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We analyze different BRST invariant solutions for the introduction of a massterm in Yang-Mills (YM) theories. First, we analyze the non-local compositegauge-invariant field $A^h_{\mu}(x)$, which can be localized by theStueckelberg-like field $\xi^a(x)$. This enables us to introduce a mass term inthe $SU(N)$ YM model, a feature that has been indicated at a non-perturbativelevel by both analytical and numerical studies. We also consider the unitaryAbelian Higgs model and investigate its spectral functions at one-loop order.This analysis allows to disentangle what is physical and what is not at thelevel of the elementary particle propagators, in conjunction with the Nielsenidentities. We highlight the role of the tadpole graphs and the gauge choicesto get sensible results. We also introduce an Abelian Curci-Ferrari actioncoupled to a scalar field to model a massive photon which, like the non-AbelianCurci-Ferarri model, is left invariant by a modified non-nilpotent BRSTsymmetry. Finally, the spectral properties of a set of local gauge-invariantcomposite operators are investigated in the $U(1)$ and $SU(2)$ Higgs modelquantized in the 't Hooft $R_{\xi}$ gauge. These operators enable us to give agauge-invariant description of the spectrum of the theory, thereby surpassingcertain incommodities when using the standard elementary fields. Thecorresponding two-point correlation functions are evaluated at one-loop orderand their spectral functions are obtained explicitly. It is shown that thespectral functions of the elementary fields suffer from a strong unphysicaldependence from the gauge parameter $\xi$, and can even exhibit positivityviolating behaviour. In contrast, the BRST invariant local operators exhibit awell defined positive spectral density.

Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions