High scale boundary conditions in models with two Higgs doublets

We investigate high scale boundary conditions on the quartic Higgs couplings and their beta functions in the type-II two Higgs doublet model and the inert doublet model. These conditions are associated with two possible UV physics scenarios: the multiple point principle, in which the potential exhibits a second minimum at M-P1, and asymptotic safety, where the scalar couplings run toward an interacting UV fixed point at high scales. We employ a renormalization group running at two loops and apply theoretical and experimental constraints to their parameter spaces. We find neither model can simultaneously accommodate the multiple point principle while also providing realistic masses for both the Higgs and the top quark. However, we do find regions of parameter space compatible with asymptotic safety.