Constraints on large scalar multiplets added to the Standard Model

We study the extension of the Standard Model (SM) by introducing a scalar multiplet with arbitrary isospin $J$ and hypercharge $Y$. We explicitly consider various possible values of the weak isospin $J$, up to and including $J=7/2$. The mass differences among the components of the multiplet originate from its coupling to the Higgs doublet of the SM, as present in the scalar potential (SP). We derive exact bounded-from-below (BFB) and unitarity (UNI) conditions for this model, even when the SP includes the most general quartic terms involving the multiplet components. We find that the upper bound on the mass differences depends not only on the UNI conditions but also on the BFB ones, thus imposing constraints on the mass differences. We compare these constraints to those derived from the oblique parameters (OPs) and from solutions of the renormalization-group equations (RGEs).