Towards Feynman rules for conformal blocks

Abstract

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 two new ones: the seven-point block in a new topology, and all 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 beyond those considered in this paper.

Dimensional reduction of higher-point conformal blocks

“…Notwithstanding, recent work on Feynman rules for higher-point conformal blocks [30] will allow us in this paper to establish these relations and obtain the exact coefficients. Notably, in this paper we will derive a simple set of Feynman-like rules for writing down the coefficients which appear in the dimensional reduction of any scalar conformal block of any topology with scalar exchanges.…”

“…Despite being seemingly more complicated objects fixed entirely by conformal symmetry, higher-point blocks share several features and properties with their simpler four-point cousins. For one, just like the four-point block they admit a power series description dictated by a Feynman-like prescription, which can essentially be read off simply from its unique unrooted binary tree graphical representation [30]. For another, they satisfy dimensional reduction relations that are very similar to those obeyed by four-point blocks.…”

“…Over the past two years a spate of new results have appeared in the literature related to higher-point blocks in d dimensions. The techniques often used include the shadow formalism [16], embedding space OPEs [18][19][20][21][22][23][24][25], 1 geodesic Witten diagrams [27][28][29] and Mellin space methods [30]. Most of the explicit results obtained in literature have been for scalar conformal blocks with scalar exchanges.…”

“…Most of the explicit results obtained in literature have been for scalar conformal blocks with scalar exchanges. The program to determine all higher-point scalar blocks recently culminated in a simple graphical Feynman rules-like prescription for directly writing down any n-point block in an arbitrary topology [24,30,31].…”

“…The superscript (t) on the constants emphasizes the dependence on the topology of the block. Naively, determining these constants for every n-point topology appears to be a hopeless task, even if the conformal blocks entering into (2.17) are known, since the number of inequivalent topologies grows rapidly with n. 10 A similar obstacle arose in a related context [24,30]. Until recently, only a few isolated examples of d-dimensional higher-point conformal blocks were known, in terms of power series representations.…”