Hypernuclei and massive neutron stars

Background: The recent accurate measurement of the mass of two pulsars close to or above $2{\mathrm{M}}_{\odot }$ has raised the question of whether such large pulsar masses allow for the existence of exotic degrees of freedom, such as hyperons, inside neutron stars.

Purpose: In the present work, we will investigate, within a phenomenological relativistic mean field approach, how the existing hypernuclei properties may constrain the neutron star equation of state and confront the neutron star maximum masses obtained with equations of state calibrated to hypernuclei properties with the astrophysical $2{\mathrm{M}}_{\odot }$ constraint.

Method: The study is performed using a relativistic mean field approach to describe both the hypernuclei and the neutron star equations of state. Unified equations of state are obtained. A set of five models that describe $2{\mathrm{M}}_{\odot }$ when only nucleonic degrees of freedom are employed. Some of these models also satisfy other well-established laboratory or theoretical constraints.

Results: The $\mathrm{\Lambda }$-meson couplings are determined for all the models considered, and the $\mathrm{\Lambda }$ potential in symmetric nuclear matter and $\mathrm{\Lambda }$ matter at saturation are calculated. Maximum neutron star masses are determined for two values of the $\mathrm{\Lambda }\text{−}\omega$ meson coupling, $g_{\omega \mathrm{\Lambda }}=2g_{\omega N}/3$ and $g_{\omega \mathrm{\Lambda }}=g_{\omega N}$, and a wide range of values for $g_{\varphi \mathrm{\Lambda }}$. Hyperonic stars with the complete baryonic octet are studied, restricting the coupling of the $\mathrm{\Sigma }$ and $\mathrm{\Xi }$ hyperons to the $\omega ,\rho ,$ and $\sigma$ mesons due to the lack of experimental data, and maximum star masses calculated.

Conclusions: We conclude that, within a phenomenological relativistic mean field approach, the currently available hypernuclei experimental data and the lack of constraints on the asymmetric equation of state of nuclear matter at high densities set only a limited number of constraints on the neutron star matter equation of state using the recent $2{\mathrm{M}}_{\odot }$ observations. It is shown that the $\mathrm{\Lambda }$ potential in symmetric nuclear matter takes a value of $\sim 30-32\mathrm{MeV}$ at saturation for the $g_{\omega \mathrm{\Lambda }}$ coupling given by the SU(6) symmetry, being of the order of the values generally used in the literature. On the other hand, the $\mathrm{\Lambda }$ potential in $\mathrm{\Lambda }$ matter varies between $-14$ and $-8$ MeV, taking for vector mesons couplings the SU(6) values, at variance with generally employed values between $-1$ and $-5$ MeV. If the SU(6) constraint is relaxed and the vector meson couplings to hyperons are kept to values not larger than those of nucleons, then values between $-13$ and $+9$ MeV are obtained.

Figure 1

For the TM1-a model, experimental values of the binding energies $B_{\mathrm{\Lambda }}$ in the $s$ and $p$ shells of single $\mathrm{\Lambda }$ hypernuclei (black circles) and modeled values (red squares and blue triangles, respectively) obtained after adjusting $R_{\sigma \mathrm{\Lambda }}$ in order to minimize the quantity ${\chi }^2$ defined in Eq. (15). For $\mathrm{\Lambda }$ in $p$ shells, $p_{1/2}$ and $p_{3/2}$ states are plotted.

Figure 2

TM1-a (top) and TM1-b (bottom) models. The solid and dashed black lines correspond to values of $(R_{{\sigma }^{*}\mathrm{\Lambda }},R_{\varphi \mathrm{\Lambda }})$ consistent with the experimental values of the bound energy of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^6\mathrm{He}$ in Eqs. (6) and (7). Color contours: values of $U_{\mathrm{\Lambda }}^{\mathrm{\Lambda }}$ at (a) and (c) saturation density $n_0$ and (b) and (d) $n_0/5$ obtained from Eq. (11).

Figure 3

(a) Neutron star maximum mass $M_{\max }$ as a function of $R_{\varphi \mathrm{\Lambda }}$ for for the TM1-a and TM1-b models and various hyperonic models. The values $R_{\sigma \mathrm{\Lambda }},R_{\varphi \mathrm{\Lambda }}$, and $R_{{\sigma }^{*}\mathrm{\Lambda }}$ are adjusted to reproduce the binding energies of single $\mathrm{\Lambda }$ hypernuclei and of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^6\mathrm{He}$ with $\mathrm{\Delta }{B}_{\mathrm{\Lambda }\mathrm{\Lambda }}=0.50$ MeV (solid lines) and 0.84 MeV (dashed lines). The dotted line indicates the SU(6) value of $R_{\varphi \mathrm{\Lambda }}$; (b) composition of the neutron star core for the TM1-a model and $R_{\varphi \mathrm{\Lambda }}$ equal to its SU(6) value and $R_{\sigma \mathrm{\Lambda }},R_{{\sigma }^{*}\mathrm{\Lambda }}$ calibrated to hypernuclei data with $\mathrm{\Delta }{B}_{\mathrm{\Lambda }\mathrm{\Lambda }}=0.50$ MeV, for $U_{\mathrm{\Sigma }}\left(n_0\right)=0$ MeV and $U_{\mathrm{\Xi }}\left(n_0\right)=-14$ (solid lines) or $U_{\mathrm{\Xi }}(2/3n_0)=-14$ MeV (dashed lines).

Figure 4

Analog of Fig. 3 for the $\mathrm{TM}2\omega \rho$ parametrization.

Figure 5

Analog of Fig. 3 for the (a) NL3 and (b) $\mathrm{NL}3\omega \rho$ parametrizations.

Figure 6

Analog of Fig. 3 for the (a) DDME2 and (b) DDME2D parametrizations.

Figure 7

DDME2 parametrization with (a) SU(6), i.e., $R_{\omega \mathrm{\Lambda }}=2/3$ and (b) $R_{\omega \mathrm{\Lambda }}=1$ obtained for $R_{\varphi \mathrm{\Lambda }}$ equal to its SU(6) value and $R_{\sigma \mathrm{\Lambda }}$ and $R_{{\sigma }^{*}\mathrm{\Lambda }}$ calibrated to hypernuclear data with $\mathrm{\Delta }{B}_{\mathrm{\Lambda }\mathrm{\Lambda }}=0.5$ MeV. Dashed lines: only $\mathrm{\Lambda }\mathrm{s}$ included; solid lines: all hyperons included with $U_{\mathrm{\Sigma }}=0$ and $U_{\mathrm{\Xi }}\left(n_0\right)=-14$ MeV.

Figure 8

Same as Fig. 7 for DDME2D parametrization.