Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree-Fock theory

Adopting the density dependent relativistic mean-field (RMF) and relativistic Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order symmetry energy $S_4$ are studied within the covariant density functional (CDF) theory. It is found that the fourth-order symmetry energies are suppressed in RHF at both saturation and supranuclear densities, where the extra contribution from the Fock terms is demonstrated, specifically via the isoscalar meson-nucleon coupling channels. The reservation of $S_4$ and higher-order symmetry energies in the nuclear equation of state then affects essentially the prediction of neutron star properties, which is illustrated in quantities such as the proton fraction, the core-crust transition density, as well as the fraction of crustal moment of inertia. Since the Fock terms enhance the density dependence of the thermodynamical potential, the RHF calculations predict systematically smaller values of density, proton fraction, and pressure at the core-crust transition boundary of neutron stars than density dependent RMF ones. In addition, a linear anticorrelation between the core-crust transition density ${\rho }_t$ and the density slope of symmetry energy $L$ is found which is then utilized to constrain the core-crust transition density as ${\rho }_t\sim [0.069,0.098]{\mathrm{fm}}^{-3}$ with the recent empirical information on $L$. The study clarifies the important role of the fourth-order symmetry energy in determining the properties of nuclear matter at extreme isospin or density conditions.

Figure 1

The nuclear fourth-order symmetry energy $S_4$ and its potential part $S_{4,\mathrm{pot}}$ and kinetic part $S_{4,\mathrm{kin}}$ as functions of the baryonic density $\rho$. The results are calculated with RHF functionals PKA1 and PKO1, in comparison with RMF ones PKDD and TW99.

Figure 2

(a) The potential part of nuclear fourth-order symmetry energy $S_{4,\mathrm{pot}}$ is decomposed into the Hartree part $S_{4,\mathrm{pot}}^D$ and the Fock part $S_{4,\mathrm{pot}}^E$ according to Eq. (10b), as functions of the baryonic density $\rho$. The results are calculated with the RHF functional PKO1 (solid lines), in comparison with the RMF one TW99 (dotted line). (b) The Hartree part $S_{4,\mathrm{pot}}^D$ and the contributions to $S_{4,\mathrm{pot}}^E$ from the $\sigma$-, $\omega$-, $\rho$-, and $\pi$-meson coupling channels are shown in detail at low densities.

Figure 3

(a) The kinetic part of nuclear fourth-order symmetry energy $S_{4,\mathrm{kin}}$ is divided into various components according to Eq. (10c), namely $S_{4,\mathrm{kin}}^{M+p+D}=S_{4,\mathrm{kin}}^M+S_{4,\mathrm{kin}}^p+S_{4,\mathrm{kin}}^D$ and the Fock part $S_{4,\mathrm{kin}}^E$, as functions of the baryonic density $\rho$. The results are calculated with the RHF functional PKO1 (solid lines), in comparison with the RMF one TW99 (dotted line). (b) The Hartree part $S_{4,\mathrm{kin}}^D$ and the contributions to $S_{4,\mathrm{kin}}^E$ from the $\sigma$-, $\omega$-, $\rho$-, and $\pi$-meson coupling channels are shown in detail at low densities.

Figure 4

The proton fraction ${\chi }_p$ as a function of the baryonic density $\rho$ in neutron star matter [panels (a)–(d)], calculated from Eq. (15), where $E(\rho ,\delta )$ is taken as its exact values ($①$, solid lines) or approximated up to the fourth ($②$, dashed lines) or the second ($③$, dotted lines) order according to Eq. (7a). The results from the RHF functionals PKA1 and PKO1 are displayed, in comparison with the RMF ones PKDD and TW99. The horizontal dashed lines give the threshold 14.8% for the occurrence of the DUrca process. For the functionals TW99 and PKO1, the effects of the fourth-order and the higher-order symmetry energies are illustrated in the panels (e) and (f), respectively, by showing the divergence of the results with different approximations to $E(\rho ,\delta )$, labeled as $②-③$ and $①-②$.

Figure 5

The thermodynamical potential $V_{\mathrm{ther}}$ as a function of the baryonic density $\rho$ in neutron star matter, calculated from Eq. (17), where $E(\rho ,\delta )$ is taken as its exact values (solid lines) or approximated up to the fourth (dashed lines) or the second order (dotted lines) according to Eq. (7a). The results are from the RHF functional PKO1, compared with the RMF one TW99.

Figure 6

Correlations between the core-crust transition densities ${\rho }_t$ of neutron stars and the density slope parameters $L$ of symmetry energies within the selected CDF functionals. The results within the Gogny density functionals taken from Ref. [43] are plotted as well. From Table 3, the transition densities ${\rho }_t$ are taken as its exact values (squares), approximations up to the fourth (empty stars) or the second order (empty circles),. With the exact ones from the eight selected CDF models (solid squares), a linear fitting is given by the black solid line. The yellow (gray) region depicts the constraint on the density slope parameter $L=58.7\pm 28.1\mathrm{MeV}$ from Ref. [33], which determines further the constraint on the core-crust transition density with the thermodynamical method, namely ${\rho }_t\sim [0.069,0.098]{\mathrm{fm}}^{-3}$, based on the linear anticorrelation between ${\rho }_t$ and $L$, as marked by the shadowed area.

Figure 7

The fraction of crustal moment of inertia $\mathrm{\Delta }I/I$ as a function of the neutron star mass (in units of the solar mass $M_{\odot }$). The results are calculated with the RHF functional PKO1 (lines with squares) and the RMF one TW99 (lines with triangles), by taking different sets of the core-crust transition pressure $P_t$ from Table 3. Two horizontal lines represent the constraints on $\mathrm{\Delta }I/I$, namely $\mathrm{\Delta }I/I\ge 0.016$ [125] or $\mathrm{\Delta }I/I\ge 0.07$ [126].