BRST-BV Quantization of Gauge Theories with Global Symmetries

Eur. Phys. J. C (2018) 78:524 https://doi.org/10.1140/epjc/s10052-018-6003-x

BRST-BV quantization of gauge theories with global symmetries

I. L. Buchbinder1,2,3,a,P.M.Lavrov1,2,b 1 Center of Theoretical Physics, Tomsk State Pedagogical University, Kievskaya St. 60, 634061 Tomsk, Russia 2 National Research Tomsk State University, Lenin Av. 36, 634050 Tomsk, Russia 3 Departamento de Física, ICE, Universidade Federal de Juiz de Fora, Campus Universitário, Juiz de Fora, MG 36036-900, Brazil

Received: 28 February 2018 / Accepted: 17 June 2018 © The Author(s) 2018

Abstract We consider the general gauge theory with case the important physical properties of the theory can be a closed irreducible gauge algebra possessing the non- violated. Example of such a model is N = 4 supersymmetric anomalous global (super)symmetry in the case when the Yang–Mills theory, where the classical rigid superconformal gauge fixing procedure violates the global invariance of clas- symmetry is conserved at quantum level (see e.g. [18]). As sical action. The theory is quantized in the framework of a result, we arrive at a general problem to study the classical BRST-BV approach in the form of functional integral over all global symmetries in BRST quantization of the gauge theory. fields of the configuration space. It is shown that the global A specific aspect of the above general problem arises symmetry transformations are deformed in the process of in supersymmetric gauge theories. These theories possess a quantization and the full quantum action is invariant under gauge symmetry and rigid or global supersymmetries. Such such deformed global transformations in the configuration theories can be formulated either in the component formalism space. The deformed global transformations are calculated or in the terms of superfields. In the first case the supersym- in an explicit form in the one-loop approximation. metry is non manifest and when we quantize the corresponding gauge theory imposing the gauge fixing conditions on vector fields we violate the rigid supersymmetry. Therefore 1 Introduction it is unclear if the quantum effective action should be supersymmetric. At first, this problem arose in N = 1 super Yang– BRST quantization procedure, initiated in the works [1–8], Mills theories and led to the study of the aspects of global which can be realized within the Hamiltonian BFV approach symmetry in quantum gauge theories [19–22] However, this [9,10] or within the Lagrangian BV approach [11,12], is a problem was automatically resolved after formulation of the powerful and universal tool to formulate the quantum gauge N = 1 supergauge theories in terms of N = 1 superfields theories and investigate their structures. This procedure is (see e.g. [23]), where the manifestly N = 1 gauges have applicable to an extremely wide class of gauge theories been used. However, the problem is still remains in extended including the (super) Yang–Mills theories, (super)gravity, super Yang–Mills theories. At present, the best formulation (super)strings and more specific gauge theories like the gauge of 4D, N = 4 rigid super Yang–Mills theory is achieved in antisymmetric field models. All these theories possess many terms of 4D, N = 2 or in terms of 4D, N = 3 harmonic common properties therefore one can talk about a general superfields [24]. In this case only part of supersymmetries are gauge theory and study its quantum aspects, including the manifest, the other remain hidden. The same situation will renormalization issues (see e.g. [13–15]), in general terms. be in 6D, N = (1, 1) rigid super Yang–Mills theory, which Besides the local symmetries, many gauge theories are is similar in many aspects to 4D, N = 4 super Yang–Mills characterized by the rigid symmetries. For example, the role theory. At present, the best formulation of such a theory is which is played by the global chiral symmetry in the Standard achieved in terms of 6D, N = (1, 0) superfields (see e.g. Model (see e.g. [16]) or the role which is played by global [29] and references therein). Again, one part of supersym- conformal group in the String Theory (see e.g. [17]) are well metries is manifest and the other part is hidden. In all such known. In many cases it is essential to preserve the classical theories the gauges preserve the manifest symmetries and global symmetry in quantum theory since in the opposite violate the hidden supersymmetries (see e.g. [25–28,30,31]). Another aspect of the same problem arises at quantization a e-mail: [email protected] of 4D, N = 2 superconformal theories where the gauges b e-mail: [email protected] 123 524 Page 2 of 8 Eur. Phys. J. C (2018) 78:524 preserve the manifest N = 2 supersymmetry but violate the 2 General gauge theory: notations and conventions global superconformal invariance. This aspect was studied in the series of the papers [32–34] where an approach to general Consider the general gauge theory with closed irreducible problem of global symmetries in quantum gauge theories was gauge algebra. It means that the initial action, S0 = S0(ϕ), i i proposed and it was shown that under some assumptions of the fields ϕ ={ϕ }, i = 1, 2,...,n,ε(ϕ) = εi is concerning the structure of the theory, the quantization leads invariant under the gauge transformations to deformation of global symmetries and, in particular, to i i i α S , (ϕ)R (ϕ) = 0,δϕ= R (ϕ)ξ , (2.1) deformation of superconformal transformations. 0 i α α In this paper we study the above problem in general terms where ξ α are the arbitrary functions of space-time coordi- 1 α i and generalize all the previous results. We consider the gen- nates with Grassmann parities ε(ξ ) ≡ εα, and Rα = i i eral gauge theory possessing the non-anomalous global sym- Rα(ϕ), ε(Rα) = εi + εα are generators of gauge transforma- metry and quantize this theory in the framework of the BRST- tions which are assumed to be linear independent in gauge BV procedure. It is assumed that gauges violate the initial indices α. It is convenient to introduce the operators of gauge ˆ ˆ global symmetry of the classical action. Under these condi- transformations, Rα = Rα(ϕ), tions we develop a maximally general approach to the struc- ←− δ ture of the global symmetries at the quantization of gauge ˆ i Rα = Rα, (2.2) theories. We show like in work [32] that the quantization pro- δϕi cedure leads to deformation of classical global transforma- so that the gauge invariance of S0 (2.1) is written in the form tions. However, we use the more general effective action then ˆ in [32] that provides more possibilities for purely algebraic S0 Rα = 0. (2.3) analysis of the global symmetries in quantum gauge theory. ˆ The algebra of the gauge generators Rα has the following We prove that although the vacuum functional is invariant form due to the closure: under classical global transformations, the effective action ˆ ˆ ˆ γ is not invariant and its invariance requires the quantum cor- [Rα, Rβ ]=−Rγ F αβ (2.4) rections to generators of global symmetry transformations. The maximally general form of such deformations is derived. or In particular, we prove that