (cid:12)(cid:12)(cid:12) Φ + δ Φ δ Φ ∗ δ l δ Φ (cid:12)(cid:12)(cid:12) Φ ∗ and δW M δ Φ ∗ A = δ Γ M δ Φ ∗ A . (3.28)Then, the differential consequence of the Ward identities for Z M (3.7) and W M (3.9) implies − (cid:16) δ Γ M δ Φ A − c M A (cid:17) δ Φ B δ Φ ∗ A = − (cid:16) δ Γ δ Φ ∗ B − c M B ∗ (cid:17) ( − ε B + i ~ h − c M A δ Γ M δ Φ ∗ A − δ Γ M δ Φ A c M A ∗ + c M A c M A ∗ , Φ B i , (3.29)with the same notation (cid:2) , (cid:3) for the supercommutator as in (3.17).17sing Eqs. (3.25), (3.28), (3.29) and the relation δ Φ B δ Φ ∗ A = ( − ε B ( ε A +1) δδJ B δWδ Φ ∗ A = − ( − ε B ( ε A +1) (Γ ′′ − ) BC δ l δ Φ C δ Γ δ Φ ∗ A , (3.30)we present the finite variation of the effective action in the form∆Γ M = exp n i ~ h ∆ M i o(cid:18) h ∆ M A i X n ≥ n ! n h ∆Ψ B i (cid:16) − b F B + i ~ δ Γ M δ Φ ∗ B − i ~ c M B ∗ (cid:17)o n − × (cid:16) − b F A + i ~ δ Γ M δ Φ ∗ A − i ~ c M A ∗ (cid:17) − X n ≥ n ! n n − X k =1 k − Y l =1 h ∆Ψ B l i (cid:16) − b F B l + i ~ δ Γ M δ Φ ∗ Bl − i ~ c M B l ∗ (cid:17) ×h ∆Ψ B k i c M B k ∗ A n − Y l = k +1 h ∆Ψ B l i (cid:16) − b F B l + i ~ δ Γ M δ Φ ∗ Bl − i ~ c M B l ∗ (cid:17)o(cid:16) − b F A + i ~ δ Γ M δ Φ ∗ A − i ~ c M A ∗ (cid:17) + X n ≥ n ! n h ∆Ψ B i (cid:16) − b F B + i ~ δ Γ M δ Φ ∗ B − i ~ c M B ∗ (cid:17)o n − n − (cid:0) Γ M , (cid:1) + c M A δδ Φ ∗ A + ( − ε A c M A ∗ δ l δ Φ A − i ~ h c M A δ Γ M δ Φ ∗ A + δ Γ M δ Φ A c M A ∗ − c M A c M A ∗ , Φ B i δ l δ Φ B o(cid:19) h ∆Ψ i + X n ≥ n ! (cid:18) i ~ (cid:19) n − h ∆ M i . (3.31)Here, we use the notation h ∆Ψ i = ∆Ψ( b Φ) · h ∆ M i = ∆ M ( b Φ , Φ ∗ ) · , (3.32)as well as the same notation for h ∆Ψ B k i , h ∆ M A i , and introduce the operator b F A , derived fromEqs. (3.28), (3.29), (3.30), as follows b F A = − δδ Φ ∗ A + ( − ε B ( ε A +1) (Γ ′′ − M ) BC (cid:16) δ l δ Φ C δ Γ M δ Φ ∗ A (cid:17) δ l δ Φ B . (3.33)Now, we can deduce from Eq. (3.31) a new representation for a finite variation of theeffective action Γ( J, Φ ∗ ) in a local form without BRST symmetry breaking terms ( M = 0),∆Γ = − X n ≥ n + 1)! (cid:16) h ∆Ψ B i h i ~ δ Γ δ Φ ∗ B − b F B i(cid:12)(cid:12) M =0 (cid:17) n (cid:0) Γ , h ∆Ψ i (cid:1) , (3.34)in the first order with respect to the variation h ∆Ψ i , identical with the previously knownrepresentation (2.19).For the first order in powers of h ∆Ψ i and h ∆ M i , the variation (3.31) of ∆Γ M ( J, Φ ∗ ) takesthe previously known [23] “local-like” form∆Γ M = − (cid:0) Γ M , h ∆Ψ i (cid:1) + (cid:18) c M A δδ Φ ∗ A + ( − ε A c M A ∗ δ l δ Φ A (cid:19) h ∆Ψ i− i ~ h c M A δ Γ M δ Φ ∗ A + δ Γ M δ Φ A c M A ∗ − c M A c M A ∗ , Φ B i δ l δ Φ B h ∆Ψ i + h ∆ M i , (3.35)18here coincidence with the final result of [23] is achieved by the change ∆ → δ .To study gauged (field-dependent) BRST transformations in a theory with broken BRSTsymmetry, we follow the result of [22], [23] and present the variation, linear in h ∆Ψ i , h ∆ M i , ofthe effective action (3.35) in an equivalent form, being the so-called non-local form, due to theexplicit presence of (Γ ′′ − M ) BC in b F A (3.33):∆Γ M = δ Γ M δ Φ A b F A h ∆Ψ i − c M A b F A h ∆Ψ i + h ∆ M i . (3.36)We now intend to revise our previous result [22], [23], which states that the variation(3.36) implies that the effective action with soft BRST symmetry breaking is generally gauge-dependent on the mass shell, since δ Γ M δ Φ A = 0 −→ ∆Γ M = 0 . (3.37)Indeed, there is a hope that the introduction of broken BRST symmetry into the field-antifieldformalism would be consistent only if the two final terms in (3.36) should cancel each other: h ∆ M i = c M A b F A h ∆Ψ i , (3.38)which, at the classical level, imposes a condition on the gauge variation of M under a changeof the gauge-fixing functional Ψ,∆ M = δMδ Φ A b F A ∆Ψ where b F A = ( − ε B ( ε A +1) ( S ′′ − ) BC (cid:16) δ l δ Φ C δSδ Φ ∗ A (cid:17) δ l δ Φ B . (3.39)Of course, despite the fact that it seems to be a strong restriction that the BRST-breakingfunctional M corresponding to the effective action should be gauge-independent on the massshell (implying the gauge-independence of the physical S-matrix), the gauge-independence (butnot invariance) can, in fact, be restored.In order to justify the above proposition, let us subject the integrand in Z M , with thegauge-fixing functional Ψ(Φ), to the change of variables (2.23), with a field-dependent odd-valued parameter Λ (cid:0) Φ , Φ ∗ | ∆Ψ (cid:1) in (2.42) being a solution of Eq. (2.40) [corresponding to thefunctional ˆΛ(Φ ′′ ) from Eq. (A.9)], which provides the gauge-independence of the vacuum func-tional Z Ψ (0 , Φ ∗ ) (2.43): Z M, Ψ ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:16) S − i ~ ln h(cid:0) s e Λ (cid:0) ∆Ψ (cid:1) (cid:1) − (cid:0) ←− s e Λ (cid:0) ∆Ψ (cid:1) (cid:1)i + s e M (Φ , Φ ∗ )Λ (cid:0) ∆Ψ (cid:1) + J A (cid:2) Φ A + ( s e Φ A )Λ (cid:0) ∆Ψ (cid:1) (cid:3)(cid:17)o . (3.40)Thus, taking into account the fact that for any variation of the gauge-fixing functional, Ψ(Φ) → (Ψ + ∆Ψ)(Φ), in view of the result obtained in Section 2.2, there exists a parameter, Λ (cid:0) ∆Ψ (cid:1) ,of finite gauged BRST transformations, being a solution of Eq. (2.40) such that the action S Ψ ,19nd therefore also the total quantum action S in the gauge determined by Ψ + ∆Ψ, takes theform S Ψ+∆Ψ + M Ψ+∆Ψ = h S Ψ − i ~ ln n(cid:0) s e Λ (cid:0) ∆Ψ (cid:1) (cid:1) − (cid:0) ←− s e Λ (cid:0) ∆Ψ (cid:1) (cid:1)oi + (cid:2) M Ψ + s e M Ψ (Φ , Φ ∗ ) Λ (cid:0) ∆Ψ (cid:1)(cid:3) , (3.41)where the first square brackets in the right-hand-side contain an expression for S Ψ+∆Ψ , whereasthe second brackets should contain an expression for M Ψ+∆ = M Ψ + ∆ M . Therefore, based onthe equivalence theorem [43], we have, first, a representation, being different form (3.22), for afinite change of the functional Z M, Ψ ( J, Φ ∗ ) under a finite change of the gauge:∆ Z M, Ψ ( J, Φ ∗ ) = − i ~ J A (cid:0) s e Φ A (cid:1) (cid:0) ~ i δδJ , Φ ∗ (cid:1) Λ (cid:0) ~ i δδJ , Φ ∗ | ∆Ψ (cid:1) Z M, Ψ ( J, Φ ∗ )= ( − ε A J A Λ (cid:0) ~ i δδJ , Φ ∗ | ∆Ψ (cid:1) (cid:18) δδ Φ ∗ A − i ~ M A ∗ Ψ (cid:19) Z M, Ψ ( J, Φ ∗ ) (3.42)which also leads to the on-shell coincidence (for J A = 0) of the generating functionals Z M ( J, Φ ∗ )calculated in the gauges Ψ and Ψ + ∆Ψ, respectively; we also obtain the form of the soft BRSTsymmetry functional M in the gauge determined by Ψ + ∆Ψ, provided that in the gauge Ψ theformer is defined by the functional M ,∆ M = ( s e M )Λ (cid:0) ∆Ψ (cid:1) . (3.43)In the approximation linear in ∆Ψ, we have, making use of (3.43) and (2.41), M Ψ+∆Ψ = M − i ~ M A S A Ψ δ Ψ . (3.44)An important particular case, which covers practically all the known gauge models, correspondsto a gauge theory of first rank with a closed algebra, when Eqs. (A.6) for the quantum actionare fulfilled. An explicit expression of the soft BRST-breaking functional similar to Eq. (3.44)in the gauge (Ψ + ∆Ψ) reads as follows: M Ψ+∆Ψ = M + M A S A Ψ ∆Ψ (cid:0) s ∆Ψ (cid:1) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ (cid:1)(cid:27) − i , (3.45)where account has been taken of Eq. (A.9).Note, first of all, that the additional contribution to M in (3.45) does not increase themaximal power in the antifields of the functional M . Second, the gauge variation of the BRST-symmetry-broken functional does not generally turn the solution of the soft BRST-breakingequation (3.3) into a solution. However, in the above case of a gauge theory with closed algebra(reducible