The current frontier of NNLO computations is represented by 2 → 3 processes with external massive legs. In this talk, I will present recent progress in NNLO QCD computations for processes where a heavy-quark pair is produced in association with a massive boson, focusing on ttH and ttW production. In view of the high-luminosity LHC phase, providing more accurate theoretical predictions for these processes is of remarkable importance. Indeed, ttH allows for a direct measurement of the top Yukawa coupling, while ttW (in the multi-leptonic signature) represents an irreducible background for BSM searches, and it is characterised by a long-standing discrepancy between experimental data and theoretical predictions. The achievement of a complete NNLO calculation for this class of processes is a challenging task since it requires the availability of the corresponding scattering amplitudes as well as an efficient method to handle and cancel infrared singularities. In this talk, I will discuss the approximations we adopted to circumvent the bottleneck of the missing double-virtual contribution, mainly focusing on a soft-boson approximation and a high-energy (small mass) expansion. Finally, I will present results for the total cross section and phenomenologically relevant distributions, obtained within the well-established framework of qT-subtraction.
