A chord diagram expansion coming from some Dyson-Schwinger equations

Marie, Nicolas; Yeats, Karen

doi:10.48550/arxiv.1210.5457

Cited by 7 publications

(62 citation statements)

References 12 publications

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“…Chord diagrams. The main result of [9], which was the topic of the author's talk at the conference for which these are the proceedings, is a solution to the particular Dyson-Schwinger equation…”

Section: Analytic Dyson-schwinger Equationsmentioning

confidence: 99%

“…The proof of the result is the body of [9]. It involves two further recurrences, one of which generalizes a classical chord diagram recurrence of Stein [10] and one of which involves going through a rooted tree construction.…”

Section: Analytic Dyson-schwinger Equationsmentioning

confidence: 99%

“…The new material is Theorem 2.18, Proposition 3.25, and Theorem 3.32, but also interesting is the overall story of nice interpretations involving these results woven together with results of Brown [2], Brown and Schnetz [3], the author with Nicolas Marie [9], and the author [19,18].…”

Section: Introductionmentioning

confidence: 95%

“…The first situation, which is discussed in Section 2, concerns combinatorial understanding of analytic aspects of Dyson-Schwinger equations. One example of this is the recent chord diagram expansion for a class of Dyson-Schwinger equations given by Nicolas Marie and the author in [9]. For those readers who came to this proceedings volume looking for information related to the conference talk, the results of [9] are summarized.…”

Section: Introductionmentioning

confidence: 99%

“…One example of this is the recent chord diagram expansion for a class of Dyson-Schwinger equations given by Nicolas Marie and the author in [9]. For those readers who came to this proceedings volume looking for information related to the conference talk, the results of [9] are summarized. Another example is the reduction process of [19,18].…”

Section: Introductionmentioning

confidence: 99%

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Some combinatorial interpretations in perturbative quantum field theory

Yeats¹

2015

Feynman Amplitudes, Periods and Motives

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This paper will describe how combinatorial interpretations can help us understand the algebraic structure of two aspects of perturbative quantum field theory, namely analytic Dyson-Schwinger equations and periods of scalar Feynman graphs. The particular examples which will be looked at are, a better reduction to geometric series for Dyson-Schwinger equations, a subgraph which yields extra denominator reductions in scalar Feynman integrals, and an explanation of a trick of Brown and Schnetz to get one extra step in the denominator reduction of an important particular graph.

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Section: Analytic Dyson-schwinger Equationsmentioning

confidence: 99%

Section: Analytic Dyson-schwinger Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some combinatorial interpretations in perturbative quantum field theory

Yeats¹

2015

Feynman Amplitudes, Periods and Motives

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$$\hbox {Next-to}{}^k$$ 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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