This thesis discusses the properties of cold quark matter as exists in the core of massive neutron stars, with baryon densities several times of nuclear saturation density, n_0 ≈ 0.16 fm^-3, at zero temperature. Specifically, we study effective quark models, the symmetry pattern of different quark matter phases, the collective modes associated with spontaneous chiral symmetry breaking, a possible realization of quark-hadron continuity in the color-flavor locked (CFL) quark superfluid, and the implications of these issues to quark matter equations of state, and thus to neutron star structure including the mass-radius (M-R) relation.In Chapter 1, we present a general overview of quark matter described by quantum chromodynamics (QCD) in the context of dense neutron star cores. We discuss the physical motivation for studying quark matter, and how the fundamental symmetry and symmetry breaking patterns of QCD guide the construction of phenomenological quark models – in particular, the Nambu–Jona-Lasinio (NJL) model. We briefly review the modern understanding of the QCD phase diagram studied via such effective quark models, and give an overview of recent progress in constructing quark-hadron crossover equation of states using quark model and nuclear matter models, and how neutron star observations constrain the parameter spaces, thus providing insight into quark matter.After the introduction, we next focus on details of the NJL model and how it can be made to reflect the QCD symmetries in Chapter 2. We describe quark matter using effective local interactions, and demonstrate how the spontaneous breaking of chiral symmetry is realized through such interactions. We work through a Hubbard-Stratonovich transformation and derive the effective quark-meson theory in a vacuum with chiral condensate, and note its structural connection to the sigma model. We then describe diquark pairing in the NJL model, which breaks chiral symmetry at high density as well. Lastly we explore the problem of meson condensation in quark matter, which is relevant to the both neutron star M-R relation and the cooling process; we show that the physics of quark matter meson condensation is very tightly connected to hadronic meson condensation studies, and discuss the criteria of condensation instability caused by quark-meson interactions.The next part of this thesis, Chapter 3, turns to the issue of connecting the chiral symmetry breaking in the vacuum and in high density color superconductors – the interplay of chiral and diquark condensates in the effective quark model. By using a schematic NJL model, we solve the phase diagram at zero temperature, and demonstrate a continuous evolution of the Goldstone bosons, i.e., the pions, from their vacuum q ̄q form to their diquark qq form. We identify all the collective modes associated with the chiral and diquark condensates and calculate the pion self-energy, deriving a generalized Gell-Mann–Oakes–Renner (GMOR) relation. We thus establish a picture of continuous chiral symmetry breaking from vacuum to high density quark matter, and discuss its implications and connection to the quark-hadron continuity conjecture.In Chapter 4 we focus on a possible realization of quark-hadron continuity in the color-flavor-locked (CFL) superfluid phase, where the CFL diquark condensates screen color charges of elementary excitations, a novel feature of the SU(3) color-flavor structure. We construct the dressing scheme inspired by the non-linear sigma model, and derive an effective theory in terms of baryons and mesons, a gauge-invariant theory that originally started with quarks and gluons. Such a mapping is a direct realization of the quark hadron continuity in both the fermion sector and the boson sector, suggesting that we may study the properties of CFL quark matter in an entirely gauge-invariant manner at lower energies. The mapping scheme also brings up the relation between the effective baryon-meson Lagrangian’s couplings, elementary excitations and collective modes, and those of quark and nuclear matter as a potential research topic, which contributes to our further understanding of the ground state of dense matter at several times n_0.Finally, in Chapter 5 we turn back to the effective quark model and try to connect it to both nuclear matter and the more fundamental QCD. We demonstrate that the explicit single gluon exchange energy can help understanding the magnitude and density dependence of the constrained value of the phenomenological vector repulsion necessary to support massive neutron stars, with a moderate strong coupling constant and gluon mass at some 5n_0. We also estimate the effect of higher order effects of introducing quark chiral masses and CFL pairing into the quark Green’s functions. Our calculation yields an approximately flavor-symmetric vector repulsion that is a monotonous decreasing function of density, which we parametrize for use in future studies of neutron star equations of state. We also discuss the potential connections of this calculation to the concept of quark-hadron continuity, based on the similarity of the quark model to chiral baryon models.
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