1. nuclear matter properties
XLVIII International Winter Meeting Bormio 2010
Phase structure of strongly interacting matter and results for heavy-ion collisions and neutron stars
OUTLINE
hadronic SU(3) model
nuclear matter properties
generating a critical endpoint in a hadronic model
molecular / hydro simulation
including quark degrees of freedom
phase diagram
excluded volume corrections
J. Steinheimer, V. Dexheimer, H. Stöcker, SWS
Goethe University, Frankfurt

2. hadronic model based on non-linear realization of chiral symmetry
degrees of freedom
SU(3) multiplets:
baryons (n,Λ, Σ, Ξ) scalars (, , 0) vectors (ω, ρ, φ) , pseudoscalars, glueball field χ
A) SU(3) interaction
~ Tr [ B, M ] B , ( Tr B B ) Tr M
B) meson interactions
~ V(M) <> = 0  <> =  0  0
C) chiral symmetry  m = mK = 0
explicit breaking ~ Tr [ c  ] ( mq q q )
 light pseudoscalars, breaking of SU(3) _ _ _ _ _ _ _
 ~ <u u + d d> <  ~ <s s> 0 ~ < u u - d d> _

3. fit parameters to hadron masses
mesons *  *   
    K* ’
p,n
  
Model can reproduce hadron spectra via dynamical mass generation  K 

4. Lagrangian (in mean-field approximation)
L = LBS + LBV + LV + LS + LSB
baryon-scalars: _
LBS = -  Bi (gi  + gi  + gi  ) Bi
baryon-vectors: _
LBV = -  Bi (gi  + gi  + gi  ) Bi / / /
meson interactions:
LBS = k1 (2 + 2 + 2 )2
+ k2/2 (4 + 2 4 + 4 + 6 2 2 ) + k3  2 
- k4 4 - 4 ln /0 +  4 ln [(2 - 2) / (020)] I
LV = g4 (4 + 4 + 4 + β 22) I
explicit symmetry breaking: LSB = c1  + c2 

5. important reality check
nuclear matter properties at saturation density
asymmetry energy
E/A (p- n)
equation of state E/A ()
binding energy E/A ~ MeV saturation (B)0 ~ .16/fm3
compressibility ~ 223 MeV
asymmetry energy ~ MeV
phenomenology: MeV MeV
+ good description of finite nuclei / hypernuclei
SWS, Phys. Rev. C66,

6. Task: self-consistent relativistic mean-field calculation
coupled 7 meson/photon fields + equations for nucleons in 1 to 3 dimensions
parameter fit to known nuclear binding energies and hadron masses
2d calculation of all measured (~ 800) even-even nuclei
error in energy  (A  50) ~ 0.21 % (NL3: 0.25 %)
 (A  100) ~ 0.14 % (NL3: 0.16 %)
relativistic nuclear
structure models
good charge radii rch ~ 0.5 % (+ LS splittings)
SWS, Phys. Rev. C66,
(2002)

7. phase transition compared to lattice simulations
heavy states/resonance spectrum is effectively described by single (degenerate) resonance with adjustable couplings
Tc ~ 180 MeV
µc ~ 110 MeV
reproduction of LQCD
phase diagram, especially Tc, μc +
successful description of
nuclear matter saturation
phase transition becomes first-order
for degenerate baryon octet ~ Nf = 3
with Tc ~ 185 MeV
D. Zschiesche et al. JPhysG 34, 1665 (2007)

8. lines of constant entropy per baryon, i.e. perfect fluid expansion
Isentropes, UrQMD and hydro evolution
lines of constant entropy per baryon, i.e. perfect fluid expansion
E/A = 5, 10, 40, 100, 160 GeV E/A = 160 GeV goes through endpoint
J. Steinheimer et al. PRC77, (2008)

9. connect hadronic and quark degrees of freedom
order parameter of the phase transition
confined phase
deconfined phase
effective potential for Polyakov loop, fit to lattice data
U = ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2]
a(T) = a0T4 + a1 µ4 + a2 µ2T2
baryonic and quark mass shift δ mB ~ f(Φ) δ mq ~ f(1-Φ) q
quarks couple to mean fields via gσ, gω q
V. Dexheimer, SWS, astro-ph
Ratti et al. PRD (2006)
Fukushima, PLB 591, 277 (2004)
minimize grand canonical potential