or not) of first rank with M = M (Φ), Eq. (3.3) for the gauge-transformed functional M Ψ+∆Ψ is valid by construction, due to the representation (3.45).Let us now check the validity of the representation (3.38) for a variation of the BRSTsymmetry breaking functional with accuracy up to the first order in the gauge variation ∆Ψ.20n fact, it is sufficient to compare two representations for a finite change of Z M, Ψ ( J, Φ ∗ ), with(3.23) obtained from a change of the gauge condition and with (3.42) obtained via a changeof variables generated by field-dependent BRST transformations with the parameter Λ (cid:0) ∆Ψ (cid:1) .Indeed, (3.23) can be presented as∆ Z M, Ψ = i ~ J A (cid:16) δδ Φ ∗ A − i ~ M A ∗ ( ~ i δδJ , Φ ∗ ) (cid:17) ∆Ψ( ~ i δδJ ) Z M, Ψ (3.46)+ i ~ h M A ( ~ i δδJ , Φ ∗ ) (cid:16) δδ Φ ∗ A − i ~ M A ∗ ( ~ i δδJ , Φ ∗ ) (cid:17) ∆Ψ( ~ i δδJ ) + ∆ M ( ~ i δδJ , Φ ∗ ) i Z M, Ψ , and for the summand in the second line we havei ~ h i ~ M A ( ~ i δδJ , Φ ∗ ) S A Ψ ( ~ i δδJ , Φ ∗ )∆Ψ( ~ i δδJ ) + ∆ M ( ~ i δδJ , Φ ∗ ) i Z M, Ψ = i ~ h i ~ ( s e M )( ~ i δδJ , Φ ∗ )∆Ψ( ~ i δδJ ) + ∆ M ( ~ i δδJ , Φ ∗ ) i Z M, Ψ = i ~ h ∆ M − ( s e M )Λ (cid:0) ∆Ψ (cid:1) i ( ~ i δδJ , Φ ∗ ) Z M, Ψ = 0 (3.47)due to the representation (3.43) for ∆ M . The last expression in terms of the average fields(3.6) for the effective action Γ M is nothing else than the representation (3.38).Thus, the coincidence of ∆ Z M, Ψ ( J, Φ ∗ ) in (3.23) with (3.42) is guaranteed due to (3.43) and(2.42).As a consequence, the finite change of the functionals W M , Γ M in the linear approximation in∆Ψ in the relations (3.27) and (3.36) should coincide (after change ∆ → δ ) with the variationsof the functionals W , Γ, respectively, in (2.18) and (2.21) without any soft BRST symmetrybreaking term M . Concerning the finite change of the W M , Γ M in (3.25) and (3.31), as a resultof the above-established correspondence between the finite change of the gauge ∆Ψ and theparameter of gauged BRST transformation Λ (cid:0) Φ , Φ ∗ | ∆Ψ (cid:1) in (2.42), the form of ∆ M should bechosen according to (3.43).Thus, we have proved the following Statement : an addition to the quantum action S Ψ ,satisfying the master equation in the BV formalism (2.7), of a term, M (Φ , Φ ∗ ), breaking theBRST symmetry softly (3.3), first, destroys the BRST invariance of the integrand in thegenerating functional of Green’s functions, Z M , and therefore also the gauge-invariance of thetotal action ( S Ψ + M ) in the tree approximation; second, this leads to an effective action, Γ M ,being gauge-independent upon a variation of the gauge condition within the class of admissiblegauges on its extremals, δ Γ M δ Φ A = 0 and h ∆ M i = c M A b F A h ∆Ψ i −→ ∆Γ M = 0 , (3.48) Of course, any such term should be admissible, in order to have a well-determined path integral, at leastin perturbation theory. Second, the requirement of soft breaking of the BRST symmetry may be weakened inorder to consider only the breaking of BRST symmetry (see Footnote 6 for details). M in the form of gauged (field-dependent) BRST symmetry transformations (3.43).In particular, this implies that if in the reference frame determined by the gauge fermion Ψthe generating functional Z M, Ψ ( J, Φ ∗ ) is described by (3.1) then in the reference frame Ψ + ∆Ψit should have the form Z M +∆ , Ψ+∆Ψ ( J, Φ ∗ ) = Z D Φ exp n i ~ (cid:16) S Ψ+∆Ψ + M Ψ + (cid:0) s e M Ψ (cid:1) Λ (cid:0) ∆Ψ (cid:1) + J A Φ A (cid:17)o . (3.49)This fact makes the procedure of Lagrangian quantization of a gauge theory with soft BRSTsymmetry breaking consistent and leads, in particular, to a gauge-independent S-matrix withinthe conventional approach [1, 2, 3]. The problem of gauge dependence considered in a non-renormalized gauge theory with BRST broken terms should now be studied in a renormalizedtheory.Let us turn to some field-theoretic examples and constructions where the concept of BRSTsymmetry breaking is realized. In this section, we apply the above results to the study of the effective average action pro-posed in [48], [49], [50], which naturally arises within the functional renormalization group(FRG) approach to the Lagrangian quantization of Yang–Mills theories and was recently ex-amined in [51].The essence of FRG is to use, instead of Γ, the so-called effective average action Γ k with amomentum-shell parameter, k , coinciding with Γ for vanishing k ,lim k → Γ k = Γ , (4.1)in such a way that the Faddeev–Popov action S F P (Φ) for Yang–Mills theories should be ex-tended by means of soft BRST symmetry breaking terms, M , having the form of the regulatoraction S k for M = S k , S k ( A, C, ¯ C ) = 12 A i A j ( R k,A ) ij + ¯ C α ( R k,gh ) αβ C β = Z d D x n A aµ ( x )( R k,A ) abµν ( x ) A bν ( x ) + ¯ C a ( x )( R k,gh ) ab ( x ) C b ( x ) o . (4.2)In (4.2), we have specified the condensed notations, so that the total configuration space M ofthe Yang–Mills theory, { Φ A } = { A i , C α , ¯ C α , B α } = { A aµ , C a , ¯ C a , B a } ( x ) ε ( C a ) = ε ( ¯ C a ) = 1 , ε ( A aµ ) = ε ( B a ) = 0 , However, another basic requirement for a quantum gauge field theory, i.e., the unitarity of the S-matrix, isdestroyed when adding to the gauge theory any soft BRST symmetry breaking terms, and thus needs a specialinvestigation. A aµ ( x ), the Grassmann-oddFaddeev–Popov (scalar) ghost and antighost fields C a and ¯ C a , as well as the Nakanishi–Lautrupauxiliary fields B a , given in the D -dimensional Minkowski space-time R ,D − and taking valuesin the adjoint representation of the Lie algebra su ( N ). In turn, the regulator quantities ( R k,A ),( R k,gh ), having no dependence on the fields, obey the property ( R k,A ) ij = ( − ε i ε j ( R k,A ) ji andvanish as the parameter k tends to zero.The initial classical action S of the Yang–Mills fields A aµ ( x ) and its gauge transformationshave the standard form (with the coupling constant g = 1, for simplicity) S ( A ) = − Z d D x F aµν F µνa for F aµν = ∂ µ A aν − ∂ ν A aµ + f abc A bµ A cν , (4.3) δA aµ = D abµ ξ b , D abµ = δ ab ∂ µ + f acb A cµ , ε ( ξ b ) = 0 , (4.4)with the Lorentz indices µ, ν = 0 , , ..., D −
1, the metric tensor η µν , η µν = diag( − , + , ..., +), thetotally antisymmetric su ( N ) structure constants f abc , for a, b, c = 1 , . . . , N −
1, the covariantderivative D abµ , and arbitrary functions ξ b in R ,D − .The corresponding set of odd momenta for the fields, i.e., antifields, reads { Φ ∗ A } = { A ∗ aµ , C ∗ a , ¯ C ∗ a , B ∗ a } ( x ) with ε ( A ∗ aµ ) = ε ( B ∗ a ) = 1 , ε ( C ∗ a ) = ε ( ¯ C ∗ a ) = 0 , whereas a solution ¯ S (Φ , Φ ∗ ) to the quantum master equation (2.7) can be presented as¯ S (Φ , Φ ∗ ) = S ( A ) + Z d D x n A ∗ aµ D abµ C b + C ∗ a f abc C b C c + ¯ C ∗ a B a o , which, in view of the identity ∆ ¯ S = 0, is also a solution to the classical master equation( ¯ S, ¯ S ) = 0. The gauge-fixed action S Ψ (Φ , Φ ∗ ) = ¯ S (Φ , Φ ∗ + δ Ψ δ Φ ) obeys the same equations witha Grassmann-odd gauge-fixing functional Ψ(Φ), which can be chosen asΨ(Φ) = ¯ C a χ a ( A, B ) with χ a = 0 (4.5)so that the non-renormalized Faddeev–Popov action S F P (Φ) is obtained from S Ψ for vanishingantifields, Φ ∗ A , S F P (Φ) = h − Φ ∗ A δδ Φ ∗ A i S Ψ (Φ , Φ ∗ ) = S ( A ) + ¯ C a K ab C b + χ a B a = S ( A ) + s Ψ(Φ) , (4.6)where K ab and s are the Faddeev–Popov operator and the Slavnov variation (2.25), written forany functional F (Φ) as follows: K ab = δχ a δA cµ D cbµ and sF (Φ) = δFδ Φ A δS Ψ δ Φ ∗ A . (4.7)23oth actions S Ψ (Φ , Φ ∗ ), S F P (Φ) are invariant with respect to the BRST transformation[compare with Eq. (2.14)] δ µ Φ A = S A Ψ µ with S A Ψ = (cid:0) D abµ C b , f abc C b C c , B a , (cid:1) , and so does the integrand in the generating functional Z k ( J, Φ ∗ ) of Green’s functions (intro-duced in [50], [51] with the obvious change of the notation Φ ∗ A ≡ K A ) for the vanishing sources, J A = (cid:0) J aµ , J aC , J a ¯ C , J aB (cid:1) ( x ) = 0, and the regulator action, S k = 0, Z k ( J, Φ ∗ ) = Z d Φ exp n i ~ (cid:2) S F P (Φ) + Φ ∗ A s Φ A + S k (Φ) + J Φ (cid:3)o = exp { i ~ W k ( J, Φ ∗ ) } . (4.8)Before taking the limit k →
