Nuclear Equation of State Constrained from Lab to Star

The 8th International Workshop on Nuclear Dynamics in

Heavy-ion Reactions

August 21^th – 24^th, 2026 Taiyuan

Yifei Niu

Shanghai Jiao Tong University

Nuclear Equation of State (EoS)

2

• Nuclear equation of state

Symmetric

NM

Nuclear matter

(NM)

symmetry energy

Nuclear incompressibility

• EoS: dynamics of heavy-ion collisions, the structure of neutron stars, and the simulation of core-collapse supernova, neutron star mergers

How to constrain EoS of nuclear matter?

B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. 464, 113 (2008)

GMR and Nuclear Incompressibility

3

• Giant Monople Resonance(GMR)

ISGMR strength distribution

Gupta et al., PLB 760, 482 (2016)

• GMR energy v.s. incompressibility

Blaizot et al., NPA 591, 435 (1995)

Garg and Colo, PPNP 101, 55 (2018)

breathing mode

Puzzle: Why are tins so soft?

4

In even-even ^112-124Sn, the ISGMR centroid energy is overestimated by about 1 MeV by the same models which reproduce the ISGMR energy well in ²⁰⁸Pb.

overestimate by 0.5-1 MeV

Li et al., PRL 99, 162503 (2007) Garg and Colo, PPNP 101, 55 (2018)

From Sn isotopes:

Our attempt to solve the puzzle

5

• QPVC effect

​

• Quasiparticle random phase approximation (QRPA)

+ quasiparticle vibration coupling (QPVC)

fully self-consistent based on Skyrme density functional

​

• Pairing effect

2 quasiparticles(qp)

Z. Z. Li., Y. F. Niu, and G. Colo, PRC 110, 064317 (2024)

Unified description of GMR in Ca, Sn and Pb

6

Z. Z. Li, Y. F. Niu, and G. Colo, PRL 131, 082501, 2023

• “Excitation of the isoscalar giant monopole resonance and incompressibility of nuclear matter: resolution of a long-standing puzzle”

​

M. N. Harakeh, Science Bulletin 68, 3081, 2023

Unified description of GMRs

7

Linear correlation of GMR energies between different nuclei

QRPA => QPVC

• Simultaneous description of Sn (or Ca) and Pb is much improved!

Z. Z. Li, Y. F. Niu, and G. Colo, PRL 131, 082501, 2023

Relativistic QPVC approach

8

E. Litvinova, PRC 107, L041302 (2023)

Different relativistic density functionals

9

• Time-dependent relativistic Hartree-Fock approach

For the description of GMR: equivalent to Relativistic RPA based RHF

Further study with QPVC effects + more relativistic EDFs is needed

J. Geng, Z. H. Wang, P. W. Zhao, Y. F. Niu and W. H. Long, PRC 111, 024305 (2025)

Charge-exchange relativistic QPVC model

10

• Development of relativistic QPVC model for charge-exchange excitations

• PCF-PK1: relativistic point-coupling interaction with density-dependent coupling constants
• Successfully applied to β-decay half-lives
• Next: Develop relativistic QPVC for non-charge-exchange excitations with various interactions

​

Nuclear Equation of State (EoS)

11

• Nuclear equation of state

Symmetric

NM

Nuclear matter

(NM)

symmetry energy

symmetry energy slope

• EoS: dynamics of heavy-ion collisions, the structure of neutron stars, and the simulation of core-collapse supernova, neutron star mergers

How to constrain EoS of nuclear matter?

symmetry energy curvature

B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. 464, 113 (2008)

Symmetry energy constrained from Lab

12

• Isovector Giant Dipole Resonances

(IVGDR)

• Constraints on symmetry energy slope

Roca-Maza, et al., PRC 88, 024316 (2013)

symmetry energy acts as the restoring force

• Pygmy Dipole Resonances (PDR)

A. Carbone, et al., PRC 81, 041301(R)(2010)

• Constraints on symmetry energy at subsaturation densities

Z. Zhang, and L. W. Chen, PRC 90, 064317 (2014); PRC 92, 031301(R)(2015)

Tension caused by PREX-II data

13

• Neutron skin thickness

• Correlations:

16 relativistic density functionals

B. T. Reed, et al., PRL 126, 172503 (2021)

• PREX-II experiment:

D. Adhikari, et al., PRL 126, 172502 (2021)

• Tension between PREX-II and others

the “Intersection” region lies outside the 1σ PREX-2 limits

Heavy-ion collisions

Neutron skin of Sn

GDR

Dipole polarizability

Nuclear mass

C. Drischler, et al., PRL 125, 202702 (2020)

Consistency between PREX-II and GW170817

14

• Tidal deformability: independent source for constraining EoS

Neutron star merger event GW170817

B. P. Abbott, et al., PRL 121, 161101 (2018)

• Relativistic energy density functionals (EDFs)

B. T. Reed, et al., PRL 126, 172503 (2021)

• Skyrme EDFs

• Linear correlation
• PREX-II and GW170817: consistent

P. Bano, et al., PRC 108, 015802 (2023)

Why

discrepancy???

SEI family

FSUGold family

15

• 33 Skyrme EDFs and 29 relativistic EDFs (RMF+RHF)

• Relativstic+Skyrme:

almost no correlation

​

• Consistent with previous findings
• Skyrme has much better correlation than relativistic ones

​

Possible reason?

