Field-dependent BRST-antiBRST Transformations in Yang-Mills and Gribov-Zwanziger Theories

We introduce the notion of finite BRST-antiBRST transformations, both global and field-dependent, with a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in Yang-Mills theories. It turns out that the finite transformations are quadratic in their parameters. At the same time, exactly as in the case of finite field-dependent BRST transformations for the Yang-Mills vacuum functional, special field-dependent BRST-antiBRST transformations, with $s_{a}$-potential parameters $\lambda_{a}=s_{a}\Lambda$ induced by a finite even-valued functional $\Lambda$ and by the anticommuting generators $s_{a}$ of BRST-antiBRST transformations, amount to a precise change of the gauge-fixing functional. This proves the independence of the vacuum functional under such BRST-antiBRST transformations. We present the form of transformation parameters that generates a change of the gauge in the path integral and evaluate it explicitly for connecting two arbitrary $R_{\xi }$-like gauges. For arbitrary differentiable gauges, the finite field-dependent BRST-antiBRST transformations are used to generalize the Gribov horizon functional $h$, given in the Landau gauge, and being an additive extension of the Yang-Mills action by the Gribov horizon functional in the Gribov-Zwanziger model. This generalization is achieved in a manner consistent with the study of gauge independence. We also discuss an extension of finite BRST-antiBRST\ transformations to the case of general gauge theories and present an ansatz for such transformations.


Introduction
Contemporary quantization methods for gauge theories [1,2,3,4] are based primarily on the special supersymmetries known as BRST symmetry [5,6,7] and BRST-antiBRST symmetry [8,9,10,11]. They are characterized by the presence of a Grassmann-odd parameter µ and two Grassmann-odd parameters (µ,μ), respectively. In the framework of the Sp (2)-covariant schemes of generalized Hamiltonian [12,13] and Lagrangian [15,16] quantization (see also * moshin@rambler.ru † reshet@ispms.tsc.ru study of [31] extends the results of [23] to first-rank theories with a closed algebra and solves the problem of gaugeindependence for gauge theories with the so-called soft breaking of BRST symmetry. This problem was raised in [32] to study the problem of Gribov copies [33] by using various gauges in the Gribov-Zwanziger approach [34]; for recent progress, see [35,36,37,38,39]. On the other hand, there emerges the problem of finding a correspondence of the quantum action in the BRST-antiBRST invariant Lagrangian quantization [15,16,17], where gauge is introduced by a Bosonic gauge-fixing functional, F , with the quantum action of the same theory in a different gauge, F + ∆F , for a finite value ∆F , by using a change of variables in the vacuum functional. This problem has not been solved even in theories of Yang-Mills type. Note that finite field-dependent antiBRST transformations in Yang-Mills theories were considered in [25] in the same way as in the case of BRST transformations [20], so as to relate the antiBRST invariant quantum action of a Yang-Mills theory in different gauges by using an ansats for a term introduced to the quantum action in order to satisfy an infinitesimal functional equation for the transformation parameter. The study of [26] proposed finite twoparametric BRST-antiBRST transformations ("mixed", by the terminology [26]): "δ m φ = ← − s a Θ 1 + ← − s ab Θ 2 " in (3.7), including field-dependent ones, which form a Lie superalgebra; however, without any parameters, constant and/or field-dependent, being quadratic in Θ 1 , Θ 2 (allowing one to consider BRST-antiBRST transformations as group transformations), which prohibits the complete BRST-antiBRST invariance of the quantum action in Yang-Mills theories and similarly in more general gauge theories. Therefore, this leads immediately to the problem of finding a solution for the above functional equation, since the latter does not "feel" the finite polynomial character of the parameters Θ 1 ·Θ 2 , and therefore prohibits the gauge independence of the vacuum functional under finite field-dependent BRST-antiBRST transformations even for functionally-dependent parameters (see footnote 6).
A similar problem in the Sp (2)-covariant generalized Hamiltonian formalism [12,13] remains unsolved 4 as well. We expect that the solution of these problems in the Lagrangian and Hamiltonian quantization schemes for gauge theories should be based on the concept of finite BRST-antiBRST transformations with an Sp (2)-doublet of Grassmann-odd parameters µ a (φ) depending on the field variables. This would allow one to generate the Gribov horizon functional by using different gauges in a way consistent with the gauge-independence of the path integral, based on the Gribov-Zwanziger prescription [34] and starting from the BRST-antiBRST invariant Yang-Mills quantum action in the Landau gauge.
Motivated by these reasons, we intend to address the following issues, paying our attention primarily to the Yang-Mills theory in Lagrangian formalism: 1. introduction of finite BRST-antiBRST transformations, being polynomial in powers of a constant Sp (2)-doublet of Grassmann-odd parameters λ a and leaving the quantum action of the Yang-Mills theory invariant to all orders in λ a ; 2. definition of finite field-dependent BRST-antiBRST transformations, being polynomial in powers of an Sp (2)doublet of Grassmann-odd functionals λ a (φ) depending on the classical Yang-Mills fields, the ghost-antighost fields, and the Nakanishi-Lautrup fields; calculation of the Jacobian related to a change of variables by using a special class of such transformations with s a -potential parameters λ a (φ) = s a Λ(φ) for a Grassmann-even functional Λ(φ) and Grassmann-odd generators s a of BRST-antiBRST transformations; 3. solution of the so-called compensation equation for an unknown functional Λ generating the Sp (2)-doublet λ a with the purpose of establishing a relation of the Yang-Mills quantum action S F in a gauge determined by a gauge Boson F with the quantum action S F +∆F in a different gauge F + ∆F ; 4 For the recent progress achieved in this area since the appearance of the present work in arXiv, see footnote 11 in Discussion. 4. explicit construction of the parameters λ a of finite field-dependent BRST-antiBRST transformations generating a change of the gauge in the path integral within a class of linear R ξ -like gauges realized in terms of Bosonic gauge functionals F (ξ) , with ξ = 0, 1 corresponding to the Landau and Feynman (covariant) gauges, respectively; 5. construction of the Gribov horizon functional h ξ in arbitrary R ξ -like gauges by means of finite field-dependent BRST-antiBRST transformations starting from a known BRST-antiBRST non-invariant functional h, given in the Landau gauge and realized in terms of the Bosonic functional F (0) .
The present work is organized as follows. In Section 2, we remind the general setup of the BRST-antiBRST Lagrangian quantization of general gauge theories and list its basics ingredients. In Section 3, we introduce the notion of finite BRST-antiBRST transformations, both global and local (field-dependent). We find an explicit Jacobian corresponding to this change of variables in theories of Yang-Mills type and show that, exactly as in the case of fielddependent BRST transformations for the Yang-Mills vacuum functional [23], the field-dependent transformations amount to a precise change of the gauge-fixing functional. In Section 4, we present the form of transformation parameters that generates a change of the gauge and evaluate it for connecting two arbitrary R ξ -like gauges in Yang-Mills theories. In Section 5, the Gribov horizon functional in an arbitrary R ξ -like gauge, and generally in any differentiable gauge, is determined with the help of respective finite field-dependent BRST-antiBRST transformations. In Discussion, we make an overview of our results and outline some open problems. In particular, we discuss an extension of finite BRST-antiBRST transformations to the case of general gauge theories and present an ansatz for such transformations. In Appendix A, we study the group properties of finite field-dependent BRST-antiBRST transformations. In Appendix B, we present a detailed calculation of the Jacobian corresponding to the finite, both global (Appendix B.1) and field-dependent (Appendix B.2), BRST-antiBRST transformations. Appendix C is devoted to calculations involving the BRST-antiBRST invariant Yang-Mills action in R ξ -gauges.
We use DeWitt's condensed notations [40]. By default, derivatives with respect to the fields are taken from the right, and those with respect to the corresponding antifields are taken from the left; otherwise, left-hand and right-hand derivatives are labelled by the subscripts "l" and "r", respectively; F, A stands for the right-hand derivative δF/δφ A of a functional F = F (φ) with respect to φ A . The raising and lowering of Sp (2) indices, s a = ε ab s b , s a = ε ab s b , is carried out with the help of a constant antisymmetric second-rank tensor ε ab , ε ac ε cb = δ a b , subject to the normalization condition ε 12 = 1. The Grassmann parity and ghost number of a quantity A, assumed to be homogeneous with respect to these characteristics, are denoted by ε (A), gh(A), respectively.

General Setup for BRST-antiBRST Lagrangian Quantization
The BRST-antiBRST Lagrangian quantization of general gauge theories [15,16,17] involves a set of fields φ A and a set of corresponding antifields φ * Aa (a = 1, 2),φ A , where the doublets of antifields φ * Aa play the role of sources to the BRST and antiBRST transformations, while the antifieldsφ A are the sources to the mixed BRST and antiBRST transformations, with the following distributions of the Grassmann parity and ghost number: The configuration space of fields φ A is identical with that of the BV formalism [30] of covariant quantization and is determined by the properties of the initial classical theory. Namely, we consider an initial classical theory of fields A i , ε(A i ) ≡ ε i , with an action S 0 (A) invariant under gauge transformations, where R i α0 (A) are generators of the gauge transformations, ε(R i α0 ) = ε i + ε α0 , and ζ α0 are arbitrary functions of the space-time coordinates, ε(ζ α0 ) = ε α0 . The generators R i α0 (A) form a gauge algebra [30] with the relations In case the vectors R i α0 (A), enumerated by the index α 0 , are linearly independent, the theory is irreducible; otherwise it is reducible. Depending on the (ir)reducibility of the generators of gauge transformations, the specific structure of the configuration space φ A is described by the set of fields where the ghost C αs|a0...as and auxiliary B αs|a1...as fields form symmetric Sp (2) tensors, being irreducible representations of the Sp (2) group, with the corresponding distribution [16] of the Grassmann parity and ghost number. These fields absorb the pyramids of ghost-antighost and Nakanishi-Lautrup fields of a given (ir)reducible gauge theory, where L in (2.4) is the corresponding stage of reducibility [30], and L = 0 stands for an irreducible theory.
In the space of fields and antifields (φ A , φ * Aa ,φ A ), one introduces the basic object of the BRST-antBRST Lagrangian scheme, being an even-valued functional S = S(φ, φ * ,φ) subject to an Sp (2)-doublet of the generating equations [15] Here, is the Planck constant, whereas the extended antibracket (·, ·) a and the operators ∆ a , V a are given by The properties of the operators ∆ a , V a ,∆ a and those of the extended antibracket (·, ·) a were investigated in [15]. The study of [17] proved the existence of solutions to (2.5) with the boundary condition S| φ * =φ= =0 = S 0 in the form of an expansion in powers of and described the arbitrariness in solutions, which is controlled by a transformation generated by the operators∆ a , connecting two solutions and describing the gauge-fixing procedure. A solution S = S(φ, φ * ,φ) of the generating equations (2.5) allows one to construct an extended (due to the antifields) generating functional of Green's functions Z J, φ * ,φ for the fields φ A of the total configuration space [15], namely, Hence, the generating functional of Green's functions Z(J) = Z J, φ * ,φ φ * =φ=0 is given by where J A , ε(J A ) = ε A , are external sources to the fields φ A , and S ext = S ext φ, φ * ,φ is an action constructed with the help of an even-valued gauge-fixing functional F = F (φ): (2.9) Due to the commutativity of∆ a andÛ , the gauge-fixing procedure retains the form of the generating equations (2.5), A possible choice of the gauge-fixing functional F (φ) has the form of the most general Sp (2)-scalar being quadratic in the ghost and auxiliary fields [16].
