Duke University

Chiho NONAKA

in Collaboration with

Masayuki Asakawa (Kyoto University)

Hydrodynamical Evolution Hydrodynamical Evolution near the QCD Critical End Pointnear the QCD Critical End Point

Hydrodynamical Evolution Hydrodynamical Evolution near the QCD Critical End Pointnear the QCD Critical End Point

June 26, 2003@HIC03, McGill University, Montreal

06/26/2003 C.NONAKA2

Critical End Point in QCD ?Critical End Point in QCD ? Critical End Point in QCD ?Critical End Point in QCD ?

Z. Fodor and S. D. Katz(JHEP 0203 (2002) 014)

dynamical staggered quarks

not fully quantitatively reliabl e

2 1

4

f

t

n

L

= +

Þ=

NJL model (Nf = 2)

Lattice (with Reweighting)

K. Yazaki and M.Asakawa., NPA 1989

Suggestions

2SC CFL

T

RHIC

GSI

Critical end point

06/26/2003 C.NONAKA3

Phenomenological Consequence ?Phenomenological Consequence ? Phenomenological Consequence ?Phenomenological Consequence ?

Divergence of Fluctuation Correlation Length

critical end point

M. Stephanov, K. Rajagopal, and E.Shuryak, PRL81 (1998) 4816

Still we need to study EOS

Focusing

Dynamics (Time Evolution)

Hadronic Observables : NOT directly reflect properties at E

Fluctuation, Collective Flow

If expansion is adiabatic.

06/26/2003 C.NONAKA4

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

EOS with CEPEOS with CEPEOS with CEPEOS with CEP

r hT

QGP

Hadronic

, T),( hr ),( T

),( hr ),( T

fieldmagnetic extermal : h

T

TTr

C

C

3d Ising ModelSame Universality Class

QCD

06/26/2003 C.NONAKA5

EOS of 3-d Ising ModelEOS of 3-d Ising ModelEOS of 3-d Ising ModelEOS of 3-d Ising ModelParametric Representation of EOS

)1(

)00804.076201.0()(~

2

5300

0

Rr

hRhRhh

RMM

8.4

326.0

Guida and Zinn-Justin NPB486(97)626

)154.1,0( R

C

C

T

TTr

h : external magnetic field

QCDMapping

T

r

h

06/26/2003 C.NONAKA6

Thermodynamical QuantitiesThermodynamical QuantitiesThermodynamical QuantitiesThermodynamical Quantities

Singular Part of EOS near Critical Point

Gibbs free energy

Entropy density

Matching

Entropy density

Thermodynamical quantities

Baryon number density, pressure, energy density

),(),( 200 gRMhrMF

)(')1()()2(2)21)((~ 2222 ggh

11.0MhrMFrhG ),(),(

T

r

r

G

T

h

h

G

T

GS

hr

C

),(),(tanh12

1),(),(tanh1

2

1),( BBcBBcB TSTSTSTSTS QHreal

model, volume excludedH :S phase QGPQ :S

r hT

QGP

Hadronic

06/26/2003 C.NONAKA7

Equation of StateEquation of StateEquation of StateEquation of State

CEP

Entropy Density Baryon number density

[MeV] 367.7 [MeV], EET 7.154

06/26/2003 C.NONAKA8

Comparison with Comparison with Bag + Excluded Volume EOSBag + Excluded Volume EOS

Comparison with Comparison with Bag + Excluded Volume EOSBag + Excluded Volume EOS

With End Point

Bag Model + Excluded Volume Approximation(No End Point)

Focused Not Focused

= Usual Hydro Calculation

n /s trajectories in T- planeB

06/26/2003 C.NONAKA9

Slowing out of EquilibriumSlowing out of Equilibrium Slowing out of EquilibriumSlowing out of Equilibrium

B. Berdnikov and K. Rajagopal,Phys. Rev. D61 (2000) 105017

Berdnikov and Rajagopal’s Schematic Argument

along r = const. line

Correlation lengthlonger than eq

h

faster (shorter) expansion

rh

slower (longer) expansion

Effect of Focusing on ?

Focusing Time evolution : Bjorken’s solution along nB/s fm, T0 = 200 MeV

eq

06/26/2003 C.NONAKA10

Correlation Length (I)Correlation Length (I)Correlation Length (I)Correlation Length (I)

1/222

eq ),(M

rgMfMr

Widom’s scaling low

eq

depends on n /s.• Max.• Trajectories pass through the region where is large. (focusing)

eq

eq

B

rh

06/26/2003 C.NONAKA11

Correlation Length (II)Correlation Length (II)Correlation Length (II)Correlation Length (II)

,00

zma

m

time evolution (1-d)

)(

1)()()(

eq

mmmd

d

1m

Model C (Halperin RMP49(77)435)17.2z

• is larger than at Tf. • Differences among s on n /s are small.• In 3-d, the difference between and becomes large due to transverse expansion.

eq

eq

B

06/26/2003 C.NONAKA12

Consequences in Experiment (I)Consequences in Experiment (I)Consequences in Experiment (I)Consequences in Experiment (I)CERES: nucl-ex/0305002 Fluctuations

CERES 40,80,158 AGeV Pb+Au collisions

No unusually large fluctuation

CEP exists in near RHIC energy region ?

