Black Hole Binary Dynamics from the Double Copy and Effective Theory

TL;DR

The paper develops a comprehensive framework to derive the conservative two-body dynamics of compact binaries from quantum scattering amplitudes, leveraging the double-copy construction and generalized unitarity within an EFT matching setup. By isolating classical contributions through careful scale separation and using both nonrelativistic and relativistic integration methods, the authors obtain the full 3PM conservative Hamiltonian for spinless black holes, including explicit velocity-dependent terms encoded in coefficients $c_1,c_2,c_3$ of the potential $V(m p^2)$. Extensive cross-checks—canonical transformations to known 4PN results, Schwarzschild-probe limits, and scattering-angle comparisons—validate the 3PM result and highlight the consistency between PM and PN frameworks. The work also discusses mass-singuarlities, the role of dimensional regularization, and the prospects for extending to higher PM orders, with implications for precision gravitational-wave modeling and insight into the classical limit of quantum gravity. All results are obtained while maintaining four-dimensional helicity methods’ efficiency, supported by $D$-dimensional checks and EFT bookkeeping that ensure infrared subtractions are correctly handled.

Abstract

We describe a systematic framework for computing the conservative potential of a compact binary system using modern tools from scattering amplitudes and effective field theory. Our approach combines methods for integration and matching adapted from effective field theory, generalized unitarity, and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. With these methods we derive the third post-Minkowskian correction to the conservative two-body Hamiltonian for spinless black holes. We describe in some detail various checks of our integration methods and the resulting Hamiltonian.

Figures (32)

• Figure 1: A summary of known results for the two-body potential for spinless black holes in the PN and PM expansions, outlined in blue and green regions respectively. The new 3PM result summarized in Ref. 3PMPRL and discussed at length in this paper is highlighted in the shaded (red) region. The overlap gives strong crosschecks on any calculations in either approach.
• Figure 2: Examples of one- and two-loop diagram that contribute to the classical potential. Wiggly lines represent gravitons and straight lines scalars. Here the diagrams are not Feynman diagrams, but demonstrating the singularity structure from propagators in the graphs.
• Figure 3: Examples of one- and two-loop diagrams that do not contribute to the classical potential. Wiggly lines represent gravitons and straight lines scalars. The meaning of diagrams here is the same as in Fig. \ref{['DiagramKeptFigure']}.
• Figure 4: Generalized two-particle cuts for a elastic scattering of two distinct scalars $\phi_1$ and $\phi_2$ with masses $m_1$ and $m_2$. The pairs of legs (1,4) and (2,3) correspond to $\phi_1$ and $\phi_2$ respectively. The blobs represent tree amplitudes, which can have several diagrams in a given blob, and exposed lines are all on shell. Cut (a) separates the two matter fields by cutting graviton lines and contributes to the classical potential. The top and bottom internal lines in cut (b) are scalar $\phi_1$ and $\phi_2$, respectively. For cut (c) the internal lines are either both $\phi_1$ or $\phi_2$. Neither of the cuts (b) and (c) contains any new classical potential contributions.
• Figure 5: Some generalized unitarity cuts for extracting the conservative two-body potential at (a) 2PM order and (b) at 3PM order. The blobs represent tree amplitudes and exposed lines are all on shell. Straight lines represent massive scalars and the wiggly lines are either gluons or gravitons, depending on whether we are considering a gauge theory or a gravity cut.