BRST-BV quantization of gauge theories with global symmetries
aa r X i v : . [ h e p - t h ] F e b BRST-BV quantization of gauge theorieswith global symmetries
I. L. Buchbinder ( a,b,c )1 , P. M. Lavrov ( a,b )2 ( a ) Center of Theoretical Physics,Tomsk State Pedagogical University,Kievskaya St. 60, 634061 Tomsk, Russia ( b ) National Research Tomsk State University,Lenin Av. 36, 634050 Tomsk, Russia ( c ) Departamento de F´ısica, ICE,Universidade Federal de Juiz de Fora,Campus Universit´ario-Juiz de Fora,36036-900, MG, Brazil
Abstract
We consider the general gauge theory with a closed irreducible gauge algebra possessingthe non-anomalous global (super)symmetry in the case when the gauge fixing procedureviolates the global invariance of classical action. The theory is quantized in the frameworkof BRST-BV approach in the form of functional integral over all fields of the configurationspace. It is shown that the global symmetry transformations are deformed in the processof quantization and the full quantum action is invariant under such deformed globaltransformations in the configuration space. The deformed global transformations arecalculated in an explicit form in the one-loop approximation.
Keywords:
BRST quantization, BRST symmetry, global symmetryPACS numbers: 11.10.Ef, 11.15.Bt E-mail: [email protected] E-mail: [email protected]
Introduction
BRST quantization procedure, initiated in the works [1, 2, 3, 4, 5, 6, 7, 8], which can berealized within the Hamiltonian BFV approach [9, 10] or within the Lagrangian BV approach[11], [12], is a powerful and universal tool to formulate the quantum gauge theories and investi-gate their structure. This procedure is applicable to an extremely wide class of gauge theoriesincluding the (super)Yang-Mills theories, (super)gravity, (super)strings and more specific gaugetheories like the gauge antisymmetric field models. All these theories possess many commonproperties therefore one can talk about a general gauge theory and study its quantum aspectsin general terms.Besides the local symmetries, the many gauge theories are characterized by the rigid sym-metries. For example, the role which is played by the global chiral symmetry in the StandardModel (see e.g. [13]) or the role which is played by global conformal group in the String Theory(see e.g. [14]) are well known. In many cases it is essential to preserve the classical globalsymmetry in quantum theory since in the opposite case the important physical properties ofthe theory can be violated. Example of such a model is N = 4 supersymmetric Yang-Millstheory, where the classical rigid superconformal symmetry is conserved at quantum level (seee.g. [15]). As a result, we arrive at a general problem to study the classical global symmetriesat BRST quantization of the gauge theory.A specific aspect of the above general problem arises in supersymmetric gauge theories.These theories possess a gauge symmetry and rigid or global supersymmetries. Such theoriescan be formulated either in the component formalism or in the terms of superfields. In thefirst case the supersymmetry is non manifest and when we quantize the corresponding gaugetheory imposing the gauge fixing conditions on vector fields we violate the rigid supersymmetry.Therefore it is unclear if the quantum effective action should be supersymmetric. At first, thisproblem arose in N = 1 super Yang-Mills theories and led to the study of the aspects of globalsymmetry in quantum gauge theories [16, 17, 18, 19] However, this problem was automaticallyresolved after formulation of the N = 1 supergauge theories in terms of N = 1 superfields (seee.g. [20]), where the manifestly N = 1 gauges have been used. However, the problem is stillremains in extended super Yang-Mills theories. At present, the best formulation of 4 D, N = 4rigid super Yang-Mills theory is achieved in terms of 4 D, N = 2 or in terms of 4 D, N = 3harmonic superfields [21]. In this case only part of supersymmetries are manifest, the otherremain hidden. The same situation will be in 6 D, N = (1 ,
1) rigid super Yang-Mills theory,which is similar in many aspects to 4 D, N = 4 super Yang-Mills theory. At present, the bestformulation of such a theory is achieved in terms of 6 D, N = (1 ,
0) superfields (see e.g. [26]and references therein). Again, one part of supersymmetries is manifest and the other part ishidden. In all such theories the gauges preserve the manifest symmetries and violate the hiddensupersymmetries (see e.g.[23, 24, 25, 26, 27, 28]).Another aspect of the same problem arises at quantization of 4 D, N = 2 superconformal2heories where the gauges preserve the manifest N = 2 supersymmetry but violate the globalsuperconformal invariance. This aspect was studied in the series of the papers [29, 30, 31] wherean approach to general problem of global symmetries in quantum gauge theories was proposedand it was shown under some assumptions concerning the structure of the theory that thequantization leads to deformation of global symmetries and, in particular, to deformation ofsuperconformal transformations.In this paper we study the above problem in general terms and generalize all the previousresults. We consider the general gauge theory possessing the non-anomalous global symmetryand quantize this theory in the framework of the BRST-BV procedure. It is assumed thatgauges violate the initial global symmetry of the classical action. Under these conditions wedevelop maximally general approach to structure of the global symmetries at the quantization ofgauge theories. We show like in work [29] that the quantization procedure leads to deformationof classical global transformations. However, we use the more general effective action thenin [29] what provides more possibilities for purely algebraic analysis of the global symmetriesin quantum gauge theory. We prove that although the vacuum functional is invariant underclassical global transformations, the effective action is not invariant and its invariance requiresthe quantum corrections to generators of global symmetry transformations. The maximallygeneral form of such deformations is derived. In particular, we prove that the full quantumaction in the functional integral is invariant under the deformed transformations. The one-loop invariance is studied in details and the corresponding one-loop deformation of the globaltransformations is calculated in explicit form. The results obtained are exclusive general andfulfill for any bosonic or fermionic gauge theory with an irreducible closed gauge algebra andwith an arbitrary (even open) non-anomalous algebra of global symmetry.The paper is organized as follows. Section 2 is devoted to description of the general gaugetheory with closed irreducible gauge algebra and fixing the notations and conventions. InSection 3 we describe the properties of the general gauge theory with global symmetry. InSection 4 we construct the quantum action of the theory under