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 → bb 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 AMBRE/PlanarityTest/MBtools/MBnumerics/ 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. Shortand long-term prospects of the MB method for multi-loop applications to LHC-and LC-physics are discussed.

Mellin–Barnes meets Method of Brackets: a novel approach to Mellin–Barnes representations of Feynman integrals

“…However, more advanced integration algorithms may help to improve the accuracy reached on a reasonable time scale even for such high dimensional MB integrals. For recent developments in that direction, see [14,15]. All numerical values also agree with independent results of the sector decomposition implementation FIESTA [31].…”

The Sudakov form factor at four loops in maximal super Yang-Mills theory

“…For crossed topologies, the situation is more complicated as one encounters cases in which the loop-by-loop approach yields polynomials in the Feynman parameters x i which are not positive definite, even in the absence of kinematic thresholds. Consequently, the MB integrals will be highly oscillating and hence their numerical evaluation will be difficult to handle, although steps in this direction have been undertaken [125][126][127][128].…”

Numerical integration of massive two-loop Mellin-Barnes integrals in Minkowskian regions

“…In the second step, a completely new software MBnumerics.m has been used [64]. In the next section some core ideas which made possible to calculate MB integrals in Minkowskian regions used in MBnumerics.m are given.…”

“…Automatic algorithms for finding the suitable shifts and contour deformations are implemented in MBnumerics.m [64]. At the moment an effective strategy is: Starting from original n−dimensional MB integrals, MBnumerics.m looks for well converging n − 1 and n − 2 integrals, and remaining n−dimensional integrals.…”