1. 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

2. 2 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 E T (  ) 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 p T < 2 GeV/c. The most power method of study such kind distributions is Discrete Wavelet Transformations (DWT).

3. 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 p T ) 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

4. 4 Content Main notions 2-dimension Discrete Wavelet Transformation Examples with ring structure Jet like structure Conclusion SINP,MSUV.L. Korotkikh, G.Eiiubova

5. Main notions Analysis— decomposition f(x) by the help of basic function wavelets Synthesis— reconstruction of function, using wavelet coefficients Wavelet decomposition has two parameters: j – scale parameter, analogous Fourier frequency k – shift parameter, which sets a wavelet location - oscillator functions, wavelet - scaling functions

6. 6 D 8 - Daubechies wavelet oscillator function, wavelet scaling functions С.Maлла «Вейвлеты в обработке сигналов», М. «Мир», 2005

7. 7 2-dimension Discrete Wavelet Transformation

8. 8 Algorithm of Fast wavelet transform is used for calculation of wavelet coefficients The 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) h[n] и g[n] — filter coefficients of the wavelet ψ D 8 - Daubechies wavelet: h[0],…,h[7]  0

9. 9 Two dimension wavelets : Here are X, Y, D — the directions on the strings, coulombs and diagonals.

10. 10 Algorithm of two dimension analysis h[n] g[n] h[n] g[n] h[n] g[n] s j+1 S j d x j+1 d y j+1 d d j+1 along strings along coulombs

11. 11 Example with ring structure

12. 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

13. 13 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.

14. 14 Final histogram after synthesis with zero d λ j at j=1-5. You see that a small peak disappears Original histogram

15. 15 WL with d λ j  0 at j = 1,2 WL with d λ j  0 at j = 6,5,4

16. 16 Jet like structure

17. 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→jet 1 +jet 2 2)Pb+Pb →particles, no jets. (dN/dy) y=0 = 6000 Our next histograms are -5 < η < 5 -3.14 < φ < 3.14  0.078  0.050

18. 18 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 λ j, which are above the certain threshold 4. Synthesis with selected coefficients d λ j Algorithm of jet reconstruction

19.  0.078  0.050 -5 < η < 5 -3.14 < φ < 3.14 Jet E T =70 GeV jet + background reconstruction by wavelet analysis

20. a)Two jet event E T (jet1) =30 GeV E T (jet2) =22GeV b) this event + background c)Background after wavelet synthesis d)Reconstruction of event a)b) c) d)

21. 21 Event reconstruction with the help of Daubechies wavelets D 8 after removing background Event projection on axis φ

22. 22 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 λ j coefficients by the special way Wavelet analysis allows to selects the jets with E T > E Tmin =20 GeV Conclusion

23. 23 Plans for future Testify wavelet analysis on simulated and reconstructed data on CMS detector Testify on real data from RHIC