Classical and quantum gravitational scattering with Generalized Wilson Lines

The all-order structure of scattering amplitudes is greatly simplified by the use of Wilson line operators, describing eikonal emissions from straight lines extending to infinity. A generalization at subleading powers in the eikonal expansion, known as Generalized Wilson Line (GWL), has been proposed some time ago, and has been applied both in QCD phenomenology and in the high energy limits of gravitational amplitudes. In this paper we revisit the construction of the scalar gravitational GWL starting from first principles in the worldline formalism. We identify the correct Hamiltonian that leads to a simple correspondence between the soft expansion and the weak field expansion. This allows us to isolate the terms in the GWL that are relevant in the classical limit. In doing so we devote special care to the regularization of UV divergences that were not discussed in an earlier derivation. We also clarify the relation with a parallel body of work that recently investigated the classical limit of scattering amplitudes in gravity in the worldline formalism.