Revisiting $Vh(\to b\bar{b})$ at the LHC and FCC-hh

We consider $Vh$ production, focusing on the $h \to b\bar b$ decay channel, whose sizeable cross section makes it accessible at the LHC.We perform an improved analysis combining the 0-, 1- and 2-lepton channels with a scale-invariant $b$-tagging algorithm that allows us to exploit events with either a boosted Higgs via mass-drop tagging or resolved $b$-jets.Combining the boosted and resolved categories leads to a 13% improvement on the most strongly bounded Wilson coefficient at the LHC Run 3 with respect to the boosted category alone (and an 8% improvement at FCC-hh).This strategy gives sensitivity to 4 dimension-6 SMEFT operators that modify the $W$ and $Z$ couplings to quarks and is competitive with the bounds obtained from global fits and other Higgs decay channels.The benefit of the $h\to b\bar b$ decay channel is that it is the only one accessible at the LHC and HL-LHC while at FCC-hh it is competitive with the effectively background free $h\to \gamma\gamma$ channel assuming $\lesssim 5\%$ systematic uncertainty.Furthermore, we show that, at the FCC, binning in the rapidity of the $Vh$ system can significantly reduce correlations between some EFT operators.The bounds we obtain transalte to a lower bound on the new physics scale or 5, 8, and 20 TeV at the LHC Run3, HL-LHC, and FCC-hh respectively assuming a Wilson coefficient of unity.Finally, we assess the impact of the $Vh$ production channel on anomalous triple gauge coupling measurements, comparing with the determination at lepton colliders