Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions
arXiv:1304.3004 · doi:10.1016/j.nuclphysb.2013.08.002
Abstract
In our previous paper [Nucl.Phys. B 870 (2013) 243], we showed that multi-fold Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions may be reduced to two-fold MB transforms. The MB transforms were written there as polynomials of logarithms of ratios of squares of the external momenta with certain coefficients. We also showed that these coefficients have a combinatoric origin. In this paper, we present an explicit formula for these coefficients. The procedure of recovering the coefficients is based on taking the double-uniform limit in certain series of smooth functions of two variables which is constructed according to a pre-determined iterative way. The result is obtained by using basic methods of mathematical analysis. We observe that the finiteness of the limit of this iterative chain of smooth functions should reflect itself in other mathematical constructions, too, since it is not related in any way to the explicit form of the MB transforms. This finite double-uniform limit is represented in terms of a differential operator with respect to an auxiliary parameter which acts on the integrand of a certain two-fold MB integral. To demonstrate that our result is compatible with original representations of UD functions, we reproduce the integrands of these original integral representations by applying this differential operator to the integrand of the simple integral representation of the scalar triangle four-dimensional integral J(1,1,1-e).
arXiv comment12 pages; section added elaborating the connection with arXiv:1303.6909, 8 references added, minor changes in the text; accepted for publication in Nucl. Phys. B
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- Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles
- Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1
- The Moments of the Hydrogen Atom by the Method of Brackets
- Alternative method of Reduction of the Feynman Diagrams to a set of Master Integrals