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The important mathematical problem of evaluating Feynman integrals arises quite naturally in elementary-particle physics when one treats various quantities in the framework of perturbation theory. Usually, it turns out that a given quantum-field amplitude that describes a process where particles participate cannot be completely treated in the perturbative way. However it also often turns out that the amplitude can be factorized in such a way that different factors are responsible for contributions of different scales. According to a factorization procedure a given amplitude can be represented as a product of factors some of which can be treated only non-perturbatively while others can be indeed evaluated within perturbation theory, i.e. expressed in terms of Feynman integrals over loop momenta. A useful way to perform the factorization procedure is provided by solving the problem of asymptotic expansion of Feynman integrals in the corresponding limit of momenta and masses that is determined by the given kinematical situation. A universal way to solve this problem is based onthe so-called strategy of expansion by regions [3,10]. This strategy can be itself regarded as a (semianalytical) method of evaluation of Feynman integrals according to which a given Feynman integral depending on several scales can be approximated, with increasing accuracy, by a finite sum of first terms of the corresponding expansion, where each term is written as a product of factors depending on different scales. A lot of details concerning expansions of Feynman integrals in various limits of momenta and/or masses can be found in my previous book [10]. In this book, however, we shall mainly deal with purely methods.

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.

Feynman parameters are very well known and often used in practical calculations. They are closely related to alpha parameters introduced in Chap. 2 so that we shall study both kinds of parametric representations of Feynman integrals in one chapter. The use of these parameters enables us to transform Feynman integrals over loop momenta into parametric integrals where Lorentz invariance becomes manifest. Using alpha parameters we shall first evaluate one and two-loop integrals with general complex powers of the propagators, within dimensional regularization, for which results can be written in terms of gamma functions for general values of the dimensional regularization parameter. We shall show then how these formulae, together with simple algebraic manipulations, enable us to evaluate some classes of Feynman integrals.

One often uses Mellin integrals when dealing with Feynman integrals. These are integrals over contours in a complex plane along the imaginary axis of a product and ratio of gamma functions. In particular, the inverse Mellin transform is given by such an integral. We shall, however, deal with a very specific technique in this field. The key ingredient of the method presented in this chapter is the MB representation used to replace a sum of two terms raised to some power by the product of these terms raised to some powers. Our goal is to use such a factorization in order to achieve the possibility to perform integrations in terms of gamma functions, at the cost of introducing extra Mellin integrations. Then one obtains a multiple Mellin integral of gamma functions in the numerator and denominator. The next step is the resolution of the singularities in ɛ by means of shifting contours and taking residues. It turns out that multiple MB integrals are very convenient for this purpose. The final step is to perform at least some of the Mellin integrations explicitly, by means of the first and the second Barnes lemma and their corollaries and/or evaluate these integrals by closing the integration contours in the complex plane and summing up corresponding series.

The next method in our list is based on integration by parts (IBP) [15] within dimensional regularization, i.e. property (2.38). The idea is to write down various equations (2.38) for integrals of derivatives with respect to loop momenta and use this set of relations between Feynman integrals in order to solve the reduction problem, i.e. to find out how a general Feynman integral of the given class can be expressed linearly in terms of some master integrals. In contrast to the evaluation of the master integrals, which is performed, at a sufficiently high level of complexity, in a Laurent expansion in ɛ, the reduction problem is solved at , and the expansion in ɛ does not provide simplifications here.

In the previous chapter, we solved IBP relations [7] in a non-systematic way. Now we are going to do this systematically following Baikov’s method [2,4,5,14].

The method of differential equations (DE) suggested in [20] and developed in [23] and later works (see references below) is a method of evaluating individual Feynman integrals. We have agreed that, at the present level of complexity of unsolved important problems, it looks unavoidable to decompose the problem of evaluating Feynman integrals of a given family into the reduction to some master integrals and the problem of evaluating these master integrals. Thus, this basic method is oriented at the evaluation of the master integrals. Moreover, in contrast to other methods of evaluating individual Feynman integrals, it is assumed within this method that a solution of the reduction problem is already known.

Each Feynman integral presented here can be evaluated straightforwardly by use of alpha or Feynman parameters. Results are presented for the ‘Euclidean’ dependence, −, of the denominators, which is more natural when the powers of propagators are general complex numbers. As usual, − is understood in the sense of − − i0, etc. Moreover, denominators with a linear dependence on are also understood in this sense, e.g. 2 2 ⋅−i0, although sometimes this i0 dependence is explicitly indicated to avoid misunderstanding.

The Gauss hypergeometric function [3] is defined by the series

Nested sums are defined as follows [17]: