Abstract
Abstract The bulk S-Matrix can be given a non-perturbative definition in terms of the flat space limit of AdS/CFT. We show that the unitarity of the S-Matrix, ie the optical theorem, can be derived by studying the behavior of the OPE and the conformal block decomposition in the flat space limit. When applied to perturbation theory in AdS, this gives a holographic derivation of the cutting rules for Feynman diagrams. To demonstrate these facts we introduce some new techniques for the analysis of conformal field theories. Chief among these is a method for conglomerating local primary operators $ {{\mathcal{O}}_1} $ and $ {{\mathcal{O}}_2} $ to extract the contribution of an individual primary $ {{\mathcal{O}}_{{\varDelta, \ell }}} $ in their OPE. This provides a method for isolating the contribution of specific conformal blocks which we use to prove an important relation between certain conformal block coefficients and anomalous dimensions. These techniques make essential use of the simplifications that occur when CFT correlators are expressed in terms of a Mellin amplitude.
Highlights
CFT correlation functions has led to progress [6,7,8,9,10] along these lines, and in particular, we recently argued [9] that bulk locality can be understood by showing how the meromorphy of the Mellin amplitude1 leads to an analytic S-Matrix
The simplest possible AdS interaction that affects only scalar conformal blocks is a λφ4 interaction, which corresponds to a Mellin amplitude that is just a constant
As we discussed in [9], conformal blocks and tree-level AdS exchanges have identical poles and residues in the Mellin amplitude; they differ only in their asymptotic behavior at large δij. This means that we can immediately read off the coefficients in the conformal block decomposition of the correlator corresponding to this 1-loop Mellin amplitude in the [OχOχ]n,0 channel, it is where
Summary
The goal of this section will be to understand how to ‘conglomerate operators’ in order to combine a pair of local primary operators into a third composite operator that appears in the OPE of the first two. We will be able to use these techniques to extract specific terms from the conformal block expansion
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