Analyticity and the Holographic S-Matrix

TL;DR

This work proves Penedones' conjecture by relating the Mellin amplitude of a large-N CFT to the bulk flat-space S-matrix via a precise α-transform in the AdS radius. It shows that Mellin amplitudes are meromorphic, with poles dictated by the OPE, and demonstrates how branch cuts and resonances of bulk scattering emerge from summing AdS exchanges and loop diagrams in the flat-space limit. By connecting conformal blocks to bulk S-matrix elements and examining the impact of Hawking evaporation, the paper bridges the conformal bootstrap and S-matrix programs, providing a nonperturbative holographic framework for locality and analyticity in quantum gravity. The results suggest a deep, general link between CFT data, Mellin-space analyticity, and the high-energy behavior of gravitational scattering, with implications for the emergence of bulk locality and the structure of conformal block decompositions at large dimensions.

Abstract

We derive a simple relation between the Mellin amplitude for AdS/CFT correlation functions and the bulk S-Matrix in the flat spacetime limit, proving a conjecture of Penedones. As a consequence of the Operator Product Expansion, the Mellin amplitude for any unitary CFT must be a meromorphic function with simple poles on the real axis. This provides a powerful and suggestive handle on the locality vis-a-vis analyticity properties of the S-Matrix. We begin to explore analyticity by showing how the familiar poles and branch cuts of scattering amplitudes arise from the holographic description. For this purpose we compute examples of Mellin amplitudes corresponding to 1-loop and 2-loop Witten diagrams in AdS. We also examine the flat spacetime limit of conformal blocks, implicitly relating the S-Matrix program to the Bootstrap program for CFTs. We use this connection to show how the existence of small black holes in AdS leads to a universal prediction for the conformal block decomposition of the dual CFT.

Figures (7)

• Figure 1: This figure shows how the AdS cylinder in global coordinates corresponds to the CFT in radial quantization. The time translation operator in the bulk of AdS is the Dilatation operator in the CFT, so energies in AdS correspond to dimensions in the CFT.
• Figure 2: Internal vertex described in the text. Internal vertices with off-shell $\delta_{ij}$'s flowing through them can be obtained by adding on external three-point vertices.
• Figure 3: This figure shows the pole prescription for the contour integrals defining the Mellin representation, as a function of $\delta_{12}$ and $\delta_{13}$, for the very simple 4-pt example in equation (\ref{['eqn:SimplestExample']}). One can see that the contour of integration lies between the poles of the $\Gamma$ functions (black) and the poles of the Mellin amplitude (red) in each variable, including $\delta_{14} = \Delta_\phi - \delta_{12} - \delta_{13}$.
• Figure 4: This figure shows how bulk scattering processes are setup in AdS/CFT. Creating the initial state involves smearing CFT operators over an annulus, which is just $S^{d-1} \times [-\tau, \tau]$ on the cylinder bounding AdS. The integration over space and (dilatation) time in the CFT is necessary to select the direction and magnitude of the bulk momenta, respectively. The regulator $\tau$ can be taken to infinity in the flat spacetime limit. The final state is measured after a time $\pi R$, so that the particles have the opportunity to scatter exactly once.
• Figure 5: This figure shows the 1-loop and 2-loop diagrams that we have computed using the Mellin space version of the Källen-Lehmann representation. Further generalizations are straightforward.