Nuclear Symmetry Energy from Neutron Stars and Heavy-Ions
Dmitry Ofengeim (Hebrew University), Peter Shternin (Ioffe Institute) �E.E. Kolomeitsev (BLTP, JINR Dubna)
Nuclear matter
Energy density of the infinite nuclear matter as function of the proton and neutron densities:
Binding energy per nucleon:
3) Impose isospin symmetry
Symmetry energy:
Energy of symmetric nucler matter (SNM):
Two definitions of the symmetry energy: (1)
(2)
local (Yp~1/2)
NS applications
If the derivative is very small, then both definitions are equivalent
Infinite nuclear matter. Symmetry energy
where
total density,
proton fraction
kinetic part of the symmetry energy
Contributions to the symmetry energy
nucleon interaction
one-boson exchange picture:
rho-meson exchange (isovector vector meson)
nn & pp – repulsion
np -- attraction
In RMF models the symmetry energy is positive!
symmetry energy
Equation of state of nuclear matter around n0
n0 vs ε0 –Coester line problem: role of TNF, �relativistic effects, chiral forces
frequently characterized by the compressibility modulus K
Giant Monopole Resonance (GMR)
nuclear matter parameters
saturation density n0 and binding energy ε0
Giant Dipole Resosnace�[Trippa et al., PRC 77, 061304 (2008)]
Isobaric analog states:
[Danielewicz et al. NPA 958, 147 (2017)]
Pb dipole polarizability: �[Roca-Maza et al., PRC 88, 024316 (2013)]
Sn neutron skin:
[Chen et al., PRC 82, 024321 (2010)]
Behind all calculation are particular models for NN interactions and many-body techniques
Manifestations of the symmetry energy
nuclear structure data
Manifestations of the symmetry energy
Isospin defusion in HIC:
[Tsang et al., PRL 102, 122701 (2009)
Famiano et al., PRL 97, 052701 (2006)]
Y(π--)/Y(π+) ratio in collisions of tin isotopes in SπRIT exp. @ 270 AMeV
[G. Jhang et al., PLB 813, 136016 (2021)]
Elliptic flow ration
FOPI-LAND exp Au+Au @ 400AMeV
[P. Russotto et al., PRC 94, 034608 (2016)]
heavy-ion collisions
Neutron and Proton Transverse Emission Ratio Measurements
@ 50 AMeV
γ=0.5
γ=1.5
[Cozma, Trautmann, IJMPE 34 (2025) 2530001]
Extrapolation of the symmetry energy
negative above some density
increases faster than linear
[Cai, BA. Li, EPJA 61, 55 (2025)]
chemical potentials:
Neutron star matter. Composition
free Fermi gas of electron and muons
Energy-density functional for n+p+e+μ matter
Neutron stars are in the beta-equilibrium
and electrically neutral
(NS is transparent for neutrinos)
Solution of these 2 equations
Energy-density of neutron star matter
Neutron star. Cooling
neutron star is transparent for neutrino
neutrino cooling
CV – heat capacity, L - luminosity
one-nucleon reactions:
direct URCA (DU)
volume neutrino radiation
Cold degenerated fermion system
two-nucleon reactions:
modified URCA (MU)
Kinematic condition
threshold for
DU reactions are much more efficient. If they occur the neutron star will be too cold!
The occurrence of DU reaction in NS depends on how fast the symmetry energy grows with the density
Questions
Can we get constraints on the symmetry energy at high densities?
Can the symmetry energy become negative above some density?
Instability wrp. proton-neutron separation;
At small Yp, proton immobilization and formation of a proton crystal
M. Kutschera, W. Wojcik, PRC 47 (1993) 1077
M. Kutschera, PLB 340 (1994) 1
Can the DU reaction occur in neutron stars and at what star masses?
Neutron star observations
Recent progress
Ofengeim, Kolomeitsev, Shternin, Neutron star equation of state. In Encyclopedia of Astrophysics; 2026; Vol. 3, pp. 160–204
Danielewicz+’02
Rutherford+ (NICER group), ApJ, 2024
Combining Astro and HIC data
Similar idea, different methods
B.-A. Li, W.-J. Xie, PLB 2020
Tsang et al, Nat. Astro 8, 2025
NS data
+ pQCD limit
HIC data
log-linear extrapolation
The data we use
Reconstruction energy density vs n
Input
max mass
The data we use
Astro
HIC flow
Inverse β-equilibrium problem
Construct the difference
Astro
HIC
known
algebraic function of εsym and n
n+p+e+μ matter
empirical inputs
The “difference function” cannot be arbitrary
Consider this quantity in the limit
Selection criterion for εβ and εSNM
Х
Х
Х
Selection criterion for εβ and εSNM
✓
Х
forbidden
Results for the Symmetry Energy
Symmetry Energy at Low Densities
DU or not DU? That is the Question!
83.9%
With the probability 83.9% there exist neutron stars in which DU reactions are allowed!
BSk24
BHF
APR
SLy4
Conclusions
Oliinychenko et al. Phys. Rev. C 108, 3, 034908 (2023)
P. Danielewicz et al, Science 298,1592–1596 (2002)