Applicability of continuous wavelet analysis to pseudorapidity and azimuthal distributions of secondary particles produced in Au + Au collisions at √ sNN = 200 GeV at Relativistic Heavy Ion Collider is studied. After corrections for boundary effects, the wavelet spectra are presented on different scales. We present a discussion whether the multi-particle collective flows produced by different physical mechanisms can be detected through extremes in the wavelet spectra or corresponding scalograms, and what additional information concerning correlation structure of events can be extracted by this analysis.Proučavamo primjenjivost analize neprekidnih valića na pseudorapiditet i azimutalne raspodjele sekundarnih čestica proizvedenih u sudarima Au + Au na √ sNN = 200 GeV u relativističkom teškoionskom sudaraču. Nakon popravaka zbog graničnih učinaka, valići se predstavljaju na različitim ljestvicama. Raspravljamo mogu li se preko ekstremnih vrijednosti u spektrima valića ili odgovarajućim skalogramima opaziti višečestični skupni tokovi proizvedeni različitim mehanizmima, te koji se daljni podaci o korelacijskoj strukturi mogu izvesti tom analizom

Fedorisin, J.

Vokál, S.

English

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7WAVELET ANALYSIS OF MULTIPARTICLE CORRELATIONSJ. FEDORIŠINa and S. VOKÁLbaDepartment of Nuclear and Subnuclear Physics of Šafárik University, Košice, SlovakiaE-mail address: fedorisi@upjs.skbVeksler and Baldin Laboratory of High Energies, JINR, Dubna, RussiaE-mail address: vokal@sunhe.jinr.ruReceived 13 November 2007; Accepted 20 February 2008Online 20 June 2008Applicability of continuous wavelet analysis to pseudorapidity and azimuthal distri-butions of secondary particles produced in Au +Au collisions at√sNN = 200 GeVat Relativistic Heavy Ion Collider is studied. After corrections for boundary ef-fects, the wavelet spectra are presented on different scales. We present a discussionwhether the multi-particle collective flows produced by different physical mecha-nisms can be detected through extremes in the wavelet spectra or correspondingscalograms, and what additional information concerning correlation structure ofevents can be extracted by this analysis.PACS numbers: 25.75.-q, 25.75.Gz, 25.75.Ld UDC 539.126Keywords: ultrarelativistic heavy-ion collisions, quark-gluon plasma, multi-scale analysis,wavelet1. IntroductionThe primary goal of ultrarelativistic heavy-ion collisions is to sqeeze and heatup nuclear matter above the critical values predicted by QCD in order to createand study exotic state of nuclear matter called quark-gluon plasma (QGP). Itsproperties are deduced through a variety of partial signals which are subsequentlycombined to produce QGP global picture [1]. Information about the partonic col-lective flows at the early stages of QGP expansion can be retained in the angularspectra of emitted particles and extracted by analysis of multi-particle correlations.The experimental data sample discussed here was obtained with the STAR de-tector [2] at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven NationalLaboratory. The main detector for the analysis is the Time Projection Chamber(TPC) [3] which is located inside a 0.5 T solenoidal magnet and measures tracksFIZIKA B (Zagreb) 17 (2008) 2, 273–278 273fedorǐsin and vokál: wavelet analysis of multiparticle correlationsand momenta of charged particles in the pseudorapidity region |η| < 1.3 with fullazimuthal coverage.As for the selection criteria, the events are selected if their primary vertices liewithin 30 cm distance from the TPC center along the z axis (the beam line) andwithin 1 cm distance along the y axis. The tracks are chosen if their distance ofclosest approach to the primary vertex is less than 2 cm, and at least 15 measuredspace points were used for their reconstruction. Furthermore, the ratio of the num-ber of measured space points to the maximum number of space points for each trackhas to be larger than 0.52 and χ2 of track fit in the xy plane has to lie between 0.1and 1.3.Small testing sample consists of approximately 1500 minimum-bias Au+Auevents taken at the collision energy√sNN = 200 GeV in the 2004 run.2. Wavelet spectraVarious techniques are employed to study multi-particle correlations. We presentthe preliminary results obtained by means of the wavelet analysis employing thesecond derivative of the Gaussian