1

Study of order parameters through Study of order parameters through fluctuation measurements by the fluctuation measurements by the

PHENIX detector at RHICPHENIX detector at RHIC

Kensuke Homma for the PHENIX collaboration

Hiroshima University

On Aug 11, 2005 at KromerizXXXV International Symposiumon Multiparticle Dynamics 2005

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MotivationsMotivations

• RHIC experiments probed the state of strongly interacting dense medium with many properties consistent with partonic medium. What about the information on the phase transition?

• Is it the first order or second order transition?• Are there interesting critical phenomena such as tricritical

point?

K. Rajagopal and F. Wilczek, hep-ph/0011333

3

Landau’s treatment Landau’s treatment for 2for 2ndnd order phase transition order phase transition

TSUG

hTuTrTgVTG 42 )()(),(/),(

0)()( aTcTaTrSince order parameter should disappear at T=Tc, assume

Valid in a limit where the fluctuation on order parameter is negligible even at T~Tc

Gibb’s free energy

g(T,)

T>Tc =0

T<Tc =a(T-Tc)/2u

120

0

)122(

urh

G

hh h

hhH T

G

V

T

T

S

V

TC

2

2

||4

1,

||2

1

TcTaTcTa

Susceptibility

Specific heat

and CH show divergence or discontinuity,while T varies around Tc.

T>Tc T<Tc

u

TcTag

4

)( 22

4

Susceptibility and density fluctuationsSusceptibility and density fluctuations

rkik

rkik erehrh

0)()(

hTuTrATgVTG 422 )()()(),(/),(

120

2 )1222( urAk

hk

kk

222

222

1

)(

|)|2(2

1

1

)(

|)|(2

1

k

T

TcTaAk

k

T

TcTaAk

k

k

Susceptibility

2/1

2/1

||2

||

TcTa

A

TcTa

A

)'())'()()(( rrTkrr B

Fluctuation-dissipation theorem

|'|

)/|'|exp())'()()((

rr

rrTkrr B

Ornstein-Zernike behavior

kB

kkkk V

Tk

))((

T>Tc

T<Tc

22

)'( 1)'(

k

rrdVerrk

k

With Fourier transformation

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• Multiplicity fluctuations (density fluctuations ) as a function rapidity gap size with as low pt particle as possible.

Correlation length and singular behavior in correlation function.

• Average pt fluctuations (temperature fluctuations)

Specific heat See PRL. 93 (2004) 092301

• In this talk, I will focus on only multiplicity fluctuation measurements.

Fluctuation measurements by PHENIXFluctuation measurements by PHENIX

7.0

lGeometrical acceptance

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E802: 16O+Cu 16.4AGeV/c at AGSmost central events

[DELPHI collaboration] Z. Phys. C56 (1992) 63[E802 collaboration] Phys. Rev. C52 (1995) 2663

DELPHI: Z0 hadronic Decay at LEP2,3,4-jets events

Universally, hadron multiplicity distributions are well described by NBD.

Charged particle multiplicity distributions and Charged particle multiplicity distributions and negative binomial distribution (NBD)negative binomial distribution (NBD)

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2

2

2

22

2

22

2

2

)(

1

)(

1)(1

)(

1

11

/1

1

/1

/

)()1(

)(

)1/(

n

nnF

Fk

nn

k

kk

k

kn

knP

P

k

n

kn

nnn

Bose-Einstein distributionμ: average multiplicity

F2 : second order normalized factorial moment

NBD correspond to multiple Bose-Einstein distribution and the parameter k corresponds to the multiplicity of those Bose-Einstein emission sources. NBD can be Poisson distribution with the infinite k value.

NBD

Negative binomial distribution (NBD)Negative binomial distribution (NBD)

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δη= 0.09 (1/8) : P(n) x 107 δη= 0.18 (2/8) : P(n) x 106

δη= 0.35 (3/8) : P(n) x 105

δη= 0.26 (4/8) : P(n) x 104

δη= 0.44 (5/8) : P(n) x 103

δη= 0.53 (6/8) : P(n) x 102

δη= 0.61 (7/8) : P(n) x 101

δη= 0.70 (8/8) : P(n)

No magnetic fieldΔη<0.7, Δφ<π/2

PHENIX: Au+Au √sNN=200GeV

Charged particle multiplicity distributions Charged particle multiplicity distributions in different din different d gap gap

| Z | < 5cm

-0.35 < η < 0.35

2.16

< φ

< 3

.73

[rad

]

The effect of dead areas have been corrected.

