Kaluza-Klein Five-Point Functions from $\textrm{AdS}_5\times S_5$ Supergravity

We continue to explore the bootstrap approach to five-point correlation functions for IIB supergravity on $AdS_5\times S^5$. Building on the result of [1], we develop an improved algorithm that allows us to more efficiently compute correlators of higher Kaluza-Klein modes. The new method uses only factorization and a superconformal twist, and is entirely within Mellin space where the analytic structure of holographic correlators is simpler. Using this method, we obtain in a closed form all five-point functions of the form $\langle pp222\rangle$, extending the earlier result for $p=2$. As a byproduct of our analysis, we also obtain explicit results for spinning four-point functions of higher Kaluza-Klein modes.

1 Introduction Recent years have seen significant progress in computing holographic correlators, which are key objects for exploring and exploiting the AdS/CFT correspondence.Traditionally, holographic correlators are computed by diagrammatic expansions in AdS.Such a method works in principle.However, in practice, it requires the precise knowledge of the exceedingly complicated effective Lagrangians and is extremely cumbersome to use.Therefore, for almost twenty years the diagrammatic approach led to only a handful of explicit results.The new developments, on the other hand, are based a totally different strategy which relies on new principles.This is the bootstrap approach initiated in [2,3], which eschews the explicit details of the effective Lagrangian altogether.The new approach works directly with the holographic correlators and uses superconformal symmetry and consistency conditions to fix these objects.The bootstrap strategy has produced an array of impressive results. 1or example, at tree level general four-point functions for 1 2 -BPS operators with arbitrary Kaluza-Klein (KK) levels have been computed in all maximally superconformal theories [2,3,5,6], as well as in theories with half the amount of maximal superconformal symmetry [7][8][9].Note that these general results are all in the realm of four-point functions.Higherpoint functions still mostly remain terra incognita.In fact, only two five-point functions have been computed in the literature for IIB supergravity on AdS 5 × S 5 [1] and SYM on AdS 5 × S 3 [10] respectively, and both for the lowest KK modes only.
However, studying higher-point holographic correlator is of great importance.Firstly, higher-point correlators allow us to extract new CFT data which is not included in fourpoint functions.For example, the OPE coefficient of two double-trace operators and one single-trace operator can only be obtained from a five-point function.Moreover, via the AdS unitarity method [11] higher-point correlators are also necessary ingredients for constructing higher-loop correlators.Secondly, via the AdS/CFT correspondence holographic correlators correspond to on-shell scattering amplitudes in AdS.Recently, there has been a lot of progress in finding AdS generalizations of flat-space properties [9,10,[12][13][14][15][16][17][18][19][20][21][22][23][24].As we know from flat space, many remarkable properties of amplitudes are only visible at higher multiplicities.To further explore the analogy between holographic correlators and scattering amplitudes it is necessary to go to higher points.Finally, it has been observed in [25] that a ten dimensional hidden conformal symmetry is responsible for organizing all tree-level four-point functions for IIB supergravity on AdS 5 × S 5 .The nature of this hidden structure is still elusive.It is an interesting question whether the 10d hidden symmetry is just a curiosity for four points or it persists even at higher points.
