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/content/aip/journal/jmp/57/8/10.1063/1.4961517
1.
F. Brown and D. Kreimer, “Angles, scales and parametric renormalization,” Lett. Math. Phys. 103, 9331007 (2013); e-print arXiv:1112.1180 [hep-th].
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11.
ϕk-theories are scalar field theories treating only one kind of particles with spin zero represented by the one-component scalar field ϕ. Those particles self-interact in groups of k which means that all vertices are k-valent.
12.
A. Connes and D. Kreimer, “Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem,” Commun. Math. Phys. 210, 249273 (2000); e-print arXiv:hep-th/9912092 [hep-th].
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13.
A more detailed description of the notion of the forest can be found in Ref. 1
14.
It should be pointed out that there are two different definitions of the notion of a forest. In the present case we define the forest (of subdivergences) in the context of renormalization and Hopf algebra. This definition is also in accordance with the forest formula introduced in Section III. Within the framework of graph polynomials (cf., Subsection II B) the forest (or k-forest) is defined as a graph without cycles/loops consisting of k connected components. That is, a k-forest is given by the disjoint union of k trees. For example, the forest set in Figure 6 follows this definition.
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Note that, by abuse of notation, 𝕀 denotes the unit map as well as the empty tree.
18.
For reminding the notation, see Definition 1.
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23.
Here and throughout this article we explicitly exclude lightlike particles.
24.
In -theory, the superficial degree of divergence is given by ωD = 2 ⋅ EΓDL since the weight of the edges and vertices is ω(e) = 2 and ω(v) = 0, respectively.
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M. Sars, “Parametric representation of Feynman amplitudes in gauge theories,” Ph.D. thesis,Humboldt-Universität zu Berlin, Berlin, Germany, 2015.
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Note that for the empty forest , the graph polynomials are defined as ψ = 1 and ϕ(Θ) = 0.
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/content/aip/journal/jmp/57/8/10.1063/1.4961517
2016-08-29
2016-09-29

Abstract

The Feynman rules assign to every graph an integral which can be written as a function of a scaling parameter . Assuming for the process under consideration is very small, so that contributions to the renormalization group are small, we can expand the integral and only consider the lowest orders in the scaling. The aim of this article is to determine specific combinations of graphs in a scalar quantum field theory that lead to a remarkable simplification of the first non-trivial term in the perturbation series. It will be seen that the result is independent of the renormalization scheme and the scattering angles. To achieve that goal we will utilize the parametric representation of scalar Feynman integrals as well as the Hopf algebraic structure of the Feynman graphs under consideration. Moreover, we will present a formula which reduces the effort of determining the first-order term in the perturbation series for the specific combination of graphs to a minimum.

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