in collaboration with c. nonaka, b. m ü ller, and s.a. bass

@ JINR 2009 @ JINR 2009 September 2009

in collaboration with C. Nonaka, B. Müller, and S.A. Bass S. Ejiri and M. Kitazawa

New Ways to Look for the Critical Point

and Phase Transition

New Ways to Look for the Critical Point

and Phase Transition

Department of Physics, Osaka UniversityDepartment of Physics, Osaka University

Masayuki AsakawaMasayuki Asakawa

M. Asakawa (Osaka University)

QCD Phase Diagram

CSC (color superconductivity)

QGPQGP ((quark-gluon plasmaquark-gluon plasma))

Hadron Phase

T

B

• chiral symmetry breaking• confinement

order ?

1st order

crossover

CEP(critical end point)160-190 MeV

5-100

RHIC

LHC


Punchlines: Quantities to Look at

• Quantities not subject to Critical Slowing Down and Final State Interactions

• Quantities sensitive to CEP or Phase Transition

• Conserved Quantities unlike correlation lengths, (usual) fluctuations...etc.

What to see? How to see?

Question


CEP = 2nd order phase transition, but...

If expansion is adiabatic ANDno final state int.

Divergence of Fluctuation Correlation Length Specific Heat ?Critical Opalescence?

CEP = 2nd Order Phase Transition Point

2SC

CFLeven if the system goes right throughthe critical end point…

Subject to Final State Interactions


Critical Slowing Down and Final State Int.

Furthermore, critical slowing down limits the size of fluctuation, correlation length !

Time Evolution along givenisentropic trajectories (nB/s : fixed)

00

1( ) ( ) ,

zAm m m

1( ) ( ) ( )

( )eq

dm m m

d

Model H (Hohenberg and Halperin RMP49(77)435)3z

and eq

big differencealmost no enhancement

1-dim Bjorken Flow Nonaka and M.A. (2004)


Locally Conserved Charges

Fluctuations of Locally Conserved Quantities

i. Net Baryon Number

ii. Net Electric Charge

iii. Net Strangeness

iv. Energy ...

Heinz, Müller, and M.A., PRL (2000)Also, Jeon and Koch, PRL (2000)

Hadron Gas Phase

• ~2/3 of hadrons carry

electric charge 1±

Net Electric Charge

QGP Phase

• only ~ 1/2 of d.o.f., i.e.

quarks and anti-quarks

carry electric charge 1 23 3,± ±


Charge Fluctuation at RHIC

Predictions

QGP phase Hadron phase

D-measure2( )

4ch

QD

N

~ 1D ~ 4D( : hadron resonance gas)~ 3D

(STAR)

(PHENIX)~ 4D

2.8 0.05D Experimental Value

Białas, PLB(2002)Nonaka, Müller, Bass, M.A., PRC (2005)

Well-Explained by Quark Recombination


Odd Power Fluctuation Moments

We are in need of observables that are not subject to final state interactionsWe are in need of observables that are not subject to final state interactions

Fluctuation of Conserved Charges: not subject to final state interactions

One of exceptions: Stephanov (2008) in conjunction with CEP

2Q• Usual Fluctuations such as : positive definite

Absolute values carry information of states

Odd power fluctuations :

Asakawa, Heinz, Müller, Jeon, Koch

On the other hand,

• In general, do not vanish (exception, )A• Sign also carry information of states• Sign also carry information of states

NOT positive definiteNOT positive definite

• Usually fluctuations such as 2Q have been consideredeven powereven power


Physical Meaning of 3rd Fluc. Moment

B : Baryon number susceptibility

in general, has a peak along phase transition

B

B

changes the sign at QCD phase boundary !

In the Language of fluctuation moments:

33

33 2

( )1(BBB)

BB

B B

Nm

V VT

22

2

( )1 B

BB

N

V VT

B

more information than usual fluctuation


(Hopefully) More Easily Measured Moments Third Moment of Electric Charge Fluctuation

3 3

3 2 3

( ) 1(QQQ)

Q

Q

Nm

VT V

2 1

3 31 1

2 2

Q u d

B I

N N N

N N

3

1 1(QQQ)

8 27 B IB

m

singular @CEP

iso-vector susceptibilitynonsingular when Isospin-symm.

Hatta and Stephanov 2002

B

I/9


Mixed Moments Energy is also a conserved charge and mesurable !

