We summarise our recent developments concerning the calculation of loop integrals, amplitudes and cross-sections through direct numerical integration in momentum space, based on the frameworks of Loop-Tree Duality and Local Unitarity. In the first part of the talk, we elaborate on the structure of threshold singularities that appear in the integration domain and show how they can be regulated by cutting loops into trees, a relation referred to as Loop-Tree Duality, together with a suitable contour deformation or subtraction through counterterms. Moreover, we demonstrate a first application of these tools for calculating loop amplitudes.

In the second part, we show how the cancellation of infrared singularities between real and virtual interference diagrams is realised locally within the Local Unitarity representation of differential cross-sections. Moreover, we describe how ultraviolet divergences can be systematically renormalised in the hybrid MSbar + on-shell scheme using an automated and local procedure based on Bogoliubov's R operation. Finally, we propose a novel representation of hadronic cross-sections in which initial-state infrared singularities are regularised in a way that is compatible with the Local Unitarity formalism