Gauge invariant renormalizability of quantum gravity

Using the Batalin-Vilkovisky technique and the background field method the proof of gauge invariant renormalizability is elaborated for a generic model of quantum gravity which is diffeomorphism invariant and has no other, potentially anomalous, symmetries. The gauge invariant renormalizability means that in all orders of loop expansion of the quantum effective action one can control deformations of the generators of gauge transformations which leave such an action invariant. In quantum gravity this means that one can maintain general covariance of the divergent part of effective action when the mean quantum field, ghosts and antifields are switched off.


Introduction
Renormalization is one of the main issues in quantum gravity. The traditional view on the difficulty of quantizing gravitational field is that the quantum general relativity is not renormalizable, while the renormalizable version of the theory includes fourth derivatives [1] and therefore it is not unitary. In the last decades this simple two-side story was getting more complicated, with the new models of superrenormalizable gravity, both polynomial [2] and non-polynomial [3] (see also earlier papers [4,5]). Typically, these models intend to resolve the conflict between non-renormalizability and non-unitarity by introducing more than four derivatives.
The main advantage of the non-polynomial models is that the tree level propagator may have the unique physical pole corresponding to massless graviton. At the same time the dressed propagator has, typically, an infinite (countable) amount of the ghost-like states with complex poles [6] and hence the questions about physical contents and quantum consistency of such a theory remains open, especially taking into account the problems with reflection positivity [7] (see further discussion in [8]). It might happen that the construction of a consistent version of quantum gravity should not go through the S-matrix approach, since the flat limit and hence well-defined asymptotic states may not exist for the theories of gravity which are consistent even at the semiclassical level [9]. In this case the central question related to ghosts is the stability of the physically relevant classical solutions, and there are positive indications for the non-local models in this respect [10].
On the other hand, within the polynomial model one can prove the unitarity of the S-matrix within the Lee-Wick approach [11] to quantum gravity in four [12] and even higher dimensional space-times [13]. Furthermore, it is possible to make explicit one-loop calculations [14] which provide exact beta-functions in these theories due to the superrenormalizability of the theory. In the part of stability, the existing investigations concerned special backgrounds, namely cosmological [15,16] and black hole cases [17,18,19]. While the black hole results are not conclusive, the results for the cosmological backgrounds provide good intuitive understanding of the problem of stability in the gravity models with higher derivative ghosts.
Independent on the efforts in better understanding the role of ghosts and instabilities in both polynomial and non-polynomial models, it would be useful to have a formal proof of that these theories are renormalizable or superrenormalizable. The existing proofs concern only fourth derivative quantum gravity [1] (see also Refs. [20,21] as an application of a general approach [22]). In the present work we present the proof of a gauge invariant renormalizability in the general models of quantum gravity, which includes second derivative and higher derivative, polynomial and non-polynomial models. The preliminary condition for the consideration which is given in the present paper is that there should be regularization which preserves the symmetries of the classical action. Thus our consideration can not be directly applied to the models with conformal or chiral symmetries where one can expect to meet the corresponding anomalies. The consideration is based on the Batalin-Vilkovisky formalism, which enables one to analyse gauge invariant renormalization of a wide class of gauge theories (including quantum gravity) without going into the details of a quantum gravity model, but using only the general structure of gauge algebra. The Batalin-Vilkovisky formalism use an algebraic approach to construct the master equation for different types of generating functionals of Green functions. In the present work we apply this formalism to establish the general structure of extended action and renormalized effective action for a quantum gravity model of a very general form within the background field formalism.
The paper is organized as follows. In Sec. 2 we formulate the Batalin-Vilkovisky formalism combined with the background field method in the case of quantum gravity. In Sec. 3 this formalism is applied to the formal proof of renormalizability in the model of quantum gravity of the general form. On the top of that we use the same formalism to briefly discuss the gauge fixing independence of the S-matrix of gravitational excitations in the theories of quantum gravity. The next Sec. 4 is included here for the sake of completeness and consists of the brief review of the power counting which enables one to classify the non-renormalizable, renormalizable and superrenormalizable models. Taking into account the contents of the previous sections, this classification is now based on a more solid background and we decided to include it here. Finally, in Sec. 5 we draw our conclusions.
Condensed DeWitt's notations [23] are used in the paper. Right and left derivatives of a quantity f with respect to the variable ϕ are denoted as δrf δϕ and δ l f δϕ , correspondingly. The Grassmann parity and the ghost number of a quantity A are denoted by ε(A) and gh(A), see e.g. Eq. (23) in the last case. The condensed notation for the space-time integral in D dimensions, dx = d D x is used throughout the text.

