Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions

TL;DR

This paper demonstrates that Mellin-Barnes representations can derive differential equations for Feynman diagrams with arbitrary propagator powers, bypassing integration-by-parts, aiding in function reduction and master integral counting.

Contribution

It introduces a method to obtain differential equations directly from Mellin-Barnes representations without using integration-by-parts techniques.

Findings

Mellin-Barnes representations enable differential equation derivation for Feynman diagrams.

The approach facilitates reduction to basic functions and counting of master integrals.

The method offers an alternative to traditional IBP techniques.

Abstract

We argue that the Mellin-Barnes representations of Feynman diagrams can be used for obtaining linear systems of homogeneous differential equations for the original Feynman diagrams with arbitrary powers of propagators without recourse to the integration-by-parts technique. These systems of differential equations can be used (i) for the differential reductions to sets of basic functions and (ii) for counting the numbers of master integrals.