NJL-Chiral Soliton and the Nucleon Equation of State at supra-saturation density: Impact of Chiral Symmetry Restoration

TL;DR

This work addresses the problem of describing dense nuclear matter by identifying the equation of state (EoS) of bulk matter with the nucleon core, modeling nucleons as topological chiral solitons from an underlying Nambu–Jona-Lasinio (NJL) framework and stabilized by vector mesons. Chiral symmetry restoration at high density is implemented dynamically through a density-dependent scalar field that modulates the soliton's scalar and vector channels, enabling a self-consistent link between nucleon structure and supra-saturation EoS. The soliton energy density $slt$ and pressure $lt$ are computed from the meson profiles and then mapped to a bulk EoS by matching a core energy density inside nucleons to the average bulk energy density; a renormalization factor $$ is employed to compare with neutron-star EoSs. The results show quarks delocalizing with increasing density, swelling of the soliton core, and a stiffening of the high-density EoS that approaches standard NS EoSs like SLy4 and QHC18, with the onset of hard deconfinement inferred from core overlap around densities of roughly $(8-12) ho_{ m sat}$ depending on the NJL parameter set. This microscopic, self-consistent treatment highlights the intricate balance between scalar and vector sector changes during chiral restoration and provides a framework to connect nucleon-scale dynamics to macroscopic neutron-star observables.

Abstract

It has been conjectured that, at sufficiently high baryon densities, the equation of state (EoS) of bulk nuclear matter can be identified with that of the nucleon core. In this work, we illustrate how the energy density and pressure distributions inside individual nucleons can be utilized to construct the EoS of supra-dense matter. In our framework, nucleons arise as topological solitons stabilized by vector mesons, which are dynamically generated through the path integral bosonization of an underlying Nambu-Jona-Lasinio (NJL) model. The restoration of chiral symmetry is implemented dynamically via a self-consistent, density-dependent scalar field, which modifies the (isovector) and (isoscalar) channels of the soliton. We analyze the resulting changes in soliton properties for different NJL parameter sets and demonstrate that the progressive restoration of chiral symmetry leads to a stiffening of the soliton-based EoS, making it compatible with existing neutron star EoSs. An EoS constructed from the solutions of the energy-density and pressure profiles at the center of the nucleon is also explored.

Figures (21)

• Figure 1: The upper panel shows the field profiles obtained with the relaxation method (solid lines) compared with those from the shooting method (grey crosses), while the lower panel displays the relative error of the relaxation results with respect to the shooting method. The colour scheme for the different fields is kept consistent across both panels.
• Figure 2: Baryon ($\rho_B(r)$) and isoscalar ($\rho_S(r)$) charge densities, scaled by $4\pi r^2$, as functions of $r$. The radial axis is limited to $r_{\rm max}=1.5$ fm for visibility and comparison with Ref. Fukushima2020. For clarity and comparison with Ref. Fukushima2020, the isoscalar charge density is shown as $\rho_S(r)=-2gF_{\pi}^2\omega(r)$.
• Figure 3: Upper panel shows the scaled (with $4\pi r^2$) energy density $\varepsilon_{\rm slt}(r)$ and pressure $p_{\rm slt}(r)$ as functions of $r$. The lower panel shows the variation of $\varepsilon_{\rm slt}(r)$ and $p_{\rm slt}(r)$ as a function of r. The displayed quantities correspond to the soliton solution with the meson parameters of Ref. Fukushima2020.
• Figure 4: Effective core EoS of the nucleon: pressure $p_{\rm slt,\chi}(r)$ as a function of energy density $\varepsilon_{\rm slt, \chi}(r)$, obtained by eliminating $r$ from the profiles shown in \ref{['fig:ener_pres_vs_r_Fukushima']}. Red circles indicate specific radial positions. For comparison, the solution Gulminelli_2015_SLy4 of the SLy4 nuclear interaction Chabanat:1998 and QHC18 Baym_QHC2018 EoS models are also shown.
• Figure 5: Solid lines in different panels show the meson field profiles for different NJL parameters as noted in the sub-panel. For comparison, the field profiles corresponding to the parameters of Ref. Fukushima2020 are also shown in dashed curves.