Effective field theory calculation of conservative binary dynamics at third post-Newtonian order

TL;DR

The authors apply effective field theory methods to gravity to re-derive the conservative two-body dynamics at $3$PN for spin-less binaries, using an automated Mathematica workflow to generate and evaluate hundreds of Feynman diagrams with potential gravitons. They organize the calculation in the KK parametrization, manage divergences through worldline shifts and double-zero techniques, and successfully reproduce the established $3$PN Lagrangian, signaling a scalable path toward $4$PN. The work demonstrates the systematic power of EFT in high-order PN computations and supports precision gravitational-wave modeling, including templates used by LIGO/Virgo and the EOB framework. Overall, the approach provides a robust, automated route to higher-order corrections in binary gravity with clear implications for future waveform accuracy.

Abstract

We reproduce the two-body gravitational conservative dynamics at third post-Newtonian order for spin-less sources by using the effective field theory methods for the gravitationally bound two-body system, proposed by Goldberger and Rothstein. This result has been obtained by automatizing the computation of Feynman amplitudes within a Mathematica algorithm, paving the way for higher-order computations not yet performed by traditional methods.

Figures (6)

• Figure 1: The 12 topologies up to $G^3$ as they are produced by a Mathematica code. The last 4 start contributing at 3PN order. Here and in the following figures, we count them form left to right, from top to bottom.
• Figure 2: The 32 $G^4$ topologies. Only the first 8 contribute at 3PN order, and they are all factorisable into simpler subtopologies.
• Figure 3: The three ${\cal O}(G)$ diagrams. The $\phi$, $A$ and $\sigma$ propagators are represented respectively by blue dashed, red dotted and green solid lines. From the left to the right, they contain $v^6$, $v^4$ and $v^2$ corrections under the form of vertices expansion and/or double time derivative insertions in the propagators. The number in the right part of each diagram is its multiplicity factor.
• Figure 4: The 16 ${\cal O}(G^2)$ diagrams. Diagrams from 2 to 9 actually start contributing at 2PN order, and must be consequently evaluated at their next-to leading order in $v^2$ (which involves an appropriate vertices expansion or one double time derivative insertion in the propagators) to catch their 3PN contribution. Analogously, the first diagram enters already at 1PN order, so it must be expanded up to $v^4$ . As in fig. \ref{['diaG1']}, the $\phi$, $A$ and $\sigma$ propagators are represented respectively by blue dashed, red dotted and green solid lines and the number in the right part of each diagram is its multiplicity factor.
• Figure 5: The 53 ${\cal O}(G^3)$ diagrams. The first 5 enter at 2PN order, and must be consequently evaluated at their next-to leading order in $v^2$, which involve an appropriate vertices expansion or one double time derivative insertion in the propagators. As in fig. \ref{['diaG1']} and \ref{['diaG2']}, the $\phi$, $A$ and $\sigma$ propagators are represented respectively by blue dashed, red dotted and green solid lines and the number in the right part of each diagram is its multiplicity factor.