Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics

Gregor Kälin, Rafael A. Porto

TL;DR

This work builds a systematic Post-Minkowskian (PM) effective-field-theory framework to compute conservative binary dynamics directly from PM scattering data, bridged to bound-state observables via a Boundary-to-Bound (B2B) dictionary. By isolating the conservative potential-region physics with a tailored energy-integral prescription and using worldline sources with finite-size operators, the authors obtain the scattering angle to 2PM and reconstruct the corresponding Hamiltonian, matching prior results and encoding tidal corrections at leading PM order. The approach highlights a gauge-invariant, diagrammatically economical route to classical two-body gravity, connecting scattering data to adiabatic invariants and orbital frequencies without relying on gauge-fixed potentials or quantum-amplitude limits, and suggests straightforward automation to higher PM orders. The inclusion of leading tidal effects demonstrates the method’s ability to incorporate finite-size physics, with results consistent with existing formalisms and compatible with EOB-like descriptions, offering a versatile platform for advancing high-precision gravitational dynamics in the PM regime.

Abstract

We develop an Effective Field Theory (EFT) formalism to solve for the conservative dynamics of binary systems in gravity via Post-Minkowskian (PM) scattering data. Our framework combines a systematic EFT approach to compute the deflection angle in the PM expansion, together with the 'Boundary-to-Bound' (B2B) dictionary introduced in [1910.03008, 1911.09130]. Due to the nature of scattering processes, a remarkable reduction of complexity occurs both in the number of Feynman diagrams and type of integrals, compared to a direct EFT computation of the potential in a PM scheme. We provide two illustrative examples. Firstly, we compute all the conservative gravitational observables for bound orbits to 2PM, which follow from only one topology beyond leading order. The results agree with those in [1910.03008, 1911.09130], obtained through the 'impetus formula' applied to the classical limit of the one loop amplitude in Cheung et al. [1808.02489]. For the sake of comparison we reconstruct the conservative Hamiltonian to 2PM order, which is equivalent to the one derived in [1808.02489] from a matching calculation. Secondly, we compute the scattering angle due to tidal effects from the electric- and magnetic-type Love numbers at leading PM order. Using the B2B dictionary we then obtain the tidal contribution to the periastron advance. We also construct a Hamiltonian including tidal effects at leading PM order. Although relying on (relativistic) Feynman diagrams, the EFT formalism developed here does not involve taking the classical limit of a quantum amplitude, neither integrals with internal massive fields, nor additional matching calculations, nor spurious ('super-classical') infrared singularities. By construction, the EFT approach can be automatized to all PM orders.

Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics

TL;DR

This work builds a systematic Post-Minkowskian (PM) effective-field-theory framework to compute conservative binary dynamics directly from PM scattering data, bridged to bound-state observables via a Boundary-to-Bound (B2B) dictionary. By isolating the conservative potential-region physics with a tailored energy-integral prescription and using worldline sources with finite-size operators, the authors obtain the scattering angle to 2PM and reconstruct the corresponding Hamiltonian, matching prior results and encoding tidal corrections at leading PM order. The approach highlights a gauge-invariant, diagrammatically economical route to classical two-body gravity, connecting scattering data to adiabatic invariants and orbital frequencies without relying on gauge-fixed potentials or quantum-amplitude limits, and suggests straightforward automation to higher PM orders. The inclusion of leading tidal effects demonstrates the method’s ability to incorporate finite-size physics, with results consistent with existing formalisms and compatible with EOB-like descriptions, offering a versatile platform for advancing high-precision gravitational dynamics in the PM regime.

Abstract

We develop an Effective Field Theory (EFT) formalism to solve for the conservative dynamics of binary systems in gravity via Post-Minkowskian (PM) scattering data. Our framework combines a systematic EFT approach to compute the deflection angle in the PM expansion, together with the 'Boundary-to-Bound' (B2B) dictionary introduced in [1910.03008, 1911.09130]. Due to the nature of scattering processes, a remarkable reduction of complexity occurs both in the number of Feynman diagrams and type of integrals, compared to a direct EFT computation of the potential in a PM scheme. We provide two illustrative examples. Firstly, we compute all the conservative gravitational observables for bound orbits to 2PM, which follow from only one topology beyond leading order. The results agree with those in [1910.03008, 1911.09130], obtained through the 'impetus formula' applied to the classical limit of the one loop amplitude in Cheung et al. [1808.02489]. For the sake of comparison we reconstruct the conservative Hamiltonian to 2PM order, which is equivalent to the one derived in [1808.02489] from a matching calculation. Secondly, we compute the scattering angle due to tidal effects from the electric- and magnetic-type Love numbers at leading PM order. Using the B2B dictionary we then obtain the tidal contribution to the periastron advance. We also construct a Hamiltonian including tidal effects at leading PM order. Although relying on (relativistic) Feynman diagrams, the EFT formalism developed here does not involve taking the classical limit of a quantum amplitude, neither integrals with internal massive fields, nor additional matching calculations, nor spurious ('super-classical') infrared singularities. By construction, the EFT approach can be automatized to all PM orders.

Paper Structure

This paper contains 24 sections, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Boundary-to-Bound
  3. From Angles to Action $\ldots$
  4. $\ldots$ to Binary Observables
  5. Hamiltonian
  6. Post-Minkowskian Effective Theory
  7. Point-Particle Sources
  8. Conservative Effective Action
  9. Potential Region
  10. Impulse & Deflection Angle
  11. Conservative Binary Dynamics to 2PM
  12. Effective Lagrangian
  13. Tree
  14. One Loop
  15. Trajectories
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Feynman diagrams which only yield singular integrals in the potential region. The divergences can be set to zero in dimensional regularization, or absorbed into counter-terms.
  • Figure 2: Feynman topologies needed for the computation of the effective action to ${\cal O}(G^3)$. The wavy line represents the propagator (or Green's function), while the black dots are the two worldlines at particle's 1 (bottom) and 2 (top), treated as external sources.
  • Figure 3: Feynman diagram with an insertion of a tidal operator represented by the square.