Feynman Rules for Scalar Conformal Blocks

We complete the proof of “Feynman rules” for constructing M -point conformal blocks with external and internal scalars in any topology for arbitrary M in any spacetime dimension by combining the rules for the blocks (based on their Witten diagram interpretation) with the rules for the construction of conformal cross ratios (based on OPE flow diagrams). The full set of Feynman rules leads to blocks as power series of the hypergeometric type in the conformal cross ratios. We then provide a proof by recursion of the Feynman rules which relies heavily on the first Barnes lemma and the decomposition of the topology of interest in comb-like structures. Finally, we provide a nine-point example to illustrate the rules.


Introduction
The interest in conformal field theories (CFTs) is multifaceted. It ranges from applications to phase transitions in condensed matter systems, through application of conformal theories to extensions of the electroweak Standard Model, to describing one side of the AdS-CFT duality. In part, this interest is driven by the many successes of the conformal bootstrap program that allowed non-perturbative treatment of CFTs [1][2][3][4]. The bootstrap program relies on the decomposition of correlators in a CFT into conformal blocks and pure numbers: operator dimensions and the operator product expansion (OPE) coefficients. Another important motivation for studying CFTs is the AdS-CFT duality, in which the CFT correlation functions describe bulk interactions that can be encoded in terms of Witten diagrams [5][6][7].
Conformal blocks are crucial for computing correlation functions as they encode kinematic constraints on the form of correlation functions imposed by the conformal symmetry, while the information about the dynamics is encoded in terms of the "conformal data:" operator dimensions and OPE coefficients. Given how fundamental conformal blocks are for CFTs significant effort has been devoted to computing such blocks and several different methods have been presented in the literature. However, the majority of this effort has been focused on four-point correlation functions and associated four-point blocks, going back to the seminal works [8][9][10][11][12][13]. There are only a handful of results on higher-point global conformal blocks that are applicable for obtaining M -point correlation functions, where M ≥ 5 . One reason is that a complete set of crossing equations, which is one of the basic principles behind conformal bootstrap, can be formulated in terms of four-point functions alone. One could utilize crossing equations for higher-point functions as well, but they would be automatically satisfied if all four-point crossing equations are satisfied. For operators with spin, higherpoint functions might provide a practical advantage for bootstrap, but for now it is not clear if this will turn out to be the case.
Another reason for the scarcity of results on higher-point blocks is that they are notoriously difficult to compute. Blocks with as few as four points require rather involved approaches, while higher-point blocks are even more complicated. The complication cer- is simpler and there are comparatively more results for conformal blocks in those dimensions [14,15,21,25,29]. For a general number of dimensions there are only a few results for specific number of points and selected topologies 1 [15, 16, 19-21, 23, 24, 27, 28, 32].
Recently, a conjecture was made on the form of an arbitrary scalar conformal block, that is a block with both external and internal scalar operators, for any M and any topology with no restriction on the number of dimensions [28]. The conjecture provides a set of rules for writing a conformal block based on its diagrammatic representation that is reminiscent of the well-known rules that convert a Feynman diagram to a scattering amplitude. A crucial ingredient for this construction was the use of Mellin amplitudes and their close connection to amplitudes in the AdS space [36][37][38][39][40][41].
The conjecture was originally verified in specific examples, by comparison with other calculations in the literature and some novel higher-point results. Here, our goal is a proof of the conjecture. The proof is inductive: we obtain an M -point block from an (M − 1)point block. We borrow methods both from work that uses the OPE directly to construct blocks [17,42] and from work that relies on Mellin amplitudes [28].
One of the key ingredients that made the proof possible was a convenient choice of the invariant cross ratios. The choice of cross ratios is clearly not unique as any function of cross ratios is invariant as well. The arbitrariness is not there because one might consider some complicated functions of the invariant cross ratios. All cross ratios we work with are plain ratios of x 2 ij = (x i − x j ) 2 , but with M external variables there is no unique choice. Already for M = 4, one could choose the familiar u and v cross ratios, or choose instead u v and 1 v . Both choices are just ratios of x 2 ij 's and there are clearly more choices in addition to the two we just mentioned. What turned out important for the proof was tailoring cross ratios for an M -point function such that they reduce straightforwardly to cross ratios for an (M −1)-point function. The original conjecture divided the Mellin variables, and in turn associated cross ratios, into three sets: U, V, and D. Set U corresponds to, what we call, u-type cross ratios, set V to v-type, while set D contains dependent variables that are eliminated by integrating over a set of Dirac delta functions that incorporate dependencies. The particular choice of cross ratios that is described in detail later on is amenable to proof by induction. Choosing cross ratios might seem mundane, and yet it was significant for our proof. We derive suitable cross ratios using a diagrammatic method that we termed flow diagrams [29]. The flow refers to a choice of internal coordinates in a diagram. Using the convergence property of the OPE in CFTs, one can systematically reduce an M -point function to an (M − 1)-point function by combining two operators and replacing them with a single operator on the right hand side of the OPE, and so on until there is a nothing left but a simple two-or three-point function. Schematically, the OPE can be written as where we neglected functions of coordinates and derivatives on the right-hand side of the OPE. What flow diagrams represent is the position at which O k appears.
One could choose O k (x i ) or O k (x j ) and such a selection needs to be made with every OPE that is used to reduce a correlator to a conformal block and the conformal data. When working in the d + 2-dimensional embedding space, no other possibilities exists for the operator position on the right-hand side of the OPE as every coordinate needs to be placed on the light cone [1,8]. We do not use the embedding space here, but simply pick one coordinate or the other for the placement of O k . The intermediate steps depend on such choices, but the validity of the proof holds for any choice. Following how a coordinate "flows" in a diagram leads to a prescription for the cross ratios. We introduce flow diagrams in Section 2. Several concrete examples are worked out in Sections 3 and 4.
With the cross ratios at hand we turn to the inductive proof in Section 5. The conjecture for the M -point blocks is given in terms of Mellin-Barnes integrals that convert a Mellin amplitude to a conformal block. We organized our calculation such that However, the conjecture and the proof do apply to the 3 −→ 4 case. In Appendix A we inspect that everything checks out when going from M = 3 to M = 4, as well as going from M = 4 to M = 5. We spelled out the 4 −→ 5 case to display a more generic procedure.
Admittedly, the proof is rather technical with quite a few steps. Therefore, we showcase in detail an example of a 9-point function in Section 6. We display how to choose the cross ratios in that case and how to construct the corresponding conformal block. We also discuss certain discrete symmetries that a block must possess. In the simplest case, any two external operators that are connected together to a vertex can be interchanged leading to a Z 2 symmetry. The block must be invariant under such a Z 2 , which provides a non-trivial consistency check as many cross ratios need to be rearranged under operator exchange. The 9-point example in Section 6 posses a Z 2 × Z 2 × Z 2 symmetry although analogous discrete symmetries can be significantly larger and more complicated for more symmetric diagrams.
Afterwards, we conclude in Section 7. We note that with this paper the dimensional reduction of higher-point blocks discussed in [31], which assumed the Feynman rules conjecture, is now proven.

