Scalar Electroweak Multiplet Dark Matter

We revisit the theory and phenomenology of scalar electroweak multiplet thermal dark matter. We derive the most general, renormalizable scalar potential, assuming the presence of the Standard Model Higgs doublet, $H$, and an electroweak multiplet $\Phi$ of arbitrary SU(2$)_L$ rank and hypercharge, $Y$. We show that, in general, the $\Phi$-$H$ Higgs portal interactions depend on three, rather than two independent couplings as has been previously considered in the literature. For the phenomenologically viable case of $Y=0$ multiplets, we focus on the septuplet and quintuplet cases, and consider the interplay of relic density and spin-independent direct detection cross section. We show that both the relic density and direct detection cross sections depend on a single linear combination of Higgs portal couplings, $\lambda_{\rm eff}$. For $\lambda_{\rm eff}\sim \mathcal{O}(1)$, present direct detection exclusion limits imply that the neutral component of a scalar electroweak multiplet would comprise a subdominant fraction of the observed DM relic density.