kensuke homma / hiroshima univ.1 ゆらぎのはなし本間謙輔松本大学にて内容 1....

Kensuke Homma / Hiroshima Univ.

1

ゆらぎのはなしゆらぎのはなし

本間謙輔松本大学にて

内容

1. 原始時代の紆余曲折。2. こんな簡単なことで臨界現象が議

論できるなんて！3. 我田引水の解釈および現象論屋さ

んに考えて頂きたいこと。


2

Why fluctuation ?Why fluctuation ?• Fluctuations carries

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

• Why don’t we use the genuine 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

Measure maximum deviation size in homogeneous flux in our method


3

Au+Au Au+Au √sNN = 200GeV at √sNN = 200GeV at PHENIXPHENIX

• Using magnetic field-off

• Charged Track Drift chamber, Pad chamber1 with BBC vertex

• Photon Cluster Electro-magnetic calorimeter

– Cluster shower shape

– Time of flight

– association cuts by tracks

• Precisely data quality assurance was necessary to reject detector effect!

2/1

35.0||


4

Application of wavelet analysisApplication of wavelet analysis

η

extract this region

High resolution

Low resolution


5

High energy cosmic ray experiment and PHENIXHigh energy cosmic ray experiment and PHENIX

Charged track

Photon cluster

PHENIX 7.24 standard deviation

J. J. Lord and J. Iwai. Int. Conference on High Energy Physics, TX, 1992

○: Photon

+ : Charged Particle

Can DCC scenario explain these events or something else?


6

Maximum differentiMaximum differential balance distributial balance distributi

onsons

Work in Progress

δBmax [arbitrary unit]

• δBmax distribution (each centrality:10%) with base line fluctuation– black : binomial sample, 100

times larger statistics than real data obtained by hit map

– red : data

明らかな離れ孤島は見つからず。

わざわざ大げさな探索しなくたって、そもそも分布は、二項分布とは明らかに異なる。

荷電 πのみを使用して、熱・統計力学的に分布を議論するほうが生産的。


7

Measuring Multiplicity Fluctuations with Negative Measuring Multiplicity Fluctuations with Negative Binomial DistributionsBinomial Distributions

km

m

k

k

km

kmmP

1!1!

!1)(

112

2

k

Multiplicity distributions in hadronic and nuclear collisions can be well described by the Negative Binom

ial Distribution.

Central 62 GeV Au+Au

UA5: sqrt(s)=546 GeV p-pbar, Phys. Rep. 154 (1987) 247.

E802: 14.6A GeV/c O+Cu, Phys. Rev. C52 (1995) 2663.

UA5

E802

62 GeV Au+Au5-10% Centr

al

25-30% Central

PHENIX Preliminary

PHENIX Preliminary

Ntracks Ntracks

dN

/Ntr

acks

dN

/Ntr

acks


8

RHIC is accessible tothis transition line

Current understanding of QCD phase diagramCurrent understanding of QCD phase diagramT

mq

B

Surface of 1st

order transitions

QGP

<qq>

tricritcal end-pointmq=0

end-point linemq!=0

2nd ordermq=0

No quantitative agreementbetween theoretical predictions

Experiments see partonic degree of freedom in collected flows at RHIC


9

PHENIX on going analysisPHENIX on going analysis• Isothermal compressibility

• Heat capacity

• Near side azimuthal correlation function

• Breaking of v2 scaling

• Disappearance of baryon anomaly

• Correlation length and susceptibility in longitudinal space


10

What is the critical behavior ?What is the critical behavior ?

A simulation based on two dimensional Ising modelfrom ISBN4-563-02435-X C3342l

Ordered T=0.995Tc Critical T=Tc Disordered T=1.05Tc

Black Black & White GrayVarious sizesfrom small to large

Sca

le t

rans

form

atio

n

Spatial pattern ofordered state

Large fluctuations of correlation sizes on order parameters: critical temperature (focus of this talk) Universality (power law behavior) around Tc reflecting basic symmetries and dimensions of the underlying system: critical exponent (future study after finding Tc)


11

Order parameter and phase transition Order parameter and phase transition

φ

g-g0

hbTaTAghTg 4220 4

1)(

2

1))((

2

1),,(

)()(

In Ginzburg-Landau theory with Ornstein-Zernike picture, free energy density g is given as

external field h causes deviation of free energyfrom the equilibrium value g0. Accordingly an orderparameter fluctuates spatially.

