  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.

High Energy Physics - Phenomenology