Gauges and functional measures in quantum gravity II: higherderivative gravity
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Abstract
We compute the oneloop divergences in a higherderivative theory of gravity including Ricci tensor squared and Ricci scalar squared terms, in addition to the Hilbert and cosmological terms, on an (generally offshell) Einstein background. We work with a twoparameter family of parametrizations of the graviton field, and a twoparameter 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.
1 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 fourparameter family of gauges and parametrizations. In this paper, we would like to extend the analysis to higherderivative gravity (HDG).

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 \(m_\mathrm{phys}^2=m^2(k=m_\mathrm{phys})\) (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 nonlocal gravity and also [14]).
In this paper we will extend previous results in several directions. Due to the complicated structure of the theory, calculations of oneloop divergences in HDG have usually been performed with a special fourderivative gaugefixing term such that the fourderivative part of the Hessian is proportional to the square of the Laplacian. In this paper we will calculate the offshell gauge dependence of the oneloop divergences by using the more conventional secondderivative gaugefixing 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 \(\nabla ^2\) as in I. Thus, comparison with I yields some information on the cutoff dependence of these nonuniversal 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].
Since the functional measure was always kept fixed, the dependence of the results on the parameters m and \(\omega \) 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)).
We conclude this introduction by listing the contents of the following sections. Section 2 contains details of our calculations of oneloop divergences. In Sect. 3 we give a formal proof (at the level of determinants) that the onshell effective action in \(d=4\) is gauge independent. (In particular, the effective action of HDG in the 4DGlimit on an Einstein space is gaugeindependent.) 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.
2 The oneloop effective action
2.1 The HDG actions and their equations of motion
2.2 Quadratic expansion
In the following we will use these expansions in the action (2.1). We note that, for given \(g_{\mu \nu }\) and \(\bar{g}_{\mu \nu }\), different values of \(\omega \) and m will give different fluctuation fields \(h_{\mu \nu }\) and conversely for given \(h_{\mu \nu }\) and \(\bar{g}_{\mu \nu }\), different values of \(\omega \) and m will give different total metrics \(g_{\mu \nu }\). 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.
2.3 Lichnerowicz Laplacians
2.4 York decomposition
2.5 The decomposed Hessian
On the other hand, this form of the Hessian has the unpleasant feature that the kinetic operator of the field s is nonlocal. One cannot generally absorb the nonlocal prefactor in a redefinition of s, because one is not allowed to perform nonlocal redefinitions of physical fields.
One notices, however, that the terms with the lowest power of \({\Delta _L}_0\) in each of the three lines in \(H_s^{ss}\) is proportional to \((1+dm)(12\omega )\). Therefore, if either \(\omega =1/2\) (exponential parametrization) or \(m=1/d\) (the “unimodular” measure), each of the square brackets in (2.42) is proportional to \({\Delta _L}_0\) and the Hessian of s becomes local.
We further observe that for \(\omega =1/2\) also the mixed term vanishes and the term \(H_s^{hh}\) becomes local, whereas for \(m=1/d\) all terms containing h vanish, as expected in the unimodular theory.
2.6 Twoderivative gaugefixing terms
Let us observe that \(Z_\mathrm{GF}\) appears in the combination \(Z_\mathrm{GF}/a\), where a is a dimensionless gauge parameter, while \(Z_\mathrm{GF}\) is a constant of dimension \(d2\). There are then two natural options regarding the constant \(Z_\mathrm{GF}\). 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 \(G\rightarrow 0\) the coefficient \(Z_N\) of the Hessian diverges. If we keep \(Z_\mathrm{GF}\) and a constant in the limit, then the gaugefixing 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 \(Z_\mathrm{GF}\) fixed and letting simultaneously \(a\rightarrow 0\). Alternatively, one can set \(Z_\mathrm{GF}=Z_N\). In this case, in the Gaussian limit \(G\rightarrow 0\), the kinetic terms of the gaugeinvariant 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 socalled “unimodular physical gauge”, where one sets \(\hat{\xi }_\mu =0\) and \(h=0\). As shown in [27], this is equivalent to the above standard gauge in the limit \(b \rightarrow \pm \infty \) and \(a\rightarrow 0\).
2.7 Fourderivative gaugefixing terms
In HDG it is customary to use gaugefixing 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.
Note also that the field \(B_\mu \) was an auxiliary field in the twoderivative gaugefixing case (2.52), but here it is dynamical. With higherderivative gauge fixing, quite often only the contribution \(\Delta ^{(gh)}_{\mu \nu }\) is incorporated but that from \(Y^{\mu \nu }\) in the FP ghost kinetic term is ignored, and then it is claimed that somehow the contribution from the “third ghost” \(\frac{1}{2} \log (\det (Y_{\mu \nu }))\) must be added. We see here that this is automatic in the BRST invariant formulation, because we have contributions \(\log (\det (Y^{\mu \nu }))\) from the FP ghost kinetic term and \(\frac{1}{2} \log (\det (Y^{\mu \nu }))\) from the field \(B_\mu \), giving the same result.
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 \({\Delta _L}_1\) in the quadratic gaugefixing term (2.52), or in other words to set \(Y^{\mu \nu }=\bar{g}^{\mu \nu }{\Delta _L}_1\). After performing the York decomposition, this yields additional factors \({\Delta _L}_1\) and \({\Delta _L}_0\) in the quadratic actions of \(\xi _\mu \) and \(\chi \), Eq. (2.57). The resulting additional determinants are offset by the determinant of the operator coming from the \(B_\mu \) sector, as will become clear in the following.
