Functional Renormalization Group Approach and Gauge Dependence

Home , Renormalization group

arXiv:1910.06068v3 [hep-th] 5 Mar 2020 eomlzbemdl uha h ipetqatmgaiybsdon based gravity quantum simplest the as such e models that expectations renormalizable long-standing a are there First, motivations. Introduction 1 ucinlrnraiaingopapoc n gauge and approach group renormalization Functional a 5 4 3 2 1 eatmnod ´sc,IE nvriaeFdrld uzd For de Juiz de Federal Universidade ICE, F´ısica, de Departamento d h neett h o-etraiefruaini unu grav quantum in formulation non-perturbative the to interest The b -alades shapiro@fisica.ufjf.br address: E-mail eareis@fisica.ufjf.br address: E-mail [email protected] address: E-mail [email protected] address: E-mail -alades [email protected] address: E-mail ´ trF Barra V´ıtor F. zto ru prah ag dependence gauge approach, group ization conditions. on-shell Keywords: the of use the under non-inv leads even transformations the dependence BRST level, fixing global quantum the under the action at regulator preserved the is symmetry preser Unfortuna gauge that action. average the functions effective the regulator of of symmetry field class ground special argu BRST-based the standard establish the can and genera formalism of field models background gravity quantum the pr for dependence action fixing average gauge effective and the symmetry gauge the investigate We eateto hoeia hsc,TmkSaePdggclUn Pedagogical State Tomsk Physics, Theoretical of Department eatmn fPyis otenUiest fSineadTech and Science of University Southern Physics, of Departament c ainlRsac os tt nvriy 300Tmk Russia Tomsk, 634050 University, State Tomsk Research National i´ rod al Netto Paula Tibério de unu rvt,bcgon edmto,fntoa renormal- functional method, field background gravity, quantum eednei rvt theories gravity in dependence a 1 , ee .Lavrov M. Peter eFr,M,Brazil MG, Fora, de hnhn China Shenzhen, os,Russia Tomsk, Abstract d b,c 4 , 2 , laL Shapiro L. Ilya dad noi o Reis dos Antonio Eduardo e h etraieynon- perturbatively the ven a,b,c ey regardless tely, 5 e h back- the ves om Using form. l vriy 634061 iversity, t a w strong two has ity ,30630Juiz 36036-330 a, ooy 518055 nology, otegauge the to eea relativity general et,one ments, prisof operties rac of ariance a 3 , may be quantum mechanically consistent due to the asymptotic safety scenario [1] (see [2, 3] for comprehensive reviews). On the other hand, there is a possibility that the non- perturbative effects may provide unitarity in the fourth derivative theory by transforming the massive unphysical pole, which spoils the spectrum of this renormalizable theory [4]. The presence of such a massive ghost violates the stability of classical solutions (see e.g. discussion and further references in [5]). At the quantum level, the perturbative information is insufficient to conclude whether in the dressed propagator the ghost pole does transform into a non-offensive pair of complex conjugate poles [6]. The perturbative renormalization group in this model is well-explored [7, 8, 9], but it is not conclusive for the discussion of the dressed propagator. In general, the attempts to explore this possibility in the semiclassical and perturbative quantum level [10, 11] has been proved non-conclusive [12], and hence the main hope is related to the non-perturbative calculations in the framework of Functional Renormalization Group approach [13] (see [14] for an extensive review). Thus, in both cases the consistency of the results obtained within Functional Renor- malization Group approach is of the utmost importance. In this respect there are two main dangers. For the quantum gravity models based on general relativity, the running of New- ton constant in four-dimensional spacetime is always obtained on the basis of quadratic divergences. These divergences are known to have strong regularization dependence. In particular, they are absent in dimensional regularization and can be freely modified in all known cut-off schemes by changing the regularization parameter. This part of the problem does not exist for the Functional Renormalization Group applied to the fourth derivative quantum gravity. However, in this case there is yet another serious problem, related to the gauge-fixing dependence of the effective average action. This problem is the subject of the present work. In the previous publications [15, 16, 17] we explored the gauge fixing dependence in Yang-Mills theories and it was shown that such dependence for the effective average action does not vanish on-shell, except maybe in the fixed point where this object becomes identical with the usual effective action in quantum field theory. Except this special point there is uncontrollable dependence on the set of arbitrary gauge fixing parameters, and thus one can expect a strong arbitrariness in the renormalization group flows which lead to the fixed point and, in fact, define its position and proper existence. The main purpose of the present work is to extend this conclusion to quantum gravity. It is remarkable that one can complete this task for the quantum gravity theory of an arbitrary form, without using the concrete form of the action. One can use this consideration, e.g. for the superrenormalizable models of quantum gravity, when the perturbative renormalization group may be exact and, moreover, completely independent on the gauge fixing [18, 19]. This example is somehow the most explicit one, since it shows that the transition

2 from standard quantum field theory to the functional renormalization group (FRG) may actually spoil the “perfect” situation, namely exact and universal renormalization group flow. In Yang-Mills theories [20], the gauge symmetry of the initial action is broken on quantum level due to the gauge fixing procedure in the process of quantization. In turn, the effective potential depends on gauge [21, 22, 23, 24]. This dependence occur in a special way, such that it disappears on-shell [25, 26], which means that it is possible to give physical interpretation to the results obtained at the quantum level. One of the well-developed non-perturbative methods in Quantum Field Theory to study quantum properties of physical models beyond the perturbation theory is the FRG approach [27, 28] (see also the review papers [29, 30, 31, 32, 33, 34]). As we have already mentioned above, when applied to gauge theories, this method leads to obstacles related to the on-shell gauge dependence of the effective average action. There are some efforts to solve this problems. One of them consists from reformulation of Yang-Mills theory with the application of a gauge-invariant cut-off dependent regulator function that is introduced as a covariant form factor into the action of Yang-Mills fields, which leads to an invariant regulator action [35, 36]. As a consequence, the effective average action is gauge invariant on-shell. A second approach [37, 38] is based in the use of the Vilkovisky-DeWitt covariant effective action [39, 40]. This technique provides gauge independence even off-shell, but it introduce other types of ambiguities. An alternative formulation was presented in [15], it consists of an alternative way of introducing the regulator function as a composite operator. When applied in gauge theories, this approach leads to the on-shell gauge invariance of the effective average action. In the present work, we apply the background field method [41, 42, 43] (recent advances for the gauge theories can be found in [44, 45, 46, 16, 47, 48, 49] and discussion for the quantum gravities case in [50]) in the FRG approach as a reformulation to the quantization procedure for quantum gravity theories to study the gauge dependence problem in this kind of theory. This method allows us to work with the effective action which is invariant under the gauge transformations of the background fields. Despite the numerous aspects of quantum properties successfully studied with the background field method, [51, 52, 53, 54, 55, 56, 57, 58, 59, 60], the gauge dependence problem remains important [16] and need to be considered in more details. We obtain restrictions on the tensor structure of the regulator functions which allows us to construct a gauge invariant effective average action. Nevertheless, the effective average action remains dependent of the gauge choice at on-shell level. The paper is organized as follows. In section ˚sect2 we introduce general considerations of quantum gravity theories through the background field method. In section ˚sect3 we

3 consider the FRG approach for quantum gravity theories and find conditions that we must impose in the regulator functions to maintain the gauge invariance of background effective average action. Based on this, we also present some interesting candidates to the regulator functions. In section ˚sect4 we prove the gauge dependence of vacuum functional for the model under consideration. Finally, our conclusions and remarks are presented in section ˚sect5. In the paper, DeWitt notations [61] are used. The short notation for integration in D dimension is dDx = dx. All the derivatives with respect to fields are left derivative unless otherwiseR stated.R The Grassmann parity of a quantity A is denoted as ε(A).

