Towards the higher point holographic momentum space amplitudes

In this paper, we calculate higher point tree level vector amplitudes propagating in AdS$_4$. We use bulk perturbation theory to compute tree level Witten diagrams. We show that when these amplitudes are written in momentum space, they reduce to relatively simple expressions. We explicitly compute four and five point correlators and also sketch a general strategy to compute the full six-point correlators.


Introduction
After twenty years of its formulation, the AdS/CFT duality remains the most concrete example of a theory of quantum gravity [1,2]. The correspondence has elucidated different features of quantum gravity and has helped us understand dynamics of strongly coupled quantum field theories.
The AdS/CFT duality is a correspondence between quantum theory of gravity in Antide Sitter spacetime and Conformal Field Theory (CFT) that lives in the boundary of such spaces. The AdS/CFT dictionary implies that each field, say φ i , in the gravitational theory corresponds to a local operator O in the CFT. Similarly, the spin of the bulk field is the same as the spin of the CFT operator, and the mass of the bulk field determines the scaling dimension of the CFT operator. The correspondence has been effective to study nonequilibrium phenomena in strongly coupled systems including condensed matter systems. 1 It is worth comparing the similarities and differences between Anti-de Sitter space correlators and the well-known flat space (Minkowski) amplitudes. The amplitudes in the Anti-de Sitter space are not defined in the usual fashion as in the case of flat space. Unlike flat space, AdS space does not admit asymptotic states that are needed for the standard definition of S-matrix. However, AdS space does bear close resemblance to flat space Smatrix: Creation and annihilation operators in AdS space can be defined by changing the boundary conditions in the conformal boundary. The resultant scattering amplitudes in AdS space are then related to CFT correlation functions. Therefore, with the AdS/CFT correspondence, correlation functions can be computed by perturbative calculations that can be diagrammatically represented by Witten diagrams. The prescription for computing these correlators is very straightforward. However, after about two decades of AdS/CFT, despite the directness of the procedure, it remains a challenge to do concrete computations of correlation functions in the boundary for correlators of four and higher points 2 .
In the last few years, there has been a renewed attention in computing AdS/CFT correlators. Many have argued that Mellin space can remarkably simplify correlation functions in conformal field theories that has weakly coupled bulk duals [10][11][12][13][14][15][16]. While Mellin spaces are interesting, it is not easy to generalize the interesting results to higher spin fields. In this paper, we will study correlators in momentum space as a complementary approach. Even though many exciting directions has been explored in momentum space [17][18][19][20][21][22][23][24][25][26], momentum space higher point correlators have not been fully explored. By computing higher point correlators, we hope to provide theoretical data that may shed light to the structures of these correlators in general.
In a concurrent development, there has been great progress in our understanding of scattering amplitudes of gauge theories and gravity. Many amplitudes whose computation once seemed untenable show remarkable simplicity and sometime can be expressed in one line [27,28]. Several new formalisms, computation tools, and insights have been developed in the last ten years to understand this simplicity; for instance, appropriate choice of physical basis such as twistors, recursion relations, and geometric interpretation of amplitudes in term of a volume of an object such as the amplitudhedron has given fresh and deep insights about locality and unitarity [29][30][31]. A natural question to ask is if one can replicate the success of scattering amplitudes in flat space of gauge theories and gravity 3 to the study of AdS/CFT correlators. In this spirit, we will focus our attention on gauge theory AdS amplitudes that are dual to the conserved current correlators in the boundary.
Here is the brief outline of the paper. We begin with a quick review of the formalism for momentum space correlators in Section 2 where we will introduce bulk to bulk and bulk to boundary propagators for scalar and vector fields as solutions to their respective equations of motions. We will also review three point Witten diagrams, construct the ingredients essential to compute higher order functions, and move on to Section 3 where we will compute all four and five point diagrams explicitly and discuss calculation of six point diagrams with an explicit example. Finally, we summarize this work and discuss possible future directions in Section 4.

Review: Momentum Space Perturbation Theory in AdS
The main emphasis of this section is to quickly review the perturbation theory in Anti-de Sitter space. Our building blocks are Witten diagrams, where a typical Witten diagram is a combination of three distinctive constituents: 1. external lines which connect the bulk point of the AdS to the boundary, 2. internal lines that propagate the fields, 3. vertices where interaction can take place.
One can find the bulk to bulk propagators (which we will denote as BtB ) by solving the Green's function. By taking one of the points to the boundary, we can obtain the external lines, i.e. the bulk-to-boundary propagators (which we will denote as Btb ) where they correspond to propagation of some field perturbation into the bulk. Lastly, one integrates over the bulk interaction points, denoted as z below.
We will first discuss the propagators and list possible solutions as equations of motion both for scalars and vectors in AdS. Then we will use them in the simplest Witten diagram: a three point amplitude.

