Six-Point Conformal Blocks in the Snowflake Channel

We compute $d$-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two ${}_3F_2$-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kamp\'e de F\'eriet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.


Introduction
The study of higher-point conformal blocks in conformal field theory (CFT) is a complicated subject without many explicit results. In a CFT, correlation functions, which are the natural observables of the theory, are given in terms of the CFT data and the conformal blocks. The CFT data, which consist of the spectrum of quasi-primary operators as well as the operator product expansion (OPE) coefficients, completely determine all correlation functions with up to three points. For higher-point correlation functions, the appearance of conformal cross-ratios, which are invariant under conformal transformations, leads to conformal blocks. The conformal blocks are functions of the conformal cross-ratios which are in principle fully constrained by conformal invariance.
Although conformal blocks are fixed by conformal invariance, they are notoriously difficult to compute in all generality. Several techniques have been developed over the years for the computation of four-point conformal blocks, which are the simplest blocks. For example, various methods use Casimir equations [1,2], the shadow formalism [3], the weight-shifting formalism [4], integrability [5], AdS/CFT [6], and the OPE [7][8][9][10][11][12][13][14]. Another important reason why fourpoint conformal blocks have been studied extensively is the conformal bootstrap [15], a way of constraining the CFT data solely from consistency of correlation functions under associativity.
Indeed, it is known that four-point conformal blocks are sufficient to implement the full conformal bootstrap.
Conformal blocks with more than four points have not been studied in great detail as of now.
Until very recently, the only results were for scalar M -point blocks in one and two spacetime dimensions as well as scalar five-point blocks in any spacetime dimensions [16][17][18][19]. Last year, the scalar M -point conformal blocks in the so-called comb channel were presented in [20,21]. They showed that the scalar M -point conformal blocks in the comb channel can be expressed as a power series expansion in the conformal cross-ratios with the coefficients containing a product of M − 4 3 F 2 -hypergeometric functions. Although the techniques employed in the two references were different, AdS/CFT versus OPE, the two results have very similar forms even though the basis of conformal cross-ratios were distinct.
An interesting feature of higher-point correlation functions is that there exist several inequivalent topologies. Indeed, starting at six points, the use of OPE among pairs of operators in different sequences of operator pairings leads to different topologies, see Figure 3. The comb channel [17] is only one of the many topologies that are possible for higher-point correlation functions.
One possible advantage of the OPE formalism used in [21] is that it is not limited to the comb channel. Indeed, it can be applied to any pairs of quasi-primary operators in the original (M − 1)-point correlation function, leading to the M -point correlation function in any channel of interest. After all, the existence of the many topologies is a consequence of the OPE, as was directly observed above. More importantly, the choice in the pairs of quasi-primary operators does not lead to any new computational complications.
With the OPE formalism [9,11,12], the OPE differential operator is applied on the known (M − 1)-point correlation functions to generate M -point correlation functions. Since the action of the OPE differential operator on arbitrary products of conformal cross-ratios has been determined [11,12], it is intuitively straightforward to proceed with the computation. It can however be technically laborious to express the final results in the most convenient way possible. Indeed, starting at five points, it is necessary to re-express the conformal cross-ratios of the original correlation function in terms of the conformal cross-ratios appropriate for the OPE differential operator, leading to several superfluous sums. Re-summations must be performed to recover a simple result for the final correlation function. For the computation of the scalar M -point conformal blocks, we found that, with an appropriate basis of conformal cross-ratios, all re-summations were easy 2 F 1 -hypergeometric function re-summations [21]. For other channels, the necessary computations within the OPE formalism are exactly the same at the technical level, hence it should be straightforward to compute higher-point correlation functions in all topologies using the OPE formalism.
In this paper, we begin the investigation of higher-point correlation functions in all topologies by computing the scalar six-point conformal blocks in the remaining channel, see the bottom part of Figure 3, which for obvious reason, we call the snowflake channel. 1 We show that the superfluous sums can be taken care of with simple 2 F 1 -hypergeometric function re-summations (as for the comb channel) and some identities for 3 F 2 -hypergeometric functions (absent in the comb channel). This paper is organized as follows: Section 2 presents a summary of the embedding space OPE formalism. The OPE is reviewed and the action of the scalar OPE differential operator is discussed.
A general form for the contribution to the correlation functions of external scalars with scalar exchanges is presented, as well as the associated scalar conformal blocks. Moreover, the recurrence relation taking (M − 1)-point correlation functions to M -point correlation functions is introduced, and, after re-considering scalar M -point conformal blocks in the comb channel, the scalar six-point conformal block in the snowflake channel is given. Section 3 provides several consistency checks of the snowflake result. First, interesting symmetry properties of the scalar conformal blocks in the snowflake channel are proven. Then, the OPE limit and the limit of unit operator are used to ascertain that the snowflake result passes those tests. Finally, we conclude in Section 4 while several appendices present most of the more technical proofs. Appendix A contains derivations of the scalar five-point conformal blocks of [20] directly from the OPE, showing that the results of [20] and [21] are equivalent for five points. Appendix B demonstrates how the snowflake result is obtained. An equivalent result is also presented and the proof of their equivalence is shown.
Appendix C proves the identities that the scalar six-point conformal blocks in the snowflake 1 In [19,20], the snowflake channel is called the OPE channel.
channel verify under the symmetry group of the snowflake.