consideration and prove thatthe quantization leads to deformation of the classical global symmetries. In section 5 we considerthe above deformation in one-loop approximation and construct the deformation in an explicitform. Section 6 summarizes the results.In the paper the DeWitt’s condensed notations are used [32]. We employ the notation ε ( A )for the Grassmann parity of any quantity A . All derivatives with respect to sources are takenfrom the left only. The right and left derivatives with respect to fields are marked by specialsymbols ” ← ” and ” → ” respectively. The symbol A ,i ( ϕ ) means the right derivative of A ( ϕ )with respect to the field ϕ i . 3 General gauge theory: notations and conventions
Consider the general gauge theory with closed irreducible gauge algebra. It means that theinitial action, S = S ( ϕ ), of the fields ϕ = { ϕ i } , i = 1 , , ..., n, ε ( ϕ i ) = ε i is invariant underthe gauge transformations S ,i ( ϕ ) R iα ( ϕ ) = 0 , δϕ i = R iα ( ϕ ) ξ α , (2.1)where ξ α are the arbitrary functions with Grassmann parities ε ( ξ α ) ≡ ε α , and R iα = R iα ( ϕ ), ε ( R iα ) = ε i + ε α are generators of gauge transformations which are assumed to be linear indepen-dent in gauge indices α . It is convenient to introduce the operators of gauge transformations,ˆ R α = ˆ R α ( ϕ ), ˆ R α = ←− δδϕ i R iα , (2.2)so that the gauge invariance of S (2.1) is written in the form S ˆ R α = 0 . (2.3)The algebra of the gauge generators ˆ R α has the following form due to the closure:[ ˆ R α , ˆ R β ] = − ˆ R γ F γαβ (2.4)or R iα,j R jβ − ( − ε α ε β R iβ,j R jα = − R iγ F γαβ , (2.5)where F γαβ = F γαβ ( ϕ ) are the structure coefficients depending in general on fields ϕ and obeyingthe symmetry properties F γαβ = − ( − ε α ε β F γβα . (2.6)For Yang-Mills theories they are constants. The Jacobi identities written in terms of the gaugegenerators and the structure coefficients read (cid:0) F σαρ F ρβγ + F σαβ,i R iγ (cid:1) ( − ε α ε γ + cycle( αβγ ) = 0 . (2.7)For irreducible gauge theories the extended configuration space is described by the fields : φ A = ( ϕ i , B α , C α , ¯ C α ) , (2.8) ε ( ϕ i ) = ε i , ε ( B α ) = ε α , ε ( C α ) = ε ( ¯ C α ) = ε α + 1 , gh( ϕ i ) = gh( B α ) = 0 , gh( C α ) = 1 , gh ¯ C α ) = − , B α are Nakanishi-Lautrup auxiliary fields, C α and ¯ C α are the ghost and anti-ghost fields.For gauge theories under consideration which belong to the rank 1 gauge theories in terminologyof BV formalism [11, 12] the total (quantum) action, S = S ( φ ), can be written in the form ofthe Faddeev-Popov action [33], S ( φ ) = S ( ϕ ) + ¯ C α χ α,i ( ϕ ) R iβ ( ϕ ) C β + χ α ( ϕ ) B α (2.9)where χ α = χ α ( ϕ ) , ε ( χ α ) = ε α , are some gauge functions lifting the degeneracy of the classicalgauge invariant action S .The action (2.9) is invariant under the following BRST transformation δ B S ( φ ) = 0 , δ B φ A = R A ( φ ) λ (2.10)with R A ( φ ) = ( R iα ( ϕ ) C α , , − ( − ε β F αβγ ( ϕ ) C γ C β , B α ) , (2.11)where λ is a constant Grassmann parameter ( ε ( λ ) = 1). Introducing the operator of BRSTtransformations, ˆ R = ˆ R ( φ ), and using abbreviation R A = R A ( φ )ˆ R = ←− δδφ A R A , ε ( ˆ R ) = 1 , (2.12)the BRST invariance of action S (2.10) can be written as S ˆ R = 0 . (2.13)With the help of ˆ R the action (2.9) rewrites in the form S = S + Ψ ˆ R, (2.14)where we have introduced the gauge fixing functional Ψ = Ψ( φ ) of the formΨ( φ ) = ¯ C α χ α ( ϕ ) , ε (Ψ) = 1 . (2.15)The quantum action in