0, the integrand in the case J = 0 is not BRST-invariant, due tothe easily verified inequality δ µ S k (Φ) = 0 , whereas in the limit k → Z k , W k take correct values, identical with the usualgenerating functionals Z, W . The average effective action Γ k = Γ k (Φ , Φ ∗ ), being the generatingfunctional of vertex functions in the presence of regulators, is introduced according to the ruledescribed by Eq. (3.6) in Section 3,Γ k (Φ , Φ ∗ ) = W k ( J, Φ ∗ ) − J Φ , Φ A = δ W k δJ A , (4.9)with the obvious consequences of the Legendre transformation (4.9), J A = ( δ Γ k ) / ( δ Φ A ). Note,first of all, that the average effective action, by analogy with Eq. (3.12), satisfies an equationand possess tree-level and one-loop approximations which are similar to those for Γ M in Eqs.(3.13), (3.14), however, with S k , instead of the functional M . Second, as to the regulatorfunctions, we suppose that they model the non-perturbative contributions to the self-energypart of the Feynman diagrams, so that the dependence on the parameter k enables one toextract some additional information about the scale dependence of the theory beyond the loopexpansion [52]. Third, the Ward (Slavnov–Tailor) identities [53], [54] for the functionals Z k , W k and Γ k are easily obtained from the general results (3.7), (3.9) and (3.10) for Z M , W M (3.1)and Γ M (3.6), and, due to the property M ∗ A ≡ S ∗ Ak ≡
0, take the form:For Z k , J A δ Z k δ Φ ∗ A + ~ i Z d D x h ( R k,A ) abµν δ Z k δJ bν δA ∗ aµ − ( R k,gh ) ab δ Z k δJ bC δ ¯ C ∗ a + ( R k,gh ) ab δ Z k δJ a ¯ C δC ∗ b i = 0 , (4.10)for W k , J A δ W k δ Φ ∗ A + ~ i Z d D x h ( R k,A ) abµν δ W k δJ bν δA ∗ aµ − ( R k,gh ) ab δ W k δJ bC δ ¯ C ∗ a + ( R k,gh ) ab δ W k δJ a ¯ C δC ∗ b + i ~ n ( R k,A ) abµν δ W k δJ bν δ W k δA ∗ aµ − ( R k,gh ) ab δ W k δJ bC δ W k δ ¯ C ∗ a + ( R k,gh ) ab δ W k δJ a ¯ C δ W k δC ∗ b oi = 0 , (4.11)24or Γ k , 12 (cid:0) Γ k , Γ k (cid:1) − Z d D x h ( R k,A ) abµν A bν δ Γ k δA ∗ aµ − ( R k,gh ) ab C b δ Γ k δ ¯ C ∗ a + ( R k,gh ) ab ¯ C a δ Γ k δC ∗ b + i ~ n ( R k,A ) abµν (cid:0) Γ ′′ − k (cid:1) bν A δ l Γ k δ Φ A δ Φ ∗ aµ − ( R k,gh ) ab (cid:0) Γ ′′ − k (cid:1) bA δ l Γ k δ Φ A δ ¯ C ∗ a + ( R k,gh ) ab (cid:0) Γ ′′ − k (cid:1) aA δ l Γ k δ Φ A δC ∗ b oi = 0 . (4.12)The supermatrix (Γ ′′ − k ) is the inverse of Γ ′′ k , with the elements determined by analogy withEqs. (2.19), (2.20), with the obvious replacement Γ → Γ k . In the limit k →
0, the identities(4.10), (4.11), (4.12) are reduced to the standard Ward identities (2.16).The consistency of the FRG method, based on the introduction of Eq. (4.2), means thatthe values of the average effective actions Γ k calculated for two different gauges determinedby χ a and χ a + ∆ χ a corresponding, in view of Eq. (4.5), to the gauge functionals Ψ andΨ + ∆Ψ, should coincide on the mass-shell for any value of the parameter k (i.e., along theFRG trajectory, but not only in its boundary points). For completeness, let us recall thatthe FRG flow equation for Γ k , which describes the FRG trajectory, reads [51] as follows, withaccount taken of the notation ∂ t = k ddk : ∂ t Γ k = ∂ t S k − ~ i n ∂ t ( R k,A ) abµν (cid:0) Γ ′′ − k (cid:1) ( aµ )( bν ) + ∂ t ( R k,gh ) ab (cid:0) Γ ′′ − k (cid:1) ab o , (4.13)which has the same form for the Φ ∗ A -independent part of Γ k , due to the parametric dependenceon Φ ∗ A of all the terms in (4.13).Due to the result (3.45) for a finite variation of the BRST symmetry breaking term, thevariation of the regulator action S k under the variation of the gauge condition has the form∆ S k = S k,A S A Ψ ∆Ψ (cid:0) s ∆Ψ (cid:1) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ (cid:1)(cid:27) − i , (4.14)with the use of the Slavnov operator s . The corresponding gauged BRST transformation,leading to the variation of S k , as applied to the generating functional Z k in the gauge Ψ(Φ)(4.5), must be characterized by the parameter Λ(Φ) given byΛ(Φ) = ∆Ψ(Φ) (cid:0) s ∆Ψ(Φ) (cid:1) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ(Φ) (cid:1)(cid:27) − i . (4.15)According to Eq. (3.31), in the case under consideration a finite variation of the average effectiveaction Γ k with allowance for the explicit form of ∆ S k (4.14) takes the form∆Γ k = exp n i ~ h ∆ S k i o(cid:18) h ∆ S kA i X n ≥ n ! n h ∆Ψ B i (cid:16) − b F B + i ~ δ Γ k δ Φ ∗ B (cid:17)o n − (cid:16) − b F A + i ~ δ Γ k δ Φ ∗ A (cid:17) + X n ≥ n ! n h ∆Ψ B i (cid:16) − b F B + i ~ δ Γ k δ Φ ∗ B (cid:17)o n − n − (cid:0) Γ k , (cid:1) + b S kA δδ Φ ∗ A − i ~ h b S kA δ Γ k δ Φ ∗ A , Φ B i δ l δ Φ B o(cid:19) h ∆Ψ i + X n ≥ n ! (cid:18) i ~ (cid:19) n − h ∆ S k i , (4.16)25here (3.32) has been taken into account, and the operator b F A is now determined as b F A = − δδ Φ ∗ A + ( − ε B ( ε A +1) (Γ ′′ − k ) BC (cid:16) δ l δ Φ C δ Γ k δ Φ ∗ A (cid:17) δ l δ Φ B . (4.17)Being linear in h ∆Ψ i and, due to (4.14), also in h ∆ S k i , the variation ∆Γ k ( J, Φ ∗ ) takes another“local-like” form; see Eq. (5.6) in [51]:∆Γ k = − (cid:0) Γ k , h ∆Ψ i (cid:1) + b S kA δδ Φ ∗ A h ∆Ψ i − i ~ h b S kA δ Γ k δ Φ ∗ A , Φ B i δ l δ Φ B h ∆Ψ i + h ∆ S k i . (4.18)There is a representation equivalent to Eq. (4.18) and similar to Eqs. (3.36), (3.33):∆Γ k = δ Γ k δ Φ A b F A h ∆Ψ i − b S kA b F A h ∆Ψ i + h ∆ S k i . (4.19)Due to the statement proved at the end of Section 3 [see Eq. (3.48)] regarding the presence inthe gauge theory of a soft BRST breaking term, we can state that the average effective actionΓ k , at least in the non-renormalized case, being evaluated at its extremals, does not depend onthe choice of the gauge condition: δ Γ k δ Φ A = 0 and h ∆ S k i = b S kA b F A h ∆Ψ i −→ ∆Γ k = 0 , (4.20)provided that, in the approximation being linear with respect to ∆Ψ, the variation of theregulators S k (4.14) takes the form∆ S k = − s ( S k ) i ~ ∆Ψ and Λ(Φ) = − i ~ ∆Ψ(Φ) + o (∆Ψ(Φ)) , (4.21)which, after averaging with respect to the mean fields Φ A by using Γ k , leads to h ∆ S k i + h s ( S k ) i ~ ∆Ψ i = 0 ⇐⇒ h ∆ S k i − b S kA b F A h ∆Ψ i = 0 . (4.22)The result given by Eq. (4.20) allows one to revise (in comparison with [51]) the statementon the gauge-dependence of the average effective action, and therefore also on the consistencyof its introduction within the Lagrangian quantization scheme for any value of the parameter k . Indeed, the gauge dependence of the vacuum functional Z k,χ and of the average effectiveaction Γ k on its extremals [51] was explicitly shown respectively in (4.12) and (5.9) therein. Atthe same time, the gauge independence of the average effective action in [51] was achieved onthe mass-shell determined in a larger space of fields (see (6.31), (6.32) therein) with additionaldegrees