Longitudinal multiplicity density fluctuation from the mean density is introduced as an order parameter in the following.

spatial correlation

a>0 a=0 a<0

disappearsat Tc

)()( 0 cTTaTa In the vicinity of Tc, must vanish, hence

b>0 for 2nd order


12

More exact free potential form More exact free potential form

)()(2

1)cosh(

)cosh(2

1

),,(

2

2

2

20

Uyyy

xddyghTgSy

In narrow midrapidity region, cosh(y)~1 and y~

Although theorists want to define U() as a power term by assuming the system is just on Tc and/or on critical end-point, most of experimentally accessible phase spaces are relatively far from the critical temperature or end-point. It is more natural to use the polynomial expansion for the potential term.

dydz

yt

yz

)cosh(

)sinh(


13

Two point correlation function Two point correlation function & Fourier transformation& Fourier transformation

22

2

12

21)(

21

21)(

212

21212

)()(

)()(

),(

)()(),(

12

12

kikyiky

yyik

yyik

dyeydyeyGY

yyy

dydyeyy

dydyeyyG

yyyyG

Two point correlation

Fourier transformation

Relative distance between two points


14

Expectation value of |Expectation value of |kk||22

from free energy deviationfrom free energy deviation

222

/

220

)()(

1||

))((

))()((||2

1)(

1/

)(

kTATaY

NTdew

Neyw

kTATadyggY

Yg

ey

kk

ikykkk

Tg

kk

k

ikyk

Statistical weight can be obtained from free energy

Fourier expression of order parameter


15

Function form of Function form of two point correlation function two point correlation function

)(

)()(

)()(2

)(

)()(

1||

)(||

0

2

)(/||22

22

)(2

2

c

Ty

k

yikk

TTa

TAT

eTTAY

NTyG

kTATaY

NT

dyeyGY

A function form of correlation function is obtained by inverse Fourier transformation.

From g (up to 2nd order)due to spatial fluctuation

Fourier transformationof two point correlationof order parameter

In 1-D case


16

/21212

2111212212

21

2

2121

/),(

)()(),(),(

1),(,

1)(

eC

C

dd

d

d

d

inelinel

)(

)()(,)(

)(

))()()((|)(|

0

2)(/||

21212

21

c

T

TTa

TATeT

TA

T

G

From two point correlation to two particle correlationFrom two point correlation to two particle correlation

Rapidity independent termmust be added, since T has afinite range in experiments.

Two point correlation function in 1-D case at fixed T

Two particle correlation function


17

Multiplicity density measurements in PHENIXMultiplicity density measurements in PHENIX

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

PHENIX Preliminary

Negative Binomial Distributioncan describe data very well.

n/<n>

small

large

Pro

babi

lity

(A.U

.)

Zero magnetic field to enhance low pt statisticsper collision event.

pt down to100MeV/c in B-OFF200MeV/c in B-ON


18

Relations between N.B.D andRelations between N.B.D andintegrated correlation functionintegrated correlation function

2

/2

20 0 21

212121

)1/(2

/),()(

e

ddCk

Integrated correlation function can be related with 1/k

n

kkk

k

kn

knP k

n

kn

,11

,/1

1

/1

/

)()1(

)(2

2)(

Negative Binomial Distribution

k →∞ corresponds to Poisson distribution.k = 1 corresponds to Bose-Einstein distribution.Intuitively k is the number of Bose-Einstein emission sources.

can be obtained from fit to k vs.


19

Intuitive summary of advocated observationIntuitive summary of advocated observation

Compare many systems by changing resolutions andfind the increase of correlationlength in the very vicinity of Tcbut not at Tc !

exponential damping of the number ofwaves with a typical correlation length

Absence of typical correlation length causes fractal nature (power law behavior)

GL applicable

GL not applicable


20

How to define initial temperature?How to define initial temperature?

Np can be a fine scan on initial temperature T, while collision energy can be a coarse scan. Tc can be investigated with the fine scan. It is a natural assumption that Np is a monotonic function of initial T.