Introducing a set of auxiliary fields π Aa and λ A , one can represent Z(J) as a functional integral in the extended space of variables [15] where dΓ = dφ dφ * dφ dλ dπ is the integration measure.
An important property of the integrand in (2.12) for J A = 0 is its invariance under the following infinitesimal transformations of global supersymmetry: where µ a is a doublet of constant anticommuting Grassmann parameters, µ a µ b + µ b µ a ≡ 0. The transformations (2.13) realize the BRST-antiBRST transformations in the extended space (φ A , φ * Aa ,φ A , π Aa , λ A ). The symmetry of the integrand in (2.12) for J A = 0 under the transformations (2.13) with constant infinitesimal µ a allows one to derive the following Ward identities in the extended space: where the expectation value of a functional O(Γ) is given in the extended space parameterized by Γ with a gauge F (φ) in the presence of external sources J A . To obtain (2.14), we subject (2.12) to a change of variables Γ → Γ + δΓ with δΓ given by (2.13) and use the equations (2.5) for S. At the same time, with allowance for the equivalence theorem [41], the transformations (2.13) permit one to establish the independence of the S-matrix from the choice of a gauge.
Indeed, suppose Z F ≡ Z(0) and change the gauge, F → F + ∆F , by an infinitesimal value ∆F . In the functional integral for Z F +∆F we now make the change of variables (2.13). Then, choosing the parameters µ a as we find that Z F +∆F = Z F , and therefore the S-matrix is gauge-independent. For the purpose of a subsequent treatment of Yang-Mills theories, we need the particular case of solutions to the generating equations (2.5) given by a functional S = S φ, φ * ,φ linear in the antifields. Namely, we assume and allows one to present S in the form where s a are generators of BRST-antiBRST transformations, and s 2 are generators of mixed BRST-antiBRST transformations, The explicit form of X Aa and Y A for theories of Yang-Mills type was found in [15] and is given in Appendix C.
For a solution of (2.5) linear in the antifields, integration in (2.12) over φ * Aa ,φ A , π Aa , λ A is trivial [15]: which can also be established directly by inserting the solution (2.16) into (2.9).
The quantum action S F (φ) can be presented in terms of a mixed BRST-antiBRST variation, where the operators s a , acting on an arbitrary functional V = V (φ) of any Grassmann parity, define a BRST-antiBRST analogue of the Slavnov variation, s a V = V ,A s a φ A . Thus defined operators s a are anticommuting, s a s b + s b s a ≡ 0, for any a, b = 1, 2, and therefore nilpotent, s a s b s c ≡ 0, which proves the invariance of S F given by (2.23) under the infinitesimal trans- by virtue of the condition s a S 0 = S 0,i X ia = 0 from (2.17), being a consequence of the Noether identities (2.2). In view of the condition X Aa ,A = 0 from (2.17), the integration measure in (2.21) is also invariant under the transformations (2.19), which ensures the invariance of the integrand in (2.21) for J A = 0 under (2.19). By analogy with the previous consideration, this allows one to establish the Ward identities for Z(J) in (2.21), as well as the independence of the S-matrix from the choice of a gauge. Indeed, suppose Z F ≡ Z(0) in (2.21) and change the gauge F → F + ∆F by an infinitesimal value ∆F . Then, making in Z F +∆F the change of variables (2.19) with the field-dependent infinitesimal parameters being a particular case of the field-dependent BRST-antiBRST transformations studied in the following section, we find Z F +∆F = Z F , which establishes the gauge-independence of the S-matrix.

Finite BRST-antiBRST Transformations and their Jacobians
Let us introduce finite transformations of the fields φ A with a doublet λ a of anticommuting Grassmann parameters, In the general case, such transformations are quadratic in the parameters, due to λ a λ b λ c ≡ 0, for certain functions Z Aa = Z Aa (φ), Z A = Z A (φ), corresponding to the first-and second-order derivatives of φ ′A (φ|λ) with respect to λ a in (3.2). In view of the obvious property of nilpotency ∆φ A1 · · · ∆φ An ≡ 0, n ≥ 3, an arbitrary functional F (φ) under the above transformations φ A → φ A + ∆φ A can be expanded as One can easily verify the consistency of definition (3.5) by considering the equation, implied by ∆S F = 0, Taking into account the fact λ a λ 2 = λ 4 ≡ 0, the invariance relations (S F ) ,A X Aa = 0, and their differential conse- , we find that the above equation is satisfied identically: Explicitly, the finite BRST-antiBRST transformations can be presented as 5 which implies that the finite variation ∆φ A includes the generators of BRST-antiBRST transformations s 1 , s 2 , as well as their commutator s 2 = ε ab s b s a = s 1 s 2 − s 2 s 1 .