Quantum gravity in the background field formalism
Our starting point is an arbitrary action of a Riemann's metric, S 0 = S 0 (g), where g = {g µν (x)}. The action is assumed invariant under the general coordinate transformations, The standard examples of the theories of our interest are Einstein gravity with a cosmological constant term, and a general version of higher derivative gravity, where Π 1,2,3 are polynomial or non-polynomial form factors and the last term represents nonquadratic in curvature terms. In quantum theory the action (3) may lead to the theory which is non-renormalizable, renormalizable or even superrenormalizable, depending on the choice of the functions Π 1,2,3 (x) and the non-quadratic terms The parameter M 2 in the form factors Π 1,2,3 /M 2 is a universal mass scale at which the quantum gravity effect becomes relevant. For instance, it can be the square of the Planck mass, but there may be other options, including multiple scale models, as analysed in [24]. For the analysis presented below the unique necessary feature is that the action should be diffeomorphism invariant.
In the infinitesimal form the transformations (1) read where The invariance of the action S 0 (g) under the transformations (5) can be expressed in the form of Noether identity In what follows we will also need the transformation rule for vector fields A µ (x) and A µ (x), Let us present the transformations (5) in the form δg µν (x) = dy R µνσ (x, y; g)ξ σ (y), where are the generators of gauge transformations of the metric tensor g µν with gauge parameters ξ σ (x). The algebra of gauge transformations is defined by the algebra of generators, which has the following form: where are structure functions of the gauge algebra which do not depend on the metric tensor g µν . Therefore, independent on the form of the action, any theory of gravity looks like a gauge theory with closed gauge algebra and structure functions independent on the fields (metric tensor, in the case), i.e., similar to the Yang-Mills theory. It proves useful to perform quantization of gravity on an external background, represented by a metric tensorḡ µν (x). In the simplest case the Riemann space may be just the Minkowski space-time with the metric tensor η µν = const. On the other hand, introducing an arbitrary background metric provides serious advantages, as we shall see in what follows. The standard reference on the background field formalism in quantum field theory is [25,26,27] (see also recent advances for the gauge theories in [28,29,30,31] and discussion for the quantum gravity case in [32]).
Within the background field method the metric tensor g µν (x) is replaced by the sum Here h µν (x) is called quantum metric and is regarded as a set of integration variables in the functional integrals for generating functionals of Green functions. The action S 0 (ḡ + h) is a functional of two variablesḡ and h and therefore it has additional symmetries because of extra degrees of freedom. Namely, it is invariant under the following transformations δḡ µν = ǫ µν and δh µν = −ǫ µν (15) with arbitrary symmetric tensor functions ǫ νµ = ǫ µν = ǫ µν (x). In particular, this means that there is an ambiguity is defining the gauge transformations forḡ and h. To fix this arbitrariness we require that the transformation of our interest has the right flat limit whenḡ µν (x) is traded for η µν . Then the gauge transformation of the quantum metric fields h µν in the presence of external (fixed) backgroundḡ should have the form while δḡ µν (x) = 0 and the action remains invariant, δS 0 (ḡ + h) = 0. Because of the similarity with the Yang-Mills field, the Faddeev-Popov quantization procedure is quite standard and the resulting action S F P = S F P (φ,ḡ) has the form [33] Taking into account the presence of an external background metricḡ, the ghost action has the form with the notation The S gf (ḡ, h) is the gauge fixing action which corresponds to the singular gauge condition. For the non-singular gauge condition the action has the form In what follows we shall use the form (20), where χ α (x;ḡ, h) are the gauge fixing functions, which are called to remove the degeneracy of the action S 0 (ḡ + h).
Let us introduce an important notation for the full set of quantum fields including quantum metric, Faddeev-Popov ghost, anti-ghost and the Nakanishi-Lautrup auxiliary fields B α . The Grassmann parity of these fields will be denoted as ε(φ i ) = ε i , such that for ghost and anti-ghost ε(C α ) = ε(C α ) = 1, while for the auxiliary fields B α and metric ε(B α ) = ε(h µν ) = 0. The conserved quantity called ghost number is defined for the same fields as For any admissible choice of gauge fixing functions χ α (x;ḡ, h) action (17) is invariant under global supersymmetry (BRST symmetry) [34,35] where µ is a constant Grassmann parameter. Let us present the BRST transformations (24) in the form where Then the BRST invariance of the action S F P reads The invariance property (27) can be expressed in a compact and useful form called Zinn-Justin equation, by introducing the set of additional variables φ * i (x). The new fields have Grassmann parities opposite to the corresponding fields φ i (x), namely ε(φ * i ) = ε i + 1. The extended action S = S(φ, φ * ,ḡ) reads It easy to note that the new variables φ * i (x) serve as the sources to BRST generators (26). Then the relation (27) takes the standard form of the Zinn-Justin equation [38] for the action (28), One can note that using left and right derivatives in the last equation is relevant due to the nontrivial Grassmann parities of the involved quantities.
