Conformal correlators as simplex integrals in momentum space

Bzowski, Adam; McFadden, Paul; Skenderis, Kostas

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.

Citation

Bzowski, A., McFadden, P., & Skenderis, K. (2021). Conformal correlators as simplex integrals in momentum space. Journal of High Energy Physics, 192, https://doi.org/10.1007/JHEP01%282021%29192

• Journal Article Type: Article
• Acceptance Date: Dec 19, 2020
• Online Publication Date: Jan 28, 2021
• Publication Date: 2021
• Deposit Date: Feb 19, 2025
• Journal: JHEP
• Print ISSN: 1126-6708
• Electronic ISSN: 1029-8479
• Publisher: Scuola Internazionale Superiore di Studi Avanzati (SISSA)
• Peer Reviewed: Peer Reviewed
• Volume: 192
• DOI: https://doi.org/10.1007/JHEP01%282021%29192
• Public URL: https://durham-repository.worktribe.com/output/2981183
• Additional Information: Available open access via DOI