NNLO final-state quark-pair corrections in four dimensions

We describe how NNLO final state quark-pair corrections are computed in FDR by directly enforcing gauge invariance and unitarity in the definition of the regularized divergent integrals. The whole procedure is strictly four-dimensional and renormalization is performed by simply fixing bare parameters in terms of physical measurements. We give details of our approach and demonstrate how virtual and real contributions can be merged together without relying on dimensional regularization. As an example, we recompute the $H\to b\bar{b}+jets$ and ${\gamma }^{\ast }\to jets$ inclusive rates at the NNLO accuracy in the large $N_F$ limit of QCD.