Holographic Three-Point Functions from Higher Curvature Gravities in Arbitrary Dimensions

We calculate the holographic three-point function parameters $\mathcal A$, $\mathcal B$, $\mathcal C$ in general $d\geqslant 4$ dimensions from higher curvature gravities up to and including the quartic order. The result is valid both for massless and perturbative higher curvature gravities. It is known that in four dimensional CFT the $a$-charge is a linear combination of $\mathcal A$, $\mathcal B$, $\mathcal C$, our result reproduces this but also shows that a similar relation does not exist for general $d>4$. We then compute the Weyl anomaly in $d = 6$ and found all the three $c$-charges are linear combinations of $\mathcal A$, $\mathcal B$, $\mathcal C$, which is consistent with that the $a$-charge is not. We also find the previously conjectured relation between $t_2$, $t_4$, $h''$ does not hold in general massless gravities, but holds for quasi-topological ones, and we obtain the missing coefficient.


Introduction
In conformal field theory (CFT), conformal invariance typically requires the correlation functions to have fairly rigid forms.For example, in flat spacetime the two-point function of the energy-momentum tensor T ab is completely determined by the parameter C T [1,2] 0|T ab (x)T cd (y)|0 = C T I (0) where d is the spacetime dimension.Similarly, after imposing further constraints that arise from the conservation of energy, the three-point function of T ab is controlled by three parameters A, B, C [1,2], i.e.
0|T ab (x)T cd (y)T ef (z)|0 = A I (1) abcdef + C I Note that the I (0) , I (1) , I (2) and I (3) are tensorial structures whose explicit forms are inessential in our discussion here.These parameters are generally independent in d 4, but for d = 3 and d = 2, the I (i) tensors become degenerate and the number of independent coefficients is two and one respectively.
In a curved spacetime background, CFT in even dimensions becomes anomalous in that the trace of T ab acquires a non-zero expectation value known as the trace anomaly [3] (4π) d/2 T a a = −aE (d) where E (d) and are the Euler density and Weyl invariants in (even) d dimensions respectively.The coefficients a and c i are known as central charges.In general even dimensions, C T is a linear combination of the c i 's [1]; it is thus simply proportional to the only c in d = 4.
Although the concept of conformal anomaly no longer applies in odd dimensions, the quantity C T , a linear combination of c i 's in even dimensions, survives.The a-charge can also be generalized to odd dimensions as the universal coefficient of the entanglement entropy across a spherical entangling surface, which coincides with a-charge in even dimensions [4].Thus the a-charge, C T , and three-point function parameters (A, B, C) are important characteristics of CFT in general dimensions as they control the energy-momentum tensor correlators up to and including three points.These parameters are not independent.It was shown by Ward identity that C T is a linear combination of A, B, C [1,2], namely In d = 4, a similar relation exists for a [1] a = π 6 2880 (13A − 2B − 40C) . ( It thus is tempting to expect that analogous relation of (4) would exist also for the a-charge.
The AdS/CFT correspondence provides a new way to study the large N limit of CFTs from the weakly-coupled bulk gravity theory in the anti-de sitter (AdS) background in D = d + 1 dimensions [5].Einstein gravity extended with higher-order curvature invariants is insightful to study as they capture new features of the dual CFT.The large number of higher-derivative terms provide unlimited data that can not only reveal the CFT in the large N limit, but also some universal properties of CFT.
There has been extensive studies of the holographic correlators in the pure gravity sector with higher curvature extensions, e.g.ref. [6,7,8,9,10].For general higher curvature gravity, the corresponding linearized equation of motion around AdS background contains at most four derivatives of the metric, the graviton spectrum thus contains two extra modes, the massive scalar mode and ghost-like spin-2 mode [11].Unitarity of the dual CFT requires the decoupling of the ghost mode, while the decoupling of the scalar mode is required by the holographic a-theorem [12].Thus we usually require the decoupling of both modes, either exactly or perturbatively, resulting in massless gravity or perturbative gravity, respectively.
Due to computational difficulties, three-point function parameters of higher curvature gravity are usually calculated indirectly via the energy flux parameters t 2 and t 4 [7,13], together with C T , they contain all the information about the parameters (A, B, C).Three-point function parameters for massless cubic gravity was calculated in ref. [9] for d = 3 and d = 4, while the results in arbitrary dimensions or with higher-order curvature invariants are still absent in the literature.
Several features of (a, C T ) and (A, B, C) have been known.It was shown that to distinguish a and c holographically in d = 4, one needs to introduce at least the Gauss-Bonnet density, in which case only two of the three three-point function parameters are independent; one needs to further consider cubic curvature invariants to obtain all the three independent parameters [10].In additional to these algebraic relations, holography also provides a hidden differential relation between the c and a charges for all massless higher curvature gravities [14,15] where L is the effective radius of AdS.Furthermore, it was conjectured that a relation exists between t 2 , t 4 , h ′′ for massless gravity [16] a where h(λ) is the AdS vacuum equation of motion, to be defined later.This relation was verified for Lovelock gravity in arbitrary dimensions and quasi-topological gravity in d = 4.In these verifications, only a (d) and b (4) were determined, since for Lovelock gravity we have t 4 = 0.
It is thus interesting to check for more general curvature invariants in diverse dimensions and, if possible, to determine the value of b (d) .
The main purpose of this work is to calculate holographically the parameters (A, B, C) of the three-point function from the general higher curvature invariants up to and including the quartic order in arbitrary D = d + 1 dimensions with d 4. We employ an indirect method by considering the one-point function of an energy flux operator parametrized by t 2 and t 4 , which are directly related to the three-point function parameters.We consider both the finite higher-derivative coupling case and perturbative case.For the former, we need to impose the massless conditions.For the latter the number of non-trivial terms can be reduced by some appropriate order-by-order field redefinitions of the metric.
From our results, we can make some interesting statements: 1.After setting d = 3 for the general d results, the value of t 4 coincides with that obtained in ref. [9], even though our method should only be valid for d 4.
2. The a-charge and (A, B, C) turn out to be in general linearly independent for d 5.In other words, there is no generalization of (5) beyond d = 4.
3. Instead, all the three c-charges in d = 6 are linear combinations of (A, B, C).
4. The conjectured relation (7) does not hold for general massless gravity, but may hold for quasi-topological gravities.We verify this up to and including the quartic order.
The three-point function, and hence (A, B, C), contains three different possible structures, which can be enumerated holographically by Einstein, quadratic, and cubic curvature terms [10,17,18].It was claimed by ref. [9] without proof that higher order curvature polynomials will not provide further information.In fact, this can be easily proved, and we present it in appendix D. Nevertheless, the quartic-order calculation provides a useful consistent check.
The paper is organized as follows.In section 2 we review higher curvature gravity and list all the Reimann curvature polynomial invariants up to and including the quartic order.In section 3 we first briefly review the method we use to calculate the energy flux parameters, following ref.[7], then we present our results and cross check them with the known special cases.Further discussions on our results are given in section 4, where we also calculate the a-charge in general dimensions and the three c-charges in d = 6.We conclude this paper and make further comments in section 5. Some lengthy expressions and digressions are given in five appendix sections.

