Study of High Multiplicity Event Topology by Wavelet Analysis in Heavy Ion Collisions

V.L. Korotkikh, G.Eiiubova

European Workshop on Heavy Ion Physics, JINR, Dubna March 2006 г.

Scobeltsyn Institute of Nuclear Physics, Moscow State University

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There are many particles in one event in Heavy Ion Collisions. It is about in Au+Au collision at RHIC energies. So we can study a structure of event angular distribution or ET () distribution as function of η and φ . Lets show one example, but we don’t analyze this experiment.

Just now in STAR collaboration in RHIC the angular correlation of two charge particles in Au+Au √s =130 GeV/c(see Fig.) were measured, which show significant and unexpected fluctuations at low pT < 2 GeV/c.

The most power method of study such kind distributions is Discrete Wavelet Transformations (DWT).

6000)/( 0 ydydN

Publications with Wavelet analysis in nuclear-nucleus collisions

1) I.M. Dremin et al, Phys. Lett. B499, p.97(2001)(Discrete Wavelets Transformation (DWT), Pb+Pb, 158 A GeV, fix target, ring structure in angular distribution of particles)

2) V.V. Uzhinskii et al., hep-ex/0206003(2002)(Continuous Wavelets, O+Em, S+Em, 60, 200 A GeV, ring structure in angular distribution of particles)

3) I. Berden et al., Phys. Rev C65, 044903(2002) (Discrete Haar wavelets, Pb+Pb, 158 A GeV, fix target, texture of events)

4) M. Kopytin, nucl-ex/0211015(2002) (Discrete Haar wavelets, Au+Au √s = 200 A GeV (RHIC, STAR). Study of event texture)

5) J. Adams et al., nucl-ex/0407001(2005) (Discrete Haar wavelets, Au+Au √s = 200 A GeV (RHIC, STAR). Study of event texture with the help of power wavelets as function of centralities and pT)

6) V.L. Korotkikh, G.Kh.Eiiubova, Preprint НИИЯФ МГУ 2005-21/787(Discrete Daubechies wavelets. Study of MC simulation of two-dimension angular distributions with ring and jet structure of event)

7) M.V. Altaisky et al, Preprint JINR E10-2001-205 WASP (Wavelet Analysis of Secondary Particles Angular Distributions) Package

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Content

• Main notions• 2-dimension Discrete Wavelet Transformation• Examples with ring structure• Jet like structure• Conclusion

SINP,MSUV.L. Korotkikh, G.Eiiubova

Main notionsAnalysis— decomposition f(x) by the help of basic function wavelets

Synthesis— reconstruction of function, using wavelet coefficients

kj

kjkj xdxsxf,

,,0,00,0 )()()(

Wavelet decomposition has two parameters:j – scale parameter, analogous Fourier frequency k – shift parameter, which sets a wavelet location

)2(2)( 2/, kxx jjkj - oscillator functions, wavelet

)2(2)( 2/, kxx jjkj - scaling functions

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D8- Daubechies wavelet

oscillator function, wavelet

scaling functions

С.Maлла «Вейвлеты в обработке сигналов», М. «Мир», 2005

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2-dimension Discrete Wavelet Transformation

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Algorithm of Fast wavelet transform is used for calculation of wavelet coefficientsThe iterative formulas, so-called pyramid algorithm— the coefficients of scale (j +1) are calculated by the coefficients of scale j (from small to large scales)

njj

njj

nskngkd

nsknhks

][]2[][

][]2[][

1

1

h[n] и g[n] — filter coefficients of the wavelet ψ

D8- Daubechies wavelet: h[0],…,h[7] 0

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Two dimension wavelets :

Here are X, Y, D — the directions on the strings, coulombs and diagonals.

)2()2(2),(

)2()2(2),(

)2()2(2),(

,,

,,

,,

lykxyxF

kxlyyxF

lykxyxF

jjjDlkj

jjjXlkj

jjjYlkj

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Algorithm of two dimension analysis

h[n]

g[n]

h[n]

g[n]

h[n]

g[n]

s j+1S j

d x j+1

d y j+1

d d j+1

along strings

along coulombs

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Example with ring structure

L

y

yyxxtgayxf

22

20

20

3241

xarctg

)()(

1

)(cos)(1

)sin(),(

L

y

yyxxtgayxf

22

20

20

3241

xarctg

)()(

1

)(cos)(1

)sin(),(

We simulate a two dimension histogram, corresponding to our example function in order to test DWT

If we used all wavelet coefficients then we get the same exactly distributions after the synthesis

N=10000

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But to reveal structure of histogram on large scale

we put to zero the coefficients at small scales j=1..5

and make the synthesis with the coefficients j=7, j=6.

Dyx

j

d j

,,

51

,0

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Final histogram after synthesis with zero d λ

j

at j=1-5.

You see that a small peak disappears

Original histogram

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})()(

1

}2

)()(exp{),(

20

20

2

21

21

yyxxb

yyxxayxf

WL with d λj 0 at j = 1,2 WL with d λ

j 0 at j = 6,5,4

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Jet like structure

HIJING

Monte Carlo Program for Parton and Particle Production in High Energy Hadronic and Nuclear Collisions

(X.N. Wang, M. Gyulassy, Phys.Rev. D44(1991)3501)

s=5500 GeV, our event is a sum :1)p+p→jet1+jet2

2)Pb+Pb →particles, no jets. (dN/dy)y=0 = 6000

Our next histograms areddEd T

2

-5 < η < 5

-3.14 < φ < 3.14

0.0780.050

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1. Wavelet analysis of an event (Daubechies wavelets)

2. Calculation of background with large scale coefficients d λ

j and subtraction from original histogram

3. Selection of coefficients d λj , which are above the

certain threshold4. Synthesis with selected coefficients d λ

j

Algorithm of jet reconstruction

0.0780.050

-5 < η < 5 -3.14 < φ < 3.14

Jet ET=70 GeV jet + background reconstruction by wavelet analysis

a)Two jet event ET (jet1) =30 GeV ET (jet2) =22GeV b) this event +

background

c)Background after wavelet synthesis

d)Reconstruction of event

a) b)

c)d)

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Event reconstruction with the help of Daubechies wavelets

D8 after removing background

Event projection on axis φ

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We demonstrate the using of Discrete Wavelets Transformation (DWT) for the analysis of many particle events in heavy ion collision

DWT software for one and two dimension distributions was made

Method is tested on the events which are simulated by the event generators PYTHIA and HIJING

It is shown that method works well for the ring and jet structure events.

The event bacground is removed if we select d λj

coefficients by the special way Wavelet analysis allows to selects the jets with ET

> ETmin=20 GeV

Conclusion

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Plans for future

Testify wavelet analysis on simulated and reconstructed data on CMS detector

Testify on real data from RHIC