Conservative binary dynamics to third post-Minkowskian order beyond General Relativity

We present the conservative dynamics of compact binaries to third order in the post-Minkowskian approximation in a theory that extends general relativity by a massless scalar field coupled to the Gauss-Bonnet invariant. We employ the effective field theory approach to construct the effective action of binary systems by integrating out the metric and scalar degrees of freedom that mediate the gravitational interactions between the two bodies. We derive analytical expressions for the scattering impulse and the deflection angle to third order in the post-Minkowskian expansion. Our results are found to be in agreement, in the overlapping regimes, with state-of-the-art calculations in the post-Newtonian/post-Minkowskian theory.

💡 Research Summary

This paper investigates the conservative dynamics of compact binary systems within a class of gravity theories that extend General Relativity (GR) by coupling a mass‑less scalar field to the Gauss‑Bonnet invariant, commonly referred to as Einstein‑Scalar‑Gauss‑Bonnet (ESGB) gravity. Using the modern effective‑field‑theory (EFT) framework, the authors integrate out the metric perturbations and the scalar degree of freedom to construct an effective action for two point‑like bodies. The calculation is performed perturbatively in the post‑Minkowskian (PM) expansion, i.e., an expansion in powers of Newton’s constant G while keeping velocities arbitrary, and is carried through to third order (3PM).

The starting point is the action
S = –2κ²∫√–g R – 2∫(∂φ)² – α∫f(φ) 𝒢,
where 𝒢 is the Gauss‑Bonnet scalar, α is a coupling constant with dimensions of length squared, and f(φ) is an arbitrary function of the scalar field. Compact objects are modeled as point particles with scalar‑dependent masses M_i(φ), whose dependence is encoded in scalar sensitivities s_i, s′_i, s″_i. The scalar and metric fields are expanded around flat space, and the world‑line couplings generate a set of Feynman rules that involve graviton propagators, scalar propagators, and mixed vertices arising from the α f(φ) 𝒢 term.

At the diagrammatic level, the 3PM calculation contains all topologies already present in pure GR (the so‑called “H‑type” and “V‑type” graphs) plus new diagrams in which scalar propagators connect to the world‑lines or to graviton lines. Remarkably, after integration‑by‑parts reduction with FIRE6.5, the master integrals required for the ESGB case are exactly the same set that appear in the GR 3PM computation; the new physics is encoded entirely in the coefficients multiplying these integrals.

The authors obtain analytic expressions for the impulse Δp_i^μ of each body and for the scattering angle χ. The impulse can be written as
Δp_i = X n ν m c_n b̂^μ + d_n w^μ J^n,
with J = G m/|b|, b̂ the unit impact‑parameter vector, and w^μ a specific linear combination of the incoming four‑velocities. The coefficients c_n and d_n are rational functions of the Lorentz factor γ = u₁·u₂, the scalar sensitivities s₁, s₂, the dimensionless Gauss‑Bonnet strength (\barα = α/(Gm)^2), and the expansion coefficients f₁, f₂ of the coupling function f(φ). At 1PM (order G) the result reduces to the familiar GR expression plus a term proportional to s₁ s₂. At 2PM (order G²) the new contributions appear linearly in (\barα) through the combination (\barα f₁), while at 3PM (order G³) both (\barα f₁) and (\barα² f₂) terms are present, reflecting next‑to‑leading and next‑to‑next‑to‑leading order effects of the Gauss‑Bonnet coupling.

The scattering angle follows from the impulse via the exact relation
(2\sin(\chi/2) = |\Delta p_\perp|/p_\infty).
Expanding in powers of J⁻¹ yields
(\chi = \frac{2J}{|\lambda|}\gamma^{-1} + \frac{c_2}{J|\lambda|} + \frac{c_3}{J^2|\lambda|} + \mathcal{O}(J^{-3})),
where (\lambda = 2\gamma^2-1 + s_1 s_2). The sign of λ depends on the product of the sensitivities and can change the qualitative behavior of the angle. The authors verify that in the high‑energy limit (γ → ∞) and the non‑relativistic limit (γ → 1) their expressions reduce respectively to known results from high‑energy scattering in GR and to the post‑Newtonian (PN) expansions previously derived for scalar‑tensor theories.

A series of consistency checks is performed: (i) gauge independence is demonstrated by repeating the calculation in de Donder gauge and in a generalized gauge with arbitrary parameters; (ii) the on‑shell condition ((m_i u_i + \Delta p_i)^2 = m_i^2) holds up to O(G³); (iii) expanding the angle in the PN regime reproduces the 3PN effective‑one‑body coefficients obtained in earlier ESGB work.

The paper concludes that the EFT + modern multi‑loop technology provides a systematic and efficient route to high‑order PM results even in theories with higher‑derivative curvature couplings. The scalar sensitivities and the Gauss‑Bonnet coupling first affect the dynamics at 2PM, with the leading (\barα f_1) term entering the impulse and angle at that order, and the quadratic (\barα^2 f_2) term appearing at 3PM. Current observational bounds on (\sqrt{α}) (of order a few kilometers for solar‑mass binaries) imply (\barα \sim \mathcal{O}(1)) for astrophysical systems, so the 3PM corrections could be phenomenologically relevant for next‑generation gravitational‑wave detectors. Moreover, because the required master integrals are already known, extending the calculation to 4PM appears feasible, opening the path toward even higher‑precision waveform modeling in ESGB and related scalar‑tensor extensions of GR.

Overall, the work delivers the first analytic 3PM conservative dynamics for a broad class of scalar‑Gauss‑Bonnet theories, providing essential building blocks for accurate gravitational‑wave templates that can test the presence of scalar hair and higher‑curvature interactions in the strong‑field regime.