Conformal Basis, Optical Theorem, and the Bulk Point Singularity

TL;DR

This paper explores the conformal basis for wavefunctions in Minkowski space, translating the optical theorem into this basis, and analyzing its implications for scattering amplitudes, conformal block decomposition, and the bulk point singularity in AdS/CFT.

Contribution

It introduces the conformal basis for Minkowski wavefunctions, translating unitarity constraints into this framework and analyzing their impact on scattering amplitudes and AdS/CFT singularities.

Findings

Optical theorem expressed as conformal block decomposition.

Explicit computations of 3- and 4-point amplitudes in (2+1) dimensions.

Connection between massless conformal basis and bulk point singularity.

Abstract

We study general properties of the conformal basis, the space of wavefunctions in $(d+2)$-dimensional Minkowski space that are primaries of the Lorentz group $SO(1,d+1)$. Scattering amplitudes written in this basis have the same symmetry as $d$-dimensional conformal correlators. We translate the optical theorem, which is a direct consequence of unitarity, into the conformal basis. In the particular case of a tree-level exchange diagram, the optical theorem takes the form of a conformal block decomposition on the principal continuous series, with OPE coefficients being the three-point coupling written in the same basis. We further discuss the relation between the massless conformal basis and the bulk point singularity in AdS/CFT. Some three- and four-point amplitudes in (2+1) dimensions are explicitly computed in this basis to demonstrate these results.