of freedom, by means of considering the regulators S k as composite fields, however,without taking into account the change of the regulators S k under a change of the gaugecondition in (6.22), (6.27), (6.30), (6.31). In this connection, note that the consideration ofthe regulators as the composite fields following to approach [69] – where their change under avariation of the gauge condition should be taken into account – allows one to provide the gaugeindependence of Γ k on the mass shell determined by the usual average fields Φ A only.26et us calculate the form of the regulator terms S k in different, but mutually related gauges,setting as S k the values in a fixed gauge; for the sake of definiteness, in the Landau gauge. Tothis end, let us consider a family of linear gauges given by the equation χ a ( A, B ) = Λ µ ( ∂, α, β, n ) A µa + ξ B a = 0 with Λ µ ( ∂, α, β, n ) = α∂ µ + β κ µν n n ν . (4.23)Here, we have three numeric, α, β, ξ , and one vector, n µ , gauge parameters. From α, β, ξ , wecan keep only two numbers, β, ξ , for instance, dividing χ a ( A, B ) by α .Particular cases of these gauges can be obtained from the general many-parameter familyunder the choices α = 1 , β = 0 → family of R ξ − gauges , (4.24) β = − α, κ µν = n ρ ∂ ρ η µν , n < , ξ = 0 → generalized Coulomb gauges , (4.25) α = 0 , κ µν = η µd − n ν , ξ = 0 → generalized axial gauges . (4.26)The Landau and Feynman gauges are obtained from the first family for the respective choices ξ = 0 and ξ = 1. The usual Coulomb, χ aC ( A, B ), and axial, χ aA ( A, B ), gauges are derived fromthe second and third families by setting, n µ = (1 , , ...,
0) and n µ = (0 , ..., ,
1) for the respectiveparameters. For completeness, we have χ aC ( A, B ) = ∂ i A ia = 0 , for µ = (0 , i ) , (4.27) χ aA ( A, B ) = A d − a = 0 . (4.28)Denoting the Landau gauge as χ a ( A, B ) (cid:12)(cid:12) α =1 ,β = ξ =0 ≡ χ a ( A ), we can examine the form ofthe regulators which arises for arbitrary values of the parameters α, β, ξ , n µ . Following Eq.(4.14), we immediately obtain the variation of the gauge fermion and its Slavnov variation,respectively,∆Ψ = ¯ C a (cid:0) χ a ( A, B ) − χ a ( A ) (cid:1) = Z d D x ¯ C a (cid:0) { ( α − ∂ µ + β κ µν n n ν } A µa + ξ B a (cid:1) , (4.29) s ∆Ψ = Z d D x n B a (cid:0) { ( α − ∂ µ + β κ µν n n ν } A µa + ξ B a (cid:1) + ¯ C a (cid:0) ( α − ∂ µ + β κ µν n n ν (cid:1) D µab C b o . (4.30)so that the expression for S k = S k + ∆ S k reads S k = S k + Z d D x n A aµ ( x )( R k,A ) abµν ( x ) D bcν C c + ( R k,gh ) ab ( x ) (cid:0) f bcd ¯ C a C c C d − C b B a (cid:1)o × ∆Ψ (cid:16) s ∆Ψ (cid:17) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ (cid:1)(cid:27) − i . (4.31)27rom Eqs. (4.29)–(4.31), we find an approximation linear in ∆Ψ, S k (Φ) = Z d D x n A aµ ( x )( R k,A ) abµν ( x ) A bν ( x ) + ¯ C a ( x )( R k,gh ) ab ( x ) C b ( x ) o − i ~ Z d D x n A aµ ( x )( R k,A ) abµν ( x ) D bcν C c + ( R k,gh ) ab ( x ) (cid:0) f bcd ¯ C a C c C d − C b B a (cid:1)o × Z d D y ¯ C e ( y ) n(cid:0) ( α − ∂ ρ + β κ ρν n n ν (cid:1) A ρe ( y ) + ξ B e ( y ) o , (4.32)depending now on all field variables and having the standard limit S k → k →
0. For α =1 , β = ξ =0, the regulators S k (Φ) are smoothlyreduced to the initial ones S k (Φ), given in the Landau gauge, whereas the expressions for S k (Φ) in any gauges described by Eqs. (4.24)–(4.28) can now be explicitly obtained from Eq.(4.32).In order to obtain the form of S k (4.32) without the terms i ~ , so that this functional shouldstart from the tree-level term, we have to perform integration with respect to the Faddeev–Popov ghost fields in the functional integral Z k (4.8), and then extract the Faddeev-Popovoperator (4.7), K ab , in the resulting gauge and exponentiate it with help of the same Faddeev–Popov ghost fields.It is interesting to investigate the consequences of the study of gauge-dependence in the caseof the Pauli–Villars regularization [55], which does not preserve the gauge and therefore alsothe BRST invariance of the regularized quantum action in the regularization scheme withouthigher derivatives introduced in [3], but we leave this study outside this paper’s scope. In this section, we apply the above general consideration developed in Section 3 and adoptedto the case of the average effective action for Yang–Mills theories in Section 4 in the case ofthe so-called Gribov–Zwanziger [9], [10] and refined Gribov–Zwanziger theories, introduced in[56] and examined in [57], [58], [59], [60]. Let us remind that the Gribov–Zwanziger theory isdetermined by the Gribov–Zwanziger action S GZ (Φ), given in the Landau gauge χ a ( A ) = 0, S GZ (Φ) = S F P (Φ) + M ( A ) , (5.1)which contains an additive non-local BRST-non-invariant summand, implying an inclusion ofthe Gribov horizon [5] and known as the Gribov horizon functional M ( A ), with suppressedcontinuous space-time coordinates x, y , M ( A ) = γ (cid:0) f abc A bµ ( K − ) ad f dec A eµ + D ( N − (cid:1) , for ( K − ) ad ( K ) db = δ ab , (5.2)which is determined by means of the Faddeev-Popov operator ( K ) ab = ∂ µ D µab and the so-calledthermodynamic (Gribov) parameter γ , introduced in a self-consistent way by the gap equation289], [10], [11] ∂∂γ (cid:18) ~ i ln h Z D Φ exp n i ~ S GZ (Φ) oi(cid:19) = ∂ E vac ∂γ = 0 . (5.3)In Eq. (5.3), we have used the definition of the vacuum energy E vac . The idea to improve theGribov–Zwanziger theory is due to the facts that, in the first place, it fails to eliminate allGribov’s copies, and, second, a non-zero value for the Gribov parameter γ is a manifestationof nontrivial properties of the vacuum [59] of the theory as a consequence of restrictions onthe Gribov horizon. The latter means that there exist additional reasons for non-perturbativeeffects, which can be encoded in a set of dimension-2 condensate, h A µa A aµ i , in the case of a non-local Gribov–Zwanziger action with the Yang–Mills gauge fields A µa only, as well as in a similarset of dimension-2 condensates, h A µa A aµ i , h ¯ ϕ µab ϕ abµ i−h ¯ ω µab ω abµ i , for a local Gribov–Zwanzigeraction, S GZ (Φ , φ ), with an equivalent local representation for the horizon functional in termsof the functional S γ , given in an extended configuration space with auxiliary variables, φ ¯ A , S GZ (Φ , φ ) = S F P (Φ) + S γ ( A, φ ) with (5.4) S γ = ¯ ϕ acµ K ab ϕ µbc − ¯ ω acµ K ab ω bcµ + f amb ( ∂ ν ¯ ω acµ )( D mpν c p ) ϕ bcµ + γ f abc A aµ ( ϕ bcµ − ¯ ϕ bcµ ) − D ( N − γ . (5.5)Here, the fields φ ¯ A contain tensors being antisymmetric with respect to the su ( N ) indices, (cid:8) φ ¯ A (cid:9) = (cid:8) ϕ acµ , ¯ ϕ acµ , ω acµ , ¯ ω acµ (cid:9) , (5.6)even for ϕ acµ , ¯ ϕ acµ (i.e., ε ( ϕ )= ε ( ¯ ϕ )=0) and odd for ω acµ , ¯ ω acµ ( ε ( ω )= ε (¯ ω )=1), which form BRSTdoublets [61], δ µ (cid:16) ϕ acν , ¯ ϕ acν (cid:17) = (cid:16) ω acν , (cid:17) µ δ µ (cid:16) ω acν , ¯ ω acν (cid:17) = (cid:16) , ¯ ϕ acν (cid:17) µ. (5.7)Both the non-local M ( A ) and local S γ horizon functionals are not BRST-invariant: sM = γ f abc f cde (cid:2) D bqµ C q ( K − ) ad − f mpn A bµ ( K − ) am K pq C q ( K − ) nd (cid:3) A eµ = 0 , (5.8) sS γ = γf adb (cid:2)(cid:0) D deµ C e ( ϕ µab − ¯ ϕ µab ) + A dµ ω µab (cid:1)(cid:3) = 0 , (5.9)where account has been taken of the relation sK ab = f acb K cd C d , with the latter Slavnovvariation , together with the representation for S γ , being different from those of [32]. Theproblem of finding the Gribov horizon functional in reference frames other than the Landaugauge has been considered in various papers. In [14], this problem was first solved in theapproximation being quadratic in the fields for the linear covariant R ξ -gauges given by Eqs.