Scattering Amplitudes and Conservative Binary Dynamics at $O(G^5)$ without Self-Force Truncation

Abstract

We compute the complete potential-graviton contributions to the conservative radial action and scattering angle for two non-spinning bodies in general relativity, accurate through fifth order in Newton's constant and including second-order self-force (2SF) effects. The calculation is carried out in the scattering-amplitude framework, combining the double copy, effective field theory, and multi-loop 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 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.

Figures (3)

• Figure 1: Four-loop sample diagrams, including 2SF terms.
• Figure 2: Representative planar diagrams defining the eikonal sums $\mathcal{E}_{2,j}$ appearing in the classical amplitude, where $j$ corresponds to the diagram label. The label $j = 12$ corresponds to the 0SF contribution, and the labels $j = \{1, 4, 6, 9, 11, 13\}$ correspond to the 1SF contributions, with the remaining contributions being 2SF. The diagrams in each eikonal sum follow from permuting the attachment points of the dashed graviton lines.
• Figure 3: Potential-region momentum-space radial actions in Einstein gravity at 5PM as a function of the velocity $v$. The left panel compares the contribution of the different SF orders. The right panel shows the convergence of the series expansions of the finite part of the 2SF radial action.