The heavy QCD phase structure from effective lattice theories

In this thesis effective theories of lattice Quantum Chromodynamics (QCD) at finite temperature $T$ and baryon chemical potential $\muBaryon$ are studied. These effective theories, derived from full lattice QCD through combined strong coupling and hopping parameter expansions, offer a computationally cheaper formulation of their mother theory. This renders these effective theories particularly suited for exploring regions of the QCD phase diagram where conventional Monte Carlo methods are hindered by the sign problem. The work at hand proceeds in three main stages: the evaluation of existing effective theories using mean field approximations, the development of resummation techniques and derivation of higher-order corrections, and benchmarking of effective theories employing our improvements. Three mean field approaches are explored: a saddle-point approximation, a leading-order fluctuation expansion, and a novel resummed method capturing all local fluctuations. Their accuracy is tested in the leading pure gauge effective theory. The most advanced method predicts critical effective couplings with an accuracy of about 1.5\%, compared to 49\% for the least accurate approach. Using these approaches we map out the first-order deconfinement lines and their critical endpoints at finite quark mass and baryon chemical potential. Deviations from full lattice QCD simulation results are attributed to neglected non-local fluctuations and truncated series expansions. Due to these sources of systematic errors predictions for critical end points are always inaccurate. At low temperature and finite $\muBaryon$ a first order nuclear liquid-gas transition is identified, although the entropy density shows an unexpected decrease at the transition and unphysical effects like Pauli principle violations occur in the liquid phase. To improve upon these issues we refine known resummation techniques, such as a generalized form of logarithmic resummation and spatial detours. The former resummation scheme partially cancels Pauli principle violating contributions to hopping-truncated effective actions. The latter scheme allows to recover some long-range effective interactions, which can be resummed within our mean field treatments. Further, we derive an expression describing a large set of hopping corrections. In particular, for three spatial dimensions the derived result correctly describes all hopping corrections up to $\mathcal{O}(\kappa^{10})$ and is applicable for non-degenerate quark masses, flavor dependent chemical potentials and arbitrary temperatures. Lastly, effective theories employing our improvements are benchmarked against both earlier effective theory formulations and full lattice QCD simulation results. Log-resummed effective theories yield more accurate estimates for the deconfinement critical end point and display improved behavior under including higher order pure gauge corrections than their non-log-resummed counterparts. Notably, only log-resummed theories show decreasing errors for the location of deconfinement critical endpoints when higher-order corrections are subsequently included.