the full quantum action in the i j − (− )εαεβ i j =− i γ , Rα, j Rβ 1 Rβ,j Rα Rγ F αβ (2.5) functional integral is invariant under the deformed transfor- γ γ mations. The one-loop invariance is studied in detail and the where F αβ = F αβ (ϕ) are the structure coefficients depend- corresponding one-loop deformation of the global transfor- ing in general on fields ϕ and obeying the symmetry proper- mations is calculated in explicit form. The results obtained ties are exclusively general and take place for any bosonic or γ ε ε γ F αβ =−(−1) α β F βα. (2.6) fermionic gauge theory with an irreducible closed gauge algebra and with an arbitrary (even open) non-anomalous For Yang–Mills theories they are constants. The Jacobi iden- algebra of global symmetry. tities written in terms of the gauge generators and the struc- The paper is organized as follows. Section 2 is devoted ture coefficients read to description of the general gauge theory with closed irre- σ ρ + σ i (− )εαεγ + (αβγ ) = . ducible gauge algebra and fixing the notations and conven- F αρ F βγ F αβ,i Rγ 1 cycle 0 (2.7) tions. In Sect. 3 we describe the properties of the general For irreducible gauge theories the extended configuration gauge theory with global symmetry. In Sect. 4 we construct space is described by the fields : the quantum action of the theory under consideration and A i α α ¯ α prove that the quantization leads to deformation of the clas- φ = (ϕ , B , C , C ), i α α ¯ α sical global symmetries. In Sect. 5 we consider the above ε(ϕ ) = εi ,ε(B ) = εα,ε(C ) = ε(C ) = εα + 1, α α α deformation in one-loop approximation and construct the gh(ϕi ) = gh(B ) = 0, gh(C ) = 1, ghC¯ ) =−1, deformation in an explicit form. Section 6 summarizes the (2.8) results. In the paper the DeWitt’s condensed notations are used where Bα are Nakanishi-Lautrup auxiliary fields, Cα and C¯ α [35]. We employ the notation ε(A) for the Grassmann parity are the ghost and anti-ghost fields. For gauge theories under of any quantity A. All derivatives with respect to sources consideration which belong to the rank 1 gauge theories in are taken from the left only. The right and left derivatives terminology of BV formalism [11,12] the total (quantum) with respect to fields are marked by special symbols “←” → (ϕ) and “ ” respectively. The symbol A,i means the right 1 In DeWitt’s condensed notation it means the index α includes space- derivative of A(ϕ) with respect to the field ϕi . time coordinates among others. 123 Eur. Phys. J. C (2018) 78:524 Page 3 of 8 524 action, S = S(φ), can be written in the form of the Faddeev- of global parameter by latin letter. We should keep in mind Popov action [36], that summation in latin letters does not include a space-time ˆ = ˆ (ϕ) ¯ α i β α integral. Let T T , S(φ) = S0(ϕ) + C χα,i (ϕ)Rβ (ϕ)C + χα(ϕ)B (2.9) ←− δ where χα = χα(ϕ), ε(χα) = εα, are some gauge functions ˆ ˆ a ˆ ˆ i T = Taω , Ta = Ta(ϕ) = T , (3.2) lifting the degeneracy of the classical gauge invariant action δϕi a S . 0 be the operator of global transformations. Then, in general The action (2.9) is invariant under the following BRST (see [22]) transformation ←− δ δ (φ) = ,δφ A = A(φ)λ ˆ ˆ ˆ c ˆ α ij B S 0 B R (2.10) [T , T ]=−T f − Rα K − λ S , , (3.3) a b c ab ab δϕi ab 0 j with or A i α R (φ) = (Rα(ϕ)C , 0, ε ε i j − (− ) a b i j 1 ε α γ β α Ta, j Tb 1 Tb, j Ta − (−1) β F βγ (ϕ)C C , B ), (2.11) 2 =− i c − i α − λij , Tc f ab Rα K ab abS0, j (3.4) where λ is a constant Grassmann parameter (ε(λ) = 1). Intro- ducing the operator of BRST transformations, Rˆ = Rˆ(φ), Here, and using abbreviation R A = R