(4.23), (4.24) for a small value of the parameter ξ ; another form of the functional M ( A, ξ ) wassuggested in [15], and also with the help of the gauged (field-dependent) BRST transformationsin the recent paper [32]. Of course, the suggested result requires a verification of the fact thatthe functional derived actually satisfies the requirement that it should single out the first Gribovhorizon region for the gauge fields A µa in the R ξ -gauge, because an extraction of this region via29he functional M ( A ) was determined non-perturbatively [9] in the Landau gauge only, whereasa corresponding rigorous proof for M ( A, ξ ), i.e., that it actually provides the restriction for thegauge fields A µa within the Gribov region Ω( ξ ),Ω( ξ ) = n A µa (cid:12)(cid:12) χ a ( A, B ) (cid:12)(cid:12) α =1 ,β =0 = 0 , K ab ( ξ ) ≥ o , (5.10)is absent in the literature in an explicit way.As we turn to the refined Gribov–Zwanziger theory, let us propose the refined Gribov–Zwanziger action in a non-local form, and, along the lines of [56], [57], [58], [59], [60], also in alocal form, as follows: S GZ (Φ) → S RGZ (Φ) = S GZ + m A aµ A µa , (5.11) S GZ (Φ , φ ) → S RGZ (Φ , φ ) = S GZ (Φ , φ ) + m A aµ A µa − M (cid:0) ϕ abµ ϕ µab − ω abµ ω µab (cid:1) , (5.12)which can, of course, be considered as theories with composite operators.The only non-vanishing Slavnov variations are those of the first composite fields: s (cid:18) m A aµ A aµ (cid:19) = m A aµ ∂ µ C a = 0 , whereas s (cid:0) M (cid:0) ϕ abµ ϕ µab − ω abµ ω µab (cid:1) (cid:1) = 0 , (5.13)so that the only new BRST-non-invariant term is m A aµ A µa .By virtue of the properties (5.8), (5.9) of the functionals M ( A ) and S γ , as well as due toEq. (5.13) with the composite fields M + m A aµ A µa , and S γ + m A aµ A µa + M ( ϕϕ − ωω ),in Eq. (5.2), these functionals trivially satisfy both the quantum (3.3) and classical (3.2)conditions of soft BRST symmetry breaking, because of the independence on antifields.To establish the gauge-independence of physical quantities in these theories, we have toexamine the models in various gauges from the many-parameter family (4.23), thus explicitlyextending the result of [32]. In this case, the Faddeev–Popov action is written as follows: S F P (Φ , α, β, n µ , ξ ) = S ( A ) + ¯ C a Λ µ ( ∂, α, β, n ) D abµ C b + Λ µ ( ∂, α, β, n ) A µa B a + ξ B a B a . (5.14)The Faddeev–Popov operator K ab = Λ µ D abµ depends on ( α, β, n ), but not on ξ , and the func-tional M should be removed from ( α, β, n, ξ )=(1 , , n, K ab cannot be Her-mitian [14], [15] the application of the Zwanziger trick developed in the Landau gauge seemsto be impossible. Now, we apply the result of the preceding Sections 3, 4 to gauged BRSTtransformations, and then, following Eqs. (3.45), (4.14), the variation of the gauge fermion ∆Ψand its Slavnov variation s ∆Ψ are given by Eqs. (4.29), (4.30), so that the form of the Gribovhorizon functional M (Φ , α, β, n, ξ ) ≡ ˜ M in the gauge under consideration reads, ˜ M = M +∆ M ,˜ M = M ( A ) + γ f abc f cde (cid:2) D bqµ C q ( K − ) ad − f mpn A bµ ( K − ) am K pq C q ( K − ) nd (cid:3) A eµ × ∆Ψ (cid:16) s ∆Ψ (cid:17) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ (cid:1)(cid:27) − i . (5.15)30n the liner approximation with respect to ∆Ψ, we have˜ M = M ( A ) − ı ~ γ f abc f cde (cid:2) D bqµ C q ( K − ) ad − f mpn A bµ ( K − ) am K pq C q ( K − ) nd (cid:3) A eµ × ¯ C h n(cid:0) ( α − ∂ ρ + β κ ρν n n ν (cid:1) A ρh + ξ B h o . (5.16)For α =1 , β = ξ =0, the Gribov horizon functional M (Φ , α, β, n, ξ ) reduces smoothly to M ( A )given in the Landau gauge, whereas the expressions for M (Φ , α, β, n, ξ ) in any linear gaugesare now described by Eqs. (4.24)–(4.28). Thus, for α =1 , β =0 , ξ =1 we deduce from Eq. (5.15)the Gribov horizon functional in the Feynman gauge as in [32], whereas in the Coulomb gauge χ aC ( A, B ) = ∂ i A ia = 0, obtained by setting n µ = (1 , , , ≡ n µ , α = β = 1 , ξ = 0 in Eq.(4.23), in which the Gribov copies were first discovered [5], the functional M (Φ , , , n , ≡ M C has the form M C = M ( A ) + γ f abc f cde (cid:2) D bqµ C q ( K − ) ad − f mpn A bµ ( K − ) am K pq C q ( K − ) nd (cid:3) A eµ × ∆Ψ C (cid:16) s ∆Ψ C (cid:17) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ C (cid:1)(cid:27) − i , (5.17) for ∆Ψ C = ¯ C a ∂ A a , s ∆Ψ C = B a ∂ A a + ¯ C a ∂ D ab C b . (5.18)For the linear γ -dependent part of the functional S γ , which is now BRST-non-invariant, exam-ined in the general gauge χ a ( A, B ) from the family (4.23), we have an expression similar to Eq.(5.15), γ ∂∂γ S γ (Φ , φ, α, β, n, ξ ) = γ ∂∂γ S γ (1 , , n,
0) + γf adb (cid:2)(cid:0) D deµ C e ( ϕ µab − ¯ ϕ µab ) + A dµ ω µab (cid:1)(cid:3) × ∆Ψ (cid:16) s ∆Ψ (cid:17) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ (cid:1)(cid:27) − i . (5.19)On the other hand, in the Coulomb gauge we have the same expression for γ ∂∂γ S γ , given by Eq.(5.19), however, with ∆Ψ C , s ∆Ψ C given by Eq. (5.18). Finally, for the BRST-non-invariantterm m A aµ A aµ , we have a presentation in the gauge (4.23) with account taken of Eqs. (4.29),(4.30), m A aµ A aµ → m A aµ A aµ + m A aµ ∂ µ C a ∆Ψ (cid:16) s ∆Ψ (cid:17) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ (cid:1)(cid:27) − i , (5.20)and also in the Coulomb gauge, m A aµ A aµ → m A aµ A aµ + m A aµ ∂ µ C a ∆Ψ C (cid:16) s ∆Ψ C (cid:17) − h exp (cid:26) − i ~ s (cid:0) ∆Ψ C (cid:1)(cid:27) − i . (5.21)Summarizing, we state that the Gribov horizon functional and the local functional S γ are nowobtained explicitly in an arbitrary gauge from the many-parameter family, described by Eq. It is formally possible to consider the Gribov horizon functional in the axial gauge χ aA (4.28) followingEq. (5.15); however, it is an algebraic gauge without a space-time derivative, which ensures that there is noproblem of Gribov copies due to Singer’s result [6]. We have elaborated a treatment of general gauge theories with arbitrary gauge-fixing in thepresence of soft breaking of the BRST symmetry in the field-antifield formalism. To this end,we have studied the concept of gauged (equivalently, field-dependent) BRST transformationsfor theories more general than the Yang–Mills theory, and calculated the exact Jacobian (2.34)of the corresponding change of variables in the path integral determining the generating func-tionals of Green’s functions, including the effective action. We have argued, on a basis ofanalyzing the non-linear functional equation (2.40) for an unknown field-dependent odd-valuedparameter, which we call the “compensation equation”, that for any finite change of the gaugecondition Ψ → Ψ + ∆Ψ there exists a gauged BRST transformation with a field-dependent pa-rameter Λ(Φ , Φ ∗ | ∆Ψ) in (2.42), depending on ∆Ψ, which permits an entire compensation ofthe finite change of the vacuum functional, i.e., Z Ψ = Z Ψ+∆Ψ .We have investigated the influence of BRST-non-invariant terms, M , added to the quantumaction constructed within the BV formalism and satisfying the so-called soft BRST symme-try breaking condition, on the properties of gauge-dependence of the corresponding effectiveaction