According to the terminology of Batalin-Vilkovisky formalism [39,40] the sources φ * i (x) are known as antifields. The fundamental notion in the Batalin-Vilkovisky formalism is the antibracket for two arbitrary functionals of fields and antifields, F = F (φ, φ * ) and G = G(φ, φ * ). The antibracket is defined as which obeys the following properties: 1) Grassmann parity relations 2) Generalized antisymmetry 3) Leibniz rule 4) Generalized Jacobi identity In terms of antibracket Eq. (29) can be written in a compact form, which is the classical master equation of Batalin-Vilkovisky formalism [39,40]. This equation will be generalized to the quantum domain and extensively used to analyse renormalizability of quantum gravity in the next section. Now we are in a position to formulate the quantum theory. The generating functional of Green functions is defined in the form of functional integral 4 where W (J,ḡ) is the generating functional of connected Green functions. In (36) the DeWitt notations are used, namely are external sources for the fields (22). The Grassmann parities and ghost numbers of these sources satisfy the relations Let us a detailed consideration of the generating functionals and their gauge dependence. As a first step, consider the vacuum functional Z Ψ (ḡ), which corresponds to the choice of gauge fixing functional (27) in the presence of external fieldsḡ, where we introduced the operator and Ψ(φ,ḡ) is the fermionic gauge fixing functional, Taking into account (40) and (41), the definition (39) becomes an expression which is nothing but (36) without the source term in the exponential. In order to take care about possible change of the gauge fixing, let Z Ψ+δΨ be the modified vacuum functional corresponding to Ψ(φ,ḡ) + δΨ(φ,ḡ), where δΨ(φ,ḡ) is an arbitrary infinitesimal functional with odd Grassmann parity. Besides from this requirement, δΨ(φ,ḡ) can be arbitrary, in particular it may be different from Eq. (41).
Taking into account (42), with the new term we get The next step is to make the change of variables φ i in the form of BRST transformations (24) but with replacement of the constant parameter µ by a functional µ = µ(φ,ḡ), In what follows we shall use short notations R i (x; φ,ḡ) = R i (x) and µ(φ,ḡ) = µ. Due to the linearity of BRST transformations, action S F P (φ,ḡ) is invariant under (44) where matrix M i j (x, y) has the form In Yang-Mills type theories due to antisymmetry properties of structure constants the following relation holds. Then from (45) and (46) it follows that Choosing the functional µ in the form one can observe that the described change of variables in the functional integral completely compensates the modification in the expression (43) compared to the fiducial formula (42).
Thus we arrive at the gauge independence of the vacuum functional One can present this identity as vanishing variations of the vacuum functionals Z and W , Due to the invariance feature (50) we can omit the label Ψ in the definition of the generating functionals (36). Furthermore, it is known that due to the equivalence theorem [43] the invariance (50) implies that if the background metricḡ µν admits asymptotic states (e.g., if it is a flat Minkowski metric), then the S-matrix in the theory of quantum gravity does not depend on the gauge fixing. It is remarkable that we can make this statement for an arbitrary model of QG, even without requiring the locality of the classical action. One can say that if the theory admits the construction of the S-matrix, the last will be independent on the choice of the gauge fixing conditions. Let us note that this is true only within the conventional perturbative approach to quantum field theory, while the situation may be opposite in other approaches. For instance, the S-matrix is not invariant if it is constructed on the basis of the concept of average effective action related to functional renormalization group [44,45,46,47]. The corresponding proof for the Yang-Mills theory is based on the general result of Ref. [43] and can be found in Ref. [48]. We believe it can be directly generalized for the case of gravity. Similar situation takes place in the standard formulation of the Gribov-Zwanziger theory [49,50,51] when the corresponding effective action depends on the choice of gauge even on-shell [52,53]. This difficulty illustrates the situation which we meet when trying to go beyond the framework of perturbative field theory, that would be especially relevant in the case of quantum gravity. The effective action Γ(Φ,ḡ) is defined by means of Legendre transformation, where Φ = {Φ i } are mean fields and J i are the solutions of the equations In terms of effective action the property (51) means the on-shell gauge fixing independence and reads i.e. the effective action evaluated on its extremal does not depend on gauge. Until now we did not assume that the background metric may transform under the general coordinate transformation. This was a necessary approach, as it was explained after the definition of the splitting (14) of the metric into background and quantum parts. However, since effective action is defined, one can perform the coordinate transformation for the background metricḡ µν together with the corresponding transformation for the quantum metric. It is important that this transformation does not lead neither to the change of the form of the Faddeev-Popov action (17) nor to the change of the transformation rules for the auxiliary and ghost fields.
