Gauges and functional measures in quantum gravity II: higher-derivative gravity

We compute the one-loop divergences in a higher-derivative theory of gravity including Ricci tensor squared and Ricci scalar squared terms, in addition to the Hilbert and cosmological terms, on an (generally off-shell) Einstein background. We work with a two-parameter family of parametrizations of the graviton field, and a two-parameter family of gauges. We find that there are some choices of gauge or parametrization that reduce the dependence on the remaining parameters. The results are invariant under a recently discovered “duality” that involves the replacement of the densitized metric by a densitized inverse metric as the fundamental quantum variable.


Introduction
In a previous paper [1], hereafter referred to as I, we have examined the properties of quantum General Relativity (GR-the theory containing only terms up to second derivatives in the action) in a general four-parameter family of gauges and parametrizations. In this paper, we would like to extend the analysis to higher-derivative gravity (HDG).
By HDG, we will always mean the theory of gravity based on the metric as fundamental variable and an action that contains up to four derivatives. 1 This theory is important because in four dimensions it is power-counting renormalizable [5] and asymptotically free [6][7][8][9][10]. In spite of these appealing 1 There has been recently some progress on theories that contain more than four, and possibly infinitely many derivatives [2][3][4]. These also deserve the name HDG, but we shall not consider them here. b e-mail: ohtan@phys.kindai.ac.jp c e-mail: percacci@sissa.it d e-mail: duarte763@gmail.com properties, it has never been accepted as a viable fundamental theory for gravity, because of the presence of ghosts in perturbation theory. Over time, there have been many proposals trying to circumvent this problem. Among these let us mention here the following possibilities: • The mass of the ghost is not a fixed parameter but is rather subject to strong (quadratic) running above the Planck threshold. Then the equation for the pole mass m 2 phys = m 2 (k = m phys ) (where m(k) is the running mass) may not have a solution [6,11,12].
• The ghost may be an artifact of expanding around the wrong vacuum. The true vacuum of quadratic gravity (in the presence also of a Hilbert term) is not flat space but rather a kind of wave with wavelength of the order of the Planck length [13]. • The quadratic term is one of an infinite series and the sum of the series is a function that has no massive ghost pole. The ghost pole is an artifact of Taylor expanding this function to second order (see the aforementioned papers on non-local gravity and also [14]).
For further proposals see also [15][16][17][18][19]. So far, none of these arguments has convinced the community at large, so the issue of the ghosts remains open for the time being. However, the hope that a way out may exist has generated new interest in these theories in recent times, also among particle physicists [20][21][22]. It seems therefore appropriate to keep investigating the quantum properties of these theories. In this paper we will extend previous results in several directions. Due to the complicated structure of the theory, calculations of one-loop divergences in HDG have usually been performed with a special four-derivative gauge-fixing term such that the four-derivative part of the Hessian is pro-portional to the square of the Laplacian. In this paper we will calculate the off-shell gauge dependence of the oneloop divergences by using the more conventional secondderivative gauge-fixing term that is commonly used in quantum GR, depending on two parameters a and b, or a fourderivative variant of the same gauge fixing, containing an extra power of a Laplacian.
Here we extend the analysis of I to HDG. We now have four independent couplings instead of two, so the expressions for the divergences are in general much more complicated than in I and so is the interpretation of the results. The expressions simplify somewhat, and they are reported explicitly, in two limits: the "four derivative gravity (4DG) limit" in which the Einstein-Hilbert terms can be neglected relative to the curvature squared terms, and the "Einstein-Hilbert (EH) limit" where the opposite holds. In the EH limit, the action is the same as the one considered in I, but the analysis that we perform here differs in two respects: first, we choose a more general Einstein background, instead of the maximally symmetric background of I. This allows us to discriminate two divergent terms quadratic in curvature, rather than a single one as in I. Secondly, the cutoff in each spin sector is chosen to depend on the corresponding Lichnerowicz Laplacian, rather than the Bochner Laplacian −∇ 2 as in I. Thus, comparison with I yields some information on the cutoff dependence of these non-universal results. Still, we find that in this limit the qualitative picture is the same. Similar calculations of divergent terms with different parametrizations in four dimensions have been given in [23].
In four dimensions and in the 4DG-limit, all divergences are universal, i.e. independent of both gauge and parametrization. In particular, the logarithmic divergences are related to the well-known universal beta functions of HDG [9,10]. As in I, we find that all divergences are invariant under a "duality" transformation that consists in the replacement Let us discuss a little more the role of the functional measure in these considerations. In I we have used the word "measure" synonymously with "choice of quantum field in the functional integral" but we have left the definition of the functional measure a bit implicit. The reason for this is that the Wilsonian cutoff k that we have used to calculate the divergences does not regulate the divergences in the functional measure, if there are any, and so the results would have been independent of this choice anyway. The consistent interpretation of the results of I is that they give the correct divergences of the functional integrals when the measure is given by (1.8) where h μν is the purely tensorial quantum field defined as in (2.14), independently of m and ω.
Since the functional measure was always kept fixed, the dependence of the results on the parameters m and ω resulted entirely from the different forms of the Hessians, which in turn was due to the different forms of the expansion of the action (see Eqs. (2.13) and (2.15)).
If one decided to use the same expansion but a different ultralocal measure, for example (1.9) then the divergences would differ by terms of the form δ(0) times the volume. Such terms would affect the power-law divergences coefficients.
We conclude this introduction by listing the contents of the following sections. Section 2 contains details of our calculations of one-loop divergences. In Sect. 3 we give a formal proof (at the level of determinants) that the on-shell effective action in d = 4 is gauge independent. (In particular, the effective action of HDG in the 4DG-limit on an Einstein space is gauge-independent.) The main results are presented in Sect. 4. In Sect. 5, we discuss how the results can be obtained for d = 4 conformal gravity. In Sect. 6, we point out that our results show that the duality found in our previous paper I is valid in HDG as well. Section 5 contains a discussion and conclusions. In the appendix, we summarize the heat kernel coefficients for Lichnerowicz Laplacians on an Einstein manifold.