Scalar Six-Point Conformal Blocks
With the knowledge of the OPE and its explicit action on any function of conformal cross-ratios, one can in principle compute any correlation function starting from the known two-point functions or, for that matter, the only non-trivial one-point function. With this technique, starting at five points and above, it is necessary to re-express the initial correlation function in terms of the conformal cross-ratios appropriate for the OPE differential operator, and then rewrite the solution in terms of the most convenient conformal cross-ratios by re-summing as many superfluous sums as possible. This method is quite powerful, allowing the computation of conformal blocks in any channel. Although straightforward, it is however not always clear a priori what is the best choice of conformal cross-ratios that will lead to the simplest final answer. In this section, we quickly review the OPE and then sketch the derivation and state the result for the scalar six-point conformal blocks in both channels. Concrete proofs are left for the appendices.

M -Point Correlation Functions from the OPE
The embedding space OPE introduced in [11,12] states that the product of two quasi-primary operators can be expressed as where the OPE differential operator and the remaining quantities (half-projectors, tensor structures, etc.) are introduced and detailed in [12]. The action of the scalar OPE differential operator on the conformal cross-ratios where we have definedq = 1≤a≤M a =i,j q a as well as The OPE (2.1) was used to determine two-, three-, and four-points in [13]. Very specific rules, somewhat reminiscent of Feynman rules, were also developed in [14] to compute all necessary ingredients to implement the full conformal bootstrap at the level of four-point correlation functions.

Scalar M -Point Correlation Functions
In general, it is possible to write the contribution to scalar M -point correlation functions, i.e.
with scalar external and exchanged quasi-primary operators, from a specific channel as The scalar M -point conformal block is given by where the cross-ratios u a and v ab are defined below in (2.7).
The full M -point correlation functions are sums of the different I M , including the ones for non-trivial exchanged representations which are not discussed here. 2 We note that the OPE differential operator used here is a simple rescaling of the one defined in [12].
In (2.4) for some convenient choice of k, l and m with the appropriate channel on the RHS to generate the desired channel on the LHS. For future convenience, we also definep a = a b=2 p b andh a = a b=2 h b . In this context, the OPE (2.1) was also put to work in the computation of scalar M -point correlation functions in the comb channel in [21], to which we now turn.

Scalar M -Point Correlation Functions in the Comb Channel
Acting repetitively with the OPE as in (2.6), we found in [21] the following scalar M -point conformal blocks (2.4) and (2.5) in the comb channel, 3 In general, FM is a function of both the vector m = (ma) and the matrix m = (m ab ). Thus the separation between CM and FM is somewhat arbitrary in (2.5). With our choice of conformal cross-ratios, F6 in the snowflake channel is only a function of the vector m, as is the case for FM in the comb channel. Moreover, we construct F 6|snowflake such that it has some interesting symmetry properties. and 9) and finally The computations were straightforward yet somewhat tedious. The choice of conformal crossratios was based on the OPE limit, a limit we use again later to check the validity of the scalar six-point conformal blocks in the snowflake channel discussed in the next subsection.

