Third Moments of Conserved Chargesas Probes of QCD Phase Structure

Masakiyo Kitazawa(Osaka Univ.)

M. Asakawa, S. Ejiri and MK,PRL103, 262301 (2009).

xQCD, Bad Honnef, June 22, 2010

Phase Diagram of QCD Phase Diagram of QCD Phase Diagram of QCD Phase Diagram of QCD

T

0

HadronsColor SC

Quark-Gluon Plasma

QCD critical point

How can we map these components of phase diagram in heavy-ion collision experiments?

Third moments of conserved charges(including skewness) would smartly do this!

Fluctuations at QCD Critical Point Fluctuations at QCD Critical Point Fluctuations at QCD Critical Point Fluctuations at QCD Critical Point

2nd order phase transition at the CP.

divergences of fluctuations of

•pT distribution•freezeout T•baryon number, proton, chage, …

Stephanov, Rajagopal, Shuryak ’98,’99

baryon # susceptibility

However,•Region with large fluctuations may be narrow.•Fluctuations may not be formed well due to critical slowing down. •Fluctuations will be blurred by final state interaction.

(Net-)Charge Fluctuations (Net-)Charge Fluctuations (Net-)Charge Fluctuations (Net-)Charge Fluctuations

2( )4

Q

ch

ND

N

D-measure:

Asakawa, Heinz, Muller, ’00Jeon, Koch, ’00

When is experimentally measured D formed?•Conserved charges can remember fluctuations at early stage, if diffusions are sufficiently slow.

NQ

NQ: net charge # / Nch: total # y

0

0

us

s

d

duu

d

gg

g g

g

hadrons:hadrons: quark-gluon:quark-gluon:

D ~ 3-4 D ~ 1largesmall

values of D:

Experimental Results for Experimental Results for DD-measure -measure Experimental Results for Experimental Results for DD-measure -measure

•Failure of QGP formation?•Is the diffusion so fast?

NO! The result does not contradict these statements.Large uncertainty in Nch. Bialas(’02), Nonaka, et al.(’05)

RHIC results: D ~ 3

PHENIX ’02, STAR ’03

•hadron gas: D ~ 3-4•free quark-gluon gas: D ~ 1

STAR, ’10

Higher Order Moments Higher Order Moments Higher Order Moments Higher Order Moments

Ratios between higher order moments (cumulants)

Higher order moments increase much faster near the CP.

Ejiri, Karsch, Redlich, ’05Gupta, ’09

Stephanov, ’09

We want much clearer signals to map the phase diagram, such as changing signs.

We want much clearer signals to map the phase diagram, such as changing signs.

RBC-Bielefeld ’09

Hadrons:1 Quarks:1/32

4th/2nd at =0 reflects the charge of quasi-particles

Take a Derivative of Take a Derivative of BB Take a Derivative of Take a Derivative of BB

B has an edge along the phase boundary

B

B

changes the sign at QCD phase boundary!

•m3(BBB) can be measured by event-by-eventanalysis if NB in y is determined for each event.

Note:33

33 2

( )1(BB )B

B

B B

B mVT

N

V

: third moment of fluctuations

22

2

( )1 B

BB V VT

N

y

NB

Impact of Negative Third Moments Impact of Negative Third Moments Impact of Negative Third Moments Impact of Negative Third Moments

Once negative m3(BBB) is established, it is evidences that

(1) B has a peak structure in the QCD phase diagram.(2) Hot matter beyond the peak is created in the collisions.

•No dependence on any specific models.•Just the sign! No normalization (such as by Nch).

Third Moment of Electric Charge Third Moment of Electric Charge Third Moment of Electric Charge Third Moment of Electric Charge

•net baryon # in y : difficult to measure•net charge # in y : measurable!

Experimentally,

3 3

3 2 3

(( QQ

1Q

))

Q

Q

mV

N

T V

Q : chemical potential associated to NQ

Under isospin symmetry,

3

1 1( )

8 2QQQ

7 B IB

m

singular @CEP isospin susceptibility(nonsingular)

Hatta, Stephanov ’02

Third Moment of Electric Charge Third Moment of Electric Charge Third Moment of Electric Charge Third Moment of Electric Charge

•net baryon # in y : difficult to measure•net charge # in y : measurable!

Experimentally,

3 3

3 2 3

(( QQ

1Q

))

Q

Q

mV

N

T V

Q : chemical potential associated to NQ

B

I/9

The Ridge of Susceptibility The Ridge of Susceptibility The Ridge of Susceptibility The Ridge of Susceptibility

= 0 at B=0 (C-symmetry)

m3(BBB) is positive for small B (from Lattice QCD)

Region with m3(BBB)<0 is limited near the critical point:

~ B at B>>QCD (since ~B4 for free Fermi gas)

T

The Ridge of Susceptibility The Ridge of Susceptibility The Ridge of Susceptibility The Ridge of Susceptibility

Analysis in NJL model:

= 0 at B=0 (C-symmetry)

m3(BBB) is positive for small B (from Lattice QCD)

Region with m3(BBB)<0 is limited near the critical point:

~ B at B>>QCD (since ~B4 for free Fermi gas)

T

m3(BBB)<0

m3(QQQ)<0

Proton # Skewness @STAR Proton # Skewness @STAR Proton # Skewness @STAR Proton # Skewness @STAR STAR, 1004.4959

Measurement of the skewnessof proton number @STARshows thatfor 19.6-200GeV.

