揺らぎ解析七転八起の道中記
広島大学
本間 謙輔
謝辞
中村智明氏（東大素粒子センター）
1.
2.
3.
4.

真空の相転移なんて存在するのか？
事象毎の揺らぎ解析（例として、DCCのようなものの探索）
統計力学的な揺らぎ解析への転身
事象毎の揺らぎ解析への回帰

2011/12/16@HIPUB

1

Phase transitions in the early universe
Central dogma of modern physics:
phase transitions are sources ofmatter creations （？？？）

1019GeV
1016GeV~~

0.1GeV
0.1MeV
0.1keV

0.1eV(baryogenesis if strong 1st order transition)

Inflation
Latent heat in 1st ordertransition => big-bang

Testable unique place
Temperature

0.1TeVEW phase transition~~~~

QCD phase transition(confinement & actual hadron mass)

Neutrino decoupling
Photon reheating
Nucleon synthesis

Recombination
Photon decoupling

PresentTime [sec]~~0.1meV

10-36 10-34

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10-101101010220

Conjectured QCD phase diagram
THigh opacity stateBulk matter flow with quark d.o.fHigh temperature state

T=221±23(stat)±18(sys)
Lattice result Tc~170MeV

RHIC can approach from well above Tc
Deconfinement p.t.(?)
Tc
Crossover mq!=0CP ?

1st order ?
Chiral p.t.
PHENIX thermal photon measurement

mBKensuke Homma / Hiroshima Univ.

3

Imaging phase transition in EO crystal
Large electro-optical coefficientof 104 pm/V （Typically order of pm/V)

Fast rise and not too long duration timecompared to effective impact time

KH2PO4(KDP)
KD2PO4(DKDP)

DT=14.2K

DT=4.2K
KH2PO4(KDP)
DT=1.3K

Soviet Physics – Solid State
Vol.8, No. 11 (1967) 2758-2760
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Ferroelectrics, 2002 Vol.272, pp. 57-624

Fourier transform in spatial frequency

DKDP crystal

Sampling by a fiber
Bundle (endoscope)

W wire electrodes

Focal plane
レンズによる局所屈折率変動のフーリエ変換像

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Electro-optic responseto static electric field
At roomtemperature
電場に対する局所応答

Sampling between two wires

VoltSampling far from wires
応答なし

Volt

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Demonstration of phase transitions
in DKDP crystal 相転移温度近傍でのDomain structureappears from around Tc

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非局所応答の出現

7

Centauro event in high-energy cosmic rays○: Photon+ : Charged Particle
J. J. Lord and J. Iwai. Int. Conference on
High Energy Physics, TX, 1992
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8

Disoriented Chiral Condensate
VVrestorationQuench Mechanism

quenchK.Rajagopal andF.Wilczek : Nucl. Phys.
B379, 395 (1993)

σππDCCσQCD vacuum

Linear sigma modeli  ( ,  )11 2 22mL   mi  i    v   H24
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Initially proposed search Strategy
fraction : f n 0n 0  n   n probability : P(f)probabilit y : P( f )df 

12 fdf

No DCC (binomial)

DCC (Centauro type)fraction : f

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Focus on asymmetry with variable domain size
N (DD )  N (DD )
Define an asymmetry betweenAI 3 (DD ) number of charged tracks and neutralN   Nclusters in event-by-event base as a
N ch (DD )  N  (DD )function of subdivided - phase
spaces normalized by one standard
N ch  N deviation for a given multiplicity class.

Domain size and domain position oflargely deviated regions can beobtained at the same time by using
Multi Resolution Analysis (MRA)
technique.

δAI3Domain Size
Deviation Size

Domain Position
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η,φ11ウェーブレット解析入門有限幅の波束を基礎にするが故に、相対関係を刻むことが可能になる。

認知による見事なスケール変換

kkk 1k 2
Mt.Fuji   ( 0 )   ( 0 )   ( 0)
aFuji
aFujiaFuji

k0Tree   ()aTree

スケールaは有限で、aFuji << aTree
無限遠まで続く波の重ね合わせで記述するフーリエ積分では、富士山の形は表現できるが、それが木々の連なりで構成されていることは表現できない。

Multi Resolution Analysis (wavelet)
Level j-1 : 2j-1 bins
0.7

Level j : 2j bins

φ(x)Scaling function

1 φ(2x)

x-0.7

＝0.7

ψ(x)

Wavelet function

1-1(＋)

1 φ(2x-1)

x1(－)

x-1x-0.7

φ(2x) = 1/√2 {φ(x) + ψ(x) }φ(2x-1) = 1/√2 {φ(x) - ψ(x) }Total number of bins is 2j
Level j represents a resolution level
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Signal Decomposition

24/2j→ Domain Sizek→ Domain Position

Signal22

per event
20φ

Djk

j: resolution levelk: k-th bin
Cjk : coefficients of φ
Djk : coefficients of ψ

Cjk

j=4

Look for a maximum Djk

j=3+

j=2+

j=1+

ψ

Cjk → Deviation Size
Djk → used to pick up k
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j=0k+

k14

QuizPink dots are distributed
around 0 based on Gaussian(mean=0, =1.0) over 28 bins.
A single domain is hiddenwith Gaussian (mean=N, =1.0)

Where is an anomalousdomain?