Scalar Six-Point Correlation Functions in the Snowflake Channel
As already mentioned, the OPE ( To reach the scalar six-point conformal blocks in the snowflake channel, we start from the fourpoint correlation functions and use the OPE appropriately. For M = 4, there is only one channel with the topology of the comb, as shown in Figure 1. One possible form for the contribution to scalar four-point correlation functions (2.4) and for scalar four-point conformal blocks (2.5) leads to [1,8] L It is clear from (2.6) and Figure 1 that there is only one channel for five-point conformal blocks since all external quasi-primary operators in four-point conformal blocks are topologically equivalent. This channel has the topology of the comb and is shown in Figure 2. It can be obtained from (2.6) with k = 3, l = 4 and m = 4, using the four-point correlation functions in the comb channel (2.11) The scalar five-point conformal blocks can be expressed as in (2.4) and (2.5) with the help of We note here that the result (2.12) for the scalar five-point conformal blocks is equivalent to but different than the results (2.7), (2.8), (2.9), and (2.10) obtained in [21]. 4 For M = 6, there is now two possible topologies that can be obtained from five-point correlation functions. Indeed, from Figure 2, it is clear that the external quasi-primary operator exchanged quasi-primary operators by appending two new external quasi-primary operators leads to the scalar six-point correlation function in the comb channel (see previous subsection). Doing the same with the quasi-primary operator O i 2 gives instead the scalar six-point correlation in the snowflake channel. The difference can be seen in Figure 3. For the scalar six-point correlation functions in the snowflake channel, we start from the scalar five-point conformal blocks (2.12) and shift the quasi-primary operators such that O ia (η a ) → O i a−1 (η a−1 ) with the understanding that , and then use (2.6) with k = 4, l = 5 and m = 5, to get as well as 15) and finally (2.16) As mentioned previously, the function F 6 (2.15) in the snowflake channel is also a double sum of the hypergeometric type, same as in the comb channel. However, the snowflake double sum does not factorize into two hypergeometric functions, unlike the comb sum, compare (2.9) with M = 6. It can however be written as a Kampé de Fériet function F 1,3,2 2,1,0 as shown in Appendix B. See (B.4) and [22] for its definition.
In the following section, we study the snowflake results (2.13), (2.14), (2.15), and (2.16). We explicitly check the symmetry properties of G 6 and verify that it behaves properly under the OPE limit and the limit of unit operator. The proof of the snowflake results as well as an alternative form for the snowflake are shown in Appendix B

Sanity Checks
The scalar six-point conformal blocks obtained in the previous section must satisfy several properties. This section investigates the identities of G 6 (2.5) in the snowflake channel from the symmetries of the associated snowflake diagram Figure 3. Then, the OPE limit and the limit of unit operator are taken to verify that the scalar six-point correlation functions reduce to the appropriate scalar five-point correlation functions.

Symmetry Properties
The scalar M -point conformal blocks must verify several identities where the conformal cross-ratios and the vectors h and p are transformed. These identities are generated from the symmetries of the scalar M -point conformal blocks in the associated topology. The symmetries of the scalar Mpoint conformal blocks in the comb channel are relatively trivial. They correspond to (Z 2 ) 2 ⋊ Z 2 , the semi-direct product of the direct product of two cyclic groups Z 2 of order two (for OPE, or dendrite, permutations depicted on the right of Figure 4) and the cyclic group Z 2 of order two (for reflection shown in the left part of Figure 4). Here we present the identities of the scalar six-point conformal blocks in the snowflake channel. The proofs are left to Appendix C.
The snowflake diagram Figure 3 is invariant under the symmetry group generated by the three transformations shown in Figure 5. Since the scalar six-point correlation functions in the snowflake channel I 6 are the same under symmetry transformations generated by the rotations, reflections and permutations described in Figure 5, there are 48 identities that the scalar six-point conformal blocks G 6 should satisfy. Although we dubbed it the snowflake channel, at first glance the symmetry group generated by rotations and reflections is only the dihedral group of order six, D 3 , which is the symmetry group of the triangle, not the hexagon expected for snowflakes. Including the OPE permutations, i.e. the permutations of the dendrites (or arms) of the snowflake, the full symmetry group of the snowflake diagram is however given by (Z 2 ) 3 ⋊ D 3 where each cyclic group of order two corresponds to dendrite permutations. Since the order of this symmetry group is |(Z 2 ) 3 ⋊D 3 | = 48, the snowflake diagram has a larger symmetry group than the hexagon, contrary to expectations.
Under this transformation, the legs and conformal cross-ratios (2.13) transform as which imply the following identity, 6 12 , v 6 23 , v 6 22 , v 6 33 , v 6 11 , v 6 13 ). (3.1) Using the decomposition (2.5) with (2.14) and (2.15), it is easy to see that C 6 does not change under this rotation generator, resulting in a non-trivial identity for F 6 [see (C.2)]. This identity can be translated into the language of Kampé de Fériet functions as discussed in the conclusion.
For reflections, we start with the generator acting as For this reflection generator, the legs and conformal cross-ratios (2.13) transform as .

