{'fraction_non_alphanumeric': 0.03432818213990799, 'fraction_numerical': 0.013094255043057685, 'mean_word_length': 4.655436957971982, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Field dependent nilpotent symmetry for gauge theories

We construct the field dependent mixed BRST (combination of BRST and anti-BRST) transformations for pure gauge theories. These are shown to be an exact nilpotent symmetry of both the effective action as well as the generating functional for certain choices of the field dependent parameters. We show that the Jacobian contributions for path integral measure in the definition of generating functional arising from BRST and anti-BRST part compensate each other. The field dependent mixed BRST transformations are also considered in field/antifield formulation to show that the solutions of quantum master equation remain invariant under these. Our results are supported by several explicit examples.


I. INTRODUCTION
The BRST transformation plays a central role in the quantization, renormalizability, unitarity and other aspects of the gauge theories [1][2][3][4][5]. Such nilpotent transformation, characterized by an infinitesimal, global and anticommuting parameter, leaves the effective action including the gauge fixing term invariant. Similar to the BRST transformation, anti-BRST transformation is also a symmetry transformation where the role of ghost and anti-ghost fields are interchanged. The anti-BRST symmetry does not play as fundamental role as BRST symmetry itself but it is a useful tool in geometrical description of BRST/anti-BRST symmetric theories [6] and widely used in the investigation of perturbative renormalization of the gauge theories [7]. We consider the mixed BRST (MBRST) transformation (defined as δ m φ = s b φ δΛ 1 + s ab φ δΛ 2 ) with infinitesimal, anticommuting but global parameters δΛ 1 and δΛ 2 corresponding to the BRST variation (s b ) and anti-BRST variation (s ab ) of the generic fields φ respectively. However, the transformations s b and s ab defined in the MBRST transformation satisfy the absolute anticommuting relation {s b , s ab }φ = 0. Such an infinitesimal mixed transformation is also a nilpotent symmetry transformation of the effective action as well as of the generating functional.
In the present work we construct the field dependent MBRST (FMBRST) transformation having finite and field dependent parameters. The usual finite field dependent BRST (FFBRST) and anti-BRST (FF-anti-BRST) transformations are the symmetry transformations of the effective action only but do not leave the generating functional invariant as the path integral measure in the definition of generating functional transforms in a nontrivial manner [8,9]. The FFBRST and FF-anti-BRST transformations for the effective theory have many applications in gauge field theory [8][9][10][11][12][13][14][15][16]. Unlike the usual FFBRST and FF-anti-BRST transformations the FMBRST transformations are shown to be the symmetry of the both effective action as well as the generating functional of the theory.
We construct the finite parameters in the FMBRST transformation in such a way that the Jacobian contribution due to FFBRST part compensates the same due to FF-anti-BRST part. Thus we are able to construct the finite nilpotent transformation which leaves the generating functional as well as the effective action of the theory invariant. We further show that the effect of FMBRST transformation is equivalent to the effect of successive operations of FFBRST and FF-anti-BRST transformations.
Our results are supported by several explicit examples. First of all we consider the gauge invariant model for single self-dual chiral boson in (1+1) dimensions [17][18][19], which is very useful in the study of certain string theoretic models [20,21] and plays a very crucial role in the study of quantum Hall effect [22]. (3+1) dimensional Abelian as well as non-Abelian Yang-Mills (YM) theory in the Curci-Ferrari-Delbourgo-Jarvis (CFDJ) gauge [23][24][25] are also considered to demonstrate the above finite nilpotent symmetry.
The Lagrangian quantization of Batalin and Vilkovisky (BV) formulation [3,4,26,27], which is also known as the field/antifield formulation, is considered to be one of the most powerful and advanced technique of quantization of gauge theories involving the BRST symmetry [3,4,28]. To study the role of FMBRST transformation in field/antifield formulation we consider the same three simple models in BV formulation. We show that the FMBRST transformation does not change the generating functional written in terms of extended quantum action in BV formulation. Hence the FMBRST transformation leaves the different solutions of the quantum master equation in field/antifield formulation invariant.
The paper is organized as follows. In Sec. II, we discuss the infinitesimal MBRST transformation. Next the FMBRST transformation is constructed in Sec. III. The nontrivial Jacobian for such FMBRST transformation is evaluated in Sec. IV. Sec. V, is devoted to the explicit examples having FMBRST symmetry transformation. In Sec. VI, we study the field/antifield formulation in the context of FMBRST transformation. Sec.
VII is reserved for concluding remarks.

