Momentum space approach to crossing symmetric CFT correlators

TL;DR

This paper constructs a crossing symmetric basis for conformal four-point functions in momentum space, leveraging factorization properties similar to scattering amplitudes, and extends Polyakov's bootstrap approach to include operators with spin.

Contribution

It explicitly constructs the momentum space crossing symmetric basis for scalar four-point functions with spinning intermediate operators, filling a gap in the conformal bootstrap framework.

Findings

Explicit basis for scalar four-point functions with spin

Manifest separation of connected and disconnected correlators

Analytic expressions for three-point functions with tensors

Abstract

We construct a crossing symmetric basis for conformal four-point functions in momentum space by requiring consistent factorization. Just as scattering amplitudes factorize when the intermediate particle is on-shell, non-analytic parts of conformal correlators enjoy a similar factorization in momentum space. Based on this property, Polyakov, in his pioneering 1974 work, introduced a basis for conformal correlators which manifestly satisfies the crossing symmetry. He then initiated the bootstrap program by requiring its consistency with the operator product expansion. This approach is complementary to the ordinary bootstrap program, which is based on the conformal block and requires the crossing symmetry as a consistency condition of the theory. Even though Polyakov's original bootstrap approach has been revisited recently, the crossing symmetric basis has not been constructed explicitly in momentum space. In this paper we complete the construction of the crossing symmetric basis for scalar four-point functions with an intermediate operator with a general spin, by using new analytic expressions for three-point functions involving one tensor. Our new basis manifests the analytic properties of conformal correlators. Also the connected and disconnected correlators are manifestly separated, so that it will be useful for the study of large $N$ CFTs in particular.