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Event Texture Search for Phase Transitions in Pb+Pb Collisions The NA44 Collaboration I. Beardena, H. Boggilda, J. Boissevainb, L. Conind, J. Doddc, B. Erazmusd, S. Esumie∗, C. W. Fabjanf, 6 D. Ferencg, D. E. Fieldsb †, A. Franzf ‡ , J. J. Gaardhojea, A. G. Hansena∗∗∗, O. Hansena, D. Hardtkei§, 6 H. van Heckeb, E. B. Holzerf, T. J. Humanici, P. Hummelf, B. V. Jacakj, R. Jayantii, K. Kaimie ¶, M. Kanetae, T. Kohamae, M. L. Kopytinej∗∗, M. Leltchoukc, A. Ljubicic, Jrg, B. Lo¨rstadk, N. Maedae ††, L. Martind, A. Medvedevc, M. Murrayh, H. Ohnishie ‡, G. Paicf ‡‡, S. U. Pandeyi, F. Piuzf, J. Plutad §§, V. Polychronakosl, M. Potekhinc, G. Poulardf, D. Reichholdi, A. Sakaguchie¶¶, J. Schmidt-Sorensenk, J. Simon-Gillob, 6 W. Sondheimb, T. Sugitatee, J. P. Sullivanb, Y. Sumie, W. J. Willisc, K. L. Wolfh ¶, N. Xub §, D. S. Zacharyi a Niels Bohr Institute,DK-2100, Copenhagen, Denmark; b Los Alamos National Laboratory, 2 Los Alamos, NM 87545, USA; 0 c Columbia University, New York, NY 10027, USA; 0 d Nuclear Physics Laboratory of Nantes, 2 44072 Nantes, France; e Hiroshima University, Higashi-Hiroshima 739, Japan; n a f CERN, CH-1211 Geneva 23, Switzerland; J g Rudjer Boscovic Institute, Zagreb, Croatia; h Texas A&M University, 5 2 College Station, Texas 77843, USA; i The Ohio State University, 2 Columbus, OH 43210, USA; v j SUNY at Stony Brook, Stony Brook, NY 11794, USA; 7 k University of Lund, S-22362 Lund, Sweden; 0 l Brookhaven National Laboratory, 0 Upton, NY 11973, USA. 7 0 (Dated: October 1, 2001) 1 0 NA44usesa512channelSipadarraycovering1.5<η<3.3tostudychargedhadronproduction / in 158 A GeV Pb+Pb collisions at theCERN SPS.Weapply a multiresolution analysis, based on x a Discrete Wavelet Transformation, to probe the texture of particle distributions event-by-event, e allowing simultaneous localization of features in space and scale. Scanning a broad range of mul- - l tiplicities, we search for signals of clustering and of critical behavior in the power spectra of local c density fluctuations. The data are compared with detailed simulations of detector response, using u heavy ion event generators, and with a reference sample created via event mixing. An upper limit n is set on the probability and magnitudeof dynamical fluctuations. : v i PACSnumbers: 25.75.-q X r a I. INTRODUCTION The main experimental challenge in relativistic heavy ∗now at KEK – High Energy Accelerator Research Organization, ion collisions is to find evidence for the expected QCD 1-1Oho,Tsukuba,Ibaraki305,Japan phase transition at high temperature. Deconfinement †nowatUniversityofNewMexico,Albuquerque,NM87131,USA ‡nowatBrookhavenNationalLaboratory,Upton,NY11973,USA and chiral symmetry restoration are expected to take ∗∗∗now at Los Alamos National Laboratory, Los Alamos, NM place during the hot, strongly interacting stage early in 87545, USA the collision. As a phase transition in such collisions is §now at Lawrence Berkeley National Laboratory, Berkeley, CA inherentlyamultiparticlephenomenon,multiparticleob- 94720, USA servables, defined on event-by-event basis, are of great ¶deceased ∗∗onanunpaidleave fromP.N.Lebedev PhysicalInstitute, Rus- interest. Recently published event-by-event analyses of sianAcademyofSciences the158GeV/APb+Pbdataeitheranalyzeasmallnum- ††nowatFloridaStateUniversity,Tallahassee,FL32306,USA berofevents[1]ingreatdetail,oranalyzepropertiesofa ‡‡affiliatedwithOhioStateUniversity,Columbus,OH43210,USA largeensembleofeventsusingasingleobservable(p )to §§Institute of Physics, Warsaw University of Technology, T comparedifferent ensemble averages[2]. In the first case, Koszykowa75,00-662Warsaw,Poland ¶¶nowatOsakaUniversity,Toyonaka,Osaka560-0043, Japan accumulationof feature informationfromlargedata sets remains an open issue. In the second case, an ensemble