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Distributional Celestial Amplitudes

Yorgo Pano, Majdouline Borji

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.

Distributional Celestial Amplitudes

Abstract

Scattering amplitudes are tempered distributions, which are defined through their action on functions in the Schwartz space 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 . In this paper, we investigate this problem by characterizing the Mellin transform of the Schwartz space . 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 . We conclude the paper with applications to tree-level graviton celestial amplitudes.
Paper Structure (10 sections, 9 theorems, 78 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Mellin Transform of the Schwartz Space
  3. The Inversion Formula for the Mellin Transform
  4. The Mellin transform of the Schwartz space $\mathcal{S}(\mathbb{R}^+)$
  5. The Mellin Transform of Tempered Distributions
  6. Parseval's Relation for the Mellin Transform
  7. Applications: Graviton Celestial Amplitudes
  8. Three-Graviton Celestial Amplitude
  9. Four-Graviton Celestial Amplitude
  10. Conclusion

Key Result

Proposition 1

Given $\phi$ in $\mathcal{S}(\mathbb{R}^+)$, the following properties hold:

Figures (1)

  • Figure 1: This figure represents the complex $\Delta$-plane. The blue dots are the poles of the functions $\tilde{f}(\Delta)$ and the red dots are the poles of $\tilde{f}(2-\Delta)$. The striped region represents the strip of holomorphy $S_c$ common to both functions.

Theorems & Definitions (18)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • ...and 8 more