Perturbation expansions at large order: results for scalar field theories revisited

McKane, Alan J.

doi:10.1088/1751-8121/aaf768

Cited by 16 publications

(18 citation statements)

References 50 publications

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“…Finally, by reevaluating Eq. (15) for N f = 1, 3, 5, 10, the position of respective UV renormalons is marked in Fig. 5 with the dashed grey vertical line.…”

Section: U (1) As a Function Of The Number Of Flavorsmentioning

confidence: 99%

Resummation in QFT with Meijer G-functions

2019

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We employ a recent resummation method to deal with divergent series, based on the Meijer G-function, which gives access to the non-perturbative regime of any QFT from the first few known coefficients in the perturbative expansion. Using this technique, we consider in detail the φ 4 model where we estimate the non-perturbative β−function and prove that its asymptotic behavior correctly reproduces instantonic effects calculated using semiclassical methods. After reviewing the emergence of the renormalons in this theory, we also speculate on how one can resum them. Finally, we resum the non-perturbative β−function of abelian and non-abelian gauge-fermion theories and analyze the behavior of these theories as a function of the number of fermion flavors. While in the former no fixed points are found, in the latter, a richer phase diagram is uncovered and illustrated by the regions of confinement, large-distance conformality, and asymptotic safety. 1 arXiv:1807.05060v3 [hep-th]

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“…Finally, by reevaluating Eq. (15) for N f = 1, 3, 5, 10, the position of respective UV renormalons is marked in Fig. 5 with the dashed grey vertical line.…”

Section: U (1) As a Function Of The Number Of Flavorsmentioning

confidence: 99%

Resummation in QFT with Meijer G-functions

2019

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“…The perturbative Hopf algebra structure of Dyson-Schwinger equations of this model was formulated in the pioneering papers [22,23]. On the non-perturbative side, this QFT has a real Lipatov instanton when g is real, for which the conventional one-instanton semi-classical analysis [21,66,78,93] of the fluctuation determinant has been studied [76,77]. Further extensions to multi-dimensional cubic interactions have many interesting applications and implications for conformal quantum field theories in general [11,19,38,39,49,52,53,54,55,56,57].…”

Section: Introductionmentioning

confidence: 99%

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

Borinsky

Dunne

Meynig

2021

SIGMA

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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.

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“…The dominance of the a k,0 would correspond to asymptotic dominance by primitive Feynman diagrams as the a k,0 are the leading terms for each primitive. What is currently known about whether or not the contributions of the primitive diagrams dominates is explained in section 4 of [12]; while it remains conjectural, McKane therein explains that in the 80s the contribution of the primitives was generally expected to dominate, and that this remains plausible but unproven from today's perspective. Dominance of the low terms, a 1,0 , a 1,1 , a 2,0 etc., corresponds to renormalon-type behavior as these terms come from small Feynman diagrams iterated into themselves repeatedly.…”

Section: Discussionmentioning

confidence: 99%

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

Courtiel

Yeats

2020

Commun. Math. Phys.

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Green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter. The leading logs are given by the main diagonal of this expansion, i.e. the subseries where the coupling and the scale parameter appear to the same power; then the next-to leading logs are listed by the next diagonal of the expansion, where the power of the coupling is decremented by one, and so on.We give a general method for deriving explicit formulas and asymptotic estimates for any nextto k leading-log expansion for a large class of single scale Green functions. These Green functions are solutions to Dyson-Schwinger equations that are known by previous work to be expressible in terms of chord diagrams. We look in detail at the Green function for the fermion propagator in massless Yukawa theory as one example, and the Green function of the photon propagator in quantum electrodynamics as a second example, as well as giving general theorems. Our methods are combinatorial, but the consequences are physical, giving information on which terms dominate and on the dichotomy between gauge theories and other quantum field theories. arXiv:1906.05139v1 [math-ph]

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Perturbation expansions at large order: results for scalar field theories revisited

Cited by 16 publications

References 50 publications

Resummation in QFT with Meijer G-functions

Resummation in QFT with Meijer G-functions

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

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

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