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Gravitational Wave Scattering in Spinless WQFT

Yilber Fabian Bautista, Mathias Driesse, Kays Haddad, Gustav Uhre Jakobsen

TL;DR

This work develops a spinless worldline quantum field theory (WQFT) framework for gravitational wave scattering off a Schwarzschild black hole and demonstrates that the exponential S-matrix, $\hat{S}=e^{i\hat{N}}$, maps, through a partial-wave transform, directly onto the black-hole perturbation theory (BHPT) phase shifts in the absence of dissipation. By computing the $N$-matrix up to two loops, the authors reproduce BHPT phase shifts to $\mathcal{O}(G^{3})$ and establish that the $N$-matrix is infrared finite and exponentiates in the relevant basis, enabling a direct projection onto BHPT observables. The paper introduces an efficient diagram-generation scheme and a detailed treatment of the required two-loop integrals via canonical differential equations, providing a robust route for incorporating non-minimal, higher-spin, and tidal effects through future Wilson-coefficient matching. Collectively, the results validate WQFT as a precise and scalable platform for gravitational-wave/BH scattering and set the stage for precision tests of GR and black-hole tidal responses at higher PM orders. This framework paves the way for including spin, finite-size effects, and Kerr backgrounds, with potential applications to gravitational-wave waveform modeling and BH spectroscopy.

Abstract

We develop the computational framework for gravitational wave - black hole scattering in worldline quantum field theory (WQFT) without spin. Crucially, we prove on general grounds that, in the absence of dissipation, the exponential representation of the $S$-matrix maps -- through a partial-wave transformation -- directly onto the scattering phase shift from black hole perturbation theory (BHPT), indicating an exponentiation of the WQFT amplitude itself in partial-wave space. Computing explicitly, we reproduce the BHPT phase shift without spin up to $O(G^{3})$ from WQFT. While this result is expected, it lays the groundwork for higher-precision analyses involving non-minimal effects. Along the way, we outline our efficient diagram generation technique and include a pedagogical discussion on the computation of the required two-loop integrals.

Gravitational Wave Scattering in Spinless WQFT

TL;DR

This work develops a spinless worldline quantum field theory (WQFT) framework for gravitational wave scattering off a Schwarzschild black hole and demonstrates that the exponential S-matrix, , maps, through a partial-wave transform, directly onto the black-hole perturbation theory (BHPT) phase shifts in the absence of dissipation. By computing the -matrix up to two loops, the authors reproduce BHPT phase shifts to and establish that the -matrix is infrared finite and exponentiates in the relevant basis, enabling a direct projection onto BHPT observables. The paper introduces an efficient diagram-generation scheme and a detailed treatment of the required two-loop integrals via canonical differential equations, providing a robust route for incorporating non-minimal, higher-spin, and tidal effects through future Wilson-coefficient matching. Collectively, the results validate WQFT as a precise and scalable platform for gravitational-wave/BH scattering and set the stage for precision tests of GR and black-hole tidal responses at higher PM orders. This framework paves the way for including spin, finite-size effects, and Kerr backgrounds, with potential applications to gravitational-wave waveform modeling and BH spectroscopy.

Abstract

We develop the computational framework for gravitational wave - black hole scattering in worldline quantum field theory (WQFT) without spin. Crucially, we prove on general grounds that, in the absence of dissipation, the exponential representation of the -matrix maps -- through a partial-wave transformation -- directly onto the scattering phase shift from black hole perturbation theory (BHPT), indicating an exponentiation of the WQFT amplitude itself in partial-wave space. Computing explicitly, we reproduce the BHPT phase shift without spin up to from WQFT. While this result is expected, it lays the groundwork for higher-precision analyses involving non-minimal effects. Along the way, we outline our efficient diagram generation technique and include a pedagogical discussion on the computation of the required two-loop integrals.
Paper Structure (14 sections, 106 equations, 3 figures)

This paper contains 14 sections, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Wave scattering in WQFT and BHPT
  3. Scattering in WQFT
  4. Scattering in BHPT
  5. Matching strategy
  6. Diagrammatics and integration
  7. Integrand generation
  8. Integration
  9. Two-loop results & BHPT matching
  10. Summary & outlook
  11. WQFT framework and classical $n$-point functions
  12. Integrals
  13. Spin-weighted spherical harmonics
  14. Euler angles

Figures (3)

  • Figure 1: Schematic of our kinematic setup. An idealized plane wave of definite momentum $k_1^\mu$ and helicity $h_1$ scatters off a black hole with momentum $p^\mu$. The scattered wave comes out at an angle $\theta$ with momentum $k_2^\mu$ and helicity $h_2$.
  • Figure 2: The six one-loop diagrams. Using retarded propagators, all causality flows from left to right. Only active (red) gravitons can go on-shell and the $i0^+$ prescription matters only for those. Graphs (4)-(6) vanish in the TT gauge implied by \ref{['eq:Kinematics']}. The six graphs have the symmetry factors (1, $1/2$, $1/2$, 1, 1, 1).
  • Figure 3: The 20 two-loop diagrams. Using retarded propagators, all causality flows from left to right. Only active (red) gravitons can go on-shell and the $i0^+$ prescription matters only for those. In TT gauge, only graphs (1)-(10) are non-zero. Graphs (2, 3, 4, 5, 7, 9, 12, 13, 15, 16) have symmetry factor 1/2, graphs (6, 8) have symmetry factor $1/6$ and all other graphs have symmetry factor 1.