Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond

TL;DR

This paper computes scalar seven-point conformal blocks in the extended snowflake channel in arbitrary dimensions, expanding the understanding of complex topologies in conformal field theories using hypergeometric functions.

Contribution

It introduces a new method to compute seven-point conformal blocks in the extended snowflake channel, extending previous work on simpler topologies.

Findings

Derived power series expansion for conformal blocks.

Expressed blocks in terms of hypergeometric functions and Kampé de Fériet functions.

Verified symmetry properties and consistency with known limits.

Abstract

Seven-point functions have two inequivalent topologies or channels. The comb channel has been computed previously and here we compute scalar conformal blocks in the extended snowflake channel in $d$ dimensions. Our computation relies on the known action of the differential operator that sets up the operator product expansion in embedding space. The scalar conformal blocks in the extended snowflake channel are obtained as a power series expansion in the conformal cross-ratios whose coefficients are a triple sum of the hypergeometric type. This triple sum factorizes into a single sum and a double sum. The single sum can be seen as originating from the comb channel and is given in terms of a ${}_3F_2$-hypergeometric function, while the double sum originates from the snowflake channel which corresponds to a Kamp\'e de F\'eriet function. We verify that our results satisfy the symmetry properties of the extended snowflake topology. Moreover, we check that the behavior of the extended snowflake conformal blocks under several limits is consistent with known results. Finally, we conjecture rules leading to a partial construction of scalar $M$-point conformal blocks in arbitrary topologies.