T

dyn

dynPT P

2

2, )sgn(

PT

n

jj

n

jx

jxj

x

N

MMN

M

1

1

2

2

N

PM T

PTdybPT

222

,

Mean PT Fluctuation

06/26/2003 C.NONAKA13

Consequences in Experiment (II)Consequences in Experiment (II)Consequences in Experiment (II)Consequences in Experiment (II)

Xu and Kaneta, nucl-ex/0104021(QM2001)

Kinetic Freeze-out Temperature

J. Cleymans and K. Redlich, PRC, 1999

?

Low T comes from large flow.

f

?

Entropy density

EOS with CEP

EOS with CEP gives the natural explanation to the behavior of T .f

06/26/2003 C.NONAKA14

CEP and Its ConsequencesCEP and Its ConsequencesCEP and Its ConsequencesCEP and Its Consequences

Realistic hydro calculation with CEP

Future task

Slowing out of equilibrium

Large fluctuation

Freeze out temperature at RHIC

Fluctuation

Its Consequences

Focusing

Back UP

06/26/2003 C.NONAKA16

Hadronic ObservablesHadronic ObservablesHadronic ObservablesHadronic Observables

Fluctuations

Mean transverse momentum fluctuation

Charge fluctuations

D-measure

Dynamical charge fluctuation

Balance function

Collective Flow

Effect of EOS

Jeon and Koch PRL85(00)2076

Pruneau et al, Phys.Rev. C66 (02) 044904

Gazdzicki and Mrowczynski ZPC54(92)127

Korus and Mrowczynski, PRC64(01)054906

Asakawa, Heinz and Muller PRL85(00)2072

Bass, Danielewicz, Pratt, PRL85(2000)2689

Rischke et al. nucl-th/9504021

06/26/2003 C.NONAKA17

Baryon Number DensityBaryon Number DensityBaryon Number DensityBaryon Number Density ),0(,),( ''

0

BBB

T

BBBB ndTT

sPTn

Critical end point

1st order

crossover

c

c

dTdm

Crossover :

st order : ),0(,),( ''

0

BBB

T

BBBB ndTT

sPTn

))(,())(,()(

CBCHCBCQC

BC TTSTTST

06/26/2003 C.NONAKA18

nnBB/S contours/S contoursnnBB/S contours/S contoursS nB

nB/S

Focusing !

[MeV] 367.7 [MeV], EET 7.154

06/26/2003 C.NONAKA19

Focusing and CEPFocusing and CEPFocusing and CEPFocusing and CEP

MeV MeV, 7.3677.154 EE MT MeV MeV, 0.6527.143 EE MT

06/26/2003 C.NONAKA20

FocusingFocusingFocusingFocusingWhat is the focusing criterion ?

r h

T

h

r

CEP

snB / contours

,0

s

n

rB

)0( 0

hs

n

hB

)0( 0

hs

n

hB

02

2

s

n

rB 0r

From our model

),(),(12

1),(),(1

2

1),( BBcBBcB TSTSTSTSTS QHreal

Dominant termsCritical behavior

,cSr

cS

h

etc.

,0h ,0r

06/26/2003 C.NONAKA21

FocusingFocusingFocusingFocusing

Scavenius et al. PRC64(2001)045202

Analyses from Linear sigma model & NJL model

They found the Critical point in T- plane.

NJL model

Sigma model

The critical point does not serve as a “focusing” point !

Sigma model

NJL model

nB/S lines in plane

06/26/2003 C.NONAKA22

Hydrodynamical evolutionHydrodynamical evolutionHydrodynamical evolutionHydrodynamical evolutionAu+Au 150AGeV b=3 fm

06/26/2003 C.NONAKA23

Relativistic Hydrodynamical ModelRelativistic Hydrodynamical ModelRelativistic Hydrodynamical ModelRelativistic Hydrodynamical Model

Relativistic Hydrodynamical Equation

Baryon Number Density Conservation Equation

Lagrangian hydrodynamics

Space-time evolution of volume element

Effect of EoS

Flux of fluid

06/26/2003 C.NONAKA24

Sound VelocitySound VelocitySound VelocitySound Velocity

• Clear difference between n /s=0.01 and 0.03 B

Effect on Time Evolution Collective flow EOS

trajectoryof length total / :TOTAL snL B

Sound velocity along n /sB

/LTOTAL

/LTOTAL