Abstract
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.
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Notes
- 1.
As is explained in textbooks on integral calculus, the method of IBP is applied with the help of the relation \(\int _a^b \mathrm{{d}} x u v^{\prime } = \left. u v\right|_a^b - \int _a^b \mathrm{{d}} x u^{\prime } v\) as follows. One tries to represent the integrand as \( u v^{\prime }\) with some \(u\) and \(v\) in such a way that the integral on the right-hand side, i.e. of \( u^{\prime } v\) will be simpler. We do not follow this idea in the case of Feynman integrals. Instead we only use the fact that an integral of the derivative of some function is zero, i.e. we always neglect the corresponding surface terms. So the name of the method looks misleading. It is however unambiguously accepted in the physics community.
- 2.
Since the Feynman integrals are rather complicated objects the word ‘multi-loop’ means the number of loops greater than one ;-)
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Smirnov, V.A. (2012). Introduction. 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_1
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