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The possible role of a first order QCD phase transition at nonvanishing quark chemical potential and temperature for cold neutron stars and for supernovae is delineated. For cold neutron stars, we use the NJL model with nonvanishing color superconducting pairing gaps, which describes the phase transition to the 2SC and the CFL quark matter phases at high baryon densities. We demonstrate that these two phase transitions can both be present in the core of neutron stars and that they lead to the appearance of a third family of solution for compact stars. In particular, a core of CFL quark matter can be present in stable compact star configurations when slightly adjusting the vacuum pressure to the onset of the chiral phase transition from the hadronic model to the NJL model. We show that a strong first order phase transition can have strong impact on the dynamics of core collapse supernovae. If the QCD phase transition sets in shortly after the first bounce, a second outgoing shock wave can be generated which leads to an explosion. The presence of the QCD phase transition can be read off from the neutrino and antineutrino signal of the supernova.
As the density of matter increases, atomic nuclei disintegrate into nucleons and, eventually, the nucleons themselves disintegrate into quarks. The phase transitions (PT's) between these phases can vary from steep first order to smooth crossovers, depending on certain conditions. First-order PT's with more than one globally conserved charge, so-called non-congruent PT's, have characteristic differences compared to congruent PT's. In this conference proceeding we discuss the non-congruence of the quark deconfinement PT at high densities and/or temperatures relevant for heavy-ion collisions, neutron stars, proto-neutron stars, supernova explosions, and compact-star mergers.