Towards Feynman rules for conformal blocks

We conjecture a simple set of"Feynman rules"for constructing $n$-point global conformal blocks in any channel in $d$ spacetime dimensions, for external and exchanged scalar operators for arbitrary $n$ and $d$. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the $n$-point comb channel blocks. We prove these rules for all previously known cases, as well as for a seven-point block in a new topology and the even-point blocks in the"OPE channel."The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block.


Introduction
Conformal blocks are theory-independent building blocks of conformal field theories (CFTs) which capture contributions to conformal correlators from entire conformal families of representations appearing in the intermediate channels of correlation functions. Via the AdS/CFT correspondence, they play an important role in the gravitational context as well; for example they provide a basis for writing down any bulk Witten diagram.
Conformal blocks also play a crucial, central role in the revived conformal bootstrap program [1][2][3] (see also the recent review [4] and references therein), which has led to significant advances in understanding properties of d-dimensional CFTs as well as holography. This has resulted in considerable interest in and a spate of new results for conformal blocks. channel, referred to as the "snowflake channel" [61]. 1 While the recent burst of activity and progress in studying higher-point functions and conformal blocks is encouraging, the situation is far from settled. A particularly troubling aspect of going to higher-point blocks is that the number of possible inequivalent channels grows very rapidly with n, thus it seems highly inefficient and impractical to work out the associated conformal blocks on a case by case basis. What would be desirable is a set of Feynman-like rules which could be determined once and for all, that enable writing down any conformal block in any topology without having to do any computations.
Motivated by these considerations, in this paper we will present a simple, conjectural prescription for writing down an arbitrary d-dimensional n-point scalar conformal block with scalar exchanges in any given channel. Even though the blocks themselves are nonperturbative objects, we call them "Feynman rules" because they are reminiscent of Feynman rules for Mellin amplitudes [68][69][70]. This conjecture was motivated by carefully studying the power-series expansions of all known examples of scalar conformal blocks in the literature, particularly as presented in refs. [53,55,57,58,60,61].
As a highly non-trivial check of these rules, we compare the predicted blocks belonging to an infinite family of blocks previously unknown in the literature against a first-principles derivation and find exact agreement. These are the n-point conformal blocks in the so-called "OPE channel" for arbitrary even n. We also test the rules in the case of a seven-point block in a topology different from the comb channel, which we simply refer to as the "mixed channel," and find perfect agreement.
The key idea which enables us to compute these new families of blocks from first principles was previously utilized in refs. [55,57,58] to obtain the holographic duals of higher-point blocks. To obtain a particular scalar conformal block, we start with a tree-level Witten diagram in a cubic ϕ 3 effective field theory whose direct channel conformal block decomposition admits the desired block as its single-trace contribution. We call such a Witten diagram the "canonical Witten diagram" for the block, and there is a unique choice for each conformal block. Then the single-trace contribution to the canonical Witten diagram is given by the desired block times a set of known mean field theory OPE coefficients. Thus the key step is to obtain the single-trace projection of the Witten diagram, as this will immediately yield the conformal block.
Firstly, in a large N bulk theory, Mellin amplitudes are meromorphic functions with poles corresponding precisely to the exchange of single-trace operators; this provides a convenient route to single-trace projections. Secondly, Mellin amplitudes for all tree-level scalar Witten diagrams in scalar effective field theories are known (thanks to the Mellin space Feynman rules [68][69][70]); this enables us to obtain an explicit single-trace projection of any n-point canonical Witten diagram. This method of projecting out the multi-trace exchanges to obtain the conformal block is quite general, efficient and constructive, so it can be used to work out any particular conformal block. The main, and often only, computationally challenging step of this procedure will be the actual evaluation of all residual Mellin integrals, which is required to obtain an explicit power-series expansion for the block. However, in all examples we attempted we were able to systematically work out all such integrals merely by repeated, and often inductive, applications of the first Barnes lemma [74].
The outline for the rest of the paper is as follows: In section 2 we propose the Feynman rules for conformal blocks, and in section 3 we illustrate how to apply them to obtain a seven-point block in the "mixed channel," the n-point comb channel block, and the n-point

OPE channel block. In section 4 we revisit all examples from section 3 and using the
Mellin-space single-trace projection technique, we prove the Feynman rules in each case. We end with some discussion and future directions in section 5. Various technical details and computations are provided in the appendices.
When this work was largely complete, we learned of parallel, independent work to appear by Fortin, Ma and Skiba [75], which has partial overlap with some results of this paper.

