$\Delta$ (1232) effects in density-dependent relativistic Hartree-Fock   theory and neutron stars
  The density-dependent relativistic Hartree-Fock (DDRHF) theory is extended to include $\Delta$-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 $\Delta$-isobars' self-energies are evaluated in details. In comparison with the relativistic mean field theory (Hartree approximation), a weaker parameter dependence is found for DDRHF. An early appearance of $\Delta$-isobars is recognized at $\rho_B\sim0.27$fm$^{-3}$, comparable with that of hyperons. Also, we find that the $\Delta$-isobars' softening of the equation of state is found to be mainly due to the reduced Fock contributions from the coupling of the isoscalar mesons, while the pion contributions are found negligibly small. We finally conclude that with typical parameter sets, neutron stars with $\Delta$-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. 
Zhen-Yu Zhu, Ang Li, Jin-Niu Hu, Hiroyuki Sagawa
nucl-th astro-ph.HE astro-ph.SR