Connected Chord Diagrams and Bridgeless Maps

Courtiel, Julien; Yeats, Karen; Zeilberger, Noam

doi:10.37236/7400

Cited by 10 publications

(31 citation statements)

References 25 publications

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“…Other connections between Feynman diagrams and rooted maps can be seen in Ref. [18]. A Rooted map is a graph that is embedded in a unique topological surface (sphere or n-holed tori) with a directed edge.…”

Section: Figmentioning

confidence: 99%

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

Castro¹,

Roditi²

2019

J. Phys. A: Math. Theor.

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In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration is the Wick theorem, permitting an easy generalization to any quantum field theory. The iterative enumeration is constructive and enables a fast computation of the number of connected Feynman diagrams for a large amount of cases. In particular, the recurrence is solved exactly for two and four external legs, leading to the asymptotic expansion of the number of different connected Feynman diagrams. FIG. 2:A rooted map embedded in the tori and the corresponding Feynman diagram. See all the correspondences for order m = 1, 2, 3 in ref. [15]

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Section: Figmentioning

confidence: 99%

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

Castro¹,

Roditi²

2019

J. Phys. A: Math. Theor.

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“…. This result has been used by Courtiel, Yeats and Zeilberger to calculate the asymptotics of terminal chord diagrams [39]. These terminal chord diagrams can be used to formulate a solution for Dyson-Schwinger equations in quantum field theory [82].…”

Section: Some Remarks On Differential Equationsmentioning

confidence: 97%

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 full description will be displayed at Theorem 2.8, but many definitions are required beforehand. Note that the statement of this theorem has been updated since [7]: some parameters intervening in the chord diagram expansions (notably the weight of a diagram) have been clarified in [4].…”

Section: Chord Diagrams and Chord Diagram Expansionsmentioning

confidence: 99%

“…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%

$$\hbox {Next-to}{}^k$$ 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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Connected Chord Diagrams and Bridgeless Maps

Abstract: we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.

Cited by 10 publications

References 25 publications

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

Graphs in Perturbation Theory

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

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