(3.2)
Again, from the decomposition (2.5) with (2.14) and (2.15), we remark that F 6 in the form (C.3) does not change under this reflection generator, implying an identity for C 6 .
for the legs and conformal cross-ratios (2.13). Thus, the corresponding identity is Once again, the decomposition (2.5) with (2.14) and (2.15) shows that F 6 (2.15) is invariant under this generator for dendrite permutations, resulting in a second identity for C 6 .
Therefore, the three symmetry transformations of Figure 5 generate the symmetry group (Z 2 ) 3 ⋊ D 3 of order 48 (see Appendix C). Each generator has an associated identity for the scalar six-point conformal blocks (2.5) in the snowflake channel, with (3.1), (3.2), and (3.3) being the identities for rotations, reflections, and dendrite permutations, respectively. From these three identities, it is straightforward to generate the remaining 45 identities of the snowflake symmetry group by composition. The proofs that our explicit solution (2.14) and (2.15) satisfies these symmetry transformations can be found in Appendix C.
We now turn to the OPE limit and the limit of unit operator. Since we have already demonstrated that the scalar six-point conformal blocks in the snowflake channel obey several identities originating from the symmetry group of the snowflake diagram, it is only necessary to check the two limits once, all the other cases are equivalent by symmetry.

OPE Limit
The OPE limit is defined as having two embedding space coordinates coincide. The two embedding space coordinates must correspond to an OPE in the associated topology. In this limit, the original M -point correlation function reduces to the proper (M − 1)-point correlation function with a pre-factor originating from the OPE (2.1), as dictated by (2.6) for the scalar case.
For scalar six-point correlation functions in the snowflake channel depicted in Figure 3, the possible OPE limits are η 2 → η 3 , η 4 → η 5 , and η 6 → η 1 . However, since I 6 is invariant under rotations as discussed above, it is only necessary to assess the behavior of I 6 in one OPE limit.
Here, we check that in the limit η 2 → η 3 , we have For this proof, we start from the alternative form of the scalar six-point correlation functions in the snowflake channel given in Appendix B, which must lead in the OPE limit (3.4) to the scalar five-point correlation functions in the comb channel of [20] discussed in Appendix A. This choice is of no consequence since we prove that these are equal to the results of Section 2 in Appendices B and A, respectively [see (2.13), (2.14), (2.15), and (2.16) for scalar six-point correlation functions in the snowflake channel and (2.12) for scalar five-point correlation functions in the comb channel].
In the OPE limit (3.4), we have where the conformal dimensions on the RHS are the ones relevant for the five-point correlation functions, i.e.
Thus, the OPE limit (3.4) corresponds to the identity with the appropriate changes for the vectors h and p. Since after expanding in the proper conformal cross-ratios u P a and (1 − v P ab ) of [20]. Here the vectors h and p are still the original ones. Thus to complete the proof, we need to evaluate the extra sums for which we did not explicitly write the indices of summation (to make the notation less cluttered) with the help of (A.1), and express the vectors h and p in terms of the five-point ones.
First, we sum over k 22 and then k 23 after changing the variable by k 23 → k 23 + m ′ 24 , which lead to We then shift m 23 by m 23 → m 23 + m ′ 24 , and then redefine m 33 = m − m 23 . With these changes, we can compute the sums over m 23 and then m, which give Finally, changing the vectors h and p by their five-point counterparts and renaming m 2 → m 1 ,