Thus, consider a variation of the background metric under general coordinates transformations of external metric tensorḡ µν , treating it as a symmetric tensor, hence The symbol (c) indicates that the transformation concerns the background metric,i.e. in the sector of classical fields.
In the quantum fields sector h µν the form of the transformations is fixed by the requirement of invariance of the action, where the symbol (q) indicates the gauge transformations in the sector of quantum fields. Then we have With these definitions, for the variation of Z(ḡ) we have Let us stress that here we consider the transformations ofḡ only, that is why the δ (q) does not enter into the last expression. Using a change of variables in the functional integral (58) one can try to arrive at the relation δ (c) ω Z(ḡ) = 0 to prove invariance of Z(ḡ) under the transformations (55). In the analysis of the gauge fixing action S gf (φ,ḡ) we can use that this action depends only on the three variables h µν , B α andḡ µν . Also, for the two of them, h µν andḡ µν , the transformation law is already defined in (55) and (56). Thus, we need to define the transformation for the remaining field B α . This unknown transformation rule δ (q) ω B α should be chosen in such a way that it compensates the variation of S gf (φ,ḡ) caused by the transformations ofḡ µν and h µν . Therefore, we have The gauge fixing functions χ α are not independent, since they are constructed from the metric, which is transformed as a tensor, according to Eq. (55). Thus the variation of the gauge fixing functions χ α has the form (7) for the vector fields, The transformation of the auxiliary field B can be chosen by the covariance arguments, following the rule (8). This gives and provides the desired relation In the same way one can analyse the variation of the ghost action and find its invariance, for the following transformation laws for the ghost fieldsC α and C α : All in all, we conclude that the Faddeev-Popov action S F P is invariant under the new version of gauge transformations, which is based on the background transformations of all fields φ andḡ including (55), (56), (61) and (64). As a consequence of (65), vacuum functional possesses gauge invariance too, The same statement is automatically valid for the background effective action, that is the effective action with switched off mean fields Φ i . As we shall see in what follows, one can use Eq. (66) to prove the gauge invariance of an important object Γ(ḡ) = Γ(Φ = 0,ḡ), that means Indeed, this relation is one of the main targets of our work. It shows that when the mean quantum fields Φ = {h, C,C, B} are switched off (later on we shall see how this should be done), the remaining effective action of the background metric is covariant. It is useful to start by exploring the gauge invariance property of generating functionals of our interest off-shell. To this end it is useful to present the background transformations (55), (56), (61) and (64) in the form where the generators R i σ (φ) are linear in the quantum fields φ and do not depend on the background metricḡ. The general form of the transformation of an arbitrary functional (let's it be Γ = Γ(φ,ḡ)) can be written in the form Consider the variation of the generating functional Z(J,ḡ) (36) , under the gauge transformations of the background metric Using the background transformations in the sector of quantum fields φ and taking into account that for the linear change of variables the Jacobian of this transformation is independent on the fields, we arrive at the relation On the other hand, from (65) and (71) follows that or In terms of the generating functional of connected Green functions, W = W (J,ḡ) = −i ln Z(J,ḡ), the relation (73) reads where we used linearity of generators R i σ (φ) with respect to φ. Once again, consider the generating functional of vertex functions (effective action), where under the variation of external metric and the mean fields (68). In terms of Γ the relation (74) becomes or, using the identity (69), simply if the variations of all variables (68) is taken into account. It is important that the relations (77) and (78) serve as a proof of the fundamental property (67). In order to see this, one has to note that the generators of quantum fields (56), (64) and (61) have linear dependence of these fields. As a result one meets the following limit for the generators R i σ (Φ) when the mean fields are switched off: Thus the effective