Distributional Celestial Amplitudes

TL;DR

This paper rigorously defines the Mellin transform of distributions in Schwartz space to properly formulate massless celestial amplitudes, with applications to graviton scattering at tree level.

Contribution

It characterizes the Mellin transform of Schwartz space distributions, enabling a rigorous definition of celestial amplitudes for massless particles.

Findings

Rigorous mathematical framework for Mellin transforms of distributions.

Proper definition of massless celestial amplitudes.

Application to tree-level graviton 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 $S'(\mathbb{R}^+)$. 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.