[Paper Review] Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory

Zvi Bern, Andrés Luna+3 more|arXiv (Cornell University)|May 6, 2020

Pulsars and Gravitational Waves ResearchPhysics and Astronomy231 references212 citations

TL;DR

This paper develops a systematic amplitude-based effective field theory framework to compute the conservative spin-dependent two-body Hamiltonian for spinning black hole binaries up to O(G²) and to all orders in velocity. By leveraging arbitrary-spin particle amplitudes and double-copy structures, it derives the impulse and spin kick in scattering processes through compact expressions involving the eikonal phase, revealing a hidden simplicity in classical observables beyond standard post-Newtonian methods.

ABSTRACT

We describe a systematic framework for finding the conservative potential of compact binary systems with spin based on scattering amplitudes of particles of arbitrary spin and effective field theory. An arbitrary-spin formalism is generally required in the classical limit. By matching the tree and one-loop amplitudes of four spinning particles with those of a suitably-chosen effective field theory, we obtain the spin1-spin2 terms of a two-body effective Hamiltonian through O(G^2) and valid to all orders in velocity. Solving Hamilton's equations yields the impulse and spin changes of the individual bodies. We write them in a surprisingly compact form as appropriate derivatives of the eikonal phase obtained from the amplitude. It seems likely this structure persists to higher orders. We also point out various double-copy relations for general spin.

Motivation & Objective

• To develop a systematic framework for computing conservative spin-dependent potentials in binary black hole systems using scattering amplitudes and effective field theory.
• To extend post-Minkowskian and effective field theory methods to include spin effects at all orders in velocity and to O(G²).
• To establish a direct link between scattering amplitudes, the eikonal phase, and classical observables such as impulse and spin kick in spinning binary systems.
• To explore double-copy relations for arbitrary-spin vertices and gravitational Compton amplitudes, generalizing known structures to spinning systems.
• To validate the framework by reproducing known results in the post-Newtonian limit and test-mass limit, ensuring consistency with existing high-precision calculations.

Proposed method

• Formulates an arbitrary-spin Lagrangian for massive particles with spin, including minimal and nonminimal couplings to gravity.
• Uses on-shell scattering amplitudes of four spinning particles to compute tree- and one-loop-level amplitudes in quantum field theory.
• Applies the modern unitarity method and generalized cuts (quadruple and triple cuts) to extract loop integral coefficients in the classical limit.
• Matches the amplitude results to an effective field theory (EFT) four-point interaction to derive the conservative two-body Hamiltonian bilinear in spin.
• Derives physical observables—impulse and spin kick—by solving Hamilton's equations and expressing them as derivatives of the eikonal phase.
• Employs double-copy structures (KLT and BCJ-type) to relate gravitational amplitudes to simpler gauge-theory amplitudes, particularly for the Compton amplitude and three-point vertices.

Experimental results

Research questions

• RQ1 How can scattering amplitudes in quantum field theory be systematically used to compute classical spin-dependent potentials in binary black hole systems?
• RQ2 Can the eikonal phase derived from scattering amplitudes fully encode classical observables like impulse and spin kick in spinning binary systems?
• RQ3 What is the role of arbitrary-spin particle formalism in capturing the correct classical limit for spinning black holes beyond low-spin approximations?
• RQ4 How do double-copy relations extend to spinning systems, and what structures emerge in the gravitational Compton amplitude and three-point vertices?
• RQ5 To what extent can the amplitude-based EFT framework reproduce known post-Newtonian results and extend them to all orders in velocity?

Key findings

• The paper derives the spin1-spin2 interaction potential in the two-body Hamiltonian through O(G²) and to all orders in velocity using amplitude-matching techniques.
• The impulse and spin kick in scattering processes are expressed as compact derivatives of the eikonal phase, revealing a hidden simplicity in classical observables.
• The results reproduce the state-of-the-art post-Newtonian spin-orbit and spin1-spin2 potentials in the overlapping region, validating the framework.
• In the test-mass limit, the framework reproduces all-orders-in-velocity results for the scattering angle, confirming consistency with existing calculations.
• The authors identify KLT-like factorizations and double-copy structures for arbitrary-spin tree-level vertices and the gravitational Compton amplitude, suggesting deeper underlying symmetries.
• The framework suggests that a general formalism may exist to map the eikonal phase directly to physical observables, potentially simplifying future calculations in gravitational wave physics.