# The background scale Ward identity in quantum gravity

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## Abstract

We show that with suitable choices of parametrization, gauge fixing and cutoff, the anomalous variation of the effective action under global rescalings of the background metric is identical to the derivative with respect to the cutoff, i.e. to the beta functional, as defined by the exact RG equation. The Ward identity and the RG equation can be combined, resulting in a modified flow equation that is manifestly invariant under global background rescalings.

## Keywords

Ghost Ward Identity Fluctuation Field Auxiliary Field Exponential Parametrization## 1 Introduction

One is to study bi-metric truncations [4, 5, 6] and impose shift-invariance only in the IR limit [7]. If one were able to calculate the whole bi-metric flow, then the background flow could be obtained by setting the classical fields to zero. Thus in practice one method to improve on single-field truncations is to keep as much as possible of the fluctuation dependence, by calculating the flow of the two-, three- and possibly four-point functions of the fluctuation [8, 9, 10, 11, 12]. Alternatively one can try to solve simultaneously the Ward identity and the flow equation. This could be achieved in the conformally reduced case [13, 14, 15]. Other related ideas have been discussed in [16, 17, 18].

A step forward has recently been made by Morris for the special case when \(\epsilon _{\mu \nu }=2\epsilon \bar{g}_{\mu \nu }\), i.e. when the background is simply rescaled by a constant factor [19]. He derived the modified Ward identity for this transformation and showed that in six dimensions the anomalous terms coming from the cutoff have the same form as the RG equation. In this way the Ward identity and the RG equation can be combined in a single equation that is amenable to explicit treatment by the methods that are in current use. The drawback of the proposed procedure is that it only seems to work in six dimensions.

The second step is to make sure that no dimensionful parameter enters the gauge-fixing term. In the Einstein–Hilbert truncation it is convenient and customary to have a prefactor \(Z_N=1/(16\pi G)\), so that the gauge-fixing terms combine smoothly with the Hessian, but this introduces and unnecessary and, as we shall see, unwanted breaking of background scale invariance. We will use a higher-derivative gauge fixing, which amounts to introducing some power of the Laplacian in the gauge-fixing term. This type of gauge fixing is often used with four-derivative gravitational actions [37, 38, 39, 40] but normally not in the Einstein–Hilbert truncation. There is, however, no fundamental reason for this, other than simplicity [31].

The third step is to similarly avoid dimensionful parameters in the cutoff term, except for the cutoff scale itself. We will use a “pure” cutoff, namely one that does not contain any running parameter [33, 41]. As with the gauge-fixing term, in the Einstein–Hilbert truncation it is convenient to have a prefactor \(1/(16\pi G)\). In the *f*(*R*) truncation the corresponding prefactor is \(-f'(R)\). This dependence of the cutoff on running couplings is, however, the source of unnecessary anomalies.

We will see that, with these choices, the gauge fixing becomes invariant and the anomalous terms in the Ward identity coming from the cutoff have the same form as the RG equation. Then the Ward identity expresses the invariance of the effective action under the transformation of the background, fluctuation and a simultaneous rescaling of the cutoff scale. This identity can be solved and results simply in the definition of new variables that are invariant under background scale transformations. The RG equation, written in these variables, no longer depends on the scale of the background metric and has the same form as the flow equation that is commonly used. Although for the time being limited to simple scalings, this points toward a practical solution of the background-field dependence.

In Sect. 2 we discuss the transformation of the fields and of the gauge-fixing and cutoff actions. In Sect. 3 we derive the Ward identity and combine it with the RG equation. Section 4 contains a short discussion.

## 2 Variations

### 2.1 Fields

*X*by the symbol

*h*, but this would give rise to ambiguities when indices are suppressed.) Note that

*X*is a linear map of the tangent space to itself, so powers of

*X*and the trace of

*X*do not require use the metric and are basis-independent.

*h*is due to the constant component, while \(h^\perp \) is invariant:

### 2.2 Gauge fixing

*h*that, used in (1.3), yield (2.19). The simplest one is the background transformation. We use again matrix notation as in the preceding section. If we treat \(\bar{\mathbf{g}}\) and \(\mathbf {X}\) as tensors under \(\delta _\eta \), i.e.

^{1}:

*C*,

*Y*from the effective action. Scale invariance is achieved provided the auxiliary field is inert: \(\delta B_\mu =0\).

We note that the procedure proposed here is by no means unique. If one is interested mainly in the application of the formalism to *f*(*R*) theories [39, 42, 43, 44, 45, 46, 47, 48, 49], where one normally considers a spherical background, then one could define \(Y^{\mu \nu }=\bar{R}^{\frac{d-2}{2}}\bar{g}^{\mu \nu }\). This achieves scale invariance without having to introduce an auxiliary field, but it would not work on a flat background. One could also have a mix of \(\bar{\Delta }\) and \(\bar{R}\), provided the overall power is \(\frac{d-2}{2}\). Yet another choice would be the “physical gauge” advocated in [24]. In this case one would just set \(h^\perp =0\) and \(\xi _\mu =0\), where \(\xi _\mu \) is the spin-one degree of freedom of \(h_{\mu \nu }\). Since \(h^\perp \) is invariant and \(\xi _\mu \) trasforms homogeneously under scaling, these conditions are scale invariant. They produce Faddeev–Popov determinants that can be taken care of by introducing suitable auxiliary fields.

### 2.3 Cutoff term

*a*and

*c*are dimensionless constants and \(R_k(0)=k^2\), with

*k*the IR cutoff scale which controls the coarse-graining procedure. Usually one defines the RG “time” as \(t \sim \ln k\). By dimensional analysis

*r*is a dimensionless function that goes rapidly to zero for \(y>1\) and \(r(0)=1\).

