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Log expansions from combinatorial Dyson-Schwinger equations. (English) Zbl 1446.81032
Summary: 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.
81T18 Feynman diagrams
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
35J08 Green’s functions for elliptic equations
81V80 Quantum optics
81V10 Electromagnetic interaction; quantum electrodynamics
Full Text: DOI
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