Feynman rules for conformal blocks
Given any n-point conformal block, let the dimensions and insertion coordinates of the external operators be respectively, ∆ i and x i for i = 1, . . . , n. Let the dimensions of the exchanged operators be enumerated ∆ δ i for i = 1, . . . , n − 3. See figure 1 for some examples of graphical representation of blocks in different channels as unrooted binary trees with n leaves (and correspondingly n − 2 internal vertices and n − 3 internal edges), which will play a central role in the Feynman rules. Different inequivalent channels/topologies correspond to different OPE structures which can contribute to a conformal correlation function.
The Feynman rules presented here give an expression for the desired conformal block in the desired channel as an n(n − 3)/2-fold power series in powers of n(n − 3)/2 independent (c) Figure 1: Graphical representation of conformal blocks: Any n-point block admits a unique representation as an unrooted binary tree with n leaves, and consequently n − 3 internal edges (colored red to guide the eye) and n − 2 internal nodes/vertices (marked in blue). All edges are labeled with conformal dimensions; the labels on external edges (edges attached to the leaves of the unrooted tree) are shown at the leaves for better presentation.
(a): The graph shows an n-point "comb channel" conformal block (for n ≥ 4) for external scalar operators O 1 (x 1 ), . . . , O n (x n ) with conformal dimensions ∆ 1 , . . . , ∆ n and insertion coordinates x 1 , . . . , x n respectively, and exchanged scalar operators O δ 1 , . . . , O δ n−3 along the internal edges with conformal dimensions ∆ δ 1 , . . . , ∆ δ n−3 , respectively. (c): The graph shows an n-point block in the "OPE channel," for even n ≥ 6. One can obtain the OPE channel topology by starting with an n 2 -point comb channel block and attaching two external edges at every leaf to get the n-point OPE channel block. (b): The graph shows a 7-point example in a "mixed channel" which is neither the comb nor the OPE channel.
cross-ratios built out of operator insertion positions x i . 2 The set of cross-ratios will be treated as input data fed into the rules to obtain the conformal block. We will assume the independent cross-ratios are u i for i = 1, . . . , n − 3 and v j for j = 1, . . . , n−2 2 , such that in the n − 3 separate OPE limits where u i ≈ 0 for a particular i, the leading u i -dependent contribution to the conformal block is given by There is a choice in picking n(n−3)/2 independent cross-ratios subject to the constraint (2.1), and also correspondingly a choice of the leg factor. The Feynman rules described here work for any such choice. From here on, we fix a choice.
Away from the limit (2.1), the conformal block admits an expansion of the form where W 0 n (x i ), which will be referred to as the "leg factor," depends only on position coordinates x i and external dimensions ∆ i . The function g(u, 1 − v) is expressed as a power series in u i and (1 − v j ) for all i, j, with the leading behaviour g(u, 1 − v) = 1 + O(u i , 1 − v j ). This function sums all descendant contributions to the conformal block. The Feynman rules provide a prescription for writing down this function in terms of "edge factors"E i and "vertex factors" V i associated respectively with each internal edge and internal vertex of the unique unrooted binary tree representation of the desired conformal block (see e.g. figure 1): 3) The position-independent edge and vertex factors depend solely on the external and exchanged conformal dimensions, as well as the non-negative integral parameters being summed over, k i and j rs , where i = 1, . . . , n − 3 and the (rs) index takes n−2 2 values. (For convenience we have also re-enumerated the v j cross-ratios as v rs ; the precise mapping will be explained shortly.) They are determined as follows: • Label each internal edge with an index i running from 1 to n−3, such that the conformal dimension of the exchanged operator running along the edge is twice the exponent of the cross-ratio u i appearing in (2.1). Associate to each such edge an integral parameter k i and a factor of where ∆ δ i is the conformal dimension of the exchanged operator running along the edge, and ℓ δ i is an integral parameter associated with the conformal dimension ∆ δ i to be determined later. Here We refer to the parameters k i as "single-trace parameters," and the parameters ℓ δ i as "post-Mellin parameters." The single-trace parameter k i also appears in the series expansion (2.3) as the exponent of the cross-ratio u i . The post-Mellin parameters are specified entirely in terms of specific positive linear combinations of the parameters j rs appearing in (2.3), which we call "Mellin parameters." The precise relation between the two will be discussed in section 2.1. In that section, we will also present an alternate prescription for assigning the appropriate single-trace parameter to each internal edge.
• Label each internal (i.e. cubic) vertex with an index i running from 1 to n − 2.
A is the Lauricella function of three variables, defined in (A.1). Here k a , k b , and k c are the respective single-trace parameters associated with each internal edge above, and ℓ a , ℓ b , and ℓ c are the post-Mellin parameters associated with ∆ a , ∆ b and ∆ c , respectively. Here and below, we are using the shorthand, for conformal dimensions ∆ i , whereas for single-trace parameters and post-Mellin parameters we are using For a vertex with M = 2 (respectively, M = 1), one (respectively, two) of the incident edges is an external edge. So far, external edges have not been assigned a single-trace parameter. It is convenient to view an external edge as an edge with its single-trace parameter set to zero. Then the vertex factor continues to be given by (2.6), but with the associated single-trace parameter(s) set to zero.
It is worth noting that the Lauricella function F A in (2.6) with say, k a = 0 reduces to the Lauricella function of two variables, Likewise if two of the attached edges are external, with say, k a = k b = 0, then the Lauricella function reduces further to the Lauricella function of one variable, In appendix A we list some identities relating these Lauricella functions to other known functions.
Modulo the relation between Mellin and post-Mellin parameters which will be explained in section 2.1, this concludes the complete set of Feynman rules for writing down an explicit power series expansion of any scalar n-point conformal block with scalar exchanges in any channel.
Readers familiar with series expansions of conformal blocks may feel puzzled by the apparent cross-ratio-basis independence of the series coefficients appearing in the expansion (2.3).
However, the explicit form of the edge and vertex factors does in fact depend on the choice of basis of cross-ratios; this dependence is encoded in the correct pairing between the singletrace parameters and cross-ratios as discussed above, as well as the precise relation between Mellin and post-Mellin parameters, which we discuss next. At the end, as noted in (2.3), one sums over all single-trace and Mellin parameters.

