On the Vilkovisky-DeWitt approach and renormalization group in effective quantum gravity
aa r X i v : . [ h e p - t h ] S e p On the Vilkovisky–DeWitt approach and renormalization groupin effective quantum gravity
Breno L. Giacchini, a ∗ Tib´erio de Paula Netto a † and Ilya L. Shapiro b ‡ (a) Department of Physics, Southern University of Science and Technology,Shenzhen, 518055, China(b) Departamento de F´ısica, ICE, Universidade Federal de Juiz de ForaJuiz de Fora, 36036-900, MG, BrazilABSTRACTThe effective action in quantum general relativity is strongly dependent on the gauge-fixing and parametrization of the quantum metric. As a consequence, in the effectiveapproach to quantum gravity, there is no possibility to introduce the renormalization-group framework in a consistent way. On the other hand, the version of effectiveaction proposed by Vilkovisky and DeWitt does not depend on the gauge-fixing andparametrization off-shell, opening the way to explore the running of the cosmologicaland Newton constants as well as the coefficients of the higher-derivative terms of thetotal action. We argue that in the effective framework the one-loop beta functionsfor the zero-, two- and four-derivative terms can be regarded as exact, that means,free from corrections coming from the higher loops. In this perspective, the runningdescribes the renormalization group flow between the present-day Hubble scale in theIR and the Planck scale in the UV. Keywords:
Unique effective action, renormalization group, one-loop divergences,quantum gravity
The effective action in quantum field theory can be used for deriving the S -matrix or otherphysically relevant quantities. In the conventional approach, the effective action has fundamentalambiguities related to the choice of parametrization of quantum field or, in gauge theories, tothe choice of the gauge-fixing condition. These ambiguities vanish on-shell, which enables oneto formulate in a consistent manner the renormalizable theory. However, this situation createsserious difficulties in the effective theory. For instance, the renormalization group framework has ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ On leave from Tomsk State Pedagogical University. E-mail address: shapiro@fisica.ufjf.br o be constructed based on the off-shell effective action. In many cases this means that there isno way to use the renormalization group to explore the running of the relevant parameters withscale in some physical situations.The quantum version of general relativity is non-renormalizable, but it is perfectly appropriatefor the effective theory approach. The reason is that the graviton is a massless field, and the nextphysical degrees of freedom (coming from higher derivatives) have masses of the Planck orderof magnitude M P ≈ GeV [1]. However, in the case of quantum general relativity, the one-loop and higher-loop divergences are strongly dependent on the gauge-fixing and parametrizationof the quantum metric. Owed to this, the beta functions for all relevant parameters are badlydefined and there is no chance to extract the unambiguous running from the effective low-energyquantum gravity.The on-shell conditions in the non-renormalizable theory can hardly be implemented, espe-cially at the non-perturbative level. Thus, it is difficult to obtain the relevant information fromthe quantum theory. On the other hand, there is an alternative definition of the effective ac-tion, introduced by Vilkovisky [2] and DeWitt [3], based on the covariant formulation in thespace of the quantum gauge fields. The Vilkovisky–DeWitt unique effective action is gauge- andparametrization-independent, paving the way to explore in a consistent manner the running ofthe parameters of the total action, including the cosmological and Newton constants and thecoefficients of the higher-derivative terms.The gauge-fixing independence of the Vilkovisky–DeWitt effective action has been proved ina general setting and was also confirmed by one-loop direct calculations [4–10]. Recently itsuniversality has been also confirmed by an explicit calculation in an arbitrary parametrizationwithin quantum gravity [11]. Let us stress that this verification is especially relevant in the caseof conformal parametrization of the metric. One of the reasons is that there is an exception inthe general proof of universality, which is mentioned in the pioneer work [2]. It is known that theVilkovisky–DeWitt effective action depends on the choice of the configuration space metric [10].However, in the two-dimensional (2 D ) quantum gravity this metric may depend on the choice ofthe gauge-fixing and therefore the unique effective action becomes gauge-fixing dependent [12].It was shown in [11] that this effect does not take place in the four-dimensional (4 D ) quantumgravity, regardless of the algebraic similarity with the 2 D case in the conformal parametrization.The important detail is that the configuration-space metric has to be chosen as the bilinear formof the classical action in the minimal gauge, in a given parametrization of the quantum metric. Inwhat follows we shall base all considerations on this assumption, which provides the universalityof the Vilkovisky–DeWitt effective action.In the present work, we shall follow the previous publication on the subject [13] and considerthe renormalization group equations in the effective quantum gravity based on general relativityin the unique effective action formalism. Indeed, we expand the analysis performed in this seminalwork in several directions. First of all, we stress the importance of the effective approach anddiscuss in more detail the corresponding area of application for the running of the parameters.2econd, we extend the analysis to the higher-derivative part of the action, which is renormalizedin a way similar to that of the semiclassical gravity. Third, treating the renormalization group inthe effective framework, we argue that the low-energy one-loop running of the Newton constant G , cosmological constant Λ and the parameters of the higher-derivative part of the action is, infact, non-perturbative and the corresponding beta functions can be viewed as exact .The manuscript is organised as follows. In the next Sec. 2 we briefly describe the generalframework of the unique effective action in quantum general relativity. The reader can consult theparallel paper [11] for further details. In Sec. 3 we construct, solve, and discuss the renormalizationgroup equations for the Newton and cosmological constants in the framework of effective quantumgravity. In Sec. 4 we extend the analysis to the higher-derivative sector of the theory and thediscuss the non-perturbative aspects of the corresponding running. The perspectives of physicalapplications of the exact effective running are briefly described in Sec. 5. Finally, in Sec. 6, wedraw our conclusions. Let us start by formulating general definitions, valid for any gauge field theory, which wesubsequently particularise for quantum gravity. As mentioned before, off the mass shell theeffective action depends on the field parametrization. This can be readily seen by recalling thatin the one-loop approximation the effective action depends on Hessian of the classical