action Γ is invariant under non-deformed background transformations and repeats the invariance property of the Faddeev-Popov action S F P . Let us come back to formulating the instruments required for the proof of renormalizability. In the renormalization program based on Batalin-Vilkovisky formalism the extended action S = S(φ, φ * ,ḡ) (28) and corresponding extended generating functionals of Green functions Z = Z(J, φ * ,ḡ), and of connected Green functions W = W (J, φ * ,ḡ), play the role of precursor for the full effective action, which satisfies the quantum version of Eq. (35). Due to the invariance of S F P under background fields transformations, the variation of S takes the special form that shows that the action is gauge invariant on the hypersurface φ * i = 0. The variations δ ω R i (φ,ḡ) are quadratic in the sector of fields h µν and C α and linear in the sector of fieldC α . Using the condensed DeWitt's notation one can write the variations of the generators δ ω R i (φ,ḡ) in the following compact form: Let us now consider the variation of the extended generating functional Z(J, φ * ,ḡ) (80) under the gauge transformations of external metricḡ, Making the change of variables φ i according to (56), (61) and (64) in the functional integral and taking into account the triviality of the corresponding Jacobian, we arrive at the relation Combining Eqs. (83) and (84) and using the gauge invariance of S F P (65) we obtain or, equivalently, In terms of the generating functional of connected Green functions W = W (J, φ * ,ḡ) the relation (86) reads where the symbol 1 means that the operator acts on the numerical unit, 1 = 1. In the case of functional derivative one has δ δφ 1 = 0, but since in many cases the expressions are non-linear, this is a useful notation.
The extended generating functional of vertex functions (extended effective action) is defined in a standard way through the Legendre transformation of W = W (J, φ * ,ḡ) introduced in Eq. (80), As usual, where we introduced a compact notation for the second variational derivatives of Γ and W . It proves useful to introduce the following notations: where the symbol (Γ ′′ −1 ) jk denotes the matrix inverse to the matrix of second derivatives of the functional Γ defined in (89), Using these notations, in terms of extended effective action the equation (87) rewrites as or, using the relation (69), in the form At this point we can draw a general conclusion from our consideration of quantum gravity theories in the background field formalism. At the non-renormalized level any covariant quantum gravity theory has the following general property: the extended quantum action S = S(φ, φ * ,ḡ) satisfies the classical master (Zinn-Justin) equation of the Batalin-Vilkovisky formalism [39,40], as we already anticipated in Eq. (35). And, moreover, the extended effective action Γ = Γ(Φ, φ * ,ḡ) also satisfies the classical master equation, The functionals S = S(φ, φ * ,ḡ) and Γ = Γ(Φ, φ * ,ḡ) are invariant under the background gauge transformations on the hypersurface φ * = 0 and, more general, satisfy the relations (81) and (93).

Gauge-invariant renormalizability
Up to now we were considering the non-renormalized generating functionals of Green functions. The next step is to prove the gauge invariant renormalizability, that is the property of renormalized generating functionals. In the framework of Batalin-Vilkovisky formalism the gauge invariant renormalizability means the preservation of basic equations (35) for the extended action S = S(φ, φ * ,ḡ) and an identical equation (94) for the extended effective action Γ = Γ(Φ, Φ * ,ḡ) after renormalization, that means (S R , S R ) = 0 and (Γ R , Γ R ) = 0.
Let us remember that the "classical" actions S and S R are nothing but zero-order approximations of the loop expansions in the parameter of the effective actions Γ and Γ R . In this sense Eq. (35) is the zero order approximation of Eq. (94) and what we have to do now is to extend these two equations to the renormalized quantities S R and Γ R . Our strategy will be to make this extension order by order in the loop expansion parameter . As a first step, consider the one-loop approximation for Γ = Γ(Φ, Φ * ,ḡ). For the uniformity of notations we use Φ * = φ * for the antifields in what follows. The effective action can be presented in the form where S = S(Φ, Φ * ,ḡ) and Γ (1) div and Γ (1) f in denote the divergent and finite parts of the one-loop approximation for Γ.
