The classical limit of the Magnus expansion
Abstract
We uncover a connection between the classical limit of the Magnus expansion in quantum field theory and the Malvenuto-Reutenauer Hopf algebra of permutations. We study a cubic scalar theory describing a massive particle coupled to massless scalar quanta, which captures the propagator structure relevant to classical scattering in gravity and scalar QED. Using a Schwinger parametrisation of the massive propagators, we compute the classical limit of Magnus amplitudes at tree level in this theory. First, we prove that all hyperclassical contributions cancel at the level of the Schwinger proper-time integrand, before integration. Then, we show that the classical Magnus amplitude precisely matches expectations from worldline quantum field theory. Both results rely on cancellations of disconnected diagrams which we relate to the Hopf algebra, specifically to the action of the adjoint first Eulerian projector, which annihilates the shuffle products associated with disconnected contributions. Moreover, the worldline diagrams include generic directed-tree topologies, whereas the field-theory diagrams only include directed chains. The correspondence we establish between the two therefore leads to a new formula for generic Murua coefficients in terms of simpler directed chain coefficients. Finally, we observe a suggestive connection between Magnus amplitudes and the web-mixing matrices governing the exponentiation of multiple Wilson lines.
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BibTeX
@misc{brandhuber2026classical,
title = {{The classical limit of the Magnus expansion}},
author = {Andreas Brandhuber and Graham R. Brown and Sebastien MacDonald and Paolo Pichini and Gabriele Travaglini and Pablo Vives Matasan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40291},
eprint = {2609.40291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40291},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The classical limit of the Magnus expansion
AU - Andreas Brandhuber
AU - Graham R. Brown
AU - Sebastien MacDonald
AU - Paolo Pichini
AU - Gabriele Travaglini
AU - Pablo Vives Matasan
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40291
UR - https://arxiv.org/abs/2609.40291
AB - We uncover a connection between the classical limit of the Magnus expansion in quantum field theory and the Malvenuto-Reutenauer Hopf algebra of permutations. We study a cubic scalar theory describing a massive particle coupled to massless scalar quanta, which captures the propagator structure relevant to classical scattering in gravity and scalar QED. Using a Schwinger parametrisation of the massive propagators, we compute the classical limit of Magnus amplitudes at tree level in this theory. First, we prove that all hyperclassical contributions cancel at the level of the Schwinger proper-time integrand, before integration. Then, we show that the classical Magnus amplitude precisely matches expectations from worldline quantum field theory. Both results rely on cancellations of disconnected diagrams which we relate to the Hopf algebra, specifically to the action of the adjoint first Eulerian projector, which annihilates the shuffle products associated with disconnected contributions. Moreover, the worldline diagrams include generic directed-tree topologies, whereas the field-theory diagrams only include directed chains. The correspondence we establish between the two therefore leads to a new formula for generic Murua coefficients in terms of simpler directed chain coefficients. Finally, we observe a suggestive connection between Magnus amplitudes and the web-mixing matrices governing the exponentiation of multiple Wilson lines.
ER -