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The signed permutation group on Feynman graphs

### Abstract

The Feynman rules assign to every graph an integral which can be written as a function of a scaling parameter L. Assuming L 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.

Published by AIP Publishing.

Received 27 April 2016
Accepted 10 August 2016
Published online 29 August 2016

Acknowledgments:
I thank Dirk Kreimer for his great supervising, strong support, and encouragement.

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
A. Graph theoretical foundations
B. Feynman graph polynomials
C. The Hopf algebra of rooted trees
III. PARAMETRIC RENORMALIZATION
IV. THE *L*-LINEAR TERM OF SIGNED GRAPH PERMUTATIONS
A. There are at least two *ψ*_{is…it+1} with *s* = *t* + 1
B. There is at most one *ψ*_{is…it+1} with *s* = *t* + 1
V. A GENERAL FORMULA
VI. A PICTORIAL APPROACH USING FLAG DIAGRAMS
VII. EXAMPLE: *r*_{Γ} = 6
VIII. CONCLUSION

/content/aip/journal/jmp/57/8/10.1063/1.4961517

4.

D. B. West, Introduction to Graph Theory, 2nd ed. (Prentice Hall, 2000).

8.

C. Kassel, “Quantum groups,” in Graduate Texts in Mathematics (Springer, New York, 1995), Vol. 155.

9.

D. Manchon, “

Hopf algebras, from basics to applications to renormalization,” in 5th Mathematical Meeting of Glanon: Algebra, Geometry and Applications to Physics Glanon, Burgundy, France, 2–6 July 2001,

https://inspirehep.net/record/580001; e-print

arXiv:math/0408405 [math-qa].

11.

ϕ^{k}-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.

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.

17.

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.

19.

L. Rotheray, “Hopf subalgebras from Green’s functions,” Master’s thesis, Humboldt-Universität zu Berlin, Berlin, Germany, 2014.

22.

E. Panzer, “Feynman integrals and hyperlogarithms,” Ph.D. thesis, Humboldt-Universität zu Berlin, Berlin, Germany, 2014.

23.

Here and throughout this article we explicitly exclude lightlike particles.

24.

In -theory, the superficial degree of divergence is given by ω_{D} = 2 ⋅ E_{Γ} − D ⋅ L since the weight of the edges and vertices is ω(e) = 2 and ω(v) = 0, respectively.

25.

M. Sars, “Parametric representation of Feynman amplitudes in gauge theories,” Ph.D. thesis,Humboldt-Universität zu Berlin, Berlin, Germany, 2015.

26.

Note that for the empty forest , the graph polynomials are defined as ψ_{0̸} = 1 and ϕ_{0̸}(Θ) = 0.

28.

L. Comtet, Advanced Combinatorics (Springer, Netherlands, 1974).

29.

D. Stanton and D. White, “Constructive combinatorics,” in Undergraduate Texts in Mathematics (Springer, New York, 1986).

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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 L. Assuming L 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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