For these reasons, in this paper we continue to explore the bootstrap strategy for computing higher-point correlators.In particular, we will focus on computing the five-point functions of the form pp222 for IIB supergravity in AdS 5 × S 5 , where three of the operators have the lowest KK level but the other two have arbitrary KK level p.Our strategy will be similar to that of [1], which computed the p = 2 case, but with important differences.In [1], the starting point is an ansatz in position space which is a linear combination of all possible Witten diagrams with unfixed coefficients.To fix the coefficients, one imposes various constraints from superconformal symmetry and consistency conditions.These includes factorization in Mellin space [26], the chiral algebra constraint [27] and the Drukker-Plefka twist [28].The first constraint is the consistency condition for decomposing the five-point function into four-point functions and three-point functions at its singularities.The second and the third conditions come from superconformal symmetry and are the statement that the appropriately twisted five-point function becomes topological.Although these conditions uniquely fix the p = 2 five-point function, the strategy of [26] suffers from a few drawbacks which make it difficult to apply efficiently to correlators with higher KK levels.Firstly, computing the higher-point Witten diagrams in the ansatz is a nontrivial task.In particular, simplifications used in [26] for computing p = 2 diagrams no longer exist for sentation ∆ = k.Via the AdS/CFT correspondence, they are dual to scalar fields in AdS with KK level k and are usually referred to as the super gravitons.A convenient way to keep track of the R-symmetry information is to contract the indices with null polarization vectors O k (x; t) = O I 1 ..More precisely, we will compute the leading connected contribution which is of order 1/N 3 and corresponds to tree-level scattering in AdS.The disconnected piece factorize into a three-point function and a two-point function, and is protected because the lower-point functions are.
Symmetry imposes strong constraints on the form the correlator.For example, conformal symmetry allows us to write the five-point function as a function of five conformal cross ratios after extracting an overall kinematic factor where are the weights of the external operators.Note the number of these monomials is finite and we will refer to them as different R-symmetry structures.In Section 2.1, we will explicitly write down these structures.
The considerations so far have only used the bosonic symmetries in the full superconformal group.The dependence on the spacetime variables x 2 ij and on the R-symmetry variables t ij are not related.However, the fermionic generators in the superconformal group 2 We are using a different, but equivalent, set of cross ratios here compared to [1].These new cross ratios have appeared before in [29].One reason why these variables are nice is that it is possible to associate some x 2 ij to u k , for example x12 only appears in u1.Another interesting property is that they are cyclically related to each other.
will impose further constraints which correlate the x 2 ij and t ij dependence.For five-point functions, a thorough analysis the full consequence of the fermionic symmetries has not been performed in the literature.However, two classes of such constraints are known.The first is the chiral algebra construction [27] which constrain the five-point function when all the operators are inserted on a two dimensional plane.The other is the Drukker-Plefka twist [28] which imposes constraints on the correlator with generic insertion positions.In this paper, we will only need the latter.We will review these conditions in Section 2.2.A systematic way to enumerate the R-symmetry structures of the pp222 five-point function is to consider the Wick contractions.Different Wick contractions are illustrated in Fig. 1 and the corresponding R-symmetry structures are explicitly given by Here (a 1 , a 2 ) is (1, 2) or (2, 1) and (a 3 , a 4 , a 5 ) can be any permutation of (3,4,5).The Wick contractions in the first row of Fig. 1 exist for all p ≥ 2 while the second row are only possible when p ≥ 3.This is a new phenomena that arises at the level of five-point functions and should be contrasted with the four-point function case.In the four-point function pp22 , the number of Wick contractions is the same irrespective of the Kaluza-Klein weight p. 3For p = 2, all the five points are on the same footing and there is no distinction between P (I) a 3 a 4 a 5 , P (II) a 3 a 4 a 5 and among T (I)

