Classical black hole scattering from a worldline quantum field theory

TL;DR

This work establishes a nonperturbative bridge between classical black hole scattering and a worldline quantum field theory by representing graviton-dressed scalar propagators as worldline path integrals and mapping S-matrix elements to WQFT expectation values in the classical limit. It introduces a comprehensive set of WQFT Feynman rules that treat $h_{\mu u}$ and worldline fluctuations on equal footing, enabling direct calculations of $3PM$ radiation and $2PM$ deflection, and revealing the eikonal phase of a $2\to2$ scalar S-matrix as the free energy $\mathcal{Z}_{\rm WQFT}=e^{i\chi}$. The paper provides explicit 2PM and 3PM results, derived via both worldline diagrams and corresponding amplitudes, and demonstrates the consistency of the framework with known PM results and Ward identities. This approach offers a unified, diagrammatic pathway to PM observables, with clear avenues for extending to higher PM orders, spin, and double-copy relations, and for connecting to amplitude-based methods in a computationally efficient way.

Abstract

A precise link is derived between scalar-graviton S-matrix elements and expectation values of operators in a worldline quantum field theory (WQFT), both used to describe classical scattering of a pair of black holes. The link is formally provided by a worldline path integral representation of the graviton-dressed scalar propagator, which may be inserted into a traditional definition of the S-matrix in terms of time-ordered correlators. To calculate expectation values in the WQFT a new set of Feynman rules is introduced which treats the gravitational field $h_{μν}(x)$ and position $x_i^μ(τ_i)$ of each black hole on equal footing. Using these both the next-order classical gravitational radiation $\langle h^{μν}(k)\rangle$ (previously unknown) and deflection $Δp_i^μ$ from a binary black hole scattering event are obtained. The latter can also be obtained from the eikonal phase of a $2\to2$ scalar S-matrix, which we show to correspond to the free energy of the WQFT.

Figures (3)

• Figure 1: Graphical representation of the Green function $G(x,x')$ of eq. \ref{['eq:BP2']} for a massive scalar moving in a weak gravitational background $g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$, excluding the $R\phi^2$ interaction in eq. \ref{['eq:Sprime']}. A closed expression for the Green's function in momentum space may be found in eq. \ref{['eq:Fromaggio']}.
• Figure 2: An example of a self-energy diagram, which does not appear in a three-body calculation at the same PM order. On support of the $\delta$-function constraints we have $\omega=0$, which gives rise to a singularity in the $1/\omega^2$ propagator.
• Figure 3: The seven diagrams contributing to the radiation $k^2\langle h^{\mu\nu}(k) \rangle_{\rm WQFT}$ at order 3PM, and their respective symmetry factors. All seven graphs represent tree-level contributions, the worldlines being drawn only as a visual aid. The outgoing momentum from each worldline is $q_i^\mu$.