Light bending at two-loop order from scattering amplitudes
Abstract
We calculate the conservative deflection of a photon by a massive spinless source, such as a black hole, at third post-Minkowskian order using scattering amplitudes. We work in the heavy-source and long-range limits and adopt the geometric-optics approximation. The required five-point photon-graviton tree amplitude is constructed by on-shell recursion and sewn to three scalar-graviton vertices across a five-fold cut. Polynomial division organizes the cut information into sixteen contributions, including the residues with fewer propagators. We perform the source energy integrations and retain the near-on-shell photon regions in the remaining spatial integrals. Within the static region in which all exchanged momenta are soft, we relate a photon pole prescription to the subtraction of the cubic tree iteration from the conservative kernel. The sum of the sixteen contributions determines the classical logarithmic term at two loops. Its transverse Fourier transform gives the interaction radial action and yields the third-order deflection angle. The result agrees with the weak-deflection expansion of Schwarzschild null geodesics and demonstrates how nonlinear classical light bending is encoded in loop amplitudes around flat spacetime.
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BibTeX
@misc{yang2026light,
title = {{Light bending at two-loop order from scattering amplitudes}},
author = {Jing Yang and Fa Peng Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08632},
eprint = {2610.08632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08632},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Light bending at two-loop order from scattering amplitudes
AU - Jing Yang
AU - Fa Peng Huang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08632
UR - https://arxiv.org/abs/2610.08632
AB - We calculate the conservative deflection of a photon by a massive spinless source, such as a black hole, at third post-Minkowskian order using scattering amplitudes. We work in the heavy-source and long-range limits and adopt the geometric-optics approximation. The required five-point photon-graviton tree amplitude is constructed by on-shell recursion and sewn to three scalar-graviton vertices across a five-fold cut. Polynomial division organizes the cut information into sixteen contributions, including the residues with fewer propagators. We perform the source energy integrations and retain the near-on-shell photon regions in the remaining spatial integrals. Within the static region in which all exchanged momenta are soft, we relate a photon pole prescription to the subtraction of the cubic tree iteration from the conservative kernel. The sum of the sixteen contributions determines the classical logarithmic term at two loops. Its transverse Fourier transform gives the interaction radial action and yields the third-order deflection angle. The result agrees with the weak-deflection expansion of Schwarzschild null geodesics and demonstrates how nonlinear classical light bending is encoded in loop amplitudes around flat spacetime.
ER -