Mellin and post-Mellin parameters
Recall that Mellin variables [71,72] are complex-valued variables γ ij (1 ≤ i, j ≤ n) which are symmetric, γ ij = γ ji with γ ii := −∆ i , satisfying the following n constraints: n j=1 γ ij = 0 (i = 1, . . . , n) . (2.11) This leads to n(n − 3)/2 independent components. These variables play a central role in the context of Mellin amplitudes of n-point bulk diagrams [68][69][70]73], which will be reviewed in section 4 in the proof of the proposed Feynman rules for conformal blocks. The constraints above can be solved in terms of auxiliary momentum variables p i (for i = 1, . . . , n) such that p i · p j := γ ij (thus individual p i are "on-shell," i.e. p i · p i = −∆ i ), when "momentum conservation," n i=1 p i = 0 is imposed. In this auxiliary space, the role of the n(n − 3)/2 independent Mellin variables is played by a choice of n(n − 3)/2 independent Mandelstam invariants s i 1 ...i k defined via (2.12) In section 4, for working out the Feynman rules, we will be interested in the following object which we call the "Mellin product," where x i (for i = 1, . . . , n) are boundary coordinates at which operators of conformal dimension ∆ i are inserted. The reason why such an object appears will be clear in section 4 where we obtain the conformal block Feynman rules starting from the Mellin representation of certain bulk Witten diagrams. This product can be recast in terms of conformal cross-ratios built out of x i coordinates, as we now describe.
For any given choice of independent cross-ratios and a given channel, there is a canonical choice of n(n − 3)/2 independent Mellin variables, which makes the Mellin product expressed in terms of cross-ratios physically intuitive. The sets U channel and V channel , of cardinalities n − 3 and n−2 2 respectively, will be defined shortly.
More precisely, given a particular channel and any choice of conformal cross-ratios {u i , v rs } consistent with (2.1)-(2.2), there exists a choice of independent Mellin variables γ ij (2.14) such that the Mellin product can be re-expressed in terms of a product over powers of the given cross-ratios, 4 where W 0 n is the leg-factor for the given choice of cross-ratios in the particular channel. The set {s i } is the set of (n − 3) independent Mandelstam invariants associated with the (n − 3) internal legs of the binary graph representation of the block. In enumerating the Mandelstam invariants, we labeled the internal edges with an index i = 1, . . . , n−3 such that the Mandelstam invariant for the edge i, given by s i , appears in the exponent of the crossratio u i . Accordingly, we can assign the single-trace parameter associated with this internal edge, appearing in the summand of (2.3), the edge factor (2.4) and the vertex factors (2.6) to be k i .
In the final product in (2.15), the set of Mellin variables appearing in the exponents determines precisely the set {γ ij : (ij) ∈ V channel }. This will be taken to be the definition of V channel . The set U channel is then defined to be the set of pairs of indices such that {γ ij : (ij) ∈ U channel } gives the residual n − 3 independent Mellin variables. It is worth noting that dependence in (2.15) on the Mellin variables from this set is encoded in the Mandelstam invariants s i . We will denote D channel to the set such that {γ ij : (ij) ∈ D channel } produces all dependent Mellin variables. Of course, the union of all these sets gives We define the set of Mellin parameters to be the set of cardinality n−2 2 . Mellin parameters make a direct appearance in the summand of the Feynman prescription for conformal blocks (2.3), where they appear in the exponents of certain cross-ratios, as well as in the edge and vertex factors (2.4) and (2.6) via the post-Mellin parameters ℓ a . To obtain the full conformal block, one sums all Mellin parameters over all integral values from 0 to ∞. We now give the prescription to compute the post-Mellin parameters ℓ a associated with the conformal dimensions ∆ a in terms of the Mellin parameters.
For an external operator with conformal dimension ∆ i = −γ ii inserted at position x i , we define the associated post-Mellin parameter to be If the set {(rs) ∈ V channel : r = i or s = i} is empty, then ℓ i = 0. Note that this definition implies that the sum over all post-Mellin parameters associated to external conformal dimensions evaluates to twice the sum over all Mellin parameters, For exchanged operators of conformal dimensions ∆ δ i , the prescription to compute the post-Mellin parameters proceeds iteratively as follows: 1. First, at all internal vertices of the binary graph with precisely two external edges and one internal edge incident, add the post-Mellin parameters associated with the external dimensions, and then drop all terms which are multiples of two (i.e. terms which are even for all integral values of the Mellin parameters). Assign this non-negative sum to be the post-Mellin parameter of the internal (exchanged) operator.
where the symbol 2J = means equality holds once one drops all terms which are even for all integral values of Mellin parameters. For example, if ℓ 1 = j 12 + j 13 + j 16 and ℓ 2 = j 12 + j 23 + j 24 , then ℓ δ 3

2J
= ℓ 1 + ℓ 2 implies ℓ δ 3 = j 13 + j 16 + j 23 + j 24 . parameters are known at a vertex with three incident exchanged operators, then the third is determined as follows: In the next section, we illustrate how to apply these rules to determine the n-point conformal block in the comb channel, the n-point conformal block in the OPE channel, and the seven-point block in the mixed channel (all depicted in figure 1). In section 4, we will reproduce these blocks from first principles which serves as a highly non-trivial check of the Feynman rules.

