Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere

TL;DR

This paper constructs conformal primary wavefunctions in 4D Minkowski space that transform under the Lorentz group as 2D conformal primaries, revealing a direct link between 4D scattering amplitudes and 2D conformal correlators.

Contribution

It introduces on-shell massive scalar wavefunctions in 4D Minkowski space with $SL(2, ext{C})$ conformal transformation properties, enabling covariant scattering amplitudes and explicit three-point function computations.

Findings

Constructed $SL(2, ext{C})$ conformal primary wavefunctions.

Demonstrated three-point amplitude reduces to 2D CFT form.

Computed coefficients using Witten-like diagrams with holographic flavor.

Abstract

The four-dimensional (4D) Lorentz group $SL(2,\mathbb{C})$ acts as the two-dimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz $SL(2,\mathbb{C})$. In this paper we construct on-shell massive scalar wavefunctions in 4D Minkowski space that transform as $SL(2,\mathbb{C})$ conformal primaries. Scattering amplitudes of these wavefunctions are $SL(2,\mathbb{C})$ covariant by construction. For certain mass relations, we show explicitly that their three-point amplitude reduces to the known unique form of a 2D CFT primary three-point function and compute the coefficient. The computation proceeds naturally via Witten-like diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.