Background Field Method and Nonlinear Gauges

Physics Letters B 797 (2019) 134882

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Physics Letters B

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Background ﬁeld method and nonlinear gauges ∗ Breno L. Giacchini a,b, , Peter M. Lavrov c,b,d, Ilya L. Shapiro b,c,d a Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China b Departamento de Física, ICE, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP: 36036-330, MG, Brazil c Department of Mathematical Analysis, Tomsk State Pedagogical University, 634061, Kievskaya St. 60, Tomsk, Russia d National Research Tomsk State University, Lenin Av. 36, 634050 Tomsk, Russia a r t i c l e i n f o a b s t r a c t

Article history: We present a reformulation of the background field method for Yang-Mills type theories, based on using Received 20 June 2019 a superalgebra of generators of BRST and background field transformations. The new approach enables Received in revised form 14 August 2019 one to implement and consistently use non-linear gauges in a natural way, by using the requirement of Accepted 19 August 2019 invariance of the fermion gauge-fixing functional under the background field transformations. Available online 27 August 2019 © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license Editor: M. Cveticˇ (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Keywords: Background field method Nonlinear gauge Yang-Mills theories Quantum gravity theories

1. Introduction counting in new non-conventional models of quantum field theory (see, for instance, [8,9]). The BFM has been the object of extensive investigation from The standard approach to the Lagrangian quantization of gauge theories [1–3] assumes the violation of the gauge invariance of the different viewpoints (see e.g. [10–19]), and in the last years the in- action. In some cases, e.g., in the semiclassical gravity theory [4], terest in this method has grown rapidly, since it is supposed to this makes all considerations and also practical calculations quite solve many important problems of gauge theories [9,20–26]. One complicated. The same certainly concerns quantum gravity. Fortu- of the standard assumptions of the BFM is related to the special nately, there is a useful approach to the quantization of gauge the- choice of gauge generators and gauge-fixing conditions, that are ories, known as the background field method (BFM) [5–7]. Within typically linear in the quantum fields. Indeed, even though the the BFM one can explicitly preserve the gauge invariance of an gauge fixing may admit a lot of arbitrariness, linearity is always effective action in all stages, and thus all physical results are repro- assumed in the framework of the BFM. At the same time, the con- duced using the effective action of background fields (background sistent formulation of the quantization method should certainly be effective action). In many cases, the calculations in gauge theories free of such a restriction, as the choice of the gauge fixing is ar- in flat and curved space-time are performed by means of the BFM, bitrary and is not related to the physical contents of the initial while the general theorems concerning gauge invariant renormal- gauge theory. In the present paper we propose a generalization of izability and gauge fixing dependence rely on the standard version the standard formulation of the BFM, in the sense that is can be of quantization. This situation makes it at least highly desirable used for a wide class of non-linear gauge-fixing conditions and to- to construct a completely consistent formulation of the BFM for gether with some types of non-linear gauge generators. This new gauge theories and quantum gravity, such that the mentioned gen- point of view on the BFM is based on the superalgebra of genera- eral theorems could be formulated directly within the formalism tors of the fundamental symmetries of this