2 Quantum gravity in the background ﬁeld formalism

Let us consider an arbitrary action S0 = S0(g), where g = gαβ(x) is the metric tensor of an arbitrary Riemann manifolds. We assume that the action is invariant under general coordinates transformations

xµ xµ = xµ(x′), (1) → which leads to the metric transformations ∂xµ ∂xν g′ (x′)= g (x) (2) αβ µν ∂x′α ∂x′β and consider the inﬁnitesimal form of these transformations, x′σ = xσ + ξσ(x), when

δg = ξσ(x)∂ g (x) g (x)∂ ξσ(x) g (x)∂ ξσ(x). (3) αβ − σ αβ − ασ β − σβ α

The diﬀeomorphism (˚eq:ten-trans) can be considered as the gauge transformation for gαβ(x)

σ δgαβ(x)= dy Rαβσ(x, y; g) ξ (y), (4) Z where

R (x, y; g)= δ(x y)∂ g (x) g (x)∂ δ(x y) g (x)∂ δ(x y) (5) αβσ − − σ αβ − ασ β − − σβ α − are the generators of the gauge transformations of the metric tensor, and ξσ(y) are the gauge parameters. The algebra of generators is closed, namely

δRαβσ(x, y; g) δRαβγ(x, z; g) du Rµνγ (u, z; g) Rµνσ(u,y; g) Z δgµν (u) − δgµν(u)

λ = duRαβλ(x, u; g)F (u,y,z), (6) − Z σγ

4 where

F λ (x, y, z)= δ(x y)δλ∂ δ(x z) δ(x z)δλ∂ δ(x y), (7) µν − ν µ − − − µ ν − with F λ (x, y, z)= F λ (x,z,y) (8) µν − νµ are structure functions of the algebra which does not depend on the metric tensor. Let us stress that the mentioned features are valid for an arbitrary action of gravity, since the algebra presented above is independent on the initial action. Therefore, any theory of gravity is a gauge theory and the structure functions are independent of the fields, that means quantum gravity is similar to the Yang-Mills theory. An useful procedure to quantize gauge theories is the so called background field formalism. In what follows, we shall perform the quantization of gravity on an arbitrary external background metricg ¯αβ(x). The standard references on the background field formalism in quantum field theory are [41, 42, 43] (see also recent advances for the gauge theories in [16] and [48] for the discussion of quantum gravity).

In the background ﬁeld formalism, the metric tensor gαβ(x) is replaced by the sum

gαβ(x)=¯gαβ(x)+ hαβ(x), (9) whereg ¯αβ(x) is an external (background) metric ﬁeld and hαβ(x) is the variable of integration, also called quantum metric. Thus, the initial action is replaced by

S (g) S (¯g + h). 0 → 0

The Faddeev-Popov SFP (φ, g¯) action is constructed in the standard way [62]

SFP (φ, g¯)= S0(¯g + h)+ Sgh(φ, g¯)+ Sgf (φ, g¯), (10) where Sgh(φ, g¯) is the ghost action and Sgf (φ, g¯) is the gauge ﬁxing action. In the presence of external metricg ¯αβ(x), they can be written as

¯α βγ σ Sgh(φ, g¯)= dxdydz g¯(x)C (x)Hα (x, y;¯g, h)Rβγσ(y, z;¯g + h)C (z), (11) Z p−

α Sgf (φ, g¯)= dx g¯(x)B (x)χα(x;¯g, h), (12) Z p− where

βγ δχα(x;¯g, h) Hα (x, y;¯g, h)= (13) δhβγ (y) and χ (x;¯g, h) are the gauge fixing functions, φi(x)= h (x), Bα(x),Cα(x), C¯α(x) is the α { αβ } set of quantum fields, Cα(x) and C¯α(x) are the ghost and anti-ghost fields, respectively, and

5 Bα(x) are the Nakanishi-Lautrup auxiliary ﬁelds. The Grassmann parity of all quantum ﬁelds are as follows,

α α α i ε(hαβ)= ε(B )=0, ε(C¯ )= ε(C )=1, ε(φ )= εi.

The ghost numbers are

gh(Bα) = gh(h )=0, gh(Cα)=1, gh(C¯α)= 1. αβ − Independently of gauge ﬁxing function choice, the action (10) is invariant under a global supersymmetry transformation, known as BRST symmetry [63, 64]. The gravitational BRST transformations were introduced in [65, 4, 66] and can be presented as

δ h (x)= (Cσ(x)∂ g (x)+ g (x)∂ Cσ(x)+ g (x)∂ Cσ(x)) λ, B αβ − σ αβ ασ β σβ α α σ α δBC (x)= C (x)∂σC (x)λ, α α δBC¯ (x)= B (x)λ, α δBB (x)=0, (14) where λ is a constant Grassmann parameter. In condensed notation, we can write the BRST transformations as

i i i δBφ (x)= R (x; φ, g¯)λ, ε(R )= εi +1, (15) where Ri = R(h), Rα , Rα , Rα and { αβ (B) (C) (C¯)} R(h)(x; φ, g¯)= Cσ(x)∂ g (x) g (x)∂ Cσ(x) g (x)∂ Cσ(x), αβ − σ αβ − ασ β − σβ α α σ α R(C)(x; φ, g¯)= C (x)∂σC (x), α α R(C¯)(x; φ, g¯)= B (x), α R(B)(x; φ, g¯)=0. (16)

The generating functional of Green functions is deﬁned as i i Z(J, g¯)= dφ exp ~[SFP (φ, g¯)+ Jφ] = exp ~W (J, g¯) , (17) Z n o n o where W (J, g¯) is the generating functional of connected Green functions. In Eq. (17) the i product of the sources Ji(x) and the ﬁelds φ (x) was written in the condensed notation of DeWitt [61]. Explicitly,

i µν (B) Jφ = dxJi(x)φ (x), where Ji(x)= J ,J (x), J¯α(x),Jα(x) (18) Z { α }

i i with the Grassmann parities ε(Ji)= ε(φ ) and ghost numbers gh(Ji) = gh(φ ).