According to (2.24), (3.4), (3.7) and λ a λ 2 = λ 4 ≡ 0, the variation ∆F (φ) of an arbitrary functional F (φ) under the finite BRST-antiBRST transformations is given by This relation allows one to study the group properties of finite BRST-antiBRST transformations (3.7), with account taken for the fact that these transformations do not form a Lie superalgebra, nor a vector superspace structure, due to the presence of the term which is quadratic in λ a . Namely, we have (for details, see Appendix A) for certain functionals ϑ a (1,2) = ϑ a (1,2) (φ) and θ (1,2) = θ (1,2) (φ), constructed explicitly in (A.7), (A.8) from the parameters of finite transformations, which are generally field-dependent, λ a (j) = λ a (j) (φ), for j = 1, 2. Therefore, the commutator of finite variations has the form where ϑ a [1,2] , θ [1,2] are given explicitly by (A.11), (A.12) and possess the symmetry properties ϑ a [1,2] = −ϑ a [2,1] , θ [1,2] = −θ [2,1] . In particular, assuming F (φ) = φ A in (3.10), we have In general, the commutator (3.11) of finite non-linear transformations (3.7) does not belong to the class of these transformations, due to the opposite symmetry properties of ϑ [1,2]a ϑ a [1,2] and θ [1,2] , which reflects the fact that a finite BRST-antiBRST transformation looks as a group element, i.e., not as an element of a Lie superalgebra; however, the linear approximation ∆ lin φ A = s a φ A λ a to a finite transformation ∆φ A = ∆ lin φ A + O λ 2 does form an algebra. Indeed, due to (A.9), (A.11), (A.12), we have Thus, the construction of finite BRST-antiBRST transformations (3.7) reduces to the usual BRST-antiBRST transformations (2.19), δφ A = ∆ lin φ A , linear in the infinitesimal parameter µ a = λ a , as one selects in (3.7) the approximation that forms an algebra with respect to the commutator. Let us now consider the modification of the integration measure dφ → dφ ′ in (2.21) under the finite transformations where the Jacobian exp (ℑ) has the form In the case of global finite transformations, corresponding to λ a = const, the integration measure remains invariant (for details, see Appendix B.1) Due to the invariance of the quantum action S F = S 0 + (1/2) s a s a F under φ A → φ ′A the above implies that the integrand with the vanishing sources , which justifies their interpretation as finite BRST-antiBRST transformations.
As we turn to finite field-dependent transformations, let us examine the particular case 6 λ a (φ) = s a Λ (φ) with a certain even-valued potential, Λ = Λ (φ), which is inspired by infinitesimal field-dependent BRST-antiBRST transformations with the parameters (2.26). In this case, the integration measure takes the form (relation (3.18) is deduced 6 Notice that the parameters λa, a = 1, 2, in the case λa = saΛ are not functionally independent: In view of the invariance of the quantum action S F (φ) under (3.7), the change φ A → φ ′A = φ A + ∆φ A induces in (2.21) the following transformation of the integrand with the vanishing sources, whence (3.20) Due to the explicit form of the initial quantum action S F = S 0 + (1/2) s a s a F , the BRST-antiBRST-exact contribution i ln (1 + s a s a Λ/2) 2 to the action S F , resulting from the transformation of the integration measure, can be interpreted as a change of the gauge-fixing functional made in the original integrand I φ , for a certain ∆F (φ), whose relation to Λ (φ) is discussed below. In other words, the field-dependent transformations with the parameters λ a = s a Λ amount to a precise change of the gauge-fixing functional. As a consequence, the integrand in (2.21) for J A = 0, corresponding to the quantum action S F +∆F = S 0 + (1/2) s a s a (F + ∆F ) with a modified gauge-fixing functional, is invariant under both the infinitesimal, δφ A , and finite, ∆φ A , BRST-antiBRST transformations, with constant parameters µ a and λ a in (2.19) and (3.7), respectively.
Let us denote by T (∆F ) the operation that transforms an integrand I , corresponding to the respective gauge-fixing functionals F and F + ∆F , which implies an additive composition law: As we denote by Λ (∆F ) the gauge-fixing functional corresponding to ∆F , there follow the properties implying relations between s 2 Λ (∆F1+∆F2) and s 2 Λ (∆Fj ) for j = 1, 2, as well as between s 2 Λ (−∆F ) and s 2 Λ (∆F ) : The relation (3.21) between the potential Λ (φ) and the variation ∆F (φ) of the gauge-fixing functional can be considered as a compensation equation (for the unknown functional ∆F (φ), with a given Λ (φ), and vice versa), whose solution, up to BRST-antiBRST-exact terms, has the form The relation (3.28) can be inverted as an equation for Λ (φ), namely, Up to BRST-antiBRST-exact terms, its solution reads In particular, the first order of λ a = µ a in powers of ∆F has the form Using (3.32), one can construct a finite BRST-antiBRST transformation that connects two quantum theories of Yang-Mills type corresponding to some gauge-fixing functionals F and F + ∆F for a given finite variation ∆F . The symmetry of the integrand in (2.21) for J A = 0 under the transformations (3.7) allows one to establish the independence of the S-matrix from the choice of a gauge. Indeed, suppose Z F ≡ Z(0) and change the gauge F → F + ∆F by a finite value ∆F . In the functional integral for Z F +∆F we now make the change of variables (3.7). Then, selecting the parameters λ a = s a Λ to meet the condition cf. (3.28), we find that Z F +∆F = Z F , whence, due to the equivalence theorem [41], the S-matrix is gauge-independent. In the particular case of an infinitesimal variation ∆F , condition (3.34) produces, in virtue of (3.33), precisely the form (2.26) of field-dependent parameters λ a = µ a in the framework of infinitesimal BRST-antiBRST transformations.
