Scattering Gravitons off General Spinning Compact Objects to $\mathcal{O}(G^2 S^4)$ (2511.10280v1)

Abstract: We compute the classical one-loop gravitational Compton amplitude describing the scattering of a graviton off a massive spinning compact object at the second post-Minkowskian order, including terms through the quartic order in spin. Our analysis includes spin-induced finite-size effects up to the hexadecapolar order, and extends recent results obtained for minimal couplings at the quadratic order in spin. From the amplitude, we determine the scattering phase in momentum space, applicable in both the eikonal and wave regimes. In the eikonal limit, we then isolate the spin-independent contribution of the graviton field, explicitly linking it to the dynamics of a massless scalar probe in a Kerr background. This constitutes the first complete description of classical one-loop Compton scattering for generic spinning compact objects at the second post-Minkowskian and hexadecapolar orders.

• The paper presents the first complete calculation of the classical one-loop gravitational Compton amplitude for generic spinning compact objects at 2PM and O(G²S⁴) order.
• It employs a generalized unitarity framework with various regularization schemes to systematically incorporate finite-size, spin-induced multipole effects.
• The results validate minimal Kerr coupling in the proper limits and extend to non-minimal couplings, providing key inputs for high-precision gravitational waveform modeling.

Classical One-Loop Gravitational Compton Scattering off Spinning Compact Objects at 2PM and $\mathcal{O}(S^4)$

Introduction and Motivation

This paper presents a detailed computation of the classical one-loop gravitational Compton amplitude describing graviton scattering off generic massive spinning compact objects, to second post-Minkowskian (2PM, $\mathcal{O}(G^2)$) order and quartic order in spin, fully incorporating spin-induced finite-size effects up to the hexadecapole (quartic) order. The calculation extends the scope of classical gravitational amplitude computations beyond minimal (Kerr) couplings to encompass general compact object structure, addressing both the amplitude construction and its observable implications.

This work is situated in the broader context of quantum field theory-inspired computational methods for classical general relativity, essential for high-precision modeling of gravitational wave sources detected by LIGO–Virgo–KAGRA and forthcoming detectors. Prior to this work, analogous computations were limited to lower orders in spin, restricted to the eikonal regime, or focused solely on black holes with minimal couplings.

Methodology: Amplitude Construction

Regularization Schemes

Three dimensional regularization schemes are explicitly discussed: ’t Hooft–Veltman (THV), dimensional reduction (DR), and conventional dimensional regularization (CDR). Notably, the THV and DR schemes yield identical results for the classical one-loop Compton amplitude, whereas the CDR scheme produces discrepancies at $\mathcal{O}(\epsilon^0)$ from quadratic order in spin upwards, due to explicit $D$-dependence in spin tensor contractions. However, these scheme sensitivities do not propagate to physical non-radiative observables derived from the scattering phase, as the relevant contributions are scheme-independent.

Tree-Level Inputs

The amplitude construction relies on a generalized unitarity framework, assembling the one-loop integrand from gauge-invariant tree-level ingredients. The essential building blocks are:

• Exponentiated three-point amplitudes for graviton emission, expanded to $\mathcal{O}(S^4)$, encoding the spin-induced multipole couplings.
• Four-point tree-level Compton amplitudes, expressed in terms of field strengths and traces over contracted spin tensors, systematically capturing the relevant spin structures.
• Pure graviton tree amplitudes, completing the unitarity cuts.

The approach employs a heavy-mass expansion to isolate classical physics. The computation tracks all explicit $D$-dependence to manage the interplay with regularization schemes appropriately.

Loop Cuts and Master Integrals

Key to the 2PM amplitude are the $t$-channel and $s$-channel unitarity cut diagrams, corresponding to physical propagation of intermediate massive and massless states. State sums over the transverse graviton polarizations are efficiently implemented using $D$-dimensional projectors consistent with the generalized Ward identities. All relevant terms are decomposed onto master integrals: cut box, triangle, and bubble. Integral coefficients are algorithmically organized using integration-by-parts (IBP) reduction (e.g., with FIRE6), ensuring that necessary consistency checks (e.g., identical coefficients from both cut channels) are manifest.

The explicit evaluation of these master integrals reveals the expected IR divergences and logarithmic structures. A pivotal consistency relation, following from Weinberg’s IR factorization theorem, is confirmed: the infrared divergence coefficient is proportional to the tree-level amplitude at each spin order. Superclassical (non-classical) and regularization-dependent terms are isolated and shown to drop out in the calculated scattering phase for non-radiative observables.

