Hard-soft-collinear factorization to all orders

We provide a precise statement of hard-soft-collinear factorization of scattering amplitudes and prove it to all orders in perturbation theory. Factorization is formulated as the equality at leading power of scattering amplitudes in QCD with other amplitudes in QCD computed from a product of operator matrix elements. The equivalence is regulator independent and gauge independent. As the formulation relates amplitudes to the same amplitudes with additional soft or collinear particles, it includes as special cases the factorization of soft currents and collinear splitting functions from generic matrix elements, both of which are shown to be process independent to all orders. We show that the overlapping soft-collinear region is naturally accounted for by vacuum matrix elements of kinked Wilson lines. Although the proof is self-contained, it combines techniques developed for the study of pinch surfaces, scattering amplitudes, and effective field theory.