A simple way to reduce the number of contours in the multi-fold Mellin-Barnes integrals
Mauricio Diaz, Ivan Gonzalez, Igor Kondrashuk, Eduardo A. Notte-Cuello
TL;DR
The paper introduces a Mellin-Barnes (MB) framework to dramatically reduce the number of contours needed in multi-fold MB representations, specifically transforming the five-fold MB form of the one-loop massless box into a two-fold MB form by applying MB to Feynman-parameter denominators and employing analytical regularization with the Cauchy integral formula. The method proceeds by first treating the triangle diagram, obtaining a two-fold MB representation, and then showing that the box diagram in four dimensions with unit propagator indices yields a five-fold MB form whose MB variables can be reorganized and regularized to yield a finite set of residues, effectively reducing to the triangle’s two-fold MB structure. Central to the approach is treating certain singular Feynman-parameter integrals as distributions, regularizing them analytically, and then evaluating the resulting beta-function contributions via residues as the regulator is removed. The results provide a general strategy for reducing MB contour counts in arbitrary diagrams and open avenues for connections with knot theory, Trotter integrals, and quantum computing, while offering a practical MB-based pathway that avoids intermediate Riemann integrations or conformal transformations. Overall, the work broadens the toolbox for analytic evaluation of Feynman diagrams by showing how MB reductions can be achieved through purely complex-analysis techniques applied to MB integrands.
Abstract
Mellin-Barnes integral representation of one-loop off-shell box massless diagram is five-fold by construction. On the other hand, it is known from the year 1992 that it may be reduced to certain two-fold Mellin-Barnes integral. We propose a way to reduce the number of the Mellin-Barnes integration contours from five to two by using the Mellin-Barnes integral representation only in combination with basic methods of mathematical analysis such as analytical regularization. We do not use any Barnes lemma to prove the reduction but we use the integral Cauchy formula instead. We recover first the well-known two-fold Mellin-Barnes representation for the one-loop triangle massless diagram and then show how the five-fold Mellin-Barnes integral representation of one-loop box diagram with all the indices 1 in four spacetime dimensions may be reduced to the two-fold Mellin-Barnes representation for one-loop triangle diagram. Singular integrals over Feynman parameters appear in the integrand of the five-fold Mellin-Barnes integral representation at the intermediate step. Such integrals should be treated as distributions with respect to certain linear combinations of the initial Mellin-Barnes integration variables in the Mellin-Barnes integrands. These distributions may be integrated out with a finite number of residues in the limit of removing the analytical regularization. We explain how to apply this strategy to an arbitrary Feynman diagram in order to reduce the number of Mellin-Barnes integration contours. On the practical side, we analyze connections between the obtained results and the knot theory, Trotter integrals, quantum computing.