  We investigate the role of the Higgs \emph{doublet} in the thermal decoupling
of multi-TeV dark matter coupled to the Weak interactions of the Standard Model
and the Higgs. The Higgs doublet can mediate a long-range force that affects
the annihilation processes and binds dark matter into bound states. More
importantly, the emission of a Higgs doublet by a pair of dark matter particles
can give rise to extremely rapid monopole bound-state formation processes and
bound-to-bound transitions. We compute these effects in the unbroken
electroweak phase. To this end, we consider the simplest renormalisable
fermionic model, consisting of a singlet and a doublet under $SU_{L}(2)$ that
are stabilised by a $\mathbb{Z}_2$ symmetry, in the regime where the two
multiplets coannihilate. In a companion paper, we use the results to show that
the formation of metastable bound states via Higgs-doublet emission and their
decay decrease the relic density very significantly.