6 The eﬀective action Γ(Φ, g¯) is deﬁned in terms of the Legendre transformation

Γ(Φ, g¯)= W (J, g¯) JΦ, (19) − where Φ = Φi are the mean ﬁelds and the J are the solution of the equation { } i δW (J, g¯) = Φi. (20) δJi It is well-known [25, 26] that the eﬀective action is gauge independent on-shell,

δΓ(Φ, g¯) =0. (21) δΓ(Φ,g¯) =0 δΦ

At this moment we have considered only the transformations of the quantum fields. However, the background metric also transform together with the quantum fields in the so- called background field transformations. The rules of such transformation can be written, in the local formulation, as

δ g¯ (x)= ∂ g¯ (x)ωσ g¯ (x)∂ ωσ g¯ (x)∂ ωσ, ω αβ − σ αβ − ασ β − σβ α δ h (x)= ∂ h (x)ωσ h (x)∂ ωσ h (x)∂ ωσ, ω αβ − σ αβ − ασ β − σβ α δ C¯α(x)= ωσ∂ C¯α(x)+ C¯σ(x)∂ ωα, ω − σ σ δ Cα(x)= ωσ∂ Cα(x)+ Cσ(x)∂ ωα, ω − σ σ δ Bα(x)= ωσ∂ Bα(x)+ Bσ(x)∂ ωα, (22) ω − σ σ where ωσ = ωσ(x) are arbitrary functions. The background ﬁeld transformations have the same structure of tensor transformations for tensors of types (0, 2) and (1, 0). The background invariance of Faddeev-Popov action for quantum gravity is known [48] and reads

δωSFP (φ, g¯)=0. (23)

A consequence of (˚eq:FPinva) is the gauge invariance of (˚eq:functional). Namely,

δωZ(J, g¯)=0. (24)

From this expression it is possible to prove that Γ(Φ, g¯) is also gauge invariant

δωΓ(Φ, g¯)=0. (25)

In the next sections we will discuss this and other features for quantum gravity theories in the framework of the FRG approach.

7 3 FRG approach for quantum gravity theories

The main idea of functional renormalization group (FRG) is to use instead of Γ an eﬀective average action, Γk, where k is a momentum-shell parameter [27], in a way that

lim Γk(φ, g¯)=Γ(φ, g¯). (26) k→0

In order to obtain Γk(φ, g¯), we introduce the average action

SkFP (φ, g¯)= SFP (φ, g¯)+ Sk(φ, g¯), (27) where Sk(φ, g¯) is the scale-dependent regulator action deﬁned in curved spacetime

Sk(φ, g¯)= dx g¯(x) k(φ, g¯) (28) Z p− L and the Lagrangian density is written as 1 (φ, g¯)= h (x)R(1)αβγδ (x;¯g)h (x)+ C¯α(x)R(2) (x;¯g)Cβ(x), (29) Lk 2 αβ k γδ kαβ

(1)αβγδ (2) where the regulator functions Rk (x;¯g) and Rkαβ(x;¯g) are dependent on the external ﬁeldsg ¯αβ(x). The regulator functions obey the properties

(1)αβγδ (2) lim Rk (x;¯g) = 0 and lim Rkαβ(x;¯g)=0, (30) k→0 k→0 which means that the average action recovers the Faddeev-Poppov action (˚eq:Sfp) in the limit when k 0. The regulator functions R(1)αβγδ (x;¯g) also obey, by construction, the → k symmetry properties

(1)αβγδ (1)βαγδ (1)αβδγ (1)γδαβ Rk (x;¯g)= Rk (x;¯g)= Rk (x;¯g)= Rk (x;¯g). (31)

We want to solve the problem of average action invariance under background ﬁeld transformations, namely

δωSkFP (φ, g¯)= δωSk(φ, g¯)=0, (32) where the relation (˚eq:FPinva) is used. In what follows, we present explicit calculation of variation of action (28). For this purpose, we write (32) as

δωSk(φ, g¯)= dx δω g¯(x) k(φ, g¯)+ g¯(x)δω k(φ, g¯) =0. (33) Z n p− L p− L o

8 For the ﬁrst term in (˚eq:twoterms) we have

σ dxδω g¯(x) k(φ, g¯)= dx∂σ( g¯(x)ω ) k(φ, g¯) Z − L − Z − L p p (34) σ = dx g¯(x)ω ∂σ k(φ, g¯), Z p− L where integration by parts was used. The variation of (φ, g¯) in second term of equation (˚eq:twoterms) can be presented as Lk 1 1 δ (φ, g¯)= δ h (x)R(1)αβγδ (x;¯g)h (x)+ h (x)δ R(1)αβγδ(x;¯g)h (x) ωLk 2 ω αβ k γδ 2 αβ ω k γδ 1 + h (x)R(1)αβγδ (x;¯g)δ h (x)+ δ C¯α(x)R(2) (x;¯g)Cβ(x) (35) 2 αβ k ω γδ ω kαβ ¯α (2) β ¯α (2) β + C (x)δωRkαβ(x;¯g)C (x)+ C (x)Rkαβ(x;¯g)δωC (x).

In terms of transformations (˚eq:backgauge), the above expression reads 1 δ (φ, g¯)= ∂ h (x)ωσ + h (x)∂ ωσ + h (x)∂ ωσ R(1)αβγδ (x;¯g)h (x) ωLk − 2 σ αβ ασ β σβ α k γδ 1 h R(1)αβγδ (x;¯g) ∂ h (x)ωσ + h (x)∂ ωσ + h (x)∂ ωσ − 2 αβ k σ γδ γσ δ σδ γ 1 (36) + h (x)δ R(1)αβγδ(x;¯g)h (x)+ C¯α(x)δ R(2) (x;¯g)Cβ(x) 2 αβ ω k γδ ω kαβ + ωσ∂ C¯α(x)+ C¯σ(x)∂ ωα R(2) (x;¯g)Cβ(x) − σ σ kαβ + C¯α(x)R(2) (x;¯g) ωσ∂ Cβ(x)+ Cσ(x)∂ ωβ . kαβ − σ σ Thus, by means of Eqs. (34) and (36) the variation of the action can be written in the compact way 1 δ S (φ, g¯)= dx g¯(x) h (x)M (1)αβγδ (x;¯g)h (x)+ C¯α(x)M (2) (x;¯g)Cβ(x) ,(37) ω k 2 αβ ωk γδ ωkαβ Z p− n o where

M (1)αβγδ (x;¯g)= δ R(1)αβγδ (x;¯g)+ ωσ∂ R(1)αβγδ(x;¯g) ∂ ωαR(1)σβγδ (x;¯g) ωk ω k σ k − σ k ∂ ωβR(1)ασγδ (x;¯g) ∂ ωγR(1)αβσδ (x;¯g) ∂ ωδR(1)αβγσ (x;¯g) − σ k − σ k − σ k and

(2) (2) σ (2) (2) σ (2) σ Mωkαβ(x;¯g)= δωRkαβ(x;¯g)+ ω ∂σRkαβ(x;¯g)+ Rkσβ(x;¯g)∂αω + Rkασ(x;¯g)∂βω . (38)

In order to ensure the invariance of (˚eq:act), it is necessary that the following conditions are satisﬁed

(1)αβγδ (2) Mωk (x;¯g) = 0 and Mωkαβ(x;¯g)=0. (39)