Feynman-like Rules
This paper uses a method of "flow diagrams" [29] in order to find u-type and v-type cross ratios for a conformal block. Flow diagrams provide a way of assigning appropriate position space coordinates for OPE-like structures of a given diagram. We can always use the OPE to decompose a conformal block into associated 3-point structures, which we call OPElike structures. Depending on the order in which one decomposes the block into OPE-like structures there is a set of possible associate position coordinates that can be assigned to each leg in the 3-point like structure. Flow diagrams are pictorial representations that assign position coordinates that satisfy the OPE structure of the block.
We say that a leg "flows" in a diagram if its position space coordinate is shared in a neighboring OPE-like structure. The collection of OPE-like structures that have the same flowing leg, naturally leads to "comb-like" structures within the flow diagram. The comb-like structures within a flow diagram provide an extremely useful method for figuring out what order to integrate the Mellin-Barnes integral representation of conformal blocks to obtain explicit series representations for conformal blocks in arbitrary topologies.
In this section we first introduce flow diagrams and explain how to flow coordinates along a given topology, effectively decomposing it in its comb-like structures. We stress that this decomposition depends on the choice of flows. Using the flow diagrams prescription, we then define rules to compute all conformal cross ratios. Finally, we combine all of the above with the known Feynman-like rules, introducing an explicit recipe for constructing scalar conformal blocks in arbitrary topologies.

