Towards Infrared QCD Correlation Functions With Functional Methods

In this thesis, we study the infrared regime of QCD with functional methods. To this end, we present the Mathematica package QMeS-Derivation. It allows for the derivation of symbolic functional equations from a given master equation. We apply this tool to Landau gauge Yang-Mills theory, where gauge invariance is implemented via BRST symmetry. We solve a self-consistent set of momentum-dependent functional equations within a vertex expansion and compare the correlation functions obtained from different functional approaches. We find good agreement of the results, hinting at gauge consistency of our setup. We proceed by using the obtained Euclidean results to compute four-gluon correlation functions, from which we extract the scalar and pseudo-scalar glueball mass by finding spectral representations of the dressings. The obtained results agree well with the masses computed from other methods. Next, we consider QCD at finite temperature and finite chemical potential and provide a setup for thermal correlation functions, where we introduce a thermal split in the quark-gluon vertex. This setup allows for a study of the phase diagram of QCD, where special emphasis is put on the investigation of the chiral phase transition