3 Universality onshell 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 gauge and parametrizationindependent. 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.
3.1 General twoderivative gauge fixing
3.2 General fourderivative gauge fixing
3.3 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” gauge \(\hat{\xi }_\mu =0\), \(h=0\), discussed in [27, 28]. It leaves only the fields \(h^{TT}\) and \(\hat{\sigma }\), and no Jacobians. The Hessians of \(h^{TT}\) and \(\hat{\sigma }\) are given by (2.35) and (2.37), respectively. There are a real scalar ghost and a real transverse vector ghost, with ghost operators \(\Delta _{L0}\) and \(\Delta _{L1} R/2\). 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.
3.4 The conformal case
Now we consider the conformal case where \(\beta /\alpha =3\). 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 spin2 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 \(\chi =\sigma \), so the decomposition of the gaugefixing and ghost actions, and the corresponding determinants, are the same as in Sect. 2.6, with \(b=0\).
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], namely \(\hat{\sigma }=0\) and \(\hat{\xi }=0\). This is equivalent to taking the Landau gauge limit \(a\rightarrow 0\). The ghosts are a real scalar and a real transverse vector and the ghost operators are \(\Delta _{L0}\bar{R}/3\) and \(\Delta _{L1}\bar{R}/2\). The effective action is given again by (3.10).
3.5 The Einstein–Hilbert case
4 The divergences
4.1 Derivation
Similar formulas hold for the spin1 and spin0 sectors and for the ghosts. In the scalar term, \(\Delta _{k}^{(0)}\) is a twobytwo matrix, and the fraction has to be understood as the product of \(\dot{\Delta }_{k}^{(0)}\) with the inverse of \(\Delta _{k}^{(0)}\). The functional trace thus involves also a trace over this twobytwo matrix. With these data one can write the expansion of (4.4) in powers of \(\bar{R}\), and comparing with (4.3) one can read off the coefficients \(A_1\), \(B_1\) and \(C_1\).
4.2 Results
Compared to I, we have to take into account the additional dependence on the couplings \(\tilde{G}\), \(\tilde{\alpha }\) and \(\tilde{\beta }\). We will consider two limiting situations. One is the limit \(\tilde{\alpha }\rightarrow 0\) and \(\tilde{\beta }\rightarrow 0\). 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 higherderivative couplings, whereas in I a maximally symmetric background was used, allowing us to calculate the coefficient of a single combination of the higherderivative couplings. We will refer to this as the “Einstein–Hilbert limit” (EH limit).^{2}
The other limit consists in taking \(\tilde{Z}_N\rightarrow 0\), which is equivalent to \(\tilde{G}\rightarrow \infty \). In this case one is left with functions of the higherderivative couplings only. We will refer to this as the “fourderivative gravity limit” (4DGlimit).
There is still the freedom of choosing between the twoderivative and fourderivative gauges. In Sects. 4.3 and 4.4 we will consider the 4DG and the EHlimits of the theory, using the twoparameter family of twoderivative gauges introduced in Sect. 2.6. In Sect. 4.5 we will discuss the changes that occur when using the twoparameter family of fourderivative gauges introduced in Sect. 2.7. It will turn out that in order to take the EH and 4DGlimits, different choices have to be made regarding the overall gaugefixing coefficient \(Z_\mathrm{GF}\). These are spelled out in detail in the following sections.
4.3 The 4DGlimit \((\tilde{Z}_N \rightarrow 0)\)
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 factor \(\tilde{Z}_N\).
4.4 The EH limit
4.5 Results with the fourderivative gauge fixing
In the preceding sections we used the twoderivative gaugefixing terms introduced in Sect. 2.6. We now discuss briefly the results when the fourderivative gaugefixing 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).
4.6 Exponential parametrization and unimodular gauge
It was found in I that, choosing the exponential parametrization \(\omega =1/2\) and the unimodular gauge \(b\rightarrow \pm \infty \), all dependence on the other two parameters m and a, as well as the dependence on the cosmological constant \(\tilde{\Lambda }\), 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 constant \(\tilde{Z}_\mathrm{GF}\). The resulting coefficients are still too cumbersome to write, so we give them again only in two limits.
5 Conformal gravity in \(d=4\)
6 Duality
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 oneloop calculations we keep the functional measure fixed at (1.8) and the origin of the \(\omega \) and mdependence 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 dualityinvariant.
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 \(\omega =0,1/2,1\). 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.
7 Concluding remarks
In this paper we have extended the analysis of I [1] from Einstein–Hilbert gravity to higherderivative 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 oneloop 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 oneloop approximation of the singlefield approximation of the exact RG equation for gravity, as first derived in [31] for the EHcase and then extended to HDG in [24, 34, 35, 36], to 3d topologically massive gravity in [40], and beyond the oneloop approximation in [41, 42, 43, 44].
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].
Different choices of the ultralocal functional measure would alter the results for the powerlaw 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 selfdual theory \(\omega =1/2\), \(m=1/d\).
Footnotes
Notes
Acknowledgements
This work was supported in part by the GrantinAid for Scientific Research Fund of the JSPS (C) No. 16K05331. ADP is grateful to CNPq for financial support.
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