Filtrations in Dyson–Schwinger equations: Next-to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mi>j</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:math>-leading log expansions systematically

Krueger, Olaf; Kreimer, Dirk

doi:10.1016/j.aop.2015.05.013

Cited by 21 publications

(36 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our results (see Eq. (2.21)) agree with [6], but differ from [7]. The reason for the mismatch is a mistake in [7], where the shuffle products of commutators were computed using a wrong algorithm.…”

Section: Photonmentioning

confidence: 99%

See 1 more Smart Citation

Log expansions from combinatorial Dyson–Schwinger equations

Krüger

2020

Lett Math Phys

Self Cite

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: Photonmentioning

confidence: 99%

“…In this way, φ R (Γ) is a scalar, although the corresponding amplitude may be of a tensorial structure (e.g. it could be proportional to a Dirac matrix γ µ ) 7. There is no problem, that three different graphs are identified with the same element I.…”

mentioning

confidence: 99%

Log expansions from combinatorial Dyson–Schwinger equations

Krüger

2020

Lett Math Phys

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our results are as found in Table 2 and Propositions 4.1, 4.2, and 4.8. See Section A.4 of [9] for an analogous statement of their results. We see a precise correspondence in the leading log and next-to leading log results.…”

Section: Comparison With Krüger and Kreimermentioning

confidence: 91%

“…Consequently, a diagram is connected when the drawing of its linear representation is "in one piece". For example, the left diagram of Figure 1 is not connected since there are two connected components ( (1,8), (6,9), (7, 10) on the one hand, (2,4), (3,5) on the other hand), while the right diagram is connected. • The root chord of C is the first chord in the intersection order.…”

Section: Definition 21 (Chord Diagram)mentioning

confidence: 99%

See 1 more Smart Citation

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

Courtiel

Yeats

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

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]

show abstract

Graphs in Perturbation Theory

Borinsky

2018

Springer Theses

View full text Add to dashboard Cite

This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, will be introduced. The Hopf algebraic structure on graphs permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories. The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together both structures provide an algebraic formulation of large graphs with constraints on the allowed subgraphs. These structures are motivated by and used to analyze renormalized zero-dimensional quantum field theory at high orders in perturbation theory. As a pure application of the Hopf algebra structure, an Hopf algebraic interpretation of the Legendre transformation in quantum field theory is given. The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems in combinatorics: The asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined in an application to zero-dimensional quantum field theory. Various quantities are explicitly given asymptotically in the zero-dimensional version of ϕ 3 , ϕ 4 , QED, quenched QED and Yukawa theory with their all order asymptotic expansions.

show abstract

Filtrations in Dyson–Schwinger equations: Next-to{j}-leading log expansions systematically

Cited by 21 publications

References 23 publications

Log expansions from combinatorial Dyson–Schwinger equations

Log expansions from combinatorial Dyson–Schwinger equations

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

Graphs in Perturbation Theory

Contact Info

Product

Resources

About