Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

TL;DR

This paper explicitly calculates multi-fold contour integrals of ratios of Euler gamma functions related to Mellin-Barnes transforms of UD functions, extending previous recursive relations and conjecturing broader applicability.

Contribution

It provides explicit calculations of complex contour integrals for UD functions and their ratios, advancing the understanding of Mellin-Barnes transforms in quantum field theory.

Findings

Explicit multi-fold contour integrals computed using Barnes lemmas.

Reproduction of recurrent relations for UD functions.

Conjecture on broader applicability to similar functions.

Abstract

In this paper we proceed to study properties of Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions. In our previous papers [Nuclear Physics B 870 (2013) 243], [Nuclear Physics B 876 (2013) 322] we showed that multi-fold Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions were obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in $d=4$ dimensions and its analog in $d=4-2\varepsilon$ dimensions exits too [Theoretical and Mathematical Physics 177 (2013) 1515]. In [Nuclear Physics B 870 (2013) 243] the chain of recurrent relations for analytically regularized UD functions was obtained implicitly by comparing the left hand side and the right hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained due to the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here we reproduce these recurrent relations by calculating explicitly via Barnes lemmas the contour integrals produced by the left hand sides of the diagrammatic relations. In such a way we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions which includes the MB transforms of UD functions.