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Six-loop gravitational interactions at the sixth post-Newtonian order

Giacomo Brunello, Manoj K. Mandal, Pierpaolo Mastrolia, Raj Patil, Matteo Pegorin, Jonathan Ronca, Sid Smith, Jan Steinhoff, William J. Torres Bobadilla

TL;DR

This work provides the first complete evaluation of the static two-body potential at 6PN order in the EFT framework of gravity, requiring the handling of six-loop, massless two-point integrals. The authors develop a sophisticated pipeline combining automatic diagram generation, spanning cuts, syzygy-based IBP reductions, and a mix of analytic and numerical techniques to reduce 1117 diagrams to 21 contributing master integrals. The resulting ${\cal O}(G_N^7)$ static potential is finite in $d=3$ and purely rational, representing a major milestone toward the full 6PN conservative dynamics and enabling future extension to velocity-dependent and hereditary effects. The methods also illustrate a fruitful connection between high-precision gravitational computations and advanced multi-loop techniques from quantum field theory, with potential applications to higher PN orders and related gauge-theory problems.

Abstract

We compute the gravitational interaction of two coalescing compact objects at sixth post-Newtonian order in the static limit, employing the diagrammatic approach within the effective field theory framework of General Relativity. The calculation requires the evaluation of six-loop Feynman diagrams that are mapped onto two-point integrals with a gauge-theory-like structure, which are computed here for the first time. The resulting seventh-order contribution in Newton's constant is finite in three space dimensions. This result provides the most technically demanding missing ingredient for the determination of the conservative dynamics of the gravitational two-body system at sixth post-Newtonian order.

Six-loop gravitational interactions at the sixth post-Newtonian order

TL;DR

This work provides the first complete evaluation of the static two-body potential at 6PN order in the EFT framework of gravity, requiring the handling of six-loop, massless two-point integrals. The authors develop a sophisticated pipeline combining automatic diagram generation, spanning cuts, syzygy-based IBP reductions, and a mix of analytic and numerical techniques to reduce 1117 diagrams to 21 contributing master integrals. The resulting static potential is finite in and purely rational, representing a major milestone toward the full 6PN conservative dynamics and enabling future extension to velocity-dependent and hereditary effects. The methods also illustrate a fruitful connection between high-precision gravitational computations and advanced multi-loop techniques from quantum field theory, with potential applications to higher PN orders and related gauge-theory problems.

Abstract

We compute the gravitational interaction of two coalescing compact objects at sixth post-Newtonian order in the static limit, employing the diagrammatic approach within the effective field theory framework of General Relativity. The calculation requires the evaluation of six-loop Feynman diagrams that are mapped onto two-point integrals with a gauge-theory-like structure, which are computed here for the first time. The resulting seventh-order contribution in Newton's constant is finite in three space dimensions. This result provides the most technically demanding missing ingredient for the determination of the conservative dynamics of the gravitational two-body system at sixth post-Newtonian order.
Paper Structure (4 sections, 5 equations, 1 figure)

This paper contains 4 sections, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Diagrammatic Structure of Gravitational Two-Body Dynamics
  3. Post-Newtonian Action
  4. Integrand generation

Figures (1)

  • Figure 1: Representative Feynman diagrams at six-loop contributing to the effective potential at 6PN, in the static limit ${\cal O}(G_N^7)$: massive compact objects, $\phi$ and $\sigma$ fields are represented by black, blue and green lines, respectively. The counting does not include diagrams that can be obtained by the exchange of the massive objects.