Writing CFT correlation functions as AdS scattering amplitudes

TL;DR

This work develops a Mellin-space framework that recasts CFT correlators as AdS scattering amplitudes, enabling a transparent link between AdS dynamics and flat-space physics. By computing Witten diagrams (contact, scalar and graviton exchange, and loops) and proving a universal flat-space limit formula, it shows that Mellin amplitudes encode bulk amplitudes through a beta-type transform in the large-sij regime, with poles dictated by single-trace data in the planar limit. The approach clarifies how two- and three-point functions fix higher-point Mellin structures, and applies the limit to N=4 SYM to connect planar correlators with type IIB string theory in ten dimensions, including explicit constraints on dilaton scattering amplitudes. Overall, the paper provides a practical, bootstrap-friendly framework for higher-dimensional CFTs and a concrete bridge between AdS/CFT and flat-space scattering.

Abstract

We explore the Mellin representation of conformal correlation functions recently proposed by Mack. Examples in the AdS/CFT context reinforce the analogy between Mellin amplitudes and scattering amplitudes. We conjecture a simple formula relating the bulk scattering amplitudes to the asymptotic behavior of Mellin amplitudes and show that previous results on the flat space limit of AdS follow from our new formula. We find that the Mellin amplitudes are particularly useful in the case of conformal gauge theories in the planar limit. In this case, the four point Mellin amplitudes are meromorphic functions whose poles and their residues are entirely determined by two and three point functions of single-trace operators. This makes the Mellin amplitudes the ideal objects to attempt the conformal bootstrap program in higher dimensions.

Figures (7)

• Figure 1: Witten diagram for a tree level $n$-point contact interaction in AdS.
• Figure 2: Witten diagram for a tree level scalar exchange in AdS.
• Figure 3: A tree level scalar exchange in AdS contributing to a $n$-point correlation function.
• Figure 4: One-loop Witten diagram contributing to the $4$-point correlation function.
• Figure 5: Poles of the integrand in (\ref{['qofc']}) in the complex plane of $c_2$. There are 8 infinite sequences of poles labeled by non-negative integers $m$. The integration contour runs along the imaginary axis.