The Scales of the Infrared

Abstract: In theories with long-range forces like QED or perturbative gravity, loop corrections lead to vanishing amplitudes. There are two well-known procedures to address these infrared divergences: dressing of asymptotic states and inclusion of soft emission. Although both yield the same IR-finite rates, we point out that they are not equivalent since they encode different infrared scales. In particular, dressing states are independent of the resolution scale of radiation. Instead, they define radiative vacua in the von Neumann space. After a review of these concepts, the goal of this paper is to present a combined formalism that can simultaneously describe both dressing and radiation. This unified approach allows us to tackle the problem of quantum decoherence due to tracing over unresolved radiation. We obtain an IR-finite density matrix with non-vanishing off-diagonal elements and estimate how its purity depends on scattering kinematics and the resolution scale. Along the way, we comment on collinear divergences as well as the connection of large gauge transformations and dressing.