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Evaluating by MB Representation

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Analytic Tools for Feynman Integrals

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 250))

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Abstract

One often uses Mellin integrals, when dealing with Feynman integrals. These are integrals over contours in a complex plane along the imaginary axis of products of gamma functions in the numerator and denominator. In particular, the inverse Mellin transform is given by such an integral. We will, however, deal with a very specific technique in this field. The key ingredient of the method presented in this chapter is the MB representationMellin–Barnes (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 with gamma functions. The next step is the resolution of the singularities in \(\varepsilon \) 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 lemmas and their corollaries and/or evaluate these integrals by closing the integration contours in the complex plane and summing up corresponding series.

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Notes

  1. 1.

    First examples of application of Mellin integrals to Feynman integrals can be found in [8, 65].

  2. 2.

    Historically, it was first advocated and applied in [16].

  3. 3.

    In some situations, e.g. in a MB integral for the Gauss hypergeometric function, the asymptotic exponents of gamma functions cancel each other so that the convergence is defined by the value of the argument \(x\) which is present in the MB integral as \(x^z\). Depending on whether \(|x|<1\) or \(|x|>1\), one has to close the integration contour to the right or to the left. Closing the contours to the different sides corresponds to an analytical continuation with respect to the argument \(x\). However, there are certainly problems with the convergence in physical regions of kinematic variables, where factors of the type \(x^z\), with \(x<0\), are present—see [17].

  4. 4.

    In [26], it was demonstrated that this Feynman integral reduces, for any values of the three indices, to a two-point function in the shifted dimension \(d-2a_3\).

  5. 5.

    Well, this is only one half of the court for singles.

  6. 6.

    See also a similar discussion in [24, 25].

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  1. Division of Theoretical High-Energy Physics, Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119992, Russia

    Vladimir A. Smirnov

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Correspondence to Vladimir A. Smirnov .

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Smirnov, V.A. (2012). Evaluating by MB Representation. 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_5

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  • DOI: https://doi.org/10.1007/978-3-642-34886-0_5

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