action. Infact, while the action S ( ϕ ) is a scalar in the space of fields ϕ i , its second functional derivativedoes not transform as a tensor. The dependence on the gauge-fixing can be understood in asimilar manner, since the effective actions calculated in different gauges are related by changes ofvariables in the form of canonical transformations [14–16].The problem can be addressed in a geometric framework via the introduction of a metric ¯ G ij and a connection T kij in the configuration space of physical fields. Accordingly, the definitionof the effective action should be modified, so that it is constructed only with scalar quantities.This programme was carried out for the first time by Vilkovisky [2], who introduced the uniqueeffective action Γ( ϕ ) throughexp i Γ( ϕ ) = Z D ϕ ′ µ ( ϕ ′ ) exp (cid:8) i (cid:2) S ( ϕ ′ ) + σ i ( ϕ, ϕ ′ )Γ ,i ( ϕ ) (cid:3)(cid:9) , (1)where σ i ( ϕ, ϕ ′ ) is the derivative (with respect to ϕ i ) of the world function [18, 19] constructedwith the connection T kij , and µ ( ϕ ′ ) is an invariant functional measure. Since σ i ( ϕ, ϕ ′ ) behavesas a vector with respect to ϕ i and as a scalar with regard to ϕ ′ i , the effective action Γ( ϕ ) in (1)is reparametrization invariant and gauge independent. We greatly appreciate the contribution of the referee of the first version of the paper [11], who gave us avaluable hint in this direction and advised to proceed this discussion independently of the parametrization-relatedcalculations. See Ref. [17] for an earlier attempt towards this in the context of non-linear σ -models. G ij in the space M of fields and the generators R iα of gauge transformations. Asour main concern here is on gravity theory, we assume that the field ϕ i is bosonic, and that thegenerators R iα are linearly independent and form a closed algebra, with structure functions whichdo not depend on the fields. Introducing the metric N αβ on the gauge group G and its inverse N αβ , N αβ = R iα G ij R jβ , N αλ N λβ = δ βα , (2)one can define the projector on M / G [2, 5], P ij = δ ij − R iα N αβ R kβ G kj . (3)The projected metric in the space M / G of physical fields is, therefore,¯ G ij = P ki G kℓ P ℓj = G ij − G ik R kα N αβ R ℓβ G ℓj . (4)Since N αβ is the inverse of a differential operator, this metric contains a non-local part whicharises from the constraints imposed by the gauge symmetry.The connection T kij can be obtained by requiring its compatibility with ¯ G ij (see e.g. [6, 10]),and it can be written in the form [2] T kij = Γ kij + T kij , (5)where Γ kij are the (local) Christoffel symbols associated to the metric G ij and T kij = − G ( i | ℓ R ℓα N αβ D | j ) R kβ + G ( i | ℓ R ℓα N αβ R mβ ( D m R kγ ) N γδ R nδ G n | j ) (6)is its non-local part. The parenthesis in the indices denote symmetrization in the pair ( i, j ) and D i indicates the covariant derivative based on the Christoffel connection Γ kij .One can proceed the loop expansion of the Vilkovisky effective action (1),Γ( ϕ ) = S ( ϕ ) + ¯Γ (1) ( ϕ ) + ¯Γ (2) ( ϕ ) + · · · , ~ = 1 , (7)which gives the one-loop contribution [2]¯Γ (1) = i G ik (cid:0) D k D j S − T ℓkj S ,ℓ − χ α,k Y αβ χ β,j (cid:1) − i Tr ln M αβ , (8)where χ α is a gauge-fixing condition, Y αβ is the weight functional and M αβ = χ α,i R iβ is the Faddeev–Popov ghost matrix. Compared to the standard effective action, the Hessian of the classicalaction has now been replaced by its covariant version, ensuring its tensor nature concerning fieldreparametrizations. Notice also that the non-local part of the connection (6) behaves as a tensoras well. The divergent part of the one-loop effective action (8) can be evaluated, e.g. , by applyingthe generalized Schwinger–DeWitt technique of Ref. [4].4evertheless, the effective action defined by (1) cannot be viewed as a final solution aimingto the off-shell universality because of two reasons. First, it might happen that the metric G ij is not uniquely defined. This issue can be solved by additional prescriptions [2], as we shallcomment later. The most serious obstacle, however, is the lack of one-particle irreducibility ofthe diagrams generated by (1), which may take place beyond one-loop. Indeed, it gives rise tonon-local divergences at the two-loop approximation in Yang–Mills theories, as shown in Ref. [7].The modification of the Vilkovisky effective action (1) proposed by DeWitt [3] can be viewedas a way out of this difficulty, inasmuch as it restores the one-particle irreducibility of the per-turbative expansion [7, 8] (see also [9] for further discussion). For example, in [7] it was verifiedby explicit calculations that the aforementioned non-local divergences which appeared in originalformulation (1) in the two-loop Yang–Mills theory are cancelled in the DeWitt approach [3].The construction introduced by DeWitt [3], usually called Vilkovisky–DeWitt effective action,consists in choosing an arbitrary point ϕ i ∗ in the configuration space and, instead of simplydefining the vector-scalar quantity σ i ( ϕ, ϕ ′ ) in terms of the mean field, one builds a system ofGaussian normal coordinates with ϕ i ∗ , according to which the covariant Taylor expansions shouldbe performed. The method involves defining an effective action and a mean field which dependon ϕ i ∗ , and only in the end, after the implicit equation is solved iteratively, this point is identifiedto the mean field. At one-loop level, the Vilkovisky–DeWitt effective action coincides to theVilkovisky one (1), therefore the Eq. (8) also holds in this more general formalism. Since for mostof the discussion in the present paper we work with one-loop results, we shall not present furthertechnical details of the Vilkovisky–DeWitt effective action. The important point here is to stressthat the formalism guarantees the one-particle irreducibility of the diagrammatic expansion.The remaining question to deal with is the aforementioned dependence of the unique effectiveaction and, in particular, its one-loop part (8), on the choice of the metric in the space of thefields. This issue is especially relevant for quantum gravity, where there is a one-parameter familyof such metrics, characterised by the parameter ¯ a = − /
4, given by [20] G µν,αβ = 12 ( δ µν,αβ + ¯ a g µν g αβ ) , where δ µν,αβ = 12 ( g µα g νβ + g µβ g να ) . (9)It was shown by explicit calculations [10] (see also [21, 22]) that the Vilkovisky–DeWitt effectiveaction depends on the choice of ¯ a . Nonetheless, already in Ref. [2] it was introduced a prescriptionto fix this ambiguity. In accordance, the field-space metric should coincide with the expressionin the highest-derivative term in the bilinear part of the classical action in the minimal gaugefixing. In the parallel paper [11] we have shown that this prescription works perfectly well evenunder changes of the parametrization of the quantum field, which also modifies the parameter ¯ a .Thus, it defines a unique off-shell effective action.The classical action of general relativity has the form S = − κ Z d x √− g ( R + 2Λ) , (10)5here κ = 16 πG and G is the Newton constant. In the effective approach, the quantum theorytakes into account only the massless modes of the quantum metric [1, 23]. For this reason, thesequantum effects are completely defined by the action (10), regardless of the presence of higher-derivative terms (to be defined below) in the full gravitational action.The one-loop divergent part of the Vilkovisky–DeWitt effective action for the Einstein gravitywas evaluated for the first time in Ref. [4], while the terms related to the cosmological constantwere calculated in [5]. We shall skip all the calculations and refer the reader to the mentionedpapers for the details. The result for the one-loop divergences is¯Γ (1)div = − µ n − ǫ Z d n x √− g (cid:26) C − E + 3136 R + 8Λ R + 12Λ (cid:27) . (11)where ǫ = (4 π ) ( n −
