Classical eikonal from Magnus expansion

TL;DR

This paper introduces a novel approach using the Magnus expansion and Hopf algebra to efficiently compute the classical eikonal in scattering problems, applicable to gravitational binaries, and demonstrates its advantages over traditional methods.

Contribution

It develops a fast algorithm leveraging the Magnus expansion and Hopf algebra to compute the classical eikonal to high order, improving efficiency and handling divergences.

Findings

Successfully computed the 3PM eikonal matching previous results.

Demonstrated the method's efficiency in calculating high-order terms.

Showed the finite nature of the Magnus-based eikonal versus divergent naive approaches.

Abstract

In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods. Importantly, the Magnus expansion yields a finite eikonal, while the na\"ive eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on.