A chord diagram expansion coming from some Dyson-Schwinger equations

Marie, Nicolas; Yeats, Karen

doi:10.4310/cntp.2013.v7.n2.a2

Cited by 24 publications

(82 citation statements)

References 17 publications

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“…Different theories with isomorphic diagrammatics only show their difference in their dependence on the coefficients of the expansions of the regularized Feynman integrals of the primitive 1 Feynman graphs of the theory contributing to the Dyson-Schwinger equation. These coefficients are the f i of [11] and our previous work [3] and are the a k,i of [7], a notation which we will also use in the present paper. The way these coefficients come in is well-controlled; for example in all cases the full H k depends only on coefficients coming from the first k terms of expansions of primitive graphs with at most k + 1 loops (this is clear and long-known in physics and is also an immediate corollary of the main results of [11] and [7]).…”

Section: Introductionmentioning

confidence: 99%

“…These coefficients are the f i of [11] and our previous work [3] and are the a k,i of [7], a notation which we will also use in the present paper. The way these coefficients come in is well-controlled; for example in all cases the full H k depends only on coefficients coming from the first k terms of expansions of primitive graphs with at most k + 1 loops (this is clear and long-known in physics and is also an immediate corollary of the main results of [11] and [7]). Furthermore, we can prove for all sufficiently large k which terms will dominate H k .…”

Section: Introductionmentioning

confidence: 99%

“…Our underlying combinatorial structures are decorated rooted connected chord diagrams. Previous work of one of us with Marie [11] and Hihn [7] showed how to solve Dyson-Schwinger equations as sums indexed by these chord diagrams. To see that the chord diagram expansions are physically interesting, compare them to the original Feynman diagram expansions of the Green functions.…”

Section: Introductionmentioning

confidence: 99%

“…Note, unusually, that these chord diagrams do not come from any sort of Wick pairing. The rigorous proofs of these chord diagram expansions [11,7] are inductive and do not explain why these objects appear.…”

Section: Introductionmentioning

confidence: 99%

“…Previous work of both of us [3] considered the original chord diagram expansion of [11] which applies only in the case of one particular shape of Dyson-Schwinger equation. We were able to prove statistical properties of various chord diagram parameters which are relevant to the expansion and we were able to understand the log expansions.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Courtiel

Yeats

2020

Commun. Math. Phys.

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Graphs in Perturbation Theory

Borinsky

2018

Springer Theses

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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.

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The Ring of Factorially Divergent Power Series

Borinsky

2018

Springer Theses

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A chord diagram expansion coming from some Dyson-Schwinger equations

Cited by 24 publications

References 17 publications

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

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

Graphs in Perturbation Theory

The Ring of Factorially Divergent Power Series

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