16

• Different K_sym v.s. L correlations in relativistic and Skyrme EDFs

Y. N. Huang, Z. Z. Li and Y. F. Niu, PRC 107, 034319 (2023)

• Relativistic EDFs: weak correlation
• Skyrme EDFs: strong correlation

• Good correlations

J v.s. L

Relativistic:

Skyrme

✓

✓

✓

✘

✓

✓

17

• 2D linear correlation:

PREX-II

GW170817

• PREX-II and GW170817: consistent

• Joint constrained values

Z. Y. Guan, and Y. F. Niu, PRL 135, 172701 (2025) 

Summary and Perspectives

18

Summary

• Nuclear EoS is constrained by observables from Lab to Star
• Nuclear incompressibility of symmetric nuclear matter

Achieve unified description of GMR in Sn and Pb, which solves long-standing puzzle “Why is Sn so soft?”

• Nuclear symmetry energy

Realize reconciliation between neutron skin thickness from PREX-II experiment and neutron-star tidal polarizability from GW170817 event

​

​

Perspectives

• Explore nuclear imcompressibility with more relativistic EDFs
• …

Acknowledgement

Thank you!

19

Collaborators:

​

LZU: Z. Z. Li Y. N. Huang Z. Y. Guan J. Geng Z. H. Wang W. H. Long

PKU: P. W. Zhao J. Meng

Milan Univ. : G. Colo

​

GMR of Ca isotopes studied by QRPA+QPVC

20

• QRPA:
• Much overestimate GMR energies

• QPVC:
• Much improve the strength function

Exp.:

Olorunfunmi et al., PRC 105, 054319 (2022)

QPVC effect on GMR energies

21

Mechanism: Self-energy

22

• QPVC energy: the energy of GMR peak
• Doorway-state energy: the energy of important doorway state 2qp⊗phonon

• Real part of self-energy for ¹²⁰Sn and ²⁰⁸Pb: determines the energy shift

Doorway state energy

Doorway state energy

¹²⁰Sn: QPVC energy > doorway state energy 🡪 larger self-energy

²⁰⁸Pb: QPVC energy < doorway state energy 🡪 smaller self-energy

Mechanism: Role of pairing gap

23

¹²⁰Sn

²⁰⁸Pb

The pairing gap makes the relative energy position of GMR and doorway state different!

Giant Monople Resonances

24

• Isoscalar giant monople resonance（breathing mode）

window into the incompressibility of nuclear matter

​

Attempts to solve the puzzle

25

• Mutually enhanced magicity (MEM) effect

Khan et al., PRC 78, 064304 (2008)

• Functionals designed to describe masses of open-shell nuclei cannot predict the masses of doubly magic nuclei, which are systematically more bound than predicted.

• ISGMR energies are practically identical in the three isotopes
• ruling out any consequences of the MEM effect in nuclear incompressibility

Lunney et al., RMP 75, 1021 (2003)

Patel et al., PLB 726, 178 (2013)

Leaving the puzzle unsolved…

26

Puzzle: Why are tins so soft?

27

In even-even ^112-124Sn, the ISGMR centroid energy is overestimated by about 1 MeV by the same models which reproduce the ISGMR energy well in ²⁰⁸Pb.

Focus issue on open problems in nuclear structure theory

​

Li et al., PRL 99, 162503 (2007)

Garg and Colo, PPNP 101, 55 (2018)

J. Piekarewicz, JPG 37, 064038 (2010)

Giant Monople Resonances (GMR)

28

• Isoscalar giant monople resonance（breathing mode）

• multipole-decomposition analysis (MDA)

• ISGMR strength distribution

Gupta et al., PLB 760, 482 (2016)

Symmetry energy constrained from Lab

29

• Pygmy Dipole Resonances (PDR)

• Neutron skin thickness

A. Carbone, G.Colo, A. Bracco, et al., PRC 81, 041301(R)(2010)

Roca-Maza, X. Vinas, X. Roca-Maza, et al., PRC 80, 024316 (2009)

Skyrme, Gogny, and RMF interactions

Symmetry energy constrained from Lab

30

• Neutron skin thickness

Roca-Maza, X. Vinas, X. Roca-Maza, et al., PRC 80, 024316 (2009)

Skyrme, Gogny, and RMF interactions

B. T. Reed, et al., PRL 126, 172503 (2021)

16 relativistic density functionals

Symmetry energy constrained from Lab

31

• Pygmy Dipole Resonances (PDR)

• Neutron skin thickness

A. Carbone, et al., PRC 81, 041301(R)(2010)

• Correlations:

16 relativistic density functionals

B. T. Reed, et al., PRL 126, 172503 (2021)

• PREX-II experiment:

D. Adhikari, et al., PRL 126, 172502 (2021)

Heavy-ion collisions

Radii from antiprotonic atoms

Constraints on symmetry energy

32

Constraints on symmetry energy

33

Symmetry energy constrained from Lab

34

• Pygmy Dipole Resonances (PDR)

a significant contributor to the dipole polarizability

• Constraints on symmetry energy slope

A. Carbone, et al., PRC 81, 041301(R)(2010)

Heavy-ion collisions

Radii from antiprotonic atoms

Z. Z. Li, Y. F. Niu, and W. H. Long, PRC 103, 064301(2021)

Symmetry energy constrained from Star

35

N. B. Zhang, B. A. Li, and J. Xu, ApJ 859: 90 (2018)

• Neutron star observables:

maxium neutron star mass, neutron star

radius, tidal deformability

Tidal deformability

GW170817

PREX-II data and astrophysical observables

36

• nonparametric EoS based on Gaussian processes

• constrain symmetry energy and neutron skin of ²⁰⁸Pb from observations of neutron stars with minimal modeling assumption

tension

cannot resolve any tension due to large uncertainties

Maxium mass of NS

Tilda deformability GW170817

Mass and radius of PSR J0030+0451