Third family of compact stars within a nonlocal chiral quark model equation of state

A class of hybrid compact star equations of state is investigated that joins by a Maxwell construction a low-density phase of hadronic matter, modeled by a relativistic mean-field approach with excluded nucleon volume, with a high-density phase of color superconducting two-flavor quark matter, described within a nonlocal covariant chiral quark model. It is found that the occurrence of a stable branch of hybrid compact stars requires a nonvanishing vector meson coupling in the quark model that exceeds a minimal value which depends on the presence of a diquark condensate. It is shown that these hybrid stars do not form a third family disconnected from the second family of ordinary neutron stars unless additional (de)confining effects are introduced with a density-dependent bag pressure. A suitably chosen density dependence of the vector meson coupling assures that at the same time the $2M_{\odot }$ maximum mass constraint is fulfilled on the hybrid star branch. A twofold interpolation method is realized which implements both the density dependence of a confining bag pressure at the onset of the hadron-to-quark matter transition and the stiffening of quark matter at higher densities by a density-dependent vector meson coupling. For three parametrizations of this class of hybrid equation of state the properties of corresponding compact star sequences are presented, including mass twins of neutron and hybrid stars at 2.00, 1.39 and $1.20M_{\odot }$, respectively, and the hybrid compact star (third) families. The sensitivity of the hybrid equation of state and the corresponding compact star sequences to variations of the interpolation parameters at the 10% level is investigated and it is found that the feature of third family solutions for compact stars is robust against such a variation. This advanced description of hybrid star matter allows us to interpret GW170817 as a merger not only of two neutron stars but also of a neutron star with a hybrid star or of two hybrid stars.

Figure 1

Quark matter EoS according to the nonlocal chiral quark model with color superconductivity for different values of the dimensionless vector meson coupling strength parameter $\eta =0.01,0.02,\dots ,0.09$ (black thin lines); $\eta =0.12,\dots ,0.19$ (magenta thin lines); and $\eta =0.21,\dots ,0.26$ (blue thin lines) compared with four nuclear matter EoS ordered by increasing stiffness: APR (violet solid line), DD2F (orange solid line), DD2 (black solid line) and DD2_p40 (magenta solid line). For a detailed discussion, see text.

Figure 2

Quark matter EoS according to the nonlocal chiral quark model without color superconductivity (diquark interaction switched off) for different values of the dimensionless vector meson coupling strength parameter $\eta =0.01,0.02,\dots ,0.09$ (black thin lines); $\eta =0.12,\dots ,0.19$ (magenta thin lines); and $\eta =0.21,\dots ,0.26$ (blue thin lines) compared with four nuclear matter EoS ordered by increasing stiffness: APR (violet solid line), DD2F (orange solid line), DD2 (black solid line) and DD2_p40 (magenta solid line). For a detailed discussion, see text.

Figure 3

Mass-radius sequences for the class of hybrid EoS emerging from Maxwell constructions of standard nuclear EoS and the color superconducting nonlocal chiral quark model EoS of the present work, as discussed in Sec. 2c and in Fig. 1.

Figure 4

Mass-radius sequences for the class of hybrid EoS emerging from Maxwell constructions of standard nuclear EoS and the nonlocal chiral quark model EoS of the present work, where the color superconducting diquark channel is switched off, as discussed in Sec. 2c and in Fig. 2.

Figure 5

Hybrid star EoS (black solid line) obtained by a Maxwell construction between the quark matter EoS for set 2 (red dashed-dotted line) and the density-dependent relativistic mean-field EoS DD2_p40 for nuclear matter in $\beta$-equilibrium with electrons and muons. The doubly interpolated quark matter EoS is based on three parametrizations of the nonlocal NJL model: a soft (low vector coupling $\eta$) one with confinement ($B\ne 0$) at low densities (red dashed line), a soft one without confinement at intermediate densities (red dashed-double-dotted line) and a stiff one (high $\eta$) at high densities (red double-dashed-dotted line). The parameters of the switching functions are given in Table 1.

Figure 6

Medium dependence of the dimensionless vector meson coupling $\eta (\mu )$ as defined by the parameters of the interpolation procedure given in Table 1 for sets 1, 2 and 3.

Figure 7

Medium dependence of the bag pressure $B(\mu )$ as defined by the parameters of the interpolation procedure given in Table 1 for sets 1, 2 and 3.

