⚛️ nuclear theory

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

This paper proposes a modified Polyakov-Nambu-Jona-Lasinio model with a chemical potential-dependent Polyakov potential to describe cold dense matter, demonstrating that repulsive vector interactions and a stiff hadronic equation of state are essential for forming stable hybrid stars with quarkyonic or deconfined cores that exceed two solar masses.

Original authors: Sk Md Adil Imam, Pedro Costa, Mariana Dutra, Odilon Lourenço, Renan Pereira, Constança Providência

Published 2026-08-14

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CC BY 4.0

Original authors: Sk Md Adil Imam, Pedro Costa, Mariana Dutra, Odilon Lourenço, Renan Pereira, Constança Providência

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic kitchen where matter is cooked up under extreme pressure. In our everyday world, matter is made of atoms, which are like tiny solar systems with a nucleus in the middle and electrons orbiting around. But if you squeeze these atoms hard enough—like crushing a grape into a seed—the electrons get squashed into the nucleus, and the atoms collapse into a soup of even smaller particles called quarks. This is the stuff inside neutron stars, the densest objects in the universe, where a teaspoon of material would weigh a billion tons. Scientists have been trying to figure out exactly what happens in this cosmic pressure cooker. Do the quarks just sit there in a messy pile, or do they break free and dance around as a super-hot, super-dense fluid? The big mystery is how matter changes from being "stuck" together (confined) to being "free" (deconfined) when the temperature is near absolute zero but the pressure is insane. Understanding this helps us solve the riddle of how heavy neutron stars can get before they collapse into black holes.

Now, meet the team of physicists who decided to build a new recipe to solve this puzzle. They created a modified mathematical model, which they call the "mPNJL" model, to simulate what happens inside these cold, heavy stars. Think of their model as a sophisticated video game engine that lets them play with the rules of the universe. In the old version of this game, the rules broke down when the temperature dropped to zero, making it impossible to study cold neutron stars. These researchers fixed the engine by adding a new "control knob" that depends on how many particles are packed together (the chemical potential). This allowed them to keep the simulation running even in the freezing cold of deep space.

What they found is a bit like discovering different types of cake layers inside a star. Depending on how they tweaked the knobs in their model, they could create stars with three distinct layers: a normal outer crust, a middle layer of "quarkyonic" matter (a weird, hybrid state where quarks are stuck but acting strangely), and a core of "deconfined" quarks (where they are completely free). They discovered that to make a star heavy enough to survive—specifically, one that weighs more than two times our Sun—they needed two things: a very stiff outer crust (so it doesn't collapse easily) and strong repulsive forces between the quarks in the core (like invisible springs pushing them apart).

The most exciting part of their simulation is that they found stable stars with masses over two solar masses. Some of these stars have a core made entirely of free-floating quarks, while others have a core made of that strange "quarkyonic" mix. In fact, for the stars with the quarkyonic core, the "speed of sound" (how fast a vibration travels through the star) actually breaks a famous speed limit known as the "conformal limit," suggesting the matter inside is incredibly stiff and resistant to being squeezed. However, they also found that if they used a softer outer crust, the stars would become unstable and collapse as soon as the quark core started to form. So, the paper suggests that while these massive, exotic hybrid stars are possible, they require a very specific and stiff set of ingredients to exist. They didn't prove these stars definitely exist in the sky, but they showed that the laws of physics allow for them, provided the right conditions are met.

Problem Statement
The behavior of strongly interacting matter under extreme density and temperature conditions, such as those found in the interiors of compact stars, remains a central challenge in nuclear physics and astrophysics. While the standard Polyakov-Nambu–Jona-Lasinio (PNJL) model successfully incorporates color confinement dynamics via the Polyakov loop, it suffers from a critical limitation in the context of cold dense matter: the standard Polyakov loop potential vanishes as temperature ( $T$ ) approaches zero. Consequently, the deconfinement order parameter becomes irrelevant in the regime of cold neutron and hybrid stars, preventing a consistent description of the confinement-deconfinement transition in these objects. Furthermore, while perturbative QCD provides constraints at very high densities, lattice QCD calculations at finite baryonic density are hindered by the fermion sign problem, necessitating effective models to explore the phase structure of cold dense matter.