The HDG actions and their equations of motion
In this paper we will consider actions of the general form , is the cosmological constant, α, β are the higher-derivative couplings. The upper sign refers to the Minkowski signature, the lower one to the Euclidean signature. This is not the most general HDG action, because we omit a term γ R 2 μνρσ ; however, in d ≤ 4 this is not a very strong restriction, because the "Gauss-Bonnet" or "Euler" combination is either zero (in d = 2, 3) or a total derivative (in d = 4). Using the identity for the Weyl tensor C μνρσ : one has Thus, modulo terms proportional to G, we can replace α R 2 + β R 2 μν by The integral of W , like the integral of C 2 , is Weyl-invariant in d = 4.
The equations of motion are where G μν is the Einstein tensor and Let us observe that in the case Z N = 0 and d = 4 the Einstein condition is enough to fulfill the equations of motion. The scalar curvature remains an undetermined constant, since the action is scale-invariant. On the other hand when Z N = 0 the equations of motion together with the Einstein condition further require In particular, in four dimensions the higher-derivative terms do not contribute and the equation of motion is the same as the trace of the Einstein equations with cosmological constant: To summarize, the choice of the Einstein condition means that the background is "almost on shell" in the sense that all the equations of motion are satisfied except, in general, for the trace equation (2.9). In the special case of pure HDG (no Einstein-Hilbert term) in d = 4 the background is completely on-shell.

Quadratic expansion
The action has to be thought of as a functional of the quantum field γ μν or γ μν defined as in (1.1) or (1.2). For w = −1/d these relations can be inverted to yield g μν = γ μν (det(γ μν )) m ; g μν = γ μν (det(γ μν )) −m , (2.11) where respectively. Conversely, m = − w/2 1+dw/2 for (1.1) and m = w/2 1−dw/2 for (1.2). We observe that the relation between m and w/2 is an involution. We choose to treat m as an independent free parameter. All dependence on m can be translated into a dependence on w if needed, using the preceding formulas. The quantum field is then expanded as in (1.3) or (1.4) or (1.5), and for the calculation of the one-loop divergences we need the terms to second order in the fluctuation. As explained in [1], we can start from the quadratic expansion of the action in δg μν , which has been given in detail in [24], and then use δg μν = δg (1) μν + δg (2) μν + · · · , (2.13) where δg (n) μν contains n powers of the tensor fluctuation For all four types of expansion considered here, we have The choice ω = 0 corresponds to the linear expansion of metric (1.3), ω = 1/2 corresponds to the exponential expansion (1.4) (and it does not matter if one starts from the metric or from the inverse metric) and ω = 1 corresponds to the linear expansion of the inverse metric (1.5).
In the following we will use these expansions in the action (2.1). We note that, for given g μν andḡ μν , different values of ω and m will give different fluctuation fields h μν and conversely for given h μν andḡ μν , different values of ω and m will give different total metrics g μν . In the following calculations the action (2.1), as a functional of the total metric, will always be kept fixed and also the functional measure for the quantum field will be kept fixed.

