объединенный ИНСТИТУТ ' ядерных исследовании

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1 объединенный ИНСТИТУТ ' ядерных исследовании Е S.V.AfanasJev, M.V.AltaiskJ, Yu.G.Zhestkov ON APPLICATION OF WAVELET ANALYSIS TO SEPARATION OF SECONDARY PARTICLES FROM NUCLEUS-NUCLEUS INTERACTIONS Submitted to «II Nuovo Ciraento» 1994

2 1 Introduction The simplest way to tackle with almost any physical problem is to build a functional basis with the same symmetry as that of original problem or close to it. That's why the Ressel functions fit the problems with cylindrical symmetry, as well as spherical functions fit the SO3 symmetrical ones. The application of this very idea to certain classes of stochastic processes found its implementation in wavelet analysis. The Brownian motion p(m ~ X{to)) ~ -^щгщ ехр \ щг^ог) - one of the most common random processes has long been known to be invariant under the scaling transformation (Sec e.g. [1] for details.) P{b l < 2 [X(bt) - Х(Ы 0 )]) = b-ip{x{t) - X{i e )) (I) Therefore, it seems quite natural to use decomposition with respect to affine group << =!±A ) namely dilatations and translations, when studying Brownian motion, as well as other similar random processes. 1

3 Technically, decomposition is performed by convolution of function ф with certain function g(t). called wavelet, with the argument shifted to b and dilated by /. It is essentia! that function g has limit supporter. Therefore, unlike to the Fourier transform, which is inherently nonlocal, wavelet analysis or synthesis can be performed locally on a signal (field). Based on the affine group representation, wavelet analysis and synthesis allow one to unfold a signal (a (ield). into space, time and direction. It works as a "microscope" discriminating different scales and a polarizer separating different angular contributions. Wavelet analysis has been applied to signal processing, image coding, turbulence data analysis and some other fields. The numerous applications of wavelets to random data analysis (See e.g. [3] and references therein.) has proved it to be a powerful tool for studying fractal signals and data on cascading processes. 2 Definitions As a decomposition based on an affine group i * ал + b, wavelet transform (VVT) of an arbitrary function f(x) can be written as \f)=j\a,b;g)dvi{a,b){a,bg\f) (3) where (a,bg\f) ~ Т я (1,Ь,в) = С;* [ f(x)g- п-(*)ч^ Г <Гх C, = {2*rJjgCk) (4) 2

4 g(k) = О)""/ <7( )е- А ЧГх (5) JR" The rotation tensor П belongs to the group S0 rotations in R n and depends on the Euler angles в. In terms of the Euler angles в and scale (length) / the reconstruction formula (3) takes the form /(j) = ГД" J R j[ L^}T g [l,b,0)<rbd, l (0) (6) where is the 50 n invariant measure. It should be noted that an arbitrary function g(x) cannot be used as a wavelet in general: the admissibility condition (4), which guarantees the existence of inverse transformation (3), is required. Historically, the wavelet transform originated from Morlet's work [4] on seismic data analysis and Zimin's [5] hierarchical basis for turbulence. Since these first works a lot of studies has been done with different wavelet functions g(x), because the admissibility condition allows a wide variety of functions: condition (4) practically means g(k = 0) =0. The most common chois of real wavelets, however, is restricted to the derivatives of Gauss d m, 2 k 2 g m (x) = (-!)- e-r, $»(*) = (гкге-~ (7) Besides, one can also use complex-valued functions as wavelets [6, 8]. 3

5 3 Detection of random signal singularities with wavelets Wavelets has long been known as an attractive tool for analyzing function regularity [7, 8] as well as for searching singularities of random signals [9]. The latter application is significant for all kinds of spectrum recognition. A very instructive example of wavelet application to singular measures is the reconstruction of singularity spectrum from so-called "devil staircase" measure (See e.g. [3] and references therein.). The basic idea of the method is the following. Let us consider a measure fi(x) (with x R 1 for simplicity). Then, considering the integral measure s(x)= l'dr(x), (8)./о one can use the following theorem [10]: Theorem 1 If s(x) is a bounded locally integrable function that satisfies s(x) - s(x 0 ) = 0(\x - x 0 \ h ), Л [0,1] at some point x 0 R, then, provided the analyzing wavelet satisfy g L 1, x h g G L 1 and the zero-mean condition its wavelet transform behaves as j g{x)dx = 0, T g {a,x) = O(a h + \x-x 0 \ h ) Thus, a way to detect a singularity is to investigate the decay rate of the amplitude of wavelet transform T g (a,x) = 0(a h ) (9) 4