averageonasetofpost-freeze-out eventsisnotrepresen- 2 tativeofthepre-freeze-out historyofthoseevents,dueto localized spot. For a pad detector, the discrete positions the dramatic non-stationarity of the open system, with of the spots correspond naturally to the pad positions, a consequent lack of ergodicity. Violations of ergodicity andthepossiblescalesaremultiplesofthepadsizes. The generally happen in the course of phase transitions[3]. scaleisananalogofaFourierfrequency. Locationhasno We concentrate ontexture,or local fluctuation observ- analoginthe Fouriertransform,anditprovidesanaddi- ables,analyzingsingleeventsindependentlytodetermine tionaldegreeofanalyticalpower,whichexplainsmuchof the scale composition of fluctuations. In the following, thesuccessthatwaveletsmetinthefieldofdataprocess- we may omit the term “local”, but we will always talk ing and pattern recognition. (Examples of Fourier-based about fluctuations in the particle density fromone point analyses of large scale azimuthal texture in the field of to another within a single event, i.e. in the local sense, relativistic heavy ion collisions exist as well [15, 16]; this as opposed to fluctuations of global quantities from one is how the elliptic flow at relativistic energies was mea- event to another. sured.) The binning of charged particle density inherent In 1985, L. Van Hove formulated a model of quark- in measurements with a segmented detector such as a Si gluon plasma hadronization[4] with a first order phase pad detector makes the Haar waveleta naturalchoice of transition. Longitudinal expansion of the colliding sys- analyzingfunction;aHaarwaveletisastepfunctionwith tem, with particle formationvia stringorcolorflux tube givenwidth, oscillatingaroundzerowithasingleperiod. breaking, can result in plasma droplets as large as a few Discrete Wavelet Transformation (DWT)[17] quanti- fm across. The droplets hadronize by deflagration[5]. fies contributions of different φ and η scales to the event This is expected to result in dN/dy distributions with texture. We use DWT to test for possible largescale en- bumps or spikes on top of an otherwise smooth struc- hancement, as a function of the collision centrality. We ture. Other models[6] also predict bubbles of one phase reporttheDWTpowerspectruminpseudorapidityηand embedded in the other. azimuthal angle φ, for different charged particle multi- In the absence of a direct, event-by-event observable- plicities. We use mixed events to remove trivial fluctua- basedtest ofthese predictions, the picture hadbeen fur- tions and backgroundeffects. ther developed [7, 8] in order to connect it with the traditional observables such as the m slope parameter T T and the baryonand strangenesschemicalpotentials: the II. EXPERIMENTAL SETUP hadron“temperatures”T intheSPSdataarehigherthan lattice QCD predictions for a phase transition temperature. Usingafirstorderphasetransitionhydrodynamical The experimental setup [18] is shown in Fig.1. The model with a sharpfront betweenthe phases,Bilic et al. NA44Sipadarray,installed10cmdownstreamfromthe [7,8]concludedthataQGPsupercooling andhadrongas target,inthe magneticfield ofthe firstdipole, measured superheating isaconsequenceofthecontinuityequations ionization energy loss of charged particles in its 512 300 and of the requirement that the entropy be increased in µm thick Si pads. The plastic scintillator T0 (two rect- the transition. In the case of bubbles in the QGP phase, angles seen in Fig.1) was used for a centrality trigger. the plasma deflagrates; otherwise, it detonates. A direct The SPS beam was collimated to a 1 2 mm profile. × measurementofthehadrontextureatfreeze-out,ifitde- T0 covered 1.4 η 3.7 for an η-dependent fraction ≤ ≤ tects presence of the droplets/bubbles, could provide an of azimuthal angle, 0.22 ∆φ/2π 0.84 respectively. ≤ ≤ argument in favor of the first