  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.
