Protoquark Stars in β-Equilibrium

• The paper introduces a modified NJL model that uses Fierz-induced vector interactions to transform a first-order chiral transition into a crossover, thereby stiffening the EoS in protoquark stars.
• It demonstrates that finite temperature and lepton fraction critically shape β-equilibrium, affecting quark flavor abundances, the cooling process, and the star’s structural evolution.
• The study predicts twin star configurations with similar masses but different radii, offering distinctive observational signatures in supernova remnants and compact merger events.

A protoquark star in β-equilibrium is a hot, high-entropy compact object formed immediately following core collapse or merger events, in which the interior is composed of deconfined quark matter—typically up, down, and strange quarks—maintained in chemical and charge equilibrium via weak interactions and lepton content adjustment. These stars evolve through early neutrino-trapped stages and cool towards stable cold configurations, with their structural and dynamical properties governed by the microphysics of the chiral and deconfinement transitions, the underlying equation of state (EoS), and the flavor–lepton correlation set by β-equilibrium.

1. The Modified NJL Model with Exchange (Fierz) Interactions

The standard Nambu–Jona-Lasinio (NJL) model describes chiral symmetry breaking via scalar and pseudoscalar contact interactions for quarks. In the modified approach, a Fierz transformation is applied to the four-fermion vertex, generating additional terms—most notably, vector and axial-vector interactions—whose relative strength is modulated by a parameter $\alpha$ ($0 \leq \alpha \leq 1$). The Lagrangian for two flavors is: $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$ where $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$ contains the Fierz-transformed vector channels. The gap and chemical potential equations acquire an explicit dependence on $\alpha$: $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$ with $\sigma_f$ and $\rho_f$ the chiral condensate and number density, respectively. Larger $\alpha$ increases the repulsive vector interaction, tending to shift the chiral phase transition from first-order to crossover and stiffening the EoS at high density.

2. Chiral Phase Transition and Equation of State under Finite Temperature

Protoquark stars feature hot and lepton-rich interiors; modeling requires consistent treatment of finite temperature and lepton content. In the mean-field thermodynamic potential $\Omega$, temperature enters via Fermi-Dirac distributions for quarks and leptons, and the pressure is obtained as $0 \leq \alpha \leq 1$0. In the (2+1)-flavor model (for strange quark matter), the gap equations become a set for the condensates $0 \leq \alpha \leq 1$1 (or their respective chiral order parameters), coupled to effective chemical potentials for each flavor.

Key equations for $0 \leq \alpha \leq 1$2-equilibrium and charge neutrality include: $0 \leq \alpha \leq 1$3

$0 \leq \alpha \leq 1$4

$0 \leq \alpha \leq 1$5

where $0 \leq \alpha \leq 1$6 is the total lepton fraction. The interplay of temperature and lepton content modifies the location and character (first-order vs. crossover) of the chiral phase transition. For nonstrange quark matter, increasing $0 \leq \alpha \leq 1$7 stiffens the EoS, while for strange quark matter at high densities, $0 \leq \alpha \leq 1$8 typically softens the EoS in the chiral-restored phase as thermal energy increases the energy density more than the pressure. Higher $0 \leq \alpha \leq 1$9 stiffens the matter due to increased leptonic pressure and suppression of strange quark content.

3. Influence of β-Equilibrium and Lepton Fraction

β-equilibrium is essential for the macroscopic stability and cooling evolution of protoquark stars. The equilibrium relations above ensure that weak processes (such as $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$0) maintain the chemical composition at each evolutionary stage. Neutrino trapping (presence of a high $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$1 and $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$2) affects the sequence of quark flavor abundances, especially retarding strange quark appearance by keeping $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$3 lower at fixed baryon density, thereby directly impacting the stiffness of the EoS. As deleptonization proceeds and neutrinos escape (after $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$4–$$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$5 s), $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$6 and the equilibrium shifts, typically increasing the central strangeness and reducing the total mass due to EoS softening.

At fixed $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$7 and high $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$8, the matter is less negatively charged (higher electron content required for neutrality), reducing the presence of negatively charged $$\mathcal{L}_{\text{eff}}^{(2f)} = \bar{\psi} (i\gamma^\mu \partial_\mu - m + \mu \gamma^0) \psi + (1-\alpha)\mathcal{L}_{\text{int}}^{(2f)} + \alpha \mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)}),$$9 and $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$0 quarks. This adjustment raises the pressure at given energy density.