A brief review on higher curvature gravities
We consider Einstein gravity extended with higher-order curvature invariant polynomials in D = d + 1 dimensions, up to and including the quartic order.The general form of action is where R (i) j is the j'th Riemann scalar polynomial of order i, coefficients e i,j are the coupling constants, and L 0 is the bare AdS radius.All the Riemann curvature polynomial invariants studied in this paper are given explicitly below In other words, there are 3, 8, 26 terms for the quadratic, cubic and quartic orders respectively.
The indices (i, j) of the coefficients e i,j in (8) are labeled based on the above order.
Before proceeding, we shall briefly review some general properties of higher curvature gravity.The equation of motion is given by E ab = 0, with The equation admits the maximally-symmetric solution ḡab whose Riemann tensor is given by R abcd = 2λ ḡa[c ḡd]b , where λ is to be determined.When evaluated on such a background, the tensor P abcd takes the following simple form [19] Substituting into (12) some useful identities emerge [19,16] The last equation determines the on-shell value of λ.Note that when taking a derivative with respect to λ, all the parameters in the theory including L 0 should be treated as being independent of λ.
The linearized theory is governed by the tensor Evaluating on the maximally-symmetric background, it takes the form [19] To be precise, the above tensor structure does not satisfy the cyclic Bianchi identity inherited from the Riemann tensor.We should impose this identity and redefine the tensor as follows The linearized theory of any higher curvature theory is completely described by the coefficients For a Lagrangian that is a polynomial of curvature invariants, these four coefficients are linear functions of coupling constants of the polynomial invariants.For a specific Lagrangian, the coefficients k i can be obtained efficiently with the method proposed in ref. [19].The effective Newton constant κ eff and the masses of the scalar mode m s and ghost-like spin-2 mode m g are given below [19] κ The quantity 1/κ eff appears as a coefficient of the linearized equation of motion of the massless graviton after the massive modes are decoupled, namely The central charge C T of the dual CFT is related to κ eff as To decouple both the massive modes, we require At the quadratic order, we have For the quartic order the constraints become too lengthy and we record them in appendix A.
An alternative approach to higher-order gravity is to treat it as an effective theory of quantum corrections to Einstein gravity.In this approach, the massive modes are decoupled perturbatively since at the zeroth order the coefficient k 1 is nonzero while k 2,3,4 are first order, making the kinetic terms of the massive modes first order.This applies to effective field theory where the first massive state appear at some high energy cutoff scale M [20].In this approach, one can perform field redefinitions of the metric order-by-order by appropriate higher curvature terms, so as to eliminate some Ricci tensor and scalar terms in the Lagrangian.This gives the following residual sets of Riemann polynomials: The coupling constants of these terms are invariant under the field redefinition.
In many of our calculations in this paper, we find that it is not necessary to impose the massless conditions explicitly.Therefore many of our results are valid for both approaches to higher-derivative gravities.