4) is the parameter of dimensional regularization and µ is the renormal-ization parameter. Also, C = R µναβ − R µν + R denotes the square of the Weyl tensor and E = R µναβ − R µν + R is the integrand of the Gauss–Bonnet invariant in the four-dimensionalspacetime. On the classical mass shell, Eq. (11) reduces to the divergent on-shell part of the usualeffective action [24, 25]¯Γ (1)div (cid:12)(cid:12) on-shell = − µ n − ǫ Z d n x √− g (cid:26) E −
585 Λ (cid:27) . (12)This expression is also gauge-fixing and parametrization independent (see e.g. [26] and referencestherein), but the advantage of the result (11) is that it is universal even off-shell. This featureopens the way for formulating consistent low-energy renormalization group equations for Λ, G [13],and for other parameters of the action, which were not included in the basic formula (10).Before proceeding to the renormalization group and the effective approach to quantum gravity,let us briefly review the power counting in the quantum theory based on general relativity. Inthis theory, the propagator behaves like k − and there are two kinds of vertices: the ones withtwo derivatives, owed to the Einstein–Hilbert term, and those with no derivative, coming withcoefficient Λ. The coupling constant (parameter of the loop expansion) is κ , with dimension ofinverse of mass-squared. Since the quantum metric h µν = g µν − η µν is dimensionless, the powercounting is especially simple. For a given p -loop diagram with n vertices with two momenta and n vertices with zero momenta, the superficial degree of divergence is ω = 2 p − n + 2 − d, (13)where d is the number of derivatives acting on the external lines of h µν . This relation defines thenumber of derivatives d = 2 p − n + 2 for the logarithmically divergent diagrams with ω = 0.It is easy to see that the expression (11) satisfies this condition; the O (Λ)-terms with n = 1 areproportional to R , that means d = 2, while O (Λ )-terms with n = 2 have d = 0.Both at the one-loop level and in higher loops, the logarithmically divergent diagrams with n = 0 satisfy the condition d = 2 p + 2, which means that the maximal number of derivativesin the counterterms grows linearly with the number of loops p . In particular, for Λ = 0 the6our-derivative terms in the one-loop formula (11) are actually exact, since they do not gainhigher-loop contributions. The same concerns the O ( R ... )-type terms at the two-loop order, andso on.On the other hand, in the real world, these terms are practically exact even if Λ = 0. Thereason is that the four-derivative terms gain p -loop contributions with coefficients proportional toΛ κ = Λ M P to the power p . In the present-day Universe, this coefficient is of the order of 10 − ,which is small enough to support the argument that the result can be regarded as exact. It isa direct exercise to extend this statement also to the lower-derivative terms in (11). We shallcome back to this reasoning and use it intensively in the next two sections when discussing therenormalization group. One can use the result (11) for analyzing the renormalization group equations in the low-energy (infrared, IR) sectors of the theory. Such a construction has a direct physical sense. Inthe high-energy domain (UV) the theory (10) cannot be applied without restrictions, as it is non-renormalizable and the contributions of massive degrees of freedom, related to higher derivativeterms, are supposed to modify the beta functions. However, since the quantum gravity basedon general relativity is a massless theory, it makes sense to explore the renormalization grouprunning in the IR. Assuming that the higher-derivative massive degrees of freedom have massesof the Planck order of magnitude, in most of the physically relevant situations these modesdecouple [27] (see also the concrete discussion of this issue in the semiclassical gravity [28,29] andqualitative discussion in quantum gravity [30–32]), such that the running is completely definedby the action (10).In other words, since the theory is massless, the quantum gravity based on general relativitycan be regarded as an effective theory of quantum gravity at the energies between the UV (Planck)scale, where the massive degrees of freedom coming from higher derivatives can become relevant,and the deep IR scale. Thus, the Vilkovisky–DeWitt unique effective action enables one toexplore the scale dependence in this vast region in a gauge-fixing and parametrization independentmanner.From the classical action (10) and the expression for the divergences (11), it is easy to obtainthe renormalization relations1 κ = µ n − h κ − π ) ( n −
4) Λ i , Λ = Λ h π ) ( n −
4) Λ κ i . (14)The bare quantities κ and Λ are µ -independent, as it is the case for the renormalized effectiveaction. Applying the operator µ dd µ to both sides of each of the relations (14), after a small algebra7e arrive at the renormalization group equations µ dd µ κ = 8Λ(4 π ) , (15) µ dΛd µ = − κ (4 π ) , (16)which are equivalent to those obtained in [13, 22].To solve Eqs. (15) and (16), we define the dimensionless quantity γ = κ Λ. Due to theuniqueness of this dimensionless combination of κ and Λ, the equation for γ gets factorized, µ d γ d µ = − γ (4 π ) . (17)The solution of this equation has the standard form γ ( µ ) = γ π ) γ ln µµ , (18)where γ = γ ( µ ) and µ marks a fiducial energy scale. We assume the initial values of therenormalization group trajectories of the cosmological constant Λ = Λ( µ ) and the gravitationalconstant G = G ( µ ) as it is useful to come back from κ to G at this stage.Now, using (18) in (15) and (16), we obtain the final solutions G ( µ ) = G (cid:2) π ) γ ln µµ (cid:3) / (19)and Λ( µ ) = Λ (cid:2) π ) γ ln µµ (cid:3) / , (20)which are certainly consistent with (18).The solutions (19) and (20) are remarkable in several aspects. First of all, such independentsolutions for the two effective charges are impossible in quantum gravity based on the usualeffective action neither in quantum general relativity nor the fourth-derivative gravity, as theindividual equations for G ( µ ) and Λ( µ ) are completely ambiguous. In the latter model, only thesolution for the dimensionless quantity in (18) is gauge-fixing and parametrization independent .Here we have a well-defined running for the two parameters only because of the use of theVilkovisky–DeWitt effective action.Let us note that the unambiguous solutions for G ( µ ) and Λ( µ ) exist in the superrenormalizablegravity model [31], but there are two relevant differences. The advantage of the equations andsolutions of [31] is that those can be exact, in the sense of