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

We give a novel combinatorial interpretation for the perturbative series solutions to 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.