Results: Kerr and Generic Compact Object Dynamics

Minimal Coupling (Kerr Limit)

The computed minimally coupled (Kerr) amplitude through $\mathcal{O}(S^4)$—i.e., up to hexadecapole deformations—satisfies expected gauge-invariance and IR-finiteness relations. An explicit symmetric kinematic parametrization allows for compact expressions in terms of Lorentz-invariant coefficients and the chosen basis $\{v^\mu, k^\mu, q^\mu, n^\mu\}$. Dimensional regularization scheme dependence first enters at quadratic order in spin; however, as argued, this affects only non-observable terms.

Crucially, the result gives access to the full, gauge-invariant classical one-loop amplitude, rather than just its eikonal or leading multipole subset. The amplitude's output is packaged in the provided ancillary file for further use in waveform or observable computation contexts.

Non-Minimal Couplings: Generic Object Structure

The construction is generalized to include spin-induced non-minimal couplings, parameterized by Wilson coefficients (e.g., $C_{\mathrm{ES}^n},\, C_{\mathrm{BS}^n}$) following the bosonic worldline (BUSY) formalism. Through careful application of effective worldline actions, the formalism systematically accounts for higher-order multipole structure in compact objects, leading to deformation of the canonical amplitudes from the Kerr case.

A subtlety arises for cubic (and higher) in spin non-minimal amplitudes: tree-level structures can violate generalized Ward identities unless vanishing Gram determinant terms (identically zero in four dimensions) are added. This fix is validated by observing complete agreement with the minimal coupling result upon setting Wilson coefficients to Kerr values, attesting to the internal consistency of the amplitude extraction pipeline.

Observables: Scattering Phase and Angle

The physical content of the amplitude is extracted through the computation of the scattering phase $\Delta$ via the $S$-matrix exponentiation. At $2$PM,

$$\Delta^{(1)} = T^{(1)} - \frac{i}{2}(T^{(0)})^\dagger T^{(0)}$$

where $T^{(1)}$ refers to the one-loop term, and unitarity-imposed subtractions eliminate IR, superclassical, and scheme-dependent terms. Thus, only (IR- and scheme-independent) physical content persists in $\langle \Delta^{(1)} \rangle$.

The computation verifies that, in the eikonal (geometric optics) regime, the monopole contribution to the graviton amplitude is in exact agreement with the dynamics of a massless scalar probe in a Kerr background for aligned-spin configurations, consistent with classical equivalence principles. Explicitly, the scattering phase and its derivatives are Fourier transformed into position space to obtain the scattering angle, providing all necessary ingredients for waveform calculations and effective potential extraction in the PM expansion.

Explicit Series

For aligned-spin configurations (relevant for equatorial-plane dynamics), the expressions for the scattering phase and angle are given explicitly through quartic order in spin and exhibit spin-shift symmetry at the monopole (Kerr) point, but not for general non-minimal couplings. This provides a diagnostic for identifying non-Kerr structure in strong-field observational signatures.

Implications, Numerical Results, and Future Directions

The paper demonstrates that the classical one-loop Compton amplitude, and hence all associated non-radiative observables, are robust under changes in regularization scheme and immune to IR divergences. Dimensional regularization scheme dependence in the presence of spin, identified for the amplitude and waveform, is isolated and clarified, providing clear guidance for future loop-level waveform computations involving spinning sources.

Numerical coefficients for the classical, aligned-spin scattering phase and angle are provided to $\mathcal{O}(S^4)$; for example, the leading phase is $\propto \omega/J$, with higher spin corrections explicitly listed. The calculations validate the amplitude framework up to the most general compact object structure currently tractable.

The work also sets a foundation for further advances:

• Extension to higher post-Minkowskian orders (beyond 2PM), required for sub-percent precision in GW waveform templates
• Incorporation of dissipative/radiative effects beyond conservative dynamics
• Detailed matching to black hole perturbation theory to fully confirm equivalence for higher-spin structures in Kerr
• Generalization to higher-order multipole moments and non-aligned configurations

Conclusions

This work achieves the first complete calculation of the classical one-loop gravitational Compton amplitude for generic spinning compact objects at 2PM and up to quartic order in spin, fully including spin-induced finite-size effects and non-minimal couplings. The results clarify the role of regularization in higher-spin calculations, provide the crucial building blocks for scattering- and waveform-based approaches to binary inspirals, and verify theoretical expectations such as equivalence with the Kerr scalar probe in the expected limits. The methodology and ancillary results offer a clear path forward for further enhancements in the field-theoretic modeling of strongly gravitating spinning systems, with direct implications for forthcoming gravitational wave data analysis and theory-experiment comparison.