Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: $\phi^3$ QFT in 6 Dimensions

TL;DR

This paper employs resurgent asymptotic analysis to derive trans-series solutions for the divergent perturbative series in a 6-dimensional massless scalar $\

Contribution

It introduces a novel trans-series solution to Dyson-Schwinger equations in $\

Findings

Identifies a complex Borel singularity structure with three resonant non-perturbative terms.

Reveals logarithmic terms arising from instanton interactions within a purely perturbative framework.

Provides a non-perturbative completion of the perturbative series in $\

Abstract

We analyze the asymptotically free massless scalar $\phi^3$ quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent perturbative solutions to the Kreimer-Connes Hopf-algebraic Dyson-Schwinger equations for the anomalous dimension. This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic approach we find a trans-series having an intricate Borel singularity structure, with three distinct but resonant non-perturbative terms, each repeated in an infinite series. These expansions are in terms of the renormalized coupling. The resonant structure leads to powers of logarithmic terms at higher levels of the trans-series, analogous to logarithmic terms arising from interactions between instantons and anti-instantons, but arising from a purely perturbative formalism rather than from a semi-classical analysis.