Quark-nuclear hybrid equation of state for neutron stars under modern observational constraints

We study a family of equations of state (EOS) for hybrid neutron star matter. The hybrid EOS are obtained by a Maxwell construction of the first-order phase transition between a hadronic phase described by the relativistic density-functional EOS of the “DD2” class with excluded volume effects and a deconfined quark matter phase modeled by an instantaneous nonlocal version of the Nambu–Jona-Lasinio model in $\text{SU}{\left(2\right)}_f$ with vector interactions and color superconductivity. The form factor in the nonlocal quark matter model is fitted to lattice QCD results in the Coulomb gauge. Owing to strong coupling in the vector meson and diquark channels, a coexistence phase of color superconductivity and chiral symmetry breaking occurs. Our results show an approximately constant behavior for the squared speed of sound with values of 0.4–0.6 in the density region relevant for neutron star interiors. To simultaneously fulfill the constraints from the Neutron Star Interior Composition Explorer radius measurement for PSR $\mathrm{J}0740+6620$ and tidal deformability from GW170817 it is necessary to consider a $\mu$-dependent bag pressure that mimics confinement.

Figure 1

LQCD data from Ref. [39] and our fit for the normalized quark effective mass.

Figure 2

$\overline{\sigma }$, $\overline{\mathrm{\Delta }}$, $\overline{\omega }$, ${\mu }_l$, and ${\mu }_8$ (in MeV) as functions of ${\mu }_B$ (in MeV).

Figure 3

QM phase diagram in the ${\eta }_D\text{−}{\mu }_B^c$ plane. Solid, dotted, and dashed lines correspond to first, second, and crossover phase transitions. The shaded band indicates the coexistence region.

Figure 4

Squared speed of sound for the nonlocal chiral QM model with ${\eta }_{\mathrm{D}}=1.1$ and ${\eta }_{\mathrm{V}}=0.1$, 0.5, and 1.1, respectively.

Figure 5

Left: (a) QM EOS for ${\eta }_D=1.1$ and Bag = 10.0 $\mathrm{MeV}/{\mathrm{fm}}^3$ for ${\eta }_V=0.6$ (green dash-dotted line), and different hadronic EOS with solid lines. (b) The corresponding $M\text{−}R$ relations for the hybrid EOS in (a). The solid dots indicate the respective phase transition points and vertical bars the location of the maximum mass (down). Right: (c) QM EOS with ${\eta }_D=1.1$, Bag = 10.0 $\mathrm{MeV}/{\mathrm{fm}}^3$, and ${\eta }_V$ from 0.5 to 0.9 (different trace lines), and hadronic EOS DD2-40 in light blue solid line. (d) The corresponding $M\text{−}R$ relation.

Figure 6

(a) Pressure vs chemical potential for the QM (dashed lines) and the hadronic (solid lines) EOS. (b) Pressure vs energy density, and we highlight the fact that NS interiors do not probe the energy densities where pQCD is applicable. (c) The squared sound speed $c_s^2$ is shown for the corresponding hybrid EOS. (d) The $M\text{−}R$ relations are given. The solid dots indicate the respective phase transition points and vertical bars show the location of the maximum mass stars. The results for the pure hadronic EOS are also shown (lighter line colors), with the star symbols indicating the points where the causal limit of the EOS is reached and the dotted lines stand for results from its acausal part. For comparison with the NS phenomenology, the mass-radius constraints from NICER observations (see Refs. [7, 68] for details) and from the merger event GW170817 are shown. (e) The tidal deformability $\mathrm{\Lambda }$ vs $M/M_{\odot }$ for the set of hybrid EOS (traced lines), hadronic EOS (solid lines, overlapped with QM ones), and comparison to the results of GW170817. (f) The comparison of ${\mathrm{\Lambda }}_2\text{−}{\mathrm{\Lambda }}_1$ calculated for our EOS in comparison to the analysis of the gravitational wave signal from the merger GW170817 [1, 69]. For the remaining mass and radius constraints in (d) see Ref. [65].

Figure 7

Same as Fig. 6, but for the ${\mu }_B$-dependent bag pressure $B\left({\mu }_B\right)$.