Nagoya UniversityChiho NONAKA

3D Hydro + 3D Hydro + UrQMD UrQMD ModelModelwithwith

QCD Critical PointQCD Critical Point

February 8, 2008 @QM2008, Jaipur, India

In collaboration with

M.ASAKAWA(Osaka) and S.A.Bass(Duke)

Chiho NONAKA

The QCD critical point search from phenomenology The QCD The QCD Critical Point in HICCritical Point in HIC

t energy scan

focusing

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

t

3D Hydro + UrQMDRealistic dynamical model for Heavy Ion Collisions

Chiho NONAKA

Relativistic Heavy Ion Collision• Schematic sketch t

hydrodynamical expansion

freeze-outhadronizationthermalizationcollision

3D Hydro + UrQMDSuccess of hydrodynamic models at RHIC:

--PT spectra, rapidity distribution, elliptic flowLimitation of Hydro

Ex. Elliptic flow at forward/backward rapidity , HBT Extension of Hydro

--Viscosity effect of hadron phase--Final state interactions

Full 3-d Hydrodynamics

EoS :1st order phase transition QGP + excluded volume model

Cooper-Fryeformula

UrQMD

t fm/c

final stateinteractions

Monte Carlo

Hadronization

TC TSWTC:critical temperature > TSW: Hydro UrQMD

3D Hydro+3D Hydro+UrQMD UrQMD ModelModelNonaka and Bass PRC75:014902(2007)

Chiho NONAKA

Highlight ofHighlight of 3D Hydro+3D Hydro+UrQMDUrQMDNonaka and Bass PRC75:014902(2007)

Chiho NONAKA

RealisticRealistic Equation of StatesEquation of States3D Hydro + UrQMD

Full 3-d Hydrodynamics

EoS :1st order phase transition QGP + excluded volume model

Cooper-Fryeformula

UrQMD

t fm/c

final stateinteractions

Monte Carlo

Hadronization

TC TSWTC:critical temperature > TSW: Hydro UrQMD

initial conditionsinitial conditions••parametrization parametrization

equation of statesequation of states••bag modelbag model

freezeout freezeout processprocess••Viscosity effect of Viscosity effect of hadron hadron phasephase••Final state interactionsFinal state interactions

Realistic EOS with QCD critical point

Chiho NONAKA

EOS with QCD Critical PointEOS with QCD Critical PointSingular part near QCD critical point + Non-singular part• Non-singular part QGP phase and hadron phase• Singular part

• Mapping (r, h) (T, µB)• Matching with known QGP and hadronic entropy density• Thermodynamical quantities

µΒ

r hT

QGP

Hadronic

!

T , µB

),( hr

!

(T,µB)fieldmagnetic extermal : h

T

TTr

C

C!

=

3d Ising ModelSame Universality Class

QCD

Nonaka and Asakawa, PRC71,044904(2005)

Chiho NONAKA

Equation of StateEquation of State

QCD critical point

entropy density baryon number density

[MeV] 367.7 [MeV], ==EE

T µ7.154

Chiho NONAKA

IsentropicIsentropic TrajectoriesTrajectories

!

TE

=154.7 MeV, µE

= 367.7 MeV MeV MeV, 0.6527.143 ==EE

T µ

•Behavior of focusing depends on the location of QCP on T-µΒ plane.•The focusing effect appears stronger in the case (T,µΒ)=(143,652).•The location of QCP is a parameter. Experiments

Isentropic trajectories: nB/s=const. line

Chiho NONAKA

Comparison withComparison with Bag ModelBag Model

With QCD critical point

Bag Model + Excluded Volume Approximation(No Critical Point)

Focused Not Focused

= Usual Hydro Calculation

Isentropic trajectories on T- µΒ plane

Chiho NONAKA

3D Hydro+3D Hydro+UrQMD UrQMD with QCPwith QCPInitial Conditions• Energy density

• Baryon number density

• Parameters

• Flow

EOS: QCP, Bag Model

Switching temperature

η0=0.5 ση=1.5

τ0=0.6 fm/c

TSW=150 [MeV]

vT=0vL=η (Bjorken’s solution);

••longitudinal direction: longitudinal direction: HH((ηη))

!

