On Feynman rules for Mellin amplitudes in AdS/CFT
Dhritiman Nandan, Anastasia Volovich, Congkao Wen
TL;DR
This work derives the tree-level Mellin-space Feynman rules for a scalar $\phi^n$ theory in AdS/CFT by explicitly evaluating all relevant Witten diagrams. It shows that the Mellin amplitude factorizes into vertex factors involving Lauricella functions, with internal-line sums over $m_i$ and momentum assignments governed by $k_i$, $\Delta_i$, and conservation laws, and confirms the rules reproduce the familiar flat-space Feynman rules in the large AdS-radius limit. The derivation proceeds from maximal off-shell vertices to general diagrams using Symanzik star formulas and a systematic treatment of the $s_i$, $\bar{s}_i$, and $c_i$ integrals, culminating in a consistent Mellin-integrand structure $\mathcal{M}(k_i,c_i)$. The results reinforce the Mellin-space approach as a holographic definition of scattering amplitudes and lay groundwork for extensions to spinning fields and loops.
Abstract
The computation of CFT correlation functions via Witten diagrams in AdS space can be simplified via the Mellin transform. Recently a set of Feynman rules for tree-level Mellin space amplitudes has been proposed for scalar theories. In this note we derive these rules by explicitly evaluating all of the relevant Witten diagram integrals for the scalar phi^n theory. We also check that the rules reduce to the usual Feynman rules in the flat space limit.