formalism (namely, the that is used for calculations and, e.g., for the analysis of power BRST [27,28] and the background field symmetries [5–7]). It is well known that a judicious choice of gauge condition can yield a considerable simplification of calculations in quantum field theory, and non-linear gauges are often used with this purpose * Corresponding author. E-mail addresses: [email protected] (B.L. Giacchini), [email protected] (see, for instance, [15,29–33]). On the other hand, such gauges (P.M. Lavrov), shapiro@fisica.ufjf.br (I.L. Shapiro). emerge naturally in the framework of the effective field theory ap- https://doi.org/10.1016/j.physletb.2019.134882 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 B.L. Giacchini et al. / Physics Letters B 797 (2019) 134882

i j − − εαεβ i j =− i γ proach, after the massive degrees of freedom are integrated out Rα, j(A)Rβ (A) ( 1) Rβ,j(A)Rα(A) Rγ (A)Fαβ , (4) and we are left with the gauge theory of relatively light quan- i = tum fields [34]. Thus, the consistent implementation of non-linear where we denote the right functional derivative by δr X/δ A X,i . i gauge fixing in all approaches to quantization looks relevant from In principle, the generators Rα(A) may be non-linear in the fields. many viewpoints. Further restrictions on the generators will be introduced in the The general analysis of renormalization of gauge theories un- next section, motivated by quantum aspects of the theory. der non-linear gauge conditions has been presented in many pa- Let us formulate the theory within the BFM by splitting the pers, including [35–40]. Nonetheless, non-linear gauges are not original fields Ai into two types of fields, through the substitution i i i frequently used in the background field formalism. A remarkable A −→ A + B in the initial action S0(A). It is assumed that the example of simultaneous use of the BFM and a non-linear gauge is fields Bi are not equal to zero only in the gauge sector.1 These the two-loop calculation in quantum gravity [15], which confirmed fields form a classical background, while Ai are quantum fields, the previous calculation of [41]. This correspondence is certainly a that means being subject of quantization, e.g., these fields are in- positive signal, making even more clear the need for a general dis- tegration variables in functional integrals. It is clear that the total cussion on this subject, which is not present in the literature. action satisfies The idea that the gauge-fixing function should transform in a covariant way under background field transformations (a condition δω S0(A + B) = 0(5) that is trivially satisfied for the usual linear gauge), as a form of i −→ i = i + i + B α preserving the background field invariance of the Faddeev-Popov under the transformation A A A Rα(A )ω . On the i action, is present in many discussions on Yang-Mills theories [21, other hand, the new field B introduces extra new degrees of free- 42]. Nevertheless, the applicability of the BFM with non-linear dom and, thence, there is an ambiguity in the transformation rule gauges is not a consensus. For example, in [43]it is mentioned for each of the fields Ai and Bi . This ambiguity can be fixed in that the Gervais-Neveu gauge [31] could be used within the BFM different ways, and in the BFM it is done by choosing the transfor- for Yang-Mills theory, while in [21]the linearity of the gauge-fixing mation laws condition and gauge generators is regarded as an important factor (q) i = i + B − i B α (c)Bi = i B α for the consistency of the quantum formalism. δω A Rα(A ) Rα( ) ω ,δω Rα( )ω , (6) As it was already mentioned above, in the present work we introduce a new geometric point of view on the BFM based on the defining the background field transformations for the fields Ai and operator superalgebra which underlies the method. As the main Bi , respectively. The superscript (q) indicates the transformation of application