Filtrations in Dyson–Schwinger equations: Next-to[formula omitted]-leading log expansions systematically

Abstract

Dyson–Schwinger equations determine the Green functions Gr(α,L) in quantum field theory. Their solutions are triangular series in a coupling constant α and an external scale parameter L for a chosen amplitude r, with the order in L bounded by the order in the coupling.Perturbation theory calculates the first few orders in α. On the other hand, Dyson–Schwinger equations determine next-to{j}-leading log expansions, Gr(α,L)=1+∑j=0∞∑MpjMαjM(u). ∑M sums for any finite j a finite number of functions M in u. Here, u is the one-loop approximation to Gr, for example, for the (inverse) propagator in massless Yukawa theory, u=αL/2.The leading logs come then from the trivial representation, e.g. M(u)=1−1−2u in massless Yukawa theory. The respective period at j=0 is p0M(u)=1. All non-leading logs are organized by corresponding suppressions in powers αj.We describe an algebraic method to derive all next-to{j}-leading log terms from the knowledge of the first (j+1) terms in perturbation theory and their filtrations. This implies the calculation of the functions M(u) and periods pjM.In the first part of our paper, we investigate the structure of Dyson–Schwinger equations and develop a method to filter their solutions. Applying renormalized Feynman rules maps each filtered term to a certain power of α and L in the log-expansion.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.