not depending on the order of the In quantum Einstein gravity based on the usual effective action, on the other hand, only by using the on-shellversion of renormalization group it is possible to define an unambiguous equation for γ [27]. i.e. , only in the trans-Planckian region. Below the Planck scalethe massive degrees of freedom decouple and we are left with the quantum effects of effectivequantum gravity, such as the ones of quantum general relativity (see e.g. [1], the review [23] andthe recent discussion of the decoupling in gravity in [29, 32]).On the contrary, the running described by (19) and (20) comes from the quantum effects ofthe purely massless degrees of freedom. Up to some extent, the running should be described bythe same equations in both UV and IR. According to the general discussion which we postponefor the next section, these equations can be seen as exact, being valid in the same form evenbeyond the one-loop approximation.It is clear that the physical interpretation of the solutions (19) and (20) depends on the signof γ . Since the positive sign of G is fixed by the positive definiteness of the theory, the sign of γ depends on the one of Λ . Due to the cosmological observations, we know that the sign ofthe observed cosmological constant is positive in the present-day Universe [33, 34]. For a positive γ the solutions (19) and (20) indicate the asymptotic freedom in the UV. In case of a moderatecosmological constant (remember κ ∝ M − P ) the value of γ is very small. This implies a veryweak running, that is irrelevant from the physical viewpoint. In particular, the running (19)and (20) is not essential for the cosmological constant problem between the electroweak scale andthe present day, low-energy, cosmic scale.On the other hand, at the electroweak energy scale, the early Universe probably passedthrough the corresponding phase transition. At that epoch, the observable value of the cos-mological constant could dramatically change because of the symmetry restoration. Does thischange Λ in the action (10)? The answer to this question is negative. Let us remember that theobservable cosmological constant is a sum of the two parts: one is the vacuum parameter in thegravitational action (10) and another is the induced counterpart, the main part of it coming fromthe symmetry breaking of the Higgs potential. The main relations are (see, e.g. , [35] or [36]) ρ obs Λ = ρ ind Λ + ρ vac Λ , ρ ind Λ = Λ ind πG ind = − λv , (21)where λ is the self-coupling and v the vacuum expectation value of the Higgs field. As far as ρ ind Λ is negative and the magnitude of ρ obs Λ is negligible, the sign of ρ vac Λ = κ is positive, independentlyof the electroweak phase transition.Thus, we conclude that the sign of γ is always positive, at least between the present-daycosmic scale in the IR and the GUT scale in the UV, where the considerations based on theMinimal Standard Model formulas, such as (21), may become invalid. In all this interval, thevalue of γ is numerically small, such that the running in (19) and (20) is not physically relevant.One can imagine a situation in which another phase transition occurs at the GUT scale (thatmeans about 10 –10 GeV), such that the new vacuum Λ between this scale and the Planck9cale M P ≈ GeV is negative. Then, the solutions (19) and (20) indicate the asymptoticfreedom in the IR. Furthermore, if the cosmological constant in this energy scale interval has theorder of magnitude of M P , these solutions describe the situation of a dramatically strong runningof both constants G and Λ, which are strongly decreasing in the IR. As we have learned in theprevious Sec. 2, for the values satisfying | Λ | ≪ M P , the higher-loop contributions cannot modifythe form of the running. In any case, the construction of the corresponding model of GUT wouldbe an interesting subject to work on in future. Here we just want to note that our results indicatethis possibility. In order to complete the discussion, let us consider the renormalization group equations forthe fourth-derivative terms in the action of gravity. To this end, we have to complement theaction (10) with at least all those terms which are present in the expression for the divergences(11). According to the power counting (13), at p -loop order it is necessary to introduce into theaction terms with up to 2 p + 2 derivatives of the metric. In this way we arrive at the well-knownaction of the higher-derivative quantum gravity [37], S tot = Z d x √− g n − κ (cid:0) R + 2Λ (cid:1) − λ C + 12 ρ E − ξ R + 12 ζ C µναβ C αβ · · ρσ C ρσµν + N X n =1 h ω n,C C µναβ (cid:3) n C µναβ + ω n,R R (cid:3) n R i + O ( R ... ) o , (22)where λ , ρ and ξ are the dimensionless parameters of the action and N = p −
1. The terms withmore than four derivatives which contribute to the propagator of the quantum metric have theforms R (cid:3) n R and C µναβ (cid:3) n C µναβ = R µναβ (cid:3) n R µναβ − R µν (cid:3) n R µν + 13 R (cid:3) n R. (23)One could also include the curvature-squared higher-derivative terms of the typeGB n = R µναβ (cid:3) n R µναβ − R µν (cid:3) n R µν + R (cid:3) n R (24)which represent the extended version of the four-dimensional Gauss–Bonnet topological invariant.Of course, these terms are not topological for n >
1, but can be shown (see e.g. [37]) to be O ( R ... )and therefore they do not contribute to the propagator. For the sake of simplicity we assumethat the sum in (22) is finite, as otherwise we arrive at the non-local actions of quantum gravity(see, e.g. , [38] for a review). In such a case, the structure of massive poles of the propagator whenloop effects are taken into account is more complicated [39] and is not relevant for the presentdiscussion. Still regarding Eq. (22), we point out that we separated one of the possible Weyl-cubicterms C ... from other terms of third and higher-order in curvatures, because in what follows weshall use it to discuss the two-loop effective low-energy beta function for the parameter ζ .10n the polynomial theories (22), the propagator can have real massive poles [37] or complexones [40], but in both cases the natural situation is that all these massive parameters have thePlanck order of magnitude [41]. Thus, in the effective approach, below the Planck scale we cancompletely ignore the quantum contributions of these massive degrees of freedom. The quantumeffects are coming only from the massless mode, associated to the Einstein–Hilbert action (10).The expression (22) includes the action of the fourth-derivative gravity [42], S four = Z d x √− g (cid:26) − λ C + 12 ρ E − ξ R − κ (cid:0) R + 2Λ (cid:1)(cid:27) , (25)as a particular case. At the one-loop level, the power counting shows that only the terms up to thefour-derivative part of the action (25) gains divergent contributions and, correspondingly, receivesthe logarithmic non-local corrections. Thus, we shall consider in details the beta functions andrenormalization group equations for the remaining parameters in this sector of the total action.The renormalization group equations for λ , ρ , and ξ were previously explored in the frame-work of the semiclassical theory, starting in [43] (see [44,45] for a formal consideration and furtherreferences), and