of this formalism, we obtain necessary conditions for the quantum fields, while that of the classical fields is labelled by (q) (c) the consistent application of the BFM [5–7]for Yang-Mills type (c). Thus, in Eq. (5)one has δω = δω +δω . Indeed, the background theories with non-linear gauge-fixing conditions and non-linear field transformation rule for the field Ai was chosen so that gauge generators. The work is organised as follows. In Sec. 2 we (c)Bi + (q) i = i + B α briefly review the BFM applied to gauge theories. The main original δω δω A Rα(A )ω . (7) results are presented in Sec. 3, including the discussion of non- linear gauges for generalised Yang-Mills theories in the BFM. In In order to apply the Faddeev-Popov quantization scheme [1] Sec. 4 some aspects of the Yang-Mills theory and quantum gravity in the background field formalism, one has to introduce a gauge- i α ¯ α are explicitly considered as examples of the general result. Finally, fixing condition for the quantum fields A , and extra fields C , C α A in Sec. 5 we draw our conclusions. The DeWitt’s condensate nota- and B . To simplify notation we denote by φ = φ the set of all tions [44]are used throughout the paper. quantum fields A = i α α ¯ α 2. Background field formalism for gauge theories φ A , B , C , C . (8) A The Grassmann parity of the fields is ε(φ ) = εA = (εi , εα, εα + As a starting point, consider a gauge theory of fields Ai , α ¯ α 1, εα + 1), while their ghost numbers are gh(C ) =−gh(C ) = 1 with Grassmann parity ε = ε(Ai ). A complete set of fields Ai = i and gh(Ai ) = gh(Bα) = 0. The corresponding Faddeev-Popov action (Aαk, Am) includes fields Aαk of the gauge sector and also fields in the BFM reads Am of the matter sector of a given theory. The classical action S0(A) is invariant under gauge transforma- SFP(φ, B) = S0(A + B) + Sgh(φ, B) + Sgf(φ, B), (9) tions where the ghost and gauge-fixing actions are defined as Ai = Ri A α (1) δξ α( )ξ . δ (A, B) B = ¯ α r χα i + B β Sgh(φ, ) C Rβ (A )C (10) Here ξ α(x) is the transformation parameter with parity ε(ξ α) = δ Ai i i = + εα, while Rα(A) (with ε(Rα) εi εα) is the generator of gauge and transformation. In our notation i is the collection of internal, α ξ β Lorentz and continuous (spacetime coordinates) indices. We as- Sgf(φ, B) = B χα(A, B) + gαβ B . (11) 2 sume that the generators are linearly independent, i.e., In this expression ξ is a gauge parameter that has to be introduced i α = =⇒ α = in the case of a non-singular gauge condition, and g is an arbi- Rα(A)z 0 z 0, (2) αβ = − εαεβ γ trary invertible constant matrix such that gβα gαβ ( 1) . We and satisfy a closed algebra with structure coefficients Fαβ that do recall that in the BFM only the gauge of the quantum field is fixed not depend on the fields, by χα(A, B), while the symmetry for the background fields may [ ] i = i α = α γ β δξ1 ,δξ2 A δξ3 A with ξ3 Fβγ ξ2 ξ1 . (3) 1 γ γ In gauge theories without spontaneous symmetry breaking one can introduce =− − εαεβ From (3)it follows that Fαβ ( 1) Fβα and background fields in the sector of gauge fields without loss of generality [22,26]. B.L. Giacchini et al. / Physics Letters B 797 (2019) 134882 3 be preserved. The standard choice of χα(A, B) in the BFM is of Apart from the global supersymmetry, a consistent formulation the type of the BFM requires that the Faddeev-Popov action be invariant under background field transformations. The former symmetry is i χα(A, B) = Fαi(B)A , (12) ensured in the representation (18)of the Faddeev-Popov action, for any choice of gauge-fixing functional . Therefore, it is pos- which is a gauge fixing condition linear in the quantum fields. sible to extend considerations to a more general case in which In the framework of Faddeev-Popov quantization, the gauge ¯ α (φ, B) = C χα(φ, B), where the gauge-fixing functions χα(φ, B) symmetry of the initial action is replaced by the global supersym- depend on all the