Lepton effects on the proto-neutron stars with the hadron-quark mixed phase in the Nambu-Jona-Lasinio model

We study the structures of hybrid stars with leptons at finite temperature under beta equilibrium. For the quark phase, we use the three flavor Nambu-Jona-Lasinio (NJL) model. For the hadron phase, we adopt nuclear equation of state (EOS) by Shen et al.. This EOS is in the framework of the relativistic mean field theory including the tree body effects. For the hadron-quark phase transition, we impose the bulk Gibbs construction or the Maxwell construction to take into account uncertainties by {\it finite size effects}. We find that the pure quark phase does not appear in stable star cores in all cases. With the phase transition, the maximum masses increase $\sim 10 %$ for high lepton fraction. On the contrary, without the transition, they decrease $\sim 10 %$. We also find that, in the NJL model, the lepton fraction is more important for structures of unstable stars than the temperature. This result is important for many astrophysical phenomena such as the core collapse of massive stars.

💡 Research Summary

This paper investigates the internal structure of proto‑neutron stars (PNSs) by constructing a hybrid equation of state (EOS) that combines a three‑flavor Nambu‑Jona‑Lasinio (NJL) model for quark matter with the relativistic mean‑field (RMF) Shen EOS for hadronic matter. The authors focus on how the lepton fraction (Yₗ) and temperature (T) influence the hadron‑quark phase transition and the resulting stellar properties.

The NJL sector employs a three‑dimensional momentum cutoff Λ = 0.602 GeV, a four‑quark coupling GₛΛ² = 1.835 and a six‑quark ’t Hooft coupling KΛ⁵ = 12.36, with current quark masses mu = md = 5.5 MeV and ms = 140.7 MeV. In β‑equilibrium and charge neutrality, the chemical potentials of u, d, and s quarks are linked to the baryon chemical potential, electron chemical potential, and trapped neutrino chemical potential. The presence of trapped neutrinos (high Yₗ) raises the electron fraction, which suppresses the appearance of negatively charged s‑quarks, thereby delaying chiral restoration for the strange sector. Consequently, the EOS becomes stiffer at high lepton fractions, a behavior absent in the MIT bag model.

The hadronic EOS is taken from Shen et al. (1998), which incorporates three‑body forces within the RMF framework and reproduces experimental nuclear masses and radii. Unlike the NJL sector, the Shen EOS softens slightly at higher Yₗ because the increased electron fraction reduces the neutron density, diminishing the repulsive nuclear interaction below saturation density.

To model the hadron‑quark transition, the authors apply two limiting constructions that bracket the uncertainties arising from finite‑size effects (surface tension and Coulomb energy). The bulk Gibbs construction assumes negligible surface tension, allowing global charge neutrality and resulting in a wide mixed‑phase region (e.g., 1.19 n₀–8.40 n₀ for T = 0 MeV, Yₗ = 0.1). The Maxwell construction assumes large surface tension, enforcing local charge neutrality and producing a sharp pressure jump with a narrow coexistence interval (e.g., 2.37 n₀–3.54 n₀ under the same conditions).

Using the Tolman‑Oppenheimer‑Volkoff (TOV) equations, the authors compute static, spherically symmetric stellar models for temperatures T = 0–30 MeV and lepton fractions Yₗ = 0.1–0.4. Stability is assessed via the condition ∂M/∂n_{B,c} ≥ 0. In all cases, a pure quark core never appears; instead, the central region is occupied by a hadron‑quark mixed phase.

Key findings include:

1. Lepton Fraction Dominance – Increasing Yₗ stiffens the NJL EOS (through suppression of s‑quarks) and softens the Shen EOS, but the net effect in the hybrid model is an increase of the maximum gravitational mass by roughly 10 % when the phase transition is present. Without the transition, the opposite trend occurs (≈ 10 % mass reduction).

2. Temperature Subordination – Variations in temperature (0–30 MeV) have a comparatively minor impact on the maximum mass and radius relative to changes in Yₗ.

3. Construction Dependence – Both Gibbs and Maxwell constructions yield similar qualitative trends (higher Yₗ → higher maximum mass), but the quantitative increase is larger for the Gibbs case (≈ 10 %) than for Maxwell (≈ 3 %). The Maxwell case also exhibits flat segments in the M–central‑density curve due to the pressure discontinuity.

4. Absence of Pure Quark Cores – Even at the highest central densities reached in the models (∼5 n₀), the mixed phase persists, indicating that a first‑order transition to a pure quark phase is unlikely under the adopted parameter set and finite‑size considerations.

These results have astrophysical implications for core‑collapse supernovae and the early evolution of neutron stars. In the immediate post‑bounce phase, trapped neutrinos (high Yₗ) can raise the maximum mass of the nascent star, potentially delaying black‑hole formation. Conversely, as the star deleptonizes, the EOS softens, possibly triggering collapse if the baryonic mass exceeds the reduced maximum mass. The study underscores the importance of incorporating realistic lepton fractions and a consistent treatment of the hadron‑quark interface when modeling proto‑neutron star evolution.