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212111

21212

22

212

212

1

2111

212

2111

212212

)()(

),(1

)(

1

:),(

:),(

:)(

1)()(

),(

)()(

),(),(

dydyyy

dydyyyCKF

k

yyC

yy

y

yy

yy

yy

yyCyyR

inclusive single particle density

inclusive two-particle density

two-particle correlation function

Relation with NBD k

Normalized correlation function

Candidates of function forms with two particle correlation length

beR

eRR

yy

eRR

yy

yy

yy

/||2

/||02

21

/||

02

21

21

21

||

HBT type correlation in E802 : failed to describe data

Empirical two component model with R0=1.0

Relation between k and Relation between k and integrated two particle correlation functionintegrated two particle correlation function

Most general form: many trials failed.

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E802 type function can not describe the dataE802 type function can not describe the data

)]1)(/(1[

2/1)(

/0

/||02

21

eRk

eRR yy

Correlation function used in E802P. Carruthers and Isa Sarcevic,Phys. Rev. Lett. 63 (1989) 1562

NBD k vs. δη at E802

Phys. Rev. C52 (1995) 2663

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Empirical two component fitEmpirical two component fit

PHENIX: Au+Au √sNN=200GeV, Δη<0.7, Δφ<π /2

2

]1/[21

)(

1:

2

/2

2/||

221

beF

kbeR yy

dependent part + independent part with R0 =1

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Participants dependence of Participants dependence of ξξ and and bb

PHENIX: Au+Au √sNN=200GeV

Two particle correlation length Correlation strength of what?

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What is the origin of the two components?What is the origin of the two components?

))'()'()()()(()',(2 rnrnrnrnrrC

2

1 12 )()()'( nrrrrrrC

N

i

N

jji

nVNrnrrrnN

ii

/)(,)()(1

Go back to Ornstein-Zernike’s theory (see Introduction to Phase Transitions and Critical Phenomena by H.E.Stanley)which explains the growth of forward scattering amplitude of lightinteracting with targets at the phase transition temperature.

)'()'()'( 22 rrnrrnrrC

)/|'|exp(

)'(/)'()'(2

rrb

rrnrrrrR

Self interactionrenormalizingsingular part ?

Long range correlation

(r-r’)

r r’

Density of fluid element at r

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ξξ vs. number of participants vs. number of participants

Two particle correlation length

PHENIX: Au+Au √sNN=200GeV

Linear behavior of the correlation length as a function of the number of participants has been obtained in the logarithmic scale.

)log()log(

||

/

3

3

part

cpart

part

N

TTN

TNdydN

One slope fit gives

α = -0.72 ± 0.03

In the case of thermalized ideal gas,

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ConclusionsConclusions• Multiplicity distributions measured in Au+Au collisions at

√SNN=200GeV can be described by the negative binomial distributions.

• Two particle correlation length has been measured based on the empirical two component model from the multiplicity fluctuations, which can fit k vs. d in all centralities remarkably well.

• Extracted correlation length behaves linearly as a function of number of participants in logarithmic scales. Assuming one slope component, the exponent was obtained as -0.72±0.03.

• The interpretation of b parameter is still ambiguous. Any criticize or different view points are more than welcome.

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Backup Slide

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Uncorrected Npart*b vs. NpartUncorrected Npart*b vs. NpartCentrality k-Map in 10% bins

0

50

100

150

200

250

300

350

400

450

50 100 200 500 1000 infinity

intrinsic k

ob

serv

ed k "0-10"

"10-20"

"20-30"

"30-40"

Npart

Npa

rt*

b

Bias on NBD kdue to finite bin size of centrality

Intrinsic k

Obs

erve

d k

18

0-5%

15-20%10-15%

0-5%

5-10%

Important HI jargon : Participants (Centrality) Important HI jargon : Participants (Centrality) peripheral central

Relate them to Npart and Nbinary (Ncoll ) using Glauber model.

Straight-line nucleon trajectories

Constant NN=(40 ± 5)mb.

Woods-Saxon nuclear density:

dRr

r o

exp1

1)(

fm

AAR

)03.065.6(

61.119.1 3/13/1

fmd )01.054.0(

b To ZDC

To BBC

Spectator

Participant

Multiplicity distribution Nch

Whether AA is a trivial sum of NN Whether AA is a trivial sum of NN or or something nontrivial ?something nontrivial ?

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Why not observing fluctuations ?Why not observing fluctuations ?• Fluctuation carries

information in early universe in cosmology despite of the only single Big-Bang event.

• Why don’t we use the event-by-event information by getting all phase space information to study evolution of dynamical system in 　　　 heavy-ion collisions ?

• We can firmly search for interesting fluctuations with more than million times of mini Big-Bangs.

The Microwave Sky image from the WMAP Mission http://map.gsfc.nasa.gov/m_mm.html