  We study the decay of the $Z$ vector boson into a photon and a massless
(invisible) dark photon in high-energy collisions. The photon can be used as
trigger for the event, while the dark photon is detected indirectly as missing
momentum in the event final state. We investigate the possibility of searching
for such a dark photon at the LHC, HL-LHC and future lepton colliders, and
compare the respective sensitivities. As expected, the best result is found for
the lepton colliders running at the $Z$ mass, FCC-ee and CEPC, with a final
sensitivity to branching ratios of order $O(10^{-11})$. We also discuss how to
use the photon angular distribution of the events in lepton collisions to
discriminate between the dark photon and a pseudo-scalar state like the axion.
