Constraints on the Symmetry Energy from Heavy Ion Collisions

Hermann WolterLudwig-Maximilians-Universität München

44th Karpacz Winter School of Theoretical Physics, and 1st ESF CompStar Workshop,

„The Complex Physics of Compact Stars“, Ladek Zdroj, Poland, 25.-29.2.08

Outline:

- the symmetry energy and its role for neutron stars

- knowledge of the symmetry energy

- Investigation in heavy ion collisions

- below saturation density: Fermi energies, diffusion, fragmentation

- high densities: relativistic energies; flow, particle production

- summary

Punchline:

- we identify several observables in heavy ion collisions which are sensitive to the symmetry energy

- however, the situation is not yet at a stage (experimentally and theoretically) to fix the symmetry energy

Collaborators:

M. Di Toro, M. Colonna, LNS, Catania

Theo Gaitanos, U. Giessen

C. Fuchs, U. Tübingen;

S. Typel, GANIL

Vaia Prassa, G. Lalazissis, U. Thessaloniki

Schematic Phase Diagram of Strongly Interacting Matter

Liquid-gas coexistence

Quark-hadron coexistence

SIS

Schematic Phase Diagram of Strongly Interacting Matter

Liquid-gas coexistence

Quark-hadron coexistence

Z/N

1

0

SIS

neutron stars

...)()()(/),( 42 IOIEEAIE BsymBB ZN

ZNI

Symmetry Energy: Bethe-Weizsäcker MassenformelE

sym

MeV

)

1 2 30

Asy-st

iff

Asy-soft

Asy-su

persti

ff

pairICsv AZNaAZZaAaAaZAE /)()1(),( 23/13/2

High density: Neutron stars

Around normal density:

Structure, neutron skins

heavy ion collisions in the Fermi energy regime

Theoretical Description of Nuclear Matter

Vij

Non-relativistic:

Hamiltonian H = S Ti + S Vij,; V nucleon-nucleon interaction

Relativistic:

Hadronic Lagrangian

y, nucleon, resonances

s,w, p,.... mesons 0

,...),,,,;(

,

ii

L

dx

dL

L

phenomenological microscopic

(fitted to nucl. matter) (based on realistic NN interactions

non-relativistic Skyrme-type Brueckner-HF (BHF)

(Schrödinger)

Relativistic Walecka-type Dirac-Brueckner HF (DB)

(Quantumhadrodyn.)

Density functional theory

Decomposition of DB self energy

...),(

ˆˆ2

1ˆˆ4

1ˆˆˆ2

1

ˆˆ

222

mesonsisovector

VVmWWΦmΦΦ

ΦgmVgiL

Density (and momentum) dependent coupling coeff.

,,,

),(),( 2

1

2

2

i

k

m

gk

i

i

i

ii

Dirac-Brueckner (DB) Density dep. RMF

(alternative: non-linear model (NL)

meson self interactions)

BF

Fsym E

Mff

E

kE

2

*

*

2*

2

2

1

6

1

No f 1.5 fFREE

f2.5 fm2 f 5f

FREE

PRC65(2002)045201

RMF Symmetry Energy: .contrib

28÷36 MeV

NL

NLρ

NLρδ

The Nuclear Symmetry Energy in different Models

The symmetry energy as the difference between symmetric and neutron matter:

stiff

soft

iso-stiff

iso-soft

empirical iso-EOS‘s cross at about

06.0

microscopic iso-EOS`s soft at low densities but stiff at high densities

C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book)

Uncertainities in optical potentials

Isoscalar Potential Isovector (Lane) Potential

data

GSI SIS

LNS, GANIL, MSU

Incident energy of Heavy Ion Collision:

Low energy (Fermi regime):

Fragmentation, liquid-gas phase transition,

Deep inelastic

High energy (relativistic):

Compression, particle production, temperature.

Modificaion of hadron properties

Transport description of heavy ion collisions:For Wigner transform of the one-body density: f(r,p;t)

,fIfUfm

p

t

fcollp

Vlasov eq.; mean field 2-body hard collisions

AN

iii tppgtrrg

Ntprf

1

))((~))((1

);,(

Simulation with Test Particles:

effective mass

Kinetic momentum

Field tensor

Relativistic BUU eq.

)1)(1()1)(1(

)()2(

1

432432

43213412

124322

3

ffffffff

ppppd

dvdpdpdp

loss term gain term

11 1

12

3 4

flucIFluctuations from higher

order corr.;

stochastic treatment

fff

Data: Famiano et al. PRL 06Calc.: Danielewicz, et al. 07

soft

stiff

SMF simulations, V.Baran 07

Central Collisions at Fermi energies:

Investigation of ratio of emitted pre-equilibrium neutrons over protons

124Sn + 124Sn112Sn + 112Sn

Peripheral collision at Fermi energies: Schematic picture of reaction phases and possible observables

pre-equilibrium emission:

Gas asymmetry

Proton/neutron ratios

Double ratios

binary events:

asymmetry of PLF/TLF

transport ratios

ternary events:

asymmetry of IMF

Velocity corr.

