Interface effects of quark matter: Light-quark nuggets and compact stars

The interface effects of quark matter play important roles in the properties of compact stars and small nuggets, such as strangelets and nonstrange quark matter ($ud\mathrm{QM}$) nuggets. By introducing a density derivative term to the Lagrangian density and adopting Thomas-Fermi approximation, we find it is possible to reproduce the results obtained by solving Dirac equations. Adopting certain parameter sets, the energy per baryon of $ud\mathrm{QM}$ nuggets decreases with baryon number $A$ and become more stable than nuclei at $A\gtrsim 300$. The effects of quark matter symmetry energy are examined, where $ud\mathrm{QM}$ nuggets at $A\approx 1000$ can be more stable than others if large symmetry energy is adopted. In such cases, larger $ud\mathrm{QM}$ nuggets will decay via fission and the surface of a $ud\mathrm{QM}$ star will fragment into a crust made of $ud\mathrm{QM}$ nuggets and electrons, which resembles the cases of a strange star’s crust. The corresponding microscopic structures are then investigated adopting spherical and cylindrical approximations for the Wigner-Seitz cells, where the droplet phase is found to be the most stable configuration with $ud\mathrm{QM}$ stars’ crusts and $ud\mathrm{QM}$ dwarfs made of $ud\mathrm{QM}$ nuggets ($A\approx 1000$) and electrons. For the cases considered here, the crust thickness of $ud\mathrm{QM}$ stars is typically $\sim 200\mathrm{m}$, which reaches a few kilometers if we neglect the interface effects and adopt Gibbs construction. The masses and radii of $ud\mathrm{QM}$ dwarfs are smaller than typical white dwarfs, which would increase if the interface effects are neglected.

Figure 1

Energy per baryon ($M/A$) of $\beta$-stable strangelets obtained with TFA (solid line), which are compared with those predicted by explicitly solving Dirac equations ($\text{symbol}+\mathrm{line}$).

Figure 2

Comparison between the density profiles in strangelets determined by wave functions with Eq. (14) (solid) and TFA with Eq. (17) (dashed).

Figure 3

Charge-to-mass ratio ($f_Z$), strangeness per baryon ($f_S$), and the ratio of root-mean-square radius to baryon number ($r_0$) for $\beta$-stable strangelets, in correspondence to those in Fig. 1.

Figure 4

Energy per baryon, charge-to-mass ratio, and ratio of root-mean-square radius to baryon number of $ud\mathrm{QM}$ nuggets. The dashed curves correspond to the properties of strangelets indicated in Figs. 1 and 3, where the same parameter sets are adopted as indicated in Table 1.

Figure 5

Same as Fig. 4, but adopting the parameter set $C=0.1$, $\sqrt{D}=150\mathrm{MeV}$, and various values of $C_I$. The curves encircled by the shaded areas indicate the properties of $ud\mathrm{QM}$ nuggets. The coefficient of the density derivative term is taken as ${\delta }_V=-1.5\times {10}^{-10}{\mathrm{MeV}}^4$, which is indicated in Table 1.

Figure 6

Density profiles of quarks and electrons in WS cells for droplet phase at $n_{\mathrm{b}}=1.3\times {10}^{-4}$ and ${10}^{-5}{\mathrm{fm}}^{-3}$. The boundary of the WS cell is indicated by the shaded region.

Figure 7

Droplet size $R_{\mathrm{d}}$, WS cell radius $R_{\mathrm{W}}$, and baryon number $A$ for the droplet phase in $ud\mathrm{QM}$ stars’ crusts. The corresponding structures in neutron stars’ crusts are indicated with red dashed curves for comparison [84].

Figure 8

Pressures of dense stellar matter in $ud\mathrm{QM}$ stars and neutron stars as functions of baryon chemical potential ${\mu }_{\mathrm{b}}$ and energy density $M/V$. The open circles mark the critical pressure and chemical potential, below which the nonuniform structures take place.

Figure 9

Energy per baryon and charge-to-mass ratio for dense stellar matter in $ud\mathrm{QM}$ stars and neutron stars as functions of baryon number density.

Figure 10

Density profiles of 1.4-solar-mass $ud\mathrm{QM}$ stars and neutron star.

Figure 11

Mass-radius relations of cold $ud\mathrm{QM}$ stars (left), neutron stars (left), $ud\mathrm{QM}$ dwarfs (right), and white dwarfs (right). The shaded areas indicate the constraints from the binary neutron star merger event GW170817 [97] and the pulse profile modeling for PSR $\mathrm{J}0030+0451$ and PSR $\mathrm{J}0740+6620$ [98, 99, 100, 101]. The four dots in the right panel are the visual binaries Sirius B, Stein 2051 B, Procyon B, and 40 Eri B [103].