function g2 as a mother wavelet function. Themethod description and its application to heavy-ion collisions data are presented atlength for instance in Refs. [4], [5] and [6] and in the publications reffered to therein.Similar approach to study multi-particle correlations in STAR data, though basedon the discrete wavelet transform, was first time applied in Ref. [7].Entries  374Mean   0.0001208RMS    0.5459η-1 -0.8-0.6-0.4-0.2 -0 0.2 0.4 0.6 0.8 1ηdN/d0246810121416182022Entries  374Mean  0.1851RMS     1.784φ-3 -2 -1 0 1 2 3φdN/d02468101214161820Fig. 1. Pseudorapidity and azimuthal angle distribution of a representative Au+Au200 GeV event.Figure 1 shows the distributions of pseudorapidities η and azimuthal angles φof particles produced in the representative experimental event with the multiplicityof charged particles close to 370 in the eta range |η| < 1. The g2 wavelet η and274 FIZIKA B (Zagreb) 17 (2008) 2, 273–278fedorǐsin and vokál: wavelet analysis of multiparticle correlationsφ spectra of both distributions seen on the three different scales a are shown inFigs. 2 and 3. Wavelet coefficients on the vertical axes are denoted as WΨ.-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1-10-50510η)η(ΨW spectrum at a=0.12gSTARPreliminary-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1-10-5051015η)η(ΨW spectrum at a=0.152gSTAR Preliminary-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1-15-10-505101520η)η(ΨW spectrum at a=0.22gSTAR PreliminaryFig. 2 (above and left).Wavelet pseudorapid-ity spectrum of the stud-ied event seen on three dif-ferent scales a.-3 -2 -1 0 1 2 3-6-4-202468φ)φ(ΨW spectrum at a=0.22gSTAR Preliminary-3 -2 -1 0 1 2 3-6-4-20246φ)φ(ΨW spectrum at a=0.42gSTAR preliminary-3 -2 -1 0 1 2 3-8-6-4-20246810φ)φ(ΨW spectrum at a=0.62gSTAR  preliminaryFig. 3 (above and left).Wavelet azimuthal anglespectrum of the studiedevent seen at the three dif-ferent scales a.Local maxima visible in the wavelet spectra imply prevailing emission angles ofgroups of secondary particles [4, 8]. Therefore, the presented plots demonstrate theFIZIKA B (Zagreb) 17 (2008) 2, 273–278 275fedorǐsin and vokál: wavelet analysis of multiparticle correlationsevolution of clusterization of particles. On the finest scales, only individual particlesor small groups of particles are observed. The relatively large numbers of maximareveal evident irregularity of the η and φ distributions on the smallest scales. Whengoing to larger scales, the particle groups gradually merge, eventually producingboth in η and φ cases two large particle flows which move in opposite directions.The relationship among the peaks observed in the η and φ wavelet spectra can bestudied only by two-dimensional wavelet method or other type of two-dimensionalcorrelation analysis (or their combination).When combining many plots, like those in the previous figures, i.e. for manyscales with sufficient density, a three-dimensional graph is constructed. It illustratesthe dependence of WΨ(a, η) and WΨ(a, φ) coefficients on η and φ, respectively, andscale a. The examples of the studied events are shown in Fig. 4.Light colors correspond to maxima, dark colors to minima. An interpretationof the structures, i.e. the maxima and the minima, is the same as in the previousfigures, only now we have much better resolution on the scale axes which allowsmore detailed study of the behaviour of the spectra in the scale domain.Fig. 4. Wavelet spectra of the η and φ distributions shown in Fig. 1.3. Extreme points in the wavelet spectraThe previous example demonstrates that the local maxima in either the waveletη or φ spectra suggest the characteristic scales and the prevailing pseudorapiditiesor azimuthal angles (in one single event). We may ask, are there any typical scalesoccurring systematically in many events or do they appear only randomly? It canbe revealed by investigating extreme points in the wavelet η and φ spectra of alarge sample of events. Motivation behind this question is that the occurrence ofdistinguished scales could hint at multi-particle flows produced by different phys-ical mechanisms. We plan to employ the method to search for the collective flowsinduced by the production of hypothetic Cherenkov gluons [9] or Mach cones [10].276 FIZIKA B (Zagreb) 17 (2008) 2, 273–278fedorǐsin and vokál: wavelet analysis of multiparticle correlationsThe local maxima found in the wavelet η and φ spectra presented in Fig. 4 areshown in Fig. 5. They indicate the presence of particle groups along with theirη (or φ) and size, therefore they could be useful to map correlation structure ofthe studied events. Additionally, it is more comfortable to work with only a fewsignificant points than with the whole wavelet spectra. We can investigate thenumber of wavelet maxima in individual events as well as their distribution in η-a(or φ-a) space.