Flat space amplitudes and conformal symmetry of the celestial sphere

Abstract

The four-dimensional (4D) Lorentz group SL(2, 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, C). In this paper we construct on-shell massive scalar wavefunctions in 4D Minkowski space that transform as SL(2, C) conformal primaries. Scattering amplitudes of these wavefunctions are SL(2, 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.1 The subleading soft theorem has a one-loop exact anomaly [9-12] whose effects remain to be understood but are recently discussed in [13,14].2 One may hope that ultimately 4D quantum gravity scattering amplitudes are found to have a dual holographic representation as some exotic 2D CFT on CS 2 , but at present there are no proposals for such a construction.

S-matrix singularities and CFT correlation functions

“…The plausibility of eq. (2.7) stems from the two sides sharing the same symmetry, as verified in [13], for the case of φ 2 ϕ interaction, where m ϕ ∼ 2m φ .…”

“…The explicit example shown in [13] is a three-point contact term, and the mass of one particle is near-extremal, i.e. it is near the sum of the other two.…”

“…The wave functions are constructed by convoluting plane waves with bulk-to-boundary propagators in AdS 3 , and thus transforms covariantly under the Mobius group [13]. Using this transformation allows one to transform Lorentz invariant scalar scattering amplitudes into Mobius covariant quantities, which can be considered as the correlation functions of some 2d CFT.…”

“…Recently Pasterski, Shao and Strominger (PSS) [13] presented a proposal for such a map, where as a first step, the four-dimensional massive momenta R 1,3 is mapped into the coordinates of hyperbolic space H 3 . Taking the metric of H 3 as, …”

BMS symmetry of celestial OPE

“…Celestial CFT is conjectured to be the holographic dual of quantum gravity in asymptotically flat space-time [4][5][6][7][8]. The observables of the celestial CFT are related to Mellin transformations of flat space scattering amplitudes [19][20][21][22][23][24][25][26].…”