10. Phase Diagram for HQM model
µc = 360 MeV
Tc = 166 MeV
µc. = 1370 MeV
ρc ~ 4 ρo
hybrid hadron-quark model
critical endpoint tuned to lattice results
V. Dexheimer, S. Schramm, astro-ph

11. Mass-radius relation using Maxwell/Gibbs construction
Gibbs construction allows for quarks in the neutron star
mixed phase in the inner 2 km core of the star
V. Dexheimer, SWS, astro-ph

12. isentropic expansion
overlap initial conditions
Elab = 5, 10, 40, 100, 160 AGeV
Csph ~ ¼ Cs,ideal
averaged Cs significantly
higher than 0.2

13. Include modified distribution functions for quarks/antiquarks
different approach – hadrons, quarks, Polyakov loop and excluded volume
Include modified distribution functions for quarks/antiquarks
Following the parametrization used in PNJL calculations
U = - ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2]
a(T) = a0T4 + a1 T0T3 + a2 T02T2 , b(T) = b3 T03 T
The switch between the degrees of freedom
is triggered by excluded volume corrections
thermodynamically consistent -
Vq = 0
Vh = v
Vm = v / 8 ~ ~ ~
µi = µ i – vi P
e = e / (1+ Σ vi ρi )
Steinheimer,SWS,Stöcker hep-ph/
D. H. Rischke et al., Z. Phys. C 51, 485 (1991)
J. Cleymans et al., Phys. Scripta 84, 277 (1993)

14. Temperature dependence of chiral condensate and Polaykov loop at µ = 0
Interaction measure e – 3p
lattice data taken from Bazavov et al. hep-lat
speed of sound shows a pronounced
dip around Tc !

15. Lattice comparison of expansion coefficients as function of T
lattice results
lattice data from
Cheng et al., PRD 79, (2009)
Steinheimer,SWS,Stöcker
hep-ph/
suppression factor peaks

16. Φ Dependence of Polyakov loop and chiral condensate on µ, T
Lines mark maximum in T derivative
σ Φ

17. quark, meson, baryon densities at µ = 0
densities of baryon, mesons and quarks
natural mixed phase,
quarks dominate beyond 1.5 Tc ρ
Energy density and pressure compared
to lattice simulations

18. hybrid simulation of a Pb-Pb collision at 40 GeV/A
red regions show the areas dominated by quarks

19. SUMMARY / OUTLOOK general hadronic model as starting point
works well with basic vacuum properties, nuclear matter, nuclei, …
phase diagram with critical end point via resonances
implement EOS in combined molecular dynamics/ hydro simulations
quarks included using effective deconfinement field
realistic phase transition line
implementing excluded volume term, natural switch of d.o.f.
strangeness distillation / modify model towards metastable hyperon matter
explore µS dependence
resonances – 1st order transition?

20. Temperature distribution from UrQMD simulation as
initial state for (3d+1) hydro calculation
Use smeared-out particle distributions as starting point for hydro simulation

21. simple time evolution including π, K evaporation (E/A = 40 GeV)
with evaporation
C. Greiner et al.,
PRD38, 2797 (1988)
E/A-mN
300
g2 = 0
If you want it exotic …
g2 = 2
200
additional coupling g2 of hyperons
to strange scalar field
100
g2 = 4
follow star calcs by J. Schaffner et al.,
PRL89, (2002)
g2 = 6
barrier at fs ~ 0.4 – 0.6 fs

22. Hypernuclei -  single-particle energies
Nuclear matter
Model and experiment agree well

23. Elab≈ 5-10 AGeV sufficient to overshoot
Evolution of the collision system
Elab≈ AGeV sufficient to overshoot
phase border, AGeV around endpoint

24. amount of volume scanning the critical endpoint (lattice)

25. Comparison of gross properties of initial conditions
Overlap of projectile and target

26. time integrated volume around the critical end point

27. effective volume sampling the critical end point
window T = Tc ± 10 MeV µ = µc ± 10 MeV
maximum shifts in time