Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: phi3 QFT in 6 Dimensions

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