Limit of Unit Operator
The limit of unit operator is defined by setting one external operator to the identity operator.
In this limit, a M -point correlation function directly becomes the corresponding (M − 1)-point correlation function.
For scalar six-point correlation functions in the snowflake channel, the symmetry properties of I 6 liberate us to verify the limit of unit operator for just one quasi-primary operator. We choose Since h 5 = p 6 = 0 and the remaining vector elements of h and p (2.16) translate directly into their five-point counterparts (2.12) in the limit of unit operator (3.5), the sums over m 3 , m 13 , m 23 , and m 33 in (2.14) are trivial due to the Pochhammer symbol (−h 5 ) m 3 +m 13 +m 23 +m 33 , forcing m 3 = m 13 = m 23 = m 33 = 0. Therefore, the conformal cross-ratios u 6 3 , v 6 13 , v 6 23 , and v 6 33 disappear in the limit (3.5).
The remaining conformal cross-ratios (2.13) relate to the conformal cross-ratios (2.12) as in and thus These observations imply that G 6 → G 5 in the limit of unit operator (3.5), which is straightforward to verify since m 3 = m 13 = m 23 = m 33 = 0.

Discussion and Conclusion
The field of research on d-dimensional higher-point correlation functions in CFT is a relatively uncharted territory. Although completely determined by conformal invariance, higher-point conformal blocks are notoriously difficult to compute in all generality. Moreover, there exists several inequivalent topologies for higher-point correlation functions, each of them having their associated set of identities originating from their topology. In this paper, we introduced all scalar six-point conformal blocks by computing the remaining topology, the scalar six-point conformal blocks in the snowflake channel.
Our results-presented in (2.13), (2.14), (2.15), and (2.16)-are obtained with the help of the embedding space OPE formalism developed in [11,12]. They show that the embedding space OPE formalism is very powerful, leading to explicit results for any conformal higher-point correlation function of interest, after straightforward (yet somewhat tedious) re-summations of the hypergeometric type.
From the symmetry group of the snowflake diagram, we showed that scalar six-point conformal blocks in the snowflake channel have symmetry groups of order 48, larger than the hexagon symmetry group, contrary to expectations. We then showed that the result verifies both the OPE limit (in which two embedding space coordinates coincide) and the limit of unit operator (where one external operator is set to the identity). Our snowflake result thus passes several non-trivial consistency checks, lending further credence to the embedding space OPE formalism.
Contrary to the scalar six-point conformal blocks in the comb channel, for which there are two extra sums that factorize into the product of two 3 F 2 -hypergeometric functions, the scalar six-point conformal blocks in the snowflake channel have two extra sums that do not factorize.
They can be written as a Kampé de Fériet function, and the snowflake invariance under rotations implies (C.2), that translates into the identity (for m 1 , m 2 , and m 3 non-negative integers and a 1 , b 1 , b 2 , and d 1 arbitrary) which is obtained from repeated use of well-known 3 F 2 -hypergeometric identities. 5 It is obvious that the embedding space OPE formalism can be used to investigate higherpoint correlation functions, including their symmetry groups. For example, starting from the scalar three-point correlation function, which has symmetry group D 3 , we conjecture that doubling the number of all external legs N times with the help of the OPE gives scalar (3× 2 N )-point correlation functions with the symmetry groups ( From the successive action of the OPE limit or the limit of unit operator, these maximally-symmetric topologies should lead to all topologies for smaller scalar higher-point correlation functions (starting from sufficiently large N ). In general, starting from a specific M -point topology with sym- 5 Kampé de Fériet functions have not been studied as extensively as standard hypergeometric functions. 6 Although this is only a conjecture, the order of the symmetry groups should be correct.
metry group H M |channel , this procedure of doubling the number of all external legs N times with the help of the OPE should generate (M × 2 N )-point topologies with symmetry groups It is less clear what happens to diagrams with fewer symmetries. For example, scalar sevenpoint correlation functions with the topology resulting from scalar six-point correlation functions in the snowflake channel (called the extended snowflake channel below) should have symmetry group of order sixteen only, smaller than for the snowflake diagram but larger than the scalar seven-point correlation functions in the comb channel. In fact, their symmetry group should be Comparing the symmetry groups and the number of topologies, we thus conclude that we should have the following identity for M -point correlation functions, This identity is in agreement with the partial results obtained above with M up to seven and we verified it for M = 8, 9 as well. It can be used to explore the orders of the symmetry groups of the different topologies.
symmetry group of the four-point conformal blocks, it is easy to generate the second equation with the third channel.
In other words, once the bootstrap equations between the s-and the t-channels are satisfied for all quasi-primary operators, then the bootstrap equations for all three channels are satisfied, therefore the u-channel is redundant.
It is unclear for now what type of generalizations occurs for the extra sums, encoded here in our function F M . In the comb channel, we argued in [21] that the M − 4 extra sums appearing in scalar M -point correlation functions were necessary for the limit of unit operator to make sense. For M = 6, this argument cannot be used for the snowflake diagram. Nevertheless, (2.15) shows that in the snowflake channel, the number of extra sums is the same, although these extra sums do not factorize as in the comb channel. Are there three extra sums that do not factorize for the other topology of the scalar seven-point correlation functions, like the scalar seven-point correlation functions in the comb channel?
An interesting avenue of research is to initiate the computation of higher-point correlation functions with spins, either for external quasi-primary operators or internal quasi-primary operators, or both. Since the embedding space OPE formalism developed in [11,12] treats all irreducible representations of the Lorentz group on the same footing, such computations should be feasible.
Finally, higher-point correlation functions can also be of use in the AdS/CFT correspondence.
Indeed, scalar six-point conformal blocks in the snowflake channel correspond to some geodesic Witten diagrams (see for example [19]) and their knowledge might elucidate some of the kinematics in AdS.