Examples
In this section, we illustrate how to apply the Feynman rules to three classes of examples: the n-point conformal block in the comb channel and the OPE channel for arbitrary n, 5 and the seven-point mixed channel block (see figure 1 for their definitions). All known ddimensional scalar conformal blocks with scalar exchanges in the literature fall into one of 5 We remind the reader that the OPE channel in this paper is only defined for even n.
the classes above. This includes the well-known four-point block, and the recently obtained five-point block [53], n-point comb channel blocks [58,60] and the six-point OPE channel block [61]. However, the seven-point example to be discussed next and the n-point OPE channel examples for n ≥ 8 are new results.
We invite the reader to test their understanding of section 2 by applying the Feynman rules in the trivial case of the four-point block and rediscover the well-known series expansion, or the slightly less non-trivial though straightforward case of the five-point block. These are special cases of the n-point comb channel block which is discussed in section 3.2.

Seven-point mixed channel block
In this section we work out the seven-point conformal block in the "mixed channel" shown in figure 1b. We use the following independent cross-ratios as input data:  In terms of the independent Mellin variables, these can be expressed as γ 57 = ∆ 57,12346 + γ 12 + γ 13 + γ 14 + γ 16 + γ 23 + γ 24 + γ 26 + γ 34 + γ 36 + γ 46 . In terms of these, the Mellin product then takes the form where the leg-factor turns can be expressed as The exponents s i in (3.4) are given by which are indeed the Mandelstam invariants attached to the internal edges of the associated binary graph, as we now describe. In the auxiliary momentum space, one assigns an incoming momentum to each external edge of the unrooted binary tree representation of the conformal block, such that the sum over all momenta is zero (see figure 2). Let p i be the momentum attached to the external edge labelled with conformal dimension ∆ i , with the on-shell condition p 2 i = −∆ i and momentum conservation. Then the Mandelstam invariants associated to each internal leg are (3.8) Using (2.12), it is easy to see this gives back (3.7).
With the cross-ratios and the set V 7,mixed in place, the only computational task remaining is determining the post-Mellin parameters. Recall that the Mellin parameters form the set (2.17) and the post-Mellin parameters for the external conformal dimensions/edges are given by (2.18). Explicitly, for the present choice of cross-ratios, this yields 1. First we consider all vertices with precisely two incident external edges and one incident internal edge: (3.10) 2. Finally, to determine ℓ δ 2 , one can choose to look at one of two possible vertices. We will work it out using both to demonstrate choice-independence. From one choice of a vertex, we get On the other hand, the choosing the following vertex yields,  13) and the (7 − 2 = 5) internal vertices of the unrooted binary tree, listed here: give the following vertex factors: In section 4.1 we will reproduce the seven-point mixed-channel conformal block of this section using holographic techniques which will involve the Mellin amplitude of a particular seven-point Witten diagram as the starting point.

n-point comb channel block
In this section we will illustrate how to apply the Feynman rules to reproduce the n-point comb channel conformal block (see figure 1a) of ref. [58]. The first step involves picking the cross-ratios; in this section, we choose those from ref. [58], 6 It turns out, the associated canonical choice of n dependent Mellin variables is given by the set Explicitly, the dependent variables take the form, 7 which, along with γ ij = γ ji , and γ ii = −∆ i for all i, j explicitly solves (2.11) as required.
After substituting in (3.18), the Mellin product, expressed in terms of the cross-ratios (3.16)  where the leg-factor W 0 n,comb is given by 20) and the s i are expressible as, The set in (3.17) is the canonical choice of dependent Mellin variables precisely because it leads directly to (3.19).
As desired, the s i are the n − 3 Mandelstam invariants associated with the n − 3 internal edges. To see this, pass again to the auxiliary momentum space, and assign an incoming momentum to each external edge of the unrooted binary tree representation of the conformal block, such that the sum over all momenta is zero. Let p i be the momentum attached to the external edge labelled with conformal dimension ∆ i , with the on-shell condition p 2 i = −∆ i and momentum conservation, with the identification γ ij := p i · p j . Then the Mandelstam invariants associated to each internal leg are (see figure 3) which precisely evaluates to (3.21). Additionally, as described in section 2.1, to each internal edge with Mandelstam invariant s i , we also assign the single-trace parameter k i .
Furthermore, from (3.19) we also identify the index set Next, applying (2.20) to the internal vertices at either extremes of the comb channel, we obtain the post-Mellin parameters ℓ δ 1 and ℓ δ n−3 : Finally to determine the remaining post-Mellin parameters, we use (2.21) on vertices with two internal edges and one external edge attached. For example, one can start with the vertex:

27)
8 For example, for n ≥ 8 (3. 24) and then proceed one vertex to the right: and so on. Proceeding iteratively, we find For the sake of completeness, we provide the explicit edge and vertex factors below. The (n − 3) edge factors are: Similarly, for the vertex factors we simply substitute all the ingredients from above into (2.6).
To facilitate comparison with the result from ref. [58] (as well as the new derivation in section 4.2) we will simplify the linear combination of post-Mellin parameters appearing in the vertex factors. Rewriting, where we used the identifications ∆ δ 0 := ∆ 1 and ∆ δ n−2 := ∆ n , simple arithmetic leads to Then the n − 2 internal vertices, for 1 ≤ i ≤ n − 4, are associated with the vertex factors Furthermore, one can check this reproduces the well-known four-point block upon setting n = 4. Finally, ref. [53] worked out the n = 5 block for a different set of cross-ratios. Starting with those cross-ratios as the input data, we checked that the Feynman rules reproduced precisely the block of ref. [53]. Generally, blocks from different choices of cross-ratios, though perhaps not manifestly identical, are still equivalent in the shared domain of convergence.
In particular it can be checked that the five-point blocks of ref. [58], ref. [30] and ref. [53] are equivalent, even though they seem slightly different.

