Systematic differential renormalization to all orders

We present a systematic implementation of differential renormalization to all orders in perturbation theory. The method is applied to individual Feynman graphs written in coordinate space. After isolating every singularity which appears in a bare diagram, we define a local subtraction procedure which consists in replacing the core of the singularity by its renormalized form given by a differential formula. The organization of subtractions in subgraphs relies on Bogoliubov′s formula, fulfilling the requirements of locality, unitarity, and Lorentz invariance. Our method bypasses the use of an intermediate regularization and automatically delivers renormalized amplitudes which obey renormalization group equations.