isospin diffusion/transport

npnp

npnpnp

DD

j

//

// ),(Isospin current due to density and isospin gradients:

drift coefficients

diffusion coeffients

npD /

npD /

symIp

In

sympn

CDD

CIDD

4

4

Differences in tranport coefficients simply connected to symmetry energy

MeVA

Esym 32)( 0

asy-stiff

asy-soft

Density range in peripheral collisions

Opposite effects on drift and diffusion for asy-stiff/soft

Isospin Transport through Neck:

Imbalance (or Rami, transport) ratio:(i = proj/targ. rapidity)(also for other isospin sens.quantities) )(

)(

21

21

LLi

HHi

LLi

HHi

mixi

iR

Limiting values: R=0 complete equilibration

R=+-1, complete trasnparency

Discussed extensively in the literature, and experimental data (MSU)

e.g. L.W.Chen, C.M.Ko, B.A.Li, PRL 94, 032701 (2005)

V. Baran, M. Colonna, e al., PRC 72 (2005)

Momentum dependence important

Isospin Transport through Neck:

exp. MSU

Asymmetry of IMF

in symm. Sn+Sn collisions

Asymmetry of IMF in symm. Sn+Sn collisions I

MF MD

MI

Stiff-soft, 124

Stiff-soft, 112

Stiff-soft, 124

Stiff-soft, 112

Asymmetry of IMF in peripheral collision rather sensitive to symmetry energy, esp. for

1. MD interactions

2. when considered as ratio relative to asymmetry of residue

3. Effects of the order of 30%, sensitive variable!

Results from Flow Analysis (P. Danielewicz, R. Lynch,R.Lacey, Science)

Flow and elliptic flow described in a model which allows to vary the stiffness (incompressibility K), and has a momentum dependence

Deduced limits for the EOS (pressure vs. density) for symmetric nm (left).

The neutron EOS (i.e. the symmetry energy) is still uncertain, thus two areas are given for two different assumptions.

v2: Elliptic flowv1: Sideward flow

...2cos),(cos),(1(),,;(: 210 ttt pyvpyvNbpyNFlow

Asymmetric matter: Differential directed and elliptic flow132132Sn + Sn + 132132Sn @ 1.5 AGeV b=6fmSn @ 1.5 AGeV b=6fm

p

n

differential directed flow

differential elliptic flow

Difference at high pt first stage

Dynamical boosting of thevector contribution

T. Gaitanos, M. Di Toro, et al., PLB562(2003)

Proton-neutron differential flow

and analogously for elliptic flow

)()1(1

,)()()(

1)(

1

protonneutronforw

wpyF

i

yN

ii

xiyN

xpn

Pion production: Au+Au, semicentral

Equilibrium production (box results)

Finite nucleus simulation:

Tpnabs

abs

/)(2exp

~ 5 (NLρ) to 10 (NLρδ)

+

+

-

-

+

+

-

-W.Reisdorf et al. NPA781 (2007) 459

Transverse Pion Flows

Simulations:V.Prassa Sept.07

Antiflow:Decoupling of thePion/Nucleon flows

OK general trend. but:- smaller flow for both - and +-not much dependent on Iso-EoS

Kaon Production:

A good way to determine the symmetric EOS (C. Fuchs, A.Faessler, et al., PRL 86(01)1974)

Also useful for Isovector EoS?

-charge dependent thresholds

- in-medium effective masses

-Mean field effects

Main production mechanism: NNBYK, pNYK

Effect of Medium-Effects on Pion (left) and Kaon (right) Ratios

Inelastic cross section

K-potential (isospin independent)K-potential (isospin dependent)

Astrophysical Implications of Iso-Vector EOS

Neutron Star Structure

Constraints on the Equation-of-state

- from neutron stars: maximum mass

gravitational mass vs.

baryonic mass

direct URCA process

mass-radius relation

- from heavy ion collisions: flow constraint

kaon producton

Equations of State tested:

Klähn, Blaschke, Typel, Faessler, Fuchs, Gaitanos,Gregorian, Trümper, Weber, Wolter, Phys. Rev. C74 (2006) 035802

Neutron star masses and cooling and iso-vector EOS

Tolman-Oppenheimer-Volkov equation to determine mass of neutron star

Proton fraction and direct URCA

cooling neutrinofast %,11 :ld thresho

:processURCA direct

)( :neutrality charge and mequilibriu

y

enp

yZ

N y

e

sym

Onset of direct URCA

Forbidden by Direct URCA constraint

Typical neutron stars

Heaviest observed neutron star (now retracted)

Dir

ect

Urc

a C

oo

lin

g l

imit

Mas

s-R

adiu

s R

elat

ion

s

Gra

vita

tio

nal

vs.

Bar

yon

M

ass

Hea

vy I

on

Co

llis

ion

o

bse

vab

les

Constraints of different EOS‘s on neutron star and heavy ion observables

Max

imu

m m

ass

Summary:

•While the Eos of symmetric NM is fairly well determined, the isovector EoS is still rather uncertain (but important for exotic nuclei, neutron stars and supernovae)

•Can be investigated in HIC both at low densities (Fermi energy regime, fragmentation) and high densities (relativistic collisions, flow, particle production)

•Data to compare with are still relatively scarce; it appears that the Iso-EoS is rather stiff.

•Effects scale with the asymmetry – thus reactions with RB are very important

•Additional information can be obtained by cross comparison with neutron star observations