Tubings, chord diagrams, and Dyson--Schwinger equations

Abstract

We give series solutions to single insertion place propagator-type systems of Dyson-Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson-Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and non-integer values of the insertion parameter s. We remark on interesting combinatorial connections and properties. * PHB thanks KY for a stay at U Waterloo in early 2022, during which many of the results of this work were first discovered. KY is supported by an NSERC Discovery grant and the Canada Research Chairs program. KY thanks the Perimeter Institute for its current support and NTNU Trondheim for its hospitality on a 2018 visit which ultimately led to this project.