Landau diagrams in AdS and S-matrices from conformal correlators

TL;DR

This work proposes a direct, position-space recipe for extracting flat-space S-matrices from boundary CFT correlators of QFTs in AdS, by mapping boundary positions to on-shell momenta and identifying conformal cross ratios with Mandelstam invariants. It shows how bulk-boundary and bulk-bulk propagators reduce to their flat-space counterparts in the $R\to\infty$ limit, and defines two related conjectures: an S-matrix conjecture (including momentum-conserving delta functions) and an Amplitude conjecture (normalized connected correlators). The analysis reveals regions where the limit fails due to AdS Landau-type singularities, motivates the concept of AdS Landau diagrams, and connects the position-space approach to Mellin-space and conformal-block formalisms, including a nonperturbative route to phase shifts via large-$\Delta$ conformal blocks. The results illuminate how conformal data encode scattering amplitudes, provide criteria for unitarity and analyticity in the flat-space limit, and open avenues for extending to massless theories and more general AdS/CFT settings.

Abstract

Quantum field theories in AdS generate conformal correlation functions on the boundary, and in the limit where AdS is nearly flat one should be able to extract an S-matrix from such correlators. We discuss a particularly simple position-space procedure to do so. It features a direct map from boundary positions to (on-shell) momenta and thereby relates cross ratios to Mandelstam invariants. This recipe succeeds in several examples, includes the momentum-conserving delta functions, and can be shown to imply the two proposals in arXiv:1607.06109 based on Mellin space and on the OPE data. Interestingly the procedure does not always work: the Landau singularities of a Feynman diagram are shown to be part of larger regions, to be called `bad regions', where the flat-space limit of the Witten diagram diverges. To capture these divergences we introduce the notion of Landau diagrams in AdS. As in flat space, these describe on-shell particles propagating over large distances in a complexified space, with a form of momentum conservation holding at each bulk vertex. As an application we recover the anomalous threshold of the four-point triangle diagram at the boundary of a bad region.

Figures (26)

• Figure 1: Analytic continuation of the boundary points. We start from a CFT on $S^{d}$ and analytically continue the insertion points of operators to complex values in order to recover the flat-space S-matrix. Geometrically this corresponds to going from a sphere to a hyperboloid.
• Figure 2: $(X_1-X_2)^2\ll R^2$
• Figure 3: $(X_1-X_2)^2\sim R^2$
• Figure 5: A 'cylinder with caps' configuration discussed in Hijano:2019qmi. We consider two Euclidean hemispheres and connect them by a Lorentzian cylinder of length $\pi$. We then insert two operators on the upper cap and the remaining two operator on the lower cap. The right figure shows a configuration of operator when viewed from the bottom of the lower cap. The angle $\theta$ depicted in the figure becomes a scattering angle in the flat-space limit.
• Figure 6: