On the Isospin Dependenceof the EoS of Nuclear Matter

D. Cozma, M. Petrovici (NIPNE, Bucharest, Romania)

Heavy-Ion Collisions from the Coulomb Barrier to the Quark-Gluon Plasma

Erice

September 22nd, 2008

– p.1

OVERVIEW

• Introduction & Motivation

• HIC ModelQMD transport model

Essential ingredients

• In-medium EffectsIn-medium NN scattering

Isospin dependence of EoS

• HIC ObservablesObservables

Case Study: Zr+Ru

• Summary and Outlook

– p.2

Introduction

Nuclear Equation of State: E/A=E(ρ)

sources: finite nuclei ρ/ρ0 ≤ 1

heavy-ions ρ/ρ0 ≤ 3

neutron stars ρ/ρ0 ≤ 10

Fuchs PRL86, 1974

– p.3

Motivation

Equation of State of asymmetric nuclear matter:Symmetry energy:

E(ρ, β) = E(ρ) + Esym(ρ) β2 + · · · β =ρn − ρp

ρn + ρp

Esym(ρ) =1

2

∂E(ρ, β)

∂β2|β=0 = a4 +

p0

ρ20

(ρ − ρ0)

- phenomenological models constrained in the low ρ region diverge at high density

– p.4

Motivation

FOPI Collaboration: RuZr and AuAu @ 400 AMeVRami et al., PRL 84,1120; Hong et al., PRC 66, 034901

– p.5

Transport Model

Transport model: Quantum Molecular DynamicsMonte Carlo cascade + Mean field + Pauli-blocking+ in medium cross sectionall 4∗ resonances below 2 GeV - 10 ∆∗ and 11 N∗

• included baryon-baryon collisions:

all elastic channels

inelastic channels NN → NN⋆, NN → N∆⋆,

NN → ∆N⋆, NN → ∆∆⋆, NR → NR′

• included pion-absorption resonance-decay channels:

∆ Nπ, ∆⋆ ∆π, ∆⋆

N1440π, N⋆ Nπ,

N⋆ Nππ, (N⋆

∆π, N⋆ N1440)

– p.6

QMD: Essential Ingredients

Inclusion of collisions:

- binary collisions: geometric criterion

- one needs consistent cross-section ↔potential parameters (mean field)

- use fit to available experimental data; if not available use detailed balance

and isospin symmetry

σ1,2→3,4(√

s) ∼ (2S3 + 1)(2S4 + 1)〈p3,4〉〈p1,2〉

1

s|M(m3,m4)|2

σf→i =p2

i

p2f

gi

gfσi→f

- Pauli blocking due to Fermi statistics: collision allowed with probability

(1 − f ′

1)(1 − f ′

2)

- angular distributions of two-body scattering: same as NN → NN

(determined from an effective model)

– p.7

Nucleon-Nucleon Interaction

Vacuum NN Interaction: - microscopical OBE model (Bonn)

T (~q′, ~q) = V (~q′, ~q) + P

Z

d3k

(2π)3V (~q′, ~k)

m2

E2k

1

2Eq − 2EkT (~k, ~q)

In-Medium NN interaction: - Dirac-Brueckner approach

G(~q′, ~q| ~P , z) = V ∗(~q′, ~q) + P

Z

d3k

(2π)3V ∗(~q′, ~k)

m2∗

E21/2~P+~k

Q(~k, ~P )

z − 2E1/2~P+~k

G(~k, ~q| ~P , z)

Li, Machleidt PRC 48, 1702 Li, Machleidt PRC 49, 566– p.8

Nucleon-Nucleon Interaction

Li, Machleidt PRC 48, 1702; C. Fuchs, PRC 64, 024003

0 30 60 90 120 150 180θ

cm(deg)

0

0.05

0.1

0.15

0.2

dσ/d

Ω (

rel)

0.050 GeV0.100 GeV0.200 GeV0.400 GeV0.600 GeV0.800 GeV1.000 GeV

NPdifferential cross-sections in vacuum

0 30 60 90 120 150 180θ

(cm) (deg)

0

0.05

0.1

0.15

0.2

dσ/d

Ω (

rel)

0.050 GeV0.100 GeV0.200 GeV0.400 GeV0.600 GeV0.800 GeV1.000 GeV

PP differential cross-section in vacuum

0 30 60 90 120 150 180θ

(cm)

0

0.05

0.1

0.15

0.2

dσ/d

Ω

vacuum0.5 ρ

0

1.0 ρ0

2.0 ρ0

3.0 ρ0

4.0 ρ0

PN differential cross-sections@ 400 MeV

0 30 60 90 120 150 180θ

(cm)

0.06

0.07

0.08

0.09

0.1

dσ/d

Ω

vacuum0.5 ρ

0

1.0 ρ0

2.0 ρ0

3.0 ρ0

4.0 ρ0

PP differential cross-sections@ 400 MeV

– p.9

Isospin dependence

EoS of isospin asymmetric nuclear mater:

V n(p)(ρ, β) = a u + b uγ + Vmdi + V pc + V

n(p)asym(ρ, β)

Ea(ρ, β) = ea ρ F (u) β2 Vn(p)asym = ∂Ea(ρ, β)/∂ ρn(p)

F1(u) =2u2

1 + uF2(u) = u F3(u) = u1/2

nucleons and resonances propagatein an isospin dependent mean field

Vasym(n∗) = Vasym(∆0) = V nasym

Vasym(p∗) = Vasym(∆+) = V pasym

Vasym(∆++) = 2V pasym − V n

asym

Vasym(∆−) = 2V nasym − V p

asym

Li, Ko, Ren PRL78, 1644 – p.10

Observables

double neutron to proton ratio (n/p)AB/(p/n)BA

Li,Li, Stocker PRC 73, 051601

neutron/proton ratio at midrapidity

Yong, Li, Chen PLB650, 344

– p.11

Observables

Elliptic flow and Differential Elliptic Flow

dN

dφ= a0 (1 + a1cos(φ) + a2cos(φ))

a2 =1

N

X

i

pi2x − pi2

y

pi2t

– p.12

Elliptic flow (EoS and in-medium NN dep)

dN

dφ= a0 (1 + a1cos(φ) + a2cos(φ))

constraints: |y|<0.50, b=5 fm, Ru+Zr @ 400 MeV

EOS + Cross-sections n+p n p

Isospin indep + Free c.s. -0.040 -0.042 -0.036

Isospin indep + Dens. Dep. c.s. -0.038 -0.038 -0.038

Isospin indep + Dens. Dep. diff. c.s. -0.038 -0.040 -0.036

Isospin dep soft + Free c.s. -0.018 -0.019 -0.017

Isospin dep soft + Dens. Dep. c.s. -0.014 -0.015 -0.013

Isospin dep soft + Dens. Dep. diff. c.s. -0.016 -0.018 -0.014

Isospin dep stiff + Free c.s. -0.014 -0.017 -0.014

Isospin dep stiff + Dens. Dep. c.s. -0.017 -0.020 -0.014

Isospin dep stiff + Dens. Dep. diff. c.s. -0.020 -0.022 -0.018

Isospin dep linear + Free c.s. -0.013 -0.017 -0.008

Isospin dep linear + Dens. Dep. c.s. -0.017 -0.022 -0.010

Isospin dep linear + Dens. Dep. diff. c.s. -0.016 -0.022 -0.009– p.13

Differential elliptic flowsensitivity to in-medium NN interaction

0 0.2 0.4 0.6 0.8 1p

t

-0.02

-0.015

-0.01

-0.005

0

v 2

free cross-sectionsin-medium cross-sectionsin-medium diff cross-sections

Isospin independent EOS

0 0.2 0.4 0.6 0.8p

t

-0.008

-0.006

-0.004

-0.002

0

v 2 (p

roto

ns)

free cross-sectionsin-medium cross-sectionsin-medium diff cross-sections

Isospin dependent (soft) EOSRuZr @ 400 MeV

sensitivity to EOS

0 0.2 0.4 0.6 0.8p

t

-0.02

-0.015

-0.01

-0.005

0

v 2 (p

roto

ns)

isospin indepisospin dep softisospin dep stiffisospin dep linear

Various EOS (free NN cs)RuZr @ 400 MeV

0 0.2 0.4 0.6 0.8p

t

-0.02

-0.015

-0.01

-0.005

0

v 2 (p

roto

ns)

isospin indepisospin dep softisospin dep stiffisospin dep linear

Various EOS (in-medium NN diff cs)RuZr @ 400 MeV

– p.14

Differential elliptic flow

splitting of the n vs. p values

0 0.2 0.4 0.6 0.8p

t

-0.02

-0.015

-0.01

-0.005

0

v 2 (p

roto

ns,n

eutr

ons)

isospin indep protonsisospin indep neutronssoft EOS protonssoft EOS neutronsstiff EOS protonsstiff EOS neutronslinear EOS protonslinear EOS neutrons

Various EOS (free NN cs)RuZr @ 400 MeV

0 0.2 0.4 0.6 0.8p

t

-0.02

-0.015

-0.01

-0.005

0

v 2 (p

roto

ns,n

eutr

ons)

isospin indep protonsisospin indep neutronssoft EOS protonssoft EOS neutronsstiff EOS protonsstiff EOS neutronslinear EOS protonslinear EOS neutrons

Various EOS (in-medium NN diff cs)RuZr @ 400 MeV

Problem ! differential elliptic flow at high pT

– p.15

Summary

• Message: - a2 sensitive to isospin dependent part of EoS; density

dependent NN cross-section of secondary importance

- no clear preference for the isospin dependent part of the equation of

state

• Consistency: - vacuum isospin dependent NN interaction → in-medium NN

cross-sections, equation of state

• Improvements: - determine the origin of the differential elliptic flow at

large pT ;

- introduce momentum dependence in the symmetry energy terms and

account for neutron-proton mass splitting

• To Be Done: - implement in transport code explicit production channels for

deuterium and compare with results from coalescence models;

- study the emission of 3H and 3He from a coalescence model

– p.16