-1 -0.8-0.6-0.4 -0.2 -0 0.2 0.4 0.6 0.800.050.10.150.20.250.3maxηη maxaSTAR Preliminary-3 -2 -1 0 1 2 300.20.40.60.811.21.41.61.8maxφφ maxaSTAR PreliminaryFig. 5. Local maxima found in the wavelet η and φ spectra shown in Fig. 4.The projections of wavelet maxima on scale axis may suggest significant scales.This can be alternatively achieved through the so-called scalogram defined as:E(a) =∫WΨ(a, b)2db where b represents either η or φ. Its local maxima or bumpsindicate relevant scales. Such spectra can be employed to sort events to variousclasses.Furthermore, the projections of wavelet maxima on the η or φ axes with respectto some distinguished direction may imply prevailing pseudorapidities or azimuthalangles. The examples of distinguished directions are reaction plane in the φ case ora direction of motion of particle triggering Mach cone or Cherenkov gluon emissionwith some characteristic opening angle or pseudorapidity, if these are measured inthe reference frame attached to the particle.4. Summary1) The continuous wavelet method is tentatively used to probe the angular dis-tributions of secondary relativistic particles produced in Au+Au collisions at√sNN = 200 GeV.2) The wavelet η and φ spectra of the small testing event sample show irregularstructure of the angular spectra on all the studied scales. Local maxima inthe wavelet spectra are associated with the groups of secondary particles andFIZIKA B (Zagreb) 17 (2008) 2, 273–278 277fedorǐsin and vokál: wavelet analysis of multiparticle correlationsimply their preferred emission angles as well as characteristic scales of theanalyzed events. When performed on sufficiently large data samples, theirthorough analysis may reveal the existence of particle collective effects ofdifferent nature.3) Though the first preliminary tests prove the capability of the method to findand study multi-particle correlations, some improvements need to be imple-mented. They include the corrections for limited acceptance and efficiency ofthe STAR detector. Also a testing of the method on the Monte Carlo dis-tributions with various shapes will be helpful for better understanding andinterpretation of the real experimental data.4) It will be necessary to develop and apply the wavelet based two-dimensionalmethod providing simultaneous access to η and φ variables.AcknowledgementsThis work was supported by the Slovak Research and Development Agencyunder the contract No. APVT-20-011704. Gratitude also goes to Subhasis Chat-topadhyay for reviewing and commenting the paper.References[1] For reviews and recent developments see Quark Matter 2006 Conference, Yu-Gang Maet al., J. Phys G: Nucl. Part. Phys. 34 (2006).[2] K. H. Ackermann, Nucl. Instr. Met. in Phys. Res. A 499 (2003) 624.[3] M. Anderson et al., Nucl. Instrum. Meth. A 499 (2003) 659.[4] V. V Uzhinskii et al., Phys. Atom. Nucl. C 67 (2004) 156.[5] J. Fedorǐsin, S. Vokál, E1-2007-4, JINR Dubna (2007).[6] J. Fedorǐsin, S. Vokál, E1-2007-66, JINR Dubna (2007).[7] M. Kopytine et al., Phys. Rev. C 71 (2005) 031901.[8] I. M. Dremin, Phys. Usp. 43 (2000) 1137.[9] I. M. Dremin, Nucl. Phys. A 767 (2006) 233.[10] J. Casalderrey-Solana, E. V. Shuryak, D. Teaney, Nucl. Phys. A 774 (2006) 577.ANALIZA VALIĆA VIŠEČESTIČNIH KORELACIJAProučavamo primjenjivost analize neprekidnih valića na pseudorapiditet i azimu-talne raspodjele sekundarnih čestica proizvedenih u sudarima Au +Au na√sNN =200 GeV u relativističkom teškoionskom sudaraču. Nakon popravaka zbog graničnihučinaka, valići se predstavljaju na različitim ljestvicama. Raspravljamo mogu li sepreko ekstremnih vrijednosti u spektrima valića ili odgovarajućim skalogramimaopaziti vǐsečestični skupni tokovi proizvedeni različitim mehanizmima, te koji sedaljni podaci o korelacijskoj strukturi mogu izvesti tom analizom.278 FIZIKA B (Zagreb) 17 (2008) 2, 273–278

Analiza valića višečestičnih korelacija

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