  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.