order phase transition. The silicon detector had inner radius 7.2 mm and outer The order of the confinement phase transition is still radius 43 mm, covering 1.5 η 3.3. The detector ≤ ≤ under debate. It is a fluctuation driven first order tran- wassplit radially into 16rings ofequalη coverage. Each sition [9, 10] in SU(3) with three massless quarks, but ring was further divided azimuthally into 32 sectors of second order in the case of finite mass [11] or infinitely equal angular coverage to form pads. The pads were massive [9, 10] strange quarks. A tricitical point may read out by AMPLEX [19] chips, one chip per sector. exist, separating the first order transition from a second δ-electrons,producedbythePbbeamtraversingthetar- order transition with the same critical exponents as the get, were swept away to one side by the dipole magnetic 3D Ising model [9]. For a second order phase transi- field ( 1.6 T). Only the δ-electron-free side was used in ≤ tion,localfluctuationsofisospinorenhancedcorrelation this analysis. Only 4 of the remaining 256channels were lengths may be observable [12, 13]. Large scale corre- inoperative. lations formed early in the collision are more likely to An amplitude distribution from a typical channel, ob- survive diffusion in the later stages. Small scale fluctu- servedinthephysicsrunanddigitizedwitha256channel ations, on the contrary, are more easily washed out by ADC is shown on Fig.2. Channel pedestals had, on the diffusiondue to secondaryscatteringamongthe hadrons average, FWHM = 0.48 < dE > of 1 MIP. In the tex- [14]. Consequently, an analysis method which can iden- ture analysis,everyevent was representedby a 2D array tify fluctuations on any scale is desirable. In this paper, of the calibrated digitized amplitudes of the channels ( we utilize a Discrete Wavelet Transformation,which has an amplitude array). Empty target runs were used to this property. measure the background, and cross-talk in the detector Awaveletis a function, 0everywhereexceptfora well was evaluated off-line. 3 s0.025 t n u o 0.02 c T0 Si d e z0.015 i l a m 0.01 r o a) N 0.005 Pb target 0 50 100 150 200 250 ADC channel FIG.2: Digitizedamplitudedistributionfromchannel1ofthe Sipadarray. Thesmoothcurveshowsaminimumχ2Landau fit performed in course of the amplitude calibration. The pedestal, thesingle and doublehitpeaksare distinguishable. b) φ = φ(η′,φ′). This makes the observable multiplicity distribution d2N/dφ′dη′ (in the presumed coordinates) differ from a simple function of η′: FIG. 1: a) The experimental setup: the target, the Si pad arrayandtheT0scintillation counter. Seetextforadescrip- tion of the detectors. b) The setup exposed to an RQMD d2N 1 dN = (1) event (GEANTsimulation). Magnetic field is on. dφ′dη′ 6 2π dη′ III. ANALYSIS TECHNIQUE Inthetruecoordinatesηandφ,theinequality1becomes an equality. However, the detector’s acceptance area in A. Detector calibration the true coordinates becomes distorted. In the following wewillrefertothis asa“Jacobianeffect”. TheJacobian effect, obviously, contributes to the event textures, espe- The NA44 spectrometer information was not used in cially on the large scale, and needs to be evaluated and this analysis, which focussed on the Si pad array data. corrected for. ADC pedestals were fitted channel by channel with a realisticfunctionalshape,determinedfromlowmultiplic- From Eq.1, the criterion of the true coordinate basis ity events in a minimum bias triggered run. Amplitude (η,φ) emerges naturally: it is the basis which makes the calibration of the Si detector was carried out channel by observable d2N/dφdη independent of φ. The minimiza- channel,byfittingtheamplitudedistributionwithasum tion problem was solved numerically with MINUIT[21], of single, double, triple, etc. (up to septuple) minimum and the resulting offsets are within the tolerance of the