The Radial Action from Probe Amplitudes to All Orders

TL;DR

This paper derives the classical relativistic radial action from scattering amplitudes to all orders in perturbation theory, including effects like winding trajectories around sources such as monopoles and NUT backgrounds.

Contribution

It introduces a method to extract all-order radial actions from scattering amplitudes in the probe limit, encompassing complex effects like trajectory winding.

Findings

All-order scattering amplitudes encode winding trajectories.

The scattering amplitude for a probe off a NUT source is given by a transcendental equation.

A systematic approach to solve the transcendental equation to any loop order.

Abstract

We extract the relativistic classical radial action from scattering amplitudes, to all orders in perturbation theory, in the probe limit. Our sources include point charges and monopoles, as well as the Schwarzschild and pure-NUT gravitational backgrounds. A characteristic relativistic effect, that scattering trajectories may wind around these sources any number of times, can be recovered when all-order amplitudes are available. We show that the amplitude for scattering a probe off a pure NUT is given by the solution of a transcendental equation involving continued fractions, and explain how to solve this equation to any desired loop order.