On Positive Celestial Geometry: ABHY in the Sky

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.