n-point OPE channel block
In the OPE channel (see figure 1c), we choose to represent the n-point conformal block (for even n ≥ 6) in terms of the following cross-ratios: 9

(3.34)
A convenient choice of dependent Mellin variables associated with the choice of crossratios above is given by the following index set The dependent Mellin variables take the form Just like for the comb channel, it is useful to consider the auxiliary momentum space in the OPE channel. We use the convention for momentum and Mandelstam invariant assignments as depicted in figure 4. In this convention, the Mandelstam invariants associated with the internal legs take the following explicit form in terms of the independent Mellin variables: With this in hand, it is straightforward to re-express the Mellin product (2.13) in terms of conformal cross-ratios (3.34): where the leg-factor W 0 n,OPE is defined to be 39) and the (rs) index in the final product in (3.38) runs over the index set where we have defined 10  we can determine the post-Mellin parameters from previously determined data, as shown: The post-Mellin parameters for the remaining internal conformal dimensions satisfy for 1 ≤ i ≤ n 2 − 3, thus they need to be determined iteratively. Working out a few explicit cases such as ℓ δ 3 , ℓ δ 5 and ℓ δ 7 allows us to conjecture, and subsequently prove by induction in appendix B.1, the general form for 1 ≤ i ≤ n 2 − 3. We now have all the ingredients to write down the full conformal block in the OPE channel. The (n − 3) edge factors are given by (2.4) with the post-Mellin parameters as determined above. The (n − 2) internal vertices of the binary graph, enumerated as follows for 2 ≤ a ≤ n/2 − 1 and 1 ≤ b ≤ n/2 − 2, correspond to the vertex factors (2.6). Explicitly 11 where to work out the linear combination of post-Mellin parameters in V (1) a we used In ref. [61], a power-series expansion was worked out for the special case of the n = 6 block. 12 Notably, the same generalized hypergeometric function makes an appearance in both their paper and the result above. For n = 6, it can be seen from (3.47) that in all exactly one factor of the Lauricella function F A appears in the vertex factors, which is directly related to the Kampé de Fériet function via (A.4). Precisely the same Kampé de Fériet function appeared in the result of ref. [61]. The choice of cross-ratios in that paper differs from the general choice made above in (3.34), so to make a precise comparison, we can start with the cross-ratios of ref. [61] and apply to them the Feynman rules of section 2.
We confirmed that doing so exactly reproduces the conformal block of ref. [61].
This section generalizes this result to any even n ≥ 6. 13 At higher n, precisely (n/2 − 2) factors of the Lauricella function of three variables F A (equivalently (n/2 − 2) factors of the Kampé de Fériet function) will appear in the power series expansion. 14

