NLO scattering amplitudes are provided by fully automated numerical tools, such as OpenLoops, for a very wide range of processes. In order to match the numerical precision of current and future collider experiments, the higher precision of NNLO calculations is essential, and their automation in a similar tool a highly desirable goal.
In our approach, D-dimensional two-loop amplitudes are decomposed into Feynman integrals with four-dimensional numerators and (D-4)-dimensional remainders. The latter are reconstructed through process-independent rational counterterm insertions into lower-loop diagrams, while the first are expressed as loop momentum tensor integrals contracted with tensor coefficients.
In this article, we describe a completely generic algorithm, first presented in [1], for the efficient and numerically stable construction of these tensor coefficients. This algorithm is fully implemented in the OpenLoops framework for QED and QCD corrections to the Standard Model. For this implementation we present performance studies on numerical stability and CPU efficiency.