Drukker-Plefka twist and chiral algebra
A highly nontrivial constraint from superconformal symmetry is given by the topological twist discovered in [28], which we will refer to as the Drukker-Plefka twist.In [28], it was found that when the operators have the following position-dependent polarization vectors (commonly referred to as a twist) the twisted correlator preserves certain nilpotent supercharge.The twisted operators are in its cohomology.More importantly, the translations of operators while keeping the polarizations twisted are exact.It then follows that the twisted correlators are topological, i.e., independent of the insertion locations Note that in terms of the variables x 2 ij and t ij , the twist condition can also be written as Let us also mention another twist for contrast, namely the chiral algebra [27].However, we will not exploit this twist in this paper.The chiral algebra twist requires that all the operators are inserted on a two dimensional plane.The coordinates therefore can be parameterized by the complex coordinates z, z.Furthermore, the polarization vectors need to be restricted to be four dimensional where t µ can be written in terms of two-component spinors Using the rescaling freedom of the polarization vector, we can write v and v as (2.12) When we twist the operators by setting wi = zi , the correlator also preserves certain nilponent supercharge.The twisted operators are in its cohomology while the antiholomorphic twisted translations are exact.Therefore, the twisted correlator are meromorphic functions of z i only.

Mellin representation
It has been commonly advertised that Mellin space [30,31] is a natural language for discussing holographic correlators.In this formalism, the connected correlators are expressed as a multi-dimensional inverse Mellin transformation where the Mellin-Mandelstam variables satisfy The function M(δ ij ) encodes the dynamical information and is referred to as the Mellin amplitude.Note that this definition is a bit schematic.To be precise, both the correlator and the Mellin amplitude also depend on R-symmetry structures.However, for the moment we will suppress this dependence to emphasize the analytic structure related to spacetime.One of the reasons that Mellin amplitudes is convenient for describing scattering in AdS is they are meromorphic functions of the Mellin-Mandelstam variables.This follows directly from the existence of the OPE in CFT.Moreover, in the supergravity limit, the poles are associated with the exchanged single-trace particles in AdS.This makes the Mellin amplitudes have similar analytic structure as tree-level scattering amplitudes in flat space and allows us to apply flat-space intuitions in AdS.
More precisely, the exchange of a conformal primary operator with spin J and dimension ∆ = τ + J in a channel is represented by a series of poles in the Mellin amplitude, labelled by m = 0, 1, 2, . .., starting from the conformal twist τ Here, the exchange channel divides the external particles into two sets which we refer to as L and R. We label the particles in L from 1 to q and the ones in R from q +1 to n. δ LR is the Mandelstam variable in this channel.The residues Q m (δ ij ) have nontrivial structures.They are related to the lower-point Mellin amplitudes M L and M R for the (q+1)and (n−q+1)point functions involving particles in L and R respectively (Figure 2).The extra external state in each lower-point amplitude is the exchanged particle which has now been put onshell.This is the basic idea of Mellin factorization [26,32].In fact, it is very similar to the factorization of amplitudes in flat space which has been studied for a long time.However, there are also important differences.In flat space, the poles are located at the squared masses of the exchanged particles.In Mellin space, as already pointed out, the squared mass is replaced by the conformal twist and there is in general a series of poles for each particle which are labelled by m in (3.3).These are related to the conformal descendants.However, in theories with special spectra such as AdS 5 × S 5 IIB supergravity, the series usually truncates.For example, for p = 2 the series truncates at m = 0 and contains just one term.Moreover, compared to flat-space amplitudes, the lower-point Mellin amplitudes also appear in the residue Q m in a more complicated way.The precise expression for the residues depends on the spin of the operator that is exchanged.The goal of the following subsection is to explain all the details of this formula.In particular, we will present the explicit residue formulas for exchanged fields with spins up to 2. We should emphasize that the structure of factorization for the general pp222 five-point functions will turn out to be far richer than for the simple case of p = 2 which was analyzed in [1].In particular, we will see poles with m ≥ 1.
Note that for the five-point function G p with p > 2 there are three non-equivalent factorization channels which we choose to be In each of them there are exchanged primary operators with spins ranging from 0 to 2 as will be discussed in the following subsection.
M R ( ) Mellin amplitudes have poles correponding to the exchange of single-trace operators.
The residues at the poles are associated with lower-point Mellin amplitudes.In the channel depicted in the figure, we have n = 5 and q = 3.The Mellin amplitude on the left has four points while the one on the right has only three.

Melllin factorization
To discuss Mellin factorization, we need to be more explicit about what fields can be exchanged as they give rise to different lower-point functions.The problem of enumerating exchanged fields reduces to finding all the possible cubic vertices s k 1 s k 2 X where s k is the scalar field dual to the superconformal primary O k and X is a field to be determined.This problem already appears in the case of four-point functions and therefore the answer is also the same.The possible cubic vertices are determined by two conditions.The first is the R-symmetry selection rule.The second is the condition that the cubic vertices cannot be extremal 4 .These determine the possible exchange fields to be [2, 3] ) Here s k is a scalar field and is dual to the superconformal primary O k which has dimension ∆ = k and transform in the [0, k, 0] representation of SU (4).A k,µ is a vector field and is dual to a spin-1 operator J k,µ which has dimension ∆ = k + 1 and transforms in the [1, k − 2, 1] representation.ϕ k,µν is a spin-2 tensor field and is dual to a spin-2 operator T k,µν which has dimension is the graviphoton and ϕ 2,µν is the graviton.Their dual operators are correspondingly the R-symmetry current and the stress energy tensor.
Let us emphasize again that in this subsection we will only focus on the Mellin-Mandelstam variable dependence.Both M L and M R in fact also depend on R-symmmtry variables.Therefore in the residues Q m there is also a gluing of the lower-point R-symmetry structures.However, this gluing is purely group theoretic.To avoid distracting the reader from the discussion of the dynamics, we will leave the details of R-symmetry gluing to Appendix A.4.Alternatively, we can view the discussion in this subsection as the Mellin factorization for each R-symmetry structure.

