Factorization of Mellin amplitudes

TL;DR

This work develops a robust factorization framework for Mellin amplitudes in conformal field theories, showing how OPE data induces pole-factor residues that decompose into products of lower-point Mellin amplitudes. It generalizes Mellin representations to tensor operators via embedding space and shadow-formalism techniques, and derives factorization formulas first through shadow projectors and then via the conformal Casimir equation, confirming scalar, vector, and spin-2 cases and proposing a general spin-$J$ seed. A key contribution is the explicit flat space limit that relates CFT Mellin amplitudes to bulk scattering amplitudes in AdS, including spinning particles, thereby connecting Mellin-space CFT data to familiar bulk dynamics and suggesting recursion-based approaches for constructing higher-point Mellin amplitudes. The results advance understanding of how bulk locality and higher-spin exchanges manifest in Mellin space, with potential implications for large-$N$ CFTs and Einstein-gravity duals.

Abstract

We introduce Mellin amplitudes for correlation functions of $k$ scalar operators and one operator with spin in conformal field theories (CFT) in general dimension. We show that Mellin amplitudes for scalar operators have simple poles with residues that factorize in terms of lower point Mellin amplitudes, similarly to what happens for scattering amplitudes in flat space. Finally, we study the flat space limit of Anti-de Sitter (AdS) space, in the context of the AdS/CFT correspondence, and generalize a formula relating CFT Mellin amplitudes to scattering amplitudes of the bulk theory, including particles with spin.

Figures (6)

• Figure 1: In a CFT correlation function, one can replace multiple operators inside a sphere by a (infinite) sum of local operators inserted at the center of the sphere.
• Figure 2: The multiple OPE of figure \ref{['fig:OPEconvergence']} leads to the factorization of the residues of the poles of the Mellin amplitude in terms of lower-point Mellin amplitudes.
• Figure 3: Scattering amplitudes have poles when the momentum $p=\sum_{a=1}^kp_a$ approaches the mass shell, $p^2+M^2=0$, of a particle in the theory. The residue of this pole factorizes in terms of lower point scattering amplitudes.
• Figure 4: Scattering amplitude of $k$ scalar particles and one particle with spin. The polarization of the spinning particle is encoded in the null vector $\varepsilon$.
• Figure 5: Euclidean AdS$_{d+1}$ embedded in Minkowski space $\mathbb{M}^{d+2}$. The tangent space $\mathbb{R}^{d+1}$ is a good local approximation to AdS in a region smaller than the AdS radius of curvature.