Abstract
We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering amplitude. The amplitudes we obtain exhibit interesting features, such as non-zero particle production, intricate high-energy and near the two-particle threshold behavior. Scattering amplitudes obtained by initializing the iteration process with zero (or small) multi-particle input end up close to saturating the S-matrix bounds derived by other methods. There is a version of the iterative algorithm that is directly related to Feynman diagrams: it effectively re-sums infinitely many two-particle reducible planar Feynman graphs in the ϕ4 theory, which remarkably produces a unitary nonperturbative scattering amplitude function. Finally, we discuss how the algorithm can be further refined by including multi-particle unitarity.
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Acknowledgments
We thank Nima Arkani-Hamed, David Atkinson, Pierre Aubert, Lucia Cordova, Miguel Correia, Joan Elias Miro, Giulia Isabella, Hofie Hannesdottir, Kelian Häring, Enrico Herrmann, Andreas Juettner, Denis Karateev, Madalena Lemos, Andrew McLeod, Sebastian Mizera, Julio Parra-Martinez, Joao Penedones, Jiaxin Qiao, Balt van Rees, Matthew Schwartz, Amit Sever, Jaroslav Trnka, Pierre Vanhove, Pedro Vieira, Matt Walters, Xiang Zhang for useful discussions. We thank the authors of [45, 46] for sharing and explaining their detailed results to us. We thank Dmitrii Vorobev and Dmitri Vasiliev for exploring the d-dimensional formulae as a part of the Physics Practicum 2022. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 949077) and from Agence Nationale de la Recherche (ANR), project ANR-22-CE31-0017. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.
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Tourkine, P., Zhiboedov, A. Scattering amplitudes from dispersive iterations of unitarity. J. High Energ. Phys. 2023, 5 (2023). https://doi.org/10.1007/JHEP11(2023)005
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DOI: https://doi.org/10.1007/JHEP11(2023)005