Collaborators: Bao-Jun Cai, Lie-Wen Chen, Chang Xu, Jun Xu, Zhi-Gang Xiao and Gao-Chan Yong

Bao-An Li

Probing High-Density Symmetry Energy with Heavy-Ion Reactions

Outline• What is symmetry energy? • Why is it so uncertain especially at high densities ?• How to probe high-density symmetry energy with heav y-ion reactions?

EOS of cold, neutron-rich nucleonic matter

18 18

12

12

12

3

0 )) (, (( ) sn ymp

nn

p pE E E ρρρ ρ

ρ ρ δρ

ρ ο 42

= + + ( )

−=

( , )n pE ρ ρ

symmetry energy

density

Energy per nucleon in symmetric matter

Energy in asymmetric nucleonic matter

δIsospin asymmetry

Esym

(ρ) = 12

∂2E∂δ 2

≈ E(ρ)pure neutron matter

− E(ρ)symmetric nuclear matter

Characterization of symmetry energy near normal density

The physical importance of L

In npe matter in the simplest model of neutron stars at ϐ-equilibrium

In pure neutron matter at saturation density of nuclear matter

Many astrophysical observables, e.g., radii, core-crust transition density, cooling rate, oscillation frequencies and damping rate of isolated neutron starsas well as gravitational waves from deformed pulsars and/or neutron star binariesare sensitive to the density dependence of nuclear symmetry energy.

Constraints on E sym (ρ0) and L based on 29 analyses of data

Bao-An Li and Xiao Han, Phys. Lett. B727, 276 (2013).

L≈ 2 Esym (ρ0)=59±16 MeV

Esym (ρ0)≈31.6±2.66 MeVFiducial values as of Aug. 2013

L=2 Esym (ρ0) if E sym=Esym (ρ0)(ρ/ρ0)2/3

Fiducial values as of Oct. 12, 2016

Constraints on E sym (ρ0) and L based on 53 analyses of data

M. Oertel, M. Hempel, T. Klähn, S. Typel arXiv:1610. 03361 [astro-ph.HE]

Assuming !(1) Gaussian Distribution of L(2) Democratic principle

(i.e., trust everyone)

L≈ 2 Esym (ρ0)

M. B. Tsang et al., Phys. Rev. C86, 015803 (2012)

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What is the high -density symmetry energy?

Xiao et al

Xiao et al

P. Russotto et al. (ASY-EOS Collaboration), Phys. Rev. C94, 034608 (2016).

Z.G. Xiao et al, Phys. Rev. Lett. 102, 062502 (2009 ).

Predictions using energy density functionalsand/or microscopic many-body theories

The next talk byWolfgang Trautmann

Can the symmetry energy become negative at high densities?Yes, it happens when the tensor force due to ρ exchange in the T=0 channel dominatesAt high densities, the energy of pure neutron matter can be lower than symmetric matter leading to negative symmetry energy

Example: proton fractions with interactions/modelsleading to negative symmetry energy

3 30 0(0.048[ / ( )] ( / )() 1 2 )symsymE Ex xρ ρ ρ ρ −=

M. Kutschera et al., Acta Physica Polonica B37 (2006)

3-body force effects in Gogny or Skyrme HF

Potential part of the symmetry energy

Both tensor force and/or 3-bodyforce can make E sym negative at high densities

Affecting the threshold of hyperon formation, kaon condensation, etc

Neutron stars as a natural testing ground of fundamental forces

Strong force

weakE&M

Connecting Quarks with the Cosmos: Eleven Science Q uestions for the New Century, Committee on the Physics of the Universe, National Research Council

• What is the dark matter? • What is the nature of the dark energy? • How did the universe begin? • What is gravity? Does Einstein’s GR have a limit?• Are there additional spacetime dimensions? • What are the masses of the neutrinos, and how have

they shaped the evolution of the universe? • How do cosmic accelerators work and what are they

accelerating? • Are protons unstable? • Are there new states of matter at exceedingly high

density and temperature? • How were the elements from iron to uranium made? • Is a new theory of matter and light needed at the

highest energies?

Contents and stiffness of the EOS of super-dense ma tter?

