Notes on flat-space limit of AdS/CFT

TL;DR

This work addresses how to recover flat-space scattering amplitudes from AdS/CFT by unifying multiple representations of the flat-space limit. It shows that the global AdS scattering smearing kernel underpins Mellin space, coordinate space, and partial-wave descriptions, while the Poincare AdS kernel provides the momentum-space formulation, with the limit realized as $\ell \to \infty$. A unified Mellin formula is derived via saddle-point analysis that applies to both massless and massive external states, and this leads naturally to bulk-point structures in coordinate space and to phase-shift relations in the partial-wave expansion. The authors also establish a momentum-coordinate duality linking the momentum-space and global smearing pictures and extend the analysis to spinning correlators, demonstrating maps to photon-like three-point amplitudes. Overall, the paper provides a coherent bridge among diverse flat-space limit frameworks, enabling systematic extraction of flat-space S-matrix data from boundary CFT correlators and extending the results to spinning cases.

Abstract

Different frameworks exist to describe the flat-space limit of AdS/CFT, include momentum space, Mellin space, coordinate space, and partial-wave expansion. We explain the origin of momentum space as the smearing kernel in Poincare AdS, while the origin of latter three is the smearing kernel in global AdS. In Mellin space, we find a Mellin formula that unifies massless and massive flat-space limit, which can be transformed to coordinate space and partial-wave expansion. Furthermore, we also manage to transform momentum space to smearing kernel in global AdS, connecting all existed frameworks. Finally, we go beyond scalar and verify that $\langle VV\mathcal{O}\rangle$ maps to photon-photon-massive amplitudes.

Figures (11)

• Figure 1: The existed frameworks describing the flat-space limit of AdS/CFT, where the question mark denotes the undiscovered relation.
• Figure 2: Massless and massive unified frameworks of the flat-space limit, where the origins are clarified.
• Figure 3: Cylinder diagram of global AdS.
• Figure 4: Poincare AdS only covers a wedge of global AdS. On LHS, the lines marked $\mathcal{B}$ meet the global AdS boundary. $\mathcal{B}$ is the boundary of Poincare AdS where CFT lives. On the RHS, we depict a local figure near $\mathcal{B}$.
• Figure 5: The original integral contour of $z$, as depicted as dotted line, picks up poles denoted as cross at positive integers, which sums to Bessel function. The contour is deformed to pass through the saddle-points in the desired limit.