A note on the total action of $4D$ Gauss-Bonnet theory

Recently, a novel four-dimensional Gauss-Bonnet theory has been suggested as a limiting case of the original $D$-dimensional theory with singular Gauss-Bonnet coupling constant $\alpha\rightarrow\alpha/(D-4)$. The theory is proposed at the level of field equations. Here we analyse this theory at the level of action. We find that the on-shell action and surface terms split into parts, one of which does not scale like $(D-4)$. The limiting $D\rightarrow4$ procedure, therefore, gives unphysical divergences in the on-shell action and surface terms in four dimensions. We further highlight various issues related to the computation of counterterms in this theory.


Introduction
Higher-order curvature terms are expected to play a central role in quantum gravity. It is generally expected that the low-energy expansion of the quantum gravity, such as string theory, will provide an effective Lagrangian containing higher-order curvature terms [1]. Finding and analysing the solutions of higher curvature Lagrangian are therefore of great physical interest.
One of the most studied higher curvature Lagrangian is the Gauss-Bonnet combination (1.1) With L GB , the Einstein equations of motion still remain second order in metric and it provides the simplest non-trivial modification of general relativity. The Einstein-Gauss-Bonnet gravity action [2][3][4][5] leads to the following field equation and admits a consistent and non-trivial solutions for D ≥ 5 ds 2 = −f (r)dt 2 + dr 2 f (r) + r 2 dΩ 2 D−2 , D ≥ 5 .
( 1.4) where dΩ 2 D−2 is the unit metric of the (D − 2)-dimensional sphere. In eq. (1.3), the Gauss-Bonnet contribution to field equations reads (1.5) Importantly, the Gauss-Bonnet term reduces to the Euler number (or to a total derivative term) in four dimensions and therefore does not contribute to the field equations. In particular, H µν vanishes identically in D = 4. Therefore, it came as a big surprise when a four-dimensional Einstein-Gauss-Bonnet theory was constructed in [6]. The authors of [6] suggested that (i) by rescaling the Gauss-Bonnet coupling parameter α →α/(D − 4), and then (ii) taking the limit D → 4, a non-trivial black hole solution in four dimensions can be obtained as a limiting case of the D-dimensional theory. Interestingly, this novel four-dimensional Gauss-Bonnet theory bypasses the Lovelock's theorem [7][8][9] and, therefore, has created a lot of excitement in the gravitational community.
The essential idea behind the work of [6] was the observation that the Gauss-Bonnet tensors H µν scale like (D − 4) in D-dimensions and this (D − 4) factor can be cancelled consistently in the field equations by modifying the coupling parameter α →α/(D − 4). At the equation of motion level, the four-dimensional Gauss-Bonnet theory was suggested as a limiting case of the original D-dimensional theory The suggested four-dimensional gravity has already intrigued a large amount of research work in applications, see for a necessarily biased selection . Recently, many works addressing various ambiguities, even at the level of field equations, when applying the method of [6] have also started appearing, in particular, see [36,37].
In this work, we want to further explore and scrutinise the four-dimensional Einstein-Gauss-Bonnet theory, however, from the action point of view. Our aim is to analyse how the total action, consisting of various surface terms and counterterms in addition to the Einstein-Gauss-Bonnet action, with singular coupling α behaves in D → 4 limit. The action analysis is essential to understand whether the theory is fundamentally in good shape or not. As is well known, the action (1.2) has to be supplemented by the surface terms to have a well defined variational problem. These surface terms although do not modify the field equation, however, they are an integral part of the gravity action and should be discussed thoroughly in any gravity system. Similarly, counterterms are needed in the total action to make it IR finite. In particular, both on-shell action and surface terms suffer from infinities as the boundary is taken to infinity and these infinities can be removed by adding local counterterms in the action.
Since the surface and counterterms live in one lower dimension, i.e. at the boundary in (D −1) dimensions, therefore one might expect that unlike various tensors (constructed from the D-dimensional metric) appearing in Einstein's field equation (1.3), not all the surface and counterterms would come with a multiplicative (D − 4) factor. If that is the case then the singular coupling constant α will make the whole action divergent in the limit as D → 4. Our analysis suggests that this is indeed the case. In particular, both on-shell and surface terms diverge in the limit D → 4. We further highlight various issues related to the computation of counterterms for the four-dimensional Einstein-Gauss-Bonnet theory.

Total action of four-dimensional Einstein-Gauss-Bonnet gravity
Before separately analysing each term of the total Einstein-Gauss-Bonnet gravity action, it is useful to first note down the expression of L GB and Ricci scalar R in D-dimensions. For the metric (1.4), we have 1 here, and in the subsequent subsections, we write results explicitly in terms of D and f . This will help to analyse the limit D → 4 in a clear and straightforward way, as the function f is well defined in this limit. For simplicity, we derive results by assuming the black hole background. The analysis can be straightforwardly generalised to pure AdS spaces.

