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

Dyson-Schwinger equations determine the Green functions $G^r(\alpha,L)$ in quantum field theory. Their solutions are triangular series in a coupling constant $\alpha$ 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 $\alpha$. On the other hand, Dyson--Schwinger equations determine next-to$^{\{\mathrm{j}\}}$-leading log expansions, $G^r(\alpha,L) = 1 + \sum_{j=0}^\infty \sum_{\mathcal{M}} p_j^{\mathcal{M}}\alpha^j \mathcal{M}(u)$. $\sum_{\mathcal{M}}$ sums a finite number of functions $\mathcal{M}$ in $u = \alpha L/2$. The leading logs come from the trivial representation \mathcal{M}(u) = \begin{bsmallmatrix}\bullet\end{bsmallmatrix}(u) at $j=0$ with p_0^{\begin{bsmallmatrix}\bullet\end{bsmallmatrix}} = 1. All non-leading logs are organized by the suppression in powers $\alpha^j$. We describe an algebraic method to derive all next-to$^{\{\mathrm{j}\}}$-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 $\mathcal{M}(u)$ and periods $p_j^\mathcal{M}$. 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 $\alpha$ and $L$ in the log-expansion. Based on this, the second part derives the next-to$^{\{\mathrm{j}\}}$-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. The reader may apply our method to any (set of) Dyson-Schwinger equation(s) appearing in renormalizable quantum field theories.