2

3 3ˆ

2

3 3ˆ

( ) ( )1E(BB )

( ) ( )1(QQE)

B B

Q Q

N Tm

VT T T

N Tm

VT T

E

T

E

T

T

ˆT T T

ˆ /T

E : total energy in a subvolume

• Result with the standard NJL parameters (Nf=2)

Regions with Negative fluctuation moments


More and More 3rd Moments3 2

ˆ3 5 3

ˆ

2ˆ

3 4

2ˆ

3 4

( ) ( )1(EEE)

( ) 1(BEE)

( ) 1(QEE)

2

B

B

Q

Q

E T Cm

VT T T

N E Cm

VT T

N E Cm

VT T

These quantities are expressed conciselyas derivatives of “specific heat” at constant ̂

22

ˆ 2 2ˆ

( )EC T

T VT

• diverges at critical end point• peaks on phase transition line

T

T

ˆT T T

ˆ /T “specific heat” at constant ̂


Comparison of Various Moments2-flavor NJLwith standard parameters

• Different moments have different regions with negative moments

By comparing the signs of various moments, possible to pin down the origin of moments

• Negative m3(EEE) region extends to T-axis (in this particular model)• Sign of m3(EEE) may be used to estimate heat conductivity

G=5.5GeV-2

mq=5.5MeV=631MeVNear SideNear SideNear SideNear Side

Far SideFar SideFar SideFar Side


Principles to Look for Other Observables

We are in need of observables that are not subject to final state interactionsWe are in need of observables that are not subject to final state interactions

After Freezeout, no effect of final state interactions

Chemical Freezeout

• usually assumed momentum independent

• but this is not right

chemical freezeout time: pT (or T) dependent

Larger pT (or T), earlier ch. freezeout

Principle I


Emission Time Distribution

Emission Time

• Larger T, earlier emission

• No CEP effect (UrQMD)


Principle II

Universality: QCD CEP belongs to the same universality class as 3d Ising ModelUniversality: QCD CEP belongs to the same universality class as 3d Ising Model

(T,) (r,h)

QCD Critical End Point 3d Ising Model End Point

same universality class

h : external magnetic fieldC

C

T

TTr

and BT

Further Assumptions• Size of Critical Region

• No general universality• Lattice calculation: not yet V limit• Effective Model Results ?

need to be treated as an input, at the moment


Singular Part + Non-singular Part

• Matched Entropy Density

( , ) :Q Bs T Hadron Phase (excluded volume model)

QGP phase

( , ) :H Bs T

• Dimensionless Quantity: Sc

D: related to extent of critical region

2 2( , ) ( , ) crit critc B c BS T s T T D

Matching between Hadronic and QGP EOS

• Entropy Density consists of Singular and Non-Singular Parts

Requirement: reproduce both the singular behavior and known asymptotic limits

• Only Singular Part shows universal behavior

critBcrit

1 1( , ) 1 tanh ( , ) ( , ) 1 tanh ( , ) ( , )

2 2real B c B H B c B Q Bs T S T s T S T s T


Isentropic Trajectories• In each volume element, Entropy (S) and Baryon Number (NB) are conserved, as long as entropy production can be ignored (= when viscosities are small)

Isentropic Trajectories (nB/s = const.)

Near CEP s and nB change rapidly

isentropic trajectories shownon-trivial behavior

Bag Model EOS case

An Example

CEP


ConsequenceFor a given chemical freezeout point,prepare three isentropic trajectories:w/ and w/o CEP

Along isentropic trajectory:

As a function of pT(T):

Principle I

: near CEP steeper

B

T• FO, CO

• QCP

B

T

B

T• FO, CO

• QCP

B

TBag Model EOS(w/o CEP, usual hydro input)

p / p ratio


Evolution along Isentropic Trajectory

p / p ~ exp 2B

T

with CEP steeper spectra at high PT

p

As a function of pT(T):

B

T

• FO, CO

• QCPB

T


Summary

p / p ratio behaves non-monotonously near CEP

Conserved Charges and Higher Moments: Conserved Charges and Higher Moments:

Ratio : Two Principles Ratio : Two Principles

p / p

p / p

Final State Interaction and Critical Slowing Down Final State Interaction and Critical Slowing DownUsual observablessuch as large fluctuation, critical opalescence, ...etc. do not work

i) Third Fluctuation Moments of Conserved Charges take negative values in regions on the FAR SIDE of Phase Transition 　　　 (more information!)

ii) Different Moments have different regions with negative moments

By comparing different moments, possible to pin down the origin of hot matter, get an idea about heat conductivity...etc.

i) Chemical Freezeout is pT(T) dependent

ii) Isentropic Trajectory behaves non-trivially near CEP (focusing)

Information on the QCD critical point:such as location, size of critical region, existence...

in collaboration with c. nonaka, b. m ü ller, and s.a. bass

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