Filtrations in Dyson-Schwinger equations: Next-to{j}-leading log expansions systematically

Based on this, the second part derives the next-to{j}-leading log expansions. Our method is general. Here, we exemplify it using the examples of the propagator in Yukawa theory and the photon self-energy in quantum electrodynamics. In particular, we give explicit formulas for the leading log, next-to-leading log and next-to-next-to-leading log orders in terms of at most three-loop Feynman integrals. The reader may apply our method to any (set of) Dyson-Schwinger equation(s) appearing in renormalizable quantum field theories.