For high-density neutron-rich nucleonic matter, the most uncertain part of the EOS is the nuclear symmetry e nergy

Strong-field gravity: GR or Modified Gravity?

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Massive neutron stars

Gravity -EOS Degeneracy in massive neutron stars

GR+[Modified Gravity]

Matter+[Dark Matter]+[Dark Energy]

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Action S=S gravity +Smatter

Why is the symmetry energy so uncertain?

(Besides the different many-body approaches used)• Isospin-dependence of short-range neutron-proton correlation due to the tensor force

• Spin-isospin dependence of the 3-body force

• Isospin dependence of pairing and clustering at low densities

(Valid only at the mean-field level)

Correlation functions

Within a simple interacting Fermi gas model

Keith A. Brueckner, Sidney A. Coon, and Janusz Dabrowski, Phys. Rev. 168, 1184 (1968)

Un/ p (k,ρ,δ) =U0(k,ρ) ±U sym1(k,ρ)�δ +U sym2(k,ρ)�δ

2 +ο(δ3)

Vnp(T0) ? Vnp (T1)fT0 ? fT1 , the tensor force in T0 channel makes them different

At saturation densityParis potential

2

pure neutron matter symmetric nuclear matter2

1( ) ( ) ( )

2symE

E E Eρ ρ ρδ

∂ = ≈ −∂

I. Bombaci and U. Lombardo PRC 44, 1892 (1991)PRC68, 064307 (2003)

Symmetry energy

Dominance of the isosinglet (T=0) interaction

BHF

Self-consistent Green’s function

Uncertainty of the tensor force at short distanceTakaharu Otsuka et al., PRL 95, 232502 (2005); PRL 97, 162501 (2006)

Cut-off=0.7 fm fornuclear structure studies

Gogny

What are Short Range Correlations (SRC) in nuclei ? (Eli Piasetzky)

1.7f

SRC ~RN LRC ~RA

, F> K1K

F> K2K

1K

2K

≤1.f

Nucleons

2N-SRC

1.7f

ρo = 0.16 GeV/fm3

~1 fm

1.8 fm

In momentum space:

large relative A pair with between the nucleons momentum

.small CM momentumand

kF ~ 250 MeV/cHigh momentum tail: 300-600 MeV/c 1.5 KF - 3 KF

Tensor force induced (1) high-momentum tail in nucleon

momentum distribution and (2) isospin dependence of SRC

Theory of Nuclear matterH.A. BetheAnn. Rev. Nucl. Part. Sci., 21, 93-244 (1971)

Fermi Sphere

Phenomenological nucleon momentum distribution n(k) including SRC effects guided by microscopic theories and experime ntal findings The n(k) is not directly measurable, but some of it s features are observable.

All parameters are fixed by (1) Jlab data: HMT in SNM=25%, 1.5% in PNM, (2) Contact C for SNM from deuteron wavefunction(3) Contact C in PNM from microscopic theories

All parameters are assumed to have a lineardependence on isospin asymmetry as indicated by SCGF and BHF calculations

B.J. Cai and B.A. Li, PRC92, 011601(R) (2015); PRC93, 014619 (2016).

The high-momentum tail in deuteron scales as 1/K 4O. Hen, L. B. Weinstein, E. Piasetzky, G. A. Miller, M. M. Sargsian and Y. Sagi, PRC92, 045205 (2015).

Stewart et al. PRL 104, 235301 (2010)Kuhnle et al. PRL 105, 070402 (2010)

Ultracold6Li and 40K

=k/kF

VMB calculations

K’=K/KF

VMB calculations

Ultracold6Li and 40K

0.00 0.05 0.10 0.15 0.200

5

10

15

20

25

30

0.00 0.01 0.020

3

6

E PN

M(

) (M

eV)

(fm-3)

Eq. (6), =0.4 0.1 Tews, et al. Gezerlis, et al. Schwenk and Pethick Epelbaum, et al. Gezerlis, et al.

