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Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: $\phi^3$ QFT in 6 Dimensions
High Energy Physics - Theory · Submitted 1 Apr 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.99 | 0.95 | 0.70 | asserted 1.00 |
| s1 | This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. | background 0.88 | 0.23 | 0.07 | asserted 1.00 |
| s2 | 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. | result 0.98 | 0.95 | 0.30 | asserted 1.00 |
| s3 | These expansions are in terms of the renormalized coupling. | result 0.56 | 0.56 | 0.03 | asserted 1.00 |
| s4 | 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. | interpretation 0.76 | 0.84 | 0.21 | asserted 1.00 |
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The paper
Original 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.
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References
How the literature uses it
14 arXiv papers cite this in 29 sentences
Papers citing it · 22 sentences without a confident label
Uses its method or tool 2 papers
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So now we recast the Broadhurst-Kreimer problem, in the manner of Borinksy, Dunne and Meynig [ref] , by setting $g(x)=γ(-3x)/x$, to obtain an ODE that is economically written as [equation] and has an unsummable formal perturbative solution [equation] The expansion coefficients behave as [equation] at large $n$, with a Stokes constant [equation] that can be determined, empirically, by considering a solution [equation] and retaining terms linear in $σ_1$ in the non-linear ODE.
Taming a resurgent ultra-violet renormalon · Trans-series and resurgent hyperasymptotics · 0.95
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Modifying Richardson extrapolation to account for logarithmic corrections (see Ref. [ref] and <ref>), the convergence is improved significantly, as shown in <ref>.
Reduction of Order and Transseries Structure of Radiation Reaction · Reduction of order and LL∞ · 0.84
Cites it as background 3 papers
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In particular, QFT has a variety of applications, including two-dimensional QFTs [ref] , Chern-Simons theory [ref] , the three-dimensional $O(2N)$ model [ref] , four-dimensional nonsupersymmetric QFTs [ref] , six-dimensional $ϕ^3$ theory [ref] , and supersymmetric gauge theories in diverse dimensions [ref] .
Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points · Introduction · 0.84
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Finally, let us note that the original two differential equations of Broadhurst and Kreimer have recently been very strikingly studied from a resurgence perspective in [ref] , and Bellon and collaborators investigated the anomalous dimension in the Wess-Zumino model [ref] in a similar context.
Some results on Landau poles and Feynman diagram cut structure by Hopf algebra · Introduction · 0.99
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To obtain the runaway solutions one would start with e.g. a trans-series-like ansatz; see e.g. the recent papers [ref] for trans-series solutions to differential equations.
Resummation of quantum radiation reaction and induced polarization · Resummation of LAD · 0.80
What it cites · 95 sentences without a confident label
Uses the method or tool of 7 papers
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We employ the trans-series approach to the resurgence properties of non-linear differential equations, along the lines of [ref] .
math.CA/0608414 · Introduction · 1.00
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We employ the trans-series approach to the resurgence properties of non-linear differential equations, along the lines of [ref] .
math.CA/0608408 · Introduction · 1.00
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The idea is to make a conformal map[Significantly higher precision can be obtained with a uniformization map [ref] , which is particularly useful if fewer perturbative terms are available.] from the cut Borel $t$ plane, based on the leading singularity at $t=+1$, to the unit disk in the $z$ plane: [equation] Note that this conformal map does not require knowledge of the nature of the leading Borel singularity, just its location.
Uniformization and Constructive Analytic Continuation of Taylor Series · Borel analysis of the perturbative expansion of the anomalous dimension · 0.93
Compares with 12 papers
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Contrast this with the massless Yukawa theory in 4 dimensions where the corresponding nonlinear equation for the anomalous dimension is quadratic in $γ(a)$, and first order in derivatives with respect to $a$ [ref] .
Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality · Perturbative Hopf-algebraic analysis of massless $ϕ^3$ theory in 6 dimensions · 1.00
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Contrast this with the massless Yukawa theory in 4 dimensions where the corresponding nonlinear equation for the anomalous dimension is quadratic in $γ(a)$, and first order in derivatives with respect to $a$ [ref] .
Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations · Perturbative Hopf-algebraic analysis of massless $ϕ^3$ theory in 6 dimensions · 1.00
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Contrast this with the massless Yukawa theory in 4 dimensions where the corresponding nonlinear equation for the anomalous dimension is quadratic in $γ(a)$, and first order in derivatives with respect to $a$ [ref] .
Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation · Perturbative Hopf-algebraic analysis of massless $ϕ^3$ theory in 6 dimensions · 1.00
Cites as background 18 papers
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The perturbative Hopf algebra structure of Dyson–Schwinger equations of this model was formulated in the pioneering papers [ref] .
Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality · Introduction · 0.81
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The perturbative Hopf algebra structure of Dyson–Schwinger equations of this model was formulated in the pioneering papers [ref] .
Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation · Introduction · 0.81
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The seminal work of Kreimer and Connes showed that there is an underlying Hopf-algebraic structure to the renormalization of quantum field theory (QFT) [ref] .
Renormalization in quantum field theory and the Riemann-Hilbert problem · Introduction · 1.00
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