fields under consideration and satisfy the condi- metry (BRST symmetry) of the Faddeev-Popov action (9), defined tions ε(χα) = εα and gh(χα) = 0. On the other hand, the presence by the transformation [27,28] of the background field symmetry is not immediate — especially in the case of non-linear gauges — as the gauge-fixing functionals A ˆ A δBRST φ = RBRST φ λ, (13) depend on the background fields. In what follows we derive necessary conditions that the fermion gauge-fixing functional should where λ is a constant anticommuting parameter. The generator satisfy to achieve the consistent application of the BFM. As it has reads been already mentioned above, during this process we shall also i δr find restrictions on the form of the generators R of gauge sym- Rˆ (φ, B) = R A (φ, B), (14) α BRST δφA BRST metry. Let us extend the transformation rule (6)to the whole set of where quantum fields, as A i 1 β B = + B α − α γ − εβ (q) α α β γ (q) α α β γ ε RBRST(φ, ) Rα(A )C , 0, Fβγ C C ( 1) , δ B =−F B ω ,δC =−F C ω (−1) γ , 2 ω γ β ω γ β − εα α (q) ¯ α =− α ¯ β γ − εγ ( 1) B . (15) δω C Fγ β C ω ( 1) . (19)

A Following the procedure used for the BRST symmetry, one can de- Regarding the Grassmann parity, one has ε(R (φ, B)) = εA + BRST fine the operator of background field transformations, 1. The BRST transformations are applied only on quantum fields, Bi = thus, δBRST 0. Moreover, it is possible to show that the BRST ˆ δr (c) i δr (q) A ˆ Rω(φ, B) = δ B + δ φ , ε(Rω) = 0. (20) operator is nilpotent, i.e., δBi ω δφA ω ˆ 2 B = The gauge invariance of the initial classical action implies that RBRST(φ, ) 0. (16) ˆ Rω(φ, B) S0(A + B) = 0. Furthermore, it is not difficult to verify Let us note that in the existing literature there is a formulation that the background gauge operator commutes with the generator of the BFM in terms of extended BRST differentials [17–20], that of BRST transformations, i.e., ˆ coincides with RBRST(φ, B) in the sector of fields φ. At the same ˆ ˆ time, the BRST variation of the background fields is equal to new [RBRST, Rω]=0. (21) ghost fields, so that in the extended field space the usual BRST algebra takes place. In contrast to this approach, in the next Section Combining this result with the representation (18)of the Faddeev- we propose a new superalgebra, underlying the BFM which enables Popov action, we get us to consider non-linear gauges and non-linear gauge generators. δ S (φ, B) = Rˆ (φ, B) S (φ, B) = 0 For the subsequent discussion, it is useful to introduce the ω FP ω FP ˆ fermion gauge-fixing functional ⇐⇒ Rω(φ, B) (φ, B) = 0. (22) ¯ α In other words, the Faddeev-Popov action is invariant under back- (φ, B) = C χα(A, B, B), with ε( ) =−gh( ) = 1. (17) ground field transformations if and only if the fermion gauge-fixing The Faddeev-Popov action can then be cast in the form functional is a scalar with respect to this transformation. The condition (22) constrains the possible forms of the (ex- B = + B + ˆ B B SFP(φ, ) S0(A ) RBRST(φ, ) (φ, ). (18) tended) gauge-fixing function χα(φ, B), as the relation B The BRST symmetry of SFP(φ, ) can be easily verified in this rep- ˆ B B = ¯ α B − α ¯ β γ − εγ B Rω(φ, ) (φ, ) C δωχα(φ, ) Fγ β C ω ( 1) χα(φ, ) resentation by applying the nilpotency property of the operator ˆ = RBRST. 0 (23) Let us point out that in order to achieve economic notations, in fixes the transformation law for χα(φ, B), Eq. (17) and hereafter we let χα to depend also on the auxiliary field Bα. This is a practically useful way of taking into account β γ δωχα(φ, B) =−χβ (φ, B)Fαγ ω . (24) the possibility of non-singular gauge conditions (see Eq. (11)). Nonetheless, further discussion and consequent results do not re- Therefore, in order to have the invariance of the Faddeev-Popov quire any kind of a priori specific dependence of the gauge-fixing action under background field transformations it is necessary that functional on Ai , Bα and Bi . the