II. THE INFINITESIMAL MBRST TRANSFORMATION
The generating functional for the Green's function in an effective theory described by the effective action S ef f [φ] is defined as where φ is the generic notation for all fields involved in the effective theory. The infinitesimal BRST (δ b ) and anti-BRST (δ ab ) transformations are defined as where δΛ 1 and δΛ 2 are infinitesimal, anticommuting but global parameters. Such transformations leave the generating functional as well as effective action invariant This implies that the effective action S ef f [φ] and the generating functional are also in- Further such MBRST transformation is nilpotent because, Now, in the next section we construct the finite field dependent version of following infinitesimal MBRST symmetry transformation

III. CONSTRUCTION OF FMBRST TRANSFORMATION
To construct the FMBRST transformation, we use to follow the similar method of constructing FFBRST transformation [8]. However, in this case unlike FFBRST transformation we have to deal with two parameters, one for the BRST transformation and the other for anti-BRST transformation. We introduce a numerical parameter κ(0 ≤ κ ≤ 1) and make all the fields (φ(x, κ)) κ-dependent in such a way that φ(x, κ = 0) ≡ φ(x) and φ(x, κ = 1) ≡ φ ′ (x), the transformed field. Further, we make the infinitesimal parameters δΛ 1 and δΛ 2 field dependent as where the prime denotes the derivative with respect to κ and Θ ′ i [φ(x, κ)](i = 1, 2) are infinitesimal field dependent parameters. The infinitesimal but field dependent MBRST transformations, thus can be written generically as Following the work in Ref. [8] it can be shown that the parameters Θ ′ i [φ(x, κ)](i = 1, 2), contain the factors Θ ′ i [φ(x, 0)] (i = 1, 2), which are considered to be nilpotent. Thus κ dependency from δ b φ(x, κ) and δ ab φ(x, κ) can be dropped. Then the Eq. (3.3) can be written as The FMBRST transformations with the finite field dependent parameters then can be constructed by integrating such infinitesimal transformations from κ = 0 to κ = 1, such are the finite field dependent parameters with i = 1, 2.
Therefore, the FMBRST transformation corresponding MBRST transformation mentioned in Eq. (2.9) is given by It can be shown that above FMBRST transformation with some specific choices of the finite parameters Θ 1 and Θ 2 is the symmetry transformation of the both effective action and the generating functional as the path integral measure is generally invariant under such transformation.

IV. METHOD FOR EVALUATING THE JACOBIAN
For the symmetry of the generating functional we need to calculate the Jacobian of the path integral measure in the definition of generating functional. The Jacobian of the path integral measure for FMBRST transformation J can be evaluated for some particular choices of the finite field dependent parameters Θ 1 [φ(x)] and Θ 2 [φ(x)]. We start with the definition, Now the transformation from φ(κ) to φ(κ + dκ) is infinitesimal in nature, thus the infinitesimal change in Jacobian can be calculated as where Σ φ sums over all fields involved in the path integral measure and ± sign refers to whether φ is a bosonic or a fermionic field. Using the Taylor expansion we calculate the above expression as The Jacobian, J(κ), can be replaced (within the functional integral) as iff the following condition is satisfied are some local functionals of fields and satisfy the initial condition The finite parameters Θ 1 and Θ 2 are arbitrary and we can construct them in such a way that the infinitesimal change in Jacobian J (Eq. (4.3)) with respect to κ vanishes Therefore, with the help of Eqs. (4.5) and (4.7), we see that It means the S 1 + S 2 is independent of κ (fields) and must vanish to satisfy the initial condition given in Eq. dimensions, and non-Abelian YM theory in (3+1) dimensions. In all these cases we construct explicit finite parameters Θ 1 and Θ 2 of FMBRST transformation such that the generating functional remains invariant.

