Conformal $n$-point functions in momentum space

TL;DR

This paper introduces a new Feynman integral representation for scalar n-point functions in conformal field theories, solving conformal Ward identities and revealing structures of momentum-space conformal cross-ratios.

Contribution

It provides a general integral representation for momentum-space scalar n-point functions in conformal field theories, including explicit analysis for 4-point functions and their singularities.

Findings

Representation solves conformal Ward identities.

Identifies conditions for singularities, anomalies, and beta functions.

Includes examples from perturbative QFT and holography.

Abstract

We present a Feynman integral representation for the general momentum-space scalar $n$-point function in any conformal field theory. This representation solves the conformal Ward identities and features an arbitrary function of $n(n-3)/2$ variables which play the role of momentum-space conformal cross-ratios. It involves $(n-1)(n-2)/2$ integrations over momenta, with the momenta running over the edges of an $(n-1)$-simplex. We provide the details in the simplest non-trivial case (4-point functions), and for this case we identify values of the operator and spacetime dimensions for which singularities arise leading to anomalies and beta functions, and discuss several illustrative examples from perturbative quantum field theory and holography.