Holographic calculation of energy flux parameters
We now turn to the main subject of this work.We shall determine holographically the threepoint function parameters (A, B, C) of the energy-momentum tensor in the dual CFT.To calculate the three-point function directly from higher curvature gravities, one needs to perturb the metric to the third order in the Lagrangian, which is quite challenging even for Einstein gravity [21].We therefore employ an alternative way to determine these parameters.
We follow ref.[7,6] by considering a specific frame and polarization in which the three-point function describes a hypothetical conformal collider experiment proposed in ref. [13].In this experiment one first creates a localized excitation with the operator O ∼ ε ij T ij where ε ij is the polarization tensor, then measures the energy flux at the null infinity of the direction indicated by the unit vector n.The energy-flux operator E( n) is Its expectation value takes the form Note that for d = 3, the coefficient of t 2 vanishes identically and we are left with only the t 4 term.To isolate the contribution from null infinity, it is convenient to define a new set of coordinate where x ± = t ± x d−1 , L is some energy scale, to be chosen as the bulk AdS radius, and ī denotes the index of the (d − 2) dimensional subspace, i.e., 1 ī d − 2. This is in fact a conformal transformation with conformal factor (y + /L) 2 , after which the energy-flux operator becomes where On the other hand, the excitation operator O takes the form where ψ(x) is some distribution that's localized at x = 0, and Eσ ≫ 1 is assumed so that the operator is localized.Thus one can see that the numerator of ( 30) is indeed the three-point function with indices contracted with specific polarizations, thus one can relate (t 2 , t 4 ) to (A, Using the above and (4) one can solve the parameters (A, B, C) in terms of (t 2 , t 4 , C T ), thus the problem is converted to the calculation of (t 2 , t 4 , C T ) quantities.
To calculate the energy flux parameters holographically, we consider the following AdS metric in Poincaré patch Inspired by (32), we define the new bulk coordinates as follows which is an isometric transformation in the bulk and reproduces (32) at the boundary u = 0.According to holographic dictionary, the energymomentum tensor is dual to the metric perturbation h ab .Specifically, the energy flux operator in ( 33) is sourced by ĥ++ = L 2 Ω 3 δ(y + )δ d−2 (y ī − y ′ ī), so that the bulk solution is where the overall factor is unimportant so we ignore it here.Remarkably, the above insertion for E( n) can be done using an exact solution instead of perturbation by considering the shockwave solution where W satisfies the equation of motion It is important to note that this equation will not be altered by higher curvature terms [22].We now only need to consider the second-order perturbation around the shockwave background, instead of the general third order around the AdS background.This simplifies the calculation greatly.Comparing (42) with (41), the desired bulk solution of W is given by One can verify that it indeed satisfies (43).
For the excitation operator O, we choose the polarization to be ε x 1 x 2 = ε x 2 x 1 = 1 with all other components vanishing, so that the only non-vanishing component of the metric perturbation is h x 1 x 2 .This implies that this particular holographic procedure requires d 4, even though the CFT energy flux (31) can be defined in d = 3.Since h ++ is localized at y + = 0, we are only interested in the behavior of h x 1 x 2 on this surface.It can be shown [13,7] that after transforming to (u, y a ) coordinate, h y 1 y 2 is also localized, namely The transformation also intruduces other components h y + y 1 , h y + y 2 , h y + y + , but as we shall see later, they can be eliminated by imposing the transverse and traceless condition Defining φ by h y 1 y 2 = (L 2 /u 2 )φ and imposing the transverse and traceless condition, the equation of motion of φ is up to interaction terms with the shockwave.
With these preliminaries, we are ready to evaluate the energy flux.This can be done by turning on the perturbations on the shockwave metric and evaluate the on-shell action, and then extract the terms of the form W φ 2 .Note that by (45) the bulk coordinate u is localized at u = L, so we do not need to consider the boundary action.After imposing the transverse and traceless condition, using the equations of motion (43), (47), and integration by parts, the on-shell effective action becomes The basis functions T 2 and T 4 depend only on the shockwave metric function W , namely where the index î covers the remaining (d − 4) directions, i.e., 3 î d − 1. Substitute the solution (44) of W into the above leads to While T 2 and T 4 are independent of the detail of the action, the coefficients ĈT , t2 , t4 are determined by the coupling constants of higher curvature gravities.Specifically, we find To obtain the final result, we need to divide the cubic action by the two-point function T 12 T 12 , which is proportional to C T .The latter can be calculated using (23) and for the case we study, we find Thus we have On the other hand, specializing to the polarization By comparing (56) to (31), we arrive at the final result where the hatted variables are given by ( 52), ( 53) and ( 54).Now we examine some known special cases.Firstly, for Lovelock gravities up to and including the quartic order, we set the coupling constants to Substituting them into our results (58), ( 52), ( 53), (54), we obtain Energy flux parameters for general Lovelock gravity was derived in ref. [10].Explicitly we have (after adapting to our conventions) In our case the function h(λ) is given by Substituting this into (63) we get exactly identical result with (62).
Secondly, after specializing our result to general massless cubic curvature gravity in d = 4 and eliminating e 3,7 , e 3,8 by the massless condition (25), we arrive at which reproduces the result of ref. [9].
We can now obtain all the three-point function parameters (A, B, C) by inverting ( 4), (36), (37).The final expressions of (A, B, C) in general dimension d are recorded in appendix B.