4. Twin Star Configurations in the Crossover Regime

A striking prediction of the modified NJL model with substantial Fierz-induced vector interactions (large $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$1) is the possible emergence of "twin star" solutions—compact stars with similar gravitational mass but distinctly different radii—within certain regions of parameter space. The underlying mechanism is as follows:

• In the (2+1)-flavor case, as the chemical potential increases, the appearance of strange quarks at lower densities due to high $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$2 or $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$3 can soften the EoS (lowering the pressure for a given energy density), while the repulsive vector interaction (large $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$4) at higher densities restiffens the EoS.
• This nonmonotonic pressure–energy relation produces two branches of stable stars: one with low central strangeness (larger radius) and one with higher strangeness content (more compact for the same mass).
• For example, for model parameters such as $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$5, low vacuum pressure $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$6 (ensuring quark matter is self-bound), $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$7, and $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$8 MeV, the model supports such twin configurations.

Such twin stars are distinguishable from conventional compact star solutions and might yield observational evidence for a strong-crossover quark matter phase transition in the cores of massive neutron stars or the direct birth of compact self-bound protoquark stars in supernovae.

5. Astrophysical Implications and Constraints

The overall EoS derived in the crossover regime (large $\mathcal{F}(\mathcal{L}_{\text{int}}^{(2f)})$9) is significantly stiffer than in the first-order chiral transition case. This feature is critical for supporting protoquark stars with gravitational masses $\alpha$0, as required by pulsar observations (e.g., PSR J0740+6620 and related objects). Additionally, the interplay between $\alpha$1, $\alpha$2, and $\alpha$3 implies that protoquark stars in the early, hot, neutrino-rich phase can be even more massive than after deleptonization and cooling, in accord with models where lepton pressure provides additional support.

The stiffness can ensure survival as a distinct compact object during the Kelvin–Helmholtz deleptonization phase. A plausible implication is that observable transitions—to a more compact configuration as the EoS softens with strangeness loading or temperature decrease—might be marked by sudden changes in emission properties, gravitational wave signatures, or neutrino burst features.

6. Key Model Relationships and Formulas

Physical Input	Key Formulae	Role
Effective Lagrangian	$\alpha$4 (above), $\alpha$5 controls Fierz weight	Vector interaction controls EoS, phase
Gap equations	$\alpha$6	Determines chiral transition, mass
Effective $\alpha$7	$\alpha$8	Shifts chemical equilibrium
$\alpha$9-equilibrium	$$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$0	Balances weak processes
Charge neutrality	$$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$1	Fixes relative quark–lepton content
Thermodynamic pressure	$$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$2	EoS for TOV integration

All model ingredients (gap equations, equilibrium and neutrality relations, and thermodynamic identities) are solved self-consistently at each $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$3, $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$4, $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$5.

7. Summary and Outlook

The modified NJL model incorporating Fierz-induced vector exchange self-consistently describes protoquark stars in β-equilibrium at finite temperature and lepton fraction. The model demonstrates that:

• Sufficiently strong vector interactions (large $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$6) convert the chiral transition from first-order to crossover, stiffening the EoS and supporting more massive protoquark stars.
• High temperature and $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$7 characteristic of early protoquark stars result in softer EoSs at low density and enhanced stiffness at high density; the combined effects determine the evolutionary trajectory of mass, radius, and strangeness content.
• In the crossover region, twin star solutions are supported—these are potentially observable either as direct supernova remnants or as end points of mass-accreting binary evolution.
• The model's treatment is natural for supernova and merger environments, satisfying both microscopic self-consistency and macroscopic observational constraints ($$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$8).

Further developments will require detailed dynamical simulations of protoquark star formation and cooling, calculation of their multimessenger signatures, and cross-comparison with observed neutron star populations, with particular attention to unique mass–radius relations and cooling timescales associated with the twin star phenomenon and the role of the Fierz-induced vector interaction parameter $$M = m - 2\left[(1-\alpha) + \frac{\alpha}{12}\right] G \sum_f \sigma_f, \qquad \tilde{\mu} = \mu - \frac{\alpha}{3} G \sum_f \rho_f,$$9 (Yuan et al., 20 Aug 2025).