gauge function χα transforms as a tensor with respect to the gauge group. This requirement can be fulfilled provided that 3. BFM compatible gauge functionals χα(φ, B) is constructed only by using tensor quantities. Thus, Eq. (24)may impose a restriction on the form of gauge-fixing func- i In this section we propose a new point of view and a gen- tions which are non-linear on the fields A . Let us anticipate that eralization of the standard BFM for gauge theories, that is based an example of this kind will be presented in Sec. 4. on using a superalgebra of generators of all the symmetries un- In order to complete the geometric point of view on the BFM ˆ ˆ derlying the method and works for a wide class of gauge fixing for gauge theories, let us introduce the generators Rα = Rα(φ, B) conditions. of the background field transformation, defined by the rule 4 B.L. Giacchini et al. / Physics Letters B 797 (2019) 134882

ˆ ˆ α ˆ Rω(φ, B) = Rα(φ, B)ω , ε(Rα) = εα (25) In order to construct the effective action (generating functional of vertex functions), let us introduce the set of mean fields ={A } It is possible to verify that the generators of BRST and background with field transformation satisfy the relations A = i α α ¯ α ˆ 2 = [ ˆ Rˆ ]= [Rˆ Rˆ ]=−Rˆ γ A , B , C , C , (34) RBRST 0, RBRST, α 0, α, β γ Fβα, (26) i ¯ that define a superalgebra of the symmetries underlying the BFM. respectively for the fields A , Bα, Cα and Cα, such that It is important to note that the last relation in (26)reproduces the ∗ i δ W (J , B,φ ) gauge algebra (4), when restricted to the sector of fields B , and A = . (35) A J provides its generalization to the whole set of fields φ . δ A At this point we can conclude that The (extended) effective action is defined as ˆ R (φ, B) S (φ, B) = 0andRˆ (φ, B) (φ, B) = 0 (27) ∗ ∗ A BRST FP α = (,φ , B) = W (J ,φ , B) − JA (36) represent necessary conditions for the consistent application of the and it satisfies the Ward identity BFM. The first relation is associated to the validity of the Ward identity (Slavnov-Taylor identities in the case of Yang-Mills fields) (, ) = 0, (37) and the gauge independence of the vacuum functional,2 while the second relation is called to provide the invariance of the effective with the anti-bracket of two functionals defined by the rule [2,3] action in the BFM with respect to deformed (in the general case) δr G δ H δ G δr H background field transformations. In what follows we shall con- (G, H) = ∗ − ∗ . (38) δφA δφ δφ δφA sider these statements explicitly. To this end, it is convenient to A A introduce the extended action Now, in order to explore the background field symmetry it is useful to switch off the antifields and deal with the traditional B ∗ = B + ∗ ˆ B A Sext(φ, ,φ ) SFP(φ, ) φA RBRST(φ, )φ , (28) generating functions, e.g., ∗ ∗ where φ ={φ } denote the set of sources (antifields) to the BRST ∗ A J B = J B | ∗ ∗ = + Z( , ) Z( , ,φ ) φ =0. (39) transformations, with the parities ε(φA ) εA 1. The correspond- A ing (extended) generating functional of Green functions reads Then, let us perform the change of the functional integration vari- A A A (q) A ables φ → ϕ (φ) = φ + δω φ in (29). If the Berezenian associ- J B ∗ = D B + J A Z( , ,φ ) φ exp i SFP(φ, ) A φ ated to this change of variables is Ber(ϕ) = 1, it follows + ∗ ˆ B A (c) φA RBRST(φ, )φ , (29) Dφ δω SFP(φ, B) − δω SFP(φ, B) (B) ¯ where JA = J i, Jα , Jα, Jα (with the parities ε(JA ) = εA ) are (q) A A + JA δω φ exp i SFP(φ, B) + JA φ = 0. (40) the external sources for the fields φ A . The BRST symmetry, to- gether with the requirement that the generators Ri of gauge trans- i α It is clear that if the generators Rα of gauge transformations formation satisfy are linear, then Ber(ϕ) = 1trivially when using the dimensional i + B regularisation, as the change of variables is linear. On the other ε δ Rα(A ) ε +1 β (−1) i + (−1) β F = 0, (30) hand, in the more general scenario