Figure 8

The equation of state for pressure vs energy density resulting from the twofold interpolation method and Maxwell construction of the deconfinement phase transition illustrated in Fig. 5 for the parameters of set 2. Results for sets 1, 2 and 3 are shown as solid, dashed and dashed-dotted lines, respectively. For comparison, the APR EoS is shown (orange dotted line) which is a standard EoS for nuclear astrophysics applications.

Figure 9

The squared speed of sound $c_s^2$ as a function of the energy density for sets 1, 2 and 3 exhibits the regions of the first-order phase transition where $c_s^2=0$ and fulfills the condition of causality $c_s^2<1$ (in units of the speed of light squared). Line styles are as in Fig. 8. For comparison, the APR EoS is shown (magenta dotted line) without the deconfinement phase transition.

Figure 10

Mass vs radius for sequences of compact stars with the hybrid EoS of this work parametrized by sets 1, 2 and 3, corresponding to different onset masses for the deconfinement transition. The dotted lines denote the unstable configurations that should not be realized in nature but guide the eye to the corresponding stable hybrid star sequence (third family) disconnected from the neutron star one (second family). For comparison, the mass-radius sequence for the APR EoS is shown (orange dashed line) which is a standard EoS for nuclear astrophysics applications, concerning only the second family, without a third family branch. The blue and red horizontal bands denote the mass measurement for PSR J0348+432 [4] and PSR J1614-2230 [3], respectively. The grey and orange bands labeled M1 and M2 are the mass ranges for the compact stars in the binary merger GW170817. From the observations of this event exclusion regions have been found (magenta hatching): the authors of Ref. [72] excluded radii of smaller than 10.68 km for stars of $1.6M_{\odot }$ and those of Ref. [73] excluded radii exceeding 13.6 km for stars of $1.4M_{\odot }$. The authors of Ref. [74] derived an upper limit of $2.16M_{\odot }$ for the maximum mass of nonrotating neutron stars. The green band denotes the mass $1.44\pm 0.07M_{\odot }$ of PSR J0437-4715, the primary target of the radius measurement by NICER [80].

Figure 11

Variation by $\pm 10\%$ of the interpolation parameters $v_1$ (${\mu }_{<}$), $v_5$ (${\eta }_{<}$) and $v_7$ ($B$), with a large effect on the EoS (upper row of panels) and the $M-R$ sequences (lower row of panels) for set 1 (high onset mass) and set 2 (low onset mass).

Figure 12

Variation by $\pm 10\%$ of the interpolation parameters $v_2$ (${\mathrm{\Gamma }}_{<}$), $v_3$ (${\mu }_{\ll }$), $v_4$ (${\mathrm{\Gamma }}_{\ll }$), and $v_6$ (${\eta }_{>}$), with a smaller effect on the EoS (left column of panels) and the corresponding $M-R$ sequences (right column of panels) for set 1 (high onset mass) and set 2 (low onset mass).

Figure 13

Baryon mass vs radius for the compact star sequences corresponding to sets 1, 2 and 3 parametrizations of the present EoS model. From this figure one can read off what drop in radius can be expected when a transition from the maximum mass of the second family branch to the third family branch takes place under baryon number conservation, triggered for instance by mass accretion from a companion star (red arrows).

Figure 14

Mass vs central energy density as solutions of the TOV equations for the three parameter sets given in Table 1. Line styles are as in the previous figures. The rising sections of the lines denote stable sequences.

Figure 15

Moment of inertia vs mass for the three hybrid EoS parametrizations introduced in this work. Unstable configurations are shown by dotted lines. Note that the relationship is multivalued in the mass twin ranges. Hybrid stars on the third family branch have generally a smaller moment of inertia. The mass of PSR J0737-3039 (A) is shown for which a measurement of $I$ is expected at the 10% level.

Figure 16

Dimensionless tidal deformability parameter $\mathrm{\Lambda }$ vs compact star mass for the three sets of hybrid EoS parametrizations introduced in this work. The constraint $\mathrm{\Lambda }<800$ derived from GW170817 [8] for the mass range $1.16-1.60M_{\odot }$ is fulfilled for the third family solutions of sets 2 and 3 while the second family sequence of set 1 cannot fulfill this constraint within a binary neutron star scenario.

Figure 17

Tidal deformability parameters ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ of the high- and low-mass components of the binary merger, constructed from the $\mathrm{\Lambda }(M)$ relation of Fig. 16 for the EoS given by sets 1–3 compared to the probability contours for the low-spin prior of the LVC analysis of GW170817 [8].