Lichnerowicz Laplacians
The evaluation of the one-loop divergences is based on the knowledge of the coefficients of the small-time expansion of the heat kernel of the kinetic operator appearing in the gauge-fixed Hessian. We will denote∇ the covariant derivative defined by the background metricḡ, =∇ 2 the d'Alembertian and −∇ 2 its Euclidean analog, known as Bochner Laplacian. In GR, the expansion of the action generates non-minimal terms that can be eliminated by choosing the de Donder gauge. The rest can be written in terms of Laplacians. Similarly, the second variation of the HDG action contains non-Laplacian terms such as∇ μ∇ν∇ρ∇σ , ∇ μ∇νḡρσ +ḡ μν∇ρ∇σ andḡ νσ∇μ ∇ ρ +ḡ μρ∇ν ∇ σ , acting on h ρσ . The heat kernel expansion for such operators is not known, so we will use the York decomposition to rewrite the Hessian as minimal operators acting on fields with definite spin. For this purpose, it is essential to rewrite all occurrences of in terms of the Lichnerowicz Laplacians acting on spin-0, spin-1 and spin-2 fields, which are defined (in Euclidean signature) as follows: These operators have the following useful properties: The York decomposition leads to significant simplifications when the background metric is an Einstein space, i.e. satisfies the condition (2.8). (As discussed in Sect. 2.1, this is not enough to put the background on-shell.) In the following we will always assume that the background is Einstein, but not that it satisfies (2.10). Now consider the second variations of the curvature squared terms in the action, as given in eqs. (B.6) and (B.7) in [24]. The terms containing the Riemann tensor can be combined with certain terms containing to give Lichnerowicz Laplacians. Further using (2.8), one arrives at and All tensor structures are provided by the background metric. We note that this procedure of eliminating the Riemann tensor does not work in the case of the second variation of the Riemann squared term, which is the reason why we do not consider such a term.

York decomposition
The York decomposition is defined by where h T T μν is transverse and tracefree, andξ μ is transverse. We will usê Employing (2.24) and (2.25) and assuming that the background is an Einstein space, one finds (2.26) Using the properties (2.17)-(2.21), we have and so on.

The decomposed Hessian
The expansion of the Euclidean action in powers of h has the following quadratic part: with where This form of the Hessian has the virtue that all the terms that contain h are proportional to the equation of motion. This shows that the field h can be completely disregarded on shell, as one would expect of a gauge-variant variable. This, however, depends on the choice of basis in the space of fields and is only true when the other scalar degree of freedom is gauge-invariant.
On the other hand, this form of the Hessian has the unpleasant feature that the kinetic operator of the field s is non-local. One cannot generally absorb the non-local prefactor in a redefinition of s, because one is not allowed to perform non-local redefinitions of physical fields.
One notices, however, that the terms with the lowest power of L 0 in each of the three lines in H ss s is proportional to (1 + dm)(1 − 2ω). Therefore, if either ω = 1/2 (exponential parametrization) or m = −1/d (the "unimodular" measure), each of the square brackets in (2.42) is proportional to L 0 and the Hessian of s becomes local.
We further observe that for ω = 1/2 also the mixed term vanishes and the term H hh s becomes local, whereas for m = −1/d all terms containing h vanish, as expected in the unimodular theory. Furthermore, H ss s becomes local and independent of m and ω for pure four-derivative gravity (Z N = 0) when d = 4. In this case the whole Hessian becomes just Finally we observe that in the conformal case d = 4, β = −3α (which corresponds to having only the term W in the action) the ss term in the Hessian vanishes too, leaving only the pure spin-2 degree of freedom, with Hessian

