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Scattering Amplitudes and Conservative Binary Dynamics at O(G5) without Self-Force Truncation

Zvi Bern1, Enrico Herrmann1, Radu Roiban2,3, Michael S. Ruf4, Alexander V. Smirnov5,6, Sid Smith7,8,9, and Mao Zeng9

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Phys. Rev. Lett. 137, 021402 – Published 6 July, 2026

DOI: https://doi.org/10.1103/zjcp-ssrw

Abstract

We compute the complete potential-graviton contributions to the conservative radial action and scattering angle for two nonspinning bodies in general relativity, accurate through fifth order in Newton’s constant and including second-order self-force effects. The calculation is carried out in the scattering-amplitude framework, combining the double-copy, effective field theory, and multiloop integration techniques based on integration by parts and differential equations. To address a major computational bottleneck, we develop improved integration-by-parts algorithms that render calculations at this order tractable. The post-Minkowskian amplitude is presented as a series expansion, following the strategy used earlier in maximal supergravity. For the first self-force sector, which involves only polylogarithmic functions, we also provide a closed-form analytic expression. For the second self-force sector, as in earlier supergravity work, we find nontrivial cancellations among contributions related to integrals supported on Calabi-Yau geometry.

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Phys. Rev. Lett. 137, 021402– Published 6 July, 2026
  • Received 30 December 2025
  • Revised 10 March 2026
  • Accepted 2 April 2026
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