9 As a result, we obtain expressions for the variation of the regulator functions, (1)αβγδ σ (1)αβγδ (1)σβγδ α (1)ασγδ β δωR (x;¯g)= ω ∂σR (x;¯g)+ R (x;¯g)∂σω + R (x;¯g)∂σω k − k k k (40) (1)αβσδ γ (1)αβγσ δ + Rk (x;¯g)∂σω + Rk (x;¯g)∂σω and δ R(2) (x;¯g)= ωσ∂ R(2) (x;¯g) R(2) (x;¯g)∂ ωσ R(2) (x;¯g)∂ ωσ. (41) ω kαβ − σ kαβ − kσβ α − kασ β If the relations (˚eq:vR1) and (˚eq:vR2) are fulfilled, then the action Sk(φ, g¯) is invariant under background field transformations. Therefore, the regulator functions should have a tensor structure in order to ensure the invariance. Thus, taking into account the symmetry properties presented in (˚eq:symmetry) we can propose the following solutions for the regulator functions (1)αβγδ αβ γδ (1) ¯ αγ βδ αδ βγ ¯ Rk (x;¯g)=¯g (x)¯g (x)Rk ( )+ g¯ (x)¯g (x)+¯g (x)¯g (x) Qk( ) (42) and (2) (2) ¯ Rkαβ(x;¯g)=¯gαβRk ( ), (43) (1,2) ¯ ¯ ¯ where Rk ( ) and Qk( ) are scalar functions and is the d’Alembertian operator defined in terms of the covariant derivatives of the background metric: ¯ =g ¯µν ¯ ¯ , (44) ∇µ∇ν with the metricity property ¯ g¯ =0. (45) ∇τ µν It is possible to show that (˚eq:nR1) and (43) presents the same variational structure of (˚eq:vR1) and (41), respectively. By using the inverse background metric variation δ g¯µν(x)= ωσ∂ g¯µν(x)+¯gµσ(x)∂ ων +¯gσν(x)∂ ωµ, (46) ω − σ σ σ we have (1)αβγδ σ αβ ασ β σβ α γδ (1) ¯ δωRk (x;¯g) = ω ∂σg¯ (x)+¯g (x)∂σω +¯g (x)∂σω g¯ (x)Rk ( ) − αβ σ γδ γσ δ σδ γ (1) ¯ +g ¯ (x) ω ∂σg¯ (x)+¯g (x)∂σω +¯g (x)∂σω Rk ( ) − σ αγ ασ γ σγ α βδ + ω ∂σg¯ (x)+¯g (x)∂σω +¯g (x)∂σω g¯ (x)Qk(¯ ) − αγ σ βδ βσ δ σδ β +g ¯ (x) ω ∂σg¯ (x)+¯g (x)∂σω +¯g (x)∂σω Qk(¯ ) − σ αδ ασ δ σδ α βγ + ω ∂σg¯ (x)+¯g (x)∂σω +¯g (x)∂σω g¯ (x)Qk(¯ ) − αδ σ βγ βσ γ σγ β +g ¯ (x) ω ∂σg¯ (x)+¯g (x)∂σω +¯g (x)δσω Qk(¯ ) − g¯αβ(x)¯gγδ(x)ωσ∂ R(1)(¯ ) g¯αγ(x)¯gβδ(x)ωσ∂ Q (¯ ) − σ k − σ k g¯αδ(x)¯gβγ(x)ωσ∂ Q (¯ ). (47) − σ k 10 (1) ¯ ¯ The derivatives in metric tensor and in functions Rk ( ) and Qk( ) can be combined to obtain

(1)αβγδ ασ β σβ α γδ (1) ¯ δωRk (x;¯g) = g¯ (x)∂σω +¯g (x)∂σω g¯ (x)Rk ( ) αβ γσ δ σδ γ (1) ¯ +g ¯ (x) g¯ (x)∂σω +¯g (x)∂σω Rk ( ) ασ γ σγ α βδ + g¯ (x)∂σω +¯g (x)∂σω g¯ (x)Qk(¯ ) αγ βσ δ σδ β +g ¯ (x) g¯ (x)∂σω +¯g (x)∂σω Qk(¯ ) ασ δ σδ α βγ + g¯ (x)∂σω +¯g (x)∂σω g¯ (x)Qk(¯ ) αδ βσ δ σγ β +g ¯ (x) +¯g (x)∂σω +¯g (x)δσω Qk(¯ ) σ αβ γδ (1) ¯ σ αγ βδ ¯ ω ∂σ g¯ (x)¯g (x)Rk ( ) ω ∂σ g¯ (x)¯g (x)Qk( ) − − σ αδ βγ ω ∂σ g¯ (x)¯g (x)Qk(¯ ) . (48) − As a result, it is possible to see that (˚eq:vR1) is satisﬁed. For the second function its variation can be expressed, after some algebra, as

δ R(2) (x;¯g) = ωσ∂ g¯ R(2)(¯ ) ∂ ωσg¯ (x)R(2)(¯ ) ∂ ωσg¯ (x)R(2)(¯ ) ω kαβ − σ αβ k − α σβ k − β ασ k ωσg¯ (x)∂ R(2)(¯ ). (49) − αβ σ k The combination of derivatives in metric tensor and in scalar function leads to

δ R(2) (x;¯g)= ωσ∂ (¯g (x)R(2)(¯ )) ∂ ωσg¯ (x)R(2)(¯ ) ∂ ωσg¯ (x)R(2)(¯ ), (50) ω kαβ − σ αβ k − α σβ k − β ασ k which has the same structure as (41). Finally, the scale-dependent regulator Lagrangian density (˚eq:lagr) in terms of (42) and (43) reads

1 αβ γδ (1) ¯ k(φ, g¯)= hαβ(x) g¯ (x)¯g (x)Rk ( ) L 2 h αγ βδ αδ βγ + g¯ (x)¯g (x)+¯g (x)¯g (x) Qk(¯ ) hγδ(x) (51) i ¯α (2) ¯ β + C (x)¯gαβ(x)Rk ( )C (x), which maintains the background ﬁeld symmetry, δωSk(φ, g¯) = 0.

11 4 Gauge dependence of eﬀective average action

In order to understand the gauge invariance and gauge dependence problems in the background ﬁeld method, we shall consider the generating functionals of Green functions

i ˆ ZkΨ(J, g¯) = dφ exp ~[S0(h +¯g)+ R(φ, g¯)Ψ(φ, g¯)+ Sk(φ, g¯)+ Jφ] Z n o i i = dφ exp ~[SkFP (φ, g¯)+ Jφ] = exp ~WkΨ(J, g¯) , (52) Z n o n o where

α Ψ(φ, g¯)= dx g¯(x)C¯ (x)χα(x; h, g¯) (53) Z p− is the fermionic gauge ﬁxing functional and δ Rˆ(φ, g¯)= dx r Ri(x; φ, g¯) (54) Z δφi(x) is the generator of BRST transformations (˚eq:BRSTcon). As far as we saw in the previous section, the regulator action (˚eq:act) does not depend on the gauge Ψ(φ, g¯). Now, we shall consider another choice of gauge ﬁxing functional Ψ Ψ+ δΨ and set J = 0 in (˚eq:Z). Thus, → i ˆ i ZkΨ+δΨ(¯g)= dφ exp ~[SkFP (φ, g¯)+ R(φ, g¯)δΨ(φ, g¯)] = exp ~WkΨ+δΨ(¯g) , (55) Z n o n o where