What is the domain size?

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Answers
N=3

j=5

N=2

j=4

N=1

j=3

N=0.5 j=2
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High energy cosmic ray experiment and PHENIXCan DCC scenario explain these events ?

PHENIX 7.24 standard deviation○: Photon+ : Charged Particle

 Charged track

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

 Photon cluster
17

Maximumdifferential balance
distributions

PHENIX Work in Progress• δBmax distribution– black : binomial sample, 100times 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、そもそもベースラインの分布は、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。

δBmax [arbitrary unit]

18

What is the critical behavior ?
Scale transformation

Ordered T=0.995Tc

Critical T=Tc

Disordered T=1.05Tc

Spatial pattern ofordered stateBlackBlack & White

Various sizes
from small to large

Gray

Search for a transitionof the correlation size from T>Tc to T=Tc

2011/12/16@HIPUB

A simulation based on two dimensional Ising modelfrom ISBN4-563-02435-X C3342l
19Spacetime evolution and Causality

Freeze outtumm  t 2  z 2 u  (t /  , 0, 0, z /  )  (cosh y, 0, 0, sinh y)

QGPGlasmaz

1  t  z  1 1  y  ln   ln 2  t  z  2  1  y → β β<< 1 (rapidity)ttLatest correlationzLarge rapidity interval
2011/12/16@HIPUBz

Small rapidity interval
20Analogue to the universe evolution
Long range rapidity
correlation

Early universe

QCD phase transition(Quark recombination)

Shorter range rapidity
correlation

Recombination

Present universe

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

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A picture of expanding medium in early stage
Initial stage

T>Tc

T=Tc

External field hLongitudinal fielddensity fluctuations fromthe mean density is anatural order parameter ( z)   ( z)  T<TcWe may expect freeze of initially embedded fluctuation
due to rapid dilution of medium in the longitudinal direction
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Density-density correlation in longitudinal space
Longitudinal space coordinate z can be transformed into rapidity
coordinate in each proper frame of sub element characterized bya formation time  at which dominant density fluctuations are embedded.

z   sinh( y )t   cosh( y )dz   cosh( y )dy
Due to relatively rapid expansion in y, analysis in y would
have an advantage to extract initial fluctuations
compared to analysis in transverse plane in high energy collision.

g (T ,  , h)  g 0   dy  d 2 xyS  1
1
2
   cosh( y ) (  )  U ( )  2
2
 2 cosh( y )  y 
2

In narrow midrapidity region like PHENIX, cosh(y)~1 and y~.
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23

Direct observable for Tc determination
GL free energy density g with  ~ 0 from high temperature side isinsensitive to transition order, but it can be sensitive to Tc

111 4 1 622g (T ,  , h)  g 0  A(T )( )  a(T )  b  c    h2246spatial correlation  disappears at Tc → a(T )  a0 (T  Tc )Fourier analysis on
G2(y)=<(0)(y)>| k |  Y  G2 ( y )e2| k |2  ik ( y )SusceptibilitydyNT
1
Y a (T )  A(T )k 2

1k   ( g  g 0 ) 1k 2h   k a 0 (T  Tc )(1  k 2 2 )2Susceptibility in long wavelength limit1
1-D two point correlation function
NTG2 ( y )  2
 (T )e | y|/  (T )2Y A(T )Correlation length (T ) 2 A(T )a0 (T  Tc )

 k 0 

a 0 (T  Tc )

TG2 (0)Product between correlation
length and amplitude can also
be a good indicator for T~Tc24

Strategy to find phase transition
Step1.
Search for increase of correlation length andsusceptibility (amplitude x correlation length)
determined by exponential form in T>Tc→T~Tc
Step2.
Search for transition of two point correlation fromexponential to power law form which needs higher
order terms in the free energy density. This would
be a stronger indication of T=Tc.

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Density measurement: inclusive dNch/d
Cu+Cu@200GeV

Centrality

Cu+Cu@62.4GeV

P(Nch)

Negative Binomial Distribution(NBD) perfectly describes
multiplicities in all collisionsystems and centralities
at RHIC.
Nch/< Nch >p+p@200GeV
Cu+Cu@22.5GeV

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Au+Au@62.4GeV

Au+Au@200GeV

26

Two point correlation via NBD
Uncorrelated
sourcessource 1Correlated
sources

k=k1

k=k1

k=k1+k2

k=k2

k!=k1+k2

k=k2

source 2

source 1+2n(k )nNBD P( n  k )  m / k 1
k=1 Bose-Einstein(n  1)(k )  1  m / k  1  m / k k k=∞ Poisson

2 1 1 2mm k
2011/12/16@HIPUBm  n  1/k corresponds to integralof two point correlation27