Two-derivative gauge-fixing terms
We now turn to the discussion of the gauge-fixing and Faddeev-Popov (FP) ghost terms. In most of the following, we use the same two-derivative gauge-fixing term as in I [1]. The gauge-fixing function is defined as with a gauge-fixing parameter −∞ <b < ∞. Here h μν is the tensorial quantum field defined above and the bars on the covariant derivatives mean that they are calculated from the background metricḡ μν . The simplest way to derive the gauge-fixing and FP ghost terms is to use the BRST transformations, which are obtained by replacing the gauge parameters with the FP ghosts. In the present case, we have where C μ is the FP ghost, and δλ is an anticommuting parameter. The BRST transformation for other fields is derived by the requirement of the nilpotency of the transformation: whereC μ is the FP anti-ghost and B μ is an auxiliary field which enforces the gauge-fixing condition.
In order to derive the gauge-fixing and FP terms, we have to know the transformations on the fluctuation field h μν . From the first relation in Eq. (2.15), we have (2.50) Anticipating that we will compute the effective action for vanishing expectation value of h μν , we can restrict this to the linear terms to obtain (2.51) The gauge-fixing and FP terms are then given as [25,26] and we have setb = (1 + dm)b and defined Here a is another dimensionless gauge parameter, and Z GF is a parameter with dimension d − 2. Since the B μ field involves no derivatives, we can simply integrate it out and then we are left with the gauge-fixing and FP ghost terms.
Using the York decomposition, Inserting this into the gauge-fixing term in (2.52), integrating by parts, rewriting in terms of Lichnerowicz Laplacians and using the York decomposition, we obtain the gaugefixing term We see that a specific combination of scalar degrees of freedom appears in the gauge-fixing term. It is sometimes convenient to write the scalar sector in terms of the gaugeinvariant variable s defined in Eq. (2.40) and this new degree of freedom which, in terms of the original fields, is given by A short calculation shows that the Jacobian of the transfor- In terms of the new variable, the gauge-fixing action is (2.57) From (2.56) we see that χ transforms in the same way as σ . Thus ξ and χ can be viewed as the gauge degrees of freedom and h T T and s as the physical degrees of freedom. We will use this parametrization later.
The ghost action (2.52) for this gauge fixing contains a non-minimal operator [1]. We also decompose the ghost into transverse and longitudinal parts 58) and the same forC μ . (This change of variables has unit Jacobian.) The ghost action then splits in two terms (2.59) Note for future reference that the gauge-fixing condition and the ghost action are pathological for b = d − 1.
Let us observe that Z GF appears in the combination Z GF /a, where a is a dimensionless gauge parameter, while Z GF is a constant of dimension d − 2. There are then two natural options regarding the constant Z GF . The first choice is to treat it as a fixed parameter (later we shall identify it with a power of the cutoff). This leads to simpler formulas for many expressions, and we shall use it extensively later. It is, however, not appropriate for the discussion of perturbative Einstein gravity. The reason is that in the limit G → 0 the coefficient Z N of the Hessian diverges. If we keep Z GF and a constant in the limit, then the gauge-fixing term becomes negligible relative to the rest of the quadratic action. The gauge fluctuations remain unsuppressed and one can anticipate divergences. This is exactly what happens, as we shall mention later on. One can compensate the behavior of Z N by keeping Z GF fixed and letting simultaneously a → 0. Alternatively, one can set Z GF = Z N . In this case, in the Gaussian limit G → 0, the kinetic terms of the gauge-invariant and gauge degrees of freedom scale in the same way and one obtains sensible results for all values of a and b.
In the following we shall discuss also different choices for the gauge fixing. One is the so-called "unimodular physical gauge", where one setsξ μ = 0 and h = 0. As shown in [27], this is equivalent to the above standard gauge in the limit b → ±∞ and a → 0.

Four-derivative gauge-fixing terms
In HDG it is customary to use gauge-fixing terms that contain four derivatives. In order to further check the gauge independence of the results, in Sect. 3.2 we consider the gauge fixing in our previous paper [24] but now with arbitrary gauge parameters.
We can simply repeat the same procedure to derive the gauge-fixing and FP terms but with additional higherderivative operator where c and f are additional gauge parameters: Note also that the field B μ was an auxiliary field in the two-derivative gauge-fixing case (2.52), but here it is dynamical. With higher-derivative gauge fixing, quite often only the contribution (gh) μν is incorporated but that from Y μν in the FP ghost kinetic term is ignored, and then it is claimed that somehow the contribution from the "third ghost" − 1 2 log(det(Y μν )) must be added. We see here that this is automatic in the BRST invariant formulation, because we have contributions − log(det(Y μν )) from the FP ghost kinetic term and 1 2 log(det(Y μν )) from the field B μ , giving the same result.
Using the York decomposition (2.24), we can calculate the contributions without fixing gauge parameters and so can check the gauge dependence directly here. We find that the terms (2.61) are cast into where we have defined transverse and longitudinal parts of theB μ and FP ghostsC μ and C μ , as usual. In Sect. 3.1 we will discuss the gauge independence of the theory in the general quadratic gauge. In Sect. 4.5 we will restrict ourselves to the class of gauges where c = f = 1 but with generic a, b. This is equivalent to inserting a Lichnerowicz Laplacian L 1 in the quadratic gauge-fixing term (2.52), or in other words to set Y μν =ḡ μν L 1 . After performing the York decomposition, this yields additional factors L 1 and L 0 in the quadratic actions of ξ μ and χ , Eq. (2.57). The resulting additional determinants are offset by the determinant of the operator coming from the B μ sector, as will become clear in the following.

Universality on-shell in d = 4
In this section we consider the theory in d = 4 on an Einstein background. As noted in Sect. 2.1, if we put Z N = 0, the equation of motion of HDG is automatically satisfied for d = 4, so one would expect the effective action to be gaugeand parametrization-independent. We will check that this is indeed the case at the formal level of determinants. In the following section we will have a more explicit check of this property in the expressions for the divergences.

General two-derivative gauge fixing
The Hessian of pure higher-derivative gravity in d = 4 has the simple form (2.45), independent of the choice of parametrization. It is expressed entirely in terms of the gaugeinvariant variables h T T and s. These fields do not appear in the gauge-fixing action, so their contribution to the one-loop effective action is given by We choose the gauge as in Sect. 2.6. It is convenient to express the gauge-fixing term as in (2.57). The fields ξ and χ contribute to the one-loop action the terms (as ordered): The ghost action (2.59) gives the determinants and finally the Jacobians of the change of variables from h μν to h T T μν , ξ μ , σ and h is Det Note that the gauge parameter a only appears in the Hessian in the prefactors and not in the operators. The only dependence on the gauge parameter b is in two determinants that cancel. This proves the gauge independence of the one-loop action. The final result is (3.5)

General four-derivative gauge fixing
If we use the general four-derivative gauge fixing of Sect. 2.7, we have again the determinants (3.1) coming from the physical degrees of freedom and (3.4) coming from the Jacobians. The determinants coming from the fields ξ μ and χ are the ghosts give (3.8) and finally we have the contributions of the B μ field (3.9) Putting everything together, we see that all the terms depending on the gauge parameters cancel out and the remaining ones give again (3.5). Thus we have explicitly shown, at this formal level, that the results are completely gauge independent.