As we identify λ a = s a Λ with a solution of (3.28), Λ (∆F ) ≡ Λ (∆F ), the representation (2.21) describes the dependence of the functional Z F (J) on a finite variation of the gauge:

The action (4.1) is invariant under the gauge transformations
with arbitrary Bosonic functions ζ n (y) in R 1,D−1 , the covariant derivative D mn µ , and the generators R mn µ (x; y) = R i α of the gauge transformations, the condensed indices being i = (µ, m, x), α = (n, y). The generators R i α in (4.2) form a closed gauge algebra with M ij αβ = 0 in (2.3), whereas the structure coefficients F γ αβ arising in (2.3) are given by The total configuration space of fields φ A and the corresponding antifields φ * Aa ,φ A of the theory are given by With allowance made for (2.1), the Grassmann parity and ghost number assume the values The generating equations (2.5) with the boundary condition S| φ * =φ=0 = S 0 are solved by a functional linear in the where the functionals X Aa = δS/δφ * Hence, the finite BRST-antiBRST transformations ∆φ A = X Aa λ a − (1/2) Y A λ 2 read as follows: where the approximation linear in λ a = µ a produces the infinitesimal BRST-antiBRST transformations δφ A = X Aa µ a = s a φ A µ a .
To construct the generating functional of Green's functions Z(J) in (2.21), we choose the gauge functional F = F (φ) to be diagonal in A µm , C ma , namely, where the gauge-fixing term S gf , the ghost term S gh , and the interaction term S add , quartic in C ma , are given by (4.14) Let us examine the choice of the coefficients α, β leading to R ξ -like gauges. Namely, in view of the contribution S gf to the quantum action S F , we impose the conditions Thus, the gauge-fixing functional F (ξ) = F (ξ) (A, C) corresponding to an R ξ -like gauge can be chosen as Explicitly, where δC ma = ε ab B m − (1/2) f mnl C la C nb µ b is the linear part of the finite BRST-antiBRST transformation (4.10), which implies In order to calculate s a s a ∆F (ξ) , we remind that Finally, the functionals λ a (φ) that connect an R ξ -like gauge to an R ξ+∆ξ -like gauge are given by (3.32) In particular, the first order of λ a = µ a in powers of ∆F (ξ) has the form (3.33) We have thus solved the problem of reaching any gauge in the family of R ξ -like gauges, starting from a certain gauge encoded in the path integral by a functional F (ξ) , within the framework of BRST-antiBRST quantization for Yang-Mills theories by means of finite BRST-antiBRST transformations with field-dependent parameters λ a in (4.24).
Generally, if the BRST-antiBRST invariant quantum action S F0 of a Yang-Mills theory is given in terms of a gauge induced by a gauge-fixing functional F 0 , then, in order to reach the quantum action S F in terms of another gauge induced by a gauge-fixing functional F , it is sufficient to make a change of variables in the path integral (2.21) with S F0 , given by a finite field-dependent BRST-antiBRST transformation with an Sp(2)-doublet of the odd-valued functionals In particular, if we choose F 0 = F (ξ) , with F (ξ) given by (4.17), then the above relation (4.26) describes the transition from an R ξ -like gauge to a gauge parameterized by an arbitrary gauge-fixing functional F = F (A, B, C).

Gribov-Zwanziger Action in R ξ -like Gauges
Let us extend the construction of the Gribov horizon [33] to the case of a BRST-antiBRST invariant Yang-Mills theory in a way consistent with the gauge-independence of the S-matrix. To this end, we examine the sum of the Yang-Mills quantum action (4.12) in the Landau gauge ∂ µ A m µ = 0 (with the gauge-fixing functional F (0) in (4.18) corresponding to the case α = 1, β = 0) and the non-local horizon functional [34] h of the Faddeev-Popov operator K induced by the gauge-fixing functional F (ξ→0) corresponding to the Landau gauge ∂ µ A m µ = 0 in the BRST approach, whereas γ ∈ R is the so-called thermodynamic, or Gribov, parameter [34], introduced in a self-consistent way by the gap equation for an analogue S h of the Gribov-Zwanziger action in the BRST-antiBRST approach: The action S h (φ) is not invariant under the finite BRST-antiBRST transformations: indeed, according to ∆φ A = s a φ A λ a + (1/4) s 2 φ A λ 2 , with allowance for (4.8)-(4.10), (A.2), we have and where we have used the identity To determine the horizon functional for a general R ξ -like gauge in the BRST-antiBRST description, we propose (5.10) Here, s a h and s 2 h are given by (5.7), (5.8), while s a ∆F (ξ) and s a s a ∆F (ξ) are given by (4.21), (4.23) for ∆ξ = ξ, whereas the Sp(2)-doublet λ a ξ (φ) of field-dependent anticommuting parameters in (4.24) relates the Landau gauge to an arbitrary R ξ -like gauge: In particular, the approximation linear in ξ implies, λ a Notice that even the approximation to h ξ (φ) being linear in powers of ξ is different from the proposal [37] for the horizon functional given by R ξ -gauges in terms of field-dependent BRST transformations, which reflects the Sp(2)symmetric character of the dependence of h ξ (φ) on the ghost and antighost fields C ma . The proposal (5.10) for the Gribov horizon functional in a general R ξ -gauge is consistent with the study of gaugeindependence for the generating functional of Green's functions, determined for a BRST-antiBRST extension of the Gribov-Zwanziger model as follows: Indeed, making in the path integral for Z GZ,F0 (J) a change of variables being a finite field-dependent BRST-antiBRST transformation with the parameters λ a ξ (φ) given by (4.24), where ∆ξ = ξ, we find, due to the fact that the Yang-Mills quantum action S F0 (φ) transforms to S F ξ (φ), with F ξ = F (ξ) , where h ξ (φ) in (5.10) corresponds to an R ξ -gauge. As a result, we have Finally, it is possible to construct a Gribov horizon functional h F (φ) in any differential gauge 8 induced by a gauge-fixing functional F (φ), starting from the horizon functional h(A) in the Landau gauge, corresponding to the gauge-fixing functional F 0 (A). To this end, it is sufficient to make a change of variables in the path integral (5.15), given by a finite field-dependent BRST-antiBRST transformation with the Sp(2)-doublet λ a (F − F 0 ) of odd-valued functionals given by (4.26). Thus, the functional h F (φ) reads as follows: Generally, a finite change F → F + ∆F of the gauge condition induces a finite change of any functional G F (φ), so that in the reference frame corresponding to the gauge F + ∆F it can be represented according to (3.8), (4.26), 7 There exist other ways to obtain the Gribov horizon functional h ξ for gauges beyond the Landau gauge, see, e.g., [35,38]; however, in view of its non-pertubative character [34], the derivation procedure faces the problem of gauge dependence. 8 Due to the result of Singer [42], Gribov copies should arise in non-Abelian gauge theories in case a differential gauge is used to fix the gauge ambiguity.