OPE Structures and Cross Ratios for Arbitrary Topologies
Any scalar contribution to a M -point correlation function where L(∆ 1 , . . . , ∆ M ) is the leg factor which ensures that the correlation function transforms covariantly under the conformal group and g(u, 1 − v) is the conformal block for that and v-type cross ratios appearing in the conformal blocks also depend on the corresponding topology and the chosen flow of coordinates, which can be found via flow diagrams presented in this paper. We now introduce the rules to build the leg factor and the cross ratios of any conformal block with arbitrary topology. The construction of the conformal block is described in the next subsection.
For the OPE decomposition of the correlation function leading to the topology of interest, we first define nI OPE vertices as OPE-like structures with n internal (or exchanged) operators. The leg factor L(∆ 1 , . . . , ∆ M ) can then be determined by looking at the 1I and 2I OPE vertices. Specifically, after defining X ab;c as and L a (∆ a ) = X ∆a 2 bc;a (2.4) to the 1I and 2I OPE vertices depicted in Figure 1, respectively. Then, the full leg factor L(∆ 1 , . . . , ∆ M ) can be built by multiplying all L a (∆ a ) for 1 ≤ a ≤ M together. We note that in Figure 1, external operators are indexed by a or b while internal (or exchanged) operators are indexed by k β or k γ .
To construct the cross ratios, it is important to first define a proper prescription for the flow of the coordinates in the OPE decomposition of the correlation function. This is    Figure 2 and its OPE decomposition shown in Figure 3 (where O σ can be an external or an internal operator), 2 we can build a u-type conformal cross ratio as following the flow of coordinates.
In Figure 2, the coordinates on the operators are fixed (obtained consistently from the flow) if the corresponding operator is external (internal). To build the flow of coordinates, it is necessary to ensure that adjacent OPE-like structures are consistent, as in Figure 3.
Indeed, in Figure 3, the internal operator O kα that pops out of the OPE decomposition appears in each OPE-like structure, and its coordinate is defined by the OPE limit. For Figure 4: The comb structure (n ≥ 4) from which one v-type conformal cross ratio can be built. It should be understood that . Equivalently, the coordinate of the right OPE-like structure is x a 2 because the OPE of the left OPE-like structure is chosen . The coordinate of any internal operator on any OPE-like structure is obtained following this flow prescription. There are obviously more than one choice of flow of coordinates per topology, and the corresponding u-type cross ratio depends on the choices made [the denominator in (2.5) clearly depends on the coordinates that flow], but every flow is an appropriate starting point for the Feynman-like rules described below.
We note here that the symbol ∼ in the OPE is a shortcut notation to indicate which operator (and its tower of descendents) is exchanged.
The v-type conformal cross ratios can be obtained similarly by selecting all pairs of OPE vertices yielding M −2 2 of such cross sections. For every pair of vertices, v cross ratios also involve all the vertices needed to connect the selected pair via internal lines. Therefore, this prescription associates each v with a comb structure inside the M -point conformal block.
The comb structures can have anywhere between 2 and M − 2 vertices that is between 4 and M points. Note that the u-type ratios involved adjacent vertex pairs only.
Specifically, for the comb structure depicted in Figure 4 and the chosen OPE decomposition depicted in Figure 5, we can build a v-type conformal cross ratio v a 1 an = x 2 a 1 an x 2 a n−1 q 2n−6 with q 2 = a 2 and q 2n−7 = a n−1 . We stress that the OPEs and O σn (x an )O σ n−1 (x a n−1 ) ∼ O kα n−3 (x a n−1 ) have been chosen to agree with (2.6). Moreover, the coordinates on the O k i must be chosen consistently following the OPE flow mentioned above. For example, x q 4 is either x a 3 if the OPE is taken as The rules for v-type conformal cross ratios (2.6) can be re-expressed in many different Figure 5: The associated OPE vertices for the comb structure in Figure 4. The coordinates x q i must be chosen in a way that is consistent with the OPE, or in other words, the flow of coordinates. Without loss of generality, we assume q 2 = a 2 and q 2n−7 = a n−1 . In other words, we take the forms. To see this, we first assume that x a 2 flows until the vertex containing O σr 1 where x σr 1 starts flowing. Concretely, this means that the coordinate of the right operator of all OPE-like structures up to and including the OPE-like structure with the top operator at Then, x σr 1 flows up to the vertex containing O σr 2 where x σr 2 starts flowing.
We keep going on until the last vertex is reached. As a result, we find that where the odd-subscript q's are not defined in terms of a's by the flow as described above.
We note however that their explicit coordinates are not needed since they always cancel in (2.6) (for every numerator containing an odd-subscript q, there is an equal denominator with the same odd-subscript q).
To proceed further, we define the boundary vertices as vertices at which the flowing coordinates change. We useT a 1 an abc to denote the boundary vertex where x a stops flowing while x b starts flowing and associate a factor w a 1 an abc = toT a 1 an abc . InT a 1 an abc , we use the upper indices a 1 a n to represent the fact that the boundary vertices are defined by focusing on the direction of the flow of coordinates from a 1 to a n in the comb structure, and the values of a, b and c depend on this choice. In the specific case described here, the flow starts from the left OPE-like structure in Figure 5 and goes toward the right vertex. After substituting (2.7) into (2.6), it is easy to check that (2.6) can be rewritten as v a 1 an = x 2 a 1 an x 2 where the product has been taken over all of boundary vertices. Similarly, focusing on the direction from a n to a 1 , (2.6) can also be rewritten as v a 1 an = x 2 a 1 an x 2 a n−1 q 2n−6 (2.10) Therefore, once the flow of coordinates for a given OPE decomposition of the correlation function in a corresponding topology has been chosen, it is straightforward to construct the u-type and v-type cross ratios. The cross ratios can then be used as a starting point for the Feynman-like rules of the conformal block. . Particularly, for the cross ratio in (2.6), the corresponding element in V will be the index pairing (a 1 a n ). This element corresponds to the indices of position coordinates of external operator insertions at the opposite extremes of the comb structure in Figure 4 which also feature in the numerator of the v-type cross-ratio (2.6) as x 2 a 1 an , in other words the coordinates that do not flow.

Feynman-like Rules for Conformal Blocks
In this section we will briefly state the Feynman-like rules for conformal blocks [28]. The complete set of rules are as follows: • For a M -point correlation function, choose an OPE decomposition (in other words, a topology), and assign a consistent flow of coordinates between every OPE-like structure.
• From the OPE flow, determine the cross ratios with the help of (2.5) and (2.6), • Assign each internal edge with an index i running from 1 to M − 3. Associate to each such edge an integer parameter m i , which we refer to as the "single-trace parameter," and a factor of where h := d/2 and ℓ k i is an integer parameter, which we call the "post-Mellin parameter," associated with the conformal dimension ∆ k i .
• Label each (cubic) vertex with an index i running from 1 to M − 2. Assuming each leg of the cubic vertex has incident conformal dimensions, ∆ a , ∆ b , and ∆ c , write the factor associated to this vertex as A is the Lauricella function given by [43][44][45] (see also ref. [40]) Here m a , m b , and m c (ℓ a , ℓ b , and ℓ c ) are the respective single-trace parameters (post-Mellin parameters) associated with each edge as mentioned in the previous point, and for conformal dimensions ∆ i we use the notation while for single-trace parameters and post-Mellin parameters we use Note that the single-trace parameter associated to an external leg is identically zero.
Thus depending on the type of vertex (1I, 2I or 3I; see the discussion around Figure 1) the vertex factor will include a Lauricella function of 1, 2, or 3 variables.
• Then the full conformal block is given by (2.1) where the leg factor is given by the discussion around equations (2.2)-(2.4) with the conformally invariant function of cross-ratios g(u, 1 − v) constructed by multiplying together all edge-and vertex-factors, including appropriate powers of the cross-ratios, and summing over all integer parameters, as follows: where V is the index set associated to the v-type cross ratios. The post-Mellin parameters ℓ i and ℓ k 1 are linear combinations of the dummy variables j rs , as we describe next.