Exchange of scalars
The simplest example of factorization is the exchange of a scalar operator with dimension ∆.The resulting M L and M R are again scalar Mellin amplitudes.Nevertheless, this example contains most of the features we shall need.In particular, the m dependence will be shared in the spinning cases.Therefore, we will first analyze this case in detail.The residue Q m introduced in (3.3) is given in [26] where L m is related to M L by5 and similarly for R m .Notice that here and in the following we will often leave the spacetime dimension d unspecified, but it should always be set to 4. This equation has several interesting consequences, which will become more evident after analyzing a few examples.
Let us start with a three-point Mellin amplitude for M L , which is just a constant c.In this case, recalling that ) and δ LR is set to ∆ + 2m by the pole condition (3.3), equation (3.6) with q = 2 immediately gives Factorizing a five-point function leads to a three-point function and a four-point function.
For pp222 , there are three inequivalent factorization channels, which can be chosen to be (12), ( 45) and (13).From (3.5), we know that the exchanged scalar operators in these three channels have twists 2, 2 and p respectively.Thus, δLR in each case is given by and the correspoding values of δ LR are 2 + m, 2 + m, p + m.After plugging these values in (3.8), it is straightforward to see that the residue vanishes for m > 0 in the channels (13) and (45), and for m ≥ p − 1 in the channel (12) 6 .Naively, one would conclude that in the (12) channel the number of poles increases with p.However, this is too fast since the other part R m can give more constraints.To see this explicitly, let us look at a four-point Mellin amplitude which has the following generic form Here we have evaluated the expression at the pole δ LR = τ + 2m.It follows that R m vanishes for this four-point Mellin M R for m ≥ 3 and therefore the number of poles does not increase for arbitrary value of p.Let us also emphasize that all four-point Mellin amplitudes that appear in the OPE of the correlator pp222 have this structure as can be checked in Appendix A.
Let us note that the absence of poles for m ≥ p − 1 can also be understood from the pole structure of the Mellin integrand.The Gamma functions in the definition of Mellin amplitude already have poles in this location and a pole in the Mellin amplitude at m ≥ p−1 would give rise to a double pole.Such double poles are associated with the appearance of anomalous dimension [2,3,31], which we do not expect at this order.On the other hand, at the moment we do not have a direct physical argument for the truncation of poles at m ≥ 3. Finally, this truncation continues to hold for the factorization formulas when the exchanged operators have spins.This will be analyzed in the following subsubsection.

Exchange of operators with spins 1 and 2
In this subsection we will be interested in studying the contribution of operators with spins.As it turns out, the analysis of the scalar case straightforwardly generalizes to the spinning case.It is convenient to get rid of the Lorentz indices of these operators by contracting them with null polarization vectors where z 2 = 0 ensures the operator is traceless (we refer the reader to Section 3 of [26] for a more detailed review).The definition of Mellin amplitudes of one spinning operator and n scalar operators is given by [26] O(x 0 , z 0 ) . . . where We have used δ δ to denote the Kronecker delta so that it can be distinguished from the Mellin-Mandelstam variables δ.The Mellin amplitudes M {a} satisfy certain linear relations that follows from the conformal invariance of the correlator, see equation ( 46) in [26].Let us first focus on the spinning generalization of (3.6) for the conserved currents which reside in the k = 2 supermultiplet.For exchanging the graviphoton, the residues are given by7 For exchanging the graviton, the residues are where ) Here we used the notation The functions L a m and L ab m (and analogously R a m , R ab m ) are defined in the same way as in (3.7).Let us also add that for m = 0 the second term in Q m for spin 2 is zero since both L0 and R0 vanish from the definition.Therefore, the appearance of the pole in m does not lead to a divergence.
These residue formulas for spinning operators clearly are not the full story as there are also non-conserved currents in the multiplets with k > 2. However, from (3.5) we can see that such non-conserved currents only appear in the channel with s 2 and s p .Similar to the scalar case (3.9), the analysis of the three-point functions requires the truncation at m = 0.The residues are The most general expressions for factorization with arbitrary external and internal dimensions and m can be found in [26].But they are not needed in this paper.
As in the scalar case, the truncation of poles also relies on the form of the spinning four-point amplitudes.They are given in Appendix B (see (B.15) and (B.20) for explicit expressions).In particular, they have the same analytic structure as the scalar four-point amplitude (3.10) except that now they carry additional indices.As a result, the truncation of poles also holds for the exchange of spinning operators.More precisely, we have the same pole locations as in (3.9)where the allowed values for m are m = 0, 1, 2 for (12) and m = 0 for (45), (13).
To summarize, the Mellin factorization formulas allow us to reconstruct all the polar part of the amplitude from the lower-point Mellin amplitudes.Furthermore, the spectrum of the theory gives rise to a further simplification where the poles truncate to a finite range independent of p.

