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

TL;DR

The work addresses the challenge of numerically evaluating Mellin-Barnes integrals in Minkowskian regions for multi-loop amplitudes. It introduces MBnumerics.m, a numerical framework that uses contour shifts, deformations, and variable mappings to tame oscillations and Gamma-function instabilities, supporting up to four-dimensional MB integrals with high accuracy. It also updates AMBRE to generate MB representations for planar and non-planar diagrams and demonstrates the approach on massive vertex and two-loop electroweak processes such as $Z \to b\bar{b}$, highlighting practical viability and precision gains in physical Minkowskian kinematics. Overall, the paper provides a workflow that combines automated MB construction with robust Minkowskian numerical integration, enabling reliable cross-checks and computations in challenging multi-scale, multi-loop problems.

Abstract

Mellin-Barnes (MB) techniques applied to integrals emerging in particle physics perturbative calculations are summarized. New versions of AMBRE packages which construct planar and nonplanar MB representations are shortly discussed. The numerical package MBnumerics.m is presented for the first time which is able to calculate with a high precision multidimensional MB integrals in Minkowskian regions. Examples are given for massive vertex integrals which include threshold effects and several scale parameters.

Figures (6)

• Figure 1: Integration contours chosen for the real part of the complex variable $z$ defined in Eqs. (\ref{['parts']}),(\ref{['zpar']}) and Eqs. (\ref{['c1']})-(\ref{['c3']}). For $C_2$$\alpha = \arctan (\frac{1}{\theta})$. Deformation from $C_1$ to $C_2$ or $C_3$ does not cross poles (black dots).
• Figure 2: Module of real part of the integral $I^{C_{1}}(s,M_{Z},{n})$ as a function of $n$.
• Figure 3: Logarithmic mapping for the integrand in Eq. (\ref{['intmap']}) . On left (right) real (imaginary) part of the integral is given.
• Figure 4: Tangent mapping for the integrand in Eq. (\ref{['intmap']}) . On left (right) real (imaginary) part of the integral is given.
• Figure 5: Real part of the integrand $J$ defined in Eq.(3.22) evaluated over contours $C_1$ and $C_2$. $T_{1}$ and $T_{2}$ are integration variables due to a tangent mapping: $t_{i}=1/\tan[-\pi T_{i}]$.