Color superconductivity in compact stellar hybrid configurations

The discovery of pulsars PSR J1614–2230 and PSR J0348+0432 with masses of around $2M_{\odot }$ imposes strong constraints on the equations of state of cold, ultradense matter. If a phase transition from hadronic matter to quark matter were to occur in the inner cores of such massive neutron stars, the energetically favorable state of quark matter would be a color superconductor. In this study, we analyze the stability and maximum mass of such neutron stars. The hadronic phase is described by nonlinear relativistic mean-field models, and the local Nambu-Jona Lasinio model is used to describe quark matter in the 2SC+s quark phase. The phase transition is treated as a Maxwell transition, assuming a sharp hadron-quark interface, and the “constant-sound-speed” (CSS) parametrization is employed to discuss the existence of stellar twin configurations. We find that massive neutron stars such as J1614-2230 and J0348+0432 can only exist on the connected stellar branch but not on the disconnected twin-star branch. The latter can only support stars with masses that are strictly below $2M_{\odot }$.

Figure 1

Square of the speed of sound, $c_{\mathrm{QM}}^2\equiv dp/d\varepsilon$, for different 2SC+s quark EoSs with fixed strange quark mass ${\mathrm{M}}_{\mathrm{s}}^0=500$ MeV (a) and 600 MeV (b). The model parameters used are the same as in Fig. 4 depicting the mass-radius relation of hybrid stars using NL3 for the hadronic EoS.

Figure 2

Diagram showing (black symbols) where phase transition between NL3 EoS and 2SC+s quark EoS with various parameter values [see Table 1 and Fig. 4] fall in the CSS parameter space. Each panel is for a different range of $c_{\mathrm{QM}}^2$. EoSs below the straight solid (red) line (regions B and C) yield a connected hybrid branch; EoSs above the (green) window (regions B and D) yield a disconnect hybrid branch. EoSs within the shaded gray area are excluded because their heaviest star is below $2M_{\odot }$.

Figure 3

Diagram showing (black symbols) where phase transition between NL3 EoS and 2SC+s quark EoS with various parameter values [see Table 2 and Fig. 4] fall in the CSS parameter space. Each panel is for a different range of $c_{\mathrm{QM}}^2$. EoSs below the straight solid (red) line (regions B and C) yield a connected hybrid branch; EoSs above the (green) window (regions B and D) yield a disconnect hybrid branch. EoSs within the shaded gray area are excluded because their heaviest star is below $2M_{\odot }$.

Figure 4

Mass-radius relationship for hybrid stars using an effective strange quark mass ${\mathrm{M}}_{\mathrm{s}}^0=500$ MeV (a), ${\mathrm{M}}_{\mathrm{s}}^0=600$ MeV (b), and the superconducting gap for the strange quark ${\mathrm{\Delta }}_{\mathrm{s}}=10$ MeV. We use NL3 for the hadronic EoS, and quantities that characterize the phase transition to the 2SC+s phase are shown in Tables 1 and 2. Variation of the speed of sound in the quark phase is shown in Fig. 1. Labels (i), (ii), and (iii) are in accordance with the corresponding (a), (b), and (c) CSS diagrams in Figs. 2 and 3.

Figure 5

Mass-radius relationship for hybrid stars using an effective strange quark mass ${\mathrm{M}}_{\mathrm{s}}^0=500$ MeV (a), ${\mathrm{M}}_{\mathrm{s}}^0=600$ MeV (b), and the superconducting gap for the strange quark ${\mathrm{\Delta }}_{\mathrm{s}}=50$ MeV. We use GM1(L) for the hadronic EoS, and quantities that characterize the phase transition to the 2SC+s phase are shown in Tables 3 and 4. Variation of the speed of sound in the quark phase is shown in Fig. 1. Labels (i) and (ii) are in accordance with the corresponding (a), (c), and (b), (d) of the CSS diagrams in Fig. 6.

Figure 6

Diagram showing (black symbols) where phase transition between GM1(L) EoS and 2SC+s quark EoS with various parameter values [(a) and (b), Table 3; (c) and (d), Table 4] fall in the CSS parameter space. Each panel is for a different range of $c_{\mathrm{QM}}^2$. EoSs below the straight solid (red) line (regions B and C) yield a connected hybrid branch; EoSs above the (green) window (regions B and D) yield a disconnect hybrid branch. EoSs within the shaded gray area are excluded because their heaviest star is below $2M_{\odot }$.

Figure 7

Hybrid star configurations based on the DD2 hadronic EoS and quark matter EoS with the effective strange quark mass ${\mathrm{M}}_{\mathrm{s}}^0=500\mathrm{MeV}$, and the superconducting gap for the strange quark, ${\mathrm{\Delta }}_{\mathrm{s}}=50\mathrm{MeV}$. (a) Mass-radius curves; (b) mass as a function of the central energy density. All curves show connected families of hybrid stars. Labels (i), (ii), and (iii) are in accordance with the corresponding (a), (b), and (c) CSS diagrams in Fig. 8. Quantities that characterize the phase transition to the 2SC+s phase are shown in Table 5.

Figure 8

Diagram showing (black symbols) where phase transition between DD2 EoS and 2SC+s quark EoS with various parameter values (see Table 5) fall in the CSS parameter space. Each panel is for a different value of $c_{\mathrm{QM}}^2\approx \langle c_{\mathrm{QM}}^2\rangle$. EoSs below the straight solid (red) line (regions B and C) yield a connected hybrid branch; EoSs above the (green) window (regions B and D) yield a disconnect hybrid branch. EoSs within the shaded gray area are excluded because their heaviest star is below $2M_{\odot }$. Only panels (b) and (c) predict phase transitions that ensure $M_{\max }>2M_{\odot }$, consistent with (ii) and (iii) curves in Fig. 7.