Five-Loop Static Contribution to the Gravitational Interaction Potential of Two Point Masses

TL;DR

The paper advances high-precision gravitational two-body dynamics by computing the static, velocity-independent contribution to the gravitational potential at five-loop order (5PN) using an effective field theory framework with a temporal Kaluza-Klein reduction. Momentum-space Feynman diagrams in $d=3-2\varepsilon$ dimensions are reduced to a small set of master integrals via IBP, with QGRAF generating diagrams and Crusher handling reductions. The main result is the static 5PN Lagrangian, obtained after canceling all $\varepsilon$-poles and showing that only rational coefficients remain, in agreement with 4PN results and independent 5PN calculations. This work strengthens the theoretical foundation for precision gravitational predictions in binary dynamics and sets the stage for incorporating velocity-dependent 5PN terms.

Abstract

We compute the static contribution to the gravitational interaction potential of two point masses in the velocity-independent five-loop (and 5th post-Newtonian) approximation to the harmonic coordinates effective action in a direct calculation. The computation is performed using effective field methods based on Feynman diagrams in momentum-space in $d = 3 - 2\varepsilon$ space dimensions. We also reproduce the previous results including the 4th post-Newtonian order.

Figures (4)

• Figure 1: Sample diagrams for the $2 \rightarrow 2$ scattering process in the static limit at 5PN order.
• Figure 2: The topologies contributing to the potential in the static limit at 5PN order.
• Figure 3: The master integral $M_{36}$ contributing to the 4th post-Newtonian approximation.
• Figure 4: The five-loop master integrals contributing to the 5th post-Newtonian approximation in the static limit.