Conformal basis, optical theorem, and the bulk point singularity

We study general properties of the conformal basis, the space of wave functions 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 operator product expansion (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.