ionizing particle Landau distributions[20] with variable detector/beamposition. Cross-talkbetweentheelectron- weights. TheLandaudistributionswerenumericallycon- icschannelsisadetector-relatedcorrelationphenomenon voluted with the pedestal shape to account for noise in andintroducesa “texture”effect ofits own. Bothglobal the fit. A typical fit from a single channel is shown in cross-talk in the AMPLEX read-out chip [19] and read- Fig. 2. Parameters of the fit were used to simulate noise out board cross-talk are expected. In our detector with ina GEANT-baseddetectorresponseMonte Carlocode. 512 channels, there are 512 (512 1)/2 = 130816 two × − An offset of the event vertex with respect to the de- channelpairs(unordered),allofwhichweresubjectedto tector’s symmetry axis results in a non-trivialfunctional covariance analysis off-line. To magnify the non-trivial dependence between the actual η and φ, and the η′, φ′ instrumental contribution to the covariance matrix ele- presumed based on the “ideal” geometry: η = η(η′,φ′), ments, we analyzed covariances not between the ampli- 4 tudes A of channels i themselves, but between of Fig. 3) shows increasing multiplicity towards midra- i pidity, as is expected. As can be seen from Fig.3, the half-ring of iAk 1 detector’s acceptance is asymmetric aroundmidrapidity. a =A =A A i i i k A correction for the cross-talk has been applied. − 1 − 16 P half-ring of i half-rXing of i P (2) Otherwise, the dominant contributor to the cov(Ai,Aj) B. Discrete Wavelet Transformation (DWT) is the trivial variation of the event’s common multiplicity [36]. Using this method, we concluded that the effec- Discrete wavelets are a set of functions, each having tivecross-talkcouplingwasnon-negligibleonlyforneigh- a proper width, or scale, and a proper location so that boring channels within the same chip; it was found to the function differs from 0 only within that width and be 8.5%. As a remedy, a chip-wise (i.e. sector-wise) around that location. The set of possible scales and lo- event mixing technique including cross-talk in the ref- cations is discrete. The DWT formalizes the two dimen- erence sample was used to construct a reference event sional particle distribution in each Pb+Pb collision in sample. pseudorapidity η and azimuthal angle φ by performing an image analysis – transforming the event into a set of 110000 functions orthogonal with respect to scale and location in the (η, φ) space. We accumulate texture information by averaging the power spectra of many events. 8800 The simplest DWT basis is the Haar wavelet, built uponthe scaling function[37] g(x)=1 for0 x<1and hhdd 6600 0 otherwise. The function ≤ // ffdd +1 : 0 x< 1 N/N/ 4400 f(x)= 1 : 1≤ x<21 (3) 22 − 2 ≤ dd 0 : otherwise  2200 is the wavelet function[38]. If the interaction vertex lies on the detector’s sym- 00 00 22 44 66 11..55 22 22..55 33 33..55 metry axis, every pad’s acceptance is a rectangle in ff hh the (φ,η) space. Then, the Haar basis is the natural choice, as its scaling function in two dimensions (2D) G(φ,η) = g(φ)g(η) is just a pad’s acceptance (modulo FIG. 3: Double differential multiplicity distributions of units). We set up a two dimensional (2D) wavelet basis: charged particles plotted as a function of azimuthal angle φ (with different symbols representing different rings) and of pseudorapidity η (with different symbols representing differ- Fλ (φ,η)=2mFλ(2mφ i,2mη j). (4) ent sectors). The φ and η are in thealigned coordinates. m,i,j − − The scaling function in 2D is G(φ,η) = g(φ)g(η). As The double differentialmultiplicity data (Fig. 3)illus- in Eq.4,we constructG (φ,η) where m is the integer m,i,j trate the quality of the detector operation, calibrations, scalefinenessindex,andiandjindexthepositionsofbin geometricalalignmentandJacobiancorrection. Thedata centers in φ and η (1 m 4 and 1 i,j 16 because