Magnusian: Relating the Eikonal Phase, the On-Shell Action, and the Scattering Generator

TL;DR

This paper clarifies the relationship between the eikonal generator and the classical eikonal phase in scattering amplitudes, revealing their differences and conditions under which they coincide, especially in integrable scattering scenarios.

Contribution

It establishes the exact relationship between the eikonal generator (Magnusian) and the classical eikonal phase, explaining their inequivalence and special cases of coincidence.

Findings

The eikonal generator and phase are generally inequivalent.

In integrable scattering, they coincide up to a Legendre transformation.

The relationship explains why the correspondence fails with spin or radiation.

Abstract

Two fundamentally distinct types of quantities are both called "eikonal" in present amplitudes literature. The unitarity of the S-matrix ensures it can be written as the exponential of a Hermitian operator. The eikonal generator or Magnusian, which is the classical limit of the expectation value of that operator, generates all scattering observables. The leading order classical behavior of the phase of an S-matrix element is called the classical eikonal phase, and it coincides with a classical on-shell action. We demonstrate that the eikonal generator (Magnusian) and the eikonal phase (classical on-shell action) are inequivalent and find the exact general relationship between them. That relationship explains the special case of integrable scattering in which the two do coincide up to a Legendre transformation and explains why such a correspondence fails in general when spin or radiation are included.