Equations of motion and propagators
For simplicity, let us restrict to the scalars. Working in Poincaré coordinates, one can derive the position space two point functions, i.e. BtB for scalars in AdS, by solving where our conventions are such that for the mostly positive convention for the metric at the boundary. The advantage of these coordinates is the manifest Poincaré invariance which makes it easy to transform the position space coordinates x i to their momentum space counterparts k i .
The solution to Eqn 2.1 is given in terms of hypergeometric functions [33]; and, one can obtain a Btb from this BtB by taking one of the end points to the boundary; say, with the limit z 1 → 0. Instead, we will focus on the corresponding momentum space expressions and restrict ourselves to the massless case for simplicity.
The expressions in what follows are based on the results of [17,20]. We first expand the Greens function in Fourier modes for which Eqn 2.1 now reads as By solving this equation, one finds that The appearance of Bessel function can be most readily observed from the massless Klein Gordon equation in AdS d+1 which reads as in momentum space. The solution depends on the chosen behavior around z = 0 and the sign of k 2 ≡ η ij k i k j . We list them in Table 1 where we define the positive definite norm k as k ≡ |k 2 |.
Vector fields: Having quickly discussed the scalars, we can turn to the main interest of the paper: gauge fields. The action for a non-abelian gauge group in AdS is written as, where, following closely the treatment of [17], we impose axial gauge. This reduces the action into 4 In momentum space, the relevant equations of motion are, (2.9) 4 Following [17] we simply add the gauge fixing term ζ a (A a 0 ) 2 and take ζ → ∞ to freeze A a 0 at zero. We also integrated by parts and ignored the boundary term S B Axial which does not alter the free equations of motion. 5 We will be mostly suppressing the momentum argument for the Fourier mode A a i (z) ≡ A a i (k, z).
Similar to the scalar case, we have different solutions for different cases which we list in Table 2. 6 Lastly, the Green's function for the vector field can be written in the following way and we have suppressed the color dependence.

A basic Witten diagram: three point amplitude
Three point amplitudes are very important as they correspond to three point correlation functions in the holographic CFT's, which contain all the dynamical data of a CFT [23]. Additionally, these amplitudes will be the building blocks for the higher point computations in the following section.
By following [18], we consider a three-point amplitude with three incoming momenta, k 1 , k 2 , and k 3 with k 1 + k 2 + k 3 = 0, and observe that the result takes a product form of some tensor structure and the so-called triple-K integral, 7 a function of three magnitudes denoted as k 1 , k 2 , and k 3 : where V ijk is the color-ordered three point vertex structure 8 . The explicit form of color- 6 Note that only the norm of momentum enters the solutions, hence we need polarization vectors ǫ. These polarization vectors satisfy transversality k ·ǫ = 0 just like the gauge fields themselves k ·A(k, z) = 0: These conditions directly follow from the axial gauge choice. 7 Similar conformal integrals have been computed in [24]. 8 We follow the convention of [34].
ordered three and four point structures can be written as (2.13) Here, we use the shorthand notation V 1jk ≡ ǫ i 1 V ijk and so on for the polarization vector ǫ 1 . One may be concerned with this shorthand notation as V 1jk can also refer to V ijk i=1 . However, we will not have this ambiguity as we will never be interested in individual components in this paper.
The structure V 's above are the same antisymmetric tensors that appears in flat space, contracted with the polarization vectors. In the above expression and what is to follow, we will suppress the color factors and indices to lighten the notation and this simplification will not affect our analysis.

Calculation of Higher Point Amplitudes
As we have seen in the previous section, one can calculate any tree level diagram with the following algorithm: 1. Take the product of elements of Witten diagram: z 4 factors for vertices, appropriately normalized 10 Bessel K functions for the sources at the boundary, and the BtB for internal lines. Integrate this product over the bulk point z's. 4. Carry out the internal momenta p integrations, which were introduced in the decomposition of BtB .
Despite the straightforwardness of the procedure, analytically carrying out the integrals of the last step can be quite cumbersome. However, we will show in this section that these integrals actually yield extremely simple results and one can use residue theorem to compute these integrations efficiently. Similarly, we will see that while the individual residues are somewhat involved the final answers are actually simple.
The validity of residue theorem lies in the fact that bulk point integrated KKK, KKJ, etc. are meromorphic functions with respect to the internal integration variable, and they vanish at infinity. Also, even though they are not individually even functions of the integration variable, the relevant integrals will always be even with respect to the internal momenta: This follows from the fact that we either have pdp or dp p and bulk point z integration of Bessel function J's bring √ p at both ends of the propagator. Therefore, we can always write the integration over internal momenta using residue theorem.

Four point function
In this section we will write the color-ordered four-point amplitude of a gauge boson propagating in Anti-de Sitter space at tree level. The amplitude is the sum of three pieces: an s−channel diagram, a t−channel diagram, and a contact diagram.

S-channel
Let us start with the s-channel diagram as show in figure 2. The important consideration that we use while computing this integral is the factorization of the BtB in terms of two Bessel functions of the first kind J n as seen in 2.10. This allows us to write down the four point function as a product of two three point function of the type KKJ (k i , k j , p, z i ) as seen in Eqn 2.14. With these considerations, we can write down an expression for the s-channel diagram of the four point amplitude, where, Recall that we have V 12k = ǫ i 1 ǫ j 2 V ijk and so on in our conventions. We would like to remind the reader that in the above expression z and z ′ are two radial coordinates where integration takes place: Note that z 4 and z ′4 are part of the appropriate volume factors in three dimensions. Writing Eqn 3.3 using the the bulk z integral performed in Eqn 2.15, we obtain M 4s =ˆp dp Let us isolate p−dependent parts of the above integral: We can rewrite it as We will compute above integrals using the method of residues. For M (1) 4s , we have Similarly, we can perform integration for M (2) 4s . The residues are given by  Figure 3: The four point contact diagram

Contact diagram
The four point contact diagram has the same integral form as the three point amplitude in Eqn 2.12 upto the replacement of KKK with KKKK: where V 1234 c is defined in Eqn 2.13. 14 From Eqn 2.15 we can immediately read the result (3.14)

Final result
From Eqn 3.12 and Eqn 3.14 we can write down the full four point amplitude as (3.15) Here the second line is the t channel contribution which simply reads as (3.16) 14 Here, we would like to remind the reader that 1, 2, . . . do not refer to components in V 1234 c .

Five point function
In this section we will write the five-point amplitude of a gauge boson propagating in Antide Sitter space at tree level. As in the case of four point, we will look at color-ordered expressions. There are two types of diagrams for the five point amplitude, as depicted in Figure 4 and Figure 5. After permuting these two topologies we can obtain diagrams which can be added to compute the total five point amplitude:

Type-b five point diagram
The amplitude is given as where V klmb was defined in Eqn 2.13 .
The integrations over z 1 and z 2 can immediately be carried out via Eqn 2.15. Then one can do the p−integration, either directly or via the method of residues. The result takes the form We can now write the full five point amplitude as the summation over different topologies and their permutations. We can compare this with the case of four point amplitude in Eqn 3.15, where we have two topologies and two permutations for the first one (first two lines there) and one permutation for the second one (last line).

Six point function
Let us move on to six point amplitudes. Unlike the previous cases, we will not be providing the explicit results here because of two reasons: Firstly, the number of calculations proliferates as we consider higher point amplitudes; in particular, we have to compute 18 different terms to fully calculate all the topologies. While doing the calculations is actually straightforward and can easily be automated, the results are not extremely illuminating and involve lengthy pieces such as M (4) 5a . The second reason for us to avoid explicitly writing six point amplitude is that we expect a pattern and we believe higher point calculations can be carried out more efficiently for generic n point amplitude 16 . We do not pursue this idea in present work; instead, we give a taste of what the six point amplitude looks like by computing one of the terms appearing in type-a topology that is to be defined below: Other expressions can be calculated in a similar fashion.

Conclusion
In this paper, we have calculated tree level momentum space amplitudes for a massless vector propagating in AdS 4 . We are interested in such correlators because of their physical relevance and their simplicity. Additionally, these momentum correlators have not been fully investigated despite their relevance in cosmology, bootstrap, and possible connection with scattering amplitudes in flat space.
We found it convenient to work in the axial gauge as it allowed us to set the radial component to zero . We explicitly computed four and five point amplitudes and provided a recipe to compute full six point expression. Remarkably, we discovered that despite the tediousness of computation, we were able to write a compact and relatively simple expression. There are several promising directions that one can consider. For instance, it would be interesting to extend this work to higher spin amplitudes. Similarly, it would be interesting to extend these calculations to beyond tree level correlators.