In the local models of quantum gravity the locality of the divergent part of effective action is guaranteed by the Weinberg's theorem [54] (see also [55] for an alternative proof). Furthermore, even if the starting action is nonlocal, the UV divergences may be described by local functionals, just because the high energy domain always corresponds to the short-distance limit. And in the case of UV divergences the energies are infinitely high, hence the distances should be infinitely short, that does not leave space to the non-localities. As it was argued in Refs. [3,56,57]), the UV divergent part of effective action for a wide class of models of quantum gravity is local, including the ones with a non-local classical action. Thus we assume that Γ (1) div is a local functional. Since it determines the form of the counterterms of the one-loop renormalized action the last is also a local functional. Furthermore, from the expansion of the divergent parts of Eqs. (94) and (97) up to the first order in follows that Γ (1) In the first order in we have a vanishing sum of the two terms, one of them is infinite and hence it has to vanish independent on another one. Therefore Let us consider Taking into account (35) and (100), we find the relation where E 2 is an unknown functional. Thus we have shown that S 1R satisfies the classical master equation (35) up to the terms of order 2 , The one-loop effective action Γ 1R can be constructed by adding a local counterterm to the O( ) part of Eq. (97). As usual, the counterterm has the divergent part which cancel the divergence of Γ (1) div , and the remaining contribution is finite and typically depends on the renormalization parameter µ. This contribution is not only finite, but also satisfies the same symmetries as the initial action S. Therefore the sum of (97) and the counterterm, that is Γ 1R , also satisfies the same symmetries. Since we are not interested in the dependence on µ in this work, we shall simply use (98) and assume that Γ 1R is constructed by following the procedure of quantization described above, with S replaced by S 1R .
Being constructed in this way, the functional Γ 1R is finite in the one-loop approximation and satisfies the equation Now we are in a position to make the second step. Consider the one-loop renormalized effective action in the form which takes into account the O( 2 )-terms, Here Γ 1,div and Γ 1,f in are divergent and finite O( 2 ) parts of the two-loop effective action constructed on the basis of S 1R instead of S. The divergent part Γ (2) 1,div of the two -loop approximation for Γ 1R determines the two -loop renormalization for S 2R according to and satisfies the equation As a third step consider We have found that S 2R satisfies the master equations up to the terms 3 E 3 , where 1,div ), The effective action Γ 2R is generated by replacing S 2R into functional integral instead of S. Therefore Γ 2R is automatically finite in the two-loop approximation, 2,div + Γ and satisfies the equation By applying the induction method we find that the totally renormalized action S R is given by the expression We assume that Γ (n) n−1,div and Γ (n) n−1,f in are the divergent and finite parts of the n-loop approximation for the effective action, which is already finite in (n − 1)-loop approximation, since it is constructed on the basis of the action S (n−1)R .
The action (110) satisfies the classical master equations exactly, while the renormalized effective action Γ R is finite in each order of the loop expansion in the powers of , and satisfies the analog of Slavnov-Taylor identities [58,59,60] in Yang-Mills theory (see also [61] for the pedagogical introduction), Therefore the renormalized action S R and the effective action Γ R satisfy the classical master equation and the Ward (or Slavnov-Taylor) identity, respectively.
As far as our main target is the symmetries of the renormalized effective action, the next stage of our consideration will be to generalize the transformation relations (81) and (93) for the renormalized functionals S R and Γ R . In the one-loop approximation from (93) follows that where we used condensed notations of (90). δ ωR i(1) div (Φ, Φ * ,ḡ) and δ ωR i (1) f in (Φ, Φ * ,ḡ) are divergent and finite parts of the one-loop approximation for the transformations δ ωR i (Φ, Φ * ,ḡ).
On the other hand, from (97) we have The comparison of the relations (113) and (114) tells us that From Eq. (115) and the definition (98) follows that the one-loop renormalized action S 1R = S 1R (Φ, Φ * ,ḡ) transforms according to where R i(1) The last relations mean that the action S 1R is invariant under the background gauge transformations with one-loop deformed gauge generators R i(1) R (118) on the hypersurface Φ * = 0. Furthermore, due to Eq. (117) the functional Γ 1R obeys the transformation rule where δ ωR 1,div (Φ, Φ * ,ḡ) and δ ωR 1,div and Γ (2) 1,f in (105) as Therefore the functional Γ 1R is finite in one-loop approximation and is invariant under the background gauge transformations up to the second order in on the hypersurface Φ * = 0.
Applying the induction method one can show that the renormalized functionals S R and Γ R satisfy the properties 6 where It is important that δ ωR i R defined in (123) are finite. The last observation is that in local theories the quantities δ ω R i R (122) are local due to the Weinberg's theorem [54], while in the non-local models of quantum gravity there are also strong arguments in favor of locality of divergences [3,57], including the transformations δ ω .