Physical gauge
In the calculation of Sect. 3.1, and even more in Sect. 3.2, there is a large number of cancellations between various determinants. Consider instead the "physical" gaugeξ μ = 0, h = 0, discussed in [27,28]. It leaves only the fields h T T and σ , and no Jacobians. The Hessians of h T T andσ are given by (2.35) and (2.37), respectively. There are a real scalar ghost and a real transverse vector ghost, with ghost operators L0 and L1 − R/2. Putting together these terms, one immediately obtains the effective action (3.5). In fact this is the most direct way of getting it, because there is no cancellation of determinants between unphysical degrees of freedom and ghosts.

The conformal case
Now we consider the conformal case where β/α = −3. The effective action in this case cannot be simply obtained as a particular case of Eq. (3.5), because the action is invariant also under Weyl transformations and this requires a separate gauge fixing. The Hessian is only nonzero in the spin-2 sector and is given by (2.46). For the Weyl invariance we can gauge fix h = 0, without any ghost because h transforms under Weyl transformations by a shift. For diffeomorphisms we choose a standard gauge fixing of the form (2.55). Since h = 0, the value of b is immaterial. Equation (2.56) is replaced by χ = σ , so the decomposition of the gauge-fixing and ghost actions, and the corresponding determinants, are the same as in Sect. 2.6, with b = 0.
We use the fieldsξ andσ (see (2.25)) in such a way that no Jacobian is needed. The one-loop effective action is (3.10) The two terms in the denominator come from the TT part. The terms in the numerator come from the ghosts (power 1) and from the gauge-fixing term (power −1/2). Note that there is no dependence on the gauge parameter a. It only appears in the prefactors of the quadratic action, which drop out. Alternatively we can choose a different gauge. For Weyl transformations we still choose h = 0, which leaves no ghost term. For diffeomorphisms we choose the second type of physical gauge explained in the end of section III.B of [27], namelyσ = 0 andξ = 0. This is equivalent to taking the Landau gauge limit a → 0. The ghosts are a real scalar and a real transverse vector and the ghost operators are L0 −R/3 and L1 −R/2. The effective action is given again by (3.10).

The Einstein-Hilbert case
Finally we consider the Einstein-Hilbert theory, where we put α = β = 0. In this case it is natural to use the twoderivative gauge fixing. In order to maintain the expressions to a manageable size we write them here for the case ω = m = 0. Then from the decompositions in Sects. 2.5 and 2.6 we find that the gauge-invariant variable h T T gives the spin-1 field ξ μ gives the scalars σ and h give 13) and finally the ghosts give (3.14) Putting all together, and making the replacements a → 1/γ and b → β, we find that it agrees with Eq. (4.4) of [29]. If we go on shell by puttingR = 4 the scalar contribution cancels the scalar ghost determinant, leaving just which is gauge independent and agrees with the classic result of [30]. A slightly more complicated calculation shows that also the dependence on the parameters ω and m automatically goes away on shell.

Derivation
The one-loop effective action contains a divergent part where k stands for a cutoff. In d = 4, the power divergences k d−4 /(d − 4) of the last three terms are replaced by terms log(k/μ), where we introduced a reference mass scale μ.
In general one would have separate Riemann squared, Ricci squared and R 2 terms, but on an Einstein space (2.8) the latter two merge into a single term proportional toR 2 . Then we will denote the coefficient of theR 2 divergence and D 1 = β γ the coefficient of theR μνρσR μνρσ divergence. We note that in I we used a maximally symmetric background, wherē Thus, the coefficient C 1 of I corresponds to the combination C 1 + 2 d(d−1) D 1 . We describe here the algorithm that we use to derive the coefficients. Instead of k we shall evaluate the derivative [31,32]: where the overdot stands for k d dk . The advantage of this procedure is that this quantity is convergent. Independently of the renormalizability properties of the theory, it defines a flow on the space of all couplings that are permitted by the symmetries of the system. The divergences of (4.1) in the limit k → ∞ are then found by integrating the differential equation (4.3) from some initial scale k 0 up to k. The coefficients A 1 , B 1 , β α , β β and β γ enter in the beta functions of the couplings , G, α, β, γ , but we postpone a discussion of this point to Sect. 7.
The technique employed to evaluate the right hand side of (4.3) is similar to the one described in I, but this time the cutoff is taken to be a function of the Lichnerowicz Laplacians instead of the Bochner Laplacian −∇ 2 , as in I. In each spin sector, the Lichnerowicz operator L is replaced by the regularized one P k Tr where k are the kinetic operators that appear in each spin sector and the numerator is˙ k =Ṙ k ( L ) = 2k 2 θ(k 2 − L ). The step function cuts off the trace to eigenvalues of L that are smaller than k 2 . For example in the spin-2 sector (4.5) where W ( and primes denote derivatives with respect toR. The coefficients Q n and the heat kernel coefficients b n are listed in Appendix A. Similar formulas hold for the spin-1 and spin-0 sectors and for the ghosts. In the scalar term, k is a two-by-two matrix, and the fraction has to be understood as the product of˙ (0) k with the inverse of (0) k . The functional trace thus involves also a trace over this two-by-two matrix. With these data one can write the expansion of (4.4) in powers ofR, and comparing with (4.3) one can read off the coefficients A 1 , B 1 and C 1 .

