1. The Density Dependence of the Symmetry Energy
from Heavy Ion Collisions
Hermann Wolter
Ludwig-Maximilians-Universität München (LMU)
collaborators:
Massimo Di Toro, Maria Colonna, V. Greco, Lab. Naz. del Sud, Catania
Theodoros Gaitanos, Univ. of Giessen
Vaia Prassa, Georgios Lalazissis, Univ. Thessaloniki
Malgorzata Zielinska-Pfabe, Smith Coll., Mass., USA
50th Winter Meeting in Nuclear Physics, Bormio, Jan , 2012

2. Points to discuss
Density (and momentum) dependence of the nuclear Symmetry Energy (SE); uncertainties, importance, astrophysics
II. Situation from many-body theory
III. Study of Symmetry Energy in heavy ion collisions,
choice of asymmetry and density regime.
transport theory
IV. Discuss specific examples:
< r0, isospin transport, Fermi energies
> r0 momentum distributions („flow“), meson production

3. Equation-of-State and Symmetry Energy
BW mass formula
density-asymmetry dep. of nucl.matt.
neutron matter EOS
Symmetry energy:
Diff. between neutron and symm matter,
rather unknown,
e.g. Skyrme-like param.,B.A.Li
asy-stiff
asy-soft
rB/r0
stiff
soft
saturation point
Fairly well fixed! Soft!
EOS of symmetric nuclear matter
density r
asymmetry d
Esympot(r) often parametrized as
useful around r=r0. Expansion around r0 :

4. SE The Nuclear Symmetry Energy in different „realistic“ models
stiff
soft
Rel, Brueckner
Nonrel. Brueckner
Variational
Rel. Mean field
Chiral perturb.
The EOS of symmetric and pure neutron matter in different many-body approaches
C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book)
The symmetry energy as the difference between symmetric and neutron matter: SE
asy-soft at r0
(asystiff very similar)
Different proton/neutron effective masses
Isovector (Lane) potential: momentum dependence
SE ist also momentum dependen t  effective mass
asy-stiff
asy-soft
Why is symmetry energy so uncertain??
->In-medium r mass, and short range tensor correlations (B.A. Li, PRC81 (2010));
-> effective mass scaling (Dong, Kuo, Machleidt, arxiv )

5. Importance of Nuclear Symmetry Energy
supernovae,neutron stars)
heavy ion collisions in the Fermi energy regime
Isospin Transport properties, (Multi-)fragmentation
(diffusion, fractionation, migration)
Esym (rB) (MeV)
rB/r0 1 2 3
Asy-stiff
Asy-soft
ASYEOS Collaboration
p, n
rel. Heavy ion collisions
Nuclear structure
(neutron skin thickness, Pygmy DR, IAS)
Slope of Symm Energy

6. Astrophysics: Supernovae and neutron stars
A normal NS (n,p,e)
or exotic NS? r0
PSR J
typical
neutron stars
heaviest neutron star
Onset of direct URCA: Yp>.11
Supernova evolution: range of densities and temperatures and asymmetries:

7. Determination of nuclear SE in heavy ion collisions  Transport theory
Z/N, asymmetry degree of freedom 1
neutron stars
Supernovae IIa
exotic
Deconfinement for asymmetric matter earlier?
Determination of nuclear SE in heavy ion collisions
 Transport theory
equation of motion in semiclass. approx: BUU equation
isospin dependent, pp,nn,pn
In-medium! 1 2 3 4
isoscalar and isovector (~10%)
EOS
2-body collisions
loss term
gain term
Approximation to a much more complicated non-equilibrium quantum transport equation (Kadanoff-Baym) by neglecting finite width of particles
Coupled transport eqs. for neutrons and protons
Isovector effects are small relative to isoscalar quantities
Relativistic transport equations: Walecka-type (RMF) models
Inelastic collisions (particle production; mesons p,K)

8. Reaction Mechanisms in Heavy Ion Collisions
Coulomb barrier to Fermi energies
Relativistic energies
peripheral
central
Isospin migration
Isospin fractionation,
multifragm
deep-inelastic
pre-equil. dipole
pre-equil. light particles
N/Z of PLF residue
= isospin diffusion
Production of particles, p,K
Flow: directed and elliptic
N/Z ratio of IMF‘s
N/Z of neck fragment and velocity correlations