Γ M . To study this problem, we have, for the first time, calculated finite changes ofthe generating functionals Z M , W M and the effective action Γ M under a finite change of thegauge condition (3.22), (3.25), (3.31) and found that, at least with accuracy up to the linearterms in the variation of the gauge-fixing functional ∆Ψ, the effective action does not dependon its extremals on the choice of gauge, provided that the change of the BRST-broken termis subject to a corresponding gauged BRST transformation with the parameter Λ(Φ , Φ ∗ | ∆Ψ)determined by (3.43) and used in (3.48), which is our principal result. Thereby, the conceptof soft BRST symmetry breaking does not violate the consistency of Lagrangian quantizationwithin the perturbation theory, so that the suggested prescription allows one, first of all, toobtain perturbatively the form of the soft BRST symmetry broken term in a different gauge by Note that the term “compensation equation” has been recently suggested [62], [63] for BRST symmetry inthe study of finite BRST–BFV and BRST–BV transformations, respectively, as well as for BRST-antiBRSTsymmetry in Yang–Mills [64] and general gauge theories in Lagrangian [65], [66] and generalized Hamiltonian[67], [68] formulations. S -matrix. We believe that these results should also be valid for a renormalized theory with softBRST symmetry breaking; however, this requires a detailed proof.We have demonstrated the applicability of our statements in the case of the functionalrenormalization group approach to the Yang–Mills and gravity theories and found, withinthe many-parameter family of linear gauges (4.23), the form of the regulator functionals inarbitrary (4.31) and linear gauges (4.32) from the same family, starting from those given, e.g.,in the Landau gauge. This construction allows one to restore the gauge-independence of theaverage effective action Γ k along the entire trajectory of a FRG flow (4.20) without havingrecourse to the composite fields technique. Finally, the general concept of the gauged BRSTtransformations related to the same gauge theory, however, given in different gauges, appears tobe very useful in constructing the Gribov–Zwanziger and the refined Gribov–Zwanziger actionsfor a many-parameter family of gauges, including the Coulomb, axial and covariant gauges(5.16), (5.21). This result extends the Gribov–Zwanziger theory with R ξ -gauges examined in[32]. At the same time, there arises a problem of comparing the form of the horizon functional inthe Coulomb gauge obtained perturbatively by means of gauged BRST transformations (5.17),(5.18) with the horizon functional obtained following to the Zwanziger non-perturbative recipe[18], which is planned to consider as a separates study. Of course, our arguments are validfor gauge theories with soft breaking of the BRST symmetry in case the transformed BRSTbreaking terms satisfy the same conditions in the final gauge as the untransformed ones in theinitial gauge, however, with a possible violation of the condition (3.3) of soft BRST symmetrybreaking. For instance, this means that for the Gribov horizon functional in a different gaugeamongst the examined family of gauges one needs to verify the validity of extracting the Gribovhorizon precisely from the configuration space of Yang–Mills fields, perhaps with the examineddimension-2 condensate.Finally, it may be hoped that, due to the appearance of the Higgs field in view of the spon-taneous breaking of the initial gauge invariance related to the group SU (2) for the electroweakLagrangian, one can examine an addition (associated with the Higgs field) to the gauge-invariant(with respect to the SU (2) group) action of a soft BRST-breaking term, so that the descriptionof the resulting model will be made consistent in the conventional Lagrangian path integralapproach developed in this paper. We consider this problem as the next one to be examined.Concluding, let us mention, first, the treatment of the Gribov horizon functional as a com-posite field [69], second, the recently obtained BRST-antiBRST extension [64] of the Gribov–Zwanziger theory in different gauges in a way consistent with the gauge independence of thephysical S -matrix, third, the concept of soft BRST-antiBRST symmetry breaking developedon a basis of finite field-dependent BRST-antiBRST transformations in [66].33 cknowledgments I thank V.A. Rubakov, S.V. Demidov and the participants of the Seminar on TheoreticalPhysics at the Institute for Nuclear Research RAS, where the results of the present study werepresented for the first time on 02.12.2013. The author also thanks V.P. Gusynin, P.M. Lavrov,O. Lechtenfeld, P.Yu. Moshin and K.V. Stepanyantz for useful discussions, as well as to I.V.Tyutin for comments on the unitarity problem. I am grateful to the authors of [11] for theircritical assessment of earlier papers, leading to a better understanding of the problem underconsideration. I thank A.D. Pereira Jr. for discussions and comments on the Gribov horizonin the Coulomb and maximal Abelian gauges. The study was supported by the RFBR grantunder Project No. 12-02-00121, and by the grant of Leading Scientific Schools of the RussianFederation under Project No. 88.2014.2.
AppendixA On Solution of Equation (2.40)
In this Appendix, we present arguments for the existence of a solution for Eq. (2.40) withrespect to an unknown field-dependent odd functional, Λ (Φ , Φ ∗ ), in the form (2.42). In doingso, we follow a strategy partially based on some previously known facts. First, any gaugetheory can be equivalently transformed to a gauge theory in the standard basis [46], with thegenerators and proper zero eigenvectors having the representation (cid:8) R iα , Z α α , . . . , Z α L − α L − , Z α L − α L (cid:9) → ((cid:0) R iα , (cid:1) , δ ¯ α B ! , . . . , δ ¯ α L − B L − ! , (cid:0) , δ ¯ α L − B L (cid:1)) (A.1)for the division of indices α s , s = 0 , ..., L being related with the rank conditions (2.2), (2.4)as α = ( α, B ), α s = ( ¯ α s , B s +1 ), for s = 1 , . . . , L − α L = B L = m L . Note that thedefinition (2.2) of an L -stage reducible gauge theory in the standard basis (A.1) looks simple, Z α s − α s Z α s α s +1 = 0, for vanishing K iα s − α s +1 , for s = 0 , ..., L −
1. Second, a transition to the standardbasis from the initial gauge theory can be realized as a non-degenerate (generally, non-local)change of variables, Φ A → Φ ′ A (Φ), in M , such that Z Ψ (0 , Φ ∗ ) = Z D Φ exp n i ~ S Ψ o = Z D Φ ′ exp n i ~ ¯ S Ψ (Φ ′ ) o , with ¯ S Ψ (Φ ′ ) = S Ψ (Φ(Φ ′ )) − i ~ Str ln (cid:13)(cid:13)(cid:13)(cid:13) δ Φ A δ Φ ′ B (cid:13)(cid:13)(cid:13)(cid:13) . (A.2)We then use the fact that any gauge theory with an open algebra of generators R iα (beingalready in standard basis) can be equivalently transformed to a theory with a closed algebra[47], so that in the new basis of the generators of gauge transformations, R ′ iα [obtained by means34f additive extension of R iα by trivial gauge generators, R ′ iα ( A ′ ) = R iα ( A ′ ) + S ,j ( A ′ ) M ijα ( A ′ )],the Lie-type structure functions F ′ γαβ ( A ′ ) in relations such as (2.27), R ′ iα , j ( A ′ ) R ′ jβ ( A ′ ) − R ′ iβ , j ( A ′ ) R ′ jα ( A ′ ) = − R ′ iγ ( A ′ ) F ′ γαβ ( A ′ ) , (cid:20) where R ′ iα , j ≡ δδA ′ j R ′ iα (cid:21) (A.3)are the only ones to survive. A transition to the gauge theory subject to relations (A.3) mayalso be effectively realized