On Positive Celestial Geometry: ABHY in the Sky

TL;DR

This paper introduces a geometric framework for celestial amplitudes using associahedra, providing a unified and boundary-geometry-based interpretation of massless scattering in high-dimensional space-times.

Contribution

It performs all Mellin integrations for arbitrary multiplicity and dimension, revealing associahedron structures in celestial space for the first time.

Findings

Celestial tree-level $$ amplitudes are described as canonical forms of associahedra.

Distributional support on the celestial sphere emerges naturally from the geometry.

The framework unifies celestial amplitudes from various bulk theories like scalars, gluons, and gravitons.

Abstract

Celestial amplitudes are multiple Mellin transforms w.r.t. conformal dimensions. For arbitrary multiplicity $n$ of massless states in sufficiently high space--time dimension $D$ we perform all Mellin integrations and find an associahedron description in celestial space. The latter expresses celestial tree--level $\phi^3$ amplitudes as the canonical forms associated with this positive geometry. This yields a geometric interpretation of celestial amplitudes in terms of the underlying boundary geometry. In particular, distributional support on the celestial sphere is not imposed but arises geometrically. Our universal treatment of Mellin integrals in $D$ dimensions also provides a unified description of celestial amplitudes arising from different bulk theories, including (scalar-scaffolded) gluons and gravitons.