9. Central collisions at Fermi energies: Ratios of emitted pre-equilibrium particles
Early emitted neutrons and protons reflect difference in potentials in expanded source, esp. ratio Y(n)/Y(p).
more emission for asy-soft, since symm potential higher
protons vs. neutrons
124Sn + 124Sn
112Sn + 112Sn
asy-soft
asy-stiff
124Sn + 124Sn
112Sn + 112Sn
„Double Ratios“
softer symmetry energy closer to data
SMF simulations, M.Pfabe IWM09
Data: Famiano, et al., PRL 06
asy-soft
asy-stiff
qualitatively as seen in ImQMD, but quantitatively weaker dependence on SymEn
Y. Zhang, et al., arXiv (2011)

10. Check momentum dependence of SE:
136,124Xe+124,112Sn, E = 32,…,150 AMeV data R. Bougault (Ganil, prelim, IWM11)
son: asysoft, mn*>mp*
stn: asystiff, „
sop: asysoft, mn*<mp*
stp: asystiff, „
Comparison of yield ratios 136Xe+124Sn, central
Prelim. data , 32 AMeV, INDRA
favors a asy-stiff with mn*<mp*
mn*<mp*
soft
n/p ratio
stiff
mn*>mp*
E=65 AMeV
E=100 AMeV
E=32 AMeV
Etrans/A [MeV]
t/3He ratio
Etrans/A [MeV]
1.5
E=32 AMeV
The single t/3He ratios seem to be a promissing observable, double ratios under study

11. Centelles, et al., PRL 102 (2009): Catania results, limits approx.
N/Z of IMF, double ratio
Catania results, limits approx.

12. r+d r+d r r Heavy Ion Collisions at Relativistic Energies: “Flow“
Fourier analysis of momentum tensor : „flow“
Global momentum space
v2: elliptic flow
v1: sideward flow
132Sn + 132Sn @ 1.5 AGeV b=6fm
r+d, stiff
dynamical boosting of vector contribution
T. Gaitanos, M. Di Toro, et al., PLB562(2003) r n p
differential elliptic flow
Proton-neutron differential flow
Elliptic flow more sensitive, since particles emitted perpendicular
r+d r
r+d r
ASYEOS Experiment
See talk by W. Trautmann

13. Particle production as probe of symmetry energy
Difference in neutron and proton potentials
p, n
„direct effects“: difference in proton and neutron (or light cluster) emission and momentum distribution
„secondary effects“: production of particles, isospin partners p-,+, K0,+
NN ND Np LK
D in-medium self-energies and width
p potential ,
NLK
in-medium in-elastic s ,
K and L potential (in-medium mass)
1. Mean field effect: Usym more repulsive for neutrons, and more for asystiff
 pre-equilibrium emission of neutron, reduction of asymmetry of residue
2. Threshold effect, in medium effective masses:
 m*N,, m*D, contribution of symmetry energy; m*K, models for K-potentials
G.Ferini et al.,PRL 97 (2006)

14. Pion ratios in comparison to FOPI data (W. Reisdorf et al
Pion ratios in comparison to FOPI data (W.Reisdorf et al. NPA781 (2007) 459)
MDI, x=0, mod. soft Xiao,.. B.A.Li, PRL 102 (09)
MDI, x=1, very soft
NLd, stiff Ferini, Gaitanos,.. NPA 762 (05)
NL, soft, no Esym
g=2, stiff Feng,… PLB 683 (10)
SIII, very soft
FOPI, exp
Contradictory results of different calculations;
Reason: Treatment of D dynamics?
Kaons better probe? , V. Prassa, et al., NPA 832 (2010)

15. Dynamics of particle production (D,p,K) in heavy ion collisions
stiff Esym
soft Esym
Central density
D and K: production in high density phase
Pions: low and high density phase
Sensitivity to asy-stiffness
NLrd
NLr NL
p and D multiplicity
Dependence of ratios on asy-stiffness
n/p
D0,-/D+,++
 p-/p+, K0/K+
 n/p ratio governs particle ratios
K0,+ multiplicity
 time [fm/c]
G.Ferini et al.,PRL 97 (2006)

16. Present constraints on the symmetry energy
p+/p- ratio,
Feng, et al. (ImIQMD)
Moving towards a better determination of the symmetry energy
Large uncertainties at higher density
Conflicting theore-tical conclusions for pion observables.
Work in exp. and theory necessary!
Au+Au, 400AMeV, FOPI
Fermi Energy HIC, MSU
p+/p- ratio
B.A. Li, et al.