Discussions
Having obtained all the three-point function parameters (A, B, C) holographically from higher curvature gravities up to and including the quartic order, we can compare our results to those in literature and study their implications.
Firstly it is interesting to mention that even though our result, based on its derivation, should be valid only for d 4, there exists a smooth d = 3 limit.In particular, if we restrict to massless cubic gravity and set d = 3, our result gives rise to the following value for t 4 t 4 = − 120 (360e 3,1 + 96e 3,2 + 27e 3,3 + 25e 3,4 + 64e 3,5 + 18e 3,6 ) L 4 − 2 (36e 3,1 + 6e 3,2 + e 3,4 + 4e 3,5 ) which is precisely the one obtained in ref. [9]that was derived using a different method.We thus expect that our new result of t 4 , arising from the quartic massless gravity, is also valid at d = 3.From our general results, setting d = 3 also gives a nonzero value for t 2 ; however, in d = 3 the symmetry group reduces to O(2), and hence there can only be one energy flux parameter t 4 .Furthermore, specializing t2 in d = 3, we find it is actually linearly independent of ĈT and t4 .Therefore, it is interesting to explore the physical meaning of the value of t 2 of general d in the d = 3 limit.
Secondly, we examine the conjecture (7).Since for massless gravity h ′ (−L −2 ) is proportional to C T [16], it is equivalent to that t2 , t4 and h ′′ (−L −2 ) are linearly dependent, i.e., However, we find that they are actually linearly independent for general massless higher curvature gravities.As mentioned earlier, the conjecture (7) already was verified for certain special cases, it is thus natural to expect that there exists a more general case that satisfies this conjecture.Although the most general and unified condition for all orders that satisfies ( 7) is hard to come by, we actually find that ( 7) is satisfied by cubic and quartic quasi-topological (QT) gravities in general dimension, with coefficients given by The values of a (d) and b (4) = −1/21 are in consistency with ref. [16] obtained from Lovelock and d = 4 QT gravities.We can then reasonably believe that all QT gravities satisfies the conjecture (7).A brief review on QT gravities can be found in appendix E, where we find that there are 15 such theories in quartic gravities.
Thirdly, we focus on identities involving the a-charge (5) and ( 6).As mentioned earlier, the a-charge can be generalized to arbitrary dimensions as the entanglement entropy across a spherical region S d−2 .It was shown with a conformal map that the entanglement entropy over the spherical region in Minkowski background equals to the thermal entropy of R × H d−1 background.The latter can be calculated holographically by the black hole entropy of a locally AdS hyperbolic topological black hole [4] The black hole entropy can be calculated from the Wald entropy in a standard way [23,24] where σ ab is the induced metric of the horizon, and ǫ = dt ∧ dr is the binormal of the horizon, satisfying ǫ ab ǫ ab = −2.For the topological black hole (70), the integrand is a constant proportional to the area of the horizon so the value diverges, thus one may assign the entropy density on the horizon to the a-charge as follows [15] a = π d/2 2πΓ(d/2) It can be shown that with this definition the value coincides with a-charge in even dimensions.
It is straightforward to calculate the a-charge from (72), we obtain With both C T and a evaluated, it follows immediately that after applying the massless condition we have (6).
Specializing our results to d = 4, we reproduce ( 5) and therefore we verify the relation holographically in higher curvature gravity up to and including the quartic order.However, C. We find that all three c-charges turn out be linear combinations of (A, B, C), namely In terms of t 2 , t 4 , we have which coincides with the CFT results derived from free Dirac fermion, real scalar, and antisymmetric two-form fields [25].Our results suggest that this is indeed a universal property of d = 6 CFT.

