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The study of the Higgs boson properties is one of the most important tasks tobe accomplished in the next years, at the Large Hadron Collider (LHC) and atfuture colliders such as the Future Circular Collider in hadron-hadron mode(FCC-hh), the potential 100 TeV follow-up of the LHC machine. In this view theprecise study of the Higgs couplings to weak gauge bosons is crucial andrequires as much information as possible. After the recent calculation of thenext-to-leading order QCD corrections to the production cross sections anddifferential distributions of a Standard Model Higgs boson in association witha pair of weak bosons, matched with parton shower in the POWHEG-BOX framework,we present the gluon fusion correction $g g\to H W^+_{} W^-_{} ( H Z Z)$ to theprocess $p p \to H W^+_{} W^-_{} (H Z Z)$. This correction can be sizeable andamounts to $+3\,\%$ ($+10\,\%$) in the $H W^+_{} W^-_{}$ process and $+5\,\%$($+18\,\%$) in the $H Z Z$ process at the LHC (FCC-hh). We also present thefirst study of the impact of the bottom--quark initiated channels $b\bar{b}\toH W^+_{} W^-_{} / H Z Z$ and find that they induce a significant $+18\,\%$correction in the $H W^+_{} W^-_{}$ channel at the FCC-hh. We present resultson total cross sections and distributions at the LHC and at the FCC-hh.

Gluon fusion and $b\bar{b}$ corrections to $H W^+ W^- / H Z Z$ production in the POWHEG-BOX