A(φ) ε ε f c = f c (ϕ), f c =−f c (−1) a b , ←− ab ab ab ba δ ε( f c ) = ε + ε + ε , (3.5) Rˆ = R A,ε(Rˆ) = 1, (2.12) ab a b c δφA α α α α εa εb K ab = K ab(ϕ), K ab =−K ba(−1) , α the BRST invariance of action S (2.10) can be written as ε(K ab) = εa + εb + εα, (3.6) ε ε ˆ λij = λij (ϕ), λij =−λij (− ) a b , SR = 0. (2.13) ab ab ab ba 1 ˆ ε(λij ) = ε + ε + ε + ε . With the help of R the action (2.9)rewritesintheform ab a b i j (3.7) ˆ S = S0 + R, (2.14) To close the algebra of gauge and global symmetries (2.5) and (3.4) we add the following relations where we have introduced the gauge fixing functional = (φ) ˆ ˆ ˆ β of the form [Rα, Ta]=−RβU αa, (3.8) ¯ α (φ) = C χα(ϕ), ε() = 1. (2.15) or

The quantum action in form of (2.14) is evidently BRST ε ε β i j − (− ) α b i j =− i , invariant due to the nilpotency of Rˆ, Rˆ2 = 0. Rα, j Tb 1 Tb, j Rα RβU αb (3.9) which means that the commutator of gauge and global transformations is some gauge transformation. The relations (2.4), 3 General gauge theory with rigid symmetry (3.3), (3.8) express the algebraic structure of the set of local and global transformations in the general gauge theory. In (ϕ) Let the initial action S0 is invariant under the global sym- particular, they show that the gauge transformations form the metry transformations as well, ideal in the commutator algebra of global and gauge trans- i i i a S , (ϕ)T (ϕ) = 0,δϕ = T (ϕ)ω (3.1) formations under consideration. The structure coefﬁcients 0 i a T a β β U αb = U αb(ϕ) obey the symmetry properties where ωa, a = 1.2,...,m are constant (not depending on β β εαεb space-time coordinates) parameters with Grassmann parities U αb =−U bα(−1) . (3.10) ε(ωa) ≡ ε i = i (ϕ) ε( i ) = ε + ε a, and Ta Ta ( Ta i a) are generators of global transformations. We want to emphasize once The Jacobi identities for global generators have the form more that the gauge transformations are characterized by i d e d i ε ε T f f + f , T (−1) a c the space-time dependent parameters while the parameters d ae bc ab i c i α β α i εa εc of the global transformations are space-time independent. + Rα U aβ K bc + K ab,i T (−1) c In the framework of the condensed notations the gauge and + i λ jk − λij k Ta, j S0,k S0, jkTc bc ab global transformations look similar. To make the difference ε ε ε ε − λij k(− ) j k (− ) a c of two these types of the transformations explicit we denote ab,k S0, j Tc 1 1 the index of gauge parameter by greek letter and the index + cycle (a, b, c) = 0. (3.11) 123 524 Page 4 of 8 Eur. Phys. J. C (2018) 78:524

λij = In the case of closed global transformations ( ab 0) the The right-hand side in (3.19) is nothing but the gauge Jacobi identity (3.11) splits into two relations transformations of with gauge parameters β (φ) = β α ε α (ϕ) (− ) a ωa ε ε U a C 1 . d e + d i (− ) a c + ( , , ) = , f ae f bc f ab,i Tc 1 cycle a b c 0 The quantum action (2.9) is not invariant under the global (3.12) transformations (3.1). Using (3.19), the variation of S = α β α i εa εc S(φ) can be presented in the form U aβ K bc + K ab,i T (−1) + cycle (a, b, c) = 0. c ˆ a ˆ a ˆ ˆ β a α (3.13) δT S = STaω = Taω R + Rβ U αaω C . (3.20) Using the Jacobi identity for two global and one gauge The first term in the right-hand side of (3.20) describes the transformations we obtain the following relations variation of gauge fixing functional under global transfor- ε ε γ mation while the second summand is the gauge transforma- i c i (− ) α a + i ( c Tc f ab,i Rα 1 Rγ U αc f ab tion of = (φ) with local parameters α = α(φ), γ γ ε ε γ ε ε + )(− ) α b + j (− ) α a F αβ K ab 1 K ab, j