Drukker-Plefka twist in Mellin space
As we reviewed in the introduction, the two superconformal constraints, namely the chiral algebra and the Drukker-Plefka twist, were both formulated and implemented in position space [1].To have a more streamlined algorithm, we would like to perform the bootstrap entirely within Mellin space and therefore need to translate such position space constraints into Mellin space.Let us first define the Mellin amplitude more precisely by restoring the R-symmetry dependence suppressed in the definition (3.1).For the pp222 correlator, we have where M(δ ij , t ij ) is a linear combination of the 29 R-symmetry structures listed in (2.7).
Usually the implementation of the twists in Mellin space is achieved by using the observation that x 2 ij monomials multiplying the Mellin transform (3.19) can be absorbed into the definition by shifting the Mellin-Mandelstam variables.This gives rise to difference equations in Mellin space.This strategy has been used, for example, in [33,34] to rewrite the superconformal Ward identities in Mellin space for four-point functions.In our case, there are extra complexities.
The issue is that the chiral algebra constraint requires all the operators to be on a two dimensional plane.When the number of operators n > 4, this cannot be achieved by a conformal transformation and there are relations among the cross ratios. 8The meromorphy of the correlator after the chiral algebra twist depends crucially on these relations.On the other hand, these relations do not hold in the definition of the Mellin ampllitude where the locations of the operators are assumed to be general.Therefore, the position space chiral algebra condition cannot be translated into Mellin space using the same strategy.
By contrast, the Drukker-Plefka twist only imposes conditions on the R-symmetry polarizations and has no restriction on the operator insertions.Therefore, we can use the same trick to implement the Drukker-Plefka twist in Mellin space.More precisely, we can extract a kinematic factor and rewrite (3.19) in terms of cross ratios (2.3), (2.4) (3.20) Here K p is a kinematic factor Performing the Drukker-Plefka twist amounts to setting t ij → x 2 ij , or equivalently σ → u for the cross ratios.To implement this in practice, we notice that doing the twist reduces to multiplying the Mellin representation of different terms of the correlator (3.23)We can absorb them by shifting δ ij and this has the effect on the Mellin amplitudes by acting with a difference operator where the explicit action of D n 1 ,...,n 5 reads  The various Pochhammer symbols come from comparing the shifted Gamma factor with the one in the Mellin representation definition.The full difference operator from the Drukker-Plefka twist, denoted as D DP , is then a sum of such operators acting on different R-symmetry structures.As we explained in Sec.2.2, the twisted correlator is just a constant in position space.Following [2,3], we should interpret its Mellin amplitude as zero.Therefore, the Drukker-Plefka twist condition becomes in Mellin space which explicitly reads The implications of this equation are discussed in the following section.

Strategy and ansatz
After introducing all the necessary ingredients, we are now ready to state our strategy.Our strategy is comprised of three steps.First, we start by formulating an ansatz in Mellin space which is based on our analysis of the analytic structure of the Mellin amplitudes.Second, we impose the Mellin factorization condition which is the statement that the pole residues should be correctly reproduced by the lower-point amplitudes.Finally, we implement the Drukker-Plefka twist in Mellin space and completely fix the ansatz.In the following, we explain the details of each step.
Step 1: Ansatz As we emphasized in the previous section, Mellin amplitudes are merophormic functions with simple poles corresponding to exchanging single-trace operators and residues related to lower-point amplitudes via factorization.Based on this, we have the following ansatz for the pp222 Mellin amplitude Here A m (δ ij , t ij ) is a rational function with possible poles in δ 34 , δ 35 , δ 45 .In particular, it includes simultaneous poles which correspond to double exchange processes in the ( 12), (34) channels etc.Similarly, B āa (δ ij , t ij ) is a rational function with possible poles in δ kl at δ kl = 1.The labels k, l need to satisfy k, l = ā, a but can be both from the set {3, 4, 5}, or belong to different sets {1, 2} and {3, 4, 5}, see equation (4.5).To avoid double counting, C jk (δ ij , t ij ) and D(δ ij , t ij ) do not have poles and they are polynomial functions of the Mellin-Mandelstam variables.Note that here we have also used our Mellin factorization analysis for the subleading poles from Section 3.1.We imposed that the poles in the ( 12) channel truncate to m = 0, 1, 2.
More concretely, the function A m (δ ij , t ij ) in the ansatz has the following form where A 34,m , A 35,m and A 45,m are polynomials of degree 2 and A ∅,m is a polynomial of degree 1. Written explicitly, A 34,m reads where {δ 23 , δ 25 , δ 45 } are chosen to be the independent Mellin-Mandelstam variables in addition to δ 12 and δ 34 which already appear in the poles.We have also used {T I } to denote collectively the 29 independent R-symmetry structures in (2.7).The expressions for A 35,m , A 45,m are similar.The polynomial A ∅,m is given by The other terms in the ansatz are similar and are given by In making the ansatz we have assumed that the degrees of various polynomials are the same as in the p = 2 correlator.This is expected from the flat-space limit which is related to the high energy limit of the Mellin amplitude [31].This can also be confirmed by Mellin factorization, which will be used in greater detail in the next step. 9 Step 2: Mellin factorization The second step of our strategy is to impose Mellin factorization.As explained in the previous section, all the polar terms of the Mellin amplitude can be completely fixed in terms of the lower-point Mellin amplitudes.For the pp222 five-point function, all these lower-point amplitudes are known and are given in Appendix B. These lower point functions depend on R-symmetry polarization vectors.One important detail which we did not discuss is how to glue together the R-symmetry structures in the lower-point functions using the representation of the exchanged fields.This step is explained in detail in Appendix A. 4. Thus all terms in the ansatz (4.1), except for the regular term D, can be fixed by using this factorization procedure.Note that the number of coefficients that remain unfixed in the ansatz is quite low as D is just a constant with respect to the Mellin-Mandelstam variables.It can depend only on the linear combination coefficients of the 29 R-symmetry structures.
Step 3: Drukker-Plefka twist The final step is to impose the Drukker-Plefka twist.As explained in Section 3.2 this 9 For example, it is straightforward to see that these are the correct degrees when exchanging scalar operators.Exchanging vector or tensor fields is a bit more nontrivial but it is possible to check that the degrees are correct.The only subtle point which avoids the factorization argument is the degree of the regular piece.However, it is natural to assume that the degree is the same as the p = 2 case so that it has the same high energy growth as the other terms.
twist can be phrased in terms of a difference operator D DP acting on the Mellin amplitude, see (3.26).This relates the regular part with the singular part already fixed by factorization and completely fixes the remaining coefficients 10 .
Using this strategy, we obtain the pp222 Mellin amplitudes in a closed form for arbitrary p.The final result for the Mellin amplitudes will be presented in the next section 11 .

