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

Courtiel, Julien; Yeats, Karen

doi:10.1007/s00220-020-03691-7

Cited by 12 publications

(16 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(2.13, 2.18 and 2.20). 18 In [6], equivalent expressions have been derived using Chord diagrams. These formulas contain the symbols a i,j , which are related to the anomalous dimension γ e as follows: with only one interaction term in the Lagrangian (we explain the redefinition of the coupling constant in Section 3.1).…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Log expansions from combinatorial Dyson–Schwinger equations

Krüger

2020

Lett Math Phys

View full text Add to dashboard Cite

We give a precise connection between combinatorial Dyson-Schwinger equations and log-expansions for Green's functions in quantum field theory. The latter are triangular power series in the coupling constant α and a logarithmic energy scale L -a reordering of terms as G(α, L) = 1± j≥0 α j H j (αL) is the corresponding log-expansion. In a first part of this paper, we derive the leading-log order H 0 and the next-to (j) -leading log orders H j from the Callan-Symanzik equation. In particular, H j only depends on the (j + 1)-loop β-function and anomalous dimensions. For the photon propagator Green's function in quantum electrodynamics (and in a toy model, where all Feynman graphs with vertex sub-divergences are neglected), our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature. In a second part of this work, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations, i.e. fixed-point relations for the Green's functions. Combining the arguments, our work provides a derivation of the log-expansions for Green's functions from the corresponding Dyson-Schwinger equations.

show abstract

Section: Mathematical Backgroundmentioning

confidence: 99%

Log expansions from combinatorial Dyson–Schwinger equations

Krüger

2020

Lett Math Phys

View full text Add to dashboard Cite

show abstract

“…The first few integers [86] as entry A051862. There is currently no known combinatorial interpretation of the integer A n coefficients, in contrast to the 4d massless Yukawa model analyzed in [18,22,23], where the corresponding perturbative expansion is the generating function for connected chord diagrams [36,69,75,91].…”

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

View full text Add to dashboard Cite

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.

show abstract

“…This sequence is listed in the OEIS [70] as entry A051862. There is currently no known combinatorial interpretation of the integer A n coefficients, in contrast to the 4d massless Yukawa model analyzed in [4,5,22], where the corresponding perturbative expansion is the generating function for connected chord diagrams [19,[89][90][91].…”

Section: Perturbative Hopf Algebraic Analysis Of Massless φ 3 Theory ...mentioning

confidence: 99%

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

Borinsky,

Dunne,

Meynig

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 12 publications

References 23 publications

Log expansions from combinatorial Dyson–Schwinger equations

Log expansions from combinatorial Dyson–Schwinger equations

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

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

Contact Info

Product

Resources

About