6 in the influence cone of singularity \x -.ro < const a. which can he naturally done by log log plotting of Т я (а,х). 4 On possible applications of wavelets in nuclear physics Гр to now the most common applications of WT belonged to either turbulence data analysis, where scaling is an inherent fealure of fluid physics, or to image processing, where the singularity detection and local reconstruction arc significant. The only known applications of WT to physics beyond turbulence, at least to the authors' knowledge, are related to spectra analysis. Such a situation seems to be rather strange for a number of reasons. Firstly, since the works of Zimin. WT proved to work efr ficiently in situations where cascade processes play significant role. Therefore, if the measure (J.{x) describes an event number at certain point x. (x R' 1 in general), then the search for j< t events can be performed with the aid of WT, in a way similar to "devil staircase" singularity reconstruction (See e.g. [3. 9] for details.) Secondly, if x is regarded as time (or energy). WT works as a tool for studying time (or energy) scaling of the process described by time (or energy) event density fi(x). Thirdly, being local in both x and Fourier space, WT can provide more information in spectral problems, where Fourier methods fail or work insufficiently. The contributions of different frequency bands to WT are kept reasonably separated. This separation is achieved 5

7 with quite insignificant loss of resolution in time variable (if a signal is considered). That's why the reconstruction is "robust" in the sense of being stable under small perturbations, which enables one to distinguish between "usefull" low bands (in Fourier space) and contributions of close high frequencies u?i -UjSsO usually generated by t he noise. The situation turns to be even more strange if we take into account the well-known facts related to Local I'arton Hudron Duality [11], i.e. the similarity between momentum spectra of hadrons and those of partons. This similarity, which is closely related to n-partcn correlations and multiplicity moments behavior in phase space, has been studied in [121. The fractal behavior of final multiparton states [14] was studied by several authors. They calculated the fractal dimension directly from multiplicity distribution moments and study the entropy of secondary particles.v=- P ln/v where P n is the probability of having "тг produced particles in the final state [15]. They found the scaling behavior, but. as the method was rather rude (see [3] for the shortcomings of the fractal dimension calculations without wavelets), the rare, but interesting events can be lost. Besides, the fractal analysis of multiparticle production in hadron-hadron collisions has been recently done by other authors [13]. That's why we are going to apply wavelet methods for separation of secondary particles (К mesons, in particular) in d+ Au interactions. 6

8 5 Secondary particle separation as an image recognition problem I he aim ol ihe present paper is not to covei all the problems mentioned above with the aid ol wavelet analvsis. In this preliminarv studv we )мм show its facilities lor energy versus time of Might data I Fig. I I obtained from <l * Ли --*... reactions in experiments earned out at I lie Xuclol ion using the internal target at deuleron moment urn of :\.H(,'< I '/< in March 1!)!(!. I'he following detectors operated at present run: Two identical telescopes containing lour scintillation counters each with 2 x 2 x {)..") <III\.'i A ' ] x ()."> cui\ I \ I x ().."> en;' and 7..") x 7.") x ()."> cm', respect iveiv. The lirsl I hree counters aimed for Л I - ", and Г01' measiireiiiets. and t he last one lor 1 he measurement of charge particle energy K. The on-line scatter plot of I OF vs F.ncrgy loss in Fig.l (iiialil at ivelv deinonsl lates 1 lie -. p. d. t and He separat ion capahilit \. In Fig.2 we present the mass spectrum obtained from the primary data bv the standard method of comparing the time of flight and I iie energ\ i i. /:'- - :n- * = - = - =,/ t it)i < >< \ I.- Due to the presence of both 7Г mesons and protons, which dominate 4, in the central (dark, see Figs. I.II ) region of the plot (/' ' ~ 250 MeV). which contains the largest part of the registered particles, it is difficult to distinguish other events in this region. Besides, high energy protons I/'. > 100 MeV) due to decreasing energy loss give contribution mainly to the low energy region about 200 MeV. 7

