We study an effective field theory
which includes the Standard Model extended by a Dark Sector consisting of
two fermionic $SU(2)_{L}$-doublets. A $Z_2$ parity guarantees that, after electroweak symmetry breaking,
the lightest neutral particle is stable, acting as a WIMP.
The dark sector interacts with the Higgs and gauge bosons
through renormalizable and non-renormalizable $d=5$ operators.
We find that a WIMP with a mass around the electroweak scale,
\textit{i.e.} accessible at the LHC,
is consistent with collider and astrophysical data only when
non-trivial magnetic dipole interactions with the gauge bosons exist.