On the addition of a large scalar multiplet to the Standard Model

We consider the addition of a single $SU(2)$ multiplet of complex scalar fields to the Standard Model (SM). We explicitly consider the various possible values of the weak isospin $J$ of that multiplet, up to and including $J = 7/2$. We allow the multiplet to have arbitrary weak hypercharge. The scalar fields of the multiplet are assumed to have no vacuum expectation value; the mass differences among the components of the multiplet originate in its coupling, present in the scalar potential (SP), to the Higgs doublet of the SM. We derive exact bounded-from-below and unitarity conditions on the SP, thereby constraining those mass differences. We compare those constraints to the ones that may be derived from the oblique parameters.