Differential multiplicity measurementsD windowΔη<0.7 integrated over Δφ<π/2 and pT>0.1GeV

Zero magnetic field toenhance low pt statistics
per collision event.
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NBD fits
at eachwindowsize inCuCu@200

L=0Large

Level (window size)
L=28(1-/DPHENIX)

16 fit examples inmost left edge in
top 10% events
out of 28/2*(1+28)
times NBD fits

L=240

small
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Extraction of  productFit with approximated functional form
10%

Parametrization oftwo particle correlation

5%kC2 (1, 2 )  2 (1, 2 )  1 (1) 1 (2 )
C2 (1, 2 ) / e2

1

 absorbs rapidity independent
bias: Npart fluctuation,reaction plane rotation, and v2

C2 (1 , 2 )   2 (1 , 2 )  1 (1 ) 1 ( 2 )
C2 (1 , 2 ) / 12  e  /   

Exact relation with NBD kn(n  1)
1k ( ) 12nLook atslopes





00

C ( , )d d21212 2 12Phys. Rev. C 76, 034903 (2007)

Approximated k ( ) functional form

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12  /   

(   )2 2 ( /   1  e  /  )

2

30

Correlation functions and correlation length
Using arbitrary R0, ξ and α.
Used in E802

C2  1  R(0,0)e | y1  y2 |/ 1
 / 2k ( ) R0 [1  ( /  )(1  e  /  )]General correlation function

C2  1 k ( ) R0
| y1  y 2 | / e| y1  y2 |





R0  y / edyyξ : correlation length, α : critical exponent
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αξ, β vs. Npart
Dominantly Npart fluctuations
and possibly correlation in azimuth

β is systematically shift to lower
values as the centrality bin width
becomes smaller from 10% to 5%.
This is understood as fluctuationsof Npart for given bin widths

β

●5%
○10%

αξ product, which is monotonicallyrelated with χk=0 indicates the nonmonotonic behavior around Npart ~90.

Au+Au@200GeV

αξ

●5%
○10%

Phys. Rev. C 76, 034903 (2007)2011/12/16@HIPUB   k 0T / 1  12

Npart2T| T T C|

Significance with Power + Gaussian:
3.98 ζ (5%), 3.21 ζ (10%)
Significance with Line + Gaussian:
1.24 ζ (5%), 1.69 ζ (10%)
32

proton-proton

How about STAR?
Analyzed 1.2M minbias 200 GeV Au+Au events, and 13M 62 GeV minbias events(not shown) Included all tracks with pT > 0.15 GeV/c, |η| < 1, full φnote: 38-46% not shown

84-93%

75-84%

65-75%

55-65%

46-55%

5-9%

0-5%ΔρρrefφΔ

ηΔ

28-38%STAR Preliminary
19-28%

9-19%Δρ
ρrefφΔ

ηΔWe see the evolution of correlation structures fromperipheral to central Au+Au
Slide from M. Daugherity, STAR
Collaboration presented at QM08
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3333

Transition
Does the transition from narrow to broad ηΔ occur quickly or slowly?
data - fit (except same-side peak)
83-94%

55-65%

Shape changes little fromperipheral to the transition

46-55%

Large change
within ~10%
centrality

0-5%

Smaller change fromtransition to most central

Low-pT manifestation ofthe “ridge”

The transition occurs quickly
Slide from M. Daugherity, STAR
Collaboration presented at QM08
2011/12/16@HIPUB

M. Daugherity, STAR Collaboration

3434m2Similarity to STAR mini jet results at low pTPeak η Width

Peak Amplitude

Au+Au@200GeV

Au+Au
200 GeV62 GeVX
Cu+Cu@200GeVSTAR Preliminary

εBJ

Au+Au@200GeV

Au+Au@62.4GeVSTAR Preliminary

M. Daugherity: QM2008

εBJ

Equivalent quantity;T ∝ m2 ∝ amplitude x widthshows similar trends to what
STAR sees.

<mc>/<mc>@AuAu200
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１事象で片付く例

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VTX, FVTX, and NCC for future runs
Central Vertex detector (VTX)
Strip pixel

Pixel

Nose Cone Calorimeter
Forward VTX (FVTX) (NCC)

PHENIX can extend both rapidity and azimuthal coverage
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37

Maximumdifferential balance
distributions

Work in Progress• ベースラインは、もう押さえられているはずであるから、この解析に戻ることが可能。より大きな検出器アクセプタンスにおいて、より多くの統計を用いて、大きな値を取る異常な揺らぎを視覚的かつ定量的に議論することができる（かも）。

2011/12/16@HIPUB

δBmax [arbitrary unit]

38まとめ： 事象毎の揺らぎ解析への回帰
Can DCC scenario explain these
events or something else?

New PHENIX？
○: Photon+ : Charged Particle
J. J. Lord and J. Iwai. Int. Conference on
High Energy Physics, TX, 1992
2011/12/16@HIPUB

 Charged track Photon cluster
39