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

TL;DR

The paper analyzes the static four-loop ($O(G^5)$) contributions to the gravitational potential between two point masses in harmonic coordinates at 4PN order. By re-evaluating the three contentious terms ${L}_{33}$, ${L}_{49}$, and ${L}_{50}$ in ${\bm x}$-space using integration-by-parts and the generalized Riesz formula, it shows that all $\pi^2$-dependent pieces cancel in the total, yielding a rational coefficient $+\dfrac{40}{3}$ for the static part, and it identifies a discrepancy with EFT results for ${L}_{50}$ that may originate from a sign or factor error in the EFT calculation. In addition, the master integral ${\cal M}_{3,6}$ is computed analytically near $d=3$ using the same techniques, confirming the presence of $\pi^2$ at the relevant order and providing an explicit $\varepsilon$-expansion. Together, these results corroborate the 4PN harmonic-coordinate action and illustrate the power of ${\bm x}$-space, saddle-point, and generalized Riesz methods for high-loop gravitational computations. The work thus strengthens confidence in the current understanding of 4PN conservative dynamics and clarifies subtle issues arising in EFT-based approaches.

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 $π^2$. 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.

Figures (10)

• Figure 1: The diagrammatic representation of the $O(a^4 s^6)$ contribution to the effective action.
• Figure 2: The spacetime diagram of $L_{33}$.
• Figure 3: The spatial projection of the diagram of $L_{33}$.
• Figure 4: The spacetime diagram of $L_{49}$.
• Figure 5: The spatial projection of the diagram of $L_{49}$.