Phase structure and observables at high densities from first principles QCD

We provide a short review of the progress made in the past decade with functional QCD in the description of the phase structure of QCD. We summarise the most important technical aspects of the framework, discuss strategies for truncations and address the problem of systematic error estimates. We detail efforts to gauge the approach systematically with lattice QCD at zero chemical potential, also including the physics of the Columbia plot at non-physical quark masses. Our main focus is, however, the high density regime of QCD. We address the predictive power of the functional approach for the appearance of new phases beyond the chiral crossover regime for chemical potentials $\mu_B/T\geq 4.5$. The onset of this regime may be signalled by a critical end point of the crossover line but may also involve a moat regime or the emergence of an instability that indicates an inhomogeneous phase. Respective results include estimates for the location of the onset of new phases, and predictions for their experimental signatures.