Mellin amplitude for p = 2
Due to the many R-symmetry structures involved, the expression for the full Mellin amplitude appears to be quite complicated at first sight.Therefore, before we present the Mellin amplitude for general p, let us first revisit the p = 2 result of [1] and present it in a simpler way.
When p = 2, the amplitude is symmetric under permutations of all the five external points.The 22 R-symmetry structures also split into two classes and within each class the structures are related by permutations.The first class is the pentagon contraction 10 At the same time the Drukker-Plefka twist provides a very non trivial consistency check for the procedure of extracting correlation functions of super-descendants and gluing of R-symmetry structures described in Appendix.
11 It would also be interesting to extend this analysis to the first correction in α .One promising candidate is the p = 2 case since it is more symmetric and we can also use the known results for the four-point function as an input [35].
It is clear that terms of the same structure are related by the permutations preserved by the R-symmetry structure.We will see that the Mellin amplitude for general p also has similar structures.

Mellin amplitudes for general p
For p > 2, we no longer have the full permutation symmetry and there are seven types of R-symmetry structures as we discussed in Section 2.1.The Mellin amplitude can be written as a sum over all the inequivalent R-symmetry structures a 3 a 4 a 5 P (I) a 3 a 4 a 5 + where the sets I 1,2,3 contain the following permutations (3,5,4), (4,3,5), (4, 5, 3), (5,3,4), (5,4, 3)} , The coefficient Mellin amplitudes are given as follows.For the structures of P (I) a 3 a 4 a 5 , the coefficients are Upon setting p = 2, the two coefficient amplitudes become degenerate up to permutations and reproduce M P 1 in (4.9).The coefficient Mellin amplitudes of a 3 a 4 a 5 and T (III) a 1 a 2 a 3 a 4 a 5 are given by M T,(I) They become M T 1 in (4.10) when p = 2. Finally, the coefficients of the two new structures N Note that they are proportional to p − 2 and therefore vanish for p = 2.
Let us also make a comment regarding the seemingly confusing bevahior at the flatspace limit.The flat-space amplitude which one obtains from holographic correlators corresponds to that of gravitons.In general, one expects that the dependence on the KK levels should factorize as different KK modes all correspond to the same particle in flat space.However, this is not the case if we naively take the high energy limit of the Mellin amplitudes.Clearly, the p-dependence is not factored out as the component amplitudes of the new R-symmetry structures for p > 2 have the same high energy scaling behavior as the other component amplitudes.To understand this, it is important to note that the flat-space amplitude from AdS is in a special kinematic configuration where the polarizations of the gravitons are perpendicular to all the momenta [9].However, such an amplitude for five points is zero in flat space.12Therefore, the high energy limit of the Mellin amplitudes is not the flat-space amplitude as one might have naively expected.In fact, in applying the prescription of [31], there is an additional power of the inverse AdS radius 1/R which renders the flat-space limit zero.In other words, the high energy limit of the Mellin amplitudes computes only the 1/R corrections.We expect these corrections to have the same power counting for different KK modes.However, we do not expect their explicit expressions to be universal.