form of (2.14) is evidently BRST invariant due to the nilpotency of ˆ R ,ˆ R = 0. Let the initial action S ( ϕ ) is invariant under the global symmetry transformations as well, S ,i ( ϕ ) T ia ( ϕ ) = 0 , δ T ϕ i = T ia ( ϕ ) ω a (3.1)5here ω a , a = 1 . , ..., m are constant parameters with Grassmann parities ε ( ω a ) ≡ ε a , and T ia = T ia ( ϕ ) ( ε ( T ia ) = ε i + ε a ) are generators of global transformations. Let ˆ T = ˆ T ( ϕ ),ˆ T = ˆ T a ω a , ˆ T a = ˆ T a ( ϕ ) = ←− δδϕ i T ia , (3.2)be the operator of global transformations. Then, in general (see [19])[ ˆ T a , ˆ T b ] = − ˆ T c f cab − ˆ R α K αab − ←− δδϕ i λ ijab S ,j , (3.3)or T ia,j T jb − ( − ε a ε b T ib,j T ja = − T ic f cab − R iα K αab − λ ijab S ,j , (3.4)Here, f cab = f cab ( ϕ ) , f cab = − f cba ( − ε a ε b , ε ( f cab ) = ε a + ε b + ε c , (3.5) K αab = K αab ( ϕ ) , K αab = − K αba ( − ε a ε b , ε ( K αab ) = ε a + ε b + ε α , (3.6) λ ijab = λ ijab ( ϕ ) , λ ijab = − λ ijba ( − ε a ε b , ε ( λ ijab ) = ε a + ε b + ε i + ε j . (3.7)To close the algebra of gauge and global symmetries (2.5) and (3.4) we add the followingrelations [ ˆ R α , ˆ T a ] = − ˆ R β U βαa , (3.8)or R iα,j T jb − ( − ε α ε b T ib,j R jα = − R iβ U βαb , (3.9)which means that the commutator of gauge and global transformations is characterized by localparameters. The structure coefficients U βαb = U βαb ( ϕ ) obey the symmetry properties U βαb = − U βbα ( − ε α ε b . (3.10)The Jacobi identities for global generators have the form T id (cid:0) f dae f ebc + f dab,i T ic (cid:1) ( − ε a ε c + R iα (cid:0) U αaβ K βbc + K αab,i T ic (cid:1) ( − ε a ε c ++ (cid:0) T ia,j λ j kbc S ,k − λ ijab S ,jk T kc − λ ijab,k S ,j T kc ( − ε j ε k (cid:1) ( − ε a ε c ++cycle ( a, b, c ) = 0 . (3.11)In the case of closed global transformations ( λ ijab = 0) the Jacobi identity (3.11) splits into tworelations (cid:0) f dae f ebc + f dab,i T ic (cid:1) ( − ε a ε c + cycle ( a, b, c ) = 0 , (3.12) (cid:0) U αaβ K βbc + K αab,i T ic (cid:1) ( − ε a ε c + cycle ( a, b, c ) = 0 . (3.13)6sing the Jacobi identity for two global and one gauge transformations we obtain thefollowing relations T ic f cab,i R iα ( − ε α ε a + R iγ (cid:0) ( U γαc f cab + F γαβ K γab )( − ε α ε b + K γab,j R jα ( − ε α ε a ++( U γαa,j T jb − U γαb,j T ja ( − ε a ε β )( − ε α ε b − ( U γaβ U βbα − U γbβ U βaα ( − ε a ε b )( − ε α ε a (cid:1) ++ (cid:0) λ ijab,k R kα ( − ε α ( ε a + ε j ) − R iα,k λ kjab ( − ε α ε b (cid:1) S ,j + λ ijab S ,jk R kα ( − ε α ε a = 0 . (3.14)For closed global transformations ( λ ijab = 0) it follows from (3.14) that the structure coefficients f cab are gauge invariant, f cab,i R iα = 0 , (3.15)and the following relations take place − (cid:0) U γaβ U βbα − U γbβ U βaα ( − ε a ε b (cid:1) ( − ε α ( ε a + ε b ) + K γab,j R jα ( − ε α ( ε a + ε b ) ++ (cid:0) U γαa,j T jb − U γαb,j T ja ( − ε a ε b (cid:1) + U γαc f cab + F γαβ K γab = 0 . (3.16)The Jacobi identity for operators ˆ R α , ˆ R β , ˆ T a lead to the relations F σαβ,j T ja − U σaγ F γαβ ( − ε a ( ε α + ε β ) ++ F σαγ U γβa − F σβγ U γαa ( − ε α ε β ++ U σβa,j R jα ( − ε α ( ε β + ε a ) − U σαa,j R jβ ( − ε a ε β = 0 . (3.17)The relations(2.1), (2.4) - (2.7), (3.1) - (3.11), (3.14) and (3.17) describe structure and propertiesof symmetry algebra of the gauge system under consideration.The operators ˆ R and ˆ T a do not commute,[ ˆ R, ˆ T a ] = − ˆ R β U βαa C α ( − ε a ++ ←− δδC α
