Derivative interactions and perturbative UV contributions in N Higgs doublet models

We study the Higgs derivative interactions on models including arbitrary number of the Higgs doublets. These interactions are generated by two ways. One is higher order corrections of composite Higgs models, and the other is integration of heavy scalars and vectors. In the latter case, three point couplings between the Higgs doublets and these heavy states are the sources of the derivative interactions. Their representations are constrained to couple with the doublets. We explicitly calculate all derivative interactions generated by integrating out. Their degrees of freedom and conditions to impose the custodial symmetry are discussed. We also study the vector boson scattering processes with a couple of two Higgs doublet models to see experimental signals of the derivative interactions. They are differently affected by each heavy field. (orig.)