  The Abelian Higgs model forms an essential part of the electroweak standard
model: it is the sector containing only Z and Higgs bosons. We present a
diagram-based proof of the tree-level unitarity of this model inside the
unitary gauge, where only physical degrees of freedom occur. We derive
combinatorial recursion relations for off-shell amplitudes in the massless
approximation, which allows us to prove the cancellation of the first two
orders in energy of unitarity-violating high-energy behaviour for any
tree-level amplitude in this model. We describe a deformation of the amplitudes
by extending the physical phase space to at least 7 spacetime dimensions, which
leads to on-shell recursion relations a la BCFW. These lead to a simple proof
that all on-shell tree amplitudes obey partial-wave unitarity.
