An Étude in non-linear Dyson–Schwinger Equations

Kreimer, Dirk; Yeats, Karen

doi:10.1016/j.nuclphysbps.2006.09.036

Cited by 90 publications

(170 citation statements)

References 8 publications

(22 reference statements)

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“…The very fact that we can renormalize by modifying the Lagrangian implies that we have a sub Hopf algebra at our disposal which has the graded elements c r k as generators which correspond to the k-th order contribution to the amplitude r [15,14].…”

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confidence: 99%

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

Kreimer

Yeats

2008

Commun. Math. Phys.

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Abstract. In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns out to be min{ρ, 1/b 1 }, where ρ is the bound on the convergent ones, the instanton radius, and b 1 the first coefficient of the β-function.1. The set-up 1.1. Introduction. In this paper we explore the recursive structure of the shortdistance sector of a renormalizable quantum field theory. Such theories have a finite number of distinct amplitudes r ∈ R ⊂ A which need renormalization. We decompose each Green function in accordance with structure functions which need renormalization, the remaining structure functions and superficially convergent Green functions contribute to amplitudes a ∈ A R in the set of all amplitudes A [14].For such a theory, the Dyson-Schwinger equations give any amplitude s ∈ A = A R ∪ R in terms of all those amplitudes r ∈ R in the theory which need renormalization, and in terms of an infinite series of integral kernels, the skeleton graphs of the theory. The very fact that we can renormalize by modifying the Lagrangian implies that we have a sub Hopf algebra at our disposal which has the graded elements c r k as generators which correspond to the k-th order contribution to the amplitude r [15,14].We can relate the integral kernels above to the primitives of the Hopf algebra structure underlying renormalization. The growth of these primitives is hence determined by the growth of integral kernels provided by overall convergent Green functions. A bound on such a growth is hence obtainable from a bound of amplitudes in A R, where results in constructive field theory are principally available. The typical example is quantum electrodynamics (QED), where the primitives for the vertex function are given by the superficially convergent four fermion e + e − → e + e − scattering kernel, two-particle irreducible in a suitable channel.To proceed from there to a bound for amplitudes in R, we need a handle on the behaviour of the singular integrations which encompass the short-distance singular sector. Here, we proceed by chosing a suitable set of primitives such that we can reduce the Dyson-Schwinger equations to recursive equations acting on onevariable Mellin transforms. A simple study of the conformal symmetries in the corresponding primitives suffice to determine the form of these Mellin transforms.

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confidence: 99%

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

Kreimer

Yeats

2008

Commun. Math. Phys.

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“…The same basic structure, although in general considerably more intricate, is seen in the nesting of subdivergent Feynman diagrams in larger Feynman diagrams. This observation is the beginning of the algebraic or combinatorial approach to Dyson-Schwinger equations as found in papers such as [11,12,17,19]. Combinatorial Dyson-Schwinger equations are equations such as (2.1) and its generalizations that describe this recursive structure for a given class of Feynman graphs.…”

Section: Dyson-schwinger Equationsmentioning

confidence: 98%

A chord diagram expansion coming from some Dyson-Schwinger equations

Marie

Yeats

2013

Communications in Number Theory and Physics

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“…ω D (Γ) determines the number of derivatives with respect to masses or external momenta needed to render a graph logarithmically divergent, and hence identifies the top-level residues which drive the iteration of Feynman integrals according to the quantum equations of motion [3]. We define these four Green functions as an evaluation by renormalized Feynman rules of a series of one-particle irreducible (1PI) Feynman graphs Γ ∈ F G i .…”

Section: Introductionmentioning

confidence: 99%

“…Note that such subtractions on skeleton kernels not only provide a means to investigate non-perturbative aspects of Green functions [3], but are also wellfounded mathematically [10].…”

Section: Introductionmentioning

confidence: 99%