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Static two-body potential at fifth post-Newtonian order

Stefano Foffa, Pierpaolo Mastrolia, Riccardo Sturani, Christian Sturm, William J. Torres Bobadilla

TL;DR

This work computes the static two-body gravitational potential at fifth post-Newtonian order by exploiting an effective field theory framework for gravity. A key result is a factorization theorem: static odd-PN diagrams decompose into products of lower-order contributions, enabling the 5PN static term to be determined from lower-order data. The authors evaluate 154 static diagrams, organize them into four classes, and obtain a finite, rational $G_N^6$ contribution to the potential, consistent with Schwarzschild behavior in the extreme mass ratio limit. The findings demonstrate the power of the EFT/factorization approach for pushing high-PN calculations and suggest efficient pathways for completing full 5PN corrections and higher orders.

Abstract

We determine the gravitational interaction between two compact bodies up to the sixth power in Newton's constant GN, in the static limit. This result is achieved within the effective field theory approach to General Relativity, and exploits a manifest factorization property of static diagrams which allows to derive static post Newtonian (PN) contributions of (2n+1)-order in terms of lower order ones. We recompute in this fashion the 1PN and 3PN static potential, and present the novel 5PN contribution.

Static two-body potential at fifth post-Newtonian order

TL;DR

This work computes the static two-body gravitational potential at fifth post-Newtonian order by exploiting an effective field theory framework for gravity. A key result is a factorization theorem: static odd-PN diagrams decompose into products of lower-order contributions, enabling the 5PN static term to be determined from lower-order data. The authors evaluate 154 static diagrams, organize them into four classes, and obtain a finite, rational contribution to the potential, consistent with Schwarzschild behavior in the extreme mass ratio limit. The findings demonstrate the power of the EFT/factorization approach for pushing high-PN calculations and suggest efficient pathways for completing full 5PN corrections and higher orders.

Abstract

We determine the gravitational interaction between two compact bodies up to the sixth power in Newton's constant GN, in the static limit. This result is achieved within the effective field theory approach to General Relativity, and exploits a manifest factorization property of static diagrams which allows to derive static post Newtonian (PN) contributions of (2n+1)-order in terms of lower order ones. We recompute in this fashion the 1PN and 3PN static potential, and present the novel 5PN contribution.

Paper Structure

This paper contains 9 sections, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Effective Field Theory approach
  3. Factorization theorem
  4. Gravity and Field Theory Diagrams
  5. Results
  6. Total 5PN static potential.
  7. Check: test-particle limit.
  8. Conclusion
  9. Acknowledgments.

Figures (2)

  • Figure 1: Examples of a prime 4PN-graph (left) and of a factorizable 5PN-graph (right): the latter can be obtained by sewing the former and the Newton potential diagram.
  • Figure 2: 5PN graphs contributing to the test-particle limit. The last graph (bottom-right) does not contribute to the 5PN potential, because its 4PN subdiagram vanishes.