12 ( − ε β F αβγ,j T ja C γ C β ( − ε a ( ε β + ε γ ) . (3.18)From (3.18) we obtain the important relationsΨ[ ˆ R, ˆ T a ] = − Ψ ˆ R β U βαa C α ( − ε a . (3.19)The right-hand side in (3.19) is nothing but the gauge transformations of Ψ with gauge param-eters Λ β ( φ ) = U βαa ( ϕ ) C α ( − ε a ω a .The quantum action (2.9) is not invariant under the global transformations (3.1). Using(3.19), the variation of S = S ( φ ) can be presented in the form δ T S = S ˆ T a ω a = (cid:0) Ψ ˆ T a ω a (cid:1) ˆ R + (cid:0) Ψ ˆ R β (cid:1) U βαa ω a C α . (3.20)7he first term in the right-hand side of (3.20) describes the variation of gauge fixing functional Ψunder global transformation while the second summand is the gauge transformation of Ψ = Ψ( φ )with local parameters Λ α = Λ α ( φ ),Λ α ( φ ) = U αβa ( ϕ ) ω a C β . (3.21)Second term in (3.20) can be presented in the form (cid:0) Ψ ˆ R β (cid:1) U βαa ω a C α = S ,α Λ α , S ,α = S ←− δδC α . (3.22)It will allow us in the next Section to analyze the (in)dependence of the effective action on theglobal symmetry transformations in the theory under consideration. In this Section we consider properties of global symmetry within the BRST quantizationtaking into account that the status of the gauge symmetry is well-known. In particular, thevacuum functional, Z, Z = Z χ = Z Dφ exp n i ~ S ( φ ) o (4.1)does not depend on the choice of admissible gauge fixing functions χ α thanks to the BRSTsymmetry of S ( φ ) (2.10), δ χ Z = 0 . (4.2)In deriving this result the following conditions( − ε β F ββα ( ϕ ) = 0 , ( − ε i −→ δδϕ i R iα ( ϕ ) = 0 (4.3)are used.Let Z T be the vacuum functional for the theory with action S ( φ ) + δ T S ( φ ) Z T = Z Dφ exp n i ~ (cid:2) S ( φ ) + δ T S ( φ ) (cid:3)o . (4.4)Making use in the functional integral (4.4) the change of variables in the form of BRST trans-formations (2.10) but with replacement λ → Λ( φ ) whereΛ( φ ) = i ~ ¯ C α χ α,i ( ϕ ) T ia ( ϕ ) ω a = i ~ Ψ( φ ) ˆ T ( ϕ ) , (4.5)and taking into account the triviality of Jacobian of such change, we arrive at the relation Z T = Z Dφ exp n i ~ (cid:2) S ( φ ) + S ,α ( φ )Λ α ( φ ) (cid:3)o . (4.6)8hen performing the change of variables C α in the form C α → C α − U αβa ( ϕ ) ω a C β (4.7)with the Jacobian equal to the unit and assuming the fulfilment of the conditions( − ε α U ααa ( ϕ ) = 0 , (4.8)we have the statement Z T = Z, (4.9)i. e. the vacuum functional is invariant under the classical global transformations.The generating functional of the Green functions, Z ( J ), and the connected Green functions, W ( J ), is represented by the functional integral Z ( J ) = Z Dφ exp n i ~ (cid:2) S ( φ ) + J A φ A (cid:3)o = exp n i ~ W ( J ) o . (4.10)As a main consequence of the BRST symmetry of S ( φ ), there exits the Ward identity for Z ( J )and W ( J ) in the form J A R A (cid:16) ~ i δδJ (cid:17) Z ( J ) = 0 , J A R A (cid:16) δWδJ + ~ i δδJ (cid:17) · . (4.11)The generating functional of vertex functions (effective action), Γ = Γ(Φ), is defined standardlythrough the Legendre transformation of W ( J ),Γ(Φ) = W ( J ) − J A Φ A , Φ A = δW ( J ) δJ A , (4.12)so that Γ(Φ) ←− δδ Φ A = − J A . (4.13)The Ward identity (4.11) rewrites for Γ(Φ) in the formΓ(Φ) ←− δδ Φ A ¯ R A (Φ) = 0 , (4.14)where ¯ R A (Φ) = R A ( ˆΦ) · , (4.15)and ˆΦ A = Φ A + i ~ (Γ ′′ − ) AB (Φ) −→ δδ Φ B . (4.16)9n (4.16) the matrix (Γ ′′ − ) AB (Φ) is inverse to(Γ ′′ ) AB (Φ) = −→ δδ Φ A (cid:16) Γ(Φ) ←− δδ Φ B (cid:17) , (Γ ′′ − ) AC (Γ ′′ ) CB = δ AB . (4.17)The Ward identity (4.14) can be