Log expansions from combinatorial Dyson\u2013Schwinger equations

Abstract

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 alpha 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) 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 beta -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.