The on-shell action
The first indication that the Einstein-Gauss-Bonnet gravity action (1.2) is not well defined in the limit D → 4 can be seen by evaluating the on-shell action, If we let α =α/(D − 4) and take the limit D → 4, then the second term in eq. (2.1) gives finite contribution and is well defined. On the other hand, the last term is although a total derivative term (hence does not contribute to the field equations), however, gives infinite contribution. Another way to see that the on-shell Einstein-Gauss-Bonnet gravity action is not well defined in the limit D → 4 is by noticing that it can also be rewritten as 2 here we have substituted the expression of L GB from the Einstein equation into the action. Notice that the above equation is independent of the singular coefficient α =α/(D − 4). Taking the limit D → 4 will definitely make the action divergent, which suggests that the on-shell action is not well defined in this limit. Expectedly, the action remains well behaved for D ≥ 5.
Here one might argue that the on-shell action does usually contain divergences and the above result may not be problematic. However, the usual divergences in the action generally appear because of the infinite extent of the space time, i.e. r → ∞, and hence those divergences are physical. The Einstein-Gauss-Bonnet theory, on the other hand, gives additional divergences in the limit D → 4, which do not seem to have any physical origin.

Surface terms
For a well-defined variational principle, one has to supplement the action (1.2) with the surface terms. These terms are required so that upon variation with metric fixed at the boundary, the action yields the Einstein equation (1.3). Though these surface terms do not modify the field equations, however, they are essential for a well defined variational problem for a gravitational system having boundaries, like the AdS space. For the Einstein-Hilbert part, the surface term is a well known Gibbons-Hawking boundary term where ω D−2 is the area of the unit (D − 2)-dimensional sphere and β is the inverse temperature. Similarly, the surface term counterpart of the Gauss-Bonnet Lagrangian is [38,41] where G ab is the Einstein tensor of boundary metric and J is the trace of the tensor Let us now explicitly evaluate S S.T GB to see whether it is well defined in the limit D → 4 or not. After a little bit of algebra, one can show that the Gauss-Bonnet surface term in D-dimensions simply reduces to We see that the surface terms associated with the Gauss-Bonnet term can be divided into two parts. Those which contain a multiplicative (D − 4) factor and those which do not. The redefinition of the coupling constant α =α/(D − 4) and the subsequent limit D → 4 are well defined for those terms which contain a multiplicative (D − 4) factor. However, the same can not be said for those terms which do not contain a multiplicative factor of (D − 4). Overall, like for the on-shell action, the limit D → 4 makes the Gauss-Bonnet surface term divergent.
One might wonder whether the sum of the on-shell and the surface terms can make the Gauss-Bonnet contribution to the total action finite in the limit D → 4. To analyse this, let us evaluate On substituting L GB from eq. (2.1) and simplifying, we get The above equation can be further simplified by evaluating the integrals. Using the integration by parts method, we get Notice that the second and fourth integral terms cancel out. Therefore, Substituting eq. (2.11) into eq. (2.10) and simplifying, we finally get Here we have used the fact that f (r h ) = 0. In eq. (2.12), we have rearranged Gauss-Bonnet terms in such a way that one can see the limiting behaviour clearly. In particular, there are no hidden (D − 4) factors in (2.12). We again see that there are terms which do not contain a multiplicative (D − 4) factor. In particular, the first term of second line in eq. (2.12). Since f (r) and f (r) are well behaved functions, the redefinition α =α/(D −4) and the subsequent limit D → 4, therefore will again give unphysical divergences in the total action.

