In this paper, we explore whether or not quark deconfinement may occur in high-mass neutron stars such as J1614 $-${} 2230 ($1.97\ifmmode\pm\else\textpm\fi{}0.04\phantom{\rule{0.16em}{0ex}}{M}_{$\bigodot${}}$) and J0348 + 0432 ($2.01\ifmmode\pm\else\textpm\fi{}0.04\phantom{\rule{0.16em}{0ex}}{M}_{$\bigodot${}}$). Our study is based on a nonlocal extension of the SU(3) Nambu\char21{}Jona-Lasinio (n3NJL) model with repulsive vector interactions among the quarks. This model goes beyond the frequently used local version of the Nambu\char21{}Jona-Lasinio (NJL) model by accounting for several key features of QCD which are not part of the local model. Confined hadronic matter is treated in the framework of nonlinear relativistic mean field theory. We find that both the local as well as the nonlocal NJL model predict the existence of extended regions of mixed quark-hadron (quark-hybrid) matter in high-mass neutron stars with masses of 2.1 to $2.4\phantom{\rule{0.16em}{0ex}}{M}_{$\bigodot${}}$. Pure quark matter in the cores of neutron stars is obtained for certain parametrizations of the hadronic lagrangian and choices of the vector repulsion among quarks. The radii of high-mass neutron stars with quark-hybrid matter and/or pure quark matter cores in their centers are found to lie in the canonical range of 12 to 13 km.

The recent direct observation of gravitational wave event GW170817 and its GRB 170817 A signal has opened up a new window to studying neutron stars and heralds a new era of astronomy referred to as the multimessenger astronomy. Both gravitational and electromagnetic waves from a single astrophysical source have been detected for the first time. This combined detection offers an unprecedented opportunity to place constraints on the neutron star matter equation of state (EoS). The existence of a possible hadron–quark phase transition in the central regions of neutron stars is associated with the appearance of g modes, which are extremely important as they could signal the presence of a pure quark matter core in the centers of neutron stars. Observations of g modes with frequencies between 1 and 1.5 kHz could be interpreted as evidence of a sharp hadron–quark phase transition in the cores of neutron stars. In this article, we shall review the description of the dense matter composing neutron stars, the determination of the EoS of such matter, and the constraints imposed by astrophysical observations of these fascinating compact objects.

We investigate the hadron-quark phase transition in cold neutron stars in light of (i) the observed limits on the maximum-mass of heavy pulsars, (ii) constraints on the tidal properties inferred from the gravitational waves emitted in binary neutron-star mergers, and (iii) mass and radius constraints derived from the observation of hot spots on neutron star observed with the Neutron Star Interior Composition Explorer instrument. Special attention is directed to the possible presence of $\mathrm{\Delta }(1232)$ baryons in neutron star matter. Our results indicate that this particle could make up a large fraction of the baryons in neutron stars and thus have a significant effect on the properties of such objects, particularly on their radii. This is partially caused by the low density appearance of $\mathrm{\Delta }\mathrm{s}$ for a wide range of theoretically defensible sets of meson-hyperon, SU(3) ESC08 model, and meson-$\mathrm{\Delta }$ coupling constants. The transition of hadronic matter to quark matter, treated in the $2\mathrm{SC}+\mathrm{s}$ condensation phase, is found to occur only in neutron stars very close to the mass peak. Nevertheless, quark matter may still constitute an appreciable fraction of the stars’ total matter if the phase transition is treated as Maxwell-like (sharp), in which case the neutron stars located beyond the gravitational mass peak would remain stable against gravitational collapse. In this case, the instability against gravitational collapse is shifted to a new (terminal) mass different from the maximum-mass of the stellar sequence, giving rise to stable compact objects with the same gravitational masses as those of the neutron stars on the traditional branch, but whose radii are smaller by up to 1 km. All models for the equation of state of our study fall comfortably within the bound established very recently by Annala et al. [ Nat. Phys., 10.1038/s41567-020-0914-9 (2020)].

We study a family of equations of state (EOS) for hybrid neutron star matter. The hybrid EOS are obtained by a Maxwell construction of the first-order phase transition between a hadronic phase described by the relativistic density-functional EOS of the “DD2” class with excluded volume effects and a deconfined quark matter phase modeled by an instantaneous nonlocal version of the Nambu–Jona-Lasinio model in $\text{SU}{\left(2\right)}_f$ with vector interactions and color superconductivity. The form factor in the nonlocal quark matter model is fitted to lattice QCD results in the Coulomb gauge. Owing to strong coupling in the vector meson and diquark channels, a coexistence phase of color superconductivity and chiral symmetry breaking occurs. Our results show an approximately constant behavior for the squared speed of sound with values of 0.4–0.6 in the density region relevant for neutron star interiors. To simultaneously fulfill the constraints from the Neutron Star Interior Composition Explorer radius measurement for PSR $\mathrm{J}0740+6620$ and tidal deformability from GW170817 it is necessary to consider a $\mu$-dependent bag pressure that mimics confinement.

In the first part of this paper, we use a nonlocal extension of the three-flavor Polyakov-Nambu-Jona-Lasinio model, which takes into account flavor-mixing, momentum dependent quark masses, and vector interactions among quarks, to investigate the possible existence of a spinodal region (determined by the vanishing of the speed of sound) in the QCD phase diagram and determine the temperature and chemical potential of the critical end point. In the second part of the paper, we investigate the quark-hadron composition of baryonic matter at zero as well as nonzero temperature. This is of great topical interest for the analysis and interpretation of neutron star merger events such as GW170817. With this in mind, we determine the composition of proto-neutron star matter for entropies and lepton fractions that are typical of such matter. These compositions are used to delineate the evolution of proto-neutron stars to neutron stars in the baryon-mass versus gravitational-mass diagram. The hot stellar models turn out to contain significant fractions of hyperons and $\mathrm{\Delta }$ isobars but no deconfined quarks. The latter are found to exist only in cold neutron stars.

The discovery of pulsars as heavy as 2 solar masses has led astrophysicists to rethink the core compositions of neutron stars, ruling out many models for the nuclear equations of state (EoS). We explore the hybrid stars that occur when hadronic matter is treated in a relativistic mean-field approximation and quark matter is modeled by three-flavor local and nonlocal Nambu–Jona-Lasinio (NJL) models with repulsive vector interactions. The NJL models typically yield equations of state that feature a first-order transition to quark matter. Assuming that the quark-hadron surface tension is high enough to disfavor mixed phases and restricting to EoSs that allow stars to reach 2 solar masses, we find that the appearance of the quark-matter core either destabilizes the star immediately (this is typical for nonlocal NJL models) or leads to a very short hybrid star branch in the mass-radius relation (this is typical for local NJL models). Using the constant-sound-speed parametrization we can see that the reason for the near absence of hybrid stars is that the transition pressure is fairly high and the transition is strongly first order.