higher-derivative quantum gravity [27, 30, 46]. In the effective approach to quan-tum gravity based on the standard effective action one can determine only the equation for ρ since the corresponding divergence survives on-shell, see Eq. (12), being unambiguous. On theother hand, the universality of the Vilkovisky–DeWitt effective action [2, 3] makes it possible thenew version of the renormalization group equations for the parameters λ and ξ . To this end, oneapplies the standard algorithm to the Eqs. (11) and (25), from which it follows the beta functions β λ = − a QG (4 π ) λ , a QG = 12130 , (26) β ξ = − b QG (4 π ) ξ , b QG = 3118 , (27) β ρ = − c QG (4 π ) ρ , c QG = 15190 . (28)We have to define the lowest possible IR scale. In flat spacetime, the running produced bythe quantum effects of the massless fields can be considered to occur for arbitrarily low energies.However, in the real applications (even in the low-energy cosmology) there is a natural IR cut-off,as it was described in Ref. [47]. In order to understand the origin of this cut-off, let us rememberthat the running of the fourth-derivative terms is related to the logarithmic form factors [48]. Forthe Weyl-squared term, for example, the corresponding term in the effective action reads a (4 π ) Z d x √− g C µναβ ln (cid:16) − ✷ µ (cid:17) C µναβ , (29)with a defined in (31) below; while the corresponding d’Alembert operator for weak perturbationsaround the cosmological (isotropic and homogeneous) background has the form (see [47] for thedetails) ✷ = ∂ t − H∂ t − H − H + . . . , (30)11here H is the Hubble parameter. The expression (30) shows that in the far future of theUniverse, with the background becoming close to the de Sitter space, there will not be physicalrunning of λ , because the theory is effectively massive. The cut-off (fictitious mass) parameteris defined by the relation H ∼ p Λ /
3. Numerically, this means that the running ends in the IRat the scale of the order H ≈ − GeV. Between this scale and the intermediate scale definedby the neutrino masses (presumably of the order m ν ≈ − GeV) the running of λ , ξ and ρ isdefined by the contributions of effective quantum gravity (26), (27) and (28) and the ones of thephoton. Starting from the neutrino scale, we have to include fermion contributions. Thus, therenormalization group equation for λ is µ dλdµ = − a (4 π ) λ , a = a QG + 15 + N f , (31)where N f is the number of fermions. The solution of this equation has the usual form λ ( µ ) = λ a (4 π ) λ ln µµ , λ = λ ( µ ) . (32)The remaining equations for ξ and ρ are µ dξdµ = − b (4 π ) ξ , b = b QG , (33) µ dρdµ = − c (4 π ) ρ , c = c QG + 3190 + 11180 N f . (34)It is worth noting that the photon and fermion contributions to b are ruled out due to theconformal invariance of these two fields in the massless versions. Another interesting point isthat the contributions of effective quantum gravity to the equations for λ and ρ have the samesign of the ones related to vector and fermion fields. We remark that the same sign pattern alsotakes place in the scalar field theory, in fourth-derivative quantum gravity [27] (see also [30, 46]for a verification) and conformal quantum gravity [27, 49, 50]. This universality of signs probablymeans there are some general rules for the quantum corrections which we do not understand yet.The solutions of Eqs. (33) and (34) have the form ξ ( µ ) = ξ b (4 π ) ξ ln µµ , ξ = ξ ( µ ) . (35)and ρ ( µ ) = ρ c (4 π ) ρ ln µµ , ρ = ρ ( µ ) . (36)Let us make an important observation based on the discussion in Sec. 2. Since the theoryis not renormalizable by power counting, the dimensional arguments show that the higher-loopcontributions to the beta functions for the parameters λ , ρ and ξ are possible only for the non-zero12osmological constant. In this case, because the coupling (parameter of the loop expansion) inthe theory (10) is κ ∼ M − P , the higher-loop corrections to the fourth-derivative terms are givenby power series in the parameter Λ M P ∼ − . Thus, in the real physical situations the higher-loop corrections for the dimensionless parameters λ , ρ and ξ are negligible. This is true at leastuntil the UV energy scale defined by the electroweak phase transition. At a higher energy scale M , the value of the induced cosmological constant density (see the discussion in the previoussection) is ρ Λ ∝ M , such that Λ ≈ M M P . Then the dimensionless parameter of expansion inloops, in the framework of effective theory, is defined by the value of the ratio Λ M P ≈ M M P . For theelectroweak phase transition, M ≈
300 GeV and this parameter is about 10 − . Assuming anotherphase transition at the GUT scale, we meet M = M X = 10 –10 GeV and the dimensionlessparameter varies between 10 − and 10 − . All these values are certainly sufficient to claim thedominance of the one-loop effects. Therefore, the running which we have just derived, based onthe effective approach to quantum general relativity within the Vilkovisky–DeWitt formalism,can be safely regarded as the exact, nonperturbative effect. Indeed, the same also concerns therunning of Λ and G , which were discussed in the previous section.Compared to other models of quantum gravity, the same level of generality can be achievedonly in the polynomial [31, 37] and non-local models of quantum gravity (see the power-countingdiscussion in [39]), which are super-renormalizable. In both cases the beta functions for λ , ρ and ξ are not present in the published literature, and for the latter it is not clear how those functionscan be derived, at least in a covariant way. Moreover, if the massive degrees of freedom in thesesuper-renormalizable models have masses of the Planck order of magnitude, the exact runningoccurs only in the trans-Planckian region. On the contrary, in the case under discussion here, theexact running is an IR effect, taking place only below the Planck scale.It is also worthwhile to make a comparison with the non-perturbative analysis in quantumgravity based on the functional renormalization group (see, e.g. , the reviews [51,52] and the morerecent [53]). It was recently shown that in this approach the effective average action remainsgauge-fixing dependent even on-shell [55]. For this reason, even within the Vilkoviksy–DeWittformulation of the off-shell effective action, regardless the last being gauge-fixing independent byconstruction, the effective average action remains gauge-fixing dependent in this case. No unam-biguous physical predictions can be extracted from the quantum calculations in this approach.The scheme of deriving the beta functions for λ , ρ and ξ described above resembles morethe running of the vacuum action parameters in the semiclassical gravity [43, 44, 56, 57] than therenormalizable gravity [27]. The similarity with the semiclassical case is based on the fact thatthe running occurs in the sector of the theory which does not define the quantum effects. In thepresent case, this sector is related to fourth-derivative terms in the action (25). At higher loopsone can meet the renormalization group running for the parameters of six- and higher-derivativeterms in the action (22). For a previous discussion on this subject in the context of Yang–Mills fields, see [54].
