Towards Feynman rules for Mellin amplitudes in AdS/CFT

TL;DR

<3-5 sentence high-level summary> This work develops a framework combining the embedding formalism and Mellin transforms to compute AdS/CFT tree-level correlators, and it provides strong evidence for a set of Feynman rules governing scalar Mellin amplitudes, with general vertices expressed via Lauricella functions. It further shows how to handle spin by factorizing tensor structures with projector-like operators D^{MA}, enabling clean expressions for current and stress-tensor amplitudes and a Yang-Mills current exchange example. The authors explicitly compute higher-point scalar amplitudes up to 12 points and demonstrate that the Mellin amplitudes decompose into sums over descendants and propagators, matching the proposed rules. They discuss conformal invariance of index structures, compare with flat-space intuition, and outline clear directions for extensions to loops and higher-spin dynamics with potential links to a higher-dimensional momentum picture.

Abstract

We investigate the use of the embedding formalism and the Mellin transform in the calculation of tree-level conformal correlation functions in $AdS$/CFT. We evaluate 5- and 6-point Mellin amplitudes in $φ^3$ theory and even a 12-pt diagram in $φ^4$ theory, enabling us to conjecture a set of Feynman rules for scalar Mellin amplitudes. The general vertices are given in terms of Lauricella generalized hypergeometric functions. We also show how to use the same combination of Mellin transform and embedding formalism for amplitudes involving fields with spin. The complicated tensor structures which usually arise can be written as certain operators acting as projectors on much simpler index structures - essentially the same ones appearing in a flat space amplitude. Using these methods we are able to evaluate a four-point current diagram with current exchange in Yang-Mills theory.