Some remarks on the spectral functions of the Abelian Higgs model

We consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows us 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 nonunitary nature directly from the spectral function viewpoint. This provides a functional analogue of the Ojima observation in the canonical formalism: them arc ghost states with nonzero norm in the BRST-invariant states of the Curci-Ferrari model.