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High Energ. Phys."],"abstract":"<jats:title>\n                    A\n                    <jats:sc>bstract<\/jats:sc>\n                  <\/jats:title>\n                  <jats:p>\n                    Two fundamentally distinct types of quantities are both called \u201ceikonal\u201d in present amplitudes literature. The unitarity of the S-matrix ensures it can be written as the exponential of a Hermitian operator. The eikonal generator or\n                    <jats:italic>Magnusian<\/jats:italic>\n                    , which is the classical limit of the expectation value of that operator, generates all scattering observables. The leading order classical behavior of the phase of an S-matrix element is called the classical eikonal phase, and it coincides with a classical on-shell action. We demonstrate that the eikonal generator (Magnusian) and the eikonal phase (classical on-shell action) are\n                    <jats:italic>inequivalent<\/jats:italic>\n                    and find the exact general relationship between them. That relationship explains the special case of integrable scattering in which the two do coincide up to a Legendre transformation and explains why such a correspondence fails in general when spin or radiation are included.\n                  <\/jats:p>","DOI":"10.1007\/jhep03(2026)241","type":"journal-article","created":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T13:52:10Z","timestamp":1774533130000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Magnusian: relating the eikonal phase, the on-shell action, and the scattering generator"],"prefix":"10.1007","volume":"2026","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3057-7906","authenticated-orcid":false,"given":"Jung-Wook","family":"Kim","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01ggx4157","id-type":"ROR","asserted-by":"publisher"}],"name":"Theoretical Physics Department, CERN"},{"id":[{"id":"https:\/\/ror.org\/03sry2h30","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7055-0345","authenticated-orcid":false,"given":"Raj","family":"Patil","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03sry2h30","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)"},{"id":[{"id":"https:\/\/ror.org\/01hcx6992","id-type":"ROR","asserted-by":"publisher"}],"name":"Humboldt Universit\u00e4t zu Berlin"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-1490-4904","authenticated-orcid":false,"given":"Trevor","family":"Scheopner","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03sry2h30","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1614-0214","authenticated-orcid":false,"given":"Jan","family":"Steinhoff","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03sry2h30","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)"}]}],"member":"297","published-online":{"date-parts":[[2026,3,25]]},"reference":[{"key":"28548_CR1","unstructured":"LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. 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