Non-Abelian Landau-Khalatnikov-Fradkin transformations & dynamically massive linear covariant gauges

After a short introduction in the world of particle physics, we arrive at the study of the strong interaction. We find a path integral formalism, the Faddeev-Popov action, that describes this strong interaction. This theory still has some issues troubling it. For instance the existence of gauge copies and the resulting overcounting in the path integral, the Gribov problem, see Section 2.4. Another problem consists of the gauge dependence introduced by gauge fixing and by the perturbative nature of the theory. As a proposition to solve the Gribov problem, we look at the $A^h$-theory introduced by cpari et al., and reviewed in Chapter 2. This model will prove essential to us, it will constantly be used throughout this thesis.In Chapter 3 we introduce Landau-Fradkin-Khalatnikov transformations (LKFT's), these show us the relation between n-point functions in different gauges. More specifically, they can be used to see how the gauge dependence can propagate between gauges. In the second chapter we sketch the working principle and propose a computation, we also test the Abelian limit to the already known results. In addition to this, the same results can be obtained from the variation of the path integral. In the fourth chapter, this non-Abelian perturbative proposal is explicitly computed up to first order, which retrieves the LKFT for the gluon propagator. A computation ''by hand'' was not feasible. Instead, we use this chapter to introduce Mathematica packages to compute the relevant diagrams semi-automatically. In the second part of the chapter the link with the Nielsen identities is shown.In these previous two chapters the perturbative character of QCD is clearly visible. In Chapter 5 we try to find some all-order results. First a one loop example will lead to a full order result where we prove that the mass only appears in the highest order terms. This result will be used to construct a specific Renormalization Group and corresponding transformation schemes. The scheme resembles the Curci-Ferrari model, but now with the explicit introduction of a dynamical mass term originating from the $A^h$- action. This allows to construct a gap equation which can be used to derive the mass from first principles, rather than through a fitting procedure.In the final chapter, we keep working within the $A^h$-model, as to model non-perturbative effects in a general linear covariant gauge. We try to compute the contribution of the extra diagrams to the gluon self energy. Here we stumble upon some divergent diagrams, which we try to resolve using the resummation of a class of diagrams. We first find converging results using numerical integration techniques, later the insights point towards an analytic solution of the problem. Motivated by this result, we look at the Renormalization Group in the second part of the chapter, giving similar results in combination with the resummation of a different class of diagrams.