Analytic Properties of Infrared-Finite Amplitudes in Theories with Long-Range Forces

Infrared divergences obscure important analytic properties of scattering amplitudes, indicating gaps in our understanding of unitarity, causality, and crossing symmetry in theories with long-range forces. Using the exactly solvable model of a charged scalar particle in a fixed Coulomb background, we demonstrate that novel analytic properties arise and can be systematically studied when long-range interactions are properly incorporated. We first canonically quantize a scalar particle in a Coulomb potential, confirming that basic conditions for unitarity and causality hold. We then examine the necessary modifications to the LSZ reduction formula, the general optical theorem, and the treatment of the disconnected components of scattering amplitudes. Next, we show that the Coulomb phase divergence is analytically related to real radiative divergences via crossing symmetry, implying that a well-defined treatment of the Coulomb phase divergence provides constraints on the real radiative divergence. In contrast to the Faddeev-Kulish approach, we propose that an effective way to eliminate infrared divergences and study these analytic properties is to fully solve the quantum theory associated with the asymptotic Hamiltonian.