All-order structure of static gravitational interactions and the seventh post-Newtonian potential

TL;DR

This paper derives a closed-form formula for static post-Newtonian corrections in two-body gravitational interactions at any odd order, and applies it to compute the seventh post-Newtonian potential.

Contribution

It introduces a novel correlation function framework and symmetry-based approach to efficiently compute high-order static gravitational corrections.

Findings

Derived a closed formula for odd-order static post-Newtonian corrections.

Computed the seventh post-Newtonian potential at seven-loop order.

Confirmed agreement with existing diagrammatic results.

Abstract

We present a closed formula for the computation of static post-Newtonian corrections to the two-body gravitational dynamics at any odd order, assuming the lower-order results are known. The formula is derived within a correlation function framework and exploits the $\mathbb{Z}_2$ symmetry of the static sector, leading to a novel theoretical interpretation of the factorization theorem. As an application, we compute the gravitational interaction of two compact coalescing objects at the seventh post-Newtonian order in the static limit, which receives contributions from seven-loop graphs at order $\mathcal{O}(G_N^8 v^0)$, and find complete agreement with the results obtained using the diagrammatic approach of the factorization theorem.