Higgs physics at the Future Circular Collider

The unique Higgs physics opportunities accessible at the CERN Future Circular Collider (FCC) in electron-positron ($\sqrt{s}$ = 125, 240, 350 GeV) and proton-proton ($\sqrt{s}$ = 100 TeV) collisions, are succinctly summarized. Thanks to the large c.m. energies and enormous luminosities (plus clean experimental conditions in the $e^+e^-$ case), many open fundamental aspects of the Higgs sector of the Standard Model (SM) can be experimentally studied: Measurement of the Higgs Yukawa couplings to the lightest fermions: u,d,s quarks (via rare exclusive $H\to(\rho,\omega,\phi)+\gamma$ decays); and e$^\pm$ (via resonant s-channel $e^+e^-\to H$ production); as well as neutrinos (within low-scale seesaw mass generation scenarios). Measurement of the Higgs potential (triple $\lambda_3$, and quartic $\lambda_4$ self-couplings), via double and triple Higgs boson production in pp collisions at 100 TeV. Searches for new physics coupled to the scalar SM sector at scales $\Lambda>$ 6 TeV, thanks to measurements of the Higgs boson couplings with subpercent uncertainties in $e^+e^-\to H\,Z$. Searches for dark matter in Higgs-portal interactions, via high-precision measurements of on-shell and off-shell Higgs boson invisible decays. All these measurements are beyond the reach of pp collisions at the Large Hadron Collider. New higher-energy $e^+e^-$ and pp colliders such as FCC are thus required to complete our understanding of the full set of SM Higgs parameters, as well as to search for new scalar-coupled physics in the multi-TeV regime.