In the Einstein–Hilbert truncation and in de Donder gauge it is very convenient to choose \(a=-1\), so that the tensor structure matches the one of the Hessian (including the gauge-fixing term). Furthermore, it is almost always assumed that \(c\, k^{d-2}=1/(16\pi G)\). Then the cutoff combines seamlessly with the Hessian resulting simply in the substitution of \(\bar{\Delta }\rightarrow \bar{\Delta }+R_k(\bar{\Delta })\), where in this specific case \(\bar{\Delta }=-\bar{\nabla }^2\). We are not committed to using any specific form of the action here, so we leave the constants *a* and *c* unspecified. Such a cutoff is then called “pure” to emphasize that it does not contain any running coupling.

*k*does not change under a variation of the background metric, we find from (2.32) \(\delta \mathcal{R}_k=-2\epsilon k^d y r'\). On the other hand \(\partial _t \mathcal{R}_k=d k^d r-2k^d y r'\), so

*h*.

## 3 The Ward identity

*j*, \(J_*^\mu \), \(J^\mu \) and \(K^\mu \) have to be interpreted as usual as functionals of these classical fields and the last three term subtracts the cutoff that had been added in the beginning to the bare action.

*k*held fixed, which can be expressed as

*h*, since the argument of \(\Gamma _k\) are always the classical expectation values and no confusion can arise.

## 4 The Ward identity and the flow equation

*k*fixed. Bringing the r.h.s. to the l.h.s. we obtain

*k*:

^{2}The solution of the Ward identity is therefore a functional

If we were able to solve the full Ward identities related to arbitrary deformations of the background, we would obtain a functional \(\hat{\Gamma }_{\hat{k}}(\hat{g}_{\mu \nu })\) that would satisfy a flow equation containing its second derivatives with respect to \(\hat{g}_{\mu \nu }\). Having only partly transferred the field dependence from the fluctuation field to \(\hat{g}_{\mu \nu }\), we will have a flow equation containing second derivatives with respect to the remaining fluctuation fields and second derivatives with respect to those deformations of \(\hat{g}_{\mu \nu }\) that have become dynamical as a result of solving the Ward identity. (In the present case, this is just the overall scale of \(\hat{g}_{\mu \nu }\).) This distinction obviously gets blurred when one uses the single-metric approximation.

## 5 Discussion

The main outcome of this paper is the generalization of the results of [19] for the background scale Ward identity in quantum gravity. Morris was able to show that in six dimensions the violation of background scale invariance is given exactly by the r.h.s. of the RG equation. This is reminiscent of the statement that in a classically scale-invariant quantum field theory in flat space, such as massless QCD, the violation of scale invariance is proportional to the beta functions. The physical meaning of the identity is different in the two cases: in QCD it is a genuine anomaly, whereas in quantum gravity the anomalous variation under a change of background can be absorbed by a change of the fluctuation field \(\underline{h}\) and of the cutoff *k*, as we have seen in the preceding section. Nevertheless, the two statements are formally the same, and one would expect such general statements to be true in any dimension. Indeed we have shown here that this is the case.

To get this result, however, one has to make certain choices that minimize the breaking of scale invariance. The main difference with [19] is the use of the exponential parametrization for the metric (1.3). When the linear split (1.1) is used, invariance of the full metric requires that the fluctuation field has a transformation opposite to the one of the background field, with the exception of the trace that has a mixed transformation consisting of a homogeneous and an inhomogeneous term. With the exponential parametrization, the fluctuation field transforms in the same way as the background metric, with the exception of the trace that transforms purely by a shift, in much the same way as a dilaton. These transformation rules merely reflect the dimensions of the fields (when the coordinates are dimensionless and a metric has given dimension of area) and the remaining choices also follow the dimensions of each field. The other differences are in the gauge-fixing and cutoff terms: one has to make sure that these do not contain dimensionful couplings that would introduce additional unwanted scale-breaking terms. Of course, it is unavoidable to break background scale invariance by introducing the cutoff scale *k*, but the main point of the present exercise has been to show that if this is the only source of scale-breaking, its effect is entirely contained in the RG flow of the couplings. To this effect, in \(d>2\) we have used a higher-derivative cutoff, such as is used in higher-derivative gravity, and a “pure” cutoff, which does not contain any Lagrangian parameter. We stress that with this gauge fixing we are able to prove invariance of the ghost action including all ghost interactions. There may be other procedures that also work well, but these three choices are sufficient to ensure that the Ward identity does not contain additional, unnecessary anomalous terms.

The Ward identity can be used to reduce the number of variables that the effective average action depends upon. Ultimately one would like to reduce the flow equation for \(\Gamma _k(h_{\mu \nu };\bar{g}_{\mu \nu })\) to a flow equation for a functional of a single field \(\hat{\Gamma }_{\hat{k}}(\hat{g}_{\mu \nu })\). Reference [19] and the present work are first steps towards background independence: we have shown how to eliminate from the RG flow the dependence on a single real degree of freedom: the overall scale of the background. This may look like a rather small step, but without it the beta functions are likely to contain spurious terms. We plan to investigate this in concrete calculations. The main value of the present work may lie in restricting the freedom of choice of parametrization, gauge and cutoff scheme. Equation (4.1) is an important statement, even if restricted to constant Weyl transformations: it is expected of any quantum field theory that is invariant under rescalings of the background metric at the classical level. One should be wary of using parametrizations and/or cutoff schemes that violate it.

## Footnotes

## Notes

### Acknowledgements

G.P.V. thanks Jan M. Pawlowski for interesting discussions.

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