Functional approaches are the only first principle quantum chromodynamics (QCD) setup to date that allow for direct computations in the strongly correlated finite density regime of QCD. Predictive power and quantitative reliability of the respective results can only be obtained within a systematic expansion scheme with controlled systematic error estimates. Here we set up such a scheme within the functional renormalization group (fRG) approach to QCD, aiming for full apparent convergence. In the current work we test this setup, using correlation functions and observables in $2+1$ flavor vacuum QCD as a natural benchmark case. While the current work includes many evolutionary improvements collected over the past two decades, we also report on three novel important developments: (i) comprehensive systematic error analysis based on the modular nature of the fRG approach. (ii) The introduction of a fully automated computational framework, allowing for unprecedented access and improvement of the fRG approach to QCD. (iii) The inclusion of the full effective potential of the chiral order parameter. This also gives access to all-order scattering events of pions and to the full momentum dependence of correlation functions, which is a first application of the automated computational framework (ii). The results compare very well to other state-of-the-art results both from functional approaches and lattice simulations, and provide data on general multiscattering events of pions and the sigma mode for the first time.

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• Functional renormalization group
• Quantum chromodynamics
• Gauge theories
• Gluons
• Mesons
• Quarks
• Spontaneous symmetry breaking
• Hydrodynamics