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Log-Expansions from Combinatorial Dyson-Schwinger Equations

We give a precise connection between combinatorial Dyson-Schwinger equations and log-expansions for Green's functions in quantum field theory. The latter are triangular power series in the coupling constant $α$ and a logarithmic energy scale $L$ --- a reordering of terms as $G(α,L) = 1 \pm \sum_{j \geq 0} α^j H_j(αL)$ is the corresponding log-expansion. In a first part of this paper, we derive the leading-log order $H_0$ and the next-to$^{(j)}$-leading log orders $H_j$ from the Callan-Symanzik equation. In particular, $H_j$ only depends on the $(j+1)$-loop $β$-function and anomalous dimensions. For the photon propagator Green's function in quantum electrodynamics (and in a toy model, where all Feynman graphs with vertex sub-divergences are neglected), our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature. In a second part of this work, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations, i.e. fixed-point relations for the Green's functions. Combining the arguments, our work provides a derivation of the log-expansions for Green's functions from the corresponding Dyson-Schwinger equations.