with non-linear generators we δ Ai βα meet a situation similar to that of BRST transformations, where it implies the Ward identity was necessary to impose the condition (30)to ensure the trivi- ∗ δ Z(J , B,φ ) ality of the Berezenian. At the first sight it seems that the back- J = 0. (31) A δφ∗ ground field symmetry would introduce further requirements on A the generators of gauge transformations. Nonetheless, when com- The relation (30)plays an important role in the derivation of puting Ber(ϕ) one finds out that the contribution from the field the Ward identity insomuch as it ensures the triviality of the Bα compensates the one from Cα (or C¯ α, which transforms under Berezenian related to the change of integration variables in the the same rule), leading to form of BRST transformations. In Yang-Mills theories, for instance, the relations (30)are satisfied due to antisymmetry properties of (q) A δ δω φ Ber( ) = exp sTr the structure constants. ϕ B ∗ δφ The generating functional W (J , B, φ ) of connected Green i + B functions is defined in a usual way, δ R (A ) + β = exp (−1)εi α ωα + (−1)εβ 1 F ωα . (41) i βα ∗ J B ∗ δ A Z(J , B,φ ) = eiW( , ,φ ), (32) Therefore, the condition (30)also ensures the triviality of the and the identity (31)canbe cast into the form A A (q) A Berezenian for the change of variables φ → φ + δω φ . This re- ∗ markable result means that, as it will be clear from what follows, δ W (J , B,φ ) JA = 0. (33) for the gauge theories considered here the consistent use of the δφ∗ A BFM does not impose an additional requirement on the generators besides those that are already needed in the framework of traditional approach (i.e., without using the BFM). 2 Let us mention that the gauge independence of the vacuum functional is needed for the gauge independent S-matrix and hence is a very important element for the After this important digression on the Berezenian, let us rewrite consistent quantum formulation of a gauge theory [2,45]. Eq. (40)in the form B.L. Giacchini et al. / Physics Letters B 797 (2019) 134882 5 (q) j α ε ε (c) A δ δ J =−J R (A)ω (−1) α i , δω Z(J , B) = iJA T , B Z(J , B) ω i j α,i ω δ(iJ ) (q) (B) =− (B) β γ δω Jα Jβ Fαγ ω , (51) ˆ ˆ + Dφ iR (φ, B)R (φ, B) (φ, B) (q) β (q) β BRST ω δ ¯J =−¯J F ωγ ,δ J =−J F ωγ . ω α β αγ ω α β αγ A × exp i SFP(φ, B) + JA φ , (42) Then, Eqs. (44) and (45)boil down to J B (c) (q) δ Z( , ) where we introduced the notation δ Z(J , B) + δ J = 0 (52) ω ω A J δ A A B = j + B − j B α α γ β Tω(φ, ) Rα(A ) Rα( ) ω , Fβγ ω B , and J B (c) (q) δ W ( , ) α γ β α γ ¯ β δ W (J , B) + δ J = 0. (53) Fβγ ω C , Fβγ ω C . (43) ω ω A δJA ˆ In other words, the invariance of the Faddeev-Popov action with If the condition Rω(φ, B) (φ, B) = 0(see Eq. (22)) for the invariance of the Faddeev-Popov action with respect to the background respect to the background field transformations implies the invari- field transformations is satisfied, Eq. (42)can be written in the ance of the generating functionals of Green functions with respect (q)J (c)B closed form to the joint transformation δω i of the external sources and δω of the background field. Moreover, in what concerns the effec- (c) δ δ Z(J , B) = iJ T A , B Z(J , B), (44) tive action, under this circumstance the transformation (47)of the ω A ω J δ(i ) mean fields Ai are not deformed, i.e., or, in terms of the generating functional of connected Green func- (q) δ Ai =[Ri (A + B) − Ri (B)]ωα. (54) tions, ω α α J B For the sake of completeness, let us compare the generating (c) δ W ( , ) δ δ W (J , B) = J T A + , B · 1. (45) functionals in the background field formalism and in the tradi- ω A ω J J δ δ(i ) tional one — and, ultimately, their relations with (B). Consider Here 1is the unite vector in the space of the fields. the generating functional of Green functions which corresponds to Finally, for the effective action, the invariance of the Faddeev- the standard quantum field theory approach, but in a very special Popov action with respect to background field transformations im- gauge fixing, plies that ˆ ¯ Z2(J ) = Dφ exp i S0(A) + RBRST(φ) (A − B, B, C, C, B) ˆ Rω(, B) (, B) = 0, (46) + J A ˆ A A A φ . (55) where Rω(, B) is defined by the substitution φ → in

