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

November, 2003@Collective Flow and QGP properties, BNL

11/19/2003 C.NONAKA2

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

NJL model (Nf = 2)

Lattice QCD

K. Yazaki and M.Asakawa., NPA 1989

Suggestions

2SC CFL

T

RHIC

GSI

Critical end point

• Imaginary chemical potential Forcrand and Philipsen hep-lat/0307020

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

11/19/2003 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.

11/19/2003 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

11/19/2003 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

11/19/2003 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

11/19/2003 C.NONAKA7

Equation of StateEquation of StateEquation of StateEquation of State

CEP

Entropy Density Baryon number density

[MeV] 367.7 [MeV], EET 7.154

11/19/2003 C.NONAKA8

Focusing and CEPFocusing and CEPFocusing and CEPFocusing and CEP

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

11/19/2003 C.NONAKA9

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

11/19/2003 C.NONAKA10

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

11/19/2003 C.NONAKA11

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

11/19/2003 C.NONAKA12

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

11/19/2003 C.NONAKA13

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

11/19/2003 C.NONAKA14

Consequences in Experiment (I)Consequences in Experiment (I)Consequences in Experiment (I)Consequences in Experiment (I)CERES:Nucl.Phys.A727(2003)97 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

11/19/2003 C.NONAKA15

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

11/19/2003 C.NONAKA16

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