α δΨ= δΨ(φ, g¯)= dx g¯(x)C¯ (x)δχα(h, g¯). (56) Z p− We will try to compensate the additional term Rδˆ Ψ in (˚eq:Z1). To do this, we change the variables in the functional integral related to the symmetries of action SFP (φ, g¯), namely the BRST symmetry and the background gauge invariance. First, we shall consider the BRST symmetry (˚eq:BRST), but trading the constant parameter λ by a functional Λ = Λ(φ, g¯). The variation of (˚eq:act) under such transformation is the following

4 δBSk(φ, g¯)= d x g¯(x) δB k(φ, g¯) , (57) Z p− n L o where 1 1 δ = δ h (x)R(1)αβγδ (x;¯g)h (x)+ h (x)R(1)αβγδ (x;¯g)δ h (x) BLk 2 B αβ k γδ 2 αβ k B γδ ¯α (2) β ¯α (2) β + δBC (x)Rkαβ(x;¯g)C (x)+ C (x)Rkαβ(x;¯g)δBC (x)

12 After some algebra, δ (φ, g¯) reads as BLk 1 δ = Cσ(x)∂ g (x)+ g (x)∂ Cσ(x)+ g (x)∂ Cσ(x) ΛR(1)αβγδ (x;¯g)h (x) BLk − 2 σ αβ ασ β σβ α k γδ 1 h (x)R(1)αβγδ (x;¯g) Cσ(x)∂ g (x)+ g (x)∂ Cσ(x)+ g (x)∂ Cσ(x) Λ − 2 αβ k σ γδ γσ δ σδ γ α (2) β ¯α (2) σ β + B (x)ΛRkαβ(x;¯g)C (x)+ C (x)Rkαβ(x;¯g)C (x)∂σC (x)Λ.

From the above expression, it is clear that the action Sk(φ, g¯) is not invariant under BRST transformations δ S (φ, g¯) = 0. The Jacobian J of such transformation can be B k 6 1 obtained in the standard way

δ(δ h (x)) δ(δ Cα(x)) δ(δ C¯α(x)) J = exp dx B αβ B B , (58) 1 α ¯α n Z h δhαβ(x) − δC (x) − δC (x) io where the functional derivatives are

δ(δBhαβ(x)) D(D + 1) σ (D + 1)(D 2) σ = δ(0)C (x)∂σΛ(φ, g¯) − δ(0)∂σC (x)Λ(φ, g¯) δhαβ(x) − 2 − 2

σ σ σ δΛ(φ, g¯) C(x) ∂σgαβ(x)+ gασ(x)∂βC (x)+ gσβ(x)∂αC (x) , (59) − h i δhαβ(x) δ(δ Cα(x)) B = (D + 1)δ(0)∂ Cσ(x)Λ(φ, g¯)+ Dδ(0)Cσ(x)∂ Λ(φ, g¯) δCα(x) σ σ δΛ(φ, g¯) + Cσ(x)∂ Cα(x) , (60) σ δCα(x) δ(δ C¯α(x)) δΛ(φ, g¯) B = Bα(x) . (61) δC¯α(x) δC¯α(x)

It is possible to choose a regularization scheme such that δ(0) = 0. As a result, the Jacobian for BRST transformations is δΛ(φ, g¯) δΛ(φ, g¯) δΛ(φ, g¯) J = exp dx R(h)(x; φ, g¯) Rα (x; φ, g¯) Rα (x; φ, g¯) . 1 αβ (C) α (C¯) ¯α n Z h δhαβ(x) − δC (x) − δC (x) io (62) where (˚eq:BRSTcon) is used. It is also interesting to consider the background gauge transformation related to expressions (˚eq:backgauge). As far as the regulator functions transform as (˚eq:vR1) and

(˚eq:vR2), the action Sk(φ, g¯) is invariant under such transformation. But now, instead of functions ω = ωσ(x) we shall consider the functional Ωσ = Ωσ(x, φ, g¯). The action (˚eq:act) remains invariant, and the corresponding Jacobian of this transformation can be obtained as before δ(δ h (x)) δ(δ Cα(x)) δ(δ C¯α(x)) J = exp dx Ω αβ Ω Ω , (63) 2 α ¯α n Z δhαβ(x) − δC (x) − δC (x) o

13 with the following functional derivatives:

δ(δΩhαβ(x)) (D + 1)(D 2) σ = − δ(0)∂σΩ (x, φ, g¯) δhαβ(x) 2 δΩσ(x, φ, g¯) ∂ h (x)+ h (x)∂ + h (x)∂ , (64) − σ αβ ασ β σβ α δh (x) αβ δ(δ C¯α(x)) δΩσ(x, φ, g¯) Ω = (D + 1)δ(0)∂ Ωσ(x, φ, g¯) ∂ C¯α(x) δC¯α(x) σ − δC¯α(x) σ α σ δΩ (x, φ, g¯) C¯ (x)∂σ , (65) − δC¯α(x) δ(δ Cα(x)) δΩσ(x, φ, g¯) Ω = (D + 1)δ(0)∂ Ωσ(x, φ, g¯) ∂ Cα(x) δCα(x) σ − δCα(x) σ δΩα(x, φ, g¯) Cσ(x)∂ . (66) − σ δCα(x) As before, δ(0) = 0. Thus, the Jacobian for background gauge transformations reads δΩσ(x, φ, g¯) J2 = exp dx ∂σhαβ(x)+ hασ(x)∂β + hσβ(x)∂α Z − δhαβ(x) n h δΩσ(x, φ, g¯) δΩα(x, φ, g¯) δΩσ(x, φ, g¯) + ∂ C¯α(x)+ C¯σ(x)∂ + ∂ Cα(x) (67) δC¯α(x) σ σ δC¯α(x) δCα(x) σ α σ δΩ (x, φ, g¯) + C (x)∂σ α . δC (x) io If it is possible to fulﬁll the condition

i ˆ J1J2 exp ~ dx[R(φ, g¯)δΨ(φ, g¯)+ δBSk(φ, g¯)] =1 (68) n Z o the generating vacuum functional ZkΨ(¯g) does not depend on the gauge ﬁxing functional Ψ. In order to verify that, let us expand the functionals Λ and Ω, with Grassmann parity ε(Λ) = 1 and ε(Ω) = 0 and ghost numbers gh(Λ) = 1 and gh(Ω) = 0 in the lower power − of ghost ﬁelds

Λ=Λ(1) +Λ(3), Ωσ = Ωσ(0) + Ωσ(2), (69) where

Λ(1) = dxC¯α(x)λ(1)(x, h, g¯), (70) Z α 1 Λ(3) = dx C¯α(x)C¯β(x)λ(3) (x, h, g¯)Cγ(x), (71) Z 2 αβγ Ωσ(0)(x) = Ωσ(0)(x, h, g¯), (72) σ(2) ¯α σ(2) β Ω (x, h, g¯)= C (x)ωαβ (x, h, g¯)C (x). (73)

