Towards 4-point correlation functions of any 1/2-BPS operators from supergravity

The quartic effective action for Kaluza-Klein modes that arises upon compactification of type IIB supergravity on the five-sphere S^5 is a starting point for computing the four-point correlation functions of arbitrary weight 1/2-BPS operators in N=4 super Yang-Mills theory in the supergravity approximation. The apparent structure of this action is rather involved, in particular it contains quartic terms with four derivatives which cannot be removed by field redefinitions. By exhibiting intricate identities between certain integrals involving spherical harmonics of S^5 we show that the net contribution of these four-derivative terms to the effective action vanishes. Our result is in agreement with and provides further support to the recent conjecture on the Mellin space representation of the four-point correlation function of any 1/2-BPS operators in the supergravity approximation.


Introduction and summary
Recently the problem of finding four-point correlation functions of 1 2 -BPS operators in N = 4 super Yang-Mills theory came into focus again. Both new demands and inspirations sparked due to interesting progress of the general bootstrap program, as well as remarkable developments concerning the integrable structure of the dual string theory. The main interest in these correlation functions is that on the one hand they involve the simplest possible operators, while on the other hand they non-trivially depend on the coupling constant and, as such, contain through the corresponding OPEs a lot of valuable information about the dynamics of unprotected operators.
It was known for some time [1]- [3] that the simplest representation of the correlation functions is achieved by transforming the latter to Mellin space, where their analytic and asymptotic properties become most transparent. Recently the authors of [4] conjectured an interesting formula for the Mellin representation of the fourpoint correlation function of arbitrary weight 1 2 -BPS operators in the supergravity limit. In addition to the expected OPE behavior in this limit, the conjecture is based on the assumption of linear asymptotic growth of the Mellin amplitude at large values of the Mandelstam variables.
Leaving technical complications out of the discussion, to derive the corresponding correlation function and in this way to prove the conjecture by [4], one has to use the quartic effective action for Kaluza-Klein modes of type IIB supergravity compactified on the five-sphere S 5 [5]. We recall that 1 2 -BPS operators of weight k are dual to supergravity scalars s I k with mass m 2 = k(k − 4) and with index I k running over the basis of an irreducible representation [0, k, 0] of SU (4). Then a four-point correlation function A comes naturally as a sum of two terms Here A exchange comprises the sum of all exchange graphs which are generated by cubic Lagrangian vertices involving two scalars s I and any other supergravity field which is allowed by representation theory. The term A contact gives a contribution of all contact graphs originating from the quartic Lagrangian vertices, i.e. the vertices which contain four scalars s I possibly with space-time derivatives.
Analysing the behaviour of exchange graphs for large values of Mandelstam variables one concludes that they grow linearly, hence the conjecture by [4]. As to the contact graphs, their linear asymptotic growth is guaranteed provided the quartic Lagrangian vertices contain fields s I with a number of derivatives not higher than two. Puzzling enough, such an expected structure of the quartic Lagrangian appears in apparent conflict with the explicit findings by [5], where it was shown that the quartic Lagrangian contains terms with four derivatives (and not more) and that these four-derivative terms cannot be removed by any on-shell field redefinition. When genuinely present, such terms would result into quadratic asymptotic growth of the Mellin amplitude, incompatible with the conjectured formula.
Obviously, the arising conundrum has a natural resolution if the four-derivative couplings actually vanish due to some internal, group-theoretic reasons. The idea that this might be the case, which implies that the corresponding Kaluza-Klein effective action is of the sigma-model type, has been put forward long ago 1 . It is essentially based on the results of evaluating concrete correlation functions. Namely, using the quartic effective action, correlation functions of weight k BPS operators of the type kkkk and 22kk were computed [6]- [11], and in all the cases one was able to demonstrate vanishing of the quartic four-derivative terms by integrating by parts together with specifying explicit group-theoretic content of the corresponding Lagrangian couplings. Unfortunately, these correlators are still rather special and they do not probe all quartic four-derivative couplings to draw a decisive conclusion on the ultimate status of the latter.
In this note we show that the quartic four-derivative terms do indeed vanish. On the one hand, this gives evidence in favour of the conjecture [4] and, on the other hand, provides a first step towards actual evaluation of the four-point correlation function of arbitrary weight 1 2 -BPS operators in the supergravity approximation. Our considerations are based on the fact that in the process of compactification the couplings in front of quartic terms with four derivatives appear in the form of weighted sums of two Clebsch-Gordan coefficients c which look schematically as g 5 c 125 c 345 . Here the legs 1, 2, 3, 4 carry representation indices of supergravity field s I with weights k 1 , k 2 , k 3 and k 4 , while summation over the fifth leg with some weight function g 5 is assumed. The Clebsch-Gordan coefficients c 125 are given in terms of integrals of spherical harmonics of S 5 , where in the fifth place occurs either a scalar harmonic Y I 5 or a vector one Y I 5 α , where α is the tangent index of the five-sphere.
By unravelling some intricate identity which involves a sum of integrals over vector harmonics, we managed to rewrite the contribution of the latter via similar sums but involving scalar harmonics only. By removing vector harmonics in favour of scalar ones in this way, we made all the couplings comparable, and further summing them up we find that they exactly cancel. Although our result on vanishing of quartic four-derivative terms provides a rigorous proof that the quartic effective action for Kaluza-Klein modes of type IIB supergravity is of the sigma-model type, a deep reason behind this finding still remains to be understood. We point out that similar manipulations might exist for the rest of the quartic effective action, which comprises terms with two derivatives and without derivatives, possibly leading to its significant simplification. We however postpone investigation of this interesting question for future work.
The paper is organised as follows. In the next section we present the known information about quartic couplings with four derivatives and collect all necessary definitions. In Section 3 we derive the main reduction formula which allows us to replace the coupling with vector spherical harmonics via new couplings with scalar harmonics only, and verify that upon this replacement the net contribution of all four-derivative couplings sums up to zero. A computation of an auxiliary integral is relegated to an appendix.

