  In the case of non-abelian gauge theories, the standard Faddeev-Popov (FP)
gauge-fixing procedure in the Landau gauge is known to be incomplete due to the
presence of gauge-equivalent field configurations. A widespread belief is that
the proper analysis of the low energy properties in this gauge requires the
extension of the gauge-fixing procedure beyond the FP recipe. This manuscript
reviews various applications of the Curci-Ferrari (CF) model, a
phenomenological proposal for such an extension, based on the decoupling
properties of Landau gauge correlators as computed on the lattice. We
investigate the predictions of the model concerning the deconfinement
transition of strongly interacting matter at finite temperature, first in the
case of pure Yang-Mills theory and then in the case of heavy-quark Quantum
Chromodynamics. We show that most qualitative aspects and also many
quantitative features of the deconfinement transition can be accounted for
within the CF model, with only one additional parameter, adjusted from
comparison to lattice simulations. Moreover, these features emerge in a
systematic and controlled perturbative expansion, as opposed to the ill-defined
perturbative expansion within the FP model in the infrared. Applications at
finite temperature and/or density require one to consider a background
extension of the Landau gauge, the so-called Landau-deWitt gauge. The
manuscript is also intended as a thorough but pedagogical introduction to these
techniques, including the rationale for introducing a background, the role of
the Weyl chambers, and the complications that emerge due to the sign problem in
the case of a real quark chemical potential. It also investigates the fate of
the correlation functions as computed in the CF model and conjectures a
specific behavior for the corresponding functions evaluated on the lattice, in
the case of the SU(2) gauge group.
