Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations

TL;DR

This paper introduces a combinatorial approach using binary tubings of rooted trees to solve Dyson-Schwinger equations, linking combinatorics with resurgent analysis and revealing effects of root multiplicity on asymptotics.

Contribution

It provides a novel combinatorial interpretation of Dyson-Schwinger solutions and explores the impact of root multiplicity on asymptotic behavior and transseries structure.

Findings

Binary tubings give series solutions to Dyson-Schwinger equations.

The structure of tubings leads to differential equations for the anomalous dimension.

Repeated roots significantly alter asymptotic and transseries properties.

Abstract

We give a novel combinatorial interpretation to the perturbative series solutions for a class of Dyson-Schwinger equations. We show how binary tubings of rooted trees with labels from an alphabet on the tubes, and where the labels satisfy certain compatibility constraints, can be used to give series solutions to Dyson-Schwinger equations with a single Mellin transform which is the reciprocal of a polynomial with rational roots, in a fully combinatorial way. Further, the structure of these tubings leads directly to systems of differential equations for the anomalous dimension that are ideally suited for resurgent analysis. We give a general result in the distinct root case, and investigate the effect of repeated roots, which drastically changes the asymptotics and the transseries structure.