Results
From now on we will deal with dimensionless variables In I, the couplingsα andβ were absent, andG only appeared in the Hessian through the overall prefactor Z N . Since this prefactor canceled between numerator and denominator in (4.4), the divergences were independent ofG. This does no longer happen in the present theory, so the coefficients A 1 , B 1 and C 1 depend in general on the dimension d, on the measure parameters m and ω, on the gauge parameters a and b, and on the couplings˜ ,G,α andβ. The general expressions for the coefficients of the divergences are extremely complicated and not instructive. We will therefore only write them in certain limits where they simplify enough. Compared to I, we have to take into account the additional dependence on the couplingsG,α andβ. We will consider two limiting situations. One is the limitα → 0 and β → 0. In this case the beta functions should reduce to those of Einstein-Hilbert gravity and match with those of I (modulo scheme dependences, due to the different form of the cutoff in this paper). Still, the present results are more general because we consider a generic Einstein background, which allows us to distinguish at least two of the higher-derivative couplings, whereas in I a maximally symmetric background was used, allowing us to calculate the coefficient of a single combination of the higher-derivative couplings. We will refer to this as the "Einstein-Hilbert limit" (EH limit). 2 The other limit consists in takingZ N → 0, which is equivalent toG → ∞. In this case one is left with functions of the higher-derivative couplings only. We will refer to this as the "four-derivative gravity limit" (4DG-limit).
There is still the freedom of choosing between the twoderivative and four-derivative gauges. In Sects. 4.3 and 4.4 we will consider the 4DG-and the EH-limits of the theory, using the two-parameter family of two-derivative gauges introduced in Sect. 2.6. In Sect. 4.5 we will discuss the changes that occur when using the two-parameter family of four-derivative gauges introduced in Sect. 2.7. It will turn out that in order to take the EH-and 4DG-limits, different choices have to be made regarding the overall gauge-fixing coefficient Z GF . These are spelled out in detail in the following sections.
4.3 The 4DG-limit (Z N → 0) For this calculation we setZ GF = 1. The coefficients A 1 , B 1 and D 1 turn out to be universal (i.e. independent of the gauge and parametrization) in any dimension: /2) , The coefficient C 1 has a complicated dependence on m, ω, a, b,α andβ, which we do not report here. A special case will be given below in (4.20). However, in d = 4 all four coefficients, including C 1 , are universal: 3πβ , This is an explicit consequence of the statement, made in Sect. 3.1, that the one-loop effective action of higherderivative gravity is independent of the parametrization and gauge choice in d = 4.
The expressions for C 1 and D 1 given above are in agreement with the standard results for the beta functions in HDG [9,10,33] (also derived by means of (4.4) in [24,[34][35][36]): 180 . (4.10) Let us consider what happens if we chooseZ GF =Z N , as is usually done in Einstein-Hilbert theory. In the 4DGlimit, the coefficients A 1 and D 1 are still given by the formulas given above, but B 1 and C 1 are different, and gaugedependent. However, for the exponential parametrization (ω = 1 2 ) in d = 4, one gets the m-independent result We note that in the limit a → 0 these reproduce the universal formulas given above. This observation agrees with the discussion in the end of Sect. 2.6. In the 4DG-limit, at fixed k, one setsZ N = 0. For fixed a and k, and with the choicẽ Z GF =Z N , this implies that the gauge-fixing term vanishes too. The situation can be fixed by letting a → 0, with the ratioZ GF /a fixed and finite. In conclusion, the gauge dependence that occurs in (4.11) for a = 0 is an artifact of a bad gauge-fixing procedure. The correct result is given by (4.9). This problem does not arise in the standard calculation of the beta functions of HDG in d = 4, because there the gauge fixing is of the type considered in Sect. 3.2. When the gaugefixing term contains four derivatives, its overall coefficient is dimensionless in d = 4 and there is never the temptation to introduce a factorZ N .

