# Some remarks on the spectral functions of the Abelian Higgs Model

## TL;DR

This paper analyzes the spectral functions of the Abelian Higgs model at one-loop, clarifying physical states, gauge effects, and non-unitarity in a modified Curci-Ferrari framework.

## Contribution

It introduces a spectral function analysis of the Abelian Higgs model and a non-unitary Curci-Ferrari extension with a scalar, revealing insights into physical states and gauge choices.

## Key findings

- Spectral functions distinguish physical from unphysical states.
- Gauge choices and tadpole graphs significantly influence spectral results.
- The Curci-Ferrari model exhibits non-unitarity through spectral analysis.

## Abstract

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.