Landau diagrams in AdS and S-matrices from conformal correlators

TL;DR

This paper presents a boundary-to-momentum mapping in AdS that extracts flat-space S-matrices from conformal correlators, introduces Landau diagrams in AdS to understand divergences, and relates these to flat-space Landau singularities.

Contribution

It introduces a simple position-space procedure to derive S-matrices from AdS correlators and defines Landau diagrams in AdS to analyze divergence regions.

Findings

The boundary-to-momentum map relates cross ratios to Mandelstam invariants.

The procedure captures momentum conservation and reproduces known proposals.

Landau diagrams in AdS describe on-shell propagation and identify divergence regions.

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.