9 6 Algorithm The main idea of implying a wavelet analysis to invesl igat ion of events in nuclear and hiiili energy olivsics is to use its good properties in separating events from noise. I'sing wavelet onecan look at experimental data with various resolution. I his can ho used to searching for tracks of particles and different kind of events, e.t.c. Го use a mult iresolnt ion analysis I Hi] one should choose a family of closed suhspaces I, С l. l (li).tn f. /.. such thai 1 /i j V <= /! 2» r it I. -, С V 2 С \\ С V С V'-i С I-j С... nv-; =о.иг. = мм ' ]. there is а о _ li. such that its linear integer translations OtuAx] = ol-r -t- n) constitute a basis in i<> (consequent ly. functions, constitute a basis in V,. 4. there exist 0 < A < В < oo, such that for all (c n ) n z <= The orthogonal projections of a function which we analyse on a chain of subspaces l', represent, snapshots of this function with different resolution. Choosing an appropriate basic function Ф. one could select different kinds of snapshots. To make the decomposition ciose. one should also define' a chain of subspaces W, ortogonal to V,. such that v; -. = v, w,. The coefficients of a projection on V, and W, an- (ii) <: = j nxwwdr, <c = //( ' )'/- :(.' )'/ ' - (i2) 8

10 where in a discrete case a sum is implied. For the simplest case of Haar wavelel (Sec e.g. il6]) the basic functions are: *" = {»'! for x /,> elsewhere (13) >-' 12 for 2 ] (k - 1 )< j- < 2'(k - 1/2), чцт) = \ -T>l' 1 for 2Hk - 1/2) < r < Tk. (14) 0 elsewhere Hie / denotes the supporter of j-\\\ level basic functions Ij = l{ = (2 J (/t- l).2 J /ti. The approximate reconstruction formula has the form i\nf = Y,s K + 4 E «П5) ч n. In our two-dimensional problem we used a pyramidal scheme with a basis taken in the form of a tensor product or explicitly. {h u h 2,h 3 } = {tt>i(x)i(>r(y),&i(x)<t>n(y),<i>i{x)il>i'(y)}- (Щ The corresponding coefficients can be easily derived from the formulae (12): =J+> -. J ^A^-l^Jt,,-! + л 2* х -1,2к» + ^гкх.гк,-! + s 2*x,: 2k, У "(1)**,*» S 2*: J -1,2*!/ -1 *2/fc r -1.2fc» '^г^.г*,,-! ^ л2*:1, 2/t "(2);*!,*» s 2fc J -l.2fc s -l _ -^fc,,2*^-1 + s 2k I -l,2k y b 2k x,2k a // J + 1 ~ <; > -1-е-' e J «* "(З);^,^ S 2fc t -l,2fc -l "+" S 2k x,2k v -\ Ь 2к х -\,2к A у 2k x,2k y > where s stand for the primary data. 9

11 7 Results The primary data /',' <ll plot for I he above meni ioned run is shown in Fig.l. I he.y-axis corresponds to ADC channel numbers. Y'-axis to TDC ones. (Moth axes are scaled by factor 1). In this plot, over a noisy background we can distinguish 1 wo contrast regions: I IK- upper, which corresponds to sccondarv protons, and the lower, which corresponds Iо ~-mesons. To clear out the eontribut ion of dominating processes we performed the wavelet analysis. Having calculated the wavelet image (the Haar wavelet was used) of the initial data plot we substracted the central domain, in which г/' 2 ' coefficients I See Fig. 1) practically wanish. The resulting mass spec! rum is presenled in fi».">. We ident ifv the central peak near ooo MeV. clcarlv distinguished on mass histogramm with the A -mesons contribution. Besides, sequentially scaling the picture, we can clearly distinguish 4 regions: upper right region: secondary deuterons two above-mentioned regions a A'-meson branch. The authors are grateful to Dr. V.Kolesnikov for invaluable technical aid in data acquisition. We are also grateful to prof. S.ltatti for stimulating discussions. One of the authors (S.A.) is grateful to Russian Fund for Fundamental Research, grant 'W H. for financial support. 10

12 20 С 16 ft Й 12 Е g '(Ml 10(1 400 Energy (McV) Fig. 1 TOFvsE plo! of primary experimenth! (!;;!: с > ^ n -.,v>' > i. ^ > *, ^ Mass (MeV) Fig. 2 Spectrum of reconstructed mass from primary data. 11

13 20 n*. зц^^^^^^^^^^^^^и г - ' /jiij т^^^^^ж 0 Н 500 McV Fig. 3.s (l) cocftiricnls plot for thrdala si-1 of Fig ns Яя&ЩфФ**:- 0 i: 500 MeV Fig. 4 U/ (,, (orfficiculs plol foi ihr dala set of Fig

14 25 20 «2»5 С > PJlO J, U i Mass (MeV) Fig. 5 Mass spectrum obtained after filtering References [1] Feder, J., Fractals, Plenum Press, New York, [2] Daubechies, I., Grossmann, A., Meyer, Y., J. Math. Phys. 27(1986)1271 [3] Muzy J.F., Васгу, Е., Arneodo. A., Phys. Rev. Lett 67(1991)3515 Muzy J.F., a.o. Phys. Rev. E47(1993)875 [4] Morlet, J Sampling theory and wave propagation. Proc. 51st Annu. Meet. Soc. Explor. Geophys., Los-Angeles. [5] Zimin V.D., Izv. Atmos. Ocean. Phys. 17(1981)941 (in Russian) 13

15 [6] Paul Т., J. Math. Phys. 25(1984)3522 [7] Holshneider, M. J. Stat. Phys. 50(1988)963 [8] Grossmann, A., Holshneider, M., Konland-Martinet, R. and Morlet, J., in Inverse Prob'tm, ed. by P. C. Sabatier, Advances in Electronics and Electron Physics, Supplement 19, Acad. Press, Orlando, [9] Combes J.M., Grossmann, A. and Tchamitchan, P., eds., Wavelets, Springer, Berlin [10] Holshneider, M. and Tchamitchan, P., in Les Ondelcts, ed by Lcmarie, P.G., Springer, Berlin, [11] Ya.I. Azimov, Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan, Z. Phys. C27( 1985)65, ibid. 031(1986)231 [12] Ochs W., Wosiek J., Phys. Lett. B305(1993)144 [13] Boca G., a.o. Nuovo Cirri. A105(1992)865 [14] G.Gustafson and A.Nilson Z. Phys. 052(1991)533 [15] A.Mikhopadhyay, P.L. Jain, and G.Singh, Phys. Rev. 047(1993)410 [16] Daubechies I., Comm. Pure. Apl. Math. 16(1988)909 [17] Beylkin G., Coifman R. and Rokhlin V., Comm. Pure. Apl. Math. 44(1991)141 Received by Publishing Department on August 18,

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18 Афанасьев СВ., Алтайский М.В., Жесткое Ю.Г. Применение вейвлет-анализа к разделению вторичных частиц в ядро-ядерных взаимодействиях Е Для выделения различных каналов реакций в d + Аи-*... взаимодействии применен вейвлет-анализ. В работе использовались экспериментальные данные, полученные на нуклотроне ОИЯИ в марте 1994 года при импульсе дейтронов 3,8 ГэВ/с. Зарегистрировано четыре канала реакции. Работа выполнена в Лаборатории высоких энергий и Лаборатории ядерных проблем ОИЯИ. Препринт Объединенного института ядерных исследований. Дубна, 1994 Afanasiev S.V., Altaisky M.V., Zhestkov Yu.G. On Application of Wavelet Analysis to Separation of Secondary Panicles from Nucleus-Nucleus Interactions E We apply wavelet analysis for separation of secondary particle from different channels of d + Aw... reaction. Using the data obtained in the experiments at Nuclotron (Dubna) in March 1994 at deuteron momentum of 3.8 GeV/c, we found 4 different regions in the time-of-flight vs energy loss plot. The investigation has been performed at the Laboratory of High Energies and Laboratory of Nuclear Problems, JINR. Preprint of the Joint Irutitute for Nuclear Reaearch. Dubna, 1994

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