4 Power counting and classification of quantum gravity models Eqs. (121) show that with the antifields switched off, for Φ * = 0, the renormalized action S R and effective action Γ R are both gauge invariant quantities. In particular, this means that if we restrict the attention by the standard non-extended generating functional of the Green functions, without introducing sources for the ghosts C,C and the auxiliary field B, the effective action will be metric-dependent and generally covariant functional. This statement concerns both divergences and the finite part of renormalized effective action.
As far as we are interested in renormalizability of the theory, our main focus should be on the structure of divergences. In this case one can use the power counting arguments to classify the theories of quantum gravity to the non-renormalizable, renormalizable and superrenormalizable models. The power counting in quantum gravity is especially simple, because the metric field is dimensionless. As a result, the dimension of a Feynman diagram is divided between the internal momenta which define divergences and the external momenta, that define the number of metric derivatives in the counterterms.
In what follows we consider diagrams with n vertices, l int internal lines and p loops. It is easy to verify that these three quantities satisfy the topological relation Another relation links the superficial degree of divergence D of the diagram and the total number of momenta external lines of the diagram d with the power of momenta in the inverse propagator of internal line r l and the number of vertices K ν with ν momenta. The formula of our interest is [1] As the first example, let us see how these two formulas work for the quantum gravity based on general relativity. In the theory without cosmological constant we have r l = 2 and K 2 = n. Replacing these numbers into (127) and using (126) we arrive at For the logarithmic divergences D = 0 and we discover that the dimension of covariant counterterms grows with the number of loops as d = 4+2p. The theory is obviously non-renormalizable.
In the presence of cosmological constant the quantity d becomes smaller d = 4 + 2p − 2K 0 with each vertex without derivatives, and the loss of dimension is compensated by the powers of the cosmological constant. The results of the previous section and locality of divergences enable one to use the quantity of d to write down all possible counterterms in any loop order p. For p = 1 there are O(R 2 ... ) and R type divergences [63], for p = 2 we meet O(R 3 ... ) [64,65], etc.
The next example is the fourth derivative quantum gravity [1]. In this case one can modify the definition of ghost action in such a way that r l = 4 for both metric and ghost propagators. Also, there are vertices with four K 4 , two K 2 and zero K 0 derivatives. Combining (127) and (126) it is easy to get The results of the previous section (for this theory the renormalizability was originally demonstrated in [1]) show that the divergences are covariant. Since they are also local, this means that if we include all terms of dimension four into the classical action, then the divergences will repeat the form of the classical action. Thus, such a theory is multiplicatively renormalizable. In Eq. (130) we used the standard (in quantum gravity) basis for the four derivative terms, with C 2 being the square of the Weyl tensor and E 4 is the integrand of the Gauss-Bonnet topological invariant, The next example of our interest is the model (3) with functions Π 1 (x), Π 2 (x) and Π 3 (x) being polynomials of the same order k ≥ 1 [2], Π 1,2,3 (x) = a 1,2,3 0 x k + a 1,2,3 1 x k−1 + · · · + a 1,2,3 k−1 x + a 1,2,3 The terms with Π 1,2,3 (x) have at most 2k + 4 derivatives of the metric. The terms + O R 3 ...
should satisfy the same restriction on the number of derivatives. Then we have r l = 2k + 4 and the maximal number of derivatives in the vertices is also ν = 2k + 4. If we are interested in the diagrams with the strongest divergences, K 4k+4 = n. Once again, combining (127) and (126) for the maximally divergent diagrams it is easy to arrive at the result This formula shows that for the logarithmic divergences at the one-loop order p = 1 and we have d = 4. Taking the covariance and locality arguments into account, the one-loop divergences repeat the form of the four-derivative action (130). Thus, the theory (3) can be renormalizable only if the coefficients a 1,2,3 k in Eq. (133) are all non-zero, and the Einstein-Hilbert action with the cosmological constant is also included.
In case of k ≥ 3 Eq. (134) tells us that there are no divergences beyond the first loop. For k = 2 we have only the cosmological constant divergences at two loop order. Finally, in the case of k = 1 there are cosmological constant -type divergences at three loop order and linear in R divergences at two loops. Obviously, the theory is superrenormalizable. Let us stress that in this case we have locality guaranteed due to the Weinber's theorem and covariance holds since we proved it in the previous section.