The EH limit
Let us now consider the EH limitα,β → 0. In this case it is not appropriate to setZ GF = 1, as we did in the preceding subsection, because of the issue in the gauge fixing discussed in the end of Sect. 2.6. Indeed if we insisted on this choice we would find that for˜ = 0 only A 1 and D 1 are universal: (4.12) whereas B 1 and C 1 would containG and diverge forG → 0. For this reason we switch now to the choiceZ GF =Z N , which is more appropriate to discuss this limit. We find that A 1 and D 1 are still given by (4.12). In d = 4 we have (4.13) We note thatG does not appear in these expressions, so that they have a smooth limitG → 0. The expression for C 1 is still too long to be written but is likewise independent ofG. We give it only for two cases: in the linear parametrization m = ω = 0 we have and in the exponential parametrization ω = 1/2, in which case the result is automatically independent of m, we have (4.15) One can plot these functions for fixed parametrization or for fixed gauge. One obtains plots that are very similar to those shown in I. They will not be repeated here.
We report for completeness the expressions B 1 and C 1 for d = 4 in the "unimodular physical" gauge b → ±∞, a → 0, forα,β,G all tending to zero: (4.16)

Results with the four-derivative gauge fixing
In the preceding sections we used the two-derivative gaugefixing terms introduced in Sect. 2.6. We now discuss briefly the results when the four-derivative gauge-fixing terms of Sect. 2.7 are used instead. The formal discussion of Sect. 3 indicates that in the 4DGlimit in four dimensions, the divergences should be universal. Explicit calculation confirms that in this case the coefficients A 1 , B 1 , C 1 , D 1 are indeed given again by (4.9). Furthermore, the coefficients A 1 , B 1 and D 1 are universal in any dimension and agree with those given in (4.8).
In the EH limit the coefficients A 1 , and D 1 in d = 4 are again as in (4.13), but the others are slightly different. For B 1 one has in general Similarly the coefficient C 1 in the linear parametrization m = ω = 0 is and in the exponential parametrization ω = 1/2: The result in the unimodular physical gauge b → ±∞, a → 0 agrees with (4.16).

Exponential parametrization and unimodular gauge
It was found in I that, choosing the exponential parametrization ω = 1/2 and the unimodular gauge b → ±∞, all dependence on the other two parameters m and a, as well as the dependence on the cosmological constant˜ , drops out. This result holds true also in the present context, in any dimension and independently of whether the gauge fixing contains two or four derivatives and independently of the treatment of the constantZ GF . The resulting coefficients are still too cumbersome to write, so we give them again only in two limits.
In the 4DG-limit the coefficients A 1 , B 1 and D 1 are given by (4.8) and In the EH limit we have We note that the formulas for B 1 and C 1 + 2 d(d−1) D 1 do not agree with Eq. (5.7) in I. This is due to the different choice of cutoff (a function of the Bochner Laplacian in I and of the Lichnerowicz Laplacians here). Specializing to d = 4, the formula for C 1 + 2 d(d−1) D 1 agrees with (5.6) in I but B 1 does not. This is a signal of the lack of universality of this coefficient.

Conformal gravity in d = 4
Independently of the choice of gauge and parametrization, the effective action for conformal gravity is given formally by (3.10). This leads to the following flow for the effective actioṅ Tr where (2) Proceeding as before we find We observe that the logarithmic divergences are given by The coefficients C 1 and D 1 agree with previous calculations reported in [37,38].

Duality
One of the main results of I was the existence of a discrete idempotent transformation leaving the functional measure, and the divergences, invariant. That invariance extends also to HDG, at least on an Einstein background. In any dimension and in any gauge we find that the divergences A 1 , B 1 , C 1 and D 1 have the duality symmetry: This duality is only manifest if the coefficient b enters in the gauge fixing (2.47) through the combinationb = b(1 + dm). With other definitions of this coefficient a form of duality will still be present but it will have a much more complicated form. The calculations reported here indicate that this duality is not limited to Einstein gravity. It will be interesting to understand more generally under what circumstances this property holds.
A transformation can be an invariance of a quantum theory if it leaves invariant the action (and its expansion) and the functional measure. As discussed in the introduction, in our one-loop calculations we keep the functional measure fixed at (1.8) and the origin of the ωand m-dependence of the results must lie in the form of the Hessians. One can indeed check explicitly that the Hessians given in Sects. 2.5 and 2.6 are duality-invariant.
Furthermore, it was observed in I that the functional measures where γ μν andĥ μν have weight w, and where γ μν andĥ μν have weight w , are equivalent. These weights are related by , which is the same as m = −m − 2 d .
Thus, if we keep the same action, the functional integrals are equivalent. Again, these measures would give rise to different power-law divergence coefficients, but duality would still hold for each choice of measure. Conversely, we could understand the invariance of the Hessian as follows. We have the functional integral where γ μν is related to the metric by [1] γ μν = g μν (det g) w/2 . (6.4) Solving (6.4) for g μν and substituting it into (6.3), we have with m = − w/2 1+dw/2 . Simply rewriting γ μν as g μν , we are lead to This is a functional integral with fixed quantum field, but the metric in the action is transformed. Viewed this way, the invariance should appear as an invariance of the action with the quantum field kept fixed. This is precisely the calculation we have given. The above discussion by the invariance of the measure suggests that duality is exact, but our calculation is done only at one loop. It would be interesting to check if the latter approach also gives the exact result. The existence of the duality is, however, more general. If we used a functional measure that contains the determinant of a "De Witt" metric in functional space, the measure itself would be invariant under arbitrary field redefinitions [39]. In particular, it would be independent of m and of the choices ω = 0, 1/2, 1. Such a measure would give rise to powerlaw divergence coefficients from the ones reported here, but duality would again appear because it is an invariance of the Hessian.

