Higher-Point Conformal Blocks in the Comb Channel

TL;DR

The paper develops $M$-point scalar conformal blocks in the comb channel for arbitrary $M$ and dimension $d$ by an embedding-space OPE approach that recursively builds blocks from lower-point data. It provides a closed-form block representation $G_M^{(d,oldsymbol{h};oldsymbol{p})}(oldsymbol{u}^M,oldsymbol{v}^M)$ with a crucial, dimension- and channel-dependent function $F_M^{(d,oldsymbol{h};oldsymbol{p})}(oldsymbol{m})$ governed by a recurrence Eq. (EqRR) and anchored by $F_4^{(d,oldsymbol{h};oldsymbol{p})}(oldsymbol{m})=1$, ultimately expressible via ${}_3F_2$ hypergeometric sums as in Eq. (EqF). The construction is validated through multiple nontrivial checks: the unit-operator limit reduces $M$-point blocks to $(M-1)$-point blocks, the $M=5$ case matches Rosenhaus (2018zqn) in arbitrary $d$, and the $d o1$ limit aligns with known $d=1$ blocks. These results enable systematic exploration of higher-point conformal structures, with potential impacts on bootstrap methods and AdS/CFT interpretations via higher-point diagrams and geodesic Witten diagrams, and they pave the way for extensions to spinning exchanges and non-comb topologies.

Abstract

We compute $M$-point conformal blocks with scalar external and exchange operators in the so-called comb configuration for any $M$ in any dimension $d$. Our computation involves repeated use of the operator product expansion to increase the number of external fields. We check our results in several limits and compare with the expressions available in the literature when $M=5$ for any $d$, and also when $M$ is arbitrary while $d=1$.

Figures (1)

• Figure 1: Conformal blocks in the comb channel.