Mellin-Barnes Representations for Feynman Integrals

Starting from Feynman integrals defined in momentum space, Feynman and Schwinger parametrizations are introduced and their equivalence is shown. Further, the properties of Feynman graphs and the corresponding F and U Symanzik polynomials are discussed. The MB master formula and a suite of useful programs for constructing and evaluating MB integrals are introduced. Peculiarities of the construction of an MB representation due to the planarity of a given diagram are examined, and the notion of the loop-by-loop (LA) and global (GA) approaches are introduced. The Cheng-Wu theorem is proved and applied to the derivation of MB representations for non-planar diagrams. The first and second Barnes lemmas are used in order to maximally simplify the structure and a number of complex variables in MB representations. Then, specific one-loop to three-loop examples of MB integral constructions are shown. An alternative "method of brackets" for constructing MB representations is also introduced. Finally, the application of the MB method to the computation of real phase space integrals is discussed.