# On the four-loop static contribution to the gravitational interaction potential of two point masses

## TL;DR

This paper calculates specific four-loop contributions to the gravitational potential between two point masses at the fourth post-Newtonian order, revealing a rational coefficient and confirming previous theoretical results.

## Contribution

It provides an analytical computation of a four-loop scalar integral and corrects prior effective-field-theory calculations, enhancing understanding of the 4PN gravitational interaction.

## Key findings

- The total four-loop contribution has a rational coefficient due to cancellation effects.
- The generalized Riesz formula enables analytical evaluation of complex integrals.
- The results confirm the current understanding of the 4PN effective gravitational action.

## Abstract

We compute a subset of three, velocity-independent four-loop (and fourth post-Newtonian) contributions to the harmonic-coordinates effective action of a gravitationally interacting system of two point-masses. We find that, after summing the three terms, the coefficient of the total contribution is rational, due to a remarkable cancellation between the various occurrences of $\pi^2$. This result, obtained by a classical field-theory calculation, corrects the recent effective-field-theory-based calculation by Foffa et al. [arXiv:1612.00482]. Besides showing the usefulness of the saddle-point approach to the evaluation of the effective action, and of x-space computations, our result brings a further confirmation of the current knowledge of the fourth post-Newtonian effective action. We also show how the use of the generalized Riesz formula [Phys. Rev. D 57, 7274 (1998)] allows one to analytically compute a certain four-loop scalar master integral (represented by a four-spoked wheel diagram) which was, so far, only numerically computed.