From Mellin amplitudes to conformal blocks
In this section we will prove the Feynman prescription for conformal blocks for the examples considered in the previous section. As outlined in section 1, our starting point will be certain canonical tree-level Witten diagrams in an effective ϕ 3 scalar field theory in AdS. We will write down their Mellin amplitudes using the Feynman rules for Mellin amplitude [68][69][70], from which we will be able to extract the desired conformal blocks via single-trace projections. 12 The authors of ref. [61] referred to it as the "snowflake channel," which is the same as the "OPE channel" above. 13 Recall that the "OPE channel" in this paper is well-defined only for even number of external operators. 14 The remaining vertex factors in (3.47) contribute one factor of the Lauricella function F A each, but this function can be trivially expressed in terms of Gamma functions or Pochhammer symbols; see (A.2).
A canonical Witten diagram is a tree-level Witten diagram of the same topology as the conformal block we are interested in computing, with generic scalar dimensions running along each edge. Since we will only be interested in the contribution coming from singletrace exchanges, and not the full amplitude, we would like to project onto the single-trace part of this Witten diagram. This is convenient to do in large N theories, since the poles of the meromorphic Mellin amplitude correspond precisely to the exchange of single-trace primaries. Evaluating the residue at these poles furnishes the required projection.
Concretely, the Mellin amplitude M n (γ ij ) for an n-point Witten diagram, whose position space ampitude is denoted A n , is defined via a multi-dimensional inverse Mellin transform, where A n is an AdS integral over products of bulk-to-bulk and bulk-to-boundary propagators which are normalized as follows: The contours of integration on the RHS of (4.1) run parallel to the imaginary axis for Re γ rs > 0 such that they separate out the semi-infinite sequence of poles running to the left or to the right. The overall normalization constant N will be fixed shortly. The set U V is the index set of n(n − 3)/2 independent Mellin variables γ ij and the set D is the index set of the n dependent Mellin variables. We have chosen to decompose the independent variable index set into a union of two disjoint subsets U and V. The precise prescription for this choice of sets was explained in section 2.1 and illustrated in section 3. Briefly, this choice will be dictated by the choice of cross-ratios and the channel (i.e. binary tree topology) for the precise conformal block we wish to extract from A n . To stress this dependence, we will call the sets U chan , V chan and D chan . The choice of cross-ratios and the index sets served as the input in section 3 for writing down the conformal block using the proposed Feynman rules. In this section, this choice will serve as the input for deriving the block from Mellin amplitudes.
The union of all three sets U chan , V chan and D chan gives the full range of indices (2.16) associated with the product over Gamma functions and powers of pairwise distances in (4.1).
This product over powers of pairwise distances was called the "Mellin product"; see (2.13).
For a canonical choice of index sets, the Mellin product admits a convenient rewriting, to be substituted in (4.1), in terms of the chosen cross-ratios as shown in (2.15) and repeated below: where s i are the Mandelstam invariants associated with the internal legs (and expressible in terms of Mellin variables drawn from the index sets U chan and V chan ), while W 0 n,chan (x i ) is the leg-factor which depends solely on external conformal dimensions and position coordinates.
We note that we have indexed the Mandelstam invariants and the cross-ratios in a manner that allows us to write (4.3) as displayed.
To evaluate the single-trace contribution to A n , one needs the Mellin amplitude M(γ ij ) for the Witten diagram. For tree-level scalar Witten diagrams, the Mellin amplitude is readily available via the "Feynman rules for Mellin amplitudes" [68][69][70]. According to these rules, in the normalization conventions we are following, the Mellin amplitude of a scalar n-point tree-level Witten diagram in a ϕ 3 theory is constructed as follows: • Label the internal lines of the Witten diagram with an index i running from 1 to n − 3, and to it associate an integer parameter k i (which will double as single-trace parameters) and a factor of where s i is the Mandelstam invariant associated to that leg, and ∆ δ i is the conformal dimension of the dual operator running along the line.
• Label each internal vertex of the diagram with an index j running from 1 to n − 2 and assign a factor of where ∆ a , ∆ b and ∆ c are the conformal dimensions incident at the vertex, and k a , k b and k c are the respective integer parameters (or single-trace parameters) associated with the internal exchanged dimensions. Set the integer parameter to zero if the conformal dimension associated to it is an external dimension.
Then for the choice of normalization constant, where ∆ i are the external conformal dimensions and ∆ δ i are the internal exchanged dimensions, the Mellin amplitude is given by With the Mellin amplitude in hand, we can proceed to evaluate the single-trace projection of the position space amplitude which correspond to putting the internal legs on-shell in the auxiliary momentum space.
These poles should be viewed as lying in the complex γ rs planes for (rs) ∈ U chan , and our task is to evaluate the residue at these poles.
Before we do so, we point out that precisely the same Lauricella functions as those in (4.5) appeared in (2.6) in the Feynman rules for conformal blocks. This is expected for the simple reason that the vertex factors (4.5) are independent of Mellin variables γ ij , so they remain unaffected through the following computation of Mellin integrals.
Turning to evaluating the residue at the "single-trace poles" (4.8) in the "U chan -plane," we obtain the following single-trace projection of the AdS diagram, denoted A s.t. n , where Γ(γ ij ) stands for Gamma functions with arguments from the index sets U chan and D chan evaluated at the poles (4.8). 15 Equation (4.9) is proportional to the desired conformal block. More precisely, To obtain an integral-free representation of the block, one must evaluate the residual n−2 2 -dimensional contour integrals in the second line of (4.10). This will introduce n−2 2 new summations. Evaluating these integrals in general for an an arbitrary n-point conformal block in an arbitrary channel is not clear to us. Instead, in the remainder of this section, we will focus on evaluating these integrals explicitly for the three classes of examples from section 3. We will reproduce the blocks as prescribed by the Feynman rules of section 2, which serves as a highly non-trivial check of the proposed Feynman rules.
Before specializing to specific examples, we can do further general manipulations. First, 15 We note that the dependent Mellin variables γ ij with (ij) ∈ D chan in the Gamma functions are assumed to have already been expressed in terms of the independent Mellin variables from the sets U chan and V chan . Still, for brevity, we prefer to use the notation in the second line of (4.9). We also emphasize the obvious fact that the Gamma functions with arguments from the index set V chan remain unaffected after taking the single-trace residues.
isolating the integral in the second line of (4.10), we rewrite each factor of v −γrs rs above by introducing an additional contour integral, as fol- where the γ rs contour runs vertically such that it separates the semi-infinite sequence of poles running to the left and to right of origin. Then we get where we switched the order of integrals. The rewriting in (4.13) makes it easier to obtain a convergent series expansion of the conformal block in powers of (1−v rs ) as desired (see (2.2)- 3)). The overall strategy now will be to repeatedly use the first Barnes lemma [74], to evaluate all γ rs integrals, which it turns out will leave us with trivial-to-evaluate γ rs contour integrals. 16 Use the Mellin-Barnes representation,

Seven-point mixed channel
To obtain the 7-point "mixed-channel" conformal block (topology shown in figure 1b), we start with the following "canonical" tree-level AdS diagram: (4.16) Here we have labeled the external scalar operators with their conformal dimensions but the vertex factors taking the explicit form: The single-trace projection of the Witten diagram (4.16), described in general terms in the discussion preceding this example, then leads to (4.9) which is proportional to the desired conformal block. This projection involves evaluating the residue at the "single-trace poles," occuring at The block itself is given by (4.10) by projecting out the theory dependent OPE coefficients: (4.20) The MFT OPE coefficients above take the well known form, The precise form of these OPE coefficients will be utilized at the end of the computation.
The non-trivial computation which needs to be done is the contour integral on the second line of (4.10), or equivalently the integral (4.14). The arguments of Gamma functions in the are expressed entirely in terms of Mellin variables from the index set V 7,mix . Four of them were shown in (4.19) which correspond to the index set U 7,mix ; the other seven, corresponding to the index set D 7,mix take the form Explicitly, I in (4.14) then takes the form (4.23) We first evaluate the 10-dimensional contour integral over the γ rs variables for (rs) ∈ V 7,mix .
We perform the contour integrations one at a time, in the following order: With this choice of ordering, we are able to make direct use of the first Barnes lemma (4.15) at every step. 17 To keep the manuscript to a reasonable length, we refrain from including the lengthy but straightforward computational details, and merely present the final result of this 10-fold contour integral: .

