VBF and VH

The production of a Standard Model Higgs boson in association with two hard jets in the forward andbackward regions of the detector, frequently quoted as the \u201cvector-boson fusion\u201d (VBF) channel, and theproduction of a Higgs boson in association with a W or Z boson, known as \u201cVH production\u201d or \u201cHiggs-strahlung\u201d, represent cornerstones in a comprehensive study of Higgs boson couplings at the LHC. Theseproduction channels do not only provide valuable information on the couplings of Higgs bosons to themassive gauge bosons by themselves, but also allow for the isolation of the Higgs boson decays into\u3c4-lepton or bottom-quark pairs, whose investigation is essential in the Higgs boson couplings analysis.In the previous reports [7\u20139], state-of-the-art predictions and error estimates for the total and dif-ferential cross sections forpp\u2192H + 2 jetsandpp\u2192HV\u2192H + 2leptons were compiled (withV= W,Z), but the process of improving and refining predictions is still ongoing, even within the Stan-dard Model. In this contribution we update the cross-section predictions for VBF and VH production,covering integrated total and fiducial cross sections as well as differential distributions. In detail, the pre-sented state-of-the-art predictions include QCD corrections up to next-to-next-to-leading order (NNLO),electroweak (EW) corrections up to next-to-leading order (NLO), and contributions from specific par-tonic channels that open at higher perturbative orders, such as photon-induced collisions or gluon-fusioncontributions. Apart from collecting numerical results, we give recommendations as to how to combinethe individual components and to assess conservative estimates of remaining theoretical uncertainties.Moreover, issues connected to the matching and the impact of parton showers (PS) are discussed