Free F

ermi Ga

s

EOS and contact C of pure neutron matter

The contact C of PNM is derived from its EOSUsing Tan’s adiabatic sweep theorem

S. Tan, Annals of Physics 323 (2008) 2971-2986

density

B.J. Cai and B.A. Li, PRC92, 011601(R) (2015).

n(k)=C/K 4

Modified Gogny Hartree-Fock energy density function al incorporating SRC- induced high momentum tail in the single nucleon momentum d istribution

Kinetic Two-body force Three-body force

Momentum-dependent potential energy due to finite- range interaction

Bao-Jun Cai and Bao-An Li (2016)

SRC-induced HMT in the single-nucleon momentum distribution affects both the kinetic energy and the momentum-dependent part of the potential energy

Readjusting model parameters to reproduce the same saturation properties of nuclear matter as well as E sym (ρ0)=31.6 MeV and L(ρ0)=58.9 MeV

Consequence: Symmetry energy gets softened at both low and high densities

The isospin-dependent part of the incompressibility at ρ0 agrees better with data

Data:

MDI+HMT-exp: -492MDI+FFG: -370

Probing the symmetry energy at high densities • π -/π + in heavy-ion collisions•Neutron-proton differential flow & n/p ratio in heavy-ion coll.•Neutrino flux of supernova explosions•Strength and frequency of gravitational waves

*2 The isovector potential for Delta resonance is c ompletely unknown

NN�� NΔ

N+π

Where does the E sym information get in, get out or get lost in pion pro duction?

*1 Isospin fractionation, i.e., the nn/pp ratio at high density is determined by the Esym(ρ)

*5.Pion mean-field (dispersion relation), S and/or P waveand their isospin dependence are poorly known, existingstudies are inconclusive.

How does the pion mean-field affect the Delta production threshold?

Other final state interactions?

*1.44 1013 G

Formation of neutron-star matter in terrestrial nuclear laboratoriesLi Ou and Bao-An Li, Physical Review C84, 064605 (2011)

Highly-magnetized, high-density, isospin-asymmetric matter formed in heavy-ion reactions

Isospin fractionation in heavy-ion reactions

low (high) density region is more neutron-rich with stiff (soft) symmetry energy

2( , ) ( ,0) ( )symE E Eρ δ ρ ρ δ= +

Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701

Pion ratio probe of symmetry energy

at supra-normal densities

+π 0π −π

nn 0 1 5 a) Δ(1232) resonance model pp 5 1 0 in first chance NN scatterings: np(pn) 1 4 1 (negelect rescattering and reabsorption)

+−

ππ 2

2

2

)(55

ZN

NZZ

NZN ≈++=

R. Stock, Phys. Rep. 135 (1986) 259.

b) Thermal model: (G.F. Bertsch, Nature 283 (1980) 281; A. Bonasera and G.F. Bertsch, PLB195 (1987) 521)

exp[2( ) / ]n p kTπ µ µπ

−

+ ∝ −

H.R. Jaqaman, A.Z. Mekjian and L. Zamick, PRC ( 1983) 2782.

c) Transport models (more realistic approach): Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701, and several papers by others

31 1( ) {ln ( ) ( )}

2

m mn p mnn p asy asy Coul m T n p

mp

mV V V kT b

m

ρµ µ δρ ρ ρλ

+− = − − + + −∑

GCCoefficients2

Probing the symmetry energy at supra-saturation den sities

Stiff Esym

density

Symmetry energy

n/p ratio at supra-normal densities

Central density

π-/ π+ probe of dense matter

2( , ) ( , 0 ) ( )symE E Eρ δ ρ ρ δ= +

n/p ?

Circumstantial Evidence for a Super-soft Symmetry Energy at Supra-saturation Densities

Z.G. Xiao, B.A. Li, L.W. Chen, G.C. Yong and M. Zha ng, Phys. Rev. Lett. 102 (2009) 062502

A super-soft nuclear symmetry energy is favored by the FOPI data!!!

W. Reisdorf et al. NPA781 (2007) 459 Data:

Calculations: IQMD and IBUU04

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What is the high -density symmetry energy?

Summary

• The study on nuclear symmetry energy and its

astrophysical impacts is Exciting!

• Symmetry energy at supra-saturation densities is

the most uncertain part of the EOS of dense

neutron-rich nucleonic matter

• Symmetry energy has significant effects on

terrestrial experiments and astrophysical

observables