  The search for dark matter (DM) at colliders is founded on the idea of
looking for something invisible. There are searches based on production and
decay processes where DM may reveal itself as missing energy. If nothing is
found, our best tool to constrain the parameter space of many extensions of the
Standard Model (SM) with a DM candidate is the Higgs boson. As the measurements
of the Higgs couplings become increasingly precise, higher-order corrections
will start to play a major role. The tree-level contribution to the invisible
decay width provides information about the portal coupling. Higher-order
corrections also gives us access to other parameters from the dark sector of
the Higgs potential that are not present in the tree-level amplitude. In this
work we will focus on the complex singlet extension of the SM in the phase with
a DM candidate. We calculate the one-loop electroweak corrections to the decay
of the Higgs boson into two DM particles. We find that the corrections are
stable and of the order of a few percent. The present measurement of the Higgs
invisible branching ratio, BR$(H \to$ invisible $) < 0.11$, already constrains
the parameter space of the model at leading order. We expect that by the end of
the LHC the experimental measurement will require the inclusion of the
electroweak corrections to the decay in order to match the experimental
accuracy. Furthermore, the only competing process, which is direct detection,
is shown to have a cross section below the neutrino floor.