which is an extension of the infinitesimal change G F → G F + δG F induced by a variation of the gauge, F → F + δF , , (5.20) corresponding, in the case G F (A), to the gauge transformations (4.2), with the functions ζ m (x) given below Due to the presence of the term with s 2 G F in a finite gauge variation of a functional G F (A) depending only on the classical fields A mµ , the representation (5.19) is more general than the one that would correspond to the usual Lagrangian BRST approach (see relation (17) in [39]), having the form similar to (5.21), and thus also to (5.20).
We emphasize that the suggested method of using the finite field-dependent BRST-antiBRST transformations with the purpose of finding the Gribov-Zwanziger horizon functional in any differential gauge, starting from the Gribov-Zwanziger theory in the Landau gauge, is valid in perturbation theory and preserves the number of physical degrees of freedom, without entering into contradiction with the result of [24] in the BRST setting of the problem.
However, it is impossible to solve this problem (in particular, in the Yang-Mills theory) in terms of finite fielddependent BRST-antiBRST transformations [26], in view of the absence of a term being quadratic in powers of the odd-valued parameters, since the corresponding Yang-Mills quantum action fails to be BRST-antiBRST invariant, and the Jacobian of the corresponding change of variables with odd-valued functionally-dependent parameters does not generate terms which are entirely BRST-antiBRST-exact. These terms change the BRST-antiBRST-exact part of the action, as well as the extremals; however, they do not affect the number of physical degrees of freedom.

Discussion
In the present work, we have proposed the concept of finite BRST-antiBRST transformations for Yang-Mills theories in the Sp(2)-covariant Lagrangian quantization [15,16], realized in the form We have determined the finite field-dependent BRST-antiBRST transformations as polynomials in the Sp (2)doublet of Grassmann-odd functionals λ a (φ), depending on the whole set of fields that compose the configuration space of Yang-Mills theories, and have also calculated the Jacobian (3.18) corresponding to this change of variables by using a special class of transformations with s a -potential parameters λ a (φ) = s a Λ(φ) for a Grassmann-even functional Λ(φ) and Grassmann-odd generators s a of BRST-antiBRST transformations.
In comparison with finite field-dependent BRST transformations in Yang-Mills theories [23], in which a change of the gauge corresponds to a unique field-dependent parameter (up to BRST-exact terms), it is only functionallydependent finite field-dependent BRST-antiBRST transformations with λ a = s a Λ(∆F ) that are in one-to-one correspondence with ∆F . We have found (3.31) a solution Λ(∆F ) to the so-called compensation equation In terms of the potential Λ inducing the finite field-dependent BRST-antiBRST transformations, we have explicitly constructed (4.24) the parameters λ a generating a change of the gauge in the path integral for Yang-Mills theories within a class of linear R ξ -like gauges related to even-valued gauge-fixing functionals F (ξ) , with ξ = 0, 1 corresponding to the Landau and Feynman (covariant) gauges, respectively. We have shown how to reach an arbitrary gauge given by a gauge Boson F within the path integral representation, starting from the reference frame with a gauge Boson F 0 by means of finite field-dependent BRST-antiBRST transformations with the parameters λ a (F − F 0 ) given by (4.26).
We have applied the concept of finite field-dependent BRST-antiBRST transformations to construct the Gribov horizon functional h ξ , given by (5.10) in arbitrary R ξ -like gauges, starting from a previously known BRST-antiBRST non-invariant functional h, as in [34], corresponding to the Landau gauge and induced by an even-valued functional F (0) .
The construction is consistent with the study of gauge-independence for the generating functionals of Green's functions Z GZ,F0 (J) in (5.15) within the suggested Gribov-Zwanziger model considered in the BRST-antiBRST approach (5.5).
There are various lines of research for extending the results obtained in the present work. First, the study of finite field-dependent BRST-antiBRST transformations for a general gauge theory in the framework of the path integral 9 (2.12). Second, the development of finite field-dependent BRST transformations for a general gauge theory in the BV quantization method 10 [30]. Third, the construction of finite field-dependent BRST-antiBRST transformations in the Sp(2)-covariant generalized Hamiltonian quantization [12,13] and the study of their properties in connection with the corresponding gauge-fixing problem. 11 Fourth, the consideration of the so-called refined Gribov-Zwanziger theory [47] in a BRST-antiBRST setting analogous to [31], and also the elaboration of a composite operator technique in the BRST-antiBRST Lagrangian quantization scheme, in order to examine the Gribov horizon functional as a composite operator with an external source, along the lines of [39]. We also mention the search for an equivalent local description of the Gribov horizon functional with a set of auxiliary set fields as in [34] such that it should be consistent with both the infinitesimal and finite BRST-antiBRST invariance. We are also interested in the study of the influence of Jacobians generated by finite field-dependent BRST-antiBRST transformations (linear and functionally-independent parameters) on the structure of transformed quantum actions and partition functions [48].
