Multi-Fold Contour Integrals Of Certain Ratios Of Euler Gamma Functions From Feynman Diagrams: Orthogonality Of Triangles

We observe a property of orthogonality of the Mellin–Barnes transformation of triangle one-loop diagrams, which follows from our previous papers (Kondrashuk and Kotikov in JHEP 0808:106, 2008; Kondrashuk and Vergara in JHEP 1003:051, 2010; Allendes et al. in J Math Phys 51:052304, 2010). In those papers it has been established that Usyukina–Davydychev functions are invariant with respect to the Fourier transformation. This has been proved at the level of graphs and also via the Mellin–Barnes transformation. We partially apply to the 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.