New relation for Witten diagrams

Abstract
 
 In this paper, we present a simple and iterative algorithm that computes Witten diagrams. We focus on the gauge correlators in AdS in four dimensions in momentum space. These new combinatorial relations will allow one to generate tree level amplitudes algebraically, without having to do any explicit bulk integrations; hence, leading to a simple method of calculating higher point gauge amplitudes.


Introduction
A great deal of developments have taken place in the last decade in the study of flat space scattering amplitudes of gauge theories. The modern amplitudes research program has led to many unexpected relations such as on shell recursion relations [1,2], the connection to mathematical structures like Grassmanian geometry, and the discovery of the amplituhedron [3][4][5]. For an introduction to these computational tools and an overview of these developments, we refer the reader to [6][7][8][9][10].
Likewise, outstanding progress has been made in our understanding of quantum gravity with the discovery of the holographic principle [11,12]. The holographic principle implies that degrees of freedom that are encoded in the boundary in d dimensions can describe the d + 1 dimension interior of the spacetime. A concrete example of holography is the gauge/gravity duality, i.e. the correspondence between Anti-de Sitter space (AdS) with Conformal Field Theory (CFT) [13,14]. By relating the boundary operators to the bulk fields, the CFT correlation functions can be interpreted as AdS scattering amplitudes, and vice versa [15,16].

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In this paper, we will outline a new diagrammatic method that will enable us to systematically compute higher point amplitudes. Crucially, this method bypasses the cumbersome bulk integrals, therefore reducing the computations of the correlators to simple algebraic relations. This framework can be utilized to calculate higher point tree level AdS amplitudes, which can be used as data points to extract physical and mathematical insights. We liken this to the similar methodology employed in flat space scattering amplitudes over the last decade, where explicit flat space amplitudes were used as data points to generate surprising relations, such as the BCJ duality, CHY relations, and the amplituhedron [4,[43][44][45].
Here is the brief outline of this paper. We begin, in section 2, with a review of the formalism for momentum space gauge theory correlators. Then, we introduce bulk to bulk and bulk to boundary propagators for gauge fields as solutions to their respective equations of motion. In section 3, we present the combinatorial rule to compute part of the scattering amplitudes for gauge fields in AdS. Subsequently in section 4, we discuss computation of the remaining part, hence obtaining the full expression. Finally, we summarize and discuss promising future directions in section 5. We provide an appendix to further illustrate the main points of this paper with an additional example.

Gauge fields in AdS
We are interested in a non-Abelian gauge theory in AdS, described by an action where z is the radial coordinate and x i approaches to the boundary coordinate as z → 0 for the AdS metric ds 2 = z −2 (dz 2 + η ij dx i dx j ) in the Poincaré patch. These coordinates make the Poincaré invariance manifest; thus, it is easy to transform the position space coordinates x i to the momentum space variables k i . In this paper, we will focus on d = 3, impose axial gauge, A a 0 = 0; and work with the coordinates {z, k i }, following closely the treatment of [46].
One can go ahead and solve the equations of motion: the normalizability at the boundary and the regularity in the bulk unambiguously determines the bulk to boundary propagator, i.e. A a i (z, k) = a i √ zE 1/2 (kz) [30]. Here is the transverse polarization vector; k is the positive norm of the momentum k, i.e. k = |k 2 |; and E is a Bessel kind function, appropriate to the sign of k 2 . 3 In this framework, the amplitude for the four point exchange diagram depicted in figure 1 takes the form , 3 In this paper, we will work with transition amplitudes and use a i 2kz π K 1/2 (kz) as our effective bulkto-boundary propagator, same as [32,47]. which is obtained by gluing bulk to bulk and bulk to boundary propagators. In this expression, and in the rest of the paper, we suppress the color dependence for notational brevity.
In the above expression, we use which is the three point vertex for color-ordered amplitudes [48]. For later convenience, we also define the four point vertex factor In order to make the notation concise, we use V 12k ≡ i 1 j 2 V ijk and likewise for other tensors. As we will not be working with individual components, such notation does not lead to any ambiguity.
One can now calculate the full amplitude by directly carrying out the bulk integrals, thus arriving at the four point expression: 4 Note that this is the amplitude associated with the s-channel diagram. Using the same method we can compute the t-channel and contact diagrams. The form of the expression above suggests that we can decompose any tree-level diagram into two parts: the vertex factors carrying the dependence on the individual vectors k i , and the rest of the amplitude that we will call the truncated diagram. For example, in eq. (2.2), we see that k 1 dependence enters into the truncated piece only through the 4 In this paper, we use the notation of [32] to denote sums of momenta, i.e. and E.g., k12345 ≡ |k1 + k2| + |k3| + |k4 + k5| and k12 ≡ k1 + k2.