"(x,y,#) ="maxW (x,y;b)H(#)

!

nB (x,y,") = nBmaxW (x,y;b)H(")

0.152.020.21.51

nBmax fm-3εmaxGeV/fm3

EE MeVMeV, 652.0143.7 ==T µQCP:

Chiho NONAKA

• T and µB in one volume element close to center

Isentropic Trajectories in HydroIsentropic Trajectories in Hydro

Hydro with Bag ModelHydro with QCD critical point

T [M

eV]

µB [MeV]

T [M

eV]

µB [MeV]

12

12RHIC12

EE MeVMeV, 652.0143.7 ==T µQCP:

•Tf=110 MeV•Behavior of isentropic trajectories in hydro with QCP is different from one in hydro with bag model. •Focusing effect appears in hydro with QCD critical point.

Chiho NONAKA

PPTT Spectra in 3D Hydro Spectra in 3D HydroHydro with Bag ModelHydro with QCD critical point

•Tf=110 MeV•PT slope is almost the same.

Chiho NONAKA

3D Hydro + 3D Hydro + UrQMDUrQMD

Hydro with Bag ModelHydro with QCD critical point

µB [MeV]T

[MeV

]

T [M

eV]

µB [MeV]

12

12

UrQMD UrQMD

Switching temperature: 150 MeV

Chiho NONAKA

PPTT Spectra Spectra

Bag Model QCD critical point

•PT slope is almost the same.•Larger difference of different initial conditions appears in the case of QCP.

Chiho NONAKA

Hadron Hadron RatiosRatiosBag Model QCD critical point

!

µB QCP

> µB BG

At TSW

!

p

" QCP

>p

" BG

Because of focusing effect

µB [MeV]

T [M

eV]

Chiho NONAKA

SummarySummary3D Hydro + UrQMD Model with the QCD critical point• Isentropic trajectories• PT spectra, hadron ratio

The QCD critical point search• Energy scan parameter sets of ε and nB in initial conditions• Switching temperature dependence• Location of QCD critical point

Physical observables• Fluctuations• Balance function

Focusing

BACKUP

Chiho NONAKA

Mean PMean PTT Fluctuation Fluctuationpreliminary

PT < 1 GeV π p K1 0.567E-02 0.266E-01 0.156E-012 0.416E-02 0.242E-01 0.115E-01

*In this calculation, definition of mean Pt fluctuation is different from CERES.

Chiho NONAKA

CEP and Its ConsequencesCEP and Its Consequences

Realistic hydro calculation with critical point

Work in Progress

Slowing out of equilibrium Large fluctuation Freeze out temperature at RHIC Fluctuation

Consequences

FocusingExistence and location of CEP in phase diagram:Collision energy dependent experiments are indispensable.

Chiho NONAKA

Sound VelocitySound VelocityEoS Hydrodynamic Expansion

sn

s

B

PC

/

2

!"

"=

ε

sn

s

B

PC

/

2

!"

"=

εH εQ

QGP

hadron Mixed

Ex. Rischke et al. nucl-th/9504021

Effect of mixed phase?

Chiho NONAKA

Shear ViscosityShear Viscosity

!

"

"0=

#eq#eq,0

$

% & &

'

( ) )

1

19* +O(* 2 )

+

, - .

/ 0

Chiho NONAKA

Lattice QCD at finite T and Lattice QCD at finite T and µµBB

Multi-parameter reweighting technique

(JHEP 0203 (2002) 014)hep-lat/0402006

)( 0!

)( 0!

)(!

)(!

),,( µ!" m=

• Allton et al. (Bielefeld-Swansea)

• Fodor and KatzOverlap problem: The lattice size is small.

