A chord diagram expansion coming from some Dyson-Schwinger equations

TL;DR

This paper presents a novel expansion method for solving propagator Dyson-Schwinger equations using rooted connected chord diagrams, refining classical recurrences and representing solutions with binary trees.

Contribution

It introduces a new expansion approach for Dyson-Schwinger equations based on chord diagrams and binary tree representations, enhancing analytical tools in quantum field theory.

Findings

Derived an explicit expansion for Dyson-Schwinger equations

Refined classical recurrence relations for chord diagrams

Connected diagram representations with binary trees

Abstract

We give an expression for the solution to propagator-type Dyson-Schwinger equations with one primitive at 1 loop as an expansion over rooted connected chord diagrams. Along the way we give a refinement of a classical recurrence of rooted connected chord diagrams, and a representation in terms of binary trees.