Equation of state for neutron stars with hyperons using a variational method

We investigate the effects of the odd-state part of bare $\mathrm{\Lambda }\mathrm{\Lambda }$ interactions on the structure of neutron stars (NSs) by constructing equations of state (EOSs) for uniform nuclear matter containing $\mathrm{\Lambda }$ and ${\mathrm{\Sigma }}^{-}$ hyperons with use of the cluster variational method. The isoscalar part of the Argonne v18 two-nucleon potential and the Urbana IX three-nucleon potential are employed as the interactions between nucleons, whereas, as the bare $\mathrm{\Lambda }N$ and even-state $\mathrm{\Lambda }\mathrm{\Lambda }$ interactions, two-body central potentials that are determined so as to reproduce the experimental data on single- and double-$\mathrm{\Lambda }$ hypernuclei are adopted. In addition, the ${\mathrm{\Sigma }}^{-}N$ interaction is constructed so as to reproduce the empirical single-particle potential of ${\mathrm{\Sigma }}^{-}$ in symmetric nuclear matter. Since the odd-state part of the $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction is not known owing to lack of experimental data, we construct four EOSs of hyperonic nuclear matter, each with a different odd-state part of the $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction. The EOS obtained for NS matter becomes stiffer as the odd-state $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction becomes more repulsive, and correspondingly the maximum mass of NSs increases. It is interesting that the onset density of ${\mathrm{\Sigma }}^{-}$ depends strongly on the repulsion of the odd-state $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction. Furthermore, we take into account the three-baryon repulsive force to obtain results that are consistent with observational data on heavy NSs.

Figure 1

Energies per baryon $E$ of hyperonic nuclear matter as functions of the baryon number density $n_{\mathrm{B}}$ for various values of $\mathrm{\Lambda }$ fractions $x_{\mathrm{\Lambda }}$ with the most attractive odd-state part of $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction (Type 1). The solid curves represent the case of $x_{\mathrm{p}}=x_{\mathrm{n}}$ while the dashed curves correspond to the case of $x_{\mathrm{p}}=0$.

Figure 2

Pressures $P$ of NS matter with interacting hyperons by the four different odd-state parts of $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction in Table 2 as functions of the baryon number density $n_{\mathrm{B}}$. The pressures without hyperons (without $Y$) and with noninteracting hyperons (free $Y$) are also shown.

Figure 3

Mass-radius relations of NSs with the four EOSs of NS matter that correspond to the different odd-state parts of $\mathrm{\Lambda }\mathrm{\Lambda }$ interactions. The results for nuclear matter without hyperons (without $Y$) and with noninteracting hyperons (free $Y$) are also shown. The horizontal green and purple bands indicate the masses of PSRs J1614-2230 [9] and J 0348+0432 [10]. The shaded region denotes the mass-radius region suggested in Ref. [36].

Figure 4

The fractions of particles $x_i$ of NS matter as functions of the baryon number density $n_{\mathrm{B}}$ for various odd-state parts of the $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction: (a) no interaction, (b) most attractive, (c) less attractive, (d) slightly repulsive, (e) most repulsive.

Figure 5

Mass-radius relations of NSs obtained from EOSs based on the most attractive and most repulsive odd-state part of the $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction (Types 1 and 4, respectively) with and without phenomenological three-baryon forces (TBF). The filled circle represents the NS for which the central density is equal to the critical density $n_{\mathrm{c}}$. The result of nuclear matter without any hyperons (without $Y$) is also shown.

Figure 6

The fractions of particles $x_i$ of NS matter as functions of the baryon number density $n_{\mathrm{B}}$ based on the most attractive odd-state part of the $\mathrm{\Lambda }\mathrm{\Lambda }$ interaction (Type 1). The results without the TBF are also shown.