Finally, let us consider an example of the non-local gravity. The main proposal of this kind of models is to avoid the presence of higher derivative massive ghost in the spectrum of tree-level theory while keeping the theory renormalizable [4,5,3]. The general analysis of how the freedom from ghosts can be achieved can be found in [3,57,56] and we will not repeat this part, since our purpose here is the study of renormalization. It is sufficient for us to give an example of the theory which satisfies the ghost-free condition. The typical Euclidean space propagator in such a theory has the form Since gravity action is always non-polynomial, this structure of propagator means that the vertices have the UV behavior which is at least proportional to The proof of the gauge-invariant renormalizability which we achieved in Sec. 3 is based only on the hypothesis of diffeomorphism invariance of the classical action. Therefore it is perfectly well applicable to the non-local models. Thus, the question of whether these theories are renormalizable depend only on power counting and locality of divergences. The power counting in this case represents a serious problem, because the expression (127) boils down to the indefinite difference of the ∞ − ∞ type. However, there is a solution [6], which is based on the topological relation (126). It is clear from Eqs. (135) and (136) that the diagrams with l int > n will be convergent, while those with l int < n will be strongly (to say the least) divergent. Thus the logarithmic divergences will be the maximal ones only if l int = n, that gives p = 1. This means that all diagrams beyond one-loop order are finite (except one-loop sub-diagrams, as usual). Furthermore, in the one-loop case all exponentials cancel out and the diagram has divergences which are of the same order as in the quantum GR. Taking covariance of divergences into account, this means that the one-loop divergences are of the four-derivative type (130).
There are two consequences of the power counting which we have described. The first is that the exponential non-local model has the power counting which is exactly the same as the polynomial model (3), (133) with k ≥ 3. In other words, such a theory is superrenormalizable by power counting. However, the theory which is free from ghosts and has one-loop divergences cannot be even renormalizable, because all the coefficients of four-derivative terms should be precisely fine-tuned to provide the structure of the propagator (135) required for absence of ghosts. The problem can be alleviated by introducing a specially fine tuned O R 3 ... terms called "killers" [57] (see also earlier discussion in [2] for the polynomial models). These terms can make the theory finite, but still do not guarantee the ghost-free structure in the dressed propagator [6]. All in all, the non-local ghost-free models meet the problem of absolutely precise fine-tuning, which can not be maintained upon (even finite) renormalization, even if the theory is superrenormalizable. Together with the problem is physical unitarity [7] this situation makes nonlocal theories less prospective, but of course they still remain very interesting models to study. Let us note that in the polynomial models (3), (133) there are no problems with locality of divergent parts of effective action, and hence the proof of gauge invariant renormalizability can be used to give solid background to the power counting arguments.

Conclusions
We described in details the general proof of that the diffeomorphism invariance can be maintained in quantum gravity theories. The main advantages of the approach of the present paper is related to the explicit form of variation of extended effective action under the gauge transformations of all fields appearing in the background field formalism. The derived form of these variations can be applied to an arbitrary gravity theory which respects diffeomorphism invariance. The variation has a very special form, providing an exact invariance of the effective action when the antifields (sources for the BRST generators) are switched off.
After switching off the mean field of quantum metric, Faddeev-Popov ghosts, auxiliary field and antifields, the divergent part of effective action possess general covariance, and this important property holds in all orders of the perturbative loop expansion. This statement is proved correct for generic models of quantum gravity, including the ones with higher derivatives and even with certain (phenomenologically interesting) models with non-localities. Starting from covariance and using power counting and locality of the counterterms one can easily classify the models of quantum gravity into non-renormalizable, renormalizable and superrenormalizable versions.
On the other hand, we have extended the usual statement concerning the gauge invariance of the background effective action up to the gauge invariance of effective action depending on the mean quantum fields. Furthermore, we extended all mentioned results from the nonrenormalizable effective action to renormalized one. The gauge invariance of renormalized extended effective under the renormalized finite gauge transformations has been proved on the hypersurface of switched off antifields. An important consequence of the last result is the gauge invariance of renormalized background effective action under deformed gauge transformations of background metric for any covariant quantum gravity theory.
It is tempting to extend the results that are achieved in this work to the non-perturbative domain. Unfortunately this can not be done for the standard versions of average effective action, since the last does not admit the consistent on-shell limit in the case of gauge fields. In this respect the most promising is the new version of functional renormalization group which is based on the composite fields, as introduced in [48] for the Yang-Mills fields. However, for this end one has to extend this new scheme to quantum gravity and, most difficult, to learn how it can be used for making practical calculations. As a reward we can hope to get a consistent non-perturbative treatment of not only vector gauge fields, but also gravity.