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

## TL;DR

This paper discusses recent advances and new prospects in the numerical calculation of Mellin-Barnes integrals in Minkowskian kinematics, highlighting methods, applications, and future potential for multi-loop Feynman diagram computations.

## Contribution

It introduces a powerful automated approach for numerical MB integral evaluation using the MB-suite, and explores alternative contour deformation methods in Minkowskian kinematics.

## Key findings

- Validated the MB method against sector decomposition in complex physical cases.
- Developed a top-bottom automated approach using the MB-suite for multidimensional integrals.
- Discussed the potential of MB methods for future high-energy physics applications.

## 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 highly non-trivial physical application (two-loop electroweak bosonic corrections to the $Z \to b \bar{{b}}$ decay), and cross-checked with the sector decomposition (SD) approach. In fact, both approaches have their pros and cons. In calculation of 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/MB/ MBtools/MBnumerics/CUBA. 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.