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

Abstract: Two fundamentally distinct 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 S-matrix elements is called the eikonal phase, and it coincides with the 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.