The gravitational Compton amplitude at third post-Minkowskian order

We compute the classical Compton amplitude for graviton interaction with a non-spinning massive body up to the third post-Minkowskian order. Our novel result utilizes the enhanced computational efficiency provided by worldline effective field theory in a non-trivial background spacetime. Physical constraints, such as infrared factorization, provide a useful cross-check of the result and we also consider its consistency with computations in black hole perturbation theory.

💡 Research Summary

The paper presents a first‑principles calculation of the classical graviton‑matter Compton scattering amplitude for a non‑spinning point mass up to third post‑Minkowskian (3PM) order. The authors adopt a worldline effective field theory (EFT) description of the massive body, coupling a minimally‑interacting point‑particle worldline to the Einstein‑Hilbert action for gravity. Dimensional regularisation (d = 4 − 2ε) is employed, with the d‑dimensional gravitational coupling defined as κ²_d = 32π G̃ μ^{2ε}. The worldline trajectory is expanded around a straight‑line motion, x^μ(τ)=v^μτ+z^μ(τ), while the metric is expanded about a curved background – the Schwarzschild–Tangherlini solution written in isotropic coordinates – as g_{μν}= \bar g_{μν}+κ h_{μν}. This choice reduces the number of Feynman diagrams relative to a flat‑space expansion and makes the recoil (deflection) vertices and n‑PM metric insertions manifest.

The perturbative expansion proceeds by constructing two elementary building blocks: (i) the deflection vertex, which encodes the absorption and re‑emission of a graviton by the massive worldline, and (ii) the n‑PM metric insertion, representing the interaction of a graviton with the static gravitational potential generated by the massive source. At 3PM order the full amplitude receives contributions from 13 distinct diagrams, which can be organised into a compact algebraic expression involving products of these blocks.

All scalar products of loop momenta and external kinematics are reduced to a two‑loop integral family \bar K_{λ}^{ν}. The family is defined by seven propagators (including velocity‑cut massive propagators) and a set of exponents λ_i, ν_i that encode powers of the propagators and derivatives of the δ‑functions enforcing v·ℓ = 0. Integration‑by‑parts (IBP) identities, implemented with LiteRed, reduce the family to eight master integrals K₁…K₈. These masters correspond to cut double‑bubble, cut double‑triangle (potential and active), cut box‑triangle, and cut double‑box topologies. The distinction between potential (purely spacelike) and active (containing time‑like graviton propagators with an i0⁺ prescription) is crucial for the infrared (IR) structure.

The master integrals are evaluated using the differential‑equation method. A canonical basis is chosen so that the ε‑expansion can be performed order‑by‑order. Boundary conditions are fixed by evaluating the integrals in the forward‑scattering limit (x → 0) and by imposing the absence of spurious singularities except for the expected Newtonian pole ∝ x⁻². The resulting ε‑expanded expressions involve logarithms, dilogarithms Li₂(±x), and transcendental constants (π², ζ₂, Catalan’s G). Notably, K₅ and K₆ contain complex logarithms and the combination G±(x)=Li₂(x)±Li₂(−x).

The amplitude is finally expressed in terms of two gauge‑invariant building blocks constructed from the graviton polarization tensors: F₁ = (v·f₁·f₂*·v)/ω², F₂ = (v·f₁·p₂)(v·f₂*·p₁)/ω⁴, where f_i^{μν}=p_i^μ ε_i^ν − p_i^ν ε_i^μ and ω is the graviton energy (conserved by the worldline). The product F₁F₂ is not independent due to a four‑dimensional identity, leaving F₁² and F₂² as a complete basis. The full 3PM amplitude takes the compact form M_{3PM}=8 ∑_{i=1}^{8} c_i K_i, with coefficients c_i being rational functions of the kinematic variable x = |q|²/(4ω²)=sin²(θ/2) and of the polarization invariants.

Infrared consistency is checked by verifying that the amplitude factorises into the expected exponential of the eikonal phase, confirming the correct soft‑graviton behaviour. The authors also compare their results with black‑hole perturbation theory (BHPT). At 2PM order the helicity‑conserving and helicity‑flipping scattering functions match exactly with BHPT calculations. At 3PM order, however, only partial agreement is found: the potential‑type contributions agree, while certain logarithmic and dilogarithmic terms differ. The discrepancy is attributed to possible missing higher‑order recoil effects or subtleties in the regularisation scheme within the worldline EFT framework.

In conclusion, the paper demonstrates that worldline EFT combined with modern multi‑loop techniques (IBP reduction, canonical differential equations) can successfully deliver the classical graviton‑matter Compton amplitude to third post‑Minkowskian order. The work provides a benchmark for future extensions, such as inclusion of spin, multi‑mass systems, and higher‑order radiative effects, and highlights the remaining challenges in achieving full agreement with black‑hole perturbation theory at high PM orders.