Landau diagrams in AdS and S-matrices from conformal correlators
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Springer
Received: August 6, 2020 Accepted: October 1, 2020 Published: November 10, 2020
Shota Komatsu,a Miguel F. Paulos,b Balt C. van Reesc and Xiang Zhaoc a
School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, U.S.A. b Laboratoire de Physique de l’Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, 75231 Paris Cedex 05, France c Centre de Physique Théorique (CPHT), Ecole Polytechnique, 91128 Palaiseau Cedex, France
E-mail: [email protected], [email protected], [email protected], [email protected] 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 [1] 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. Keywords: AdS-CFT Correspondence, Conformal Field Theory, Field Theories in Higher Dimensions, Scattering Amplitudes ArXiv ePrint: 2007.13745
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)046
JHEP11(2020)046
Landau diagrams in AdS and S-matrices from conformal correlators
Contents 1 Introduction
1 3 3 4 6 7 10 12 12 14 15
3 Illustrative examples 3.1 Two-point functions 3.2 Contact diagram and momentum conservation 3.2.1 Vertex momenta and vertex Mandelstam invariants 3.2.2 Boundary momenta vs. vertex momenta 3.2.3 General momenta, n = 3 3.2.4 General momenta, n = 4 3.3 Exchange diagram and geodesic networks 3.3.1 Contribution from the saddle 3.3.2 Contribution from the pole 3.3.3 Exchange of dominance
17 17 18 18 20 21 22 24 24 26 29
4 Landau diagrams in AdS 4.1 Comparison with flat space Landau diagrams 4.2 Anomalous thresholds and the triangle diagram
30 32 35
5 Mellin space 5.1 The saddle point 5.2 The steepest descent contour 5.3 The exchange diagram revisited 5.4 A bound on anomalous thresholds?
37 38 40 41 43
6 S-matrices from conformal block expansions 6.1 Preliminaries 6.2 Conformal block expansion at large ∆O 6.2.1 Identity and bound states 6.3 The phase shift formula
45 45
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