Methodology
To address the zero-temperature limitation, the authors propose a modified Polyakov-loop Nambu–Jona-Lasinio (mPNJL) model. The core modification involves reformulating the effective Polyakov loop potential, $U(\Phi, T)$ , to include an explicit dependence on the quark chemical potential ( $\mu_f$ ). This is achieved by:

Replacing the global $T^4$ dependence in the potential with a Stefan–Boltzmann pressure term that includes chemical potential contributions.
Introducing a phenomenological regularization for the ratio $T_0/T$ (where $T_0$ is the deconfinement temperature) to prevent divergence at $T \to 0$ . This involves a substitution $T_0/T \to T_0 / \sqrt{T^2 + g(\mu_f)}$ , where $g(\mu_f)$ is a power series in $\mu_f$ .
Deriving analytical solutions for the Polyakov loop $\Phi$ under these modified conditions, allowing for the identification of confined ( $\Phi=0$ ), quarkyonic ( $0 < \Phi < 1$ ), and deconfined ( $\Phi \to 1$ ) phases even at zero temperature.

The quark sector of the mPNJL model is coupled to various hadronic equations of state (EOS) for the low-density phase (specifically SFHo, DD2, DD2hyp, and NL3 $\omega\rho$ ) using a Maxwell construction. The study systematically varies model parameters, including the deconfinement temperature $T_0$ , the parameter $\alpha_0$ (which influences the chemical potential dependence), the vector coupling constants $G_V$ (4-quark) and $G_{vv}$ (8-quark), and a bag constant $B_0$ . The resulting hybrid EOSs are then used to solve the Tolman–Oppenheimer–Volkoff (TOV) equations to determine the global properties (mass-radius relations) of hybrid stars.

Key Contributions and Results
The paper establishes the qualitative role of each model parameter in shaping hybrid star structure and identifies the conditions required to support massive cold hybrid stars ( $M \gtrsim 2M_\odot$ ) with quark cores.

Phase Transitions: Depending on the parameter set, the model yields either a single transition (hadronic to deconfined quark matter) or a sequence of two transitions (hadronic to confined/quarkyonic matter, followed by a transition to deconfined quark matter).
Role of Vector Interactions: The study finds that repulsive vector interactions are essential for obtaining stable quark cores. Specifically, the 4-quark vector coupling ( $G_V$ ) primarily controls the location of the confinement-deconfinement transition, while the 8-quark coupling ( $G_{vv}$ ) significantly influences the stiffness of the quark EOS and the speed of sound at the stellar center.
Impact of Hadronic EOS: The choice of the low-density hadronic EOS is critical. Using the softer SFHo EOS, the authors could not obtain stable hybrid stars with masses exceeding $2M_\odot$ , as the mass-radius sequence became unstable immediately upon the onset of the hadron-quark transition. However, by employing stiffer hadronic EOSs (DD2, DD2hyp, NL3 $\omega\rho$ ), stable hybrid stars with masses above $2M_\odot$ were successfully constructed.
Configurations: The authors identified three distinct stable configurations:
1. Deconfined Core: Stars with a direct transition from hadronic matter to a deconfined quark core (using DD2).
2. Quarkyonic Core: Stars with a transition to a quarkyonic phase followed by instability upon entering the deconfined phase (using DD2hyp and NL3 $\omega\rho$ ). In these configurations, the squared speed of sound ( $c_s^2$ ) in the core exceeds the conformal limit ( $1/3$ ).
3. Three-Phase Structure: A genuine, mechanically stable three-phase structure (hadronic envelope, quarkyonic shell, and deconfined core) was found using the NL3 $\omega\rho$ EOS. In this scenario, the maximum mass reaches $\sim 2.1M_\odot$ , and the speed of sound remains below the conformal limit in the deconfined core, with the necessary stiffening provided by the hadronic and quarkyonic phases.

Significance and Claims
The paper claims to provide a consistent framework for describing the confinement-deconfinement transition in cold dense matter by modifying the Polyakov loop potential to remain finite at zero temperature. The primary significance lies in demonstrating that massive hybrid stars with quark cores are achievable within this framework, provided that:

The modified Polyakov potential parameters are tuned to allow for the existence of quarkyonic and/or deconfined phases.
Repulsive vector interactions among quarks are sufficiently strong to stiffen the EOS.
A sufficiently stiff hadronic EOS is used at low densities to support the high masses required by observational constraints (e.g., PSR J0740+6620).

The authors emphasize that this work establishes the qualitative dependence of hybrid star structure on model parameters rather than performing a quantitative fit to observational data. They note that the speed of sound behavior differs between configurations: exceeding the conformal limit in quarkyonic-core stars but remaining below it in deconfined-core and three-phase stars, suggesting different mechanisms for supporting massive stars in these scenarios. Future work is proposed to utilize Bayesian inference with multimessenger observations to quantitatively determine preferred parameter ranges and assess the statistical evidence for the different structural scenarios.

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