  We present a formalism for new $U(1)$ interactions involving weak
hypercharge, baryon, and lepton numbers, and a possible axial symmetry
generator $F_A$ in the presence of a second Brout-Englert-Higgs doublet. The
resulting $U$ boson, after mixing with the $Z$, interpolates between a
generalised dark photon, a dark $Z$, and an axially coupled gauge boson. We
especially focus on the axial couplings originating from $F_A$ or from mixing
with the $Z$, determined by the scalar sector via parameters like $\tan\beta$
and the v.e.v. of an extra dark singlet. We explore the distinctive features of
axially coupled interactions, especially in the ultrarelativistic limit, where
the $U$ boson behaves much as an axion-like particle, with enhanced
interactions to quarks and leptons. This enhancement is particularly relevant
for future muon beam dump experiments, since the muon mass considerably
increases the effective coupling, proportional to $2m_\mu/m_U$, compared to
analogous experiments with electrons. We also analyse the shape of the expected
beam dump exclusion or discovery regions, influenced by $U$ boson interactions
and the experiment geometry. Different situations are considered, limited in
particular by cases for which the $U$ decays before reaching the detector, or
has too small couplings to produce detectable events. We also compare to
vectorially coupled bosons and axion-like pseudoscalars, highlighting the
importance of understanding the parameter space for future experiment design
and optimisation.
