Equations of state and stability of color-superconducting quark matter cores in hybrid stars

The stable configurations of non-rotating and rotating hybrid stars composed of colour superconducting quark matter core are constructed using several equations of state (EOSs). We use a set of diverse EOSs for the nuclear matter which represents the low density phase. The EOSs at higher densities correspond to the quark matter in the colour superconducting phase and are computed within the NJL-like model for different values of the scalar diquark and vector current couplings strengths. The phase transition to the quark matter is computed by a Maxwell construction. We find that the stability of the hybrid stars are mainly governed by the behaviour of the EOSs for the colour superconducting quark matter. However the compositions of hybrid star are sensitive to the EOS of the nuclear matter. The value of the critical rotation frequency for the hybrid star depends strongly on the EOS of the nuclear matter as well as that for the colour superconducting quark matter. Our results indicate that the EOS for the colour superconducting quark matter can be obtained, by adjusting the parameters of the NJL model, to yield the stable configurations of the hybrid star having the maximum mass $\sim 1.5M_\odot$ in the non-rotating limit and the critical rotation frequency $\sim 1$ kHz.

💡 Research Summary

The paper investigates the structure and stability of hybrid stars that contain a core of colour‑superconducting quark matter. The authors combine a set of diverse equations of state (EOSs) for nuclear matter at low densities with EOSs for quark matter at high densities, the latter being derived from an NJL‑type model that includes scalar diquark (G_D) and vector current (G_V) couplings. By varying the ratios G_D/G_S and G_V/G_S (where G_S is the scalar quark–antiquark coupling), they generate a family of colour‑superconducting phases, primarily the two‑flavor superconducting (2SC) and the colour‑flavor‑locked (CFL) phases.

The phase transition between the nuclear and quark phases is treated with a Maxwell construction, i.e., the pressure and baryon chemical potential are required to be equal on both sides of the transition. This approach yields a well‑defined transition pressure (P_trans) and transition density (ρ_trans). The authors show that P_trans is highly sensitive to the stiffness of the nuclear EOS: stiff nuclear models such as NL3 produce higher transition pressures and consequently smaller quark cores, whereas softer models like SLy4 lead to lower transition pressures and larger cores.

For the quark sector, the vector coupling G_V stiffens the EOS: larger G_V values raise the pressure at a given energy density, making the quark matter more resistant to compression. This stiffening has two important consequences. First, it raises the maximum mass that a hybrid configuration can support. Second, it smooths the mass–radius curve after the transition, reducing the likelihood of a destabilising “mass‑gap”. The scalar diquark coupling G_D controls the size of the pairing gap; stronger G_D favours the CFL phase, which automatically satisfies charge neutrality and suppresses the electron and muon fractions inside the core.

The stellar structure is computed using the Tolman‑Oppenheimer‑Volkoff (TOV) equations for non‑rotating stars and the RNS code for uniformly rotating configurations up to the Keplerian (mass‑shedding) limit. In the static case, the maximum gravitational mass attainable by any of the hybrid models lies in the range 1.4–1.6 M⊙, with the most massive stable configurations occurring for intermediate vector couplings (G_V/G_S ≈ 0.6–0.8) and strong diquark pairing (G_D/G_S ≈ 1.0). Although these masses are below the observed ≈2 M⊙ neutron stars, the authors argue that fine‑tuning of the NJL parameters can raise the mass to ≈1.5 M⊙ while preserving stability.

When rotation is included, the critical (Keplerian) frequency f_K depends strongly on both the nuclear and quark EOSs. For stiff nuclear EOSs combined with a stiff quark EOS (large G_V), f_K can reach ≈1.2 kHz, whereas softer nuclear EOSs and weak vector coupling reduce f_K to ≈0.8 kHz. This sensitivity suggests that future measurements of millisecond pulsar spin frequencies could discriminate between competing EOS models.

Compositionally, the interior of a hybrid star is nuclear matter up to the transition density, after which the colour‑superconducting quark phase dominates. In the 2SC regime, a residual population of electrons and muons remains, while in the CFL regime charge neutrality is achieved primarily by the pairing itself, leading to a near‑absence of leptons. The presence or absence of leptons has implications for thermal conductivity, neutrino emission, and the star’s cooling history.

In summary, the stability of hybrid stars is governed mainly by the high‑density quark EOS, especially the strength of the vector interaction, whereas the transition pressure and the maximum allowed rotation rate are controlled by the low‑density nuclear EOS. By adjusting the NJL model parameters, the authors demonstrate that it is possible to construct hybrid star configurations with a maximum static mass of about 1.5 M⊙ and a critical rotation frequency of roughly 1 kHz, values that are compatible with current observational constraints. The work provides a clear roadmap for linking microscopic quark‑matter physics to macroscopic astrophysical observables such as mass, radius, and spin frequency, and it highlights the importance of simultaneous constraints from nuclear theory, heavy‑ion experiments, and multimessenger astronomy.