Critical density and impact of $\mathrm{\Delta }\left(1232\right)$ resonance formation in neutron stars

The critical densities and impact of forming $\mathrm{\Delta }\left(1232\right)$ resonances in neutron stars are investigated within an extended nonlinear relativistic mean-field (RMF) model. The critical densities for the formation of four different charge states of $\mathrm{\Delta }\left(1232\right)$ are found to depend differently on the separate kinetic and potential parts of nuclear symmetry energy, the first example of a microphysical property of neutron stars to do so. Moreover, they are sensitive to the in-medium $\mathrm{\Delta }$ mass $m_{\mathrm{\Delta }}$ and the completely unknown $\mathrm{\Delta }\text{−}\rho$ coupling strength $g_{\rho \mathrm{\Delta }}$. In the universal baryon-meson coupling scheme where the respective $\mathrm{\Delta }$-meson and nucleon-meson coupling constants are assumed to be the same, the critical density for the first ${\mathrm{\Delta }}^{-}\left(1232\right)$ to appear is found to be ${\rho }_{\mathrm{\Delta }}^{\mathrm{crit}}=(2.08\pm 0.02){\rho }_0$ using RMF model parameters consistent with current constraints on all seven macroscopic parameters usually used to characterize the equation of state of isospin-asymmetric nuclear matter at saturation density ${\rho }_0$. Moreover, the composition and the mass-radius relation of neutron stars are found to depend significantly on the values of the $g_{\rho \mathrm{\Delta }}$ and $m_{\mathrm{\Delta }}$.

Figure 1

The maximum mass of $\mathrm{\Delta }$ resonances produced in the $\mathit{NN}\to N\mathrm{\Delta }$ process in a free Fermi gas of nucleons at density $\rho$.

Figure 2

Dependence of the critical density ${\rho }_{{\mathrm{\Delta }}^{-}}^{\mathrm{crit}}$ for ${\mathrm{\Delta }}^{-}$ formation in neutron stars on the relative $\mathrm{\Delta }\text{−}\rho$ coupling strength $x_{\rho }=g_{\rho \mathrm{\Delta }}/g_{\rho \mathrm{N}}$.

Figure 3

The $\mathrm{\Delta }$ mass $m_{\mathrm{\Delta }}$ dependence of the critical density ${\rho }_{{\mathrm{\Delta }}^{-}}^{\mathrm{crit}}$ for ${\mathrm{\Delta }}^{-}$ formation in neutron stars (blue) and the Breit-Wigner mass distribution of $\mathrm{\Delta }$ resonances in free space (red).

Figure 4

Critical density for the formation of ${\mathrm{\Delta }}^{-}$ resonance from the nonlinear RMF model by varying individually ${\rho }_0$ (a), $E_0\left({\rho }_0\right)$ (b), $m_{\mathrm{dirac}}^{*0}$ (c), $K_0$ (d), $J_0$ (e), $E_{\mathrm{sym}}\left({\rho }_0\right)$ (f), and $L\left({\rho }_0\right)$ (g).

Figure 5

Critical density for the formation of ${\mathrm{\Delta }}^{-}$ resonance as a function of the scaled density slope of symmetry energy $L\left(w{\rho }_0\right)/w$ at three reduced densities of ${\rho }_{\mathrm{r}}/{\rho }_0=w=0.7,1$ and 2, respectively.

Figure 6

Fractions of different species in neutron stars composed of neutrons, protons, $\mathrm{\Delta }$ resonances, electrons, and muons within the nonlinear RMF model using two sets of model parameters with $x_{\rho }=g_{\rho \mathrm{\Delta }}/g_{\rho \mathrm{N}}=1$ and 2, respectively.

Figure 7

The mass-radius correlation of neutron stars without and with $\mathrm{\Delta }$ resonances of different masses using $x_{\rho }=g_{\rho \mathrm{\Delta }}/g_{\rho \mathrm{N}}=1$ and 2, respectively.