Numerical evaluation of multi-loop integrals by sector decomposition
T. Binoth, G. Heinrich
TL;DR
The paper tackles the challenge of numerically evaluating divergent multi-loop integrals with multiple scales. It extends the iterative sector decomposition method to handle arbitrary propagator powers and massive internal lines, producing Laurent expansions in $\epsilon$ with coefficients given by parameter integrals. The authors apply the method to diverse topologies, including two-loop massless 4-point functions with off-shell legs, Bhabha-scattering master integrals, massless double boxes (planar and non-planar) with off-shell legs, and propagator graphs up to five loops, providing extensive numerical results. Cross-checks against known analytic results (e.g., the massless planar 3-loop box TB) and consistent outputs demonstrate the method's reliability, while results in non-physical kinematics suggest it as a robust tool for independent verification and future extension to physical regions.
Abstract
In a recent paper we have presented an automated subtraction method for divergent multi-loop/leg integrals in dimensional regularisation which allows for their numerical evaluation, and applied it to diagrams with massless internal lines. Here we show how to extend this algorithm to Feynman diagrams with massive propagators and arbitrary propagator powers. As applications, we present numerical results for the master 2-loop 4-point topologies with massive internal lines occurring in Bhabha scattering at two loops, and for the master integrals of planar and non-planar massless double box graphs with two off-shell legs. We also evaluate numerically some two-point functions up to 5 loops relevant for beta-function calculations, and a 3-loop 4-point function, the massless on-shell planar triple box. Whereas the 4-point functions are evaluated in non-physical kinematic regions, the results for the propagator functions are valid for arbitrary kinematics.