A comment on consistency
Let us make a comment regarding the consistency of our result.In Section 3 we proved the truncation of the poles in δ 12 by using factorization in the (12) channel which only exploits the general analytic structure of the resulting four-point amplitude.Here we point out that the truncation can also be seen from a different point of view when it involves simultaneous poles with another channel.For concreteness, let us focus on the residue of the amplitude at the pole δ 45 = 1.The residue is, via the factorization in the (45) channel, related to a fourpoint function pp2X where the first three operators are 1, 2, 3 respectively.As we know from (3.5), the operator X belongs to the k = 2 multiplet and can be the superprimary O 2 , the R-symmetry current J µ or the stress tensor T µν .The Mellin amplitude of pp2X contains poles in δ 12 due to the operator exchanges in the (12) channel.These four-point Mellin amplitudes are given explicitly in Appendix B and we observe a truncation of the subleading poles in δ 12 for m ≥ 3.This gives another derivation of the structure of the simultaneous poles in δ 12 and δ 45 .
Similar consistency checks have also been performed in other channels (e.g., in the (13) and (45) channel), as well as for the R-symmetry gluing (see Appendix A.4 for details).

Comments on position space
Up to this point, all of our discussions are exclusively in Mellin space.This is mainly because of the simplified analytic structure of Mellin amplitudes, as can be seen from our main result (4.10).However, it is also sometimes convenient to have position space expressions as some information is difficult to extract from the Mellin space representation.This has to do with the fact that certain nonzero expressions in position space may naively vanish in Mellin space.More precisely, different inverse Mellin transformations can only be added up if their contours can be smoothly deformed from one to another.Usually the contour part is ignored for simplicity and one just adds up the Mellin amplitudes.This causes some information to be lost in the process.In fact, we have already encountered such an example in this paper: The Drukker-Plefka twisted correlator is a constant in position space but has zero Mellin amplitude. 13The existence of the ambiguities makes a direct translation of Mellin space results into position space difficult.
One could also try to directly extend the position space algorithm of [1] to the pp222 correlators.However, as explained in the introduction, this is technically difficult.Here we propose a hybrid approach.As explained in [1,10], all five-point Witten diagrams can be expressed as a linear combinations of five-point D-functions by using integrated vertex identities 14 .It is then natural to construct an ansatz in position space directly in terms of the D-functions.This will avoid directly computing the Witten diagrams which is a nontrivial task.More concretely, we propose that the ansatz for G p in position space should have the following form where the coefficients c {β} (t ij ) are linear combinations of all possible R-symmetry structures.The summation over β ij are subjected to the constraints ∆i + Let us now unpack these constraints a little.The first condition (4.20) ensures that the external operators have the correct weights under conformal transformations.The constraint (4.21) imposes a bound on the sum of weights in each D-functions. 15This is expected if we use the integrated vertex identities 16 to reduce the exchange Witten diagrams to contact Witten diagrams.Exchanging single-trace operators leads to singularities in position space.
The condition (4.22) is the statement that particle exchanges have to be in the compatible channels.The constraint (4.23) arises because the exchanged single-trace operator operators have maximal spin 2. To understand this more precisely, let us notice the following translation between position and Mellin space The condition (4.23) ensures in Mellin space that the numerator associated with an exchange pole is at most quadratic.Finally, the constraint (4.24) controls the twists of the exchanged single-trace operators.Let us emphasize that this position ansatz, as it stands, does not manifest the truncation of poles seen in (4.1).Nevertheless, this truncation can still be imposed in position space, though in a more intricate manner (this is in stark contrast with Mellin space).We notice that a given negative power (x 2 12 ) −α will lead to poles in Mellin space at all the locations δ 12 = 1, 2, . . ., α.Therefore, even though the δ 12 poles in Mellin space truncate according to (4.1), in position space the result will necessarily involve all negative powers of (x 2 12 ) −α with α = 1, 2, . . ., p − 1. Truncation only implies that the negative powers are related but cannot just simply eliminate a subset of them.This is another instance where we can see explicitly that Mellin space is simpler.
To fix the coefficients in the ansatz, one can translate the ansatz back into Mellin space and compare with the Mellin amplitude (4.10).This can be achieved by using the rule (4.25).However, as explained above, only some of the coefficients can be fixed due to the ambiguities of the translation.One may wonder if implementing the Drukker-Plefka twist and the chiral algebra condition in position space 17 will give rise to additional constraints.But unfortunately we find that this is not the case.There still remains the possibility that one can fix the remaining coefficients using the recently derived higher-point lightcone conformal blocks [37] to impose factorization in position space.But we have not found a very efficient way to implement this.Therefore, we will postpone the task of finding the expressions in position space and leave it to future work.