Maezawa san’s talk

Chiho NONAKA

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 Hadronic Observables : NOT directly reflect properties at CEP

Fluctuation, Collective Flow EOS

Focusing in nB/s trajectories Dynamics (Time Evolution)

If expansion is adiabatic.

Chiho NONAKA

Time Evolution Time Evolution

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

Berdnikov and Rajagopal’s Schematic Argument

Correlation Lengthlonger than Leq

• Realistic Hydro Calculation with Realistic EOS

slower (longer) expansion

faster (shorter) expansion

rh

along r = const. line

h

Ε What’s missing:

Chiho NONAKA

3-d Hydrodynamic Model3-d Hydrodynamic ModelHydrodynamic equation

• Baryon number density conservation

• Coordinates

Lagrangian hydrodynamics• Tracing the adiabatic path of each volume element• Effects of phase transition of observables

Algorithm• Focusing on conservation law

)( : !µµ!!µµ!µ! " uugpuuTT ##=velocity local : pressure : density energy : µ! up

Flux offluid

Chiho NONAKA

Trajectories on the phase diagramTrajectories on the phase diagramLagrangian hydrodynamics

Effect of phase transition

temperature and chemical potential of volume element of fluid

C.N et al., Eur. Phys.J C17,663(2000)

Chiho NONAKA

Jets in mediumJets in mediumJet quenching mechanisms

arXiv:0705.2575(hep-ph)with Qin, Ruppert, Turbide, Gale

nucl-th/0703019with Majumder

nucl-th/0611027with Renk and Ruppert

Ex. Nuclear modification factor in 3D hydro

•pQCD •Mixed and hadron phase

•AMY •gluon bremsstrahlung

•parton path

Chiho NONAKA

Soft + HardSoft + HardSoft

Hard

• Full 3-d Hydrodynamic Model• QGP formation, EoS

•Hard scattering & jet production• Propagation of jet in medium, energy loss

• First schematic attempt

t fm/c

Interaction between Soft and Hard

Hirano & NaraPRC66:041901,2002, PRL91:082301,2003

Hadronization

Fragmentation

Improved Cooper-Fryeformula (Reco)

FinalInteractions

• Dynamical effect on jets• Jet correlations

UrQMD

Chiho NONAKA

Non-Singular PartNon-Singular PartHadron Phase

• Excluded volume model

QGP Phase!

P(T,µB) = P

i

ideal(T,µ

Bi"V0P(T,µBi

))i

#

!

= Pi

ideal(T, ˜ µ

Bi)

i

"

!

P(T,µB ) =(32 + 21N f )"

2

180T4 +

N f

2

µB

3

#

$ %

&

' (

2

T2 +

N f

4" 2

µB

3

#

$ %

&

' (

4

) B

Nf=2B:Bag constant (220 MeV)4

T

PPQGP

PH

BTc

µB=0

Chiho NONAKA

Isentropic Trajectories in HydroIsentropic Trajectories in HydroConservation law in ideal fluid

Hydrodynamic expansion: nB/s =const.

nB/s =const. lines on T-µB plane: behavior of expansion

Chiho NONAKA

Trajectories on the phase diagramTrajectories on the phase diagramLagrangian hydrodynamics

Effect of phase transition

temperature and chemical potential of volume element of fluid

C.N et al., Eur. Phys.J C17,663(2000)

Chiho NONAKA

Hadron ratios near CEPHadron ratios near CEP

•Hadron ratio (ex. p/π) is not sensitive to collision energy near CEP.

•Near chemical freeze-out temperature the contribution from recombination, resonances is small.

Chemical Freeze-outHeinz :hep-ph/0109006

•Statistical Model Free resonance gas model

•At chemical freeze-out Quasi-particle state

Chiho NONAKA

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

)1(

)00804.076201.0()(~

2

53

00

0

!

!!!!

!

"#"

"

$=

+$==

=

Rr

hRhRhh

RMM

8.4

326.0

=

=

!

