Towards a unified hadron-quark equation of state for neutron stars within the relativistic mean-field model

TL;DR

This work tackles the uncertain dense-matter equation of state inside neutron stars by developing EVA–01, a unified hadron–quark EoS built on a density-dependent relativistic mean-field framework augmented with a Polyakov-like field $\Phi$ that dynamically mediates deconfinement. The model extends the DDRMF–SW4L basis to include $u,d,s$ quarks, baryons (including the full octet and $\Delta$ resonances), and leptons, all coupled through scalar, vector, and isovector mesons, with a Polyakov potential $U_\Phi$ that links deconfinement to effective masses and thermodynamics. The phase diagram reveals three distinct first-order transitions: a nuclear liquid–gas transition at low densities, a sharp hadron–quark deconfinement boundary, and an intra-quark chiral transition that remains subleading and does not generate a CEP; the model yields a pseudo-critical temperature at zero chemical potential of $T_\mathrm{pc} = 149.8$ MeV. Applied to proto-neutron-star evolution, EVA–01 predicts a stiff, lepton-rich Stage 1 with large radii, a hotter Stage 2 with significant hyperon and Delta populations softening the EoS, and a cold Stage 3 that satisfies current astrophysical constraints with $M_{\text{max}} \approx 2.01\,M_\odot$ and a slow-hybrid-star branch facilitated by a large energy-density jump. This unified framework bridges microphysical modeling and multimessenger observations, offering a robust tool for exploring SSHS dynamics and potential future refinements such as crust inclusion and color-superconducting phases, while highlighting the current absence of a CEP within the finite-temperature phase structure.

Abstract

The equation of state of dense matter remains a central challenge in astrophysics and high-energy physics, particularly at supra-nuclear densities where exotic degrees of freedom like hyperons or deconfined quarks are expected to appear. Neutron stars provide a unique natural laboratory to probe this regime. In this work, we present EVA--01, a novel equation of state that provides a unified description of dense matter by incorporating both hadron and quark degrees of freedom within a single relativistic mean-field Lagrangian, from which the equation of state is derived at finite temperature. The model extends the density-dependent formalism by introducing a Polyakov-loop-inspired scalar field to dynamically govern the hadron-quark phase transition, following the approach of chiral mean-field models. The resulting model is consistent with a wide range of theoretical and observational constraints, including those from chiral effective field theory, massive pulsars, gravitational-wave events, and NICER data. We analyze its thermodynamic properties by constructing the QCD phase diagram, identifying the deconfinement, chiral, and nuclear liquid-gas transitions. As a first application, we model the evolution of proto-neutron stars using isentropic snapshots and explore the implications of the slow stable hybrid star hypothesis. Our findings establish EVA--01 as a robust and versatile framework for exploring dense matter, bridging the gap between microphysical models and multimessenger astrophysical observations.

Figures (7)

• Figure 1: Phase diagram predicted by our model. The green curve indicates the deconfinement phase transition, given by a first order transition between hadron and quark matter. The blue curve corresponds to the chiral phase transition within the quark sector, as detailed in Fig. \ref{['fig:condensados']}. The light gray region between both curves is discussed in the main text. The small orange curve indicates the nuclear liquid–gas coexistence line in the hadron phase, which terminates at a critical point (orange dot).
• Figure 2: $P$–$\mu_B$ diagram, panels $(a)$ and $(c)$, and light chiral condensate $\langle \bar{u}u + \bar{d}d \rangle$, panels $(b)$ and $(d)$, for the quark phase at two representative temperatures, $T = 30$ MeV and $T = 140$ MeV. In panels $(a)$ and $(c)$, the crossing of two stable branches (green curves) indicates a first-order phase transition. In panels $(b)$ and $(d)$, the sharp drop in the chiral condensate at the same $\mu_B$ value confirms the partial restoration of chiral symmetry. The vertical dashed line marks the transition point in each case.
• Figure 3: Pressure-energy density relationship for the three PNS snapshots: stage $1$ (orange), stage $2$ (red), and stage $3$ (blue, dashed). The shaded regions indicate the theoretical constraints from chiral Effective Field Theory (cEFT) at low densities Drischler:2021lma and perturbative QCD (pQCD) at high densities Kurkela:2009cqmGorda:2018nopAnnala:2020efq. A vertical gray line marks the approximate crust-core boundary at $0.5\, n_0$, and small circles denote the maximum central density reached in stable stellar configurations. Note that the logarithmic scale, necessary to display the wide range of values, obscures many of the differences between the three EoS, particularly in the transition plateaus and throughout the intermediate and high-density regions.
• Figure 4: Particle abundances ($Y_i$) as a function of normalized baryon number density ($n_B/n_0$) for three PNS snapshots: $(a)$ stage $1$, $(b)$ stage $2$, and $(c)$ stage $3$. In each case, the hadron phase is separated from the quark phase by a first-order phase transition, indicated by the hatched region representing the density gap.
• Figure 5: Temperature $(a)$ and squared speed of sound, $(c_s/c)^2$$(b)$, as a function of normalized baryon number density for the three PNS snapshots: stage $1$ (orange line), stage $2$ (red line), and stage $3$ (blue, dashed line). Panel $(a)$ shows the isothermal crusts at low density and the thermal continuity at the phase transition for the hot stages. Panel $(b)$ illustrates the behavior of the speed of sound, including the convergence to the conformal limit ($(c_s/c)^2 = 1/3$) at high densities.