13s a further illustration of the method, let us derive the two-loop beta function for the uniquetwo-loop divergence derived until now [58, 59], namely for the C -term in the total action (22),¯Γ (2)div = µ n − (4 π ) ( n −
4) 2091440 κ Z d n x √− g C µναβ C αβ · · ρσ C ρσµν . (37)Using the standard routine, we arrive at the beta function for the parameter ζ in the action (22), β ζ = − a W (4 π ) κ ζ , a W = 209720 . (38)We shall skip the discussion of possible matter-gravity contributions in this case and restrict theconsideration to the pure quantum gravity model, where the results are available. In the effectivequantum theory with Λ = 0, the expression (38) is exact, while in the case Λ = 0 it gains third-and higher-loop corrections in the form of a power series in Λ M P . As we already discussed above,in the physically relevant situations these contributions are strongly suppressed compared to theleading two-loop term.It is interesting to notice that the beta function (38) depends on G (via κ ). This is a generalfeature that occurs with all the divergent terms whose number of derivatives is different thanfour, and it is related to the fact that the coupling κ has negative mass dimension—or, in otherwords, to the non-renormalizability of the theory. Here, however, the situation is different fromthe Eq. (16) defining the running of the cosmological constant. In fact, the Eq. (15) for G alsodepends on Λ, but it does not depend on ζ . Therefore, we can use the solution for G alreadyestablished in (19) to determine the one of ζ . For the other massive parameters in the totalaction (22) we have a qualitatively similar picture.All in all, the running of ζ between the H scale in the IR and the Planck scale in the UV isdescribed by the equation (38) and the solution has the form ζ ( µ ) = ζ − a W π ) ζ Λ h − (cid:0) π ) γ ln µµ (cid:1) / i , ζ = ζ ( µ ) . (39)As in the previous cases, we have chosen the sign of the term in the action (22) such that therunning is the asymptotic-freedom type in the UV for a positive value of the correspondingparameter. Formula (39) also shows that the running of ζ depends on γ , thus the situation isvery similar to what happens with G and Λ. Since the value of γ is very small the running issupposed to be weak; the same qualitative behaviour ought to occur for the other parametersassociated to higher-order curvature terms in the total action. The last observation is that thesingularity in the limit Λ → G does not occur within the effective approach. In the special case Λ = 0, Eq. (38) has thestandard form of solution. 14 Physical applications
Let us briefly discuss about the possible physical applications of the running of the parametersΛ, G , λ , ρ and ξ . Certainly it would be interesting to apply the solutions (19) and (20), andalso the solutions for the dimensionless parameters, to both cosmology and astrophysics. Theirdetailed elaboration, nonetheless, is beyond the scope of the present work.First of all, the use of the running of Λ and G requires fixing a physical identification of thescale µ from the Minimal Subtraction scheme of renormalization. In cosmology the most well-motivated identification is with the Hubble parameter, µ ∼ H (see e.g. [36,60]). In astrophysics, itwas originally used the identification µ ∼ r − for objects like stars, galaxies and their clusters, with r being the distance from the center of the object [61,62]. Further detailed analysis led to the moreintricate identification of Ref. [63], which was phenomenologically successful. Nowadays, thereare some publications on the systematic derivation and covariant forms of the scale identification,see e.g. [64, 65], which enable one to apply the solution (19).In the case of the four-derivative terms we meet the explicit non-local form factors given byEq. (29) and b (4 π ) Z d x √− g R ln (cid:16) − ✷ µ (cid:17) R . (40)An observation concerning the cosmological applications of these two logarithmic form factors isin order. The Weyl-squared term in the action (25) affects the gravitational wave type cosmicperturbations, but not the background solution or density perturbations. There are no reasonswhy the numerical coefficient of this term should have a particularly large value. Thus, thepresence of the logarithmic form factor (29) can give an effect of the IR running, similar to whathas been previously described as a consequence of a photon effect in [47]. It is remarkable thatusing the unique effective action one can report on the same IR running in effective quantumgravity. At low energies the effect related to the fourth-derivative term is weak and no essentialobservational manifestations should be expected. At the same time, close to the Planckian scale,when the initial seeds of the tensor modes of cosmic perturbations are formed, there might besome effects of the logarithmic form factor in (29). This issue may deserve a detailed study, butit is beyond the scope of the present work.On the other hand, the coefficient of the classical R -term in the action (25) can be eitherunconstrained or fixed by the observational data. The last is the situation in the Starobinskyinflation [66], where one can show that this value should be as large as 5 × [67]. In thiscase, even at the Planck scale, the effect of the form factor (40) is enhanced by eight orders ofmagnitude. This situation is in sharp contrast with other models, including the Higgs inflationand inflaton-based models, which are otherwise equivalent to the R -based model of Starobinsky.Thus, using quantum gravity we might gain a possibility to distinguish this among the otherinflationary models.It is important to stress that all these expectations become possible only because of the use15f the Vilkovisky–DeWitt unique effective action. In the usual formulation of effective quantumgravity both beta functions associated to the terms C and R are dependent on the choiceof gauge-fixing and parametrization of quantum metric [26], preventing their use in reasonableapplications. Using the effective approach and Vilkovisky–DeWitt unique effective action in quantum gen-eral relativity, we constructed the renormalization group equations for the Newton and cosmo-logical constants and for the parameters of the fourth-derivative terms in the extended actionof gravity. The part of Newton and cosmological constants has been considered earlier in [13],but our analysis is done from a different perspective. In particular, we show that in the effec-tive approach all the mentioned one-loop beta functions can be regarded as exact, meaning theydo not gain significant higher-loop corrections. The same concerns the renormalization groupequation for the coefficient of the six-derivative term. This equation is derived on the basis ofthe two-loop divergences calculated in the well-known works [58, 59] and does not require theVilkovisky–DeWitt approach to be universal.The one-loop equations come from the quantum effects of the purely massless modes and,therefore, are valid in both UV and IR. In the UV, the renormalization group trajectories canbe used only until the scale where the massive degrees of freedom associated to higher-derivativeterms become active. However, in the IR there are no restrictions except the extremely low-energyHubble scale IR cut-off.In this respect, the renormalization group equations under discussion strongly differ from theones in renormalizable and superrenormalizable models of quantum gravity. In fact, those arevalid only in the UV regime, usually with respect to the Planck scale.
Acknowledgements
The authors are grateful to the referee of the original version of the work [arXiv:2006.04217] for theadvice to explore the nonperturbative aspects of the renormalization group in the Vilkovisky–DeWitt effective action. The work of I.Sh. is partially supported by Conselho Nacional deDesenvolvimento Cient´ıfico e Tecnol´ogico - CNPq under the grant 303635/2018-5.
