  Gauge theories have been a cornerstone of the description of the world at the
level of the fundamental particles. The Lagrangian or the action describing the
corresponding interactions is invariant under certain gauge transformations.
This symmetry is reflected in terms of the Ward-Green-Takahashi (or the
Slavnov-Taylor) identities which relate various Green functions among each
other, and the Landau-Khalatnikov-Fradkin transformations which relate a Green
function in a particular gauge to it in an arbitrary covariant one. As an
outcome, all physical observables should be independent of the choice of the
covariant gauge parameter. The most systematic scheme to solve quantum field
theories (QFT) is perturbation theory where the above-mentioned identities are
satisfied at every order of approximation. This feature is exploited highly
usefully as a verification of the results obtained after time- and
effort-consuming exercises. As it stands, not all natural phenomena are
realized in the perturbative regime of QFTs. Therefore, one is inevitably led
to make efforts to solve these theories in a non perturbative fashion. A
natural starting point for such studies in the continuum are the
Schwinger-Dyson Equations (SDE). One of the most undesirable features
associated with their non perturbative truncation is the loss of highly and
rightly esteemed gauge invariance rendering the predictions less reliable.
Arguably, the only hope we have to recuperate the credibility of the results is
to employ only such truncation schemes which yield predictions that match onto
the blindly reliable perturbative ones at every order of approximation. It is a
highly difficult task but not impossible.
