Worldline formalism, eikonal expansion and the classical limit of scattering amplitudes

We revisit the fundamentals of two different methods for calculating classical observables: the eikonal method, which is a scattering amplitude-based method, and the worldline quantum field theory (WQFT) method. The latter has been considered an extension of the worldline effective field theory. We show that the eikonal and WQFT methods are equivalent and that calculations can be translated freely between them. We also show how the scattering amplitude can serve as the generating function of all the WQFT diagrams. This offers another computational avenue for the WQFT diagrams, which can be useful at higher loop orders. Concretely, we focus on 2-into-2 scattering processes mediated by massless force carriers. On one hand, taking the classical limit of the QFT scattering amplitude leads to the eikonal method. On the other hand, since in the classical limit the scattering particles are almost on-shell throughout the scattering process, the worldline, a first quantized formalism, is the most efficient framework to study the scattering amplitude. This is an alternate but equivalent formalism to the quantum field theoretical (QFT) framework. By taking the classical limit of the scattering amplitude computed in the worldline, we can derive the WQFT rules of Mogull, Plefka and Steinhoff [1]. In WQFT, the Feynman diagrams are reorganized into a new set of diagrams that facilitate the ℏ expansion. Unlike the QFT eikonal method, which identifies the eikonal phase recursively, the worldline-based computation allows to target and systematically extract the classical contributions directly through a specific set of WQFT diagrams. In worldline formalism the perturbative expansion of the scattering amplitude is naturally organized in diagrams which factorize (reducible) and diagrams which are new to that order (irreducible), in a one-to-one map with the structure of the amplitude in the eikonal method. This opened up the possibility to investigate and prove the conjectured exponentiation of the eikonal phase in Du et al.[2].