The S-matrix bootstrap. Part I: QFT in AdS

Miguel F. Paulos,Joao Penedones,Jonathan Toledo,Balt C. van Rees,Pedro Vieira

doi:10.1007/jhep11(2017)133

Abstract

We propose a strategy to study massive Quantum Field Theory (QFT) using conformal bootstrap methods. The idea is to consider QFT in hyperbolic space and study correlation functions of its boundary operators. We show that these are solutions of the crossing equations in one lower dimension. By sending the curvature radius of the background hyperbolic space to infinity we expect to recover flat-space physics. We explain that this regime corresponds to large scaling dimensions of the boundary operators, and discuss how to obtain the flat-space scattering amplitudes from the corresponding limit of the boundary correlators. We implement this strategy to obtain universal bounds on the strength of cubic couplings in 2D flat-space QFTs using 1D conformal bootstrap techniques. Our numerical results match precisely the analytic bounds obtained in our companion paper using S-matrix bootstrap techniques.

Highlights

This work aims to use the exact same conformal field theory (CFT) structures to constrain non-conformal quantum field theories
In particular we show that our construction allows for the extraction of upper bounds on the residues of poles in a 2-to-2 elastic scattering amplitude of massive particles, which must be obeyed by any unitary two-dimensional Quantum Field Theory (QFT)
By putting a QFT in an AdS background we can define a set of boundary ‘conformal theory’ observables which are near-identical to the correlation functions of a CFT; they

Summary

Introduction

This work aims to use the exact same CFT structures to constrain non-conformal quantum field theories. We provide an alternative connection by expressing the flat-space phase shift directly in terms of (a limit of) the spectrum and OPE coefficients in the boundary correlation functions This formula works only for physical values of the Mandelstam variables but has the significant advantage of making unitarity manifest. In this initial exploration we have focused on two-dimensional QFTs in order to simplify the numerical analysis. We consider it highly nontrivial that such a result for massive QFTs follows from an analysis of conformal crossing symmetry equations

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Conclusion