(4.24)
Next, we evaluate the remaining integrals via the Cauchy residue theorem. We close all γ rs 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, at γ rs = j rs for j rs ∈ Z ≥0 for each (rs) ∈ V 7,mix . 18 These poles come from the poles of Γ(− γ rs ) in the first line of (4.24), and 17 Notably, after the γ 15 integral, we need to do a linear change of variables γ 14 → γ 14 − γ 24 − γ 34 . 18 We recall that the contour for the γ rs integrals was chosen such that it separates the semi-infinite series of poles running to the left from those running to the right; see (4.12).
the residues, which are elementary to compute, introduce n−2 2 additional infinite sums: . This leads to the following final expression for the conformal block:

Comb channel
To derive the comb channel conformal block obtained in section 3.2 via Feynman rules, we start with the Mellin amplitude of the following tree-level n-point AdS diagram (n ≥ 4): where the external scalar operators of dimensions ∆ i are inserted at coordinates x i . In total (n − 3) single-particle bulk fields are exchanged in the interior, which are dual to single-trace operators with conformal dimensions ∆ δ i for 1 ≤ i ≤ n − 3. The ellipses in the middle 12   To reproduce the block from section 3.2, we will utilize the same input data as before.
This data comprises of a set of n(n − 3)/2 independent cross-ratios (  Using the Feynman rules for Mellin amplitudes, it is trivial to write down Mellin amplitude for the diagram (4.27), where each of the (n − 3) edge factors is given in terms of the Mandelstam invariants s i and the associated single-trace parameters k i by (4.4), and the (n − 2) vertex factors follow directly from (4.5): One can now obtain the single-trace projection of the AdS diagram (4.27) by evaluating the residue at the "single-trace poles" in the "U comb -plane." The poles are situated at (4.8), which in the U comb -plane corresponds to γ (n−1)n = ∆ (n−1)n,δ n−3 − k n−3 . (4.32) The conformal block (4.10) is then obtained by projecting out the following OPE coefficients: for 1 ≤ i ≤ n − 4, whose general form was given in (4.21).
We will now evaluate the second line of (4.10), or more precisely, the equivalent form in (4.14). Substituting (4.31) and (4.32) in (4.14), we get where we employed the additional identifications ∆ δ n−2 := ∆ n and k n−2 := 0 to write I compactly.
To evaluate the n−2 2 -dimensional contour integral over γ rs variables, we will employ a multi-dimensional variant of the first Barnes lemma which can be easily proven by a repeated application of the first Barnes lemma (4.15).
Looking forward, our strategy will be to evaluate the contour integrals in the order shown below (see figure 6 for color key): Concretely, for 2 ≤ m ≤ n − 1, define (4.37) Then for 2 ≤ m ≤ n − 2 integrating J m over the elements in the m-th row of the index set which is proven in appendix B.2. Moreover, at m = n − 1, J m reduces to a γ ij -independent expression, In terms of J m , the original contour integral (4.34) can be written as where in the second step onward, we made repeated use of (4.38) to perform all γ rs integrals in the manner indicated in (4.36).
Now we turn to the γ rs integrals. For carrying out the contour integrals, it is convenient to rewrite I in terms of Gamma functions as Examining the pole structure of the integrand (4.42), we notice that just like in the example of the seven-point block in the previous subsection, we can evaluate the remaining γ rs contour integrals by closing the contours to the right. In the process, each integral picks up a semiinfinite sequence of poles originating from Γ(− γ rs ) at γ rs = j rs for non-negative integers j rs , for each (rs) ∈ V comb . All other poles lie to the left of the contour and the contribution from the arc at infinity vanishes. This immediately leads to It is now suggestive to re-express I in terms of Pochhammer symbols, as shown here: × Γ(∆ (n−1)δ n−3 ,n )Γ(∆ (n−1)n,δ n−3 )Γ(∆ nδ n−3 ,n−1 ) .
Just like in the previous two examples, we will use the same cross-ratios (3.34) as input data as used for Feynman rules. Let us recall the associated index sets which will be important in the computations to follow. The associated choice of dependent and independent Mellin index sets will also be identical. The dependent set D OPE was given in (3.35) which allowed a rewriting of the Mellin product as shown in (3.38), with the Mandelstam invariants for each internal leg as defined in (3.37), and also determined the set V OPE as shown in (3.40 It is useful to represent the index sets visually as shown in figure 7. According to the Feynman rules for Mellin amplitudes, the AdS diagram (4.45) has the Mellin amplitude, come directly from (4.5): for 2 ≤ a ≤ n/2 − 1 and 1 ≤ b ≤ n/2 − 2.
Substituting (4.47) into (4.1), we proceed to obtain the single-trace projection of the AdS diagram as described around (4.8). This leads to (4.9) which is the desired conformal block times a set of known OPE coefficients, for 2 ≤ a ≤ n/2−1 and 1 ≤ b ≤ n/2−2, which can be factored out to obtain the block (4.10).
This single-trace projection is obtained by evaluating the residue at the poles (4.8), which in the U OPE -plane occur at   The direction of the dotted arrow (right to left) indicates the order in which we integrate over the connected elements of the set V OPE . Each connected chain corresponds to a subset of contour integrals that will be evaluated with the help of the inductive first Barnes lemma (4.35). While any ordering works, the precise ordering chosen here makes it possible to set up an inductive step. The strategy will be as follows: We will first evaluate integrals (4.14) associated with the two (right-most) black-colored chains in (4.54). Using the resulting expression from the black-colored chain integrals, we will establish a two-step induction; the green and magenta colored chains above suggest how the induction will work.
In appendix B.3.1, we present the computation of the integrals marked as black-colored chains above. The end result of this computation is given in (B.24). To set up induction, we define a new contour integral I n−2K−1 such that where I is given by (B.24), and I n−2K−1 is defined to be  57) and (4.59) 21 We note that W j at j = n − 4 coincides with (B.25). and L n−2K−1 (4.60) It can be checked that (4.55) holds. This will serve as the base case for an inductive argument which we develop next.
This will turn out to be associated with integrating out a green-colored chain in (4.54) in an intermediate step, where this computation is described in appendix B.3.2. Here we rewrite the result of this computation, given in (B.31) as follows:  Now, we would like to integrate over γ 2(n−2K−3) , γ 1(n−2K−2) , γ 3(n−2K−2) , γ 4(n−2K−2) , . . . , γ (n−2K−4)(n−2K−2) . This will be associated with integrating over the magenta-colored chain in (4.54) immediately to the left of the green-colored chain we previously integrated out. This computation is This establishes the inductive step, and together with the base case (4.55) furnishes the following chain of equalities: As we move progressively to the right down the chain of equalities above, we account for evaluations of more and more contour integrals from the set V OPE , until we are left with just one integral. At the right-most equality at K = n 2 − 2, the original contour integral I (see (B.12)) reduces to where the W j for 2 ≤ j ≤ n − 3 are given in (4.57)-(4.58), M 3 is given by (4.59) which at K = n 2 − 2 simplifies to and L 3 is obtained by setting K = n 2 − 2 in (4.60), (4.67) The contour integral in (4.65), which corresponds to the lone black dot in (4.54) at the left-most extreme, can be evaluated using the first Barnes lemma (4.15), to give where we identified a factor of W 1 above by comparing with (4.58) at j = 1, thus extending the regime of validity of the W j coefficients in (4.57)-(4.58) to 1 ≤ j ≤ n − 3. A careful comparison between the block found using the Feynman rules for conformal blocks, and the one found using the Mellin formalism in this section confirms that there is full agreement between the n-point conformal block of this section and section 3.3, thus confirming the Feynman rules for n-point blocks in the OPE channel.