14 The terms that vanish in (˚eq:condition) and do not depend on ghost ﬁelds, leads to

δΩσ(0)(x, h, g¯) ∂ h (x)+ h (x)∂ + h (x)∂ =0. (74) σ αβ ασ β σβ α δh (x) αβ Analyzing the terms which are linear in the anti-ghost ﬁelds, which contains the auxiliary ﬁelds B(x), we obtain i λ(1) x, h, g δχ x, h, g α ( ¯)= ~ α( ¯) (75) and

λ(3) (x, h, g¯)= λ(1)(h, g¯)R(2) (x;¯g) λ(1)(h, g¯)R(2) (x;¯g), (76) αβγ α kβγ − β kαγ where

λ(1)(h, g¯)= dxλ(1)(x, h, g¯). (77) α Z α Now, the vanishing terms with structure C¯(x)C(x) can be related to the second order σ σ(2) of Ω (x, h, g¯) functional leading to a differential equation for ωαβ (x, h, g¯), i ∂ ωσ(2) x, h, g ∂ g x λ(1) h, g R(1)τσγδ x g h x σ αβ ( ¯) = ~ β τσ( ) α ( ¯) ( ;¯) γδ( ) 2 h (1)τσγδ (1) + hτσ(x)R (x;¯g)∂βgγδ(x)λα (h, g¯) . (78) i From Eq. (74) since Ωσ(0)(x, h, g¯) is an arbitrary function we can not have just one (3) particular solution. In addition, the λαβγ relation in (76) creates non-local term of structure BC¯CCC¯ which can be only eliminated if we consider new functional Λ of higher orders in ghost fields. Even so, this process would repeat endlessly. The only case left for us is to consider the simple solution when Ωσ =0andΛ=Λ(1) we have the result i ZkΨ+δΨ(¯g) = dφ exp ~[SkFP (φ, g¯)+ δSkFP (φ)] , Z n o Z (¯g) = Z (¯g) . (79) kΨ 6 kΨ+δΨ As final result, the vacuum functional in the FRG approach for gravity theories depends on the gauge fixing even on-shell, which leads to a gauge dependent S-matrix.

5 Conclusions and Perspectives

We explored the problem of gauge invariance and the gauge-ﬁxing dependence using the background ﬁeld formalism, for gravity theories in the FRG framework. It was shown

15 that the background field invariance is achieved when the regulator functions are choosen to have the tensor structure. Nevertheless, even in this case the on-shell gauge dependence cannot be cured in the standard FRG approach which we dealt with. In this respect the situation is qualitatively the same as in the Yang-Mills theories, as it was discussed in [15]. The on-shell gauge dependence takes place due to the fact that the regulator action (˚eq:act) is not BRST-invariant. It turns out that this is a fundamentally important feature, that can be changed only by trading the standard and conventional FRG framework to an alternative one, which is based on the use of composite operators for constructing the regulator action. Unfortunately, until now there is no way to perform practical calculations in this alternative formulation. For this reason, taking our present results into account, it remains unclear whether the quantum gravity results obtained within the FRG formalism can have a reasonable physical interpretation. One can expect that all predictions of this formalism will depend on an arbitrary choice of the gauge fixing. Thus, one can, in principle, provide any desirable result, but the value of this output is not clear. Alternatively, there should be found some physical reason to claim that one special gauge fixing is “cor- rect” or “preferred” for some reason, but at the moment it is unclear how this reason can look like, since the original theory is gauge (diffeomorphism) invariant. It is worthwhile to comment on the effect of the on-shell gauge fixing dependence on the non-universality of the renormalization group flow in the framework of the Functional Renormalization Group approach. First of all, the complete understanding of this issue requires the detailed analysis and, in principle, explicit calculations in the sufficiently general model of quantum gravity. This calculation is a separate problem that is beyond the scope of the present work. At the same time, we can give some strong arguments in favor of such a gauge fixing dependence for both renormalization group flow and the fixed point, which is obtained from this flow. One can classify the models of quantum gravity into two classes. The first class is the quantum general relativity, where the renormalization group equation for the Newton constant G does not exist at the perturbative level, at least without the cosmological constant term. In the presence of the cosmological constant there is a perturbative renormalization group equation for the Newton constant, but it is strongly gauge-fixing dependent, as discussed e.g. in [7] and more recently, with a very general calculations, in [67] (see more references therein). Let us add that in the Functional Renormalization Group approach the renormalization group flow for G is extracted from the quadratic divergences, and for this reason the corresponding results do not have the perturbative limit. On the other hand, the quadratic divergences are also gauge-fixing dependent [67] and, furthermore, suffer from the total ambiguity in defining the cut-off parameter Ω. For instance, if we start from the

16 total expression for the one-loop divergences in the Schwinger-DeWitt formalism,

1 1 Ω2 (1) d4x g A 4 A 2 A , Γdiv = √ 0Ω + 1Ω + 2 log 2 (80) −2 Z 2 µ where A0, A1 and A2 are the algebraic sums of the contributions of quantum metric and ghosts, given by the functional traces of the coincidence limits of the corresponding Schwinger-DeWitt coefficients. Without the cosmological constant only A1 has the term linear in the scalar curvature R, and hence only this term can contribute to the renormalization group flow for G. Now, changing the cut-off as

Ω2 Ω2 + αR, (81) −→ with an arbitrary coeﬃcient α, we observe that the quartic and logarithmic divergences do not change, while the quadratic ones modify according to6

A A + αA R, (82) 1 −→ 1 0 making it completely ambiguous. Thus, in our opinion, it is difficult to take the results extracted from A1 as a physical prediction, even without gauge fixing ambiguity. The theories of quantum gravity of the second kind are those with higher derivatives, being fourth derivative [4] models, the superrenormalizable models with six or more derivatives [18] or the nonlocal superrenormalizable models. In all these cases the renormalization group flow in the Functional Renormalization Group has a well-defined perturbative limit (see e.g. [3]) and hence it should be supposed that the invariance or non-invariance of the renormalization group flow should be the same in both usual perturbative and Functional Renormalization Group approaches. On the other hand, without the on-shell invariance condition there is absolutely no way to consider the renormalization group flows in any one of these models in a consistent way. This issue was analyzed in detail in [7, 68, 69, 18] and [19], thus there is no point to repeat these considerations here. Thus, we can conclude that in those versions of Functional Renormalization Group where one can expect a minimally reasonable interpretation of the application to quantum gravity, it is difficult to expect the gauge-invariant flow in the framework of the Functional Renormalization Group, if this approach is not reformulated in a new invariant way. Finally, we mention two recent papers. In [70] it was shown that the renormalization group flow in quantum general relativity should be compatible with the Ward identities related to the possibility to change the conformal frame. Certainly, this is not the symmetry that we discussed in the previous sections. Also, even if the mentioned compatibility is achieved, this does not make the whole approach sufficiently unambiguous, as we explained

6One of the authors (I.Sh.) is grateful to M. Asorey for explaining this point.

