Fusing vectors into scalars at high energy lepton colliders

We study vector boson fusion production of new scalar singlets at high energy lepton colliders. We find that CLIC has the potential to test single production cross-sections of a few tens of attobarns in di-Higgs and di-boson final states. In models with a sizeable singlet-Higgs mixing, these values correspond to a precision in Higgs couplings of order 0.1% or better. We compare our sensitivities with those of the LHC and interpret our results in well-motivated models like the Twin Higgs, the NMSSM and axion-like particles. Looking forward to even higher energy machines, we show that the reach of muon colliders like LEMMA or MAP overcomes the one of future hadron machines like FCC-hh. We finally study the pair production of the new scalar singlets via an off-shell Higgs. This process does not vanish for small mixings and will constitute a crucial probe of models generating a first order electro-weak phase transition.