Sequential deconfinement of quark flavors in neutron stars

A scenario is suggested in which the three light quark flavors are sequentially deconfined under increasing pressure in cold asymmetric nuclear matter as found, for example, in neutron stars. The basis for this analysis is a chiral quark matter model of Nambu–Jona-Lasinio (NJL) type with diquark pairing in the spin-1 single-flavor, spin-0 two-flavor, and three-flavor channels. Nucleon dissociation sets in at about the saturation density, $n_0$, when the down-quark Fermi sea is populated ($d$-quark drip line) because of the flavor asymmetry induced by $\beta$ equilibrium and charge neutrality. At about $3n_0$, $u$-quarks appear and a two-flavor color superconducting (2SC) phase is formed. The $s$-quark Fermi sea is populated only at still higher baryon density, when the quark chemical potential is of the order of the dynamically generated strange quark mass. Two different hybrid equations of state (EOSs) are constructed using the Dirac-Brueckner Hartree-Fock (DBHF) approach and the EOS of Shen et al. [H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Nucl. Phys. A637, 435 (1998)] in the nuclear matter sector. The corresponding hybrid star sequences have maximum masses of 2.1 and 2.0 ${\mathrm{M}}_{\odot }$, respectively. Two- and three-flavor quark-matter phases exist only in gravitationally unstable hybrid star solutions in the DBHF case, whereas the Shen-based EOSs produce stable configurations with a 2SC phase component in the core of massive stars. Nucleon dissociation via $d$-quark drip could act as a deep crustal heating process, which apparently is required to explain superbursts and cooling of x-ray transients.

Figure 1

Schematic of chemical potentials (columns) and sequential deconfinement of quarks with increasing baryon density (from left to right). The flavor-dependent thresholds for chiral symmetry restoration (deconfinement) are approximately given by the dynamically generated quark masses $m_f$, $f=u,d,s$ (dashed lines). With increasing quark chemical potential, $\mu =({\mu }_u+{\mu }_d)/2$, the $d$-quark chemical potential is the first to reach the threshold in isospin asymmetric matter. Nucleon dissociation therefore sets in as $d$-quarks are deconfined. Still higher $\mu$ is needed to form two-flavor and three-flavor quark matter phases.

Figure 2

Solution of the NJL gap equations for isospin-asymmetric charge-neutral matter. The upper (lower) panel corresponds to the hybrid EOS based on the DBHF (Shen) nuclear EOS. The asymmetry at a given value of the baryon chemical potential, ${\mu }_B$, is different in the two cases because the charge density of nuclear matter depends on the model used.

Figure 3

Pressure of matter in $\beta$ equilibrium for different nuclear-matter models (DBHF, upper panel; Shen, lower panel) and phase transition constructions with color superconducting quark matter (NJL model). Here DBHF (or Shen) $+$ NJL refers to the mixed phase of nuclear matter and quark matter. At low chemical potential the quark-matter phase component is the negatively charged d-CSL phase, which lowers the asymmetry of the system and thereby gives a higher pressure for the mixed phase. At higher chemical potentials (1370 MeV for DBHF$\hspace{0.5em}+\hspace{0.5em}$NJL and 1230 for Shen$\hspace{0.5em}+\hspace{0.5em}$NJL), there is a transition in the quark sector to a two-flavor color superconducting (2SC) phase component.

Figure 4

Phase diagrams in the plane of baryon and charge chemical potential. The dash-dotted line denotes the border between oppositely charged phases. The nuclear-matter EOSs are DBHF (upper panel) and Shen et al. (lower panel). Only one- and two-flavor solutions are displayed, because three-flavor matter is charge neutral for ${\mu }_Q~0$. The transition to the nearly symmetric three-flavor CFL phase occurrs at ${\mu }_B~1600$ MeV (see Fig. 2). At low densities the mixture of nuclear matter with one-flavor quark matter (d-CSL) is favored. At higher densities, beyond the up-quark threshold, a mixture with two-flavor quark matter is favored. The two-flavor phases considered here are the normal quark-matter (NQ) and the superconducting (2SC) phases.

Figure 5

(a) Compact star sequences and (b) hybrid EOSs. The phase structure at the center, $r=0$, changes with increasing density, as indicated in the figures. Constraints on the compact-star mass come from 4U 1636 [2] and on the mass-radius relation from RX J1856 [4].

Figure 6

Direct Urca process in quark matter (a) and triangle of momentum conservation for it (b).

Figure 7

Density profiles of two stars with masses $1.4$ and $2.0$ ${\mathrm{M}}_{\odot }$. In the model adopted here, the mixed phase of d-CSL quark matter with nuclear matter extends up to the crust-core boundary.