Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically

TL;DR

This work tackles the numerical evaluation of multi-loop Feynman diagrams that exhibit infrared and threshold singularities across kinematic regions. It combines sector decomposition, which disentangles overlapping infrared poles, with a contour deformation in Feynman-parameter space to handle threshold singularities while preserving the $-i\delta$ prescription, enabling finite, computable integrals in $d \to 4$ dimensions. The authors implement three independent codes that perform sector decomposition, contour deformation (via $z_i = x_i - i\lambda x_i(1-x_i)\frac{\partial {\cal G}_s}{\partial x_i}$) and numerical integration (Cuba library), and validate the method by reproducing known two-loop results for $gg \to h$ with heavy-quark/squark loops and by producing new SUSY-QCD master integral evaluations with high accuracy. This approach offers a general, automated tool for complex multi-loop amplitudes in collider physics, complementing other numerical methods and enabling reliable threshold computations.

Abstract

We present a method to evaluate numerically Feynman diagrams directly from their Feynman parameters representation. We first disentangle overlapping singularities using sector decomposition. Threshold singularities are treated with an appropriate contour deformation. We have validated our technique comparing with recent analytic results for the gg->h two-loop amplitudes with heavy quarks and scalar quarks.