Finally, the suggested Gribov horizon functionals beyond the Landau gauge allow one to study such quantum properties as renormalizability and confinement within the BRST-antiBRST extension of the Gribov-Zwanziger theory in a way consistent with the gauge independence of the physical S-matrix. We intend to study these problems in our forthcoming works.
Concluding, let us outline an ansatz for finite field-dependent BRST-antiBRST transformations of the path integral (2.12), corresponding to the case of a general gauge theory. To this end, notice that the construction (3.5), (3.7) of finite BRST-antiBRST transformations in Section 3, in fact, applies to any infinitesimal symmetry transformations δφ A = X Aa µ a = s a φ A µ a , with anticommuting parameters µ a , a = 1, 2, for a certain functional S F (φ), such that δS F (φ) = 0, and does not involve any subsidiary conditions on X Aa and the corresponding s a , since the construction is achieved only by using Y A = (1/2) X Aa ,B X Bb ε ba in (3.7), according to (2.20). Let us apply this to the vacuum functional Z(0) of a general gauge theory, given by the path integral (2.12) in the extended space where the integrand I 2) 9 We have solved this problem in our recent works [43,44]. 10 Shortly after the publication of the present work, we have become aware of the more recent study [45] of finite BRST transformations in the BV formalism. 11 We have solved this problem in detail [46], including the case of Yang-Mills theories.
In this connection, let us determine finite BRST-antiBRST transformations, Γ p → Γ p + ∆Γ p , parameterized by anticommuting parameters λ a , a = 1, 2, as follows: Thus determined finite BRST-antiBRST symmetry transformations for the integrand I (F ) Γ in a general gauge theory have the form (X pa = σ a Γ p and 4) or, in terms of the components, with the notation U V ≡ U a V a = −U a V a for pairing up any Sp(2)-vectors U a , V a , we obtain and Therefore, ∆ (1) ∆ (2) F is given by Hence, the commutator of finite variations reads [1,2] . (A.9) Finally, using the identity In particular, the linear approximation ∆ lin F = (s a F ) λ a , ∆F = ∆ lin F + O λ 2 , implies (3.13).

B Calculation of Jacobians
In this Appendix, we present the calculation of the Jacobian (3.14), (3.15), induced in the functional integral (2.21) by the finite BRST-antiBRST transformations (3.7) with an Sp (2)-doublet of anticommuting parameters λ a , considering the global case, λ a = const, and the case of field-dependent functionals λ a (φ) of a special form, λ a (φ) = s a Λ (φ).

B.1 Constant Parameters
Let us assume λ a to be constant parameters in (3.7) and consider an even matrix M in (3.15) Notice the fact that Q 1 ∼ λ a , R ∼ λ 2 , which, in view of the nilpotency properties λ a λ 2 = λ 4 ≡ 0, implies Indeed, due to the relations X Aa ,A = 0 in (2.17), we have Next, let us examine Str Q 2 1 : Then, due to the relation X Aa ,A = 0 in (2.17), we have and therefore Thus, the Jacobian exp (ℑ) in (3.15) is given by which proves (3.16).

B.2 Field-dependent Parameters
In the case of field-dependent parameters λ a (φ) = s a Λ (φ) from (3.7), given by an even-valued potential Λ (φ), let us consider an even matrix M in (3.15) with the elements M A B , Using the property Str (AB) = Str (BA) , (B.10) which takes place for any even matrices A, B, and the fact that the occurrence of R ∼ λ 2 in Str (M n ) more than once yields zero, λ 4 ≡ 0, we have Furthermore, Str (P + Q + R) n = Str (P + Q) n + nStr (P + Q) n−1 R = Str (P + Q) n + nStr P n−1 R , (B.12) since any occurrence of R ∼ λ 2 and Q ∼ λ a simultaneously entering Str (M ) n yields zero, owing to λ a λ 2 = 0, as a consequence of which R can only be coupled with P n−1 .
Considering the contribution Str (P + Q) n in (B.16), we notice that an occurrence of Q ∼ λ a more then twice yields zero, λ a λ b λ c ≡ 0. A direct calculation for n = 2, 3 leads to Next, starting from the case n = 4, Str M 4 = Str P 4 + 4P 3 Q + 4P 2 Q 2 + 2P QP Q , one can prove that for any n ≥ 4 we have Str (P + Q) n = Str P n + nP n−1 Q + nP n−2 Q 2 + K n P n−3 QP Q , where the coefficients 12 K n are given by (in particular, n = 4, C 2 4 = 6, K 4 = C 2 4 − 4 = 2) The proof of (B.18) goes by induction. To this end, suppose that (as in the case n = 4) (P + Q) n = P n + A (1) n (P, Q) + B (2) n (P, Q) + C (2) n (P, Q) , where A (1) n = a kl P k QP l , a n ≡ a k0 = 1 , B (2) n = b kl P k Q 2 P l , C (2) n = c kml P k QP m QP l , m ≥ 1 , and Str A (1) n = nStr P n−1 Q , Str B (2) n = nStr P n−2 Q 2 , Str C (2) n = K n Str P n−3 QP Q . (B.21) Then, due to the vanishing of the terms containing Q more than twice, we have Due to the contraction property P 2 = f · P =⇒ P l = f l−1 · P in (B.34), the above implies which proves the induction.
