Geometrical methods for the analytic evaluation of multiple Mellin-Barnes integrals

Abstract

Two recently developed techniques of analytic evaluation of multifold Mellin-Barnes (MB) integrals are presented. Both approaches rest on the definition of geometrical objets conveniently associated with the MB integrands, which can then be used along with multivariate residues analysis to derive series representations of the MB integrals. The first method is based on introducing conic hulls and considering specific intersections of the latter, while the second one rests on point configurations and their regular triangulations. After a brief description of both methods, which have been automatized in the MBConicHulls.wl Mathematica package, we review some of their applications. In particular, we show how the conic hulls method was used to obtain the first analytic calculation of complicated Feynman integrals, such as the massless off-shell conformal hexagon and double-box. We then show that the triangulation method is even more efficient, as it allows one to compute these nontrivial objects and harder ones in a much faster way.

Figures (3)

• Figure 1: The integration contour of Eq. (\ref{['1foldEx']}) (in blue). It separates the poles of $\Gamma(-z)$ (in red) from the poles of $\Gamma(-1/2+z)$ (in green).
• Figure 2: Whittaker and Watson, "A course of Modern Analysis" 1927, p.286.
• Figure 3: A two-dimensional conic hull.