Towards Spinning Mellin Amplitudes

TL;DR

This work develops a spinning Mellin amplitude framework for four-point conformal correlators with external spinning operators by focusing on symmetric traceless exchanges. It extends the scalar Mellin formalism through discrete spin variables $a_{ij},b_{ij}$ and introduces a generalized Mack polynomial $\tilde{P}_{\nu,J}^{({\bf n}_L,{\bf n}_R)}$, enabling a kinematical basis for spinning partial waves via ${\mathcal{M}}_{\nu,J}^{({\bf n}_L,{\bf n}_R)}(s,t;a_{ij},b_{ij})$ and a spectral function $b^{({\bf n}_L,{\bf n}_R)}_J(\nu)$. The construction leverages AdS/CFT/Witten diagram techniques, including geodesic Witten diagrams, and generalizes the Symanzik star-formula to include discrete spin data, ensuring that double-trace poles are appropriately separated or canceled, leaving a clean single-trace spinning Mellin amplitude. The framework reduces in 3D to the full spinning Mellin amplitude for symmetric traceless exchanges and provides a systematic, tensor-structure-aware basis for computing spinning conformal partial waves in Mellin space, with potential extensions to higher-point functions and broader representations. Overall, the paper advances practical computation and conceptual understanding of spinning CFT correlators and their holographic duals in Mellin space.

Abstract

We construct the Mellin representation of four point conformal correlation function with external primary operators with arbitrary integer spacetime spins, and obtain a natural proposal for spinning Mellin amplitudes. By restricting to the exchange of symmetric traceless primaries, we generalize the Mellin transform for scalar case to introduce discrete Mellin variables for incorporating spin degrees of freedom. Based on the structures about spinning three and four point Witten diagrams, we also obtain a generalization of the Mack polynomial which can be regarded as a natural kinematical polynomial basis for computing spinning Mellin amplitudes using different choices of interaction vertices.

Figures (2)

• Figure 1: Three point geodesic Witten diagram where the orange curve describes the geodesic $\Gamma_{12}$, and the interaction vertex moves only on the geodesics.
• Figure 2: A four point exchange diagram can be constructed by gluing two three point diagrams through the cutting identity of bulk-bulk propagator.