3( ) 0PN

Proton # Skewness @STAR Proton # Skewness @STAR Proton # Skewness @STAR Proton # Skewness @STAR

Remark: Proton number, NP, is not a conserved charge.

No geometrical connection b/w 2nd & 3rd moments.3 2

2

( ) ( )P P

PVT V

N

T

N

Measurement of the skewnessof proton number @STARshows thatfor 19.6-200GeV.

STAR, 1004.4959

3( )BBB B

B

m

3( ) 0PN

Derivative along Derivative along TT Direction Direction Derivative along Derivative along TT Direction Direction

Signs of m3(BBE) and m3(QQE) change at the critical point, too.

T

T

ˆT T T

ˆ /T

2

3 3ˆ

2

3 3ˆ

( ) ( )1( )

( ) ( )1

E

EQQ( )

BBB B

QQ

Tm

VT T T

Tm

VT

N

N

T

E

T

E

E : total energy in a subvolume measurable experimentally

More Third Moments More Third Moments More Third Moments More Third Moments 3 2

ˆ3 5 3

ˆ

2ˆ

3 4

2ˆ

3 4

( ) ( )1( )

( ) 1( )

(

B

EEE

EE

EE) 1

(2

Q )Q

B

B

B

E

E

T Cm

VT T T

Cm

VT T

Cm

VT T

N

N

E

T

T

ˆT

ˆ /T

22

ˆ 2 2ˆ

( )TC

V T V

E

T

“specific heat” at constant ̂•diverges at critical point•edge along phase boundary

More Third Moments More Third Moments More Third Moments More Third Moments 3 2

ˆ3 5 3

ˆ

2ˆ

3 4

2ˆ

3 4

( ) ( )1( )

( ) 1( )

(

B

EEE

EE

EE) 1

(2

Q )Q

B

B

B

E

E

T Cm

VT T T

Cm

VT T

Cm

VT T

N

N

E

Signs of these three moments change, too!

T

T

ˆT

ˆ /T

22

ˆ 2 2ˆ

( )TC

V T V

E

T

“specific heat” at constant ̂•diverges at critical point•edge along phase boundary

Model Analysis Model Analysis Model Analysis Model Analysis

•Regions with m3(*EE)<0 exist even on T-axis. This behavior can be checked

2-flavor NJL; G=5.5GeV-2, mq=5.5MeV, =631MeV

•on the lattice•at RHIC and LHC energies

Trails to the Edge of Mountains Trails to the Edge of Mountains Trails to the Edge of Mountains Trails to the Edge of Mountains

m3(EEE) on the T-axis

2 3

3 3 3

1 ( ) 1E( )E E

C S

T

T Tm

T T T T

•Experimentally: RHIC and LHC

•On the lattice:

Trails to the Edge of Mountains Trails to the Edge of Mountains Trails to the Edge of Mountains Trails to the Edge of Mountains

m3(EEE) on the T-axis

2 3

3 3 3

1 ( ) 1E( )E E

C S

T

T Tm

T T T T

•Experimentally: RHIC and LHC

•On the lattice:

•Experimentally: energy scan at RHIC

•On the lattice: ex.) Taylor expansion

2 4 62 4 6B B Bc c c

3 4 63( ) ~BBB 5B Bm c c

Cheng, et al. ‘08

c4 c6

m3(QQQ), etc. at >0

Summary 1 Summary 1 Summary 1 Summary 1

Seven third moments

all change signs at QCD phase boundary near the critical point.

To create a contour map of the third moments on the QCD phase diagram should be an interesting theoretical subject.

m3(BBB), m3(BBE), m3(BEE), m3(EEE), m3(QQQ), m3(QQE), and m3(QEE)

Negative moments would be measured and confirmed bothin heavy-ion collisions and on the lattice. In particular,

(1) m3(EEE) at RHIC and LHC energies,(2) m3 (QQQ)=0 at energy scan,

are interesting!

Summary 2 Summary 2 Summary 2 Summary 2

Let’s go see the scenery over the ridge!Let’s go see the scenery over the ridge!But, do not forget to first draw a mapBut, do not forget to first draw a map

for a safe expedition.for a safe expedition.

Critial Point

Loreley, photo by MK, 2005

Derivative along Derivative along TT direction direction Derivative along Derivative along TT direction direction

simple T-derivative:2 3 2

3

2 3 2

3

( ) ( ) ( )

( ) ( ) ( )Q

B

I

B

Q Q

B BT

T VT

E

E T

T V

N

T

N N

N N N

E : total energy in a subvolume

2 2

3 33 3

( ) ( )( ) , ( )BQEQ EB

Q ENm m

VT VT

E EN

measurable experimentally

Problem: T and can not be determined experimentally.

mixed 3rd moments:

Further Possibility Further Possibility Further Possibility Further Possibility

•If measured moments originate from a narrow region in the T- plane, and •if experimental resolution is sufficiently fine,

3 22ˆ

3

3 3

( ) ( )( )

( )EEE E( )BE

BE ET C

T VT

m

N

mT

exp. exp.

lattice

T

ˆT

This formula is used to determine /T experimentally.

Moreover, third moments provide the divergence vector of and C .These information may enable us to pin down the initial state of fireballs.

Loreley