17. Summary and Outlook Thank you
While the EOS of symmetric NM is now fairly well determined, the density (and momentum) dependence of the Symmetry Energy is still rather uncertain, but important for exotic nuclei, neutron stars and supernovae.
Constraints come from neutron star observables and from HIC both at sub-saturation (Fermi energy regime) and supra-saturation densities (relativistic collisions).
At subsaturation densities the constraints become increasingly stringent (g~1),
but constraints are largely lacking at supra-saturation densities.
Observables for the suprasaturation symmetry energy
N/Z of pre-equilibrium light clusters,
difference flows, (first hints -> ASYEOS)
part. production rations p-/p+ , K0/K+ (FOPI,HADES)
More work to do in exp. (more data)
and theory (consistency of transport codes, p,D dynamics)
Nuclear symmetry energy: an interesting field that connects
areas of nuclear structure, reactions and astrophysics
Thank you

18. Neutron star masses and cooling and the symmetry energy
Proton fraction and direct URCA
Onset of direct URCA
Forbidden by Direct URCA constraint
Tolman-Oppenheimer-Volkov equation to determine mass of neutron star
PSR-J1614_2230
Typical neutron stars
Klähn, Blaschke, Typel, Faessler, Fuchs, Gaitanos,Gregorian, Trümper, Weber, Wolter, Phys. Rev. C74 (2006)

19. stiff stiff stiff soft soft soft
Careful study of consistency between nuclear structure and astrophysics:
Than, Khoa, Van Gai, PRC80 (2009); Loan, Tan, Khoa, Margeron, PRC83 (2011)
Two classes of eff. NN interactions: M3Y-Pn, Gogny-D1S, asy-soft very good fit to nuclear struct
CDM3Yn, SLy5, asy-stiff, fit to nuclei struct not so good
Mass-Radius relationship for NS, compared to evaluations by Özel. et al (left) and Steiner et al. (right)
Symmetry energy
Flow constraints from HIC (Danielewicz, et al.,)
stiff
stiff
stiff
soft
soft
soft
EOS‘s that fit better nuclear structure have difficulties with high density behaviour of SE and with neutron star observables

20. around normal density: Structure
Expansion of SE around r0:
Symmetry pressure
strong linear correlation between neutron skin thickness and SE slope L~p0 (or also a4)
strong correlation between Dipole strength of Pygmy resonance and SE slope L
 possibility to determine slope of SE, but also neutron skin thickness
PDR
neutron skin of Pb
as ~ slope L (MeV)

21. Transport approaches
Transport theory describes the non-equilibrium aspects of the temporal evolution of a collision. The central quantitiy is the phase space density (coordinate and momentum distribution).
Demonstrate two aspects:
Evolution in coordinate space:
 movies curtesy T. Gaitanos, T.Chossy
2. Evolution in momentum space
non-equilibrium,
non-sphericity of local momentum distributions
time (fm/c)
density (fm-3)
 Transport theory!!!

22. I.8 Importance of Symmetry energy: neutron stars
dependence of maximum neutron star mass on symmetry energy.
However, other constraints, e.g. coling due to URCA process
Klähn, Blaschke, Typel, Faessler, Fuchs, Gaitanos,Gregorian, Trümper, Weber, Wolter, Phys. Rev. C74 (2006)
Onset of direct URCA
Typical neutron stars
Heaviest observed neutron star
Forbidden by Direct URCA constraint

23. Neutron Stars: a Laboratory for the High-Density Equation-of-State
A normal NS (n,p,e)
or exotic NS?
PSR J
typical
neutron stars
Onset of direct URCA
(fast cooling)
core
crust
Composition of NS matter
Klähn, Blaschke, Typel, Faessler, Fuchs, Gaitanos,Gregorian, Trümper, Weber, Wolter, Phys. Rev. C74 (2006)

24. The Nuclear Symmetry Energy in different „realistic“ models
stiff
soft
Rel, Brueckner
Nonrel. Brueckner
Variational
Rel. Mean field
Chiral perturb.
The EOS of symmetric and pure neutron matter in different many-body approaches
C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book)
The symmetry energy as the difference between symmetric and neutron matter:
asy-stiff
asy-soft
Esym(r)
BHF
Skyrme
RMF
many more in: B.A. Li et al., Phys. Rep. 464 (2008)