Discussions and outlook
In this paper we continued our journey of exploring the structure of five-point functions of 1  2 -BPS operators of 4d N = 4 SYM in the strongly coupled regime which is dual to AdS 5 × S 5 IIB supergravity.We improved the bootstrap approach of [1] which relies only on superconfromal symmetry and consistency with factorization.The important difference compared to the old approach is that both constraints are now implemented in Mellin space.Moreover, in the new method we only need to use the Drukker-Plefka twist and the chiral algebra condition is not needed.Using this approach, we obtained in a closed form the Mellin amplitudes for the infinite family of correlators of the form pp222 .
Compared to the simplest 22222 case studied in [1], the pole structure of the Mellin amplitudes of operators with higher KK levels is in general more complicated.However, an important simplifying feature we observed in this paper is a new type of pole truncation phenomenon.We find that the residues of certain poles associated with conformal descendants vanish.Morevoer, in the pp222 case the number of poles does not grow with respect to p when p is large enough.Consequently, the pole structure of the Mellin amplitudes is much simpler than what is naively expected.This property played an important role in obtaining the pp222 amplitudes and also gives us hope to bootstrap in closed forms more general families of five-point functions with different KK levels.
Note that in deriving the pole truncation conditions, we have only used general properties of Mellin factorization.The same argument holds in many other theories and we expect similar simplifications in the pole structure.This leads to a number of possible extensions of our results in different setups.A prime example to consider is the gluon sector of certain 4d N = 2 SCFTs which is dual to SYM in AdS 5 × S 3 .The first five-point function for the lowest KK level has been computed in [10].To make further progress in computing amplitudes of higher KK levels, one can adapt the strategy used here.One important ingredient which still needs to be worked out is the relations between different component correlators of the super four-point functions (see [21] for progress in this direction).This would be the input for exploiting the full power of the Mellin factorization.However, this will be a direct generalization to what we have done in Appendix A. Another interesting application is to the 6d N = (2, 0) theory which is dual to eleven dimensional supergravity in AdS 7 × S 4 .
Going beyond five-point functions, an exciting future direction is to compute the super graviton six-point function of AdS 5 × S 5 IIB supergravity.This will provide a new benchmark for the program of holographic correlators at higher points.The results in this paper can already help us gain a nontrivial amount of knowledge of the structure of this new correlator.Moreover, much of the technology developed here, in particular the Mellin Drukker-Plefka twist, can also be straightforwardly applied to that problem.It appears to be a feasible target and we hope to report progress in this direction in the near future.
Finally, let us mention that the pp222 five-point functions we computed in this paper contain a wealth of new data of 4d N = 4 SYM.Through OPE, we can extract various non-protected three-and four-point functions.In [1] we constructed five-point conformal blocks (see [29,[38][39][40][41] for progress in higher-point conformal blocks) and explained how to use them to extract data from the p = 2 five-point correlator.It would be interesting to perform a similar analysis here for the pp222 correlators.The expression we have for general p will be helpful for solving the mixing problem for the CFT data which is similar to the one appearing in four-point functions.It would also be interesting to extract the chiral algebra correlator from our supergravity result and compare with the field theory calculation.The four-point function case has been analyzed in [42,43].
The relevant superdescendants are obtained extracting the appropriate component by acting with certain differential operators: Given the charges and symmetries of those operators the ansatz for the differential operators needs to be 18 and Before fixing the coefficients let us quote two simple identities which are very useful in the following 19   18 These differential operators depend on p through the coefficients µ, ν1, ν2.This dependence is not explicit in the notation. 19The second identity is obtained as follows
For the spinning correlators we can also write inverse Mellin transforms as follows −1 β (k,l) p 1 ,p 2 (u, v; y ij ) .

(B.13)
As explained in the previous section, the functions α  .

(B.20)
There are once again some satellite poles, but the explanation follows exactly the same reasoning as before.The relevant Gamma factors in M

Figure 1 .
Figure 1.Inequivalent R-symmetry structures in the pp222 five-point function.Here (a 1 , a 2 ) is (1, 2) or (2, 1) and (a 3 , a 4 , a 5 ) can be any permutation of (3, 4, 5).Each thin line represents a single contraction.The thick line represents the multi-contraction t a 12 with the power a given by the number next to the line.The R-symmetry structures in the first row have counterparts in the 22222 five-point correlator.For pp222 they are simply obtained by multiplying the p = 2 structures with t p−2 12 .The R-symmetry structures in the second row are new and do not appear in 22222 .

I 1 M 2 Ma 3 a 4 a 5 + I 3 M 1 M
P,(II) a 3 a 4 a 5 P (II) a 3 a 4 a 5 + M T,(II) a 3 a 4 a 5 T (II) T,(III) a 1 a 2 a 3 a 4 a 5 T (III) a 1 a 2 a 3 a 4 a 5 + M N,(II) a 3 a 4 a 5 N (II) a 3 a 4 a 5 , (4.10)