  We consider the unitary Abelian Higgs model and investigate its spectral
functions at one-loop order. This analysis allows to disentangle what is
physical and what is not at the level of the elementary particle propagators,
in conjunction with the Nielsen identities. We highlight the role of the
tadpole graphs and the gauge choices to get sensible results. We also introduce
an Abelian Curci-Ferrari action coupled to a scalar field to model a massive
photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a
modified non-nilpotent BRST symmetry. We clearly illustrate its non-unitary
nature directly from the spectral function viewpoint. This provides a
functional analogue of the Ojima observation in the canonical formalism: there
are ghost states with nonzero norm in the BRST-invariant states of the
Curci-Ferrari model.
