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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]
==
==
==
==
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