set is composed of two pieces, obtained by switching the we use 16 = 24 rings≤and≤16 sectors≤). Diff≤erent values magnetic field polarity: a negative polarity run is used of λ (denoted as φ, η, and φη) distinguish, respectively, for sectors 9 to 24 (range of π/2 < φ < 3π/2); a posi- functions with azimuthal, pseudorapidity, and diagonal tive polarity run is used for sectors 1 to 8 and 25 to 32 texture sensitivity: (range of 0 <φ< π/2 and 3π/2<φ <2π). The reason to disregard one side of the detector is additional oc- Fφ =f(φ)g(η), Fη =g(φ)f(η), Fφη =f(φ)f(η) (5) cupancy due to δ-electrons, as was explained in section II. Figure 3 demonstrates the quality of alignment as Then, Fλ with integer m, i, and j are known [17] to m,i,j well, since the η and φ along the horizontal axes are the form an orthonormal basis in the space of all measur- alignedcoordinates. Any geometricaloffset ofthe detec- able functions defined on the continuum of real numbers tormakesacceptancesofdifferentpadsnon-equalandde- L2(R). Fig. 4showsthewaveletbasisfunctionsF intwo pendentonthepadposition. Theacceptanceofeachpad dimensions. Atfirstglanceitmightseemsurprisingthat, has been calculated in the aligned coordinates, and the unlike the 1D case, both f and g enter the wavelet basis d2N/dφdη uses the actual acceptances dφ. The shape in 2D. Fig. 4 clarifies this: in order to fully encode an of the φ dependence of d2N/dφdη (left panel of Fig. 3) arbitrary shape of a measurable 2D function, one con- is flat as it should be for an event ensemble with no re- siders it as an addition of a change along φ (f(φ)g(η), action plane selection. The η dependence (right panel panel (b)), a change along η (g(φ)f(η), panel (c)), and 5 FIG. 4: Haar wavelet basis in two dimensions. The three modesofdirectionalsensitivityare: a)diagonalb)azimuthal c)pseudorapidity. Forthefinestscaleused,theacceptanceof a Si pad would correspond to the white rectangle drawn “on top” of the function in panel a). Every subsequent coarser scale is obtained by expanding the functions of the previous scale by a factor of 2 in both dimensions. a saddle-pointpattern (f(φ)f(η), panel (c)), added with appropriate weight (positive, negative or zero), for a variety of scales. The finest scale available is determined by the detector segmentation, while the coarser scales correspond to successively rebinning the track distribu- FIG. 5: Understanding the analyzing potency of the DWT tion. The analysis is best visualized by considering the powerspectra: a)foracheckerboardpatternb)forasmooth scaling function G (φ,η) as binning the track distri- m,i,j gradient patternc) forasample ofa thousandrandom white bution ρ(φ,η) in bins i,j of fineness m, while the set of noise images – in this case the average power spectrum is waveletfunctionsFmλ,i,j(φ,η)(or,tobeexact,thewavelet shown. expansioncoefficients ρ,Fλ )givesthedifferencedis- h m,i,ji tribution between the data binned with given coarseness and that with binning one step finer. ingtoreformulatethispropertyforwavelets,wherescale While the DWT analyzes the object (an image, a se- plays the same role as frequency in Fourier analysis. quence of data points, a data array) by transforming it, To dothat, we link scaleswith frequencies,orin other thefull informationcontentinherentintheobjectispre- words, we must understand the frequency spectra of the served in the transformation. wavelets. TheFourierimagesof1Dwaveletfunctionsoc- We adopt the existing [22] 1D DWT power spectrum cupy a set of wave numbers whose characteristic broad- analysistechniqueandexpanditto2D.Thetrackdensity ness grows with scale fineness m as 2m; 22m should be in an individual event is ρ(φ,η) and its local fluctuation used in the 2D case. Discrete wavelets of higher orin a given event is σ2 ρ ρ¯,ρ ρ¯ , where ρ¯ is the ders have better frequency localization than the Haar