Conformal correlators as simplex integrals in momentum space

TL;DR

This paper derives a general solution for scalar conformal correlators in momentum space using simplex integrals, revealing recursive structures and simplifying the representation of contact Witten diagrams in holography.

Contribution

It introduces a novel simplex integral representation for conformal correlators in momentum space, with recursive properties and improved integration efficiency over Mellin representations.

Findings

Provides a general solution to conformal Ward identities in momentum space.

Establishes a recursive structure for n-point functions based on (n-1)-point functions.

Simplifies the integral representation of holographic contact diagrams with fewer integrations.

Abstract

We find the general solution of the conformal Ward identities for scalar $n$-point functions in momentum space and in general dimension. The solution is given in terms of integrals over $(n-1)$-simplices in momentum space. The $n$ operators are inserted at the $n$ vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where $n$-point functions are built in terms of $(n-1)$-point functions. To illustrate our discussion, we derive the simplex representation of $n$-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves $(n-2)$ integrations, which is an improvement (when $n>4$) relative to the Mellin representation that involves $n(n-3)/2$ integrations.