Hybrid stars using the quark-meson coupling and proper-time Nambu–Jona-Lasinio models

Background: At high density deconfinement of hadronic matter may occur leading to quark matter. The immense densities reached in the inner core of massive neutron stars may be sufficient to facilitate the transition.

Purpose: To investigate a crossover transition between two phenomenological models which epitomize quantum chromodynamics in two different regimes, while incorporating the influence of quark degrees of freedom in both.

Method: We use the Hartree-Fock quark-meson coupling model and the proper-time regularized three-flavor Nambu–Jona-Lasinio model to describe hadronic and quark matter, respectively. Hybrid equations of state are obtained by interpolating the energy density as a function of total baryonic density and calculating the pressure.

Results: Equations of state for hadronic, quark, and hybrid matter and the resulting mass versus radius curves for hybrid stars are shown, as well as other relevant physical quantities such as species fractions and the speed of sound in matter.

Conclusions: The observations of massive neutron stars can certainly be explained within such a construction. However, the so-called thermodynamic correction arising from an interpolation method can have a considerable impact on the equation of state. The interpolation dependency of and physical meaning behind such corrections require further study.

Figure 1

Species fractions as a function of density for hadronic matter in generalized beta equilibrium for (a) standard or baseline scenario and (b) $\mathrm{\Lambda }=1.3$ GeV.

Figure 2

Pressure as a function of energy density for hadronic matter in generalized beta equilibrium.

Figure 3

Constituent quark masses of light (blue lines) and strange (purple lines) quarks as a function of chemical potential ($\mu ={\mu }_{\ell }={\mu }_s$) for the parameter set (a) and (d) PS1, (b) and (e) PS2, and (c) and (f) HK. The constant light (blue) and strange (purple) current quark masses are shown by the horizontal dashed lines. In the first row the vector coupling is set to zero, and in the second it is $G_{\mathrm{V}}=G_{\mathrm{S}}$. The vertical red dashed line in the top row indicates the critical chemical potential.

Figure 4

Species fractions as a function of density for beta-equilibrium quark matter for parameter set PS2 (solid) and HK (dashed). Each of the particle number densities is divided by the total quark density ${\rho }_{\mathrm{tot}}={\rho }_d+{\rho }_u+{\rho }_s=3\rho$. The down quark fraction is red, up green, and strange purple. The electron fraction (blue) is multiplied by 100 so as to be visible on the same plot. Note that the electron fraction defined here differs by a factor of $1/3$ from the figures in Sec. 2. Plot (a) zero vector coupling and nonzero flavor-independent vector coupling, (b) flavor-dependent vector interaction with $G_{\mathrm{V}}=G_{\mathrm{S}}/2$, and (c) flavor-dependent vector interaction with $G_{\mathrm{V}}=G_{\mathrm{S}}$. Here we use the saturation density ${\rho }_0=0.17{\mathrm{fm}}^{-3}$.

Figure 5

Chemical potentials (solid) and constituent quark masses (dashed) as a function of total baryonic density ($\rho$) for both flavor-independent and flavor-dependent vector interactions. The line colors for the quarks are up (orange), down (green), and strange (blue). Plots (a)–(c) PS2 model with $G_{\mathrm{V}}=0,G_{\mathrm{S}}$ and $g_{\mathrm{V}}=G_{\mathrm{S}}$, respectively. Plots (d)–(f) HK model with $G_{\mathrm{V}}=0,G_{\mathrm{S}}$ and $g_{\mathrm{V}}=G_{\mathrm{S}}$, respectively. Here we use the saturation density ${\rho }_0=0.17{\mathrm{fm}}^{-3}$.

Figure 6

Pressure as a function of energy density for the (a) PS2 and (b) HK parameter sets. Results using the flavor-dependent interaction $G_{\mathrm{V}}$ [i.e., use Eq. (4.1)] and flavor-independent interaction $g_{\mathrm{V}}$ [i.e., use Eq. (4.8)] for different values of the vector coupling. Here we normalize the energy density with ${\epsilon }_0=140{\mathrm{MeV}\mathrm{fm}}^{-3}$.

Figure 7

Thermodynamic correction $\mathrm{\Delta }P$ as a function of total baryonic density as arising from the interpolation. For plot (a), the interpolation is between the “Standard” or baseline scenario of the HF-QMC model and the proper-time regularized PS2 NJL model with flavor-dependent vector interaction. Similarly for plot (b), but with the three-momentum regularized NJL model with flavor-dependent vector interaction. The crossover region is chosen to be $\left(\overline{\rho },\mathrm{\Gamma }\right)=\left(3{\rho }_0,{\rho }_0\right)$. Specific curves for both plots are indicated in the key of plot (a).

Figure 8

Pressure as a function of energy density. For plots (a) and (b), the interpolation is between the “Standard” or baseline scenario of the HF-QMC model and the proper-time regularized PS2 model with flavor-dependent vector interaction. Similarly for plots (c) and (d), but with the three-momentum regularized model with flavor-dependent vector interaction. Plots (a) and (c) do not include the thermodynamic correction $\mathrm{\Delta }P$, whereas plots (b) and (d) include the correction for thermodynamic consistency. The crossover region is chosen to be $\left(\overline{\rho },\mathrm{\Gamma }\right)=\left(3{\rho }_0,{\rho }_0\right)$. The line type colors indicate the strength of vector coupling used in the quark model, (blue) $G_{\mathrm{V}}=0$, (orange) $G_{\mathrm{V}}=G_{\mathrm{S}}/2$, (green) $G_{\mathrm{V}}=G_{\mathrm{S}}$. Solid curves are the interpolated EoS and colored dashed curves the quark EoS. Black dot-dot line is the hadronic EoS.

Figure 9

Speed of sound squared (in units of $c$) as a function of energy density. For plots (a) and (b), the interpolation is between the “Standard” or baseline scenario of the HF-QMC model and the proper-time regularized PS2 model with flavor-dependent (solid) and -independent (dashed) vector interactions. Similarly for plots (c) and (d), but with the three-momentum regularized model with flavor-dependent (solid) and -independent (dashed) vector interactions. Plots (a) and (c) do not include the thermodynamic correction $\mathrm{\Delta }P$, whereas plots (b) and (d) include the correction for thermodynamic consistency. The crossover region is chosen to be $\left(\overline{\rho },\mathrm{\Gamma }\right)=\left(3{\rho }_0,{\rho }_0\right)$. Specific curves for all plots are indicated in the key of plot (a).

Figure 10

Neutron star mass as a function of radius. For plots (a) and (b), the interpolation is between the “Standard” or baseline scenario of the HF-QMC model and the proper-time regularized PS2 model with flavor-dependent (solid) and -independent (dashed) vector interactions. Similarly for plots (c) and (d), but with the three-momentum regularized model with flavor-dependent (solid) and -independent (dashed) vector interactions. Plots (a) and (c) do not include the thermodynamic correction $\mathrm{\Delta }P$, whereas plots (b) and (d) include the correction for thermodynamic consistency. The crossover region is chosen to be $\left(\overline{\rho },\mathrm{\Gamma }\right)=\left(3{\rho }_0,{\rho }_0\right)$. Specific curves for all plots are indicated in the key of plot (a).