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Three-Body Effective Potential in General Relativity at Second Post-Minkowskian Order and Resulting Post-Newtonian Contributions

Florian Loebbert, Jan Plefka, Canxin Shi, Tianheng Wang

TL;DR

This work advances the three-body problem in general relativity by combining the post-Minkowskian and post-Newtonian expansions within a worldline EFT framework. The authors derive the $2\mathrm{PM}$ three-body potential as a differential operator acting on a central three-point integral $I_{3\delta}$, whose Yangian-invariant structure is bootstraped to obtain the $\epsilon$-expanded master integrals and their PN series. They recover the $1\mathrm{PN}$ result, reproduce the known $2\mathrm{PN}$ contributions, and present new $G^2 v^4$ terms at $3\mathrm{PN}$, outlining a method to obtain higher-order corrections via integral bootstrap. The study reveals a deep interplay between PM/PN approaches, Yangian symmetry in a dual momentum space, and potential applications to hierarchical three-body systems and gravitational-wave source modeling. These results provide a concrete pathway to systematically improve three-body GR dynamics in both analytical and phenomenological contexts.

Abstract

We study the Post-Minkowskian (PM) and Post-Newtonian (PN) expansions of the gravitational three-body effective potential. At order 2PM a formal result is given in terms of a differential operator acting on the maximal generalized cut of the one-loop triangle integral. We compute the integral in all kinematic regions and show that the leading terms in the PN expansion are reproduced. We then perform the PN expansion unambiguously at the level of the integrand. Finding agreement with the 2PN three-body potential after integration, we explicitly present new $G^2v^4$-contributions at order 3PN and outline the generalization to $G^2v^{2n}$. The integrals that represent the essential input for these results are obtained by applying the recent Yangian bootstrap directly to their $ε$-expansion around three dimensions. The coordinate space Yangian generator that we employ to obtain these integrals can be understood as a special conformal symmetry in a dual momentum space.

Three-Body Effective Potential in General Relativity at Second Post-Minkowskian Order and Resulting Post-Newtonian Contributions

TL;DR

This work advances the three-body problem in general relativity by combining the post-Minkowskian and post-Newtonian expansions within a worldline EFT framework. The authors derive the three-body potential as a differential operator acting on a central three-point integral , whose Yangian-invariant structure is bootstraped to obtain the -expanded master integrals and their PN series. They recover the result, reproduce the known contributions, and present new terms at , outlining a method to obtain higher-order corrections via integral bootstrap. The study reveals a deep interplay between PM/PN approaches, Yangian symmetry in a dual momentum space, and potential applications to hierarchical three-body systems and gravitational-wave source modeling. These results provide a concrete pathway to systematically improve three-body GR dynamics in both analytical and phenomenological contexts.

Abstract

We study the Post-Minkowskian (PM) and Post-Newtonian (PN) expansions of the gravitational three-body effective potential. At order 2PM a formal result is given in terms of a differential operator acting on the maximal generalized cut of the one-loop triangle integral. We compute the integral in all kinematic regions and show that the leading terms in the PN expansion are reproduced. We then perform the PN expansion unambiguously at the level of the integrand. Finding agreement with the 2PN three-body potential after integration, we explicitly present new -contributions at order 3PN and outline the generalization to . The integrals that represent the essential input for these results are obtained by applying the recent Yangian bootstrap directly to their -expansion around three dimensions. The coordinate space Yangian generator that we employ to obtain these integrals can be understood as a special conformal symmetry in a dual momentum space.

Paper Structure

This paper contains 17 sections, 86 equations.

Table of Contents

  1. Introduction
  2. Effective field theory
  3. 2PM potential
  4. The 1PN Expansion
  5. Post-Newtonian Expansion and Integral Bootstrap
  6. Order $c^0$: $I_3[{\frac{1}{2}},{\frac{1}{2}},{\frac{1}{2}}]$.
  7. Order $c^{-2}$: $I_3[{\frac{1}{2}},{\frac{1}{2}},-{\frac{1}{2}}]$.
  8. Order $c^{-4}$: $I_3[-{\frac{1}{2}},-{\frac{1}{2}},{\frac{1}{2}}]$ and $I_3[{\frac{1}{2}},{\frac{1}{2}},-{\frac{3}{2}}]$.
  9. The 2PN Expansion
  10. New Contributions at 3PN
  11. Conclusions and Outlook
  12. The $3\delta$ Integral
  13. The case $\sigma^2>0$.
  14. The case $\sigma^2<0$.
  15. The case $\sigma^2=0$.
  16. ...and 2 more sections