Acknowledgments
The authors would like to thank Valentina Prilepina for useful discussions. The work of JFF is supported by NSERC and FRQNT. The work of WJM is supported by the China Scholarship Council and in part by NSERC and FRQNT.

A. Scalar Five-Point Conformal Blocks and the OPE
The scalar higher-point correlation functions in the comb channel were obtained very recently in [20,21]. In this appendix, we show how to obtain the scalar five-point conformal blocks of [20] from the OPE approach of [21]. The proof is a straightforward application of the re-summation for n a non-negative integer. Other useful identities used in the proof are the binomial identity Fig. 6: Scalar five-point conformal blocks of [20].
The scalar five-point conformal blocks (2.12), which is our starting point to compute the scalar six-point conformal blocks in the snowflake channel, can be obtained similarly.

A.1. Proof of the Equivalence
The comb channel of [20] is depicted in Figure 6. To get Figure 6, we need to shift the quasiprimary operators in [20]  where the legs L 5 and the vectors h and p are defined in [21]. For notational simplicity, we omit the indices of summation n 1 , n 11 , and s 2 on the sum. Here C 4 and K 5 are given by the analog of (2.11) found in [21] and (2.3), respectively, i.e.
Thus, by combining the powers of the conformal cross-ratios, G P 5 (2.5) is given by where all the superfluous sums must be appropriately taken care of to reach the result of [20].
Using the last identity of (A.2) for the sum over m 23 leads to We can now make a redefinition of the variable such that s 2 → s 2 + l and re-sum over s 2 using (A.1) again. Expressing C 4 in terms of Pochhammer symbols, the scalar five-point conformal blocks become Changing variables again as in n 11 → n 11 + r 23 + l, we can first evaluate the summation over n 11 using (A.1), and then the summation over l with the help of the second identity in (A.3) after changing k 23 → k 23 − l, and finally replace n 1 = m 1 − r 23 , leading to At this point, we are left with only one extra sum (over r 23 ), as expected. We now use the following relations, to re-sum the summation over r 23 into a 3 F 2 -hypergeometric function, given by With the help of the identity (A.3), this can be rewritten as which translates into the result of [20], i.e.
This computation shows that the results of [20] and [21] are equivalent. Moreover, once a choice of conformal cross-ratios has been made, the OPE approach does lead to the correct result, after several re-summations.
The steps highlighted here can be repeated to obtain the scalar five-point conformal blocks in the comb channel discussed in (2.12). The proof of their equivalence at the level of (A.4) and (2.12) follows the one for the scalar six-point conformal blocks in the snowflake channel shown in Appendix B.

