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.
Highlights
The importance of the differential-equation approach to the description of the analytical properties of Feynman diagrams has been recognized a long time ago [1]
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
We argue that linear systems of homogeneous differential equations may be derived for Feynman diagrams starting from their Mellin-Barnes representations without resorting to IBP relations [6]
Summary
The importance of the differential-equation approach to the description of the analytical properties of Feynman diagrams has been recognized a long time ago [1]. [13], within analytical regularization [14], Feynman diagrams satisfy holonomic systems of linear differential equations under the condition that all particles have different masses. This statement is the basis of the algorithm proposed in Ref. It has been shown [18] recently that Mellin-Barnes integrals satisfy systems of differential equations corresponding to Gelfand-Kapranov-Zelevisky hypergeometric equations [19] Another necessary condition is that two contours differing by a translation by one unit along the real axis are equivalent. The technique advocated here may allow one to gain deeper insights into the mathematical structures of multi-scale Feynman diagrams
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