Momentum space approach to crossing symmetric CFT correlators II: General spacetime dimension

TL;DR

The paper generalizes a momentum-space, crossing-symmetric basis for conformal four-point functions from three dimensions to general spacetime dimensions, leveraging spherical harmonics to handle symmetric traceless tensors. It develops a complete helicity framework by decomposing spinning operators into spin sectors with the Funk-Hecke formula, derives momentum-space two- and three-point functions, and analyzes their analytic properties to define cubic vertices for factorization. The s-channel Polyakov block is then constructed for arbitrary spin using spinning cubic vertices and bulk-to-bulk propagators, with the addition theorem summing over spin components to yield a manifestly crossing-symmetric representation. This framework provides a versatile, dimension-agnostic description of scalar four-point functions and lays the groundwork for incorporating external spinning operators and connections to de Sitter correlators in future work.

Abstract

Our previous work [1] constructed, in three-dimensional momentum space, a manifestly crossing symmetric basis for scalar conformal four-point functions, based on the factorization property proposed by Polyakov. This work extends this construction to general dimensional conformal field theory. To facilitate the treatment of symmetric traceless tensors, we exploit techniques of spherical harmonics in general dimensions.