B. Snowflake and the OPE
In this appendix we expound the proof of the scalar six-point conformal blocks in the snowflake channel. We also present an alternative form for the scalar six-point conformal blocks in the snowflake channel, and prove that it is equivalent to the one introduced in the main text.

B.1. Proof of the Snowflake
Starting from the scalar five-point correlation functions (2.12) , the legs and the conformal cross-ratios transform into Using the recurrence relation (2.6) with k = 4, l = 5, and m = 5, it is clear that the resulting scalar six-point correlation functions are in the snowflake channel. To proceed, we must act with the OPE differential operator as in (2.  , such that the action of the OPE differential operator on I 5 (2.12) gives G 6 = n 11 s 11 n 12 s 12 × (−1) s 11 +s 12 +s 22 +l 2 +l 3 +l 4 +m 11 +m 12 +m 22 +m 13 +m 23 +m 33 s 12 − r 45 −r 4 − l 4 m 33 with the proper legs (2.13) and m 3 =r 5 +r. Here again, we omit the indices of summation on the sum for notational simplicity. Equation (B.3) corresponds to the scalar six-point conformal blocks in the snowflake channel, and we now aim to re-sum as many superfluous sums as possible to get to our final results (2.14) and (2.15).
After changing the variables by s 22 → s 22 + m 22 − k 22 and s 11 → s 11 + m 11 − k 11 , and also defining s 22 = s − s 11 , we can evaluate the summation over s 11 , leading to Using the following identity, and changing the index of summation s by s → s+m 23 +r 23 −j, the sum over s can be performed, leading to With n 22 = n − n 11 , it is now possible to sum over n 11 , n, j, k 11 , and k 22 , giving after redefining We are thus left with three extra sums, two sums over t 1 and t 2 , respectively, and one sum from F 5 , which is given by

Using (A.3) twice such that
and re-summing over t 2 using (A.1) gives (2.14) and (2.15) (with t 1 → t 2 and the index of summation from the 3 F 2 -hypergeometric function chosen to be t 1 ). This completes the proof of the scalar six-point conformal blocks in the snowflake channel.
To express F 6 in terms of Kampé de Fériet functions [22], which are defined as where (a) m+n = (a 1 ) m+n · · · (a p ) m+n , we first rewrite (2.15) as by re-summing over t 2 . Using (A.3) on the 3 F 2 -hypergeometric function, expanding the resulting 3 F 2 -hypergeometric function, and combining Pochhammer symbols, we get to as stated in (2.15).

B.2. An Alternative Form
Instead of starting from the scalar five-point correlation functions (2.12), it is also possible to repeat the steps above starting from (A.4). With the legs and the conformal cross-ratios L * 6 = η 13 η 12 η 23 with F * 6 given by (2.15). We note here that although the alternative forms for F M are the same (F * 6 = F 6 ), the alternative forms for C M (B.6) and (2.14) are not the same function (C * 6 = C 6 ). However, the two results must be equivalent as shown below.
First, it is easy to see that the conformal cross-ratios (B.5) and (2.13)  .
Hence, since the scalar six-point correlation functions I 6 must be the same, we get the identity for the scalar six-point conformal blocks in the snowflake channel (2.5). To prove (B.7), we reexpress G * v6 in terms of the conformal cross-ratios for G 6 , expand in the conformal cross-ratios of the latter, and evaluate the superfluous sums.
Using the fact that F * 6 = F 6 , we first obtain in terms of the conformal cross-ratios (2.13). Expanding in terms of u 6 a and 1 − v 6 ab , G * v6 becomes We only need to evaluate the extra sums now to recover C 6 (2.14) and thus prove (B.7).
We start by observing that the terms containing k 11 are given by Thus the summation over k 11 can be done with the help of (A.1). The same can be said for the summations over all of the k ab , except for k 23 . 11 After these steps, G * v6 becomes We can now proceed with the summation over m 11 , getting to To evaluate the sums over k12 and k13, we must first change the variables by k12 → k12 +m ′ 11 and k13 → k13 +m ′ 23 , respectively.
To evaluate the summation over k 23 , we first change the variable by k 23 → k 23 + m ′ 33 . Then, using the following identity we can re-sum over k 23 , leading to At this point, we sum over m 22 and m 12 after we change the variable such that m 12 → m 12 + m ′ 11 . This gives Changing the variable by m 23 → m 23 + m ′ 33 + j and then redefining m 23 = m − m 33 , we can finally perform the re-summations over m 33 , j, m, and m 13 (where we first make the substitution m 13 → m 13 + m ′ 23 ). Redefining m ′ ab by m ab , we are thus left with G * v6 = G 6 , completing the proof.

