Duke University

1
Hydrodynamical Evolution near the QCD Critical
End Point

• Duke University
• Chiho NONAKA

in Collaboration with
Masayuki Asakawa (Kyoto University)
June 26, 2003_at_HIC03, McGill University, Montreal
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Critical End Point in QCD ?
Suggestions

• NJL model (Nf 2)

K. Yazaki and M.Asakawa., NPA 1989

• Lattice (with Reweighting)

Z. Fodor and S. D. Katz (JHEP 0203 (2002) 014)
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Phenomenological Consequence ?
M. Stephanov, K. Rajagopal, and E.Shuryak, PRL81
(1998) 4816
critical end point
Divergence of Fluctuation
Correlation Length
If expansion is adiabatic.
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EOS with CEP

• How to Construct EOS with CEP?
• Assumption
• Critical behavior dominates in a large region
  near end point
• Near QCD end point singular part of EOS
• Mapping
• Matching with known QGP and
• Hadronic entropy density
• Thermodynamical quantities

3d Ising Model
QCD
Same Universality Class
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EOS of 3-d Ising Model

• Parametric Representation of EOS

Guida and Zinn-Justin NPB486(97)626
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Thermodynamical Quantities

• Singular Part of EOS near Critical Point
• Gibbs free energy
• Entropy density
• Matching
• Entropy density
• Thermodynamical quantities
• Baryon number density, pressure, energy
  density

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Equation of State
CEP
Baryon number density
Entropy Density
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Comparison with Bag Excluded Volume EOS

• n /s trajectories in T- m plane

B
Bag Model Excluded Volume Approximation (No
End Point)
With End Point
Usual Hydro Calculation
Not Focused
Focused
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Slowing out of Equilibrium

• Berdnikov and Rajagopals Schematic Argument

B. Berdnikov and K. Rajagopal, Phys. Rev. D61
(2000) 105017
slower (longer) expansion
Correlation length longer than xeq
xeq
x
along r const. line

• Effect of Focusing on x?

E
h
Time evolution Bjorkens solution along nB/s
t0 1 fm, T0 200 MeV
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Correlation Length (I)

• x

eq
Widoms scaling low
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Correlation Length (II)

• x time evolution (1-d)

Model C (Halperin RMP49(77)435)
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Consequences in Experiment (I)
CERES nucl-ex/0305002

• Fluctuations
• CERES
• 40,80,158 AGeV PbAu
• collisions

Mean PT Fluctuation
No unusually large fluctuation
CEP exists in near RHIC energy region ?
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Consequences in Experiment (II)

• Kinetic Freeze-out Temperature

Low T comes from large flow.
?
f
Xu and Kaneta, nucl-ex/0104021(QM2001)
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CEP and Its Consequences
Future task

• Realistic hydro calculation with CEP

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Back UP
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Hadronic Observables

• Fluctuations
• Mean transverse momentum fluctuation
• Charge fluctuations
• D-measure
• Dynamical charge fluctuation
• Balance function
• Collective Flow
• Effect of EOS

Gazdzicki and Mrowczynski ZPC54(92)127
Korus and Mrowczynski, PRC64(01)054906
Asakawa, Heinz and Muller PRL85(00)2072
Jeon and Koch PRL85(00)2076
Pruneau et al, Phys.Rev. C66 (02) 044904
Bass, Danielewicz, Pratt, PRL85(2000)2689
Rischke et al. nucl-th/9504021
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Baryon Number Density
Crossover

1 st order
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nB/S contours
nB
S
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Focusing and CEP
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Focusing

• What is the focusing criterion ?

h
r
CEP
From our model
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Focusing

• Analyses from Linear sigma model NJL model

Scavenius et al. PRC64(2001)045202
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Hydrodynamical evolution
AuAu 150AGeV b3 fm
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Relativistic Hydrodynamical Model

• Relativistic Hydrodynamical Equation
• Baryon Number Density Conservation Equation
• Lagrangian hydrodynamics
• Space-time evolution of volume element
• Effect of EoS

Flux of fluid
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Sound Velocity

• Effect on Time Evolution
• Collective flow

EOS