Eur. Phys. J. Spec. Top.
https://doi.org/10.1140/epjs/s11734-026-02124-1

Review

Beyond ${\rho }^{2/3}$ scaling: microscopic origins and multimessengers of high-density nuclear symmetry energy

Department of Physics and Astronomy, East Texas A&M University, 75429, Commerce, TX, USA

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Received: 10 October 2025
Accepted: 6 January 2026
Published online: 20 January 2026

Abstract

The nature and Equation of State (EOS) of dense neutron-rich matter are still very poorly known, while they have broad impacts on many interesting issues in both astrophysics and nuclear physics. In particular, the nuclear symmetry energy $E_{\text{sym}}(\rho )$ encoding the cost to make nuclear matter more neutron-rich has been the most uncertain component of the EOS of dense neutron-rich nucleonic matter. It significantly affects the radii, tidal deformations, cooling rates, and frequencies of various oscillation modes of isolated neutron stars as well as the strain amplitude and frequencies of gravitational waves from their mergers, besides its many effects on the structures of nuclei as well as the dynamics and observables of their collisions. Siemens (1970s) observed that $E_{\text{sym}}(\rho )$ scales as ${(\rho /{\rho }_0)}^{2/3}$ near the saturation density ${\rho }_0$ of nuclear matter, since both the kinetic part and the potential contribution (quadratic in momentum) exhibit this dependence. The scaling holds if: (1) the nucleon isoscalar potential is quadratic in momentum, and (2) the isovector interaction is weakly density-dependent. After examining many empirical evidences and understanding theoretical findings in the literature, we conclude that: (1) Siemens’ ${\rho }^{2/3}$ scaling is robust and serves as a valuable benchmark for both nuclear theories and experiments up to $2{\rho }_0$ but breaks down at higher densities, (2) Experimental and theoretical findings about $E_{\text{sym}}(\rho )$ up to $2{\rho }_0$ are broadly consistent, but uncertainties remain large for its curvature $K_{\text{sym}}$ and higher-order parameters, (3) Above $2{\rho }_0$, uncertainties grow due to poorly constrained spin-isospin-dependent tensor and three-body forces as well as the resulting nucleon short-range correlations. Looking forward, combining signals from both observations of neutron stars and terrestrial heavy-ion reaction experiments is the most promising path to finally constraining the high-density $E_{\text{sym}}(\rho )$ and the EOS of supradense neutron-rich matter. Multiple examples of community efforts to further constrain the high-density $E_{\text{sym}}(\rho )$ using both real and mocked data of present and future high-precision observations of neutron stars, as well as heavy-ion collisions involving high-energy rare isotopes, are briefly reviewed.