References [1] J.F. Donoghue,
General relativity as an effective field theory: The leading quantum correc-tions , Phys. Rev. D , 3874 (1994), arXiv:gr-qc/9405057.162] G.A. Vilkovisky, The Unique Effective Action in Quantum Field Theory , Nucl. Phys. B , 125 (1984);
The Gospel according to DeWitt , in:
Quantum Theory of Gravity , ed.S.M. Christensen (Adam Hilger, Bristol, 1984).[3] B.S. DeWitt,
The effective action , in:
Quantum Field Theory and Quantum Statistics , essaysin honor of the sixtieth birthday of E.S. Fradkin, Vol. 1: Quantum Statistics and methods ofField Theory, ed. C.J. Isham, I.A. Batalin and G.A. Vilkovisky, (Hilger, Bristol, 1987);
Theeffective action , in:
Architecture of fundamental interactions at short distances , ed. P. Ra-mond and R. Stora (North-Holland, Amsterdam, 1987).[4] A.O. Barvinsky and G A. Vilkovisky,
The generalized Schwinger-DeWitt technique and theunique effective action in quantum gravity , Phys. Lett. , 313 (1983);
The general-ized Schwinger-Dewitt technique in gauge theories and quantum gravity , Phys. Rept. , 1(1985).[5] E.S. Fradkin and A.A. Tseytlin,
On the new definition of off-shell Effective Action , Nucl.Phys. B , 509 (1984).[6] G. Kunstatter,
Vilkovisky’s unique effective action: an introduction and explicit calculation ,in:
Super Field Theories , proceedings of NATO Advanced Research Workshop on SuperfieldTheories, ed. H.C. Lee, V. Elias, G. Kunstatter, R.B. Mann and K.S. Viswanathan, NATOASI Series B, Vol. 160 (Plenum Press, New York, 1987).[7] A. Rebhan,
The Vilkovisky-DeWitt Effective Action and its application to Yang-Mills Theo-ries , Nucl. Phys. B , 832 (1987).[8] A. Rebhan,
Feynman rules and S -Matrix equivalence of the Vilkovisky-deWitt effective ac-tion , Nucl. Phys. B , 726 (1988).[9] P. Ellicott and D. Toms, On the New Effective Action in Quantum Field Theory , Nucl. Phys.B , 700 (1989).[10] S.R. Huggins, G. Kunstatter, H.P. Leivo and D.J. Toms,
The Vilkovisky-de Witt EffectiveAction for Quantum Gravity , Nucl. Phys. B , 627 (1988).[11] B.L. Giacchini, T. de Paula Netto, and I.L. Shapiro,
On the Vilkovisky unique effective actionin quantum gravity, arXiv:2006.04217.[12] A.T. Banin and I.L. Shapiro,
Gauge dependence and new kind of two - dimensional gravitytheory with trivial quantum corrections.
Phys. Lett.
B327 , 17 (1994).[13] T. Taylor and G. Veneziano,
Quantum gravity at large distances and the cosmological con-stant , Nucl. Phys. B , 210 (1990). 1714] B. Voronov and I. Tyutin,
Formulation of gauge theories of general form. II. Gauge invariantrenormalizability and renormalization structure , Theor. Math. Phys. , 628 (1982).[15] B.L. Voronov and I.V. Tyutin, On renormalization of R gravitation , Sov. Nucl. Phys. ,998 (1984) [Yad. Fiz. , 998 (1984)].[16] B.L. Voronov, P.M. Lavrov and I.V. Tyutin, Canonical transformations and the gauge depen-dence in general gauge theories , Sov. Nucl. Phys. , 498 (1982) [Yad. Fiz. , 498 (1982)].[17] J. Honerkamp, Chiral multiloops , Nucl. Phys. B , 130 (1972).[18] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965).[19] J.L. Synge,
Relativity: the general theory (North-Holland, Amsterdam, 1960).[20] B.S. DeWitt,
Quantum Theory of Gravity. 1. The Canonical Theory , Phys. Rev. , 1113(1967).[21] S.D. Odintsov,
Does the Vilkovisky-De Witt effective action in quantum gravity depend onthe configuration space metric? , Phys. Lett. B , 394 (1991).[22] J.F. Barbero and J. P´erez-Mercader,
Superspace dependence of the Vilkovisky-DeWitt effec-tive action for quantum gravity , Phys. Rev. D , 3663 (1993).[23] C.P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory,
Living Rev. Rel. , 5 (2004), arXiv:gr-qc/0311082.[24] G. ’t Hooft and M. Veltman, One loop divergencies in the theory of gravitation , Ann. Inst.H. Poincare Phys. Theor. A , 69 (1974).[25] S. Christensen and M. Duff, Quantizing Gravity with a Cosmological Constant , Nucl. Phys.B , 480 (1980).[26] J.D. Gon¸calves, T. de Paula Netto and I.L. Shapiro,
Gauge and parametrization ambiguityin quantum gravity , Phys. Rev. D , 026015 (2018), arXiv:1712.03338.[27] E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of grav-ity,
Nucl. Phys. B , 469 (1982).[28] E.V. Gorbar and I.L. Shapiro,
Renormalization Group and Decoupling in CurvedSpace , JHEP , 021 (2003), arXiv:hep-ph/0210388; Renormalization Group and De-coupling in Curved Space: II. The Standard Model and Beyond
JHEP , 004 (2003),arXiv:hep-ph/0303124.[29] S.A. Franchino-Vi˜nas, T. de Paula Netto, I.L. Shapiro and O. Zanusso, Form factors anddecoupling of matter fields in four-dimensional gravity , Phys. Lett. B , 229 (2019),arXiv:1812.00460. 1830] G. de Berredo-Peixoto and I.L. Shapiro,
Higher derivative quantum gravity with Gauss-Bonnet term,
Phys. Rev. D , 064005 (2005), arXiv:hep-th/0412249.[31] L. Modesto, L. Rachwa l and I.L. Shapiro, Renormalization group in super-renormalizablequantum gravity,
Eur. Phys. J. C , 555 (2018), arXiv:1704.03988.[32] P.d. Teixeira, I.L. Shapiro and T.G. Ribeiro, One-loop effective action: nonlocal form factorsand renormalization group, arXiv:2003.04503, to be published in Gravitation and Cosmology (2020).[33] S. Perlmutter et al. [Supernova Cosmology Project], Discovery of a supernova explosion athalf the age of the Universe , Nature , 51 (1998), arXiv:astro-ph/9712212.[34] A.G. Riess et al. [Supernova Search Team],
Observational Evidence from Supernovae foran Accelerating Universe and a Cosmological Constant , Astron. J. , 1009 (1998),arXiv:astro-ph/9805201.[35] S. Weinberg,
The cosmological constant problem,
Rev. Mod. Phys. , 1 (1989).[36] I.L. Shapiro, J. Sol`a, Scaling behavior of the cosmological constant: Interface between quan-tum field theory and cosmology,
JHEP , 006 (2002), arXiv:hep-th/0012227.[37] M. Asorey, J.L. L´opez and I.L. Shapiro, Some remarks on high derivative quantum gravity,
Int. Journ. Mod. Phys.