Concluding remarks
In this paper we have extended the analysis of I [1] from Einstein-Hilbert gravity to higher-derivative gravity containing the squares of the Ricci scalar and Ricci tensor. In four dimensions the analysis is essentially complete, because the remaining independent invariant is a total derivative. In higher dimensions this is not so. The analysis was also limited to Einstein backgrounds, which nearly solve the equations of motion, but also in this way is more general than the analysis in I, which was limited to maximally symmetric backgrounds.
We have obtained formulas for the one-loop divergences up to quadratic terms in the curvature. These are all the divergences that arise at one loop in d = 4. The method we have used is a one-loop approximation of the single-field approximation of the exact RG equation for gravity, as first derived in [31] for the EH-case and then extended to HDG in [24,[34][35][36], to 3d topologically massive gravity in [40], and beyond the one-loop approximation in [41][42][43][44].
The coefficients are related to the beta functions for the couplings. Comparing the action (2.1) with our results (4.3), we find the beta functions for the dimensionless couplings (4.6) as As already remarked, working on an Einstein space prevents us from disentangling the beta functions ofα andβ. On the other hand, even though we did not write a term γ R μνρσ R μνρσ in the action, the divergence D 1 gives rise to a beta function βγ = −(d − 4)γ + D 1 . (7. 2) The signs of the coefficients imply that α, β, γ are asymptotically free in d = 4 (for a suitable range of parameters) [6][7][8][9][10] whereasG goes in the UV to a fixed pointG = −(d −2)/B 1 .
In order for this nontrivial fixed point to make sense, B 1 must be negative. In spite of the non-universality of the beta functions, many calculations indicate that B 1 is indeed negative for the linear split. For example, B 1 in (4.13) is negative for m = ω = 0 and a wide range of a and b. With the exponential parametrization discussed in Sect. 4.6, B 1 in the EH limit given in (4.21) is negative for 3 ≤ d ≤ 13. We refer the reader to the literature for a more detailed discussion. The explicit coefficients of the divergences, in arbitrary dimension, gauge and parametrization, are too complicated to write, and we have exhibited only some special cases. The universal values agree with the literature. The divergences also agree, in d = 4 and in the EH limit, with earlier calculations in general gauges [29,[45][46][47][48]. For further discussions on the use of the exponential parametrization see [27,[49][50][51][52][53][54].
Most of the general features noted in I persist in the theories considered here, as we have observed. In particular, one striking feature that we have checked in full generality is the existence of a "duality" symmetry under the change of parametrization (1.7), or As observed in I, this transformation preserves the dimension of the quantum field and is also an invariance of the functional measure. 3 However, the parametrization dependence of the divergences is not due to the actual choice of the functional measure. Instead, it comes entirely from the different form of the Hessians in different parametrizations. Since these Hessians only differ by terms that are proportional to the equations of motion, the parametrization dependence, as well as the gauge dependence, goes away on shell. Different choices of the ultralocal functional measure would alter the results for the power-law divergences coefficients. We refer to [57] and the references therein for a discussion of this point and to [58,59] for more general results using Pauli-Villars regularization.
Whether the duality extends also to other classes of actions, to other backgrounds and to higher loops are all questions that we leave for further investigation. Also left for future work is the calculation of divergences in the unimodular case m = −1/d, which contains in particular the unique self-dual theory ω = 1/2, m = −1/d.
These formulas can be obtained by the methods described for example in Appendix B of [60]. Here we do not take into account isolated modes that do not contribute to the fluctuation h μν . Thus these formulas hold in the case when the manifold has no Killing or conformal Killing vectors, or else if the manifold is noncompact and has a continuous spectrum, so that the spurious isolated modes are of measure zero. In order to get the heat kernel coefficients of shifted Lichnerowicz Laplacians + a R, one can use