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

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