17 above. However, qualitatively, the importance of the Ward identities to the consistency of the renormalization group flow is certainly coherent with our analysis. On the other hand, Ref. [71] there was proposed a new formulation of the Functional Renormalization Group for quantum gravity. In this new version, there is not one scale parameters, but two: one for regularizing the UV sector and another one for regularizing the IR sector. The idea is certainly interesting for quantum gravity since the two limits in all known models are very much different. However, since the UV part is essentially the same as in the standard formulation, it is difficult to expect that the gauge fixing dependence on-shell in this new formulation will be a smaller problem than it is in the standard versions with one scale parameter. We can state that the preliminary analysis of this model confirms this qualitative conclusion. The full consideration is possible, but would be quite involved and is beyond the scope of the present work. The results of the considerations which we described above make more interesting the discussion of the possible ways to solve the problem of on-shell gauge fixing dependence, such as the ones suggested in [35, 36, 34] or in [15]. In our opinion, the last approach is more transparent and physically motivated, but (as we have already mentioned above) there is no well-developed technique of using it for practical calculations.

Acknowledgments

E.A.R. and V.F.B. are grateful to Coordena¸cão de Aperfei¸coamento de Pessoal de N´ıvel Superior - CAPES for supporting their Ph.D. and Ms. projects. P.M.L. is grateful to the Department of Physics of the Federal University of Juiz de Fora (MG, Brazil) for warm hospitality during his long-term visit, when this work has been initiated. The work of P.M.L. was partially supported by the Ministry of Science and Higher Education of the Russian Federation, grant 3.1386.2017 and by the RFBR grant 18-02-00153. I.L.Sh. was partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico - CNPq under the grant 303635/2018-5 and Funda¸cão de Amparo àPesquisa de Minas Gerais - FAPEMIG under the project APQ-01205-16.

References

[1] S. Weinberg, Ultraviolet divergences in quantum theories of gravitation. (In S. W. Hawking; W. Israel (eds.). General Relativity: An Einstein centenary survey. Cam- bridge University Press, 1979).

18 [2] M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5, gr-qc/0610018.

[3] R. Percacci, Asymptotic Safety, (In D. Oriti - Editor: Approaches to quantum gravity, 111-128, Cambridge University Press, 2007), arXiv:0709.3851.

[4] K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D16 (1977) 953.

[5] F. de O. Salles, and I.L. Shapiro, Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?. Phys. Rev. D89 (2014) 084054; Erratum: Phys. Rev. D90 (2014) 129903, hep-th/1401.4583; A.M. Pelinson, F. de Oliveira Salles, and I.L. Shapiro, Gravitational waves and perspectives for quantum gravity, Mod. Phys. Lett. A29 (2014) 1430034 (Brief Review), gr-qc/1410.2581; P. Peter, F. de Oliveira Salles, and I.L. Shapiro, On the ghost-induced instability on de Sitter background, Phys. Rev. D97 (2018) 064044, gr-qc/1801.00063; A. Salvio, Metastability in Quadratic Gravity, Phys. Rev. D99 (2019) 103507, arXiv:1902.09557; S. Castardelli dos Reis, G. Chapiro and I. L. Shapiro, Beyond the linear analysis of stability in higher derivative gravity with the Bianchi-I metric, Phys. Rev. D100 066004 (2019) arXiv:1903.01044.

[6] A. Salam and J. Strathdee, Remarks on High-energy Stability and Renormalizability of Gravity Theory, Phys. Rev. D18 (1978) 4480.

[7] E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B201 (1982) 469.

[8] I.G. Avramidi and A.O. Barvinsky, Asymptotic Freedom In Higher Derivative Quan- tum Gravity Phys. Lett. B159 (1985) 269.

[9] G. de Berredo-Peixoto and I.L. Shapiro, Higher derivative quantum gravity with Gauss- Bonnet term, Phys. Rev. D71 (2005) 064005, hep-th/0412249.

[10] E. Tomboulis, 1/N Expansion and Renormalization in Quantum Gravity, Phys. Lett. B70 (1977) 361; Renormalizability and Asymptotic Freedom in Quantum Gravity, Phys. Lett. B97 (1980) 77; Unitarity in Higher Derivative Quantum Gravity, Phys. Rev. Lett. 52 (1984) 1173.

[11] I. Antoniadis and E.T. Tomboulis, Gauge Invariance And Unitarity In Higher Deriva- tive Quantum Gravity, Phys. Rev. D33 (1986) 2756.

19 [12] D.A. Johnston, Sedentary Ghost Poles In Higher Derivative Gravity, Nucl. Phys. B297 (1988) 721.

[13] A. Codello and R. Percacci, Fixed points of higher derivative gravity, Phys. Rev. Lett. 97 (2006) 221301, hep-th/0607128; A. Codello, R. Percacci, L. Rachwal and A. Tonero, Computing the Eﬀective Action with the Functional Renormalization Group, Eur. Phys. J. C76 (2016) 226, hep-th/1505.03119.

[14] A. Codello, R. Percacci and C. Rahmede, Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation, Annals Phys. 324 (2009) 414, arXiv:0805.2909.

[15] P.M. Lavrov and I.L. Shapiro, On the Functional Renormalization Group approach for Yang-Mills ﬁelds, JHEP 1306 (2013) 086, arXiv:1212.2577.

[16] P.M. Lavrov, Gauge (in)dependence and background ﬁeld formalism, Phys. Lett. B791 (2019) 29; arXiv:1805.02149.

[17] P.M. Lavrov, E. A. dos Reis, T. de Paula Netto and I.L.Shapiro, Gauge invariance of the background average eﬀective action, Eur. Phys. J. C79 (2019) 661, arXiv:1905.08296.

[18] M. Asorey, J.L. L´opez and I.L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A12 (1997) 5711, hep-th/9610006.

[19] L. Modesto, L. Rachwaland I.L. Shapiro, Renormalization group in superrenormalizable quantum gravity, Eur. Phys. J. C78 (2018) 555, arXiv:1704.03988.

[20] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invari- ance, Phys. Rev. 96 (1954) 191.

[21] R. Jackiw, Functional evaluation of the eﬀective potential, Phys. Rev. D9 (1974) 1686.

[22] L. Dolan and R. Jackiw, Gauge invariant signal for gauge symmetry breaking, Phys. Rev. D9 (1974) 2904.

[23] N.K. Nielsen, On the gauge dependence of spontaneous symmetry breaking in gauge theories, Nuc. Phys. B101 (1975) 173.

[24] R. Fukuda and T. Kugo, Gauge invariance in the eﬀective action and potential, Phys. Rev. D13 (1976) 3469.

20 [25] P.M. Lavrov and I.V. Tyutin, On the generating functional for the vertex functions in Yang-Mills theories, Sov. J. Nucl. Phys. 36 (1981) 474.

[26] B.L. Voronov, P.M. Lavrov and I.V. Tyutin, Canonical transformations and gauge dependence in general gauge theories, Sov. J. Nucl. Phys. 36 (1982) 292.

[27] C. Wetterich, Average action and the renormalization group equations, Nuc. Phys. B352 (1991) 529.

[28] C. Wetterich, Exact evolution equation for the eﬀective potential, Phys. Lett. B301 (1993) 90, arXiv:1710.05815.

[29] J. Berges, N. Tetradis and C. Wetterich, Non-perturbative renormalization ﬂow in quantum ﬁeld theory and statistical physics, Phys. Rept. 363 (2002) 223, hep-ph/0005122.

[30] C. Bagnuls and C. Berviller, Exact renormalization group equations: an introductory review, Phys. Rept. 348 (2001) 91, hep-th/0002034.

