We propose a modified Polyakov-loop Nambu–Jona-Lasinio 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 quark matter and subsequently to deconfined quark matter, or a single transition directly from hadronic to deconfined quark matter or from hadronic to confined quark matter. Stable massive cold hybrid stars with only confined 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 confined and/or a deconfined core can reach maximum masses above $2M_{\odot }$, 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 shows variations at finite baryon densities, with ${\mathrm{c}}_s^2(\mu )$ departing from the asymptotic conformal value ${\mathrm{c}}_s^2=1/3$ in confined core stars. These variations serve as a diagnostic of the equation-of-state stiffness while remaining fully consistent with causality and thermodynamic stability. This work establishes the qualitative role of each model parameter in shaping hybrid-star structure.

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• Equations of state
• Phase transitions
• Quantum chromodynamics
• Quark matter
• Neutron stars & pulsars
• Chiral symmetry
• Composition of astronomical objects