Abstract
Scattering amplitudes are tempered distributions, which are defined through their action on functions in the Schwartz space \( \mathcal{S}\left(\mathbb{R}\right) \) 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 \( {\mathcal{S}}^{\prime}\left({\mathbb{R}}^{+}\right) \). In this paper, we investigate this problem by characterizing the Mellin transform of the Schwartz space \( \mathcal{S}\left({\mathbb{R}}^{+}\right) \). 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 \( \mathcal{S}\left({\mathbb{R}}^{+}\right) \). We conclude the paper with applications to tree-level graviton celestial amplitudes.
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Acknowledgments
We would like to thank Guillaume Bossard and Andrea Puhm for useful discussions. YP is supported by the PhD track fellowship of Ecole Polytechnique.
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Borji, M., Pano, Y. Distributional celestial amplitudes. J. High Energ. Phys. 2024, 120 (2024). https://doi.org/10.1007/JHEP07(2024)120
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DOI: https://doi.org/10.1007/JHEP07(2024)120