Tubings, chord diagrams, and Dyson--Schwinger equations

TL;DR

This paper introduces a combinatorial approach using binary tubings of rooted trees to solve complex Dyson--Schwinger equations, extending previous methods to more general systems and parameters.

Contribution

It presents a new series solution method for Dyson--Schwinger equations using binary tubings, broadening applicability beyond chord diagram techniques.

Findings

Solutions are combinatorially transparent and straightforward to interpret.

Extended the class of Dyson--Schwinger equations solvable by combinatorial methods.

Identified interesting combinatorial connections and properties.

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.