Rα 1 α α a β (φ) = U β (ϕ)ω C . (3.21) γ j γ j ε ε ε ε a + (U α , T − U α , T (−1) a β )(−1) α b a j b b j a γ β γ β εa εb εαεa Second term in (3.20) can be presented in the form − (U aβU bα − U bβU aα(−1) )(−1) ←− ε (ε +ε ) ε ε + λij k (− ) α a j − i λkj (− ) α b δ ab,k Rα 1 Rα,k ab 1 S0, j ˆ β ωa α = α, = . Rβ U αa C S,α S,α S α (3.22) ε ε δ + λij k (− ) α a = . C abS0, jk Rα 1 0 (3.14) It will allow us in the next section to analyze the λij = (in)dependence of the effective action on the global sym- For closed global transformations ( ab 0) it follows from c (3.14) that the structure coefficients f ab are gauge invariant, metry transformations in the theory under consideration. c i = , f ab,i Rα 0 (3.15) 4 Structure of global symmetries on quantum level and the following relations take place γ β γ β εa εb εα(εa +εb) − U aβU bα − U bβU aα(−1) (−1) In this section we consider properties of global symmetry

γ j εα(εa +εb) within the BRST quantization taking into account that the + K ab, j Rα(−1) status of the gauge symmetry is well-known. In particular, γ γ ε ε + j − j (− ) a b U αa, j Tb U αb, j Ta 1 the vacuum functional, Z, + γ c + γ γ = . U αc f ab F αβ K ab 0 (3.16) i Z = Zχ = Dφ exp S(φ) (4.1) ˆ ˆ ˆ h¯ The Jacobi identity for operators Rα, Rβ , Ta lead to the relations does not depend on the choice of admissible gauge ﬁxing functions χα thanks to the BRST symmetry of S(φ) (2.10), σ j σ γ εa (εα+εβ ) F αβ, j Ta − U aγ F αβ (−1) σ γ σ γ εαεβ δχ Z = 0. (4.2) + F αγ U βa − F βγ U αa(−1)

σ j εα(εβ +εa ) σ j εa εβ + U βa, j Rα(−1) − U αa, j Rβ (−1) = 0. In deriving this result the following conditions −→ (3.17) δ εβ β ε i (−1) F βα(ϕ) = 0,(−1) i Rα(ϕ) = 0 (4.3) The relations (2.1), (2.4)–(2.7), (3.1)–(3.11), (3.14) and δϕi (3.17) describe structure and properties of symmetry alge- are used. bra of the gauge system under consideration. ˆ ˆ Let ZT be the vacuum functional for the theory with action The operators R and Ta do not commute, S(φ) + δT S(φ) ˆ ˆ ˆ β α εa [R, Ta]=−RβU αaC (−1) i Z = Dφ exp S(φ) + δ S(φ) . (4.4) ←− T ¯ T δ 1 ε α γ β ε (ε +ε ) h + (− ) β j (− ) a β γ . α 1 F βγ,j Ta C C 1 δC 2 Making use in the functional integral (4.4) the change of (3.18) variables in the form of BRST transformations (2.10)but with replacement λ → (φ) where From (3.18) we obtain the important relations i i β α ε ¯ α i a ˆ [ ˆ, ˆ ]=− ˆ (− ) a . (φ) = C χα, (ϕ)T (ϕ)ω = (φ)T (ϕ), (4.5) R Ta RβU αa C 1 (3.19) h¯ i a h¯ 123 Eur. Phys. J. C (2018) 78:524 Page 5 of 8 524

and taking into account the triviality of Jacobian of such In (4.16) the matrix ( −1)AB() is inverse to change, we arrive at the relation −→ ←− δ δ ( ) () = () , i α AB δA δB Z = Dφ exp S(φ) + S,α(φ) (φ) . (4.6) T h¯ −1 AC A ( ) ( )CB = δ . (4.17) Then performing the change of variables Cα in the form B α α α a β The Ward identity (4.14) can be interpreted as the invariance C → C − U β (ϕ)ω C (4.7) a of effective action () under the quantum BRST transfor- ¯ with the Jacobian equal to the unit and assuming the fulfil- mations of A with generators R A(). ment of the conditions In the functional integral (4.10) we make the change of integration variables ϕi in the form of global transformations (− )εα α (ϕ) = , 1 U αa 0 (4.8) (3.1). Then, using the conditions we have the statement −→ ε δ (− ) i i (ϕ) = , 1 i Ta 0 (4.18) ZT = Z, (4.9) δϕ i. e. the vacuum functional is invariant under the classical we obtain global transformations. i ( ) Z(J) = Dφ exp S(φ) + δ S(φ) + J φ A + j T i (ϕ)ωa , The generating functional of the Green functions, Z J , h¯ T A i a ( ) and the connected Green functions, W