Discussion
In this paper we proposed a simple set of rules for constructing any scalar conformal block with scalar exchanges, given the appropriate cross-ratios as input data. The rules are summa- examples, is also expected to work in exactly the same manner for any particular choice of conformal block beyond those considered in this paper. However, proving it for an arbitrary choice of a conformal block will presumably require more work. Nevertheless, it would be useful to prove these rules in generality for arbitrary blocks.
Finally, it should also be possible to generalize these rules to arbitrary-point conformal blocks for external and/or exchanged operators in arbitrary representations of the Lorentz group. This would be especially useful from the point of view of setting up an n-point conformal bootstrap for external scalars where internal exchanges can still involve spinning operators. Weight-shifting operators [25] and differential operators [32,33]

A Lauricella functions
The Lauricella function F A of ℓ variables is a generalized hypergeometric sum of ℓ variables [81][82][83] (see also ref. [69]) defined as One can always perform one of the sums in the (A.1) to re-express F

(ℓ)
A in terms of functions involving ℓ − 1 summations. For example, we present some identities for ℓ ≤ 3: where F p,r,u q,s,v is the Kampé de Fériet function [82,84] (see also ref. [61]), defined by the following hypergeometric series, In this appendix we will compute ℓ δ 2i+1 which satisfies the recursion relation where the even-indexed ℓ δ 2i+2 and the smallest odd-indexed ℓ δ 1 are known from (3.43).
We will now prove by induction that ℓ δ 2i+1 takes the form Let's first establish the base case. For i = 1, For the inductive step, assume (B.2) is true for i = K, where 0 ≤ K < n/2 − 3. We will now show that (B.2) holds for i = K + 1. That is, assuming the following: we can now compute where we can use which is the same as (B.2) for i = K + 1, as needed.

B.2 Proof of (4.38)
We notice that the LHS of (4.38) is of the form: where the integration variables are s i = γ mi , and the coefficients are with the overall factor (B.10) Using the inductive Mellin Barnes lemma (4.35), we can easily evaluate (B.8) to obtain Substituting in the explicit form for K m and using the definitions (4.37) and (4.39), we recognize (B.11) to be precisely the RHS of (4.38).

B.3.3 Integrals over a magenta-colored chain
Integrating ( which agrees with W j in (4.58) if we set j = n − 2K − 4 (i.e. for even j).