Determining the post-Mellin Parameters
The Feynman-like rules collected above leave the edge-and vertex-factors in terms of post-Mellin parameters ℓ i , one associated to each conformal dimension. We will now complete the Feynman-like prescription by specifying how these parameters are to be calculated.
For an external operator with conformal dimension ∆ i inserted at position x i ,the associated post-Mellin parameter is defined to be edge (which we refer to as 1I vertices). In this case one simply adds post-Mellin parameters of the external edges together and subtract all even multiples of j rs variables appearing in the sum. That is, for the following 1I vertex, where the symbol 2J = means equality holds once all even multiples of j rs variables have been dropped. If there are any post-Mellin parameters that have still yet to be computed, solve for them iteratively using the procedure above by adding together the known post-Mellin parameters of two legs of a cubic vertex. That is, if two post-Mellin parameters are known at a shared vertex, the unknown post-Mellin parameter is computed as (2.20) Repeat this process until all post-Mellin parameters have been determined.

Examples of Cross Ratios
We first show how to use (2.5) for an arbitrary four-point structure and (2.6) for a specific comb-like structure inside a nine-point topology.

u-type Cross Ratios
In this section we will demonstrate how to use (2.5) to solve for the u-type cross ratios given a flow diagram. The construction of these ratios is sketched in four steps.
1. Identify the pairs of neighboring OPE structures in a flow diagram. The pairs should result in two OPE-like structures that have one O k i between them 3 so that any given structure should be of the form below.
2. Next, identify which legs do not flow in the pair of OPE-like structures. We say a leg does not flow if we do not assign those factors in the OPE flow diagram for that structure. In this paper we circled the legs that do not flow in the u-type diagrams.
Also identify the lower script labeling of O k i . We simply label the u-type cross ratios with the subscript corresponding to the O k i in the diagram that is denote them as u i .
3. Once the legs which do not flow are identified, in this case x a and x d , we draw a dashed green line to the leg that flows from a vertex to its neighbor, in this case x b and x c respectively. The two x p and x q factors that are connected by the green dotted lines in a single vertex become subscripts on the numerator of u i as an x 2 pq factor.
4. Finally, the denominator of the expression for u i is obtained from the numerator by exchanging the labels of the vertices that do flow. Namely, For sufficiently complicated blocks, this prescription might occasionally lead to some confusion. A useful checkup on the expressions for the u-type cross ratios can be obtained by drawing a dashed red line to the leg that flows into a vertex to its neighbor, in this case x c and x b respectively. The two x p and x q factors that are connected by the red dotted lines in a single vertex become subscripts on the denominator of u i as an x 2 pq factor yielding again (3.6)

v-type Cross Ratios
In this section we will demonstrate how to use (2.6) to construct v-type cross ratios given an appropriate choice of flow diagram.
1. To solve for the v-type cross ratios, we need to find the comb-like structure that results for any vertex pair, V i V j . The subscripts of the V i V j pair correspond to the labels of the two OPE vertices that are the end vertices of the corresponding comb-like structure.
This will result in M −2 2 v rs cross ratios. For example if we consider the following flow diagram, x 4 x 9 x 9 V 3 x 4 x 5 x 2 V 4 x 6 x 9 x 2 V 5 x 7 x 9 x 7 V 6 x 8 x 9 Figure 6: A possible flow diagram for a 9-point asymmetrical block. then we will have V i V j pairs found in the upper right triangle of the matrix 4 below 4 Note that this matrix corresponds to M −2 2 in any given diagram.
2. Now that we have identified all V i V j pairs, draw the associated comb-like structure between these vertices. For reference, for the rest of this example we will consider the x 4 x 9 (3.8) 3. Next, in each structure associated to the vertex pair, identify which legs do not flow.
We say a leg does not flow if we do not assign those factors in the OPE flow diagram for that structure. The two legs that do not flow will become the subscripts for a x 2 rs term in the numerator of the v-type cross ratio. Moreover, we now circle the two x i x j pairs that are between vertices. The x i and x j subscripts from within the blue circle will become x 2 ij factors in the numerator.
In the case of the V 3 V 7 comb structure x 1 and x 5 do not flow. Thus, the numerator contains x 2 15 and the x i x j pairs circled in blue, x 23 , x 24 , x 49 .
4. Once we have identified which x r and x s legs do not flow, we draw a dashed orange line to show the shortest path from x r to x s assuming it must go through the other points labeled in the diagram, a geodesic between x r to x s . The two x i and x j factors that are connected by the orange dotted lines in a single vertex become subscripts on an x 2 ij factor in the denominator.
Using our example, x 1 and x 5 do not flow, thus the denominator contains the pairs of the geodesic between x 1 and x 5 , which are x 2 13 , x 2 24 , x 2 24 , x 2 95 .
Put the method all together to get the full v-type cross ratio. 5 We now turn to the construction of the u-type and v-type cross ratios for 4-point and 5-point conformal blocks.

Low-point examples
In this section we demonstrate how to find the cross ratios for a conformal block using the 4-point and 5-point blocks respectively. This section directly corresponds to Section 2.1 and is used to complement the equations there through illustrative examples. For a non-trivial example of how to find u-type and v-type cross ratios in a higher-point block see Section 6.