"

Guida and Zinn-Justin NPB486(97)626

)154.1,0( 1.154!!"# $!!R

C

C

T

TTr

!=

h : external magnetic field

QCDMapping

µΒ

T

rh

Order parameter

qqM !

Chiho NONAKA

Gibbs Free Energy

Entropy Density for Singular Part

Singular Part of EOSSingular Part of EOS

MhrMFrhG != ),(),(

rM

Fh !

"

#$%

&

'

'=

!

F(M,r) = h0M

0R2"#g($)

T

r

r

G

T

h

h

G

T

Gs

hr

c

!

!

!

!"

!

!

!

!"=

!

!"=

µ

Mh

G

r

!="

"

!

"G

"rh

="F

"rh

#"M

"rh

h

mapping

Free energy:

Guida and Zinn-Justin NPB486(97)626

11.0=!

critΔµBcrit

!

(r,h)" (T,µB)

Chiho NONAKA

Singular Part + Non-singular PartSingular Part + Non-singular Part Entropy Density

!

Sreal(T,µ

B) =1

21" tanh[S

c(T,µ

B)]{ }SH (T,µB

) +1

21+ tanh[S

c(T,µ

B )]{ }SQ (T,µB)

Hadron Phase (excluded volume model)

QGP phase

• Dimensionless parameter: Sc

Critical domain( ) ( ) DTsTS

cBc!"+"=

22),(

critcritµµ

• Choice of parameters:Thermodynamical inequalities

DT ,,critcrit

µ!!

0,0,0

,,,

!"#

$%&

'

(

(!"

#

$%&

'

(

(!"

#

$%&

'

(

(

VTNTNVNV

P

T

S µ!!!!

Critical exponent near CEPkeeps correctly.

• crit

ΔµBcrit

!

SH(T,µ

B)

!

SQ(T,µ

B)

Chiho NONAKA

Thermodynamical Thermodynamical QuantitiesQuantities

1st order

Baryon number density

!

nB(T,µ

B) =

"P

"µB

="s

"µB0

T

# T',µ

B( )dT ' + nB(0,µ

B)

Pressure

Energy Density

! +=

T

BBBPdTTSTP

0

),0('),'(),( µµµreal

BBnPTS µ! ""=

real

CEP

1st order

crossover

!

"Tc

"µc

!

Sreal(T,µ

B) =1

21" tanh[S

c(T,µ

B)]{ }SH (T,µB

) +1

21+ tanh[S

c(T,µ

B )]{ }SQ (T,µB)

!

+"T

C(µ

B)

"µC

SQTC,µ

BTC( )( ) # SH T

C,µ

BTC( )( )( )

Chiho NONAKA

DimensionlessDimensionless ParameterParameter

( ) ( ) DTsTScBc

!"+"=22

),(critcrit

µµ

T

r

r

G

T

h

h

G

T

Gs

hr

c

!

!

!

!"

!

!

!

!"=

!

!"=

µ

Phase transition region

Chiho NONAKA

Focusing and QCDFocusing and QCD Critical PointCritical Point

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

The fine-tuning of the collision energy is not necessary not only on the high energy side but also on the low energy side.

If the critical region is large enough

Chiho NONAKA

FocusingFocusing

Scavenius et al. PRC64(2001)045202

nB/s trajectories on T-µ plane

NJL model

Linear Sigma model

Hydrodynamic expansion entropy, baryon number conservation

• CEP is not an attractor for nB/s trajectories.• Critical phenomena near CEP ?

Chiho NONAKA

Sound VelocitySound VelocityEoS Hydrodynamic Expansion

sn

s

B

PC

/

2

!"

"=

ε

sn

s

B

PC

/

2

!"

"=

εH εQ

QGP

hadron Mixed

Ex. Rischke et al. nucl-th/9504021

Effect of mixed phase?

Chiho NONAKA

Sound Velocity in Sound Velocity in EoS EoS with CEPwith CEPEffect on Time Evolution

Collective flow EOS

trajectoryof length total / :TOTAL

snLB

sn

s

B

PC

/

2

!"