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terms k 12 and k 12 . Note that apparently different pieces are combinations of such terms, e.g. k 123412 = k 12 + k 12 + k 34 . We can further decompose the truncated diagram into two distinct scalar diagrams, straight and crossed, where we define the projectors To be specific, p 1 p 2 m n k denotes the bulk point integrated-propagator in the momentum space, which coincides with the truncated four point amplitude. We will use this graph as a basis element to construct higher point truncated diagrams: working with these bulk-point integrated diagrams will allows us to efficiently extract several different Witten diagrams from their common truncated graph. For example, we can connect two of these basis elements to construct the amputated graph necessary for the five point Witten diagram, as can be seen in figure 2.
The advantage of the decomposition in eq. (2.3) is the simplicity of the scalar graphs: they are fully agnostic to what is attached at the vertices as long as we know the sum of norms of the momenta that flows into that vertex. This simply follows from the form of our effective bulk to boundary propagators as their bulk-point dependencies at the vertices are merely additive. For example, p 1,2 in eq. (2.3) represents the sum of the norms of bulk to boundary momenta, i.e. p 1 = k 1 + k 2 + . . . . This framework suggests that any tree-level Witten diagram can be decomposed into sums and products of vertex factors, projectors, and several scalar graphs; thus, the calculation of a Witten diagram reduces to a calculation of scalar graphs. E.g., for the five point diagram given in figure 2, we can obtain the full expression once we compute the corresponding scalar graphs.
The calculations of graphs with crossed lines and graphs without crossed lines are different: we will deal with them separately. We will present the algorithm to compute the graphs of the straight lines in the next section: this algorithm exists in the literature albeit in a completely different context and theory [49][50][51][52]. After that, we will establish the connection between the crossed and straight lines in section 4. Such a relation will enable us to compute the complete amplitude. examine the explicit expression corresponding to the straight line: where B(z 1 , p 1 ; z 2 , p 2 ) encodes the contribution of other graphs that connect to this propagator at the bulk points z 1 and z 2 . As we discussed above, these contributions are additive and are represented by the letter p in the diagram. The remarkable feature of G s is that it is proportional to the cosmological propagator derived for the conformally coupled scalar; specifically where {z, k} → {−η νe , −iE e } in the notation of [51]. In that paper, the authors show that one can compute similar graphs using algebraic means. The nice feature of our bulk-point integrations is that they do not extirpate this formalism, hence we can also use graph-wise calculations in our setting. 5 Here, we summarize our prescription to calculate any tree-level Witten diagram for AdS 4 gauge bosons in momentum space in axial gauge:  We have already explained the first two items, let us now move on to the third point. For that, we need to review the algorithm of [51] which we will use to procure expressions for straight-only scalar graphs.
One starts with the full diagram and considers the ways to decompose it into different subdiagrams by cutting the lines. Then, one associates an expression to each decomposition and sums these partial amplitudes.
The partial amplitude for a particular decomposition is simply the product of the expressions associated with the subgraphs. For a particular subgraph, the associated expression is inverse of the sum of all vertex norms within that subgraph and line norms going out of that subgraph. With this rule, we associate the corresponding expressions to all subgraphs, starting from the full graph itself.
Let us clarify this rather formal explanation with an explicit example: a single straight line, depicted in figure 3. As we see, there is only one possible decomposition because there is only one line to cut! For this partial-amplitude, there are three subgraphs: the full graph indicated by the blue rectangle and its subgraphs indicated by the green rectangles. Since there is no line crossing the blue rectangle, its corresponding expression is inverse of the sum of the vertex norms, i.e. (k 12 + k 34 ) −1 = k −1 1234 . On the contrary, the green rectangles get contributions from the line norm as well, hence they are k −1 1212 and k −1 3412 . The full partial amplitude is simply the product of these three terms: The expression of the blue rectangle satisfies the generic feature of our algorithm: since there is always a graph than encapsulates the full diagram, all amplitudes will have a factor in the denominator, which is the sum of all vertex norms, i.e. k 1 + k 2 + · · · + k n . As this factor will always be multiplicative, our algorithm guarantees that the amplitudes for straight-only scalar graphs will have a pole where k 123...n → 0 for any tree-level n-point Witten diagram.
The appearance of such poles can be understood in the context of flat-space limit due to a nifty relation:

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This simply follows from our choice of momenta variables and the momentum conservation condition in flat space [31,51,53]; hence flat space limits of our expressions are immediately manifest. A more non-trivial example is the graph with two straight lines, which is relevant for five point Witten diagrams. In this case we have two lines to cut, indicating two distinct decompositions. We calculate the partial amplitude for each decomposition and take their sum: which yields a surprisingly compact final amplitude This is the same expression computed by brute force calculation in [32].

