Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

Abstract

We observe a property of orthogonality of the Mellin-Barnes transformation of the triangle one-loop diagrams, which follows from our previous papers [JHEP {\bf 0808} (2008) 106, JHEP {\bf 1003} (2010) 051, JMP {\bf 51} (2010) 052304]. In those papers it has been established that Usyukina-Davydychev functions are invariant with respect to Fourier transformation. This has been proved at the level of graphs and also via the Mellin-Barnes transformation. We partially apply to one-loop massless scalar diagram the same trick in which the Mellin-Barnes transformation was involved and obtain the property of orthogonality of the corresponding MB transforms under integration over contours in two complex planes with certain weight. This property is valid in an arbitrary number of dimensions.

arXiv comment12 pages, 2 figures

References in corpus (11)

• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Towards the two-loop Lcc vertex in Landau gauge
• Solution to Bethe-Salpeter equation via Mellin-Barnes transform
• New four-dimensional integrals by Mellin-Barnes transform
• Fourier transforms of UD integrals
• Triangle UD integrals in the position space
• Transformations of triangle ladder diagrams
• Gluon self-interaction in the position space in Landau gauge
• Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions
• Belokurov-Usyukina loop reduction in non-integer dimension
• Box ladders in non-integer dimension

Cited by in corpus (3)

• Mellin-Barnes and the method of brackets
• Finiteness of N =4 super-Yang-Mills effective action in terms of dressed N =1 superfields
• Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals