On the extension of the SM through a scalar quadruplet

We consider the extension of the Standard Model (SM) through a scalar quadruplet with hypercharge either $1\mathrm{/}2$ or $3\mathrm{/}2$; in the first case, we assume $CP$ invariance of the scalar potential (SP). We write down the unitarity conditions on the SP. We use a partly numerical method to find the bounded-from-below conditions on the SP. We determine the masses of the new scalars of the model. We compute the three- and four-Higgs couplings and compare them to the ones of the SM.