Generalized chord diagram expansions of Dyson-Schwinger equations

TL;DR

This paper introduces a novel method for solving Dyson-Schwinger equations using expansions over decorated rooted connected chord diagrams, linking combinatorial structures with analytic solutions.

Contribution

It provides a new combinatorial expansion framework for Dyson-Schwinger equations based on decorated chord diagrams, enhancing analytic solution control.

Findings

New series solutions for Dyson-Schwinger equations using chord diagrams

Explicit connection between diagram decorations and primitive graph expansions

Improved understanding of the combinatorial structure of Dyson-Schwinger solutions

Abstract

Series solutions for a large family of single equation Dyson-Schwinger equations are given as expansions over decorated rooted connected chord diagrams. The analytic input to the new expansions are the expansions of the regularized integrals for the primitive graphs building the Dyson-Schwinger equation. Each decorated chord diagram contributes a weighted monomial in the coefficients of the expansions of the primitives and so indexes the analytic solution in a tightly controlled way.