4-point Block
Consider a 4-point block with the following labeling (4.1)

Flow Diagrams
In order to create an associated flow diagram for (4.1), it is convenient to expand out the conformal block into a set of 3-point OPE-like structures. In the case of (4.1) we see the following 3-point structures, In (4.2) we assigned a label of V 1 , and V 2 in order to keep track of individual vertices in the diagram. To complete a flow diagram we have to assign position coordinates which satisfy the OPE structure of the block. For the 4-point conformal block, we see four possible flow diagrams Flow diagrams are not unique. Each flow diagram generates a corresponding unique set of cross ratios. For example, using (2.5) and (2.10) we can recreate all u-type and v-type cross ratios that are possible for a 4-point block with our chosen labeling.
In the case of the 4-point block there is simply only one u-type cross ratio per any flow diagram because there are only two vertices.
x 3 x 4 : x 3 x 4 : x 3 x 4 : x 3 x 4 : Next, we demonstrate how to use flow diagrams to solve the v-type cross ratios. In the below diagrams we show the resulting v-type cross diagrams and their cross ratios based on every possible choice of 4-point flow diagrams.

5-point Block
Consider a 5-point block with the following labeling

Flow Diagrams
In order to create an associated flow diagram for Figure 7, it is convenient to expand out the conformal block into a set of 3-point OPE-like structures. In the case of Figure 7 we see the 3-point structures of Figure 8.

u-type and v-type Cross Ratios
To finish solving for the u-type and v-type cross ratios we choose to use the flow diagram in the top left of (4.6). In this paper, the inductive proof in Section 5 makes the choice to assume the O 1 (x 1 ) leg never flows. To enforce the choice that O 1 (x 1 ) never flows, we restricted our labeling to require the O 1 (x 1 ) leg to always connect to the O 2 (x 2 ) leg. 8 Solving for the u-type cross ratios x 4 x 5 : x 4 x 5 : and solving for the v-type cross ratios, It is worth re-noting that since x 2 ij is symmetric that we tend to use the convention of writing i < j. In this section we outlined the methodology for obtaining u-type and v-type cross ratios for any conformal block using two examples, the 4-point and 5-point blocks.

Proof of Feynman Rules
In this section we provide the full proof of the Feynman rules based on the OPE flow and the associated conformal cross ratios discussed above. The proof proceeds by induction.
Consider the M -point bulk diagram in AdS d+1 shown in Figure 9a. It is a canonical AdS diagram, which is constructed only with the help of cubic vertices, and its topology is the same as the topology of the associated conformal block we are interested in. The only explicit cubic vertices, which correspond to OPEs in the CFT language, shown in Figure 9a are denoted by green dots. All the remaining cubic vertices are included inside the gray blobs. space as, In writing (5.2), we have defined the "diagonal" Mellin variables γ aa := −∆ a . 9 Together, 9 Mellin variables are symmetric, complex variables so that the constraints (5.2) can be solved in terms of auxiliary momentum variables p a , such that γ ab = p a · p b , along with momentum conservation M a=1 p a = 0, the index sets constitute the off-diagonal upper-triangular matrix indices The decomposition into dependent and independent sets of variables is not unique. For the set V, we will be employing the choice presented in Section 2.1.
In our convention, the normalization constant in (5.1) is given by , where ∆ a are the external conformal dimensions, ∆ ka are the internal exchanged dimensions and we have defined Then, according to the Feynman rules for Mellin amplitudes of tree-level AdS diagrams [39][40][41], the Mellin amplitude M M in (5.1) is given by The edge factors take the form where s a is the Mandelstam invariant associated to the internal leg ∆ ka , and m a is an integral parameter associated to the same leg to be summed over as shown in (5.7). All vertices of the canonical M -point AdS diagram are cubic vertices; the associated vertex factors are where p a · p a = −∆ a on-shell. We can then define Mandelstam variables which we will use in the proof later.
A is the Lauricella function of type A of three variables, 10 given by (2.13).

Setting Up Induction
To proceed with the induction, we first write: where in the second equality we defined B M −1 as As will become clear later in the proof, the choice of s 1 in (5.14) allows us to set which immediately yields In (5.11) and (5.12), s a can be determined as follows: We cut the ∆ ka -internal line inside the M -point diagram, leading to two disconnected parts. One part contains the external operator O 1 (x 1 ) while the other part does not. We denote the external operators in the , and we define the corresponding set J ka ∋ b.
As shown in Figure 9a, it is clear that J ka is a subset of {3, 4, . . . , M }. Indeed our choice in cutting the ∆ ka -internal line is such that one part contains the external operator , while the remaining internal operators from the other part are contained in J ka . As a result, s a can be computed from 12 To get the corresponding conformal blocks, we want to evaluate B M at the "single-trace poles," by evaluating the residue at the simple poles at which the Mandelstam invariant goes "on-shell." Let us call that quantity B s.t. M . We will obtain B s.t. M inductively starting with .

(5.21)
The Feynman-like rules for conformal blocks given in Section 2.2 are equivalent to the 12 The Mandelstam variable s a can also be obtained from the part which contains O 1 (x 1 ). Let us denote the external operators in the part with following expression for B s.t.
M : whereV a andÊ a are "Gamma-vertex" and "Gamma-edge" factors for the M -point conformal blocks. Per the rules,V a andÊ a are given by what remains is precisely the Gamma-edge factor in (5.23). We will return to what happens to this excess factor shortly. The explanation of the Gamma-vertex factor is slightly more involved. First, note that to obtain the theory-independent conformal block, after taking the single-trace projection of (5.10) one must divide out by theory-specific OPE coefficients. The  Consequently, where the leg factor is given by It is straightforward to check that this is consistent with

Proof by induction
We will now prove ( We refer the reader to Appendix A where these steps are carried out for the first two non-trivial cases, corresponding to M = 4 and M = 5, for illustrative purposes. In the following, we will proceed in full generality for an arbitrary M in the arbitrary topology associated with the canonical diagram shown in Figure 9a.

