Distributional Celestial Amplitudes

Abstract

Scattering amplitudes are tempered distributions, which are defined through their action on functions in the Schwartz space $S(\mathbb{R})$ by duality. For massless particles, their conformal properties become manifest when considering their Mellin transform. Therefore we need to mathematically well-define the Mellin transform of distributions in the dual space $S^{\prime }({\mathbb{R}}^{+})$. In this paper, we investigate this problem by characterizing the Mellin transform of the Schwartz space $S({\mathbb{R}}^{+})$. This allows us to rigorously define the Mellin transform of tempered distributions through a Parseval-type relation. Massless celestial amplitudes are then properly defined by taking the Mellin transform of elements in the topological dual of the Schwartz space $S({\mathbb{R}}^{+})$. We conclude the paper with applications to tree-level graviton celestial amplitudes.

27 pages, 1 figure