Flat space spinning massive amplitudes from momentum space CFT

TL;DR

This paper explores how CFT correlators in momentum space transition to flat space scattering amplitudes in the limit of large AdS radius and operator dimensions, exemplified through a 3-point function involving a conserved current and massive vectors.

Contribution

It provides a detailed method to obtain flat space scattering amplitudes from CFT correlators using the momentum space representation and triple-K integrals, with explicit calculations for spin-1 fields.

Findings

Flat space limit of CFT correlators reproduces scattering amplitudes.

Triple-K integrals simplify to delta functions in the flat space limit.

Method applied to electromagnetic form factors of massive vectors.

Abstract

We discuss the flat space limit of AdS using the momentum space representation of CFT correlators. The flat space limit involves sending the AdS radius and the dimensions of operators dual to massive fields to infinity while also scaling appropriately the sources of the dual operators. In this limit, d-dimensional CFT correlators become (d+1)-dimensional scattering amplitudes. We exemplify our discussion with the computation of the flat-space limit of the CFT 3-point function of a conserved current, a non-conserved charged vector operator and its conjugate. The flat-space limit should yield the scattering amplitude of an Abelian gauge field with two massive vector fields. This scattering amplitude computes the electromagnetic form factors of the electromagnetic current in a spin-1 state, and these form factors encode the electromagnetic properties of the massive vector field (charge, magnetic moment and quadruple moment). In terms of the CFT, the flat-space limit amounts to zooming in the infrared region of the triple-K integrals that determine the 3-point function, while also scaling to infinity the order of (some of) the Bessel functions that feature in the triple-K integrals. In this limit the triple-K integral becomes proportional to the energy-preserving delta function, and the flat space limit correctly yields the corresponding flat space scattering amplitude in complete detail.