A. Bosonized chiral model
We start with the generating functional for the bosonized self-dual chiral model as [17,18] Z CB = Dφ e iS CB , where Dφ is the path integral measure in generic notation. The effective action S CB in (1+1) dimensions is given as where the fields ϕ, ϑ, u, B, c andc are the self-dual field, Wess-Zumino field, multiplier field, auxiliary field, ghost field and anti-ghost field respectively. The nilpotent BRST and anti-BRST transformations for this theory are BRST: anti-BRST: The FMBRST transformations corresponding to the above MBRST transformations are constructed as where Θ 1 and Θ 2 are finite field dependent parameters and are still anticommuting in nature. We construct the finite parameters Θ 1 and Θ 2 as where γ is an arbitrary parameter.
Using Eq. The infinitesimal change in Jacobian J 1 for the FFBRST transformation with the parameter Θ 1 is calculated as To write the Jacobian J 1 as e iS 1 in case of BRST transformation, we make following ansatz for S 1 as where ξ i (i = 1, 2, 3) are arbitrary κ-dependent constants and satisfy the initial conditions The essential condition in Eq. (4.5) satisfies with Eqs. (5.11) and (5.12) iff where prime denotes the derivative with respect to κ. Equating the both sides of the above equation, we get the following equations 14) The solution of above equations satisfying the initial conditions is Then the expression for S 1 in terms of κ becomes On the other hand the infinitesimal change in Jacobian J 2 for the FF-anti-BRST parameter Θ 2 is calculated as Similarly, to write the Jacobian J 2 as e iS 2 in the anti-BRST case, we make ansatz for S 2 as where arbitrary κ-dependent constants ξ i (i = 4, 5, 6) have to be calculated.
The essential condition in Eq. (4.5) for the above Jacobian J 2 and functional S 2 Comparing the L.H.S. and R.H.S. of the above equation, we get following equations Solving the above equations, we get the following values for ξ i 's Putting these values in expression of S 2 , we get Thus under successive FFBRST and FF-anti-BRST transformations the generating functional transformed as

B. Maxwell's theory
The generating functional for Maxwell theory, using Nakanishi Lautrup type auxiliary field (B), can be given as where the effective action in covariant (Lorentz) gauge with the ghost term is The infinitesimal off-shell nilpotent BRST and anti-BRST transformations under which the effective action S M ef f as well as generating functional Z M remain invariant, are given as BRST: The nilpotent FMBRST symmetry transformation for this theory is then constructed as where Θ 1 and Θ 2 are finite, field dependent and anticommuting parameters. We choose particular Θ 1 and Θ 2 in this case as where γ is an arbitrary parameter. The infinitesimal change in Jacobian using Eq. (4.3) for the FMBRST transformation with the above finite parameters vanishes. It means that the path integral measure and hence the generating functional is invariant under FMBRST transformation. Now, the infinitesimal change in Jacobian for the FFBRST transformation with the parameter Θ 1 is calculated as To write the Jacobian J 1 as e iS 1 in the BRST case, we make following ansatz for S 1 as The essential condition in Eq. (4.5) is satisfied subjected to where prime denotes the derivative with respect to κ. Equating the both sides of the above equation, we get the following Eqs.
The solution of above equations satisfying the initial conditions ξ i = 0, (i = 1, 2) is Putting these value in the Eq. (5.33), the expression of S 1 becomes However, the infinitesimal change in Jacobian J 2 for the FF-anti-BRST transformation with the parameter Θ 2 is calculated as Similarly, to write the Jacobian J 2 as e iS 2 in the anti-BRST case, we make the following ansatz for S 2 as The essential condition in Eq.