[31] J. Polonyi, Lectures on the functional renormalization group method, Central Eur. J. Phys. 1 (2003) 1, hep-th/0110026.

[32] J.M. Pawlowski, Aspects of the functional renormalization group, Annals Phys. 322 (2007) 2831, hep-th/0512261.

[33] B. Delamotte, An introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852 (2012) 49, cond-mat/0702365.

[34] O.J. Rosten, Fundamentals of the exact renormalization group, Phys. Rep. 511 (2012) 177, arXiv:1003.1366.

[35] T.R. Morris, A gauge invariant exact renormalization group. 2. JHEP 12 (2000) 012, hep-th/0006064.

[36] S. Arnone, T.R. Morris, and O.J. Rosten, A generalized manifestly gauge invariant exact renormalization group for SU(N) Yang-Mills, Eur. Phys. J. C50 (2007) 467, hep-th/0606181.

[37] V. Branchina, K.A. Meissner and G. Veneziano, The prize of an exact, gauge invariant RG ﬂow equation, Phys. Lett. B 574 (2003) 319, hep-th/0309234.

[38] J.M. Pawlowski, Geometrical eﬀective action and Wilsonian ﬂows, hep-th/0310018.

21 [39] G.A. Vilkovisky, in B.S. DeWitt Sixtieth Anniversary Volume, S. Christensen eds., Hilger, Bristol U.K. (1983).

[40] G.A. Vilkovisky, in The unique eﬀective action in quantum ﬁeld theory, Nucl. Phys. B234 (1984) 124.

[41] B.S. DeWitt, Quantum theory of gravity. II. The manifestly covariant theory, Phys. Rev. 162 (1967) 1195.

[42] I.Ya. Arefeva, L.D. Faddeev and A.A. Slavnov, Generating functional for the s matrix in gauge theories, Theor. Math. Phys. 21 (1975) 1165 ( Teor. Mat. Fiz. 21(1974) 311-321).

[43] L.F. Abbott, The background ﬁeld method beyond one loop, Nucl. Phys. B185 (1981) 189.

[44] A.O. Barvinsky, D. Blas, M. Herrero-Valca, S.M. Sibiryakov and C.F. Steinwachs, Renormalization of gauge theories in the backgrorund-ﬁeld approach, JHEP 1807 (2018) 035, arXiv:1705.03480.

[45] I.A. Batalin, P.M. Lavrov and I.V. Tyutin, Multiplicative renormalization of Yang- Mills theories in the background-ﬁeld formalism, Eur. Phys. J. C78 (2018) 570, arXiv:1806.02552.

[46] J. Frenkel and J.C. Taylor, Background gauge renormalization and BRST identities, Annals Phys. 389 (2018) 234, arXiv:1801.01098.

[47] I.A. Batalin, P.M. Lavrov and I.V. Tyutin, Gauge dependence and multiplicative renormalization of Yang-Mills theory with matter ﬁelds, Eur. Phys. J. C79 (2019) 628, arXiv:1902.09532.

[48] P.M. Lavrov and I.L. Shapiro, Gauge invariant renormalizability of quantum gravity, Phys. Rev. D100 (2019) 026018, arXiv:1902.04687.

[49] B.L. Giacchini, P.M. Lavrov and I.L. Shapiro, Background ﬁeld method and nonlinear gauges, Phys. Lett. B797 (2019) 134882, arXiv:1906.04767.

[50] R. Percacci and G.P. Vacca, The background scale Ward identity in quantum gravity, Eur. Phys. J. C77 (2017) 52, arXiv:1611.07005.

[51] G. ’t Hooft, An algorithm for poles at dimension four in the dimensional regularization procedure, Nucl. Phys. B62 (1973) 444.

22 [52] H. Klusberg-Stern and J.B. Zuber, Renormalization of non-Abelian gauge theories in a background-ﬁeld gauge. I. Green’s functions, Phys. Rev. D12 (1975) 482.

[53] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Background field method versus normal field theory in explicit examples: One loop divergences in S matrix and Green’s functions for Yang-Mills and gravitational fields, Phys. Rev. D12 (1975) 3203.

[54] D.M. Capper and A. MacLean, The background ﬁeld method at two loops: A general gauge Yang-Mills calculation, Nucl. Phys. B203 (1982) 413.

[55] S. Ichinose and M. Omote, Renormalization using the background-ﬁeld formalism, Nucl. Phys. B203 (1982) 221.

[56] M.H. Goroﬀ and A. Sagnotti, The ultraviolet behavior of Einstein gravity, Nucl. Phys. B266 (1986) 709.

[57] A.E.M. van de Ven, Two-loop quantum gravity, Nucl. Phys. B378 (1992) 309.

[58] P.A. Grassi, Algebraic renormalization of Yang-Mills theory with background ﬁeld method, Nucl. Phys. B426 (1996) 524.

[59] C. Becchi and R. Collina, Further comments on the background ﬁeld method and gauge invariant eﬀective action, Nucl. Phys. B562 (1999) 412, hep-th/9907092.

[60] R. Ferrari, M. Picariello and A. Quadri, Algebraic aspects of the background ﬁeld method, Annals Phys. 294 (2001) 165, hep-th/0012090.

[61] B.S. DeWitt, Dynamical theory of groups and ﬁelds, (Gordon and Breach, 1965).

[62] L.D. Faddeev and V.N. Popov, Feynman diagrams for the Yang-Mills ﬁeld, Phys. Lett. B25 (1967) 29.

[63] C. Becchi, A. Rouet and R. Stoura, The abelian Higgs Kibble Model, unitarity of the S-operator, Phys. Lett. B52 (1974) 344.

[64] I.V. Tyutin, Gauge invariance in ﬁeld theory and statistical physics in operator formalism, Lebedev Inst.preprint N 39 (1975); arXiv:0812.0580.

[65] R. Delbourgo and M. Ramon-Medrano, Becchi-Rouet-Stora gauge identities for gravity, Nucl. Phys. B110 (1976) 467.

[66] P.K. Townsend and P. van Nieuwenhuizen, BRS gauge and ghost ﬁeld supersymmetry in gravity and supergravity, Nucl. Phys. B120 (1977) 301.

23 [67] J.D. Gon¸calves, T. de Paula Netto and I.L. Shapiro, On the gauge and parametrization ambiguity in quantum gravity, Phys. Rev. D97 (2018) 026015, arXiv:1712.03338.

[68] I.G. Avramidi, Covariant methods for the calculation of the eﬀective action in quantum ﬁeld theory and investigation of higher-derivative quantum gravity, (PhD thesis, Moscow University, 1986). hep-th/9510140.

[69] I. L. Shapiro and A. G. Jacksenaev, Gauge dependence in higher derivative quantum gravity and the conformal anomaly problem, Phys. Lett. B324 (1994) 284.

[70] P. Labus, T. R. Morris and Z. H. Slade, Background independence in a background dependent renormalization group, Phys. Rev.D94 (2016) 024007, arXiv:1603.04772.

[71] S.P. de Alwis, Exact RG Flow Equations and Quantum Gravity, JHEP 1803 (2018) 118, arXiv:1707.09298.