Analytic One-loop Scattering Waveform in General Relativity

TL;DR

This work delivers an analytic next-to-leading order gravitational waveform in General Relativity for scattering at all velocities, computed in impact-parameter space via a unitarity-based, KMOC framework. By treating Fourier and loop integrals on equal footing and reducing to a minimal basis of 28 combined master integrals, the authors obtain a robust, numerically friendly representation of the waveform and its frequency-domain structure. They separate infrared, tail, and finite contributions, verifying consistency with known momentum-space results and providing a numerical path to the NLO power spectrum through Lebedev quadrature. The approach extends to spinning systems and offers a practical route to high-precision GW templates in the scattering regime, with potential applications to higher-order radiative observables. Overall, the paper demonstrates the power of amplitude-based methods to yield controlled classical gravitational observables, bridging quantum amplitudes, Fourier analysis, and gravitational-wave phenomenology.

Abstract

Leveraging the computational framework presented in reference [JHEP 07, 062 (2024)], we evaluate the analytic scattering waveform in General Relativity to second order, $G^3 M^3 /r b^2$ and to all orders in velocity. This new representation of the next-to-leading order waveform is well-suited for numerical evaluation. Integrating the [modulus square of the] waveform over the angles on the celestial sphere, we also compute the power spectrum of the radiation to order $G^4$ numerically.

Figures (13)

• Figure 1: Generalized unitarity cuts probing the singularities at $q_i^2=0$ at one-loop. Left: double graviton exchange. Right: single graviton exchange with the one-loop gravitational Compton amplitude on one side of the cut. This figure has been copied from Ref. Brunello:2024ibk.
• Figure 2: When considering one-particle and two-particle generalized unitarity cut in general relativity, there are overlapping contributions due to the nonlinear nature of General Relativity. These diagrams give non-zero contributions in taking residues and discontinuities at $q_i^2=0$.
• Figure 3: Integrals arising from the application of IBP identities to the one-loop amplitude for combined Fourier and loop integrals can be reorganized into one of these three topologies, ordered as in Eq. \ref{['eq:subtopo_fl']}. The integral family of Eq. \ref{['eq:topo_fl']} includes additional master integrals, but they do not contribute to the final result.
• Figure 4: One-loop diagrams with a pinched graviton propagator $D_8$ arise when applying IBPs solely to the loop integration, but vanish in the final result after both Fourier and loop IBP reduction.
• Figure 5: Plots of the finite part of the NLO frequency-domain waveform $\mathcal{W}_{h,\text{fin}}^{(1)}$, for the $[+]$ polarization. Each plot show different values of the boost $\gamma$ with fixed symmetric mass ratio $\nu$, which are, respectively $\{ \frac{1}{4}, \frac{2}{9},\frac{5}{36}\}$. The plots are at the fixed angle $(\theta,\phi)= \left(\frac{7 \pi}{5},\frac{7\pi}{10} \right)$. Solid lines represent the real part, while dashed lines represent the imaginary part.