Log expansions from combinatorial Dyson–Schwinger equations

Abstract

AbstractWe 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 $$\alpha $$ α and a logarithmic energy scale L—a reordering of terms as $$G(\alpha ,L) = 1 \pm \sum _{j \ge 0} \alpha ^j H_j(\alpha L)$$ G ( α , L ) = 1 ± ∑ j ≥ 0 α j H j ( α L ) is the corresponding log expansion. In a first part of this paper, we derive the leading log order $$H_0$$ H 0 and the next-to$$^{(j)}$$ ( j ) -leading log orders $$H_j$$ H j from the Callan–Symanzik equation. In particular, $$H_j$$ H j only depends on the $$(j+1)$$ ( j + 1 ) -loop $$\beta $$ β -function and anomalous dimensions. In two specific examples, our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature: 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. 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.