Dimensional renormalization: Ladders and rainbows

Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive the factors and renormalized Green's functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering) diagrams at zero momentum transfer, in the context of dimensional regularization, and we prove that the correct anomalous dimensions for those processes emerge in the limit $D\to 4$.