J , is represented by (4.19) the functional integral δ (φ) i where T S is defined in (3.20) and ji are external sources Z(J) = Dφ exp S(φ) + J φ A ϕi h¯ A to . The conditions (4.18) lead to that the Jacobian of change of variables in the functional integral (4.19) is equal i = exp W(J) . (4.10) to unit. By performing the change of variables φ A in the form h¯ φ A → φ A + A(φ)(φ) As a main consequence of the BRST symmetry of S(φ), there R (4.20) ( ) ( ) exits the Ward identity for Z J and W J in the form with (φ) given in (4.5), and then additionally the transfor- a h¯ δ mations (4.7), we find in the first order in ω J R A Z(J) = 0, A i δ J Dφ j T i (ϕ)+ J R A(φ) (φ) δW h¯ δ i a A a A + · = . JA R 1 0 (4.11) δ J i δ J α β ε i + (ϕ) (− ) a (φ)+ φ A = , Jα(C) U aβ C 1 exp S JA 0 The generating functional of vertex functions (effective h¯ action), = (), is defined standardly through the Legen- (4.21) dre transformation of W(J), where we took into account that the Jacobian of change of δW(J) α A A variables is trivial and Jα(C) are sources to fields C .The () = W(J) − JA ,= , (4.12) δ JA equation (4.21) rewrites so that h¯ δ h¯ δ h¯ δ ←− j T i + J R A δ i a i δj A i δ J a i δ J () =−JA. (4.13) δA h¯ h¯ δ δ α εa + Jα( ) U aβ (−1) Z(J)=0, a =1, 2,...,m. C δ δ β The Ward identity (4.11) rewrites for () in the form i i j J (C) ←− (4.22) δ () R¯ A() = 0, (4.14) δA In terms of the functional W = W(J) the relations (4.22) where read

¯ A() = A()ˆ · , δW h¯ δ δW h¯ δ δW h¯ δ R R 1 (4.15) j T i + + J R A + + i a δj i δj A δ J i δ J a δ J i δ J and α δW h¯ δ δW ε + α β + (− ) a = , = , ,..., . −→ J (C) U a 1 0 a 1 2 m δj i δj δ Jβ( ) − δ C ˆ A = A + ih¯ ( 1)AB() . (4.16) δB (4.23) 123 524 Page 6 of 8 Eur. Phys. J. C (2018) 78:524

Then in terms of () the relations (4.23) can be presented side of (5.2), which is non-analytical in h¯ , does not contribute in the form in the relations (4.26). As a result in zero loop approximations ←− ←− ←− δ δ δ these relation take the form i A α () T () + M () + α U a() = 0, δi a δA a δ (C) δS() δ () T¯ A () + 1 R A()()˜ = 0. (5.3) δA (0)a δA (4.24) In the next order we get where the notations ←− ←− i () = i (ϕ)| · , A() = A()ˆ ()ˆ · , δ δ Ta Ta ϕi →ˆ i 1 Ma R a 1 () ¯ A () + () ¯ A () = . S A T(1)a 1 A T(0)a 0 (5.4) α α β ε δ δ () = (ϕ)| (C) (− ) a , U a U aβ ϕi →ˆ i 1 Now let us take into account the Ward identity for () = , , , , JA ji Jα(B) Jα(C) Jα( ¯ ) ¯ C (4.14), then in the first order in h we have α α α A = i , (B) , (C) , (C¯ ) , (4.25) ←− δ () R A() = 0. (5.5) were used. 1 δA The relations (4.24) mean that the effective action is invari- With this result the relations (5.3) rewrites ant under the quantum global transformation, ←− ←− δ δ () ¯ A () = . () T¯ A() = 0 (4.26) S T(0)a 0 (5.6) δA a δA with the deformed generators These relations demonstrate that the quantum action S() is one-loop invariant under the deformed global symmetry T¯ A() = T i () + Mi (), 0 , a a a transformations. In particular, the generators of this global α α (C) () + α(), (C¯ ) () . i Ma Ua Ma (4.27) symmetry in the sector of fields have the form The proof of invariance (4.26) is based on using the change ¯ i () = ¯ i (φ)| , ¯ i (φ) = i (ϕ) T(0)a T(0)a φ→ T(0)a Ta of variables (4.5) which are not analytical in loop expansion α −Ri (ϕ)α (φ) − Ri (ϕ) j (φ), parameter h¯ . Therefore, unlike the study of gauge depen- α (0)a α, j (0)a (5.7) dence of effective action in the framework of BRST-BV for- where the following notations malism, the derivation of (4.27) is not related to the change α α − β (φ) = ( 1) (C¯ ) j (φ) of gauge functions. In