Classical eikonal from Magnus expansion
Abstract In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum me...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2025-01-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP01(2025)111 |
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| author | Joon-Hwi Kim Jung-Wook Kim Sungsoo Kim Sangmin Lee |
| author_facet | Joon-Hwi Kim Jung-Wook Kim Sungsoo Kim Sangmin Lee |
| author_sort | Joon-Hwi Kim |
| collection | DOAJ |
| description | Abstract In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods. Importantly, the Magnus expansion yields a finite eikonal, while the naïve eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on. |
| format | Article |
| id | doaj-art-67175c61ad314a5a946d0dab1ad21a1b |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-67175c61ad314a5a946d0dab1ad21a1b2025-02-09T12:07:09ZengSpringerOpenJournal of High Energy Physics1029-84792025-01-012025114810.1007/JHEP01(2025)111Classical eikonal from Magnus expansionJoon-Hwi Kim0Jung-Wook Kim1Sungsoo Kim2Sangmin Lee3Walter Burke Institute for Theoretical Physics, California Institute of TechnologyMax Planck Institute for Gravitational Physics (Albert Einstein Institute)Department of Physics and Astronomy, Seoul National UniversityDepartment of Physics and Astronomy, Seoul National UniversityAbstract In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods. Importantly, the Magnus expansion yields a finite eikonal, while the naïve eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on.https://doi.org/10.1007/JHEP01(2025)111Black HolesClassical Theories of GravityScattering Amplitudes |
| spellingShingle | Joon-Hwi Kim Jung-Wook Kim Sungsoo Kim Sangmin Lee Classical eikonal from Magnus expansion Journal of High Energy Physics Black Holes Classical Theories of Gravity Scattering Amplitudes |
| title | Classical eikonal from Magnus expansion |
| title_full | Classical eikonal from Magnus expansion |
| title_fullStr | Classical eikonal from Magnus expansion |
| title_full_unstemmed | Classical eikonal from Magnus expansion |
| title_short | Classical eikonal from Magnus expansion |
| title_sort | classical eikonal from magnus expansion |
| topic | Black Holes Classical Theories of Gravity Scattering Amplitudes |
| url | https://doi.org/10.1007/JHEP01(2025)111 |
| work_keys_str_mv | AT joonhwikim classicaleikonalfrommagnusexpansion AT jungwookkim classicaleikonalfrommagnusexpansion AT sungsookim classicaleikonalfrommagnusexpansion AT sangminlee classicaleikonalfrommagnusexpansion |