Eq. (20)with the (deformed) transformations of the mean fields In the last expression all the dependence of the quantity Z2(J ) (cf. (6) and (19)) on the external field is only through the gauge-fixing functional. Thus, this functional depends the external field Bi , but since this (q) i = i B α (q) α =− α β γ J δω A Rα (A, )ω ,δω B Fγ β B ω , (47) dependence is not of the BFM type, Z2( ) is nothing else but (q) α α β γ ε (q) ¯ α α ¯ β γ ε the conventional generating functional of Green functions of the δ C =−F C ω (−1) γ ,δC =−F C ω (−1) γ . i ω γ β ω γ β theory, defined by S0 in a specific B -dependent gauge. One of

i the consequences is that any kind of physical results does not de- In the sector of fields A , the generator is given by i ˆ pend on B . Furthermore, in Eq. (55) RBRST(φ) is defined by setting Bi = 0in Eq. (15). The arguments of are written explicitly, show- Ri B = Ri ˆ + B · 1 − Ri B with α (A, ) α(A ) α( ), ing that we assume that Ai only occurs in a specific combination δ Bi ˆ A = A + −1 AB with . We stress that, being formulated in the traditional way i , (48) J δB (i.e., not in the BFM), Z2( ) per se does not impose any constraint − on the linearity of the gauge-fixing fermion with respect to the where 1 is the inverse matrix of second derivatives of the ef- quantum field Ai . fective action, Making some change of variables in the functional integral, it is − AC δ δr (, B) easy to verify that (39)is related to (55)in the following way: 1 = δ A , where (, B) = . CB B AB A B δ δ J B = J − Bi (49) Z( , ) Z2( ) exp iJi . (56) Let us stress that the relation (46) follows from (44), which only Accordingly, for the generating functional of connected Green func- ˆ holds if the condition Rω(φ, B) (φ, B) = 0is satisfied. The prop- tions one has erty (46)is crucial to the BFM inasmuch as, when the mean fields J B = J − Bi are switched off, it ensures the gauge invariance of the func- W ( , ) W 2( ) J i , (57) tional (B) = (, B)| A = , namely, iW2(J ) 0 where Z2(J ) = e . Recall that, according to (35), (c) J B B = 0 (50) i δ W ( , ) δω ( ) . A = . (58) δ J i It is worth mentioning that the previous expressions can be i Similarly, simplified if the generators Rα of gauge transformations are linear in fields Ai . In this case one can define the background field δ W (J ) i ≡ 2 = i + Bi transformation of all the external sources as A2 A (59) δ J i 6 B.L. Giacchini et al. / Physics Letters B 797 (2019) 134882

ˆ a abc b c Following the same line, let us define the effective action associ- Rω(φ, B)χ (A, B, B) = f χ (A, B, B)ω . (66) ated to Z (J ), as 2 This requirement can be fulfilled provided that χ a(A, B, B) is con- a a i α α ¯ α = J − i − (B) α structed only by using vectors (in the group index) such as B , Aμ, 2(A2, B , C , C ) W 2( ) J iA2 Jα B F a (B), Dab(B)Ab , Dab(B)Dbc(B)Ac and so on. The most simple − ¯ α − ¯ α μν μ ν μ ν α JαC JαC . (60) gauge-fixing function compatible with this condition is A moment’s reflection shows that a B = ab B b χ (A, B, ) Dμ ( )Aμ, (67) B = i α α ¯ α (, ) 2(A2, B , C , C ). (61) that turns out to be the most popular gauge-fixing function for Yang-Mills theory. In principle, nonetheless, non-linear gauges In other words, the effective action (, B) in the background which satisfy (66)can also be used. One simple example is field formalism is equal to the initial effective action in a partic- i = i + Bi a a bc c bd d ular gauge with mean field A2 A — or, switching off the χ (A, B, B) = F (B) D (B)A D (B)A , (68) B = | μν μ ν λ λ mean fields, ( ) 2(A2) i =Bi . We point out that the afore- A2 mentioned particularity of the gauge is not associated to its