Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere

TL;DR

This work develops a framework to recast 4D flat-space scattering into a 2D conformal structure on the celestial sphere by constructing conformal primary wavefunctions for massive scalars that transform covariantly under SL(2,C). The authors provide an explicit integral construction using the H3 bulk-to-boundary propagator, establishing SL(2,C) covariance at the level of amplitudes and clarifying the mass- and dimension-label separation. In the near-extremal three-point case, they show the amplitude reduces to the standard 2D CFT primary three-point function with a computable coefficient obtained via a Witten-like diagram on hyperbolic slices, highlighting a holographic flavor of flat-space scattering. The results illuminate how 4D amplitudes can mirror 2D conformal correlators on the celestial sphere, offering a concrete bridge toward flat-space holography and potential extensions to broader classes of fields and interactions.

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.