"=

•The system may not expand uniformly.

Collective flow & HBT

Chiho NONAKA

Slowing out of Equilibrium Slowing 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 ξeq

hΕ

faster (shorter) expansion

rh

slower (longer) expansion

Effect of Focusing on correlation length?

Focusing Time evolution : Bjorken’s scaling solution along nB/sτ0 = 1 fm, T0 = 200 MeV

ξξeq

Chiho NONAKA

Correlation LengthCorrelation Length in Equilibriumin Equilibrium

!!

"

#

$$

%

&=

'

(

()*1

/222

eq ),(M

rgMfMr

Widom’s scaling low ξ eq

r h

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

eq

eq

B

•nB/s non-critical component of the EOS

Chiho NONAKA

Evolution of Correlation LengthEvolution of Correlation Length

!

" m#( ) =a

$0

m#$0( )z

,

ξ : time evolution (1-d)

!

d

d"m# (") = $% m# (")[ ] m# (") $

1

&eq (")

'

( ) )

*

+ , ,

!

m" #( ) = 1$ #( )

Model H (Halperin RMP49(77)435)

!

z = 3.0

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

eq

eqB

Critical slowing down

Chiho NONAKA

FluctuationFluctuationCube of the equilibrium correlation length

• There is the possibility that the fluctuation shows some enhancement.

Fluctuation

!

"# 3

Chiho NONAKA

Fluctuations (I)Fluctuations (I)CERES:Nucl.Phys.A727(2003)97

Fluctuations CERES 40,80,158 AGeV Pb+Au collisions

No unusually large fluctuation or non-monotonic behavior

Mean PT Fluctuation

CEP: attractor of isentropic trajectoriesSimilar correlation length and fluctuation is observed near CEP.

!

"PT# sgn($PT ,dyn

2)

$ dyn

2

PT!

Mx

2 "

N j Mx

j # Mx( )2

j=1

n

$

N j

j=1

n

$!

"PT ,dyb

2# $MPT

2%$PT

2

N

Chiho NONAKA

Fluctuations (II)Fluctuations (II) NA49

• Suggestion of QCD critical end point?

• Fluctuation Conservation

Chiho NONAKA

Kinetic Freeze-out TemperatureKinetic Freeze-out Temperature

Xu and Kaneta, nucl-ex/0104021(QM2001)BRAHMS nucl-ex/0404011

Low T comes from large flow.

f

?

Entropy density

EOS with CEP• Kinetic Freeze-out mean (elastic) collision rate expansion rate

SPS ~ 0.09RHIC ~ 0.09

!

(p + p ) /(" ++ "#

)

• π−N cross section

• Collision rate SPS ~ RHIC

• Expansion rate RHIC > SPS

RHIC > SPS

•Tf

Chiho NONAKA

CEP at RHIC?CEP at RHIC?

Lacey et al, Phys.Rev.Lett.98:092301,2007

Au+Au Collisions

!

"

s~ T# f cs

!

T =165 ± 3 MeV

!

cs

= 0.35 ± 0.05

!

" f = 0.3± 0.03fm

!

"

s~ 0.09 ± 0.015

Chiho NONAKA

Where is CEP on T-Where is CEP on T-µµBB plane? plane?• NJL/I, NJL/II Asakawa & Yazaki• CO (composite operator) Barducci et al.• NJL/inst (instanton NJL) Berges & Rajagopal• RM (random matrix) Halaz et al.• LSM (linear sigma model)/ NJL Scavenius et al.• CJT(effective potential) Hatta & Ikeda• HB (hadronic bootstrap) Antoniou et al.Stephanov:hep-ph/0402115

experiments

Chiho NONAKA

Where is the Critical Point ?Where is the Critical Point ?

Science,312,190(2006)

Energy Scan:• Necessity of lower energy experiments at RHIC• SPS