Massive cold hybrid stars in a modified Polyakov-Nambu-Jona-Lasinio model

We propose a modified Polyakov-loop Nambu–Jona-Lasinio (mPNJL) model in which the Polyakov potential is given by an explicit dependence on the quark chemical potential, allowing it to remain finite at zero temperature and thus to describe the confinement-deconfinement transition in cold dense matter. Combining this modified quark sector with hadronic equations of state via a Maxwell construction, we find that, depending on the model parameters, the equation of state can exhibit either two phase transitions, from hadronic matter to confined (quarkyonic) quark matter and subsequently to deconfined quark matter, or a single transition directly from hadronic to deconfined quark matter or from hadronic to quarkyonic quark matter. Stable massive cold hybrid stars with only quarkyonic and/or deconfined quark phase are obtained. We systematically examine how the parameters of the modified Polyakov potential and the quark vector interactions control the location of these transitions, and find that repulsive vector interactions are essential to obtain a stable quark core. Hybrid stars with quarkyonic and/or a deconfined core can reach maximum masses above 2M⊙, provided a sufficiently stiff hadronic equation of state is used at low density. In the core of the maximum-mass configurations, the speed of sound exceeds the conformal limit, c2s = 1/3, for the quarkyonic core stars. This work establishes the qualitative role of each model parameter in shaping hybrid-star structure.

Improvements to AI systems:

1. Physics-constrained equation-of-state (EoS) generator

• Train a neural network to generate hybrid star EoSs that automatically satisfy the mPNJL model’s constraints (e.g., Polyakov potential finiteness at zero T, Maxwell construction continuity).

• The AI can then rapidly sample the full parameter space (Polyakov potential coefficients, vector coupling strength, hadronic EoS stiffness) to produce physically valid EoSs, replacing slow numerical solvers.

2. Phase-transition classifier and predictor

• Build a classifier that takes model parameters as input and predicts whether the system exhibits two transitions (hadronic → quarkyonic → deconfined) or one (hadronic → deconfined or hadronic → quarkyonic).

• The AI can output the exact transition densities and chemical potentials, enabling fast scans of parameter space for astrophysical applications.

3. Stable hybrid star mass–radius emulator

• Use a deep surrogate model to map (Polyakov potential parameters, vector coupling, hadronic EoS) → (maximum mass, radius, central density, speed-of-sound profile).

• This emulator can instantly identify which parameter combinations yield >2 M⊙ stars and whether the core exceeds the conformal limit (c2s > 1/3), without solving the Tolman-Oppenheimer-Volkoff equations each time.

4. Inverse design tool for observational matching

• Implement an inverse model that, given observed mass–radius constraints (e.g., from NICER or gravitational-wave events), retrieves the most probable mPNJL parameters.

• The AI can quantify which parameter regions are ruled out and which produce quarkyonic cores, directly linking microphysics to astrophysical data.

5. Speed-of-sound anomaly detector

• Train a regression model to predict the speed-of-sound profile from the EoS shape.

• The AI can flag configurations where c2s > 1/3 in the core, helping identify which parameter sets lead to exotic (quarkyonic) behavior—useful for testing QCD-inspired models against lattice or perturbative constraints.

6. Uncertainty-aware parameter sampler

• Use Bayesian neural networks to propagate uncertainties from the hadronic EoS and Polyakov potential into the hybrid star observables.

• The AI can output posterior distributions for maximum mass and transition densities, providing robust predictions for multi-messenger astronomy.

What the improved AI system can do:

• Generate millions of valid hybrid EoSs in seconds, covering all possible phase-transition scenarios.

• Predict the exact phase structure (quarkyonic vs. deconfined) from any parameter set.

• Instantly compute mass–radius curves and identify stable configurations with >2 M⊙ cores.

• Invert observational data to infer the underlying QCD parameters, including vector interaction strength and Polyakov potential shape.

• Automatically detect and flag stars with superluminal or conformal-limit-violating cores, aiding in falsifying or confirming quarkyonic matter existence.

We propose a modified Polyakov-loop Nambu--Jona-Lasinio (mPNJL) model in which the Polyakov potential is given by an explicit dependence on the quark chemical potential, allowing it to remain finite at zero temperature and thus to describe the confinement-deconfinement transition in cold dense matter. Combining this modified quark sector with hadronic equations of state via a Maxwell construction, we find that, depending on the model parameters, the equation of state can exhibit either two phase transitions, from hadronic matter to confined (quarkyonic) quark matter and subsequently to deconfined quark matter, or a single transition directly from hadronic to deconfined quark matter or from hadronic to quarkyonic quark matter. Stable massive cold hybrid stars with only quarkyonic and/or deconfined quark phase are obtained. We systematically examine how the parameters of the modified Polyakov potential and the quark vector interactions control the location of these transitions, and find that repulsive vector interactions are essential to obtain a stable quark core. Hybrid stars with quarkyonic and/or a deconfined core can reach maximum masses above 2M, provided a sufficiently stiff hadronic equation of state is used at low density. In the core of the maximum-mass configurations, the speed of sound exceeds the conformal limit, c s squared = 1/3, for the quarkyonic core stars. This work establishes the qualitative role of each model parameter in shaping hybrid-star structure.