Quartic couplings with four derivatives
In [5] the quartic Lagrangian for the fields s I with four derivatives was found to be of the following form Here ∇ a is a covariant derivative along AdS space and each summation label j = 1, . . . , 4 stands for a concise notation for the representation index I j running over a basis of of an irreducible representation [0, k j , 0] of SU(4). The couplings A 1234 and S 1234 have the following symmetry properties Explicitly, A 1234 is given by the sum of the following individual terms . . .
The symmetric coupling is In the formulae above and summation over the index 5 is assumed. The couplings a 123 and t 123 are given as the following integrals over the five-sphere of the spherical harmonics Here Y k are scalar spherical harmonics and Y k α are vector spherical harmonics satisfying the irreducibility condition ∇ α Y k α = 0. Both Y k and Y k α are eigenvalues of the sphere Laplacian ∇ 2 with the following eigenvalues In what follows we will also need the following product formulae which follow from the orthogonality relation for scalar harmonics where the coefficients are 2 This completes our discussion of the known results on the quartic Lagrangian with four-derivative vertices, for further information and derivation of the above formulae we refer the reader to [5].
To proceed with proving our main result, we employ the same strategy as in [12], where the vanishing of quartic four-derivative-vertices were shown for the so-called sub-extremal and sub-sub-extremal cases. Recall that we are ultimately interested in the four-point function of BPS operators corresponding to arbitrary weights k 1 , . . . k 4 .
We can therefore restrict the infinite sum in (2.1) to representations which correspond to these weights. The sum in (2.1) is not ordered and, therefore, there are 24 ordered sets of the indices k 1 , . . . , k 4 which split into 3 equivalence classes due to the symmetries (2.2). Further, integrating by parts and using (2.2), we represent the part of the Lagrangian (2.1) contributing to the four-point function k 1 k 2 k 3 k 4 in the form similar to that in [12] Since we are interested here in the four-derivative vertices only, in the above formula we have omitted the contribution of two-derivative terms and terms without derivatives which arise upon integrating by parts and using equations of motion. There terms however should be taken into account in subsequent analysis of the remaining part of the quartic effective action. We also note that the meaning of the sums in (2.7) is different from that in (2.1) -in (2.7) the sums are ordered, i.e. summation over 1 means summation over index I 1 corresponding to the representation with a given weight k 1 and so on. It is now obvious that it is enough to analyse the coupling 1234 + A 1234 , as the latter involves vector spherical harmonics. Its contribution into (2.8) comes in the combination (2.9) Our further strategy will be to reduce this combination to structures of the type f n 5 a 125 a 345 and permutations thereof. After this is done, all the couplings become comparable and we can add them up according to (2.8).

Reduction formula
The reduction of (2.9) is based on the following formula [5] In what follows it appears advantageous to split (3.1) into anti-symmetric and symmetric part with respect to indices 1 and 2, namely, Acting on (3.2) with the Laplacian and taking into account that we obtain Now we multiply this relation with a similar one where indices 1, 2, 5 are replaced by 3, 4, 6 and integrate over the five-sphere. The orthogonality relation for the vector spherical harmonics together with (3.4) used upon integrating the Laplacian by parts leads to the following formula which for (2.9) implies the following relation where K 1234 is the following integral with the symmetry properties Thus, we have reduced evaluation of our main quantity W 1234 to the computation of the integral K 1234 . In order not to overload our discussion with heavy formulae we perform the computation of K 1234 in the appendix and here present only the final reduction formula for W 1234 : Finally, we sum − 3 δ W 1234 with the remaining couplings in (2.8) and observe that all the terms are neatly cancelled delivering thereby C 1234 = 0. In this way we have shown that the quartic four-derivative couplings of the supergravity effective action vanish.

A Evaluation of K 1234
In this appendix we compute the integral (3.6) in terms of structures f n 5 a 125 a 345 and permutations thereof. In the computation process the following identity valid for any will be heavily used. Here g αβ is the metric of the unit five-sphere. Note also that on a scalar function two covariant derivatives commute.
We start with computing The latter formula gives rise to the following identity and a similar one with indices 3 and 4 interchanged. To simplify our presentation, in the sequel we will drop the integration sign and always identify expressions differing by a total derivative. Using (A.3) we then get and We continue our further treatment with evaluating the quantity U 1234 . To this end, we compute a il5 a jk5 − a ik5 a jl5 ) , a il5 a jk5 − a ik5 a jl5 ) .
As the result, for the quantity U 1234 we get a 135 a 245 − a 125 a 345 ) In this way U 1234 has been reduced to the desired structure. Now we look for a similar reduction of the quantity V 1234 . Here we perform a sequence of the following transformations. First, we have 3] entering in the 2nd and 5th terms are anti-symmetric in α and β and, therefore, in these terms one can replace ∇ β ∇ α ∇ γ with 1 2 [∇ β , ∇ α ]∇ γ and then apply identity (A.1). In this way we get As the next step, we consider the first line in the expression above and transform it in the following way The resulting expression undergoes further transformation which finally results into Comparing the first line in the above formula with the original expression (A.6) for V 1234 we observe that it coincides with − 1 2 V 1234 . This allows us to find the following answer for V 1234 All the terms in the right hand side of the last formula are reducible, i.e. by using eqs.(2.4), (2.5), (2.6), (3.4) they can be written via f n 5 a 125 a 345 and permutations thereof. For instance, Proceeding in a similar manner, after tedious computation we find