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

Abstract

We analyze the asymptotically free massless scalar φ 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.

Noise Effects on Pade Approximants and Conformal Maps

“…Broadhurst and Kreimer showed that in this approach the perturbative expansion of the anomalous dimension γ(a) is characterized by a quartically nonlinear third order ODE for γ(a) [35]. The resurgent trans-series structure of this function has recently been analyzed in detail in [36,37], revealing an intricate Borel Riemann surface structure. On the first sheet there is a single dominant Borel branch point singularity with exponent 1 12 , in addition to two further resonant collinear Borel singularities, and all three of these singularities are repeated in integer multiples.…”

“…Here we show that if noise is introduced to this computation, the result is dominated by the leading Borel singularity, so the slope relating the breakdown order N c to the logarithm of the noise strength, as in (34), can be well approximated by the one-cut situation. of the anomalous dimension in the Hopf algebraic analysis of the ϕ 3 scalar quantum field theory in 6 dimensions [35,36]. The blue dots show the average of multiple realizations of the random noise, and the blue line shows the general one-cut scaling relation in (18).…”

“…Figure7: This plot shows the critical truncation order N c at which Padé breaks down in the presence of noise, as a function of the logarithm of the noise strength, for the Borel transform (37) of the anomalous dimension in the Hopf algebraic analysis of the ϕ 3 scalar quantum field theory in 6 dimensions[35,36]. The blue dots show the average of multiple realizations of the random noise, and the blue line shows the general one-cut scaling relation in(18).…”

Resummation of quantum radiation reaction and induced polarization

“…a trans-series-like ansatz; see e.g. the recent papers [27,28] for trans-series solutions to differential equations.…”

Reduction of order, resummation and radiation reaction

“…While it is known that there is information encoded in such asymptotic growth rates[46], it is out of scope for our present purposes, as we will obtain accurate resummations with only a handful terms.…”