C. Non-Abelian YM theory in CFDJ gauge
The generating functional for non-Abelian YM theory in CFDJ gauge can be written as where φ is generic notation for all the fields in the effective action S CF and h a is the Nakanishi Lautrup type auxiliary field. The effective action as well as the generating functional are invariant under following infinitesimal BRST and anti-BRST transformations BRST:

47)
Anti-BRST: where δΛ 1 and δΛ 2 are infinitesimal, anticommuting and global parameters. The infinitesimal MBRST symmetry transformation (δ m = δ b + δ ab ) in this case is written as: Corresponding FMBRST symmetry transformation is constructed as: with two arbitrary finite field dependent parameters Θ 1 and Θ 2 . The generating functional Z CF Y M is made invariant under the above FMBRST transformation by constructing appropriate finite parameters Θ 1 and Θ 2 . We construct the finite nilpotent parameters Θ 1 and Θ 2 as We end up the section with conclusion that in all the three cases the FMBRST transformation with appropriate finite parameters is the finite nilpotent symmetry of the effective action as well as the generating functional of the effective theories. Here we also note that the successive operations of FFBRST and FF-anti-BRST also leave the generating functional as well as effective action invariant and hence equivalent to FMBRST transformation.

VI. FMBRST SYMMETRY IN FIELD/ANTIFIELD FORMULATION
In this section we consider the field/antifield formulation using MBRST transformation.
Unlike BV formulation using either BRST or anti-BRST transformations, we need two sets of antifields in BV formulation for MBRST transformation. We construct FMBRST transformation in this context. The change in Jacobian under FFBRST transformation in the path integral measure in the definition of generating functional is used to adjust with the change in the gauge-fixing fermion Ψ 1 [28]. Hence the FFBRST transformation is used to connect the generating functionals of different solutions of quantum master equation [10,13]. However in case of BV formulation for FMBRST transformation we

A. Bosonized chiral model in BV formulation
We recast the generating functional in Eq. (5.1) for (1+1) dimensional bosonized chiral model using both BRST and anti-BRST exact terms as here Lagrange multiplier field u is considered as dynamical variable and expression for gauge-fixing fermions for BRST symmetry (Ψ 1 ) and anti-BRST symmetry (Ψ 2 ) respectively are 3) The effective action S CB is invariant under combined BRST and anti-BRST transformations given in Eq. (5.5). The generating functional Z CB can be written in terms of antifields φ ⋆ 1 and φ ⋆ 2 corresponding to all fields φ as where φ ⋆ i (i = 1, 2) is a generic notation for antifields arising from gauge-fixing fermions Ψ i . The above relation can further be written in compact form as where W Ψ 1 +Ψ 2 [φ, φ ⋆ i ] is an extended action for the theory of self-dual chiral boson corresponding the gauge-fixing fermions Ψ 1 and Ψ 2 .
This extended quantum action, W Ψ 1 +Ψ 2 [φ, φ ⋆ i ] satisfies certain rich mathematical relations commonly known as quantum master equation [4], given by The generating functional does not depend on the choice of gauge-fixing fermions [3] and therefore extended quantum action W Ψ i with all possible Ψ i are the different solutions of quantum master equation. The antifields φ ⋆ 1 corresponding to each field φ for this particular theory can be obtained from the gauge-fixed fermion Ψ 1 as Similarly, the antifields φ ⋆ 2 can be calculated from the gauge-fixing fermion Ψ 2 as where the expressions for gauge-fixing fermions Ψ 1 and Ψ 2 are The generating functional for such theory can further be expressed in fields/antifields formulation as In the compact form above generating functional is written as where W Ψ 1 +Ψ 2 [φ, φ ⋆ i ] is an extended action for the Maxwell's theory corresponding to the gauge-fixing fermions Ψ 1 and Ψ 2 .
The antifields for gauge-fixed fermion Ψ 1 are calculated as The antifields φ ⋆ 2 can be calculated from the gauge-fixed fermion Ψ 2 as