25. Momentum Dependence of the Symmetry Energy
asy-soft at r0
(asystiff very similar)
Different proton/neutron effective masses
-> crossing around Fermi momentum!
data
m*n < m*p
m*n > m*p
Isovector (Lane) potential: momentum dependence
Why is symmetry energy so uncertain??
->In-medium r mass, and short range tensor correlations (B.A. Li, PRC81 (2010));
-> effective mass scaling (Dong, Kuo, Machleidt, arxiv )

26. Astrophysical Connection
Ranges of density, temperature, asymmetry in SN event
Liebendörfer, et al., NIX XI, 2010
Composition of the matter
collapse
post-bounce
density: log10n [g/cm3]
entropy: log10s[kB/baryon]
not uniform, but light and heavy clusters (nuclei embedded in the medium)
[g/cm3]
Hillebrandt, et al., Scient. Americ. 2006
Symmetry energy plays an essential role in astrophysics

27. Check momentum dependence of SE in detail:
136,124Xe+124,112Sn, E = 32,…,150 AMeV data R. Bougault (Ganil, prelim, IWM11)
Xe+Sn, neutron-rich
neutron poor
E=32 AMeV, central
Protons, neutron-rich
neutron-poor
asy-soft at r0
(asystiff very similar)
Protons, neutron-rich
data neutron-poor
Mass3 data: neutron-rich t He
neutron-poor t He
Mass3 calc: neutron-rich t He
Effect of asystifffnes as large as m*- effect

28. Comparison of SE determinations at lower energy
From isospin diffusion and p/n emission ratios
isospin diffusion, Ri(b), peripheral
N/Z of IMF, central
pre-equ p/n double ratio, central
N/Z of evap. LCP from PLF, peripheral
asy-stiff
asy-soft
Note: for Catania results: Limits approximate!
B. Tsang, et al., PRL 102, (2009)
Current state:
Generally consistent with each other , but still uncertainties. More work necessary, also on consistency of codes

29. Differential elliptic flow
Au+Au, 400 AMeV
t-3He pair similar but weaker
Au+Au, 600 AMeV
asystiff
asysoft
m*n>m*p
Inversion of elliptic flows because of inversion of n,p-potentials with effective mass
m*n<m*p
V.Giordano, M.Colonna et al., arXiv , PRC81(2010)
W. Reisdorf,
Experimental indication of splitting of ell- flow with effective mass
Elliptic flow more sensitive, since determined by particles that are emitted perp to the beam direction

30. ASY-EOS: Hunting the high density symmetry energy with v2
Russotto, et al., PLB 697, 471 (2011)
FP g=0.98(35)
inversion of neutron
and hydrogen flows
Indication of an asy-stiff symmetry energy at suprasaturation density
SIS experiment S394
First beamtime May 2011

31. r+d r Heavy Ion Collisions at Relativistic Energies: “Flow“ p n r+d r
Fourier analysis of momentum tensor : „flow“
Global momentum space
v2: elliptic flow
v1: sideward flow
132Sn + 132Sn @ 1.5 AGeV b=6fm p n
r+d r
for 124Sn “asymmetry” I = 0.2
neutron
proton
Asy-stiff
Asy-soft
Proton/neutron potentials
Proton-neutron differential flow
r+d r
dynamical boosting of vector contribution
T. Gaitanos, M. Di Toro, et al., PLB562(2003)
and analogously for elliptic flow

32. In-medium Klein-Gordon eq. for Kaon propagation:
Two models for medium effects tested:
One-Boson Exchange (Schaffner-Bielich et al.)
Chiral perturbation (Kaplan, Nelson et al.)
ChPT
OBE
Splitting for K0,+
In-medium K energy (k=0)
Isospin-dependence
DPG08_kaon 5

33. Kaon potential models sfree seff Two models for medium effects tested:
1.Chiral perturbation (Kaplan, Nelson, et al.) (ChPT)
2.One-boson-exch. (Schaffner-Bielich, et al.,) (OBE)
density and isospin dependent
and two models for NN sin-med
sfree
K+ multiplicity/Apart
seff
In-Medium K energy (k=0)
w/o pot
ChPT
OBE
Splitting for K0,+
and NLr and NLrd
K0/K+ ratios
depend. on
 sin-med (no)
iso-EOS
K potential
isosp. dep of K potential
V.Prassa, et al.,NPA832(2010)88