New prospects for the numerical calculation of Mellin-Barnes integrals in Minkowskian kinematics

Abstract

During the last several years, remarkable progress has been made in numerical calculations of dimensionally regulated multi-loop Feynman diagrams using Mellin Barnes (MB) representations. The bottlenecks were non-planar diagrams and Minkowskian kinematics. The method has been proved to work in a highly non-trivial physical application (two-loop electroweak bosonic corrections to the Z -> b (b) over bar decay), and cross-checked with the sector decomposition (SD) approach. In fact, both approaches have their pros and cons. For multidimensional integrals, depending on masses and scales involved, they are complementary. A powerful top bottom approach to the numerical integration of multidimensional MB integrals is automatized in the MB-suite AM B RE/ Pla na rityTest/ M Btools/M Bnumerics/CUBA. New key elements are: a dedicated use of the Cheng Wu theorem for non-planar topologies and of shifts and deformations of the integration contours. An alternative bottom up approach starting with complex 1-dimensional MB integrals, based on the exploration of steepest descent integration contours in Minkowskian kinematics, is also discussed. Short and long-term prospects of the MB method for multi-loop applications to LHC- and LC-physics are discussed.