Hadron-quark phase transition in a hadronic and Polyakov--Nambu--Jona-Lasinio models perspective

In this work we study the hadron-quark phase transition matching relativistic hadrodynamical mean-field models (in the hadronic phase) with the more updated versions of the Polyakov-Nambu-Jona-Lasinio models (on the quark side). Systematic comparisons are performed showing that the predicted hadronic phases of the matching named as RMF-PNJL, are larger than the confined phase obtained exclusively by the Polyakov quark models. This important result is due to the effect of the nuclear force that causes more resistance of hadronic matter to isothermal compressions. For sake of comparison, we also obtain the matchings of the hadronic models with the MIT bag model, named as RMF-MIT, showing that it presents always larger hadron regions, while shows smaller mixed phases than that obtained from the RMF-PNJL ones. Thus, studies of the confinement transition in nuclear matter, done only with quark models, still need nuclear degrees of freedom to be more reliable in the whole $T\times\mu$ phase diagram.

💡 Research Summary

The paper investigates the hadron‑to‑quark phase transition by coupling relativistic mean‑field (RMF) models for the hadronic phase with modern versions of the Polyakov‑Nambu‑Jona‑Lasinio (PNJL) model for the quark phase. The authors first motivate the study by pointing out that most previous works have relied solely on quark‑only descriptions (e.g., MIT bag or simple PNJL) to draw the transition line in the temperature–chemical‑potential (T‑μ) plane. Such approaches neglect the strong nuclear forces that dominate the low‑temperature, high‑density region, potentially leading to an underestimation of the hadronic domain.

In the theoretical framework, several calibrated RMF parameter sets (NL3, TM1, FSUGold, etc.) are employed. These models incorporate scalar σ, vector ω, and isovector ρ meson exchanges, reproducing saturation properties, binding energies, and compressibilities of nuclear matter. Within the mean‑field approximation, the pressure P_H(μ,T) and energy density ε_H(μ,T) are derived analytically. On the quark side, the PNJL model is used in its most recent incarnation, which couples the NJL four‑fermion interaction (including scalar, pseudoscalar, and vector channels) with a Polyakov‑loop effective potential that mimics confinement. Temperature‑ and chemical‑potential‑dependent Polyakov parameters are chosen to reproduce lattice QCD thermodynamics at μ=0, while the vector coupling g_V is varied to explore its impact on the stiffness of quark matter.

The matching between the two phases is performed using the Gibbs criteria: at a given temperature the transition occurs where the pressures of the two phases are equal, P_H(μ,T)=P_Q(μ,T), and the baryon chemical potentials coincide. This yields a transition line in the T‑μ plane and defines a mixed‑phase region where both phases coexist. The authors systematically compare three scenarios: (i) pure PNJL (no hadronic degrees of freedom), (ii) RMF‑PNJL (the main focus), and (iii) RMF‑MIT, where the MIT bag model provides the quark EOS.

Results show that the RMF‑PNJL transition line lies at higher chemical potentials (or lower temperatures) than the pure PNJL line. The shift is attributed to the additional repulsive contribution from the nuclear force, which raises the isothermal compressibility of hadronic matter and makes it more resistant to compression. Consequently, the hadronic region in the RMF‑PNJL diagram is significantly larger than the confined region obtained from PNJL alone. When the MIT bag model is used, the RMF‑MIT transition line is pushed even further toward high μ, producing the largest hadronic domain among the three cases. However, the mixed‑phase width in RMF‑MIT is smaller than in RMF‑PNJL because the bag model’s pressure is a simple constant (the bag constant), leading to a sharper transition.

The discussion connects these findings to astrophysical and experimental contexts. In neutron‑star interiors, a larger hadronic domain implies that quark matter would appear only in the most massive stars, affecting predictions for maximum mass, radius, and tidal deformability—quantities now probed by gravitational‑wave observations. In relativistic heavy‑ion collisions, the location of the transition line influences the expected signatures of deconfinement (e.g., fluctuations of conserved charges, flow anisotropies). The authors stress that any realistic mapping of the QCD phase diagram must incorporate nuclear degrees of freedom, especially in the region of high baryon density and moderate temperature.

Finally, the paper acknowledges limitations: the mean‑field treatment neglects correlations and fluctuations; the PNJL model, while improved, still lacks a fully dynamical confinement mechanism; and the MIT bag model is overly simplistic. Future work is suggested to combine RMF with more sophisticated quark models (e.g., Dyson‑Schwinger approaches, functional renormalization group) and to benchmark the results against lattice QCD calculations at finite μ (via extrapolation or reweighting techniques). The overall conclusion is that matching hadronic RMF models with updated PNJL descriptions yields a more reliable and physically consistent picture of the hadron‑quark transition across the entire T‑μ plane.