Constant-sound-speed parametrization for Nambu–Jona-Lasinio models of quark matter in hybrid stars

The discovery of pulsars as heavy as 2 solar masses has led astrophysicists to rethink the core compositions of neutron stars, ruling out many models for the nuclear equations of state (EoS). We explore the hybrid stars that occur when hadronic matter is treated in a relativistic mean-field approximation and quark matter is modeled by three-flavor local and nonlocal Nambu–Jona-Lasinio (NJL) models with repulsive vector interactions. The NJL models typically yield equations of state that feature a first-order transition to quark matter. Assuming that the quark-hadron surface tension is high enough to disfavor mixed phases and restricting to EoSs that allow stars to reach 2 solar masses, we find that the appearance of the quark-matter core either destabilizes the star immediately (this is typical for nonlocal NJL models) or leads to a very short hybrid star branch in the mass-radius relation (this is typical for local NJL models). Using the constant-sound-speed parametrization we can see that the reason for the near absence of hybrid stars is that the transition pressure is fairly high and the transition is strongly first order.

Figure 1

EoS, $\varepsilon \left(p\right)$, for dense matter without vector interaction, showing the energy density discontinuity $\mathrm{\Delta }\varepsilon$ between hadronic matter (NL3) and quark matter. For quark matter modeled by nonlocal NJL SET III (a), the discontinuity is greater than $\mathrm{\Delta }{\varepsilon }_{\mathrm{crit}}$ (15), so there is no stable connected hybrid branch. In contrast, the local NJL SET IV (b) has a smaller discontinuity $\mathrm{\Delta }\varepsilon <\mathrm{\Delta }{\varepsilon }_{\mathrm{crit}}$, which leads to a connected but very short hybrid branch (see Table 4).

Figure 2

Squared speed of sound for the nonlocal NJL quark matter model as a function of pressure. NL3 to quark-matter transitions are marked by dots; $\mathrm{GM}1\to \mathrm{quark}$ matter is marked by triangles. Solid lines show the behavior without vector interactions, while dashed and dotted lines represent the results for a finite vector interaction coupling constant. Left, middle, and right panels correspond to the nonlocal NJL model with parameter set I, panel (a); set II, panel (b); and set III, panel (c).

Figure 3

Squared speed of sound for the local NJL quark-matter model as function of pressure. The symbols and line styles are similar to those in Fig. 2. Left and right panels correspond to the local NJL model with parameter set IV, panel (a); and set IV, panel (b). Note that the local NJL model shows a more density-independent speed of sound and thus is more accurately characterized by the CSS parametrization than in the nonlocal case.

Figure 4

Schematic phase diagram of the possible topologies of $M\left(R\right)$ [11]. The thick straight (red) line [Eq. (15)] separates EoSs with a connected hybrid branch (regions B and C below the line) from those without a connected hybrid branch (regions A and D above the line). The dashed black lines delimit the regions (B and D) where disconnected hybrid branches occur. The insets show $M\left(R\right)$ in each region, with a thin dashed (red) line indicating unstable hybrid stars and a thin solid (red) line indicating stable hybrid stars.

Figure 5

Diagram showing (black symbols) where different quark matter parametrizations, for the local and nonlocal models, combined with GM1 nuclear matter, fall in the CSS parameter space. Panels (a), (b), (c), and (d) are for $c_{\mathrm{QM}}^2\simeq 0.2,c_{\mathrm{QM}}^2\simeq 0.25,c_{\mathrm{QM}}^2\simeq 0.3$, and $c_{\mathrm{QM}}^2=0.46$, respectively. EoSs below the straight solid (red) line (regions B and C) yield a connected hybrid branch. EoSs within the shaded gray area are excluded because their heaviest star is below $2M_{\odot }$. The hatched area at densities $n_{\mathrm{trans}}<n_0$ is excluded because uniform nuclear matter is not stable in that region. See Tables 3 and 4.

Figure 6

Diagram showing (black symbols) where different quark matter parametrizations, for the local and nonlocal models, combined with NL3 nuclear matter, fall in the CSS parameter space. Panels (a), (b), (c), and (d) are for $c_{\mathrm{QM}}^2\simeq 0.2,c_{\mathrm{QM}}^2\simeq 0.25,c_{\mathrm{QM}}^2\simeq 0.3$, and $c_{\mathrm{QM}}^2\simeq 0.35$, respectively. EoSs below the straight solid (red) line (regions B and C) yield a connected hybrid branch. EoSs within the shaded gray area are excluded because their heaviest star is below $2M_{\odot }$. The hatched area at densities $n_{\mathrm{trans}}<n_0$ is excluded because uniform nuclear matter is not stable in that region. See Tables 3 and 4.