Flat-space Partial Waves From Conformal OPE Densities

TL;DR

This paper explores the flat-space limit of conformal four-point functions, showing how OPE densities relate to scattering partial waves and how conformal inversion formulas simplify to familiar scattering formulas.

Contribution

It demonstrates that the OPE density converges to scattering partial waves in the flat-space limit and connects conformal inversion formulas to standard scattering amplitude formulas.

Findings

OPE density reduces to scattering partial waves in the flat-space limit

Conformal inversion formulas simplify to scattering formulas

Identifies conditions for divergence of the OPE density

Abstract

We consider the behavior of the OPE density $c(\Delta,\ell)$ for conformal four-point functions in the flat-space limit where all scaling dimensions become large. We find evidence that the density reduces to the partial waves $f_\ell(s)$ of the corresponding scattering amplitude. The Euclidean inversion formula then reduces to the partial wave projection and the Lorentzian inversion formula to the Froissart-Gribov formula. The flat-space limit of the OPE density can however also diverge, and we delineate the domain in the complex $s$ plane where this happens. Finally we argue that the conformal dispersion relation reduces to an ordinary single-variable dispersion relation for scattering amplitudes.