Transverse energy ET

21

0-5%

15-20%10-15%

0-5%

5-10%

Number of participants, Np and Centrality Number of participants, Np and Centrality peripheral central

b To ZDC

To BBC

Spectator

Participant Np

Multiplicity distribution

Np depends on centrality bin width


22

N.B.D. N.B.D. kk vs. d vs. d

PHENIX PreliminaryAu+Au @√sNN=200GeV


2

/2 )1/(2

)(

1 e

k

10 % centralitybin width

5% centralitybin width

k()

One correlation lengthassumption is reasonable.

23

PHENIX preliminary resultsPHENIX preliminary results can absorb all rapidity independent fluctuations or offset contributions like;

1. finite centrality bin width( Np or initial temperature fluctuations )2. reaction plain rotations andelliptic flows due to partialsampling in azimuth by PHENIX

PHENIX Preliminary

PHENIX Preliminary


10% cent. bin width 5% cent. bin width

Shift to smaller fluctuations

Np

)(

)()(

0 cTTa

TAT

Divergence of correlation length canbe a signature of critical temperature.


24

Correlation length Correlation length and static susceptibility and static susceptibility

)(

)()(

0 cTTa

TAT

PHENIX Preliminary

PHENIX Preliminary

Np

Np

T

GTTTa

kTTa

gg

h

k

ck

c

k

kk

21

0

20

0

220

1

20

2

)0()(

1

)1)((

1

)(

k=

0 *

T

Divergence of susceptibility is theindication of 2nd order phase transition.

Divergence of correlation length is theindication of a critical temperature.

Co

rrel

atio

n le

ng

th

T~Tc?

Au+Au √sNN=200GeV

Au+Au √sNN=200GeV


25

What about relation with <qq>What about relation with <qq>

PHENIX dNch/dcorresponding to Np~90PRC 71, 034908 (2005)

PHENIX BJ (=1fm/c)corresponding to Np~90PRC 71, 034908 (2005)

NA50, Eur. Phys. J. C39 (2005):355

Accidental coincidence ?Need more simultaneous observations!


26

SummarySummary

1. Multiplicity density distribution in Au+Au collisions at √SNN=200 GeV can be well approximated by N.B.D..

2. Two point correlation lengths have been extracted based on the function form by relating pseudo rapidity density fluctuations to the GL theory up to the second order term in the free energy. One correlation length assumption is enough accurate.

3. Absolute scale of correlation length is 0.001~0.01.

4. The static susceptibility as a function of Np indicate a non monotonic increase at Np~90. The corresponding Bjorken energy density is 2.4GeV/fm3 with =1.0 fm and transverse area=60fm2

5. It is interesting to note the coincidence with the energy density at which J/y suppression begins at SPS. Need more simultaneous observations by independent methods in order to conclude whether the non monotonic behavior is related with critical temperature or not.


27

QuestionsQuestions

1. What can be an external field to the longitudinal density correlation?

2. Why the correlation length can be so small ? Is it possible to cause temperature like correlations by using particles each of which occupies 1/1000 of unit rapidity (dNch/dEta ~ 1000 at RHIC)? Is there another type of thermalizaion before hadron formation?

3. Why the non monotonic behavior of correlation length looks like a peak rather than a step? 　 Why is the behavior so sensitive to initial energy densities rather than the density at the critical temperature.

4. How rapid is the longitudinal expansion?

5. Are there plural formation times? Is there a time for fluctuations are embed before hadron formation time?

6. How fluctuations in very early stage survive in the final state, while the collision system is rapidly expanding in longitudinal space?


28

The answer might be Hawking-Unruh effect in The answer might be Hawking-Unruh effect in background condensed color electric fieldbackground condensed color electric field

2k

T

sss

e

gQQgQa

meEa

aT

22 /

/2

Hawing Unruh

(QED)

(QCD)2/3.0~6/1 sAQT s

kensuke homma / hiroshima univ.1 ゆらぎのはなし 本間謙輔 松本大学にて 内容 1....

Documents

kensuke homma / hiroshima univ.1 ゆらぎのはなし本間謙輔松本大学にて内容 1....