Abstract
In this chapter, basic definitions for Feynman integrals are given, ultraviolet (UV), infrared (IR) and collinear divergences are characterized, and basic tools such as alpha parameters are presented. Various kinds of regularizations, in particular dimensional one, are presented and properties of dimensionally regularized Feynman integrals are formulated and discussed.
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Notes
- 1.
When dealing with graphs and Feynman integrals one usually does not bother about the mathematical definition of the graph and thinks about something that is built of lines and vertices. So, a graph is an ordered family \(\{\mathcal{V}, \mathcal{L},\pi _{\pm }\}\), where \(\mathcal{V}\) is the set of vertices, \(\mathcal{L}\) is the set of lines, and \(\pi _{\pm }:\mathcal{L}\rightarrow \mathcal{V}\) are two mappings that correspond the initial and the final vertex of a line. By the way, mathematicians use the word ‘edge’, rather than ‘line’.
- 2.
called also the sunrise diagram, or the London transport diagram.
- 3.
In fact, the matrix \(A\) involved here equals \(e \beta e^+\) with the elements of an arbitrarily chosen column and row with the same number deleted. Here \(e\) is the incidence matrix of the graph, i.e. \(e_{il}=\pm 1\) if the vertex \(i\) is the beginning/end of the line \(l\), \(e^+\) is its transpose and \(\beta \) consists of the numbers \(1/\alpha _l\) on the diagonal—see, e.g., [23].
- 4.
An alternative definition of algebraic character [19, 35, 38] (see also [13]) exists and is based on certain axioms for integration in a space with non-integer dimension. It is unclear how to perform the analysis within such a definition, for example, how to apply the operations of taking a limit, differentiation, etc. to algebraically defined Feynman integrals in \(d\) dimensions, in order to say something about the analytic properties with respect to momenta and masses and the parameter of dimensional regularization. After evaluating a Feynman integral according to the algebraic rules, one arrives at some concrete function of these parameters but, before integration, one is dealing with an abstract algebraic object. Let us remember, however, that, in practical calculations, one usually does not bother about precise definitions. From the purely pragmatic point of view, it is useless to think of a diagram when it is not calculated. On the other hand, from the pure theoretical and mathematical point of view, such a position is beneath criticism.
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Smirnov, V.A. (2012). Feynman Integrals: Basic Definitions and Tools. In: Analytic Tools for Feynman Integrals. Springer Tracts in Modern Physics, vol 250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34886-0_2
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