A word about the counter terms
We saw above that both on-shell action and surface terms are divergent in four-dimensional Einstein-Gauss-Bonnet theory. There are mainly two different types of divergence (i) the IR divergence because the volumes of both M and ∂M are infinite, and (ii) divergences due to the limit D → 4. To make sense of the total action one therefore has to regularise the action by eliminating these divergences. Remarkably, for the AdS spacetime, the IR divergences that arise in the total action are all proportional to the boundary metric. By subtracting suitable combinations of curvature scalars constructed from the boundary metric, called counterterms, one can accordingly make the total action finite. This counterterm regularization procedure has a physical interpretation in the AdS/CFT context and leads to a well-defined meaning to the notions of energy and momentum in AdS [42]. One therefore might try to regularise the four-dimensional Einstein-Gauss-Bonnet action by a similar counterterm procedure. However, this is not as straightforward as it seems and there are many subtleties in implementing the counterterm method. In particular, even for the Einstein action, the expression of the counterterms explicitly depend on D and it changes from dimension to dimension. The situation is even more complicated with the Gauss-bonnet action. As far as we know, the general expression of counterterms for the Gauss-bonnet action in arbitrary dimension D is not known 3 . There are numerous counterterms proposals to handle higher-derivative terms, but all of them seem to work in specific dimensions. Moreover, the number of terms and their complexity severally enhance with D [41]. Since the whole idea of [6] is based on the fact that one must first do the computation in D dimensions and then take the limit D → 4, therefore, to obtain consistent counterterms for the four-dimensional Einstein-Gauss-Bonnet theory one must first evaluate them in D dimensions. This is an extremely non-trivial task, as infinitely many terms can contribute to the counterterms in general D. We can certainly evaluate the counterterms for a fixed D (say D = 8), however, then it would not make sense to let it go to four.
One might also try to find the counterterms using an ad-hoc way, for example by guessing them, such that all the divergences in the four-dimensional Einstein-Gauss-Bonnet action cancel out. This is the usual working procedure for gravity theories in AdS space. However, this ad-hoc procedure can not be called physical and considered seriously in the context of four-dimensional Einstein-Gauss-Bonnet theory, as the counterterms are then not obtained from a consistent D → 4 limit, i.e. this ad-hoc procedure will be against the very philosophy of [6]. 3 A different counterterm regularization method, called Kounterterm regularization, for the Gauss-Bonnet gravity in D-dimensions has been suggested in [39,40]. In this method, the total action is rewritten as S = SEGB + cD−1 ∂M d D−1 x BD−1, without explicitly adding the surface terms. Here, cD−1 is a dimension dependent constant and function BD−1 is made up of boundary intrinsic and extrinsic curvatures. The explicit form of BD−1, however, depends on whether D is odd or even. While applying this method, our preliminary analysis suggests that depending upon whether we start from odd or even D, the D → 4 limit might not give a unique answer for the total action of four-dimensional Einstein-Gauss-Bonnet theory. This again sounds problematic for the theory, though more work is needed for confirmation. It will certainly be interesting to perform a detailed analysis of the total action using the Kounterterm regularization method. We leave this exercise for future work.

Concluding remarks
Recently, a novel Einstein-Gauss-Bonnet theory in four dimensions was suggested which not only bypasses the Lovelock's theorem but also contains the same number of massless spin-2 degrees of freedom as the Einstein-Hilbert term. This theory was defined as a limiting case of the original D-dimensional Einstein-Gauss-Bonnet theory with rescaled coupling constant α =α/(D − 4). The main idea was that the (D − 4) factor in the singular coefficient α can cancel the (D − 4) factor that generally appears in the Einstein equations. In this note, we further scrutinised this idea at the action level. We investigated the on-shell action and the corresponding surface terms and showed that these terms are not finite in the limit D → 4. In particular, the singular coefficient α makes the total Einstein-Gauss-Bonnet divergent in the D → 4 limit. The four-dimensional Einstein-Gauss-Bonnet theory therefore seems to be imprecise at least at the action level. We further highlighted various issues related to the counterterms regularisation in the fourdimensional Einstein-Gauss-Bonnet theory.
An interesting question one might ask is, does the novel four-dimensional Einstein-Gauss-Bonnet theory in AdS space exhibit a dual boundary theory. In the AdS D /CF T D−1 context, the gravity and the dual boundary theory are connected in the semiclassical approximation via Z CF T = Z AdS = e −S AdS . (3.1) where e −S AdS is the classical gravitational action. In this approximation, the AdS action becomes the generating function of the connected correlation functions of dual CFT.
Since the gravity action is directly related to the physical observables of the dual CFT theory, it is desirable that S AdS remains free from any divergences. Note that the usual IR divergences in the gravity side correspond to UV divergences in the dual CFT side and therefore have a precise meaning. However, the same can not be said for the divergences that appear due to D → 4 limit. In particular, the introduction of Gauss-Bonnet term in the gravity action corresponds to next to leading order corrections to the 1/N (N being the number of colours) expansion of the dual CFT [43]. Therefore, the four-dimensional Einstein-Gauss-Bonnet action should not contain any divergence whose dual counterpart in CF T 3 does not exist. As we have shown in this work, unless the counterterms miraculously cancel out D → 4 divergences, it seems difficult to make a dual CFT connection of the novel four-dimensional Einstein-Gauss-Bonnet theory.
Note added: While this work was in the final stage of preparation, a way of regularising the action via the addition of a scalar degree of freedom was presented in [35].