$\mathrm{\Delta }\left(1232\right)$ effects in density-dependent relativistic Hartree-Fock theory and neutron stars

The density-dependent relativistic Hartree-Fock (DDRHF) theory is extended to include $\mathrm{\Delta }$ isobars for the study of dense nuclear matter and neutron stars. To this end, we solve the Rarita-Schwinger equation for spin-3/2 particle. Both the direct and exchange terms of the $\mathrm{\Delta }$ isobars' self-energies are evaluated in detail. In comparison with the relativistic mean field theory (Hartree approximation), a weaker parameter dependence is found for DDRHF. An early appearance of $\mathrm{\Delta }$ isobars is recognized at ${\rho }_B\sim 0.28{\mathrm{fm}}^{-3}$, comparable with that of hyperons. Also, we find that the $\mathrm{\Delta }$ isobars' softening of the equation of state is mainly due to the reduced Fock contributions from the coupling of the isoscalar mesons, while the pion contributions are negligibly small. We finally conclude that with typical parameter sets, neutron stars with $\mathrm{\Delta }$ isobars in their interiors could be as heavy as the two massive pulsars whose masses are precisely measured, with slightly smaller radii than normal neutron stars.

Figure 1

$\mathrm{\Delta }$ critical density ${\rho }_{\mathrm{\Delta }}^{\mathrm{crit}}$ as a function of $x_{\rho }$, for $(x_{\sigma },x_{\omega })=(0.8,1.0),(1.0,1.0),(1.0,1.2)$. Previous RMF calculations [4] with $(x_{\sigma },x_{\omega })=(1.0,1.0)$ are also shown for comparison.

Figure 2

Effective baryonic mass $M^{*}$ as a function of the baryon density ${\rho }_B$ for both $Ne\mu$ matter and $N\mathrm{\Delta }e\mu$ matter. The calculations are done for one representative parameter set (PKO1) at fixed meson coupling of $(x_{\sigma },x_{\omega },x_{\rho })=(0.8,1.0,0.5)$.

Figure 3

Relative fraction $Y_i={\rho }_i/{\rho }_B$ of $N\mathrm{\Delta }e\mu$ matter as a function of the baryon density ${\rho }_B$ with the representative parameter set (PKO1) and fixed meson coupling of $(x_{\sigma },x_{\omega },x_{\rho })=(0.8,1.0,0.5)$.

Figure 4

Total pressure $P$ for $N\mathrm{\Delta }e\mu$ matter as a function of the baryon density ${\rho }_B$ with all four available DDRHF parameter sets (PKO1, PKO2, PKO3, PKA1), at fixed meson-$\mathrm{\Delta }$ coupling of $(x_{\sigma },x_{\omega },x_{\rho })=(0.8,1.0,0.5)$. The results without $\mathrm{\Delta }$ isobars with PKO1 are also shown for comparison.

Figure 5

Total pressure $P$ as a function of the baryon density ${\rho }_B$ for both $Ne\mu$ matter (in dashed lines) and $N\mathrm{\Delta }e\mu$ matter (in solid lines). The corresponding contributions from the Hartree channels (in blue) and Fock channels (in red) also shown. The calculations are done for one representative parameter set (PKO1) and at fixed meson coupling of $(x_{\sigma },x_{\omega },x_{\rho })=(0.8,1.0,0.5)$.

Figure 6

Pressure contributions from different mesons for both the Hartree channels (left panel) and Fock channels (right panel), with (in solid lines) or without (in dashed lines) $\mathrm{\Delta }$ isobars. The calculations are done for one representative parameter set (PKO1) and at fixed meson coupling of $(x_{\sigma },x_{\omega },x_{\rho })=(0.8,1.0,0.5)$.

Figure 7

Gravitational mass $M$ as a function of the central density ${\rho }_c$ (the stellar radius $R$) in the left (right) panel. The calculations with PKO1 parameter set are done for three cases of $\mathrm{\Delta }$-meson coupling of $(x_{\sigma },x_{\omega })=(0.8,1.0),(1.0,1.0),(1.0,1.2)$ at fixed $x_{\rho }=0.5$. The calculations with PKA1, PKO2, PKO3 parameter sets are done for one case of $\mathrm{\Delta }$-meson coupling of $(x_{\sigma },x_{\omega })=(0.8,1.0)$. The results without $\mathrm{\Delta }$ isobars for PKO1 and PKA1 are also shown for comparisons, together with two recent massive stars: PSR J1614–2230 [14, 15] and PSR J0348+0432 [16]. The cross in the $(x_{\sigma },x_{\omega })=(1.0,1.2)$ curve is the point beyond which no physical solution is found.