Evaluating multiloop Feynman integrals by Mellin-Barnes representation

When calculating physical quantities that describe a given process one needs to evaluate a lot of Feynman integrals. After a tensor reduction based on some projectors (see, e.g., [1]) a given Feynman graph generates various scalar Feynman integrals that have the same structure of the integrand with various distributions of powers of propagators. A straightforward analytical strategy is to evaluate, by some methods, every scalar Feynman integral generated by the given graph. If the number of these integrals is small such strategy is reasonable. In non-trivial situations, where the number of different integrals can be at the level of hundreds and thousands, it is reasonable to follow a well-known advanced strategy: to derive, without calculation, and then apply integration by parts (IBP) [2] and Lorentz-invariance (LI) [3] identities between the given family of Feynman integrals as recurrence relations. The goal of this procedure is to express a general integral from the given family as a linear combination of some basic (master) integrals. Therefore the whole problem of evaluation is decomposed into two parts: solution of the reduction procedure and evaluation of the master Feynman integrals. There were several recent attempts to make the reduction procedure systematic: (i) Using the fact that the total number of IBP and LI equations grows faster than the number of independent Feynman integrals one can sooner