Studies of the hybrid star structure within $2+1$ flavors NJL model

In this paper, we use the equation of state (EOS) of the $2+1$ flavors Nambu-Jona-Lasinio (NJL) model to study the structure of compact stars. To avoid the ultraviolet divergence, we employ the proper-time regularization (PTR) with an ultraviolet cutoff. For comparison, we fix three sets of parameters, where the constraints of chemical equilibrium and electric charge neutrality conditions are taken into consideration. With a certain interpolation method in the crossover region, we construct three corresponding hybrid EOSs but find that the maximum masses of hybrid stars in the three different cases do not differ too much. It should be pointed out that the results we get are in accordance with the recent astro-observation $\mathrm{PSR}\mathrm{J}0348+0432$, $\mathrm{PSR}\mathrm{J}1614-2230$, $\mathrm{PSR}\mathrm{J}1946+3417$.

Figure 1

Quark number density of $u$ and $d$ quarks as a function of $\mu$ at $T=0$ with parameters fixed at $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$.

Figure 2

Quark number density of $s$ quark as a function of $\mu$ at $T=0$ with parameters fixed at $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$.

Figure 3

the EOS of hadrons which is depicted as the blue dotted line and the EOSs of quark matter which are depicted as the black solid line, the green dot-dashed line and the red dashed line corresponding to $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$, $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(250\mathrm{MeV}{)}^3$, $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(270\mathrm{MeV}{)}^3$, respectively.

Figure 4

When $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$, the binding energy of quarks and hadrons is shown with the black solid line and the green dashed line, respectively.

Figure 5

When $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$, the EOS of the hybrid system is shown with the green dashed line which is obtained by an interpolation function of hadronic matter EOS and quark matter EOS in the crossover region.

Figure 6

When $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$, the relation between energy density and baryon chemical potential in the hybrid star (blue dotted line), the quark matter (black solid line), and hadronic matter (green dashed line), respectively.

Figure 7

When $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$, the energy density as a function of the pressure in the hybrid star (blue dotted line), quark matter (black solid line), and hadronic matter (green dashed line).

Figure 8

The sound velocity of hadron system in APR model and our hybrid system when $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$ which are depicted as blue dashed line and green solid line, respectively.

Figure 9

M-R relation of hybrid stars given by three hybrid EOSs of the system; that is, the black solid line corresponds to the case of $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(230\mathrm{MeV}{)}^3$, the dotted line corresponds to the case of $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(250\mathrm{MeV}{)}^3$, and the dashed line corresponds to the case of $⟨\overline{u}u⟩{|}_{T=0,\mu =0}=-(270\mathrm{MeV}{)}^3$.