Phase structure and thermodynamics of cold and dense quark matter

The phase structure of quantum chromodynamics at low temperatures and finite density challenges our understanding ever since the theory of the strong interaction was formulated for the first time. While the vacuum phase structure is already non-trivial, i.e., governed by chiral symmetry breaking and the dynamical generation of hadrons, the ground state at high densities and low temperatures is by now commonly accepted to be a color superconductor. In this high-density regime, the thermodynamic properties are strongly affected by the color-superconducting gap in the excitation spectrum of the quarks. To gain a deeper understanding of the phase structure of quantum chromodynamics at finite density, we develop advanced calculation techniques to treat the limit of vanishing temperature and finite chemical potential in a well-defined manner. In particular, we focus on a careful ordering of mathematical operations. In addition to that, we develop the Mathematica tool TensorBases, which simplifies the systematic derivation of the correlation functions required to study non-perturbative phenomena. Using these computational techniques, we study quantum chromodynamics with two massless quark flavors from first principles using the functional renormalization group approach. This includes the characterization of the phase structure at zero temperature over a wide range of chemical potentials, solely via the identification of resonances in four-quark interaction channels generated by the fundamental quark-gluon interactions. Allowing for the dynamical formation of composite diquark and meson states in the renormalization group flow, which mimics the continuous transition in effective degrees of freedom from high to low energies, we calculate both the chiral condensate in the vacuum and the chirally symmetric two-flavor color-superconducting gap at large chemical potential in a unified framework. We demonstrate that gap-induced corrections to the thermodynamic properties of quantum chromodynamics can be sizable at high densities. For this purpose, we expand the pressure about the color-superconducting ground state and the strong coupling constant up to the next-to-leading order. At very high densities, we find that the speed of sound and the pressure approach the conformal limit associated with the free quark gas from below. This is in agreement with standard perturbative calculations. Towards lower densities, we then find that the presence of a gap pushes the speed of sound above the conformal value. Taking into account results from chiral effective field theory, our findings suggest a maximum in the speed of sound at supranuclear densities that exceeds the conformal limit. The astrophysically more relevant case of three-flavor quantum chromodynamics at high densities is expected to be a color superconductor in the color-flavor-locked phase. We extend our framework to this case and implement neutron-star conditions in terms of β-equilibrium, charge and color neutrality, within an expansion of the pressure in terms of the gap, the strong coupling, and the strange quark mass up to next-to-leading order. Utilizing a Bayesian analysis and taking into account constraints from chiral effective field theory and astrophysical observations, we constrain the size of the color-flavor-locking gap at perturbative densities. Even in this scenario, the color-superconducting ground state has a substantial effect on the thermodynamic properties of three-flavor quantum chromodynamics at high densities and enhances both the pressure and the speed of sound in accordance with constraints from astrophysical observations.