interpreted as the invariance of effective action Γ(Φ) underthe quantum BRST transformations of Φ A with generators ¯ R A (Φ).In the functional integral (4.10) we make the change of integration variables ϕ i in the formof global transformations (3.1). Then, using the conditions( − ε i −→ δδϕ i T ia ( ϕ ) = 0 , (4.18)we obtain Z ( J ) = Z Dφ exp n i ~ (cid:2) S ( φ ) + δ T S ( φ ) + J A φ A + j i T ia ( ϕ ) ω a (cid:3)o , (4.19)where δ T S ( φ ) is defined in (3.20) and j i are external sources to ϕ i . The conditions (4.18) leadto that the Jacobian of change of variables in the functional integral (4.19) is equal to unit. Byperforming the change of variables φ A in the form φ A → φ A + R A ( φ )Λ( φ ) (4.20)with Λ( φ ) given in (4.5), and then additionally the transformations (4.7), we find in the firstorder in ω a Z Dφ (cid:16) j i T ia ( ϕ )+ J A R A ( φ )Λ a ( φ )+ J α ( C ) U αaβ ( ϕ ) C β ( − ε a (cid:17) exp n i ~ (cid:2) S ( φ )+ J A φ A (cid:3)o = 0 , (4.21)where we took into account that the Jacobian of change of variables is trivial and J α ( C ) aresources to fields C α . The equation (4.21) rewrites h j i T ia (cid:16) ~ i δδj (cid:17) + J A R A (cid:16) ~ i δδJ (cid:17) Λ a (cid:16) ~ i δδJ (cid:17) + ~ i J α ( C ) U αaβ (cid:16) ~ i δδj (cid:17) δδJ β ( C ) ( − ε a i Z ( J ) = 0 , (4.22) a = 1 , , ..., m. In terms of the functional W = W ( J ) the relations (4.22) read j i T ia (cid:16) δWδj + ~ i δδj (cid:17) + J A R A (cid:16) δWδJ + ~ i δδJ (cid:17) Λ a (cid:16) δWδJ + ~ i δδJ (cid:17) ++ J α ( C ) U αaβ (cid:16) δWδj + ~ i δδj (cid:17) δWδJ β ( C ) ( − ε a = 0 , a = 1 , , ..., m.. (4.23)Then in terms of Γ(Φ) the relations (4.23) can be presented in the formΓ(Φ) (cid:16) ←− δδ Φ i T ia (Φ) + ←− δδ Φ A M Aa (Φ) + ←− δδ Φ α ( C ) U αa (Φ) (cid:17) = 0 , (4.24)10here the notations T ia (Φ) = T ia ( ϕ ) | ϕ i → ˆΦ i · , M Aa (Φ) = R A ( ˆΦ)Λ a ( ˆΦ) · ,U αa (Φ) = U αaβ ( ϕ ) | ϕ i → ˆΦ i Φ β ( C ) ( − ε a , (4.25) J A = (cid:0) j i , J α ( B ) , J α ( C ) , J α ( ¯ C ) (cid:1) , Φ A = (cid:0) Φ i , Φ α ( B ) , Φ α ( C ) , Φ α ( ¯ C ) (cid:1) , were used.The relations (4.24) mean that the effective action is invariant under the quantum globaltransformation, Γ(Φ) ←− δδ Φ A ¯ T Aa (Φ) = 0 (4.26)with the deformed generators¯ T Aa (Φ) = (cid:0) T ia (Φ) + M ia (Φ) , , M α ( C ) a (Φ) + U αa (Φ) , M α ( ¯ C ) a (Φ) (cid:1) . (4.27)The proof of invariance (4.26) is based on using the change of variables (4.5) which are notanalytical in loop expansion parameter ~ . Therefore unlike the study of gauge dependence ofeffective action in the framework of BRST-BV formalism, the derivation of (4.27) is not relatedto change of gauge functions. In the next Section we will however proof the correctness of(4.26) in loop expansion procedure. As we pointed out at the end of Section 4, the change of variables (4.5) is non-analyticalin ~ and the use of loop expansion looks doubtful. However, we will show that the one-loopapproximation still works.Consider the relations (4.26) in loop approximation. For the effective action we haveΓ(Φ) = S (Φ) + ~ Γ (Φ) + O ( ~ ) . (5.1)The generators ¯ T Aa (Φ) are written in the same approximation as follows¯ T Aa (Φ) = 1 ~ R A (Φ) ˜Λ(Φ) + ¯ T A (0) a (Φ) + ~ ¯ T A (1) a (Φ) + O ( ~ ) , ˜Λ(Φ) = ~ Λ(Φ) . (5.2)Due to the invariance of quantum action S (Φ) (2.9) under the BRST transformations (2.10),the first item in the right-hand side of (5.2), which is non-analytical in ~ , does not contributein the