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Abstract

This thesis discusses the quantitative description of the phase structure of Quantum Chromo- dynamics (QCD). We find that, in strongly correlated theories such as QCD, even a qualitative investigation of the phase structure can require highly quantitative methods. Hence, the de- velopment of a method with systematic error control is essential. In the present work, we use functional renormalisation group (fRG) method to this aim. This work focusses on three ideas: Firstly, we identify quantitatively dominating and sub-leading scattering-processes in our approximations. This allows a formulation of low energy effective theories of the four-quark interaction, as well as the description of gluon condensation. For the former, we present results for meson and quark masses. The latter provides an estimate of the Yang-Mills mass gap. Secondly, we further develop the use of highly precise numerical methods from fluid-dynamics in the fRG. In particular we use Discontinuous Galerkin methods, which are able to capture shock-development. Shock-waves are found to play a big role in a possible creation-mechanism of first-order phase transitions. Lastly, we focus on general RG-transformations (gRGt). For example, they allow a real time formulation of fRG flows and hence give access to spectral functions. Furthermore, we use them to formulate complex RG-flows, which enables us to locate Lee-Yang singularities in the complex plane and extrapolate the position of (real) phase transitions. Finally, we also use gRGts to formulate significant qualitative improvements of current fRG approximation schemes by means of dynamical field transformations.