the next section we will however prove (0)a C S that the correctness of (4.26) in loop expansion procedure is − α β ε (ε + ) + ( 1) (C) j (φ) ¯ (− ) j β 1 (ϕ) correct. S C 1 Nβa, j −1 α(C)β( ¯ ) j + (S ) C (φ)χβ,kj(ϕ)Ta (ϕ) α β − ε (ε +ε + ) + ¯ χ (ϕ)( 1)kl(φ) j (ϕ)(− ) l j a 1 , C C β,jk S Ta,l 1 5 Deformation of the global transformations in one-loop (5.8) approximation α − α β j (φ) = ( 1) j (C) (φ) ¯ (0)a S C − β α (ε + )(ε + ) As we pointed out at the end of Sect. 4, the change of variables 1 j (C¯ ) α 1 β 1 + (S ) (φ)C (−1) Nβa(ϕ) ¯ (4.5) is non-analytical in h and the use of loop expansion − α β ε (ε +ε ) + ( 1) jl(φ) ¯ (ϕ)(− ) l α β , looks doubtful. However, we will show that the one-loop S C C Nβa,l 1 (5.9) (ϕ) = χ (ϕ) k (ϕ). approximation still works. Nβa β,k Ta (5.10) Consider the relations (4.26) in loop approximation. For are introduced. The deformation of global generators T i (ϕ), the effective action we have a defined in initial configuration space {ϕ}, looks very non- 2 () = S() + h¯ 1() + O(h¯ ). (5.1) trivial already in the one-loop approximation. Such a defor- ¯ A() mation is done in the full configuration space of the gauge The generators Ta are written in the same approximation i theory (2.8) with the help of generators Rα(ϕ) of gauge sym- as follows i (ϕ) metry of initial action (2.1) and its derivatives Rα, j .The ¯ A 1 A ˜ ¯ A T () = R ()() + T( ) () relations (2.1), (3.1) and (5.7) lead to a h¯ 0 a ¯ i i j α + ¯ A () + ( 2), ()˜ = (). S ,i (ϕ)T( ) (φ) =−S ,i (ϕ)Rα, (ϕ) (φ) = 0. (5.11) h¯ T(1)a O h¯ h¯ (5.2) 0 0 a 0 j (0)a

Due to the invariance of quantum action S() (2.9) under the It means non invariance of an initial action S0(ϕ) under the BRST transformations (2.10), the ﬁrst item in the right-hand deformed global transformations. 123 Eur. Phys. J. C (2018) 78:524 Page 7 of 8 524

6 Summary to incorporate the global symmetries into its solutions but they did not discuss the deformations of the global symme- In the present paper we have studied the general problem try generators. It is clear that this approach differs from our of global (rigid) symmetries in quantum general gauge the- consideration since from the beginning we work only in the ory with closed irreducible gauge algebra. Using the BRST- configuration field space and derive the above deformations. BV quantization procedure [11,12] we have constructed a The general enough approach to the problem of global deformation of global symmetry generators so that the full symmetries in the quantum gauge theory was proposed in quantum action of the initial gauge theory and hence the the work [32] under assumptions that the initial classical effective action are invariant under such deformed global theory consists of only bosonic fields and the algebra of transformations. This statement is valid for any bosonic or the classical global transformations is closed. In our paper fermionic gauge theory with an irreducible and closed alge- we have considered more general case of the theory with bra of gauge transformations and with an arbitrary (even bosonic and fermionic fields and open algebra of the global open) non-anomalous algebra of global symmetry. Form (super)symmetries. However, what is very important, we of the deformed global symmetry generators in one-loop work with more general effective action than in [32]. In our approximation is calculated in the explicit form. Note that case, the sources are introduced not only for initial fields but from algebraic point of view the deformation of global sym- also for ghost and auxiliary fields. This allowed us to describe metry generators, studied in the present paper, is similar to the deformation of the generators completely in the algebraic the known phenomena of the deformation of gauge genera- terms. tors under renormalization procedure (see e.g. [13,37,38]), The global (super)symmetric and gauge transformations although the mechanisms of deformation differ. have been studied in the framework of singular gauge fixing In general the BRST-BV technique involves introduction procedure. In practice it is more convenient to use a non- for every field of the full configuration space the correspond- singular gauges. All our basic statements are still valid in ing antifield with opposite Grassmann parity. It allows to this case as well. We should only modify the quantum action study the many properties of the gauge theory on quantum (2.9) in the form level in general terms. However, to simplify the considera- (φ) = (ϕ) + ¯ αχ (ϕ) i (ϕ) β tion in this paper we derived all the results in the configura- S S0 C α,i Rβ C tion space of fields only (2.8). Generalization of the results α ξ α α +χα(ϕ)B + B B , (6.1) obtained for extended configuration space of fields and anti- 2 fields can be done in the same method. where ξ is a gauge parameter. Due to the property of BRST Problem of global symmetries in quantum general gauge α transformations (2.11) B Rˆ = 0, the action (6.1) remains theories is discussed in earlier papers [19–22] however the invariant under the BRST transformations, SRˆ = 0. Thus, accents were aimed on the other aspects. First, in [19]itwas from a principled point of view the results obtained will be shown the possibility to construct the Lagrangian models the same. possessing the gauge and supersymmetry invariance under the assumption that the global supersymmetry transforma- Acknowledgements The authors thank I.V. Tyutin for useful discus- tions obey the closed algebra. In our paper we work with sions and S.M. Kuzenko for correspondence. The research was sup- an arbitrary (including open) global (super)symmetry of a ported in parts by Russian Ministry of Education and Science, Project given gauge theory. Also we suppose the absence of anoma- No. 3.1386.2017. The authors are also grateful to RFBR Grant, Project No. 18-02-00153 for partial support. lies in combined gauge and global algebra of generators but in principle it is possible to study the problem of coexistence of Open Access This article is distributed under the terms of the Creative gauge and global symmetries in presence of anomalies using Commons Attribution 4.0 International License (http://creativecomm the approach of work [20]. Of course the implementation ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit of this program requires a separate independent study. Sec- to the original author(s) and the source, provide a link to the Creative ond, in paper [21] it was shown that the attempts to construct Commons license, and indicate if changes were made. an action invariant under BRST- and closed rigid symmetries Funded by SCOAP3. lead to the breakdown of non-degeneracy of the full quantum action. In our paper we proved that this problem is resolved on the base of quantum deformation of global generators. References Third, in paper [22] it was shown that the global symmetries of an initial gauge action in the field-antifield formalism can 1. C. Becchi, A. Rouet, R. Stora, Renormalization of the abelian be extended to include the all fields and the all antifields. Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975) 2. 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