An effective theory on the light shell

Abstract

We describe work on the construction of an effective field theory on a spherical light shell. The motivation arises from classical electromagnetism: If a collision produces charged particles with zero net charge emerging simultaneously from a point and instantaneously accelerating to the speed of light, then the electromagnetic fields due to these charges lie entirely on a spherical shell expanding at the speed of light. We show that this also applies to classical color radiation from high-energy collisions that produce colored particles. Specifically, the color fields produced in such a process are associated with a non-linear σ-model on the 2D light shell with specific symmetry-breaking terms. The quantum version of such a picture exhibits asymptotic freedom and should therefore be a useful starting point for a light-shell effective theory for QCD. We start in the simplified context of zero-flavor scalar quantum electrodynamics. Our effective theory has 3 major ingredients: breaking down the fields into soft and hard sectors with the large energy of the hard fields in the radial direction scaled out, a special gauge called light-shell gauge in which the picture simplifies, and a gauge-invariant source defined on a spherical light shell having infinitesimal radius. We match the fields between the effective theory and the full theory, meaning zero-flavor scalar QED. This allows us to compute the amplitude for the production of any number of scalars from the gauge-invariant source. We then find the tree-level amplitude for the emission of a photon using our effective theory and show that our result agrees with the full theory. To calculate loop effects in our effective theory, we need the photon propagator in light-shell gauge. We derive this propagator and use it to calculate the 1-loop correction to the amplitude for the production of a scalar and anti-scalar pair arising from virtual photon effects. This reduces to a pair of purely angular integrals in the effective theory and reproduces the familiar double logs of the full theory subject to an appropriate interpretation of an angular cutoff.