We analyze different BRST invariant solutions for the introduction of a mass
term in Yang-Mills (YM) theories. First, we analyze the non-local composite
gauge-invariant field Aμh​(x), which can be localized by the
Stueckelberg-like field ξa(x). This enables us to introduce a mass term in
the SU(N) YM model, a feature that has been indicated at a non-perturbative
level by both analytical and numerical studies. We also 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. Finally, the spectral properties of a set of local gauge-invariant
composite operators are investigated in the U(1) and SU(2) Higgs model
quantized in the 't Hooft Rξ​ gauge. These operators enable us to give a
gauge-invariant description of the spectrum of the theory, thereby surpassing
certain incommodities when using the standard elementary fields. The
corresponding two-point correlation functions are evaluated at one-loop order
and their spectral functions are obtained explicitly. It is shown that the
spectral functions of the elementary fields suffer from a strong unphysical
dependence from the gauge parameter ξ, and can even exhibit positivity
violating behaviour. In contrast, the BRST invariant local operators exhibit a
well defined positive spectral density.Comment: PhD thesis, september 202