Three-Body Effective Potential in General Relativity at Second Post-Minkowskian Order and Resulting Post-Newtonian Contributions

TL;DR

This paper derives the three-body gravitational potential at second Post-Minkowskian order, connecting it with Post-Newtonian expansions, and introduces new higher-order contributions using advanced integral techniques based on Yangian symmetry.

Contribution

It provides the first explicit computation of the three-body potential at 2PM order and introduces a novel method using Yangian bootstrap for integral evaluation in gravitational calculations.

Findings

Agreement with known 2PN three-body potential

Explicit new $G^2v^4$-contributions at 3PN order

Generalization to higher powers of $v$ outlined

Abstract

We study the Post-Minkowskian (PM) and Post-Newtonian (PN) expansions of the gravitational three-body effective potential. At order 2PM a formal result is given in terms of a differential operator acting on the maximal generalized cut of the one-loop triangle integral. We compute the integral in all kinematic regions and show that the leading terms in the PN expansion are reproduced. We then perform the PN expansion unambiguously at the level of the integrand. Finding agreement with the 2PN three-body potential after integration, we explicitly present new $G^2v^4$-contributions at order 3PN and outline the generalization to $G^2v^{2n}$. The integrals that represent the essential input for these results are obtained by applying the recent Yangian bootstrap directly to their $\epsilon$-expansion around three dimensions. The coordinate space Yangian generator that we employ to obtain these integrals can be understood as a special conformal symmetry in a dual momentum space.