Recall that the Jacobian exp (ℑ) in (3.15) is given by where, according to the previous considerations, 30) or, in detail, Str (P ) + Str (Q) + Str (R) , n = 1 , Str (P n ) + C 1 n Str P n−1 Q + C 2 n Str P n−2 Q 2 , n = 2, 3 , Str (P n ) + C 1 n Str P n−1 Q + C 2 n − K n Str P n−2 Q 2 + K n Str P n−3 QP Q , n > 3 . It has also been established (Appendix B.1) that the quantity Str (R) in (B.16) cancels the contribution Str Q 2 1 to the Jacobian, where these contributions enter in the first and second orders, Str M 1 and Str M 2 , respectively, thus summarily producing an identical zero: Therefore, we can exclude Str (R) and Str Q 2 1 from further consideration. Recalling that λ a = s a Λ, we can deduce the additional properties where the quantity f is given by Indeed, As a consequence, we have QP = (1 + f ) · Q 2 , namely, in view of X Aa ,B X Bb = ε ab Y A from (2.17), where the term Str Q 2 1 has been omitted according to the previous considerations related to (B.33). We further notice that Str (Q 1 Q 2 ) ≡ 0. Indeed, due to X Aa ,B X Bb = ε ab Y A and Y A ,B X Bb = 0 in (2.17), we have Besides, Indeed, Therefore, ℑ in the expression (B.28) for the Jacobian exp (ℑ) has the general structure Let us examine A (f ), namely, Let us examine the explicit structure of the series related to b 1 (f ): the quantity Str (Q 2 ) derives from Str P n−1 Q for n ≥ 1 in (B.38), and is coupled with the combinatorial coefficient C 1 n . The part of ℑ containing Str (Q 2 ) is given by Let us examine the explicit structure of the series related to b 2 (f ): the quantity Str 2 (Q 2 ) derives from Str P n−2 Q 2 for n ≥ 2 in (B.38), coupled with the combinatorial coefficients C 2 n for n = 2, 3 and C 2 n − K n for n > 3, and also derives from Str P n−3 QP Q for n > 3 in (B.38), coupled with the combinatorial coefficients K n . The part of ℑ containing Str 2 (Q 2 ) reads By virtue of (B.20), this implies the vanishing of b 2 (f ), namely, Let us examine the explicit structure of the series related to c (f ): the quantity Str (Q 1 Q 2 ) derives from Str P n−2 Q 2 for n ≥ 2 in (B.38), and is coupled with the combinatorial coefficients C 2 n , for n = 2, 3, and C 2 n − K n , for n > 3. The part of ℑ containing Str (Q 1 Q 2 ) is given by By virtue of (B.20), this implies the vanishing of c (f ), namely, and therefore the Jacobian exp (ℑ) is finally given by which is identical with (3.17).

C BRST-antiBRST Invariant Yang-Mills Action in R ξ -like Gauges
In this Appendix, we present the details of calculations used in Section 4 to establish a correspondence between the gauge-fixing procedures in the Yang-Mills theory described by a gauge-fixing function χ(φ) = 0 from the class of R ξ -gauges in the BV formalism [30] and by a gauge-fixing functional F in the BRST-antiBRST quantization [15,16].
The Yang-Mills theories belong to the class of irreducible gauge theories of rank 1 with a closed algebra, which implies that M ij αβ = 0 in (2.3) and that any solution of the equation R i α X α = 0 has the form X α = 0. The corresponding space of fields and antifields φ A , φ * Aa ,φ is given by as we take into account (2.1) and the following distribution of the Grassmann parity and ghost number: ε(φ A ) ≡ (ε i , ε α , ε α + 1) , gh(φ A ) = 0, 0, (−1) whereas a solution to the generating equations (2.5) with a vanishing right-hand side can be found in the linear form (2.16), S = S 0 + φ * Aa X Aa +φ A Y A , obviously satisfying the boundary condition S| φ * =φ=0 = S 0 . Here, the functionals X Aa and Y A can be chosen as [15] X Aa = X ia 1 , X αa 2 , X αab By construction, the functionals X Aa = δS/δφ * Aa and Y A = δS/δφ A obey the properties S 0,i X ia = 0, X Aa ,B X Bb = ε ab Y A , Y B ,A X Aa = 0. Besides, in Yang-Mills theories the explicit form (4.2), (4.3) of the gauge generators R i α and structure coefficients F γ αβ = const is such that X Aa = X ia 1 , X αa 2 , X αab 3 in (C.4) possess the properties X Aa ,A = 0, so that the entire set of relations (2.17) is fulfilled, and the solution given by (C.4) satisfies the generating equations (2.5) identically.
Choosing the gauge-fixing functional F (A, C) in the quadratic form (4.11) and using the identities (for arbitrary su(N )-vectors F m and G m ) we have (C.16) Next, d D x δF δC ma C ma = βε ab d D x C mb C ma = βε ba d D x C ma f mnl B l C nb + 1 6 f mnl f lrs C sd C rb C nc ε cd . (C.18) At the same time, δ C δF δC mc (x) = βε cd δC md (x) = βε cd d D y δ mn δ (y − x) δC nd (y) , δ δC nd (y) δF δC mc (x) = βε cd δ mn δ (y − x) , Therefore, d D x δF δC ma C ma − 1 2 ε ab C mac δ δC nd δF δC mc C nbd = −βε ab d D x C ma f mnl B l C mb + 1 6 f mnl f lrs C sd C rb C nc ε cd (C.21) − β 2 ε ab ε cd d D x ε ac B m + 1 2 f mnl C lc C na ε bd B m + 1 2 f mrs C sd C rb . where By virtue of the identity f lmn C nb C ma ε ab ≡ 0, the quantum action (C.22) equals to (4.12).