From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing

Charlotte Sleight,Massimo Taronna

doi:10.1103/physrevd.104.l081902

Abstract

We present a simple general relation between tree-level exchanges in AdS and dS. This relation allows to directly import techniques and results for AdS Witten diagrams, both in position and momentum space, to boundary correlation functions in dS. In this work we apply this relation to define Mellin amplitudes and a spectral representation for exchanges in dS. We also derive the conformal block decomposition of a dS exchange, both in the direct and crossed channels, from their AdS counterparts. The relation between AdS and dS exchanges itself is derived using a recently introduced Mellin-Barnes representation for boundary correlators in momentum space, where (A)dS exchanges are straightforwardly fixed by a combination of factorisation, conformal symmetry and boundary conditions.

Highlights

The last decade has seen significant progress in our understanding of scattering in anti-de Sitter (AdS) space
Through the AdS=CFT correspondence [1] we can reformulate scattering processes in AdS in terms of correlation functions in conformal field theory (CFT), which are sharply defined by the requirements of conformal symmetry, unitarity and a consistent operator product expansion (OPE)
Numerous highly effective techniques for the study of scattering in AdS have been developed within the conformal bootstrap program, including Mellin amplitudes [2,3], spectral representation [4,5], large spin expansion [6,7,8], dispersion relations, and inversion formulas [9,10,11]

Summary

INTRODUCTION

The last decade has seen significant progress in our understanding of scattering in anti-de Sitter (AdS) space. For Bunch-Davies initial conditions, we present a simple general relation (28) between exchanges in dS and AdS at tree level, which allows us to directly import AdS techniques (both in position and momentum space [29]) to in-in correlators in dS, which we briefly explore in the final section of this paper. The latter scaling dimensions will be kept generic throughout

Bulk-Boundary Propagators

Bulk-Bulk propagators

Spectral representation

Mellin amplitudes

Conformal block decomposition and crossing