Diagram
Our first non-trivial step is to relate (5.21) with the corresponding (M − 1)-point quantity.
Since the Mellin parameters γ 1a should not appear in the (M − 1)-point AdS diagram, they must be properly absorbed into other quantities. Specifically, to absorb γ 1a appearing in Thus s a can be re-expressed as 14 where we define Moreover, we also define As a result, all γ 1a in (5.12) can be absorbed into ∆ ′ a and ∆ ′ ka , leading to .
(5.32) 14 Since J c ka contains O 1 (x 1 ) and the latter does not appear in the (M −1)-point AdS diagram, we compute s a from J ka .
For the inductive step, we assume that B s.t. M −1 satisfies the Feynman-like rules (5.22), i.e. Schematically, the Gamma-vertex and Gamma-edge factors take the form where ℓ ′ a are the post-Mellin parameters of the (M − 1)-point topology, and σ 1 , σ 2 , σ 3 label the operators incident on the internal vertex. It is worth noting that ℓ ′ a do not contain any j 1b which appear in the post-Mellin parameters ℓ a for the M -point block. Moreover, it can be proved that theV ′ a (Ê ′ a ), which are not inside the initial comb structure, 15 can be obtained from the corresponding M -point quantitiesV a (Ê a ) through replacing all j 1b inV a (Ê a ) by −λ b . To prove this, we first note that the following relations hold where the 2J -equality is defined above (2.19), leads to when 3 is not in J ka . Since all j 1b on the right-hand side of the above equality only appear once, we conclude that when 3 is not in J ka , then ℓ ka can be written as It is worth noting that when 3 is not in J sa , thenĵ sa can be built from λ sa through replacing all λ b in λ sa by j 1b . With the help of (5.35) and (5.38), theV i andÊ i , which are not inside 15 The definition of the initial comb structure is discussed in more detail in Section 5.2.6.
the initial comb structure, are given bŷ .

(5.41)
Sinceĵ σa are the same as λ σa , except with all λ b in λ σa replaced by j 1b , we can conclude that theV ′ a (Ê ′ a ) which are not inside the initial comb structure can be obtained from the corresponding M -point quantitiesV a (Ê a ) through replacing all j 1b inV a (Ê a ) by −λ b .

Turning to the M -point Block: Projecting Out the Single-trace Contribution
Substituting (5.33) into (5.20), we find that B s.t.
M can be written as .

(5.42)
Evaluating the residue at the final single-trace pole ∆ k 1 + 2m 1 = s 1 = ∆ 1 + ∆ 2 − 2γ 12 gets essentially all instances of γ 12 replaced with Let us also perform the integral over γ 13 , the delta function forces all instances of γ 13 to become where (5.43) has been used to replace γ 12 . We define the substitution S by As a consequence, B s.t. M can be written as  Figure 11: The new 1I OPE vertex.

Recovering the M -point Cross Ratios
To proceed further, we first prove that H s.t. M can be rewritten as   Figure 12: The diagram that defines the cross ratios in (5.54) (left) and the OPE vertex (right). Here the arrows represent the fact that x a 1 flows all the way to the last vertex on the rightmost side.
where (5.29) has been used and X ab,c is defined in (2.2). Substituting (5.31), (5.50), and (5.51) into (5.47), we find that proving (5.48) is equivalent to proving the following identity where we defined K ′ as (5.53) To compute K ′ , we note that terms in K ′ can be classified according to their powers. Specifically, in K ′ there are three types of terms with powers γ 12 , γ 13 , and γ 1a for 4 ≤ a ≤ M , respectively. To prove (5.52), we need to compute each type of terms. Before computing these terms, we introduce the following identity  L ′ (−γ 12 , . . . , −γ 1M ) has contribution where we assume that the vertex involving O ′ 3 (x 3 ) is as depicted in Figure 13. Moreover, due to the fact that λ ka = b∈J ka γ 1b with J ka ⊆ {3, . . . , M }, those u ′ cross ratios related to internal lines which directly connect O ′ 3 (x 3 ) and O ′ 2 (x 2 ) have powers γ 13 (see Figure 14). In other words, the following terms with powers γ 13 can be extracted where (5.54) has been used. Together with (x 2 12 ) −γ 12 (x 2 13 ) −γ 13 , we find that terms with powers γ 12 are given by The last group we need to compute includes terms with powers γ 1a for 4 ≤ a ≤ M . To compute those terms, we assume that the external operator O ′ a (x a ) is inside the circle labeled at x q 1 . 16 After expanding the circle labeled by 3 ′ as depicted in Figure 15, we find that terms with powers γ 1a include: 1. The leg factor L ′ a (−γ 1a ) given by  Figure   15, given by 3. And the spacetime coordinates (x 2 1a ) −γ 1a .
Combining the three contributions, we find that the terms with powers γ 1a are given by We note that X a 1 q 1 ,a p+q+1 i=1 u ′ α i can be computed through repeated use of (5.54). Specifically, we assume that the boundary vertices in Figure 15 are given byT ′a2 ar i ar i+1 p i+1 [as described around (2.8)] for 0 ≤ i ≤ σ a − 1 with the understanding that a r 0 ≡ a and a rσ a ≡ 3. In other words, x a flows until the vertex involving circle a ′ r 1 at which point x ar 1 starts flowing. Then x ar 1 flows until the vertex involving the circle a ′ r 2 at which point x ar 2 starts flowing. By continuing this procedure until the last boundary vertex which contains the circle 3 ′ is 16 The following computations can be easily generalized to the case when O ′ a (x a ) is inside the circle labeled by M ′ . Figure 16: The comb structure which can be used to construct the cross ratio v 1j following our rules.
reached, we find that where w ab abc was defined in (2.8) and (5.54) has been used. Thus the terms with powers γ 1a for 4 ≤ a ≤ M are given by Multiplying (5.57), (5.58), and (5.61) together leads to the final expression for K ′ , given by After performing the substitution S (5.45), we find that We note that ar i ar i+1 p i+1 is exactly v 1a which can be built from our rules by looking at the comb structure in Figure 16 inside the M -point conformal block. As a consequence, we find that which completes the proof of (5.52) and thus leads to a proof of (5.48).

