A Natural Language for AdS/CFT Correlators

TL;DR

The paper argues that Mellin space provides a natural, highly structured framework for AdS/CFT correlators in large $N$ theories, with OPE data appearing as meromorphic poles whose residues are fixed by lower-point correlators. It derives a universal factorization formula across AdS propagators and introduces diagrammatic Mellin-space rules that reconstruct tree-level Witten diagrams from vertices and propagators, with a functional equation ensuring consistency. The authors prove that these rules reproduce the factorization results and satisfy the conformal Casimir-based functional equation, and they show that in the flat-space limit Mellin amplitudes reduce to the bulk S-matrix with standard Feynman rules. This work suggests a holographic, non-perturbative definition of gravitational scattering via the flat-space limit of Mellin amplitudes and points to exciting extensions to loops, higher spins, and more general CFTs.

Abstract

We provide dramatic evidence that `Mellin space' is the natural home for correlation functions in CFTs with weakly coupled bulk duals. In Mellin space, CFT correlators have poles corresponding to an OPE decomposition into `left' and `right' sub-correlators, in direct analogy with the factorization channels of scattering amplitudes. In the regime where these correlators can be computed by tree level Witten diagrams in AdS, we derive an explicit formula for the residues of Mellin amplitudes at the corresponding factorization poles, and we use the conformal Casimir to show that these amplitudes obey algebraic finite difference equations. By analyzing the recursive structure of our factorization formula we obtain simple diagrammatic rules for the construction of Mellin amplitudes corresponding to tree-level Witten diagrams in any bulk scalar theory. We prove the diagrammatic rules using our finite difference equations. Finally, we show that our factorization formula and our diagrammatic rules morph into the flat space S-Matrix of the bulk theory, reproducing the usual Feynman rules, when we take the flat space limit of AdS/CFT. Throughout we emphasize a deep analogy with the properties of flat space scattering amplitudes in momentum space, which suggests that the Mellin amplitude may provide a holographic definition of the flat space S-Matrix.

Figures (8)

• Figure 1: By acting with the conformal Casimir on a Witten diagram with a bulk to bulk propagator, we collapse the propagator into a delta function. We derive the functional equation by looking at this process in Mellin space.
• Figure 2: A pictorial representation of the derivation of the factorization formula.
• Figure 3: Structure of poles of the integrand of the factorization equation \ref{['factorMellin']} in the $c$ complex plane. When $\delta_{LR}=\Delta+2m$, the integration contour is pinched between poles at two places.
• Figure 4: Four-point and five-point Witten diagrams in cubic scalar theory.
• Figure 5: Left (Right): Six-point linear (star) Witten diagram in cubic scalar theory.