Towards Feynman rules for Mellin amplitudes in AdS/CFT

TL;DR

This paper develops a set of Feynman rules for scalar Mellin amplitudes in AdS/CFT, enabling efficient computation of higher-point functions and amplitudes involving spinning fields, with potential applications to Yang-Mills theory.

Contribution

It introduces conjectured Feynman rules for scalar Mellin amplitudes in AdS/CFT, including vertices expressed via hypergeometric functions and a method to handle spinning fields.

Findings

Evaluated 5- and 6-point Mellin amplitudes in scalar theories.

Derived Feynman rules for Mellin amplitudes involving spins.

Successfully computed a four-point current diagram in Yang-Mills theory.

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 $\phi^3$ theory and even a 12-pt diagram in $\phi^4$ 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.