Conclusion
In this work we considered general higher curvature gravity up to and including the quartic order and calculated holographically the three-point function parameters (A, B, C) of the dual field theory in general d dimensions.We adopted an indirect method by calculating the central charge C T and the energy flux parameters t 2 , t 4 , which are known to be directly related to the three-point function parameters.We therefore obtained the complete list of the holographic results of (a, C T ) and (A, B, C) for general d dimensions.
Despite of the fact that our method should be valid only for d 4, we not only reproduce the previously known result in d = 4, but also the correct value of t 4 of ref. [9] C Central c-charges in d = 6 In d = 6 there are three Weyl invariants I i [26] where C abcd is the Weyl tensor, and the explicit form of ∇ a J a is irrelavent since it is a total divergence and can be canceled by a local counterterm.This gives three c-charges in d =6.
The central charges of cubic curvature gravity in d = 6 was computed in ref. [27], we therefore extend the result to quartic order.We employ the reduced Fefferman-Graham expansion trick to calculate the central charges [28,12].We find that the a-charge is given by (73) specialized to d = 6, and the three c-charges are D General structure of three-point function Holographic three-point functions can be extracted from the cubic effective action of graviton on AdS background.For higher curvature gravities whose the Lagrangian depends on the metric g ab and R abcd , the metric dependence becomes implicit if one chooses R ab cd as the independent variable [29], i.e., L = L(R ab cd ).The general form of the cubic effective action can then be obatined by varying the action to the third order where ḡab is the AdS metric, tensors P abcd and C abcdef gh are given by ( 12) and ( 17) respectively.
The tensor G abcdef ghijkl is defined by Note that we have also imposed the transverse and traceless condition so that δ |g| = when evaluated on this background, these three tensors can only be built from the metric ḡab , thus their forms are all rigidly fixed, with theory-dependent coefficients, e.g., (13) and (18).
For Einstein gravity only the contribution from P abcd is non-zero, while for quadratic gravity the G abcdef ghijkl contribution vanishes.All three tensors are non-zero for cubic gravity, which can therefore enumerate all the possible structures.For the quartic or higher orders, no new structures arise; they just modify the coefficients in these three tensors.

E Quasi-topological gravity
We shall briefly review QT gravity and present dimension-generic cubic and quartic QT combinations.There has been an extensive study on QT gravity (e.g.ref. [7,30,31,32]).A quasi-topological (QT) gravity is a type of gravity theories whose equation of motion on the special spherically symmetric metric ansatze is algebraic in f (r), i.e., does not involve derivatives of f (r).This condition is equivalent to [33] ∇ a P abcd f = 0 (90) where . . .| f denotes evaluating on the metric ansatze (89).This makes the black hole solutions of QT gravities easy to obtain, thus QT gravity serves as a simplified model of general higher curvature gravity.

For d 3 ,
by O(d − 1) invariance the most general form of the energy flux can be determined by two parameters t 2 and t 4[13]

d 5 ,
we find that a and (A, B, C) are in general linearly independent.In other words, the d = 4 relation (5) does not have a higher-dimensional generalization.This somewhat unexpected result instructs us to further consider central charges in d = 6, where there are three c-charges.Details and explicit values of the c-charges in d = 6 can be found in appendix