All orders factorization and the Coulomb problem

In the limit of large nuclear charge, Z 1, or small lepton velocity, β 1, Coulomb corrections to nuclear beta decay and related processes are enhanced as Zα/β and become large or even non-perturbative (with α the QED fine structure constant). We provide a constructive demonstration of factorization to all orders in perturbation theory for these processes and compute the all-orders hard and soft functions appearing in the factorization formula. We clarify the relationship between effective field theory amplitudes and historical treatments of beta decay in terms of a Fermi function.