Applying the First Barnes Lemma
To prove the Feynman-like rules, our next step is to compute I. As we will see later, all λ a integrals can be evaluated through repeated use of the first Barnes lemma leaving us with trivial-to-evaluateλ a contour integrals.
Before doing real computations, we note that any (M − 1)-point topology can be reached by gluing a set of comb-like structures. 17 Specifically, starting with the OPE vertex con- , we can glue 2I and 3I OPE vertices in the proper order until we reach another OPE vertex containing O ′ 2 (x 2 ), where this procedure stops. This procedure produces a comb-like topology, dubbed the "initial comb-like structure", which connects O ′ 3 (x 3 ) and O ′ 2 (x 2 ), but some of the teeth of this comb correspond to internal lines that need to be glued further.
For 2I OPE vertices included in this initial comb-like structure, there is nothing further to do since the corresponding tooth represents an external operator. However, this is not the case for 3I OPE vertices. From this initial comb-like topology, we select one of the 3I OPE vertices and repeat the procedure above by gluing 2I and 3I OPE vertices in the correct order until we reach another 1I OPE vertex. To obtain the desired (M − 1)-point conformal block, we continue this procedure with each additional comb-like structure until all the 3I OPE vertices have been completely glued, i.e. until the number of 3I OPE vertices added in the corresponding comb-like structure is zero. We dub the comb-like structure, which is not the initial comb-like structure and does not contain 3I OPE vertices, the "final comb-like structures". We stress that even when the initial comb-like structure does not contain any 3I OPE vertex, i.e. the conformal block is in the comb topology, it is not a final comb-like structure by our definition. Now, we are ready to compute I. Let us start with one final comb-like structure, depicted in Figure 17. Since we want to evaluate the λ a integral, we also depict the λ a appearing in this final comb-like structure in Figure 18. There are n Gamma-verticesV ′ a and n − 1 Figure 17: The final comb-like structure. Figure 18: The λ j dependence that comes from ∆ ′ ia = ∆ ia − λ j as well as ∆ ′ kα a = ∆ kα a − λ kα a in the final comb-like structure Figure 17.
Gamma-edgesÊ ′ a associated with this final comb-like structure. Without loss of generality, we label the Gamma-vertices from the leftmost side to the rightmost side in Figure 17 bŷ With these conventions, using (5.34), we find that , where we used the relations (5.71) We note that since O ′ i 1 and O ′ i 2 dwell at the same 1I OPE vertex, λ ka for 1 ≤ a ≤ M − 3 diagram will be exclusively considered in the computations for I. either do not contain λ i 1 and λ i 2 , or contain λ kα 1 = λ i 1 + λ i 2 . Thus, after changing the λ i 2 integral to the λ kα 1 integral by using the terms in I which contain λ i 1 are given by With the help of the first Barnes lemma we can evaluate the integral over λ i 1 , leading to , Substituting the above result into I and noting that Γ(∆ kα 1 − λ kα 1 + 2m α 1 + ℓ ′ kα 1 ) in the numerator of (5.75) cancelÊ ′ α 1 , we find that I becomes
can also defineṼ i 1 andẼ α 1 byṼ For future use, we generalize the above definition and defineṼ a for 1 ≤ a ≤ M − 2 andẼ a As a consequence, I can be written as Comparing (5.80) with (5.68), we find that the integrand in (5.80) corresponds to the initial integrand but with the leftmost vertex in Figure 18 removed, as shown in Figure 19. Now, we can change the λ i 3 integral to λ kα 2 integral by noting that and then evaluate the λ kα 1 integral. Again, the λ kα 1 integral just replace all −λ kα 1 and −λ i 3 Figure 19: The final comb-like structure after evaluating the λ i 1 integral.
Thus evaluating the integral over λ σ 1 ≡ λ M leads to where (5.83) has been used to get M a=4λ a . Using (5.35) and noting that we find that ℓ 12,k 1 = 0,

Recovering the M -point Edge-and Vertex-factors
Finally, we evaluate the remainingλ a integrals via the Cauchy residue theorem. We close all λ a contours to the right to be able to drop the contribution from the arc at infinity, picking the lone semi-infinite sequence of poles starting at the origin, atλ a = j 1a for j 1a ∈ Z

Nine-point Scalar Block and its Symmetries
In this section we put to use our Feynman-like rules in a concrete, previously unknown higher-point example: the nine-point conformal block for arbitrary external and internal scalars in an asymmetric topology shown in Figure 21.