linear- which is quadratic on the quantum field. Note, however, that the a B a ity with respect to the quantum fields, but to its dependence on B gauge resultant from the substitution of Fμν ( ) by Fμν(A) in (68) (see Eq. (55)). is not admissible, as it violates the transformation law (66). We also point out that the specific dependence of the gauge condition a 4. Two particular cases on the auxiliary field B (and also on the ghost fields, in the gen- eralisation proposed in Sec. 3) is not critical, as it already satisfies In this section we present the applications of the formalism de- the vector transformation law. scribed above, to the Yang-Mills and quantum gravity theories. We conclude that, in principle, it is possible to use non-linear gauge fixing conditions for the quantum fields in Yang-Mills the- 4.1. Yang-Mills theory ories formulated in the BFM, provided that the relation (66)is satisfied. Such procedure may introduce parameters with negative mass dimension, which could affect renormalization; however, a As an example of the results presented in the previous section, general analysis on this issue is beyond the scope of the present let us consider the case of the pure Yang-Mills theory, defined by work. the action

1 a a 4.2. Quantum gravity S0(A) =− F (A)F (A), (62) 4 μν μν As a second example, consider the case of quantum gravity where F a (A) = ∂ Aa − ∂ Aa + f abc Ab Ac is the field strength for μν μ ν ν μ μ ν theories, defined by the action S0(g) of a Riemann metric g = the non-Abelian vector field A , taking values in the adjoint rep- μ {gμν(x)} with ε(g) = 0, and which is invariant under general co- resentation of a compact semi-simple Lie group. Being a particular ordinate transformations. The generator of such transformation is case of the more general theory described above, it is instructive linear and reads to present the correspondence with the notations used in Sec. 3, namely Rμνσ (x, y; g) =−δ(x − y)∂σ gμν(x) − gμσ (x)∂νδ(x − y) − − i → a Bi → Ba α → abc gσν(x)∂μδ(x y). (69) A Aμ, μ, Fβγ f , α α = i → ab = ab + acb c Therefore, for an arbitrary gauge function ω with ε(ω ) 0one Rα(A) Dμ (A) δ ∂μ f Aμ. (63) σ has δω gμν = Rμνσ (g)ω , or, writing all the arguments explicitly,

Here the structure coefficients f abc of the gauge group are con- σ stant. The action (62)is invariant under the gauge transformations δω gμν(x) = dy Rμνσ (x, y; g)ω (y). (70) ab defined by the generator Dμ (A) with an arbitrary gauge function ωa with ε(ωa) = 0. In the Faddeev-Popov quantization, the Grass- In this case, the structure functions are given by A = a a a ¯ a mann parity of the fields φ (Aμ, B , C , C ) is, respectively, α α (x) α (x) F (x, y, z) = δ(x − y)δ ∂ δ(x − z) − δ(x − z)δ ∂γ δ(x − y), εA = (0, 0, 1, 1). βγ γ β β The background field formalism for Yang-Mills theory com- (71) prises the definition of the background field transformation (see α =− α which satisfy Fβγ (x, y, z) Fγ β (x, z, y), as usual. Eqs. (6) and (19)) In terms of the notation used in Sec. 3 one has the correspondence (q) a = abc b c (c)Ba = ab B b δω Aμ f Aμω ,δω μ Dμ ( )ω , i i i (q) a abc b c (q) a abc b c A → gμν(x), B → g¯μν(x), R (A) → Rμνσ (x, y; g), δω B = f B ω ,δω C = f C ω , α α α (q) ¯ a abc ¯ b c F → F (x, y, z). (72) δω C = f C ω . (64) βγ βγ According to (6) and (19), in the BFM one defines the background Note that, in agreement to (6), the generator of the transformation a field transformation in the sector of fields Aμ reads (c) σ δ g¯μν = Rμνσ (g¯)ω ab + B − ab B = acb c ω Dμ (A ) Dμ ( ) f A , (65) σ σ σ =−ω ∂σ g¯μν − g¯μσ ∂νω − g¯σν∂μω , (73) and thus all the quantum fields transform according the same rule. (q) δ g =[R (g + g¯) − R (g¯)]ωσ = R (g)ωσ , (74) Also, the condition (22)for the background field invariance of the ω μν μνσ μνσ μνσ (q) α =− α γ β =− σ α + σ α Faddeev-Popov action reads δω V Fβγ V ω ω ∂σ V V ∂σ ω , (75) B.L. 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