  An electroweak multiplet stable due to a new global symmetry is a simple and
well-motivated candidate for thermal dark matter. We study how direct searches
at a future linear collider, such as the proposed CLIC, can constrain scalar
and fermion triplets, quintets and septets, as well as a fermion doublet. The
phenomenology is highly sensitive to charged state lifetimes and thus the mass
splitting between the members of the multiplet. We include both radiative
corrections and the effect of non-renormalisable operators on this splitting.
In order to explore the full range of charged state lifetimes, we consider
signals including long-lived charged particles, disappearing tracks, and
monophotons. By combining the different searches we find discovery and
exclusion contours in the mass-lifetime plane. In particular, when the mass
splitting is generated purely through radiative corrections, we can exclude the
pure-Higgsino doublet below 310 GeV, the pure-wino triplet below 775 GeV, and
the minimal dark matter fermion quintet below 1025 GeV. The scenario where the
thermal relic abundance of a Higgsino accounts for the whole dark matter of the
Universe can be excluded if the mass splitting between the charged and neutral
states is less than 230 MeV. Finally, we discuss possible improvements to these
limits by using associated hard leptons to idenify the soft visible decay
products of the charged members of the dark matter multiplet.
