Hybrid neutron stars with the Thomas-Fermi approximation and nonlocal Nambu–Jona-Lasinio model

We probe the outcomes of the baryon-quark phase transition in the hybrid neutron star (HNS) structure with the help of the Gibbs and Maxwell constructions, adopting a semiclassical mean-field (MF) model for the equation of state (EOS) of baryonic matter based on the Thomas-Fermi (TF) approximation and a nonlocal extension of the Nambu–Jona-Lasinio (NJL) model for the EOS of the deconfined quark phase. We find that the repulsive vector contribution of the nonlocal NJL (NNJL) EOS plays an inevitable role in modeling a stable $2M_{\odot }$ HNS. Our results exclude the emergence of the pure quark phase in the inner core of a stable HNS. Within the Gibbs construction, as the quark vector interaction becomes stronger, the contribution of the baryon-quark coexisting phase in the total HNS mass is reduced. On the other hand, a stable HNS is not predicted within the Maxwell construction because it does not include a pure quark core. A comparison is made to the corresponding results employing the local (standard) NJL (LNJL) model of quark matter. Fulfilling the observational constraints, our model indicates that a neutron star (NS) with canonical mass of around $1.4M_{\odot }$ is not massive enough to be described as an HNS.

Figure 1

Pressure of SNM (left panel) and PNM (right panel) as a function of baryonic density. The results are compared with the ones obtained form the flow data analysis of heavy-ion collisions [54].

Figure 2

Pressure as a function of baryonic chemical potential for the quark interactions of NNJL I (left panel) and NNJL II (right panel) at ${\eta }_{{}_V}=0,0.04,0.08$, together with the baryonic interactions of TF[90] (short dashes) and TF[96] (long dashes).

Figure 3

Baryonic chemical potential as a function of baryonic density (left panels) and energy density (right panels) for the quark interactions of NNJL I (upper panels) and NNJL II (lower panels) at ${\eta }_{{}_V}=0,0.04,0.08$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs construction, as the plateaus correspond to the Maxwell construction.

Figure 4

Pressure as a function of baryonic density (left panels) and energy density (right panels) for the quark interactions of NNJL I (upper panels) and NNJL II (lower panels) at ${\eta }_{{}_V}=0,0.04,0.08$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs construction, as the plateaus correspond to the Maxwell construction.

Figure 5

Pressure as a function of baryonic and electron chemical potentials for the quark interactions of NNJL I (upper panels) and NNJL II (lower panels) at ${\eta }_{{}_V}=0,0.04,0.08$, combined with the baryonic interactions of TF[90] (left panels) and TF[96] (right panels) via the Gibbs construction, as the plateaus correspond to the Maxwell construction.

Figure 6

Gravitational mass of HNSs (in units of the solar mass $M_{\odot }$) as a function of central baryonic density (left panels) and central energy density (right panels) for the quark interactions of NNJL I (upper panels) and NNJL II (lower panels) at ${\eta }_{{}_V}=0,0.04,0.08$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs and constructions, as the plateaus correspond to the Maxwell construction. The circles indicate the maximum mass configurations, as the EOS above the respective baryonic densities and energy densities is excluded. The onset of the baryon-quark phase transition in the star center is shown with the triangles and vertical slashes corresponding to the Gibbs and Maxwell constructions, respectively. The horizontal band shows the observational mass constraint from PSR $\mathrm{J}0740+6620$ [25].

Figure 7

Gravitational mass of HNSs (in units of the solar mass $M_{\odot }$) as a function of mixed-phase mass fraction for the quark interactions of NNJL I (left panel) and NNJL II (right panel) at ${\eta }_{{}_V}=0,0.04,0.08$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs construction. The circles indicate the maximum mass configurations. The horizontal band shows the observational mass constraint from PSR $\mathrm{J}0740+6620$ [25].

Figure 8

Quark volume fraction as a function of baryonic density for the quark interactions of NNJL I (left panel) and NNJL II (right panel) at ${\eta }_{{}_V}=0,0.04,0.08$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs construction. The circles indicate the baryonic densities and quark volume fractions in the center of the respective maximum mass configurations.

Figure 9

Number fraction of baryons, leptons, and quarks as a function of baryonic density for the quark interaction of NNJL I at ${\eta }_{{}_V}=0$ (upper panel), ${\eta }_{{}_V}=0.04$ (middle panel), ${\eta }_{{}_V}=0.08$ (lower panel), combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs construction. The solid (dashed) vertical lines indicate the central baryonic densities of the maximum mass configurations obtained by the baryonic EOS of TF[90] (TF[96]).

Figure 10

Same as Fig. 9 but for the NNJL II interaction.

Figure 11

Pressure as a function of baryonic density (left panels) and energy density (right panels) for the quark interactions of RKH (upper panels) and HK (lower panels) given by the LNJL model at ${\eta }_{{}_V}=0,0.22,0.44$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs construction, as the plateaus correspond to the Maxwell construction. The circles indicate the maximum mass configurations, as the EOS above the respective baryonic densities and energy densities is excluded. The onset of the baryon-quark phase transition in the star center is shown with the triangles and vertical slashes corresponding to the Gibbs and Maxwell constructions, respectively.

Figure 12

Mass-radius diagrams for the quark interactions of NNJL I (upper left panel) and NNJL II (upper right panel) at ${\eta }_{{}_V}=0,0.04,0.08$, together with the quark interactions of RKH (lower left panel) and HK (lower right panel) given by the LNJL model at ${\eta }_{{}_V}=0,0.22,0.44$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs and Maxwell constructions. The circles and triangles (vertical slashes) indicate the maximum mass configurations, and onsets of the baryon-quark phase transition in the star center within the Gibbs (Maxwell) construction, respectively. The horizontal bands show the observational mass constraint from PSR $\mathrm{J}0740+6620$ [25] and the gravitational mass inferred for the lighter component of GW190814 [59]. The areas of the mass-radius limits inferred from the NICER measurements of PSR $\mathrm{J}0030+0451$ [29, 30] are also shown. The region on the top left of each panel is excluded by causality [63].

Figure 13

Tidal deformability of HNSs as a function of gravitational mass (in units of the solar mass $M_{\odot }$) for the quark interactions of NNJL I (upper left panel) and NNJL II (upper right panel) at ${\eta }_{{}_V}=0,0.04,0.08$, together with the quark interactions of RKH (lower left panel) and HK (lower right panel) given by the LNJL model at ${\eta }_{{}_V}=0,0.22,0.44$, combined with the baryonic interactions of TF[90] (solid lines) and TF[96] (dashed lines) via the Gibbs and Maxwell constructions. The circles and triangles (vertical slashes) indicate the maximum mass configurations and onsets of the baryon-quark phase transition in the star center within the Gibbs (Maxwell) construction, respectively. The vertical bar shows the tidal deformability range of a $1.4M_{\odot }$ NS, obtained by the analysis of the GW170817 signal [27].