u-type Cross Ratios
To illustrate how to use the rules for generating the cross ratios as described in Section 2.1 consider the 9-point asymmetrical conformal block with the topology and labeling shown in Figure 21. In the case of solving for both the u-type cross ratios and the v-type cross ratios, our method of generating the cross ratios requires making a flow diagram as laid out in Section 2.1. Flow diagrams are not unique, and any diagram that meets the conditions above is valid.
x 4 x 9 x 9 V 3 x 4 x 5 x 2 V 4 x 6 x 9 x 2 V 5 x 7 x 9 x 7 V 6 x 8 x 9 Figure 22: Flow diagram using rules in Section 2 for the 9-point asymmetrical block.
The first step is choosing an OPE flow. For this conformal block we use the flow diagram as shown in Figure 22 where we labeled our vertices V i and show the flow by depicting which coordinates continue moving throughout the block. We say a coordinate does not flow when it does not have another comb structure to connect to that contains the same coordinate (for example x 1 never flows in this diagram). According to Figure 2 in order to write down the u-type cross ratios, we first need to identify all of the four-point structures within our diagram. The figures below outline the corresponding four-point structures for Figure 21 and the respective u-type cross ratios that arise from each structure [see Section 2.1 for the general discussion and Sections 3 and 4 for a detailed explanation of the diagrammatic notations that follow]: 19 x 9 x 4 x 9 V 3 x 4 x 5 : x 4 x 9 x 2 V 4 x 6 x 9 : x 6 x 9 x 2 V 5 x 7 x 9 : x 7 x 9 x 7 V 6 x 8 x 9 : x 2 29 x 2 87 (6.3)

v-type Cross Ratios
Using our flow diagram Figure 22 we can also write down the v-type cross ratios. To do that we first find the comb-like structures for any V i V j pair. To illustrate the rules explained around Equation ( x j 's connecting the flow diagram. In this case it is only x 2 and x 3 . Finally, we draw geodesics between the non-flowing legs and assign these pairwise to the denominator, in this case x 3 13 and x 2 24 , 2. V 7 V 2 : As is clear from the following comb structure, x 1 and x 9 do not flow. Thus the numerator contains x 2 19 and the x i x j pairs circled in blue, x 2 23 and x 2 24 , while the denominator contains the pairs of the geodesic between x 1 and x 9 , which are x 2 13 , x 2 24 and x 2 29 . Note that for this block we see a cancellation of the x 2 24 factor in the final cross ratio. , while the denominator contains the pairs of the geodesic between x 1 and x 5 , which are x 2 13 , x 2 24 , x 2 24 and x 2 59 . We also see a cancellation of the x 2 24 factor, though the numerator and denominator of the final simplified cross ratio still contains three x 2 ij factors each.

2J
= ℓ k 5 + ℓ 6 = j 16 + j 17 + j 18 + j 19 + j 28 + j 36 + j 37 + j 38 + j 39 + j 46 + j 47 + j 48 + j 56 + j 57 + j 58 Having computed the post-Mellin parameters ℓ i and ℓ k i , we can use the Feynman rules As an example of an edge factor, we write down the edge factor associated to the exchange of the operator O k 5 of dimension ∆ k 5 :

Discussion
In this paper we provided an inductive proof of the Feynman-like rules for scalar conformal blocks with scalar exchanges found in [28] using the OPE flow diagrams developed in [29].
These two results turned out to be complementary-their union was necessary to obtain a complete proof. On the one hand, the Feynman-like rules led to an easy recipe for writing down conformal blocks, without however providing an efficient technique to generate the appropriate cross ratios. On the other hand, the OPE flow diagrams by themselves did not directly lead to rules for conformal blocks but gave a straightforward and intuitive method for generating conformal cross ratios. For completeness, we also showed the four-to five-point induction and we presented an explicit example for an asymmetric nine-point topology. In the latter case, we also gave the identities the conformal blocks must satisfy from the symmetries of the associated topology, which we verified to low order in a power series expansion of the blocks. Further research is needed to generalize this prescription to higher-point blocks with spinning operators, either exchanged or external. Another avenue would be to explore the analytic structure of the higher-point conformal blocks. This could be done with the help of new cross ratios reminiscent of the Dolan-Osborn cross ratios that greatly simplify four-point conformal blocks in even spacetime dimensions [11]. From the point of view of AdS, it would be of interest to generalize the induction of the single trace part of Witten diagrams presented here to the full Witten diagrams. We hope to return to these problems in the near future.

A First Few Cases of Induction
In this section we will instantiate the proof of the Feynman-like rules presented in Section 5 by working out the first two non-trivial cases. The methods provided here may seem overkill for these instances. The strategies presented here, on the other hand, are critical in proving the general M -point case in Section 5.

A.1 Three-point to Four-point
We consider the four-point scalar exchange Witten diagram shown in Figure 23. It is wellknown that the conformal block decomposition of this diagram contains a single-trace component, as well as contributions coming from the exchange of double-trace operators. To obtain the four-point conformal block, we need only isolate the single-trace contribution to the diagram [16,19]. We will achieve this single-trace projection in Mellin space following the method utilized in Ref. [28].
The position space amplitude for the exchange diagram admits the following Mellin representation, where the normalization constant is and the Mellin amplitude M 4 is given by where the "edge-" and "vertex-factors" take the form with the Mandelstam invariant s 1 = ∆ 1 + ∆ 2 − 2γ 12 . 20 The functions F A appearing above 20 Here γ ab (= γ ba ) are the Mellin variables satisfying so that Furthermore, we define satisfy p a · p b = γ ab for all a, b with momentum conservation  so that We can easily evaluate B 3 since the three delta functions eat up the three contour integrals, enforcing the following constraints (A.14) Consequently, where we have defined the leg factors To proceed, we substitute (A.15) into (A.13) and then calculate its single-trace projection 21 While precise knowledge of the individual primed sets is not important in this subsection, to make connection with the general inductive proof to follow in Section 5, we note that U ′ = ∅, V ′ = ∅, and .

(A.18)
Then the conformal block is obtained as where we have projected out the (known) theory-dependent mean field theory OPE coeffi- to extract the theory-independent conformal block.

(A.27)
It is easily verified that the expression above reproduces the Feynman-like rule prescription given in (5.22) for M = 4.