as a non-degenerate change of variables, Φ ′ A → Φ ′′ A (Φ) in M : Z Ψ (0 , Φ ∗ ) = Z D Φ ′ exp n i ~ ¯ S Ψ (Φ ′ ) o = Z D Φ ′′ exp n i ~ ˆ S Ψ (Φ ′′ ) o , with ˆ S Ψ (Φ ′′ ) = ¯ S Ψ (Φ ′ (Φ ′′ )) − i ~ Str ln (cid:13)(cid:13)(cid:13)(cid:13) δ Φ ′ A δ Φ ′′ B (cid:13)(cid:13)(cid:13)(cid:13) . (A.4)Notice that the transformations Φ → Φ ′ , Φ ′ → Φ ′′ have a more general form than the gaugedBRST transformations (2.15) and can be equivalently realized by a set of operations (2.9) withdefinite respective functionals, X i (Φ , Φ ∗ ), for i = 1 , S Ψ ,ˆ S Ψ = ~ i ln (cid:20) exp {− [∆ , X ] } · exp {− [∆ , X ] } exp (cid:26) i ~ S Ψ (cid:27)(cid:21) . (A.5)Since the transformed action ˆ S Ψ (A.4) has a form being linear in the antifields, ˆ S Ψ (Φ ′′ , Φ ∗ ) =Φ ∗ A ˆ S A Ψ (Φ ′′ ), we now obtain the relations (derivatives with respect to the fields in ˆ S A Ψ , B and ˆ S Ψ , B are understood as taken for Φ ′′ B , and we omit the Jacobi matrices of the above changes ofvariables for the sake of simplicity) (cid:0) ˆ S AB Ψ = 0 , ∆ ˆ S A Ψ = 0 (cid:1) = ⇒ ˆ S A Ψ , B ˆ S B Ψ = 0 , (A.6)which, first of all, imply the nilpotency of the Slavnov variation, ˆ s e = 0, in the new basis of thegauge algebra and, second, allow one to present the equation (2.40) for a gauge theory with aclosed algebra as an equation for the parameter ˆΛ, i ~ n ln (cid:0) s ˆΛ (cid:1)o = ˆ s (cid:0) ∆Ψ(Φ ′′ ) (cid:1) , (A.7)where account has been taken of the fact that the generator ˆ s coincides with ˆ s e , being, however,expressed in terms of the action ˆ S Ψ and fields Φ ′′ A .Using the functional equation (A.7), we can express the variation ∆Ψ(Φ ′′ ) with accuracyup to BRST exact terms, ˆ sR (Φ ′′ ),∆Ψ(Φ ′′ ) = i ~ ˆΛ(Φ ′′ )(ˆ s ˆΛ) − n ln (cid:0) s ˆΛ (cid:1)o , (A.8) For simplicity, we use notation for the gauge fermion Ψ and its variation ∆Ψ in the case of a theory witha closed algebra which is the same as the notation used for a theory with an open algebra and the action S Ψ . ′′ ) reads as follows:ˆ s ˆΛ(Φ ′′ ) = exp (cid:26) − i ~ ˆ s (cid:0) ∆Ψ(Φ ′′ ) (cid:1)(cid:27) − ⇒ ˆΛ(Φ ′′ ) = ∆Ψ(Φ ′′ ) (cid:0) ˆ s ∆Ψ(Φ ′′ ) (cid:1) − h exp (cid:26) − i ~ ˆ s (cid:0) ∆Ψ(Φ ′′ ) (cid:1)(cid:27) − i . (A.9)Finally, in order to obtain a solution of the initial equation (2.40) which equivalently maybe rewritten as (cid:0) s e Λ (cid:1) − (cid:0) ←− s e Λ (cid:1) = exp (cid:20) i ~ (cid:16) exp n − [∆ , ∆Ψ] o − (cid:17) S Ψ (cid:21) (A.10)we have to make the inverse transformations Φ ′′ → Φ ′ → Φ for ˆΛ(Φ ′′ ) with respect to thoseused for the transition to the standard basis (A.1) and then to the gauge theory with a closedalgebra (A.3), described in Eqs. (A.2), (A.4), and therefore a solution, Λ(Φ , Φ ∗ ), of Eq. (2.40)does exist and is expressed by the variation ∆Ψ(Φ) in the form (2.42). References [1] M. Henneaux and C. Teitelboim,
Quantization of gauge systems ,Princeton University Press, 1992;S. Weinberg,
The quantum theory of fields, Vol. II , Cambridge University Press, 1996;D.M. Gitman and I.V. Tyutin,
Quantization of fields with constraints , Springer, 1990.[2] N.N. Bogolyubov and D.V. Shirkov,
Introduction to theory of Quantized Fields , John Wileyand Sons, New York, 1980.[3] L.D. Faddeev and A.A. Slavnov,
Gauge Fields, Introduction to Quantum Theory , seconded., Benjamin, Reading, 1990.[4] C. Becchi, A. Rouet and R. Stora,
Renormalization of the abelian Higgs-Kibble model ,Commun. Math. Phys. 42 (1975) 127;I.V. Tyutin,
Gauge invariance in field theory and statistical physics in operator formalism ,Lebedev Inst. preprint N 39 (1975), [arXiv:0812.0580[hep-th]].[5] V.N. Gribov,
Quantization of nonabelian gauge theories , Nucl.Phys. B139 (1978) 1.[6] I.M.Singer,
Some remarks on the Gribov ambiguity , Comm.Math.Phys. 60 (1978) 7-12. In (A.10) the action of group-like element g (Λ(Φ , Φ ∗ )) = (cid:0) ←− s e Λ (cid:1) , being trivial for nilpotent s e , measuresdifference of (2.40) with the equation (A.7) for the gauge theory with closed algebra Gauge fields beyond perturbation theory , [arXiv:1310.8164[hep-th]].[8] J. Serreau, M. Tissier and A. Tresmontant,
Covariant gauges without Gribov ambiguitiesin Yang-Mills theories , [arXiv:1307.6019[hep-th]].[9] D. Zwanziger,
Action from the Gribov horizon , Nucl. Phys. B321 (1989) 591.[10] D. Zwanziger,
Local and renormalizable action from the Gribov horizon ,Nucl. Phys. B323 (1989) 513.[11] M.A.L. Capri, A.J. G´omes, M.S. Guimaraes, V.E.R. Lemes, S.P. Sorellao andD.G. Tedesko,
A remark on the BRST symmetry in the Gribov-Zwanzider theory ,Phys. Rev. D82 (2010) 105019, arXiv:1009.4135 [hep-th];L. Baulieu, M.A.L. Capri, A.J. Gomes, M.S. Guimaraes, V.E.R. Lemes, R.F. Sobreiroand S.P. Sorella,
Renormalizability of a quark-gluon model with soft BRST breaking in theinfrared region , Eur. Phys. J. C66 (2010) 451, arXiv:0901.3158 [hep-th];D. Dudal, S.P. Sorella, N. Vandersickel and H. Verschelde,
Gribov no-pole condition,Zwanziger horizon function, Kugo-Ojima confinement criterion, boundary conditions,BRST breaking and all that , Phys. Rev. D79 (2009) 121701, arXiv:0904.0641 [hep-th];L. Baulieu and S.P. Sorella,
Soft breaking of BRST invariance for introducing non-perturbative infrared effects in a local and renormalizable way ,Phys. Lett. B671 (2009) 481, arXiv:0808.1356 [hep-th];M.A.L. Capri, A.J. G´omes, M.S. Guimaraes, V.E.R. Lemes, S.P. Sorella and D.G. Tedesko,
Renormalizability of the linearly broken formulation of the BRST symmetry in presence ofthe Gribov horizon in Landau gauge Euclidean Yang-Mills theories ,arXiv:1102.5695 [hep-th];D. Dudal, S.P. Sorella and N. Vandersickel,
The dynamical origin of the refinement of theGribov-Zwanziger theory , arXiv:1105.3371 [hep-th].[12] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker, and A. Sternbeck,
Lattice gluody-namics computation of Landau gauge Green’s functions in the deep infrared , Phys. Lett.B676 (2009) 69, arXiv:0901.0736[hep-lat];V. Bornyakov, V. Mitrjushkin, and M. Muller-Preussker,
SU(2) lattice gluon propaga-tor: Continuum limit, finite-volume effects and infrared mass scale m(IR) ,Phys. Rev. D81(2010) 054503, arXiv:0912.4475[hep- lat].[13] V.G. Bornyakov, V.K. Mitrushkin and R.N. Rogalyov,
Gluon propagators in 3D SU(2)theory and effects of Gribov copies , arXiv:1112.4975[hep-lat].3714] R.F. Sobreiro and S.P. Sorella,
A study of the Gribov copies in linear covariant gauges inEuclidean Yang-Mills theories , JHEP 0506 (2005) 054, [arXiv:hep-th/0506165].[15] P. Lavrov and A. Reshetnyak,
Gauge dependence of vacuum expectation values of gaugeinvariant operators from soft breaking of BRST symmetry. Example of Gribov-Zwanzigeraction , to appear in Proc. of QUARKS’2012, arXiv:1210.5651[hep-th].[16] D. Dudal, M.A.L. Capri, J.A. Gracey et al.,
Gribov Ambiguities in the Maximal AbelianGauge , Braz. J. Phys. 37 (2007) 320-324, [arXiv:hep-th/0609160].[17] Sh. Gongyo and H. Iida,
Gribov-Zwanziger action in SU (2) Maximally Abelian Gauge with U (1) Landau Gauge
Phys.Rev. D 89 (2014) 025022, arXiv:1310.4877[hep-th].[18] D. Zwanziger,
Equation of State of Gluon Plasma from Local Action , Phys.Rev. D 76(2007) 125014, [arXiv:hep-th/0610021].[19] M. de Cesare, G. Esposito and H. Ghorbani, Size of the Gribov region in curved spacetime,Phys. Rev. D 88 (2013) 087701, arXiv:1308.5857[hep-th].[20] P.M. Lavrov and I.V. Tyutin.
On the structure of renormalization in gauge theories ,Sov. J. Nucl. Phys. 34 (1981) 156;P.M. Lavrov and I.V. Tyutin.
