https://www.physi.uni-heidelberg.de/~fschney/2008SS-Preseminar/buballarev.pdf

NJL-model analysis of dense quark matter

The current status of theoretical studies on the quantum chromodynamics (QCD) phase diagram at finite temperature and baryon chemical potential is reviewed with special emphasis on the origin of various phases and their symmetry breaking patterns. Topics include quark deconfinement, chiral symmetry restoration, order of the phase transitions, QCD critical point(s), colour superconductivity, various inhomogeneous states and implications from QCD-like theories.

https://iopscience.iop.org/article/10.1088/0034-4885/74/1/014001/pdf

The phase diagram of dense QCD

Strange quark matter and compact stars

Phases of cold, dense quarks at large N(c)

We show that a hybrid (nuclear+quark matter) star can have a mass-radius relationship very similar to that predicted for a star made of purely nucleonic matter. We show this for a generic parameterization of the quark matter equation of state and also for an MIT bag model, each including a phenomenological correction based on gluonic corrections to the equation of state. We obtain hybrid stars as heavy as 2 M☉ for reasonable values of the bag model parameters. For nuclear matter, we use the equation of state calculated by Akmal and coworkers using many-body techniques. Both mixed and homogeneous phases of nuclear and quark matter are considered.

/pdf/hybrid-stars-that-masquerade-as-neutron-stars-25wwa0o0o3.pdf

Hybrid stars that masquerade as neutron stars

In any context in which color superconductivity arises in nature, it is likely to involve pairing between species of quarks with differing chemical potentials. For suitable values of the differences between chemical potentials, Cooper pairs with nonzero total momentum are favored, as was first realized by Larkin, Ovchinnikov, Fulde, and Ferrell (LOFF). Condensates of this sort spontaneously break translational and rotational invariance, leading to gaps which vary periodically in a crystalline pattern. Unlike the original LOFF state, these crystalline quark matter condensates include both spin-zero and spin-one Cooper pairs. We explore the range of parameters for which crystalline color superconductivity arises in the QCD phase diagram. If in some shell within the quark matter core of a neutron star (or within a strange quark star) the quark number densities are such that crystalline color superconductivity arises, rotational vortices may be pinned in this shell, making it a locus for glitch phenomena.

Crystalline color superconductivity

We develop the Ginzburg-Landau approach to comparing different possible crystal structures for the crystalline color superconducting phase of QCD, the QCD incarnation of the Larkin-Ovchinnikov-Fulde-Ferrell phase. In this phase, quarks of different flavor with differing Fermi momenta form Cooper pairs with nonzero total momentum, yielding a condensate that varies in space like a sum of plane waves. We work at zero temperature, as is relevant for compact star physics. The Ginzburg-Landau approach predicts a strong first-order phase transition (as a function of the chemical potential difference between quarks) and for this reason is not under quantitative control. Nevertheless, by organizing the comparison between different possible arrangements of plane waves (i.e., different crystal structures) it provides considerable qualitative insight into what makes a crystal structure favorable. Together, the qualitative insights and the quantitative, but not controlled, calculations make a compelling case that the favored pairing pattern yields a condensate which is a sum of eight plane waves forming a face-centered cubic structure. They also predict that the phase is quite robust, with gaps comparable in magnitude to the BCS gap that would form if the Fermi momenta were degenerate. These predictions may be tested in ultracold gases made of fermionic atoms. In a QCD context, our results lay the foundation for a calculation of vortex pinning in a crystalline color superconductor, and thus for the analysis of pulsar glitches that may originate within the core of a compact star.

Crystallography of color superconductivity

We survey the non-locked color-flavor-spin channels for quark-quark (color superconducting) condensates in QCD, using an NJL model. We also study isotropic quark-antiquark (mesonic) condensates. We make mean-field estimates of the strength and sign of the self-interaction of each condensate, using four-fermion interaction vertices based on known QCD interactions. For the attractive quark pairing channels, we solve the mean-field gap equations to obtain the size of the gap as a function of quark density. We also calculate the dispersion relations for the quasiquarks, in order to see how fully gapped the spectrum of fermionic excitations will be. We use our results to specify the likely pairing patterns in neutral quark matter, and comment on possible phenomenological consequences.

Single color and single flavor color superconductivity

After a brief review of the phenomena expected in cold dense quark matter, colour superconductivity and colour-flavour locking, we sketch some implications of recent developments in our understanding of cold dense quark matter for the physics of compact stars. We give a more detailed summary of our recent work on crystalline colour superconductivity and the consequent realization that (some) pulsar glitches may originate in quark matter.

Color Superconductivity in Compact Stars

We present a derivation of the gap equation for the crystalline color superconducting phase of QCD which begins from a one-loop Schwinger-Dyson equation written using a Nambu-Gorkov propagator modified to describe the spatially varying condensate. Some aspects of previous variational calculations become more straightforward when rephrased beginning from a diagrammatic starting point. This derivation also provides a natural base from which to generalize the analysis to include quark masses, nontrivial crystal structures, gluon propagation at asymptotic densities, and nonzero temperature. In this paper, we analyze the effects of nonzero temperature on the crystalline color superconducting phase.

Jeffrey A. Bowers

Papers

Crystalline color superconductivity

Crystallography of color superconductivity

Single color and single flavor color superconductivity

Color Superconductivity in Compact Stars

Diagrammatic approach to crystalline color superconductivity