Unitarity and the On-Shell Action of Worldline Quantum Field Theory

TL;DR

This work develops an on-shell action formalism within Worldline Quantum Field Theory (WQFT) to describe spinning-body scattering in General Relativity at post-Minkowskian (PM) order. The real on-shell action, derived from vacuum diagrams with retarded propagators, generates scattering observables via Poisson brackets on background data, establishing a concrete link to amplitude methods through generalized unitarity. The authors construct the full on-shell action up to formal 3PM, including dissipative terms and up to quartic spin on both bodies, and prove the equivalence with scattering-amplitude on-shell actions up to two loops. This framework integrates unitarity-based techniques into WQFT, enabling efficient high-spin, high-PM calculations and setting the stage for higher-loop and radiation-reaction analyses in gravitational scattering.

Abstract

We develop the on-shell action formalism within Worldline Quantum Field Theory (WQFT) to describe scattering of spinning compact bodies in General Relativity in the post-Minkowskian (PM) expansion. The real on-shell action is constructed from vacuum diagrams with causal (retarded) propagators from which scattering observables such as momentum impulse and spin kick follow via Poisson brackets of the initial scattering data. Furthermore, we explore the implications of unitarity at the level of the worldline and show how generalised unitarity techniques can be adapted to WQFT to efficiently compute multi-loop contributions. Our work establishes a concrete link between WQFT and amplitude-based methods, elucidating how unitarity cuts ensure equivalence between the on-shell action derived from either approach. Extending the state-of-the-art, we complete the full on-shell action -- including dissipative terms -- at (formal) 3PM order and up to quartic spin interactions on both massive bodies.