On the generating functional for the vertex functions inYang-Mills theories , Sov. J. Nucl. Phys. 34 (1981) 474.[21] B.L. Voronov, P.M. Lavrov and I.V. Tyutin,
Canonical transformations and gauge depen-dence in general gauge theories , Sov. J. Nucl. Phys. 36 (1982) 292.[22] P. Lavrov, O. Lechtenfeld and A. Reshetnyak,
Is soft breaking of BRST symmetry consis-tent? , JHEP 1110 (2011) 043, arXiv:1108.4820 [hep-th].[23] P. Lavrov, O. Radchenko and A. Reshetnyak,
Soft breaking of BRST symmetry and gaugedependence , MPLA A27 (2012) 1250067, arXiv:1201.4720 [hep-th].[24] M. Vasiliev,
Higher spin gauge theories in various dimensions , Fortsch. Phys. 52 (2004)702–717, [arXiv:hep-th/0401177];D. Sorokin,
Introduction to the classical theory of higher spins , AIP Conf. Proc. 767 (2005)172–202, [arXiv:hep-th/0405069];N. Bouatta, G. Comp`ere, A. Sagnotti,
An introduction to free higher-spin fields ,[arXiv:hep-th/0409068];X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev,
Nonlinear higher spin theories invarious dimensions , [arXiv:hep-th/0503128];A. Fotopoulos, M. Tsulaia,
Gauge Invariant Lagrangians for Free and InteractingHigher Spin Fields. A review of BRST formulation , Int.J.Mod.Phys. A24 (2008) 1–60,[arXiv:0805.1346[hep-th]]; 38.L. Buchbinder and A. Reshetnyak,
General Lagrangian Formulation for Higher SpinFields with Arbitrary Index Symmetry. I. Bosonic fields , Nucl. Phys. B 862 (2012) 270-323, [arXiv:1110.5044[hep-th]];A. Reshetnyak,
General Lagrangian Formulation for Higher Spin Fields with Arbitrary In-dex Symmetry. 2. Fermionic fields
Nucl. Phys. B 869 (2013) 523-597, [arXiv:1211.1273[hep-th]].[25] I.A. Batalin and G.A. Vilkovisky,
Gauge algebra and quantization , Phys. Lett. 102B (1981)27;[26] I.A. Batalin and G.A. Vilkovisky,
Quantization of gauge theories with linearly dependentgenerators , Phys. Rev. D28 (1983) 2567.[27] O. Radchenko and A. Reshetnyak,
Notes on soft breaking of BRST symmetry in theBatalin-Vilkovisky formalism , Russ.Phys.J. 55 (2013) 1005-1010, arXiv:1210.6140 [hep-th].[28] P. Lavrov and O. Lechtenfeld,
Field-dependent BRST transformations in Yang-Mills the-ory , Phys.Lett. B725 (2013) 382-385, arXiv:1305.0712[hep-th].[29] S.D. Joglekar and B.P. Mandal,
Finite field dependent BRS transformations , Phys. Rev.D51 (1995) 1919.[30] S.D. Joglekar,
Connecting Green’s functions in an arbitrary pair of gauges and an applica-tion to planar gauges , IJMPA 16 (2001) 5043.[31] S. Upadhyay, S.K. Rai and B.P. Mandal,
Off-Shell Nilpotent Finite BRST/Anti-BRSTTransformations , J. Math. Phys. 52 (2011) 022301, arXiv:1002.1373hep-th].[32] P. Lavrov and O. Lechtenfeld,
Gribov horizon beyond the Landau gauge , Phys.Lett. B725(2013) 386-388, arXiv:1305.2931[hep-th].[33] B.S. DeWitt,
Dynamical theory of groups and fields , Gordon and Breach, 1965.[34] O.M. Khudaverdian and A.P. Nersessian,
On the geometry of the Batalin-Vilkovsky for-malism
Mod.Phys.Lett. A8 (1993) 2377-2386, [arXiv:hep-th/9303136].[35] I.A. Batalin and I.V. Tyutin,
On possible generalizations of field - antifield formalism ,Int.J.Mod.Phys. A8 (1993) 2333-2350, [arXiv:hep-th/9211096];
On the multilevel generalization of the field - antifield formalism
Mod.Phys.Lett. A8 (1993)3673-3682, [arXiv:hep-th/9309011];
On the multilevel field - antifield formalism with the most general Lagrangian hypergauges
Mod.Phys.Lett. A9 (1994) 1707-1716, [arXiv:hep-th/9403180].[36] A.S. Schwarz,
Geometry of Batalin-Vilkovisky quantization , Commun.Math.Phys. 155(1993) 249-260, [arXiv:hep-th/9205088];M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky,
The geometry of the masterequation and topological quantum field theory , Int. J. Modern Phys. A 12 (1997) 1405-1429.3937] P.M. Lavrov, P.Yu. Moshin and A.A. Reshetnyak,
Superfield formulation of theLagrangian BRST quantization method , Mod.Phys.Lett. A10 (1995) 2687-2694,[arXiv:hep-th/9507104].[38] D.M. Gitman, P.Yu. Moshin and A.A. Reshetnyak,
Local superfield Lagrangian BRSTquantization
J.Math.Phys. 46 (2005) 072302, [arXiv:hep-th/0507160];
An Embedding of the BV quantization into an N=1 local superfield formalism , Phys.Lett.B621 (2005) 295-308, [arXiv:hep-th/0507046].[39] A.A. Reshetnyak,
The Effective action for superfield Lagrangian quantization in reduciblehypergauges , Russ.Phys.J. 47 (2004) 1026-1036, [arXiv:hep-th/0512327].[40] I.A. Batalin and K. Bering,
On generalized gauge-fixing in the field-antifield formalism ,Nucl.Phys. B739 (2006) 389-440, [arxiv:hep-th/0512131].[41] A. Kiselev,
The geometry of variations in Batalin-Vilkovisky formalism , Journal of Physics:Conference Series 474 (2013) 012024, 1-51 [arXiv:1312.1262 [math-ph]].[42] G. Leibbrandt,
Introduction to the technique of the dimensional regularization , Rev. Mod.Phys. 47 (1975) 849.[43] K.E. Kallosh and I.V. Tyutin,
The equivalence theorem and gauge invariance in renormal-izable theories , Sov. J. Nucl. Phys. 17 (1973) 98.[44] I.V. Tyutin,
Once again on the equivalence theorem , Phys. Atom. Nucl. 65 (2002) 194-202,[arxiv:hep-th/0001050].[45] S.R. Esipova, P.M. Lavrov and O.V. Radchenko, Int. J. Mod. Phys. A 29 (2014) 1450065,arXiv:1312.2802[hep-th].[46] I.A. Batalin, P.L. Lavrov and I.V. Tyutin,
An Sp(2)covariant quantization of gauge theorieswith linearly dependent generators , J. Math. Phys. 32, (1991) 532.[47] B.L. Voronov, I.V. Tyutin,
Formulation of gauge theories of general form. I , Theor. Math.Phys. 50 (1982) 218-225.[48] C. Wetterich,
Average Action And The Renormalization Group Equations.
Nucl. Phys.B352 (1991) 529.[49] M. Reuter and C. Wetterich,
Average action for the Higgs model with abelian gauge sym-metry,
Nucl. Phys. B391 (1993) 147.[50] M. Reuter and C. Wetterich,
Effective average action for gauge theories and exact evolutionequations,
Nucl. Phys. B417 (1994) 181.[51] P. Lavrov and I. Shapiro,
On the Functional Renormalization Group approach for Yang-Mills fields , JHEP, 1306 (2013) 086, [arXiv:1212.2577[hep-th]].4052] J. Polchinski,
Renormalization and effective lagrangians,
Nucl. Phys. B231, 269 (1984).[53] A.A. Slavnov,
Ward identities in gauge theories , Theor. Math. Phys. 10 (1972) 99.[54] J.C. Taylor,
Ward identities and charge renormalization of the Yang-Mills field , Nucl.Phys. B33 (1971) 436.[55] W. Pauli, F. Villars,
On the Invariant Regularization in Relativistic Quantum Theory , Rev.Mod. Phys, 21 (1949) 434-444.[56] D. Dudal, J. A. Gracey, S.P. Sorella et all,
A refinement of the Gribov-Zwanziger approachin the Landau gauge: infrared propagators in harmony with the lattice results , Phys.Rev.D78 (2008) 065047, arXiv:0806.0348[hep-th].[57] D. Dudal, J.A. Gracey, S.P. Sorella et all ,
The Landau gauge gluon and ghost propagatorin the refined Gribov-Zwanziger framework in 3 dimensions
Phys.Rev. D78 (2008) 125012,arXiv:0808.0893[hep-th].[58] D. Dudal, S. Sorella, N. Vandersickel, H. Verschelde,
A Renormalization group invariantscalar glueball operator in the (Refined) Gribov-Zwanziger framework , JHEP 0908 (2009)110, [arXiv:0906.4257[hep-th]].[59] S. Sorella, D. Dudal, S. Guimaraes, N. Vandersickel,
Features of the Refined Gribov-Zwanziger theory: Propagators, BRST soft symmetry breaking and glueball masses , PoSFACESQCD (2010) 022, [arXiv:1102.0574[hep-th]].[60] D. Dudal, S. Sorella, N. Vandersickel,
The dynamical origin of the refinement of the Gribov-Zwanziger theory , Phys.Rev. D84 (2011) 065039, [arXiv:1105.3371[hep-th]].[61] D. Dudal, S.P. Sorella and N. Vandersickel,
More on the renormalization of the horizonfunction of the Gribov-Zwanziger action and the Kugo-Ojima Green function(s) , Eur. Phys.J. C (2010) 283, [arXiv:1001.3103 [hep-th]].[62] I.A. Batalin, P.M. Lavrov and I.V. Tyutin, A systematic study of finite BRST-BFV trans-formations in generalized Hamiltonian formalism , arXiv:1404.4154[hep-th].[63] I.A. Batalin, P.M. Lavrov, I.V. Tyutin,
A systematic study of finite BRST-BV transfor-mations in field-antifield formalism , arXiv:1405.2621[hep-th].[64] P.Yu. Moshin and A.A. Reshetnyak,
Field-dependent BRST-antiBRST Transformationsin Yang-Mills and Gribov-Zwanziger Theories , Nucl. Phys. B 888C (2014) 92-128,arXiv:1405.0790 [hep-th].[65] P.Yu. Moshin and A.A. Reshetnyak,
Finite BRST-antiBRST Transformations in La-grangian Formalism , arXiv:1406.0179[hep-th].[66] P.Yu. Moshin and A.A. Reshetnyak,
Field-Dependent BRST-antiBRST Lagrangian Trans-formations , arXiv:1406.5086[hep-th]. 4167] P.Yu. Moshin and A.A. Reshetnyak,
Finite BRST-antiBRST Transformations in Gener-alized Hamiltonian Formalism , Int. J. Mod. Phys. A (2014), arXiv:1405.7549 [hep-th].[68] I.A. Batalin, P.M. Lavrov and I.V. Tyutin,
A systematic study of finite BRST-BFV Trans-formations in Sp(2)-extended generalized Hamiltonian formalism , arXiv:1405.7218[hep-th].[69] A. Reshetnyak,