C. Symmetry Properties
This appendix presents the proofs of the symmetry properties of the scalar six-point conformal blocks in the snowflake channel, which are generated by rotations, reflections and dendrite permutations as in (3.1), (3.2), and (3.3), respectively. Before proceeding, we first verify that they indeed generate (Z 2 ) 3 ⋊ D 3 , the semi-direct product of (Z 2 ) 3 (for dendrite permutations) and the dihedral group of order six.
By defining the action of the symmetry generators on the external quasi-primary operators as R for the rotation (3.1), S for the reflection (3.2), and P for the permutation (3.3), it is easy to see that the dihedral part of symmetry group of the snowflake diagram has for presentation r, s|r 3 = s 2 = (rs) 2 = 1 , (C.1) with r = R −1 and s = S. The presentation (C.1) corresponds to D 3 , with r and s representing rotations by 2π/3 and reflections with respect to one of the three different axes, respectively. The (Z 2 ) 3 = Z 2 × Z 2 × Z 2 part of the symmetry group is generated by p 1 = P , p 2 = R −1 P R, and It is trivial to check that the p i 's commute and that they correspond to dendrite permutations. To exclude the direct nature of the product, it suffices to observe that the generators p i do not commute with the generators r and s. Having excluded (Z 2 ) 3 × D 3 , it is easy to verify that the snowflake diagram has the symmetry group (Z 2 ) 3 ⋊ D 3 of order 48. 12

C.1. Rotations of the Triangle
To prove invariance under the rotation generator R (3.1), it is only necessary to check that (2.15) verifies To simplify the notation, we denote (C.2) as F 6 = F 6R .
First, we rewrite (2.15) as and then use the identity to express F 6 as We now modify the sum over t 1 with the help of (A.3), This is the simplest form to check the invariance of F 6 under the reflection generator (3.2). We can use (A.3) again to obtain It is now possible to expand the 3 F 2 -hypergeometric function as a summation over t 1 and evaluate the sum over t 2 with the help of (A.1), leading to Expressing the sum over t 3 as a 3 F 2 -hypergeometric function, we get Using (A.3) once more leads to where in the last equality we simply changed the order of the two bottom parameters of the which is nothing else than F 6 = F 6R , completing the proof of (C.2) and (3.1).

C.2. Reflections of the Triangle
Invariance under the reflection generator S implies the identity (3.2), which we rewrite as G 6 = G 6S to simplify the notation. Before proceeding, we observe that, under the reflection generator, F 6 → F 6 from the definition (C.3). Expressing G 6S in terms of the original conformal cross-ratios and expanding, we get where we must re-sum all the extra sums.
We start by evaluating the summations over k 11 , k 12 , and k 23 using (A.1), getting to We then change the variables by k 13 → k 13 + m ′ 33 , k 22 → k 22 + m ′ 22 , and k 33 → k 33 + m ′ 13 , and use the identity to re-sum over k 13 , leading to Similarly, using the identity we can compute the sum over k 33 and we get Once again, we introduce and sum over k 22 to obtain
We then change the variable m 13 by m 13 → m 13 + m ′ 33 + j 1 and compute the summation over j 1 . As a result, we get We now redefine m 23 = n − m 13 and evaluate the summation over m 13 with (A.1), which implies We are thus left with two extra sums (over m 12 and j 2 ). However, they are both trivial.