The LHC experiments have, so far, measured many of the Higgs couplings and found excellent agreement with the minimally-realized electroweak symmetry breaking (EWSB) mechanism in the Standard Model. Nevertheless, there are important couplings that are currently out of reach which test the nature of EWSB and fermion mass generation. This talk will focus on two of these couplings: the charm Yukawa and the hhVV couplings. A measurement of the first would confirm that the 125 GeV Higgs which gives mass to third generation fermions also gives mass the second generation. To this end, I will describe recent ideas to probe the charm Yukawa coupling; in particular, by using Higgs differential distributions. While, in the second case, deviations of the hhVV coupling from the SM would signal non-linearity and herald new physics at higher energies. As I will show, double Higgs production in VBF at the LHC can provide such a test at the 20% level by the end of the high luminosity run while a percent level constraint can be obtained at a future circular collider.