A12 , 5711 (1997).[38] L. Modesto,
Super-renormalizable quantum gravity,
Phys. Rev.
D86 , 044005 (2012),arXiv:1107.2403;L. Modesto and L. Rachwa l,
Super-renormalizable and finite gravitational theories,
Nucl.Phys.
B889 , 228 (2014), arXiv:1407.8036;
Nonlocal quantum gravity: A review,
Int. J.Mod. Phys.
D26 , 1730020 (2017).[39] I.L. Shapiro,
Counting ghosts in the “ghost-free” non-local gravity.
Phys. Lett.
B744 , 67(2015), arXiv:1502.00106.[40] L. Modesto and I.L. Shapiro,
Superrenormalizable quantum gravity with complex ghosts,
Phys. Lett.
B755 , 279 (2016), arXiv:1512.07600.[41] A. Accioly, B.L. Giacchini and I.L. Shapiro,
On the gravitational seesaw in higher-derivativegravity , Eur. Phys. J. C , 540 (2017), arXiv:1604.07348.[42] K.S. Stelle, Renormalization of higher derivative quantum gravity , Phys. Rev.
D16 , 953(1977).[43] B.L. Nelson and P. Panangaden,
Scaling behavior of interacting quantum fields in curvedspace-time,
Phys. Rev.
D25 , 1019 (1982). 1944] I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro,
Effective action in quantum gravity , (IOPPublishing, Bristol, 1992).[45] I.L. Buchbinder and I.L. Shapiro,
Introduction to Quantum Field Theory with Applicationsto Quantum Gravity, (Oxford University Press, to be published).[46] I.G. Avramidi and A.O. Barvinsky,
Asymptotic freedom in higher derivative quantum gravity,
Phys. Lett.
B159 , 269 (1985).[47] G. Cusin, F. de O. Salles and I.L. Shapiro,
Tensor instabilities at the end of the Λ CDMuniverse,
Phys. Rev.
D93 , 044039 (2016), arXiv:1503.08059.[48] A.O. Barvinsky and G.A. Vilkovisky,
Covariant perturbation theory. 2: Second order in thecurvature. General algorithms,
Nucl. Phys. , 471 (1990).[49] I. Antoniadis, P.O. Mazur and E. Mottola,
Conformal symmetry and central charges infour-dimensions,
Nucl. Phys.
B388 , 627 (1992), arXiv:hep-th/9205015.[50] G.B. Peixoto and I.L. Shapiro,
Conformal quantum gravity with the Gauss-Bonnet term ,Phys. Rev.
D70 , 044024 (2004), arXiv:hep-th/0307030.[51] M. Niedermaier and M. Reuter,
The Asymptotic Safety Scenario in Quantum Gravity,
LivingRev. Rel. , 5 (2006), arXiv:gr-qc/0610018.[52] R. Percacci, Asymptotic Safety , in:
Approaches to Quantum Gravity: Toward a New Un-derstanding of Space, Time and Matter (pp. 111-128), ed. D. Oriti (Cambridge UniversityPress, Cambridge, 2007), arXiv:0709.3851.[53] A. Bonanno, A. Eichhorn, H. Gies, J.M. Pawlowski, R. Percacci, M. Reuter, F. Saueressigand G. P. Vacca,
Critical reflections on asymptotically safe gravity, arXiv:2004.06810.[54] P.M. Lavrov and I.L. Shapiro,
On the Functional Renormalization Group approach for Yang-Mills fields,
JHEP , 086 (2013), arXiv:1212.2577.[55] V.F. Barra, P.M. Lavrov, E.A. dos Reis, T. de Paula Netto and I.L. Shapiro,
Functionalrenormalization group approach and gauge dependence in gravity theories,
Phys. Rev.
D101 ,065001 (2020), arXiv:1910.06068.[56] I.L. Buchbinder,
On Renormalization group equations in curved space-time,
Theor. Math.Phys. , 393 (1984).[57] D.J. Toms, The Effective Action And The Renormalization Group Equation In Curved Space-Time,
Phys. Lett.
B126 , 37 (1983).[58] M.H. Goroff and A. Sagnotti,
Quantum gravity at two loops,
Phys. Lett.
B160 , 81 (1985);
The ultraviolet behavior of Einstein gravity,
Nucl. Phys.
B266 , 709 (1986).2059] A.E.M. van de Ven,
Two loop quantum gravity,
Nucl. Phys.
B378 , 309 (1992).[60] A. Babic, B. Guberina, R. Horvat and H. Stefancic,
Renormalization group running of thecosmological constant and its implication for the Higgs boson mass in the standard model,
Phys. Rev.
D65 , 085002 (2002), arXiv:hep-ph/0111207;B. Guberina, R. Horvat and H. Stefancic,
Renormalization-group running of the cosmologicalconstant and the fate of the universe , Phys. Rev.
D67 , 083001 (2003), arXiv:hep-ph/0211184.[61] T. Goldman, J. P´erez-Mercader, F. Cooper, and M.M. Nieto,
The Dark matter problem andquantum gravity,
Phys. Lett.
B 281 , 219 (1992);O. Bertolami, J.M. Mour˜ao, and J. P´erez-Mercader,
Quantum gravity and the large scalestructure of the universe,
Phys. Lett.
B 311 , 27 (1993).[62] I.L. Shapiro, J. Sol`a and H. Stefancic,
Running G and Λ at low energies from physicsat M X : possible cosmological and astrophysical implications , JCAP , 012 (2005),arXiv:hep-ph/0410095.[63] D.C. Rodrigues, P.S. Letelier and I.L. Shapiro, Galaxy rotation curves from General Rela-tivity with Renormalization Group corrections,
JCAP , 020 (2010), arXiv:0911.4967.[64] S. Domazet and H. Stefancic,
Renormalization group scale-setting in astrophysical systems,
Phys. Lett.
B703 , 1 (2011), arXiv:1010.3585.[65] D.C. Rodrigues, B. Chauvineau, and O.F. Piattella,
Scalar-Tensor gravity with system-dependent potential and its relation with Renormalization Group extended General Relativity,
JCAP , 009 (2015), arXiv:1504.05119;N.R. Bertini, W.S. Hip´olito-Ricaldi, F. de Melo-Santos, and D.C. Rodrigues, Cosmologicalframework for renormalization group extended gravity at the action level,
Eur. Phys. J. C , 479 (2020), arXiv:1908.03960.[66] A.A. Starobinski, A new type of isotropic cosmological models without singularity , Phys.Lett.
B91 , 99 (1980).[67] A.A. Starobinsky,
The perturbation spectrum evolving from a nonsingular initially de-Sittercosmology and the microwave background anisotropy,
Sov. Astron. Lett.9