relations (4.26). As a result in zero loop approximations these relation take the form δS (Φ) δ Φ A ¯ T A (0) a (Φ) + δ Γ (Φ) δ Φ A R A (Φ) ˜Λ(Φ) = 0 . (5.3)11n the next order we get S (Φ) ←− δδ Φ A ¯ T A (1) a (Φ) + Γ (Φ) ←− δδ Φ A ¯ T A (0) a (Φ) = 0 . (5.4)Now let us take into account the Ward identity for Γ(Φ) (4.14), then in the first order in ~ wehave Γ (Φ) ←− δδ Φ A R A (Φ) = 0 . (5.5)With this result the relations (5.3) rewrites S (Φ) ←− δδ Φ A ¯ T A (0) a (Φ) = 0 . (5.6)These relations demonstrate that the quantum action S (Φ) is one-loop invariant under the de-formed global symmetry transformations. In particular, the generators of this global symmetryin the sector of fields Φ i have the form¯ T i (0) a (Φ) = ¯ T i (0) a ( φ ) | φ → Φ , ¯ T i (0) a ( φ ) = T ia ( ϕ ) − R iα ( ϕ )Σ α (0) a ( φ ) − R iα,j ( ϕ )Θ j α (0) a ( φ ) , (5.7)where the following notationsΣ α (0) a ( φ ) = (cid:16) C α ( S ′′ − ) β ( ¯ C ) j ( φ ) + ( S ′′ − ) α ( C ) j ( φ ) ¯ C β ( − ε j ( ε β +1) (cid:17) N βa,j ( ϕ ) ++( S ′′ − ) α ( C ) β ( ¯ C ) ( φ ) χ β,kj ( ϕ ) T ja ( ϕ ) ++ C α ¯ C β χ β,j k ( ϕ )( S ′′ − ) kl ( φ ) T ja, l ( ϕ )( − ε l ( ε j + ε a +1) , (5.8)Θ j α (0) a ( φ ) = (cid:16) ( S ′′ − ) j α ( C ) ( φ ) ¯ C β + ( S ′′ − ) j β ( ¯ C ) ( φ ) C α ( − ( ε α +1)( ε β +1) (cid:17) N βa ( ϕ ) ++( S ′′ − ) j l ( φ ) C α ¯ C β N βa,l ( ϕ )( − ε l ( ε α + ε β ) , (5.9) N βa ( ϕ ) = χ β,k ( ϕ ) T ka ( ϕ ) . (5.10)are introduced. The deformation of global generators T ia ( ϕ ), defined in initial configurationspace { ϕ } , looks very non-trivial already in the one-loop approximation. Such a deformation isdone in the full configuration space of the gauge theory (2.8) with the help of generators R iα ( ϕ )of gauge symmetry of initial action (2.1) and its derivatives R iα,j ( ϕ ). The relations (2.1), (3.1)and (5.7) lead to S ,i ( ϕ ) ¯ T i (0) a ( φ ) = − S ,i ( ϕ ) R iα,j ( ϕ )Θ j α (0) a ( φ ) = 0 . (5.11)It means non invariance of an initial action S ( ϕ ) under the deformed global transformations. In the present paper we have studied the general problem of global (rigid) symmetries inquantum general gauge theory with closed irreducible gauge algebra. Using the BRST-BV12uantization procedure [11, 12] we have constructed a deformation of global symmetry genera-tors so that the full quantum action of the initial gauge theory and hence the effective action areinvariant under such deformed global transformations. This statement is valid for any bosonicor fermionic gauge theory with an irreducible and closed algebra of gauge transformationsand with an arbitrary (even open) non-anomalous algebra of global symmetry. Form of thedeformed global symmetry generators in one-loop approximation is calculated in the explicitform. Note that from algebraic point of view the deformation of global symmetry generators,studied in the present paper, is similar to the known phenomena of the deformation of gaugegenerators under renormalization procedure (see e.g. [34, 35, 36]), although the mechanisms ofdeformation differ.In general the BRST-BV technique involves introduction for every field of the full configu-ration space the corresponding antifield with opposite Grassmann parity. It allows to study themany properties of the gauge theory on quantum level in general terms. However, to simplifythe consideration in this paper we derived all the results in the configuration space of fieldsonly (2.8). Generalization of the results obtained for extended configuration