Evaluating by MB Representation

Abstract

One often uses Mellin integrals¹ 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.