From straight graph to crossed graph
We have presented an elegant formalism in the previous section, and showed how one can easily extract the amplitude for a straight-only graph. An astute reader may object that despite its efficiency this formalism only yields a small part of the full amplitude. For instance, we need to calculate three more graphs if we want to obtain the amplitude for a five point Witten diagram, as seen in figure 2. The situation appears to deteriorate at higher point computations as there are exponentially more graphs to calculate. Below, we will demonstrate the opposite: all graphs are actually tied to the straightonly scalar graph hence one does not need to explicitly compute them once the straight-only scalar graph is obtained. This follows from the form of the gauge propagator in our settings: This split form of the propagator is exactly what motivated us to introduce the decomposition in eq. (2.3) in the first place. Also, this reveals that the straight and crossed lines are related in a beautiful way:

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This simple relation of straight and crossed lines explains why the apparent problem of exponential increase in the number of total graphs poses no issue in the actual calculations: one can simply write the full expression as a simple operator acting on the straight-only graph: Alternatively, first we can find the amplitude for each graph and then combine them. In case of figure 2, we write A (11) (4.3b) whose explicit expression is given in (3.5). Incorporating all A (ij) with the projectors, we get the full two line truncated diagram: One should keep in mind that M mn and M mnpr are actually not the full amplitudes; these expressions sill need to be contracted with the appropriate vertex factors V ijk or V ijkl c . The different choices of these terms yield different Witten diagrams; for example, which match the respective amplitudes calculated by brute force in [32].

Discussion and future directions
In this paper we discussed AdS amplitudes for gauge fields and developed a formalism that considerably simplifies the calculation of any tree level Witten diagram. This formalism

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is based on two observations: first, the calculation of truncated amplitudes reduce to computations of scalar graphs in a judiciously chosen basis, and second, the amplitudes for the scalar graphs can be extracted by mere algebraic means. With these observations, we can obtain any tree level amplitude for gauge fields in a systematic and elegant fashion.
The advantage of our procedure is twofold. Working in the appropriate basis, we can relate several calculations to each other, drastically simplifying the overall complexity. Indeed, in the conventional approach, the number of integrals required for calculation increases exponentially. 6 The other advantage of our technique is that it is purely algebraic, which allows us to bypass the bulk integrations altogether. However, it is an open question how to extend this formalism beyond the gauge boson and to other dimensions. 7 We hope that our formalism can be utilized to generate more data points in the study of amplitudes in Anti-de Sitter space. Knowledge of higher point amplitudes may result in unraveling deeper physical and mathematical insights, similar to what the flat space program has achieved over the last decade. In this sense, we see our work as a complementary approach to those developments; for instance, it would be interesting to explore a possible connection between AdS amplitudes and geometric structures like the amplituhedron.

A A bestiary for star triangle topology
In this appendix, we will make a detailed analysis of figure 4. The truncated amplitude can be written as  [32]. In our formalism, we only did one explicit calculation, eq. (3.5), and obtained the rest trivially by eq. (4.3). 7 The algorithm we used to compute scalar graphs is applicable only for AdS4. It is intriguing to know whether analogous algorithms exist in other dimensions.

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The other pieces such as A (112)  This reduces the whole calculation to that of the straight-only graph. We will carry out that computation using our algorithm as an explicit demonstration. Our diagram satisfies a neat symmetry hence it is sufficient to calculate only one subgraph. More explicitly, at the top layer, we have the decomposition where each term is related to one another by permutation of k 12 , k 34 , and k 56 .
As we go further, subgraphs in deeper layers also satisfy similar symmetries. For example, the decomposition of the second graph above can be written as  where ab → cd stands for {k ab , k ab } → {k cd , k cd }. Here, I denotes the first diagram in the right hand side of eq. (A.6). We can immediately read off its value by our algorithm: I = 1 k 123456 k 1212 k 123456 k 3434 k 125634 k 5656 k 12 34 56 . and substitute them into eq. (A.1) to get the full truncated amplitude.
One naively sees a divergence in the calculation of the piece A . This poses no issue, as that term does not contribute to the result since it vanishes once it is contracted with the vertex factor. This becomes transparent if we rewrite the three point vertex as a sum of three projectors, i.e.
As the crossed lines come with factors of k i 1 , k j 2 , and k k 3 , the number of non-vanishing terms in the vertex factor decreases per crossed line entering the vertex; e.g., there are only two pieces for A (112) , only one piece for A (122) , and no piece for A (222) .
With the truncated star triangle diagram at hand, we can calculate several different Witten diagrams by attaching different vertex structures. The simplest such diagram is the six-point amplitude shown in figure 4.
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