space of fields andantifields can be done in the same method.Problem of global symmetries in quantum general gauge theories is discussed in earlier pa-pers [16, 17, 18, 19] however the accents were aimed on the other aspects. First, in [16] itwas shown the possibility to construct the Lagrangian models possessing the gauge and su-persymmetry invariance under the assumption that the global supersymmetry transformationsobey the closed algebra. In our paper we work with an arbitrary (including open) global (su-per)symmetry of a given gauge theory. Also we suppose the absence of anomalies in combinedgauge and global algebra of generators but in principle it is possible to study the problemof coexistence of gauge and global symmetries in presence of anomalies using the approachof work [17]. Of course the implementation of this program requires a separate independentstudy. Second, in paper [18] it was shown that the attempts to construct an action invariantunder BRST- and closed rigid symmetries lead to the breakdown of non-degeneracy of the fullquantum action. In our paper we proved that this problem is resolved on the base of quantumdeformation of global generators. Third, in paper [19] it was shown that the global symmetriesof an initial gauge action in the field-antifield formalism can be extended to include the allfields and the all antifields. After that the authors of [19] introduced a constant ghost for eachglobal symmetry and modified the master-equation to incorporate the global symmetries intoits solutions but they did not discuss the deformations of the global symmetry generators. It isclear that this approach differs from our consideration since from the beginning we work onlyin the configuration field space and derive the above deformations.The general enough approach to the problem of global symmetries in the quantum gaugetheory was proposed in the work [29] under assumptions that the initial classical theory consistsof only bosonic fields and the algebra of the classical global transformations is closed. In our13aper we have considered maximally general case of the theory with bosonic and fermionic fieldsand open algebra of the global (super)symmetries. However, what is very important, we workwith more general effective action then in [29]. In our case, the sources included not only toinitial fields but also to ghost and auxiliary fields. This allowed us to describe the deformationof the generators completely in the algebraic terms.The global (super)symmetric and gauge transformations have been studied in the frameworkof singular gauge fixing procedure. In practice it is more convenient to use a non-singulargauges. All our basic statements still will be valid in this case as well. We should only modifythe quantum action (2.9) in the form S ( φ ) = S ( ϕ ) + ¯ C α χ α,i ( ϕ ) R iβ ( ϕ ) C β + χ α ( ϕ ) B α + ξ B α B α , (6.1)where ξ is a gauge parameter. Due to the property of BRST transformations (2.11) B α ˆ R = 0,the action (6.1) remains invariant under the BRST transformations, S ˆ R = 0. Thus, from aprincipled point of view the results obtained will be same. Acknowledgments
The authors thank I.V. Tyutin for useful discussions and S.M. Kuzenko for correspondence.The research was supported in parts by Russian Ministry of Education and Science, projectNo. 3.1386.2017. The authors are also grateful to RFBR grant, project No. 18-02-00153 forpartial support.
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