average ρ (over the acce≡ptahnc−e) in t−he giiven event[39]. wavelets. Despite this advantage, we use Haar because Using completeness of the basis, we expand only Haar allows one to say that the act of data taking withthe (binned !) detectorconstitutes the firststageof ρ ρ¯= ρ,Fλ Fλ ρ¯,Fλ Fλ (6) the wavelet transformation. − h m,i,ji m,i,j −h m,i,ji m,i,j In2D,wefinditmostinformativetopresentthethree Notice that ρ¯, being constant within the detector’s modes of a power spectrum with different directions of rectangular acceptance, is orthogonal to any Fλ sensitivityPφη(m),Pφ(m),Pη(m)separately. Wedefine m,i,j with m 1. Due to the orthonormality condition the power spectrum as nhFenmλt,si,jfo,rFdm≥λi′′ff,ie′,rje′nit=scδaλle,λs′dδmo,nmo′tδif,oi′rδmj,jc′r,otshs-eteρrm−sρ¯incothmepσo2- Pλ(m)= 1 ρ,Fλ 2, (7) sum,andthesumcontainsnocross-termsbetweenρand 22m h m,i,ji i,j X ρ¯forthefourobservablescales. Insteadofa ρ,Gm=5,i,j h i set, the Si detector energy amplitude array – its closest wherethedenominatorgivesthemeaningofspectralden- experimentallyachievableapproximation–isusedasthe sity totheobservable. Sodefined,thePλ(m)ofarandom DWT input. We usedthe WAILI [23]softwarelibraryto white noise field is independent of m. obtain the wavelet decompositions. In order to illustrate the sensitivity of the wavelet The Fourier power spectrum of a random white noise transformationto texture features ofthe differentscales, field is known to be independent of frequency [24]. We we have applied the wavelettransform to three test pat- arelookingfordynamicaltexturesinthedata,andthere- terns, shown in Fig. 5. All patterns are 16 16 pixel × fore would like to treat the random white noise case as matrices. The left hand side shows the test pattern, and a “trivial” one to compare with. Therefore it is interest- the right shows the power spectrum resulting from the 6 ducedtextureasstatisticalfluctuationscancel(shownas 2) 10 2inFig.6). Averageeventsretainthetextureassociated æP withtheshapeof d2N/dφdη,withthedeadchannelsand MI the finite beam geometrical cross-section (though this is E 1 d onlypartiallyvisibleintheaverageevent,duetothefact 2Æ/ that event averaging is done without attempting to se- -1 sm) ( 10 lpercotpeovretinotnsaalctcoortdhiengvatroiatnhcee,voerrtseqxupaoresditiflonu)c.tuPatλio(mn)σ2is. ( lP Therefore, for Poissonian statistics of hits in a pad, the 000 222 444 000 222 444 000 222 444 eventaveragingoverM eventsshoulddecreasePλ(m)by fhfhfh fffiiinnneeennneeessssss fff fffiiinnneeennneeessssss hhh fffiiinnneeennneeessssss a factorof M. The averageeventwhose powerspectrum is shown on Fig. 6 is formed by adding 7 103 events, however its Pλ(m) is down less than 7 1×03 compared FIG. 6: Power spectra of 7×103 events in the multiplicity × to that of the single events. This demonstrates that the bin326< dN/dη<398(between≈6%and10%centrality). average event’s texture is not due to statistical fluctua- (cid:13) – true events,△ – mixed events, 2– the average event. tions, but rather, predominantly due to the systematic uncertainties listed. Consequently, we can use the averageevent’sPλ(m)toestimatethemagnitudeofthestatic wavelettransform. Patterna),acheckerboard,hasstruc- texture-relatedsystematics. AsseenfromFig. 6,thesys- ture only on the finest scale and all power components tematics are far below the Pλ(m) of single events (true of scales coarser than 4 are zero. Pattern b) has exactly or mixed), with the exception of pseudorapidity, where the opposite scale composition; the slow gradation be- non-constancy of dN/dη over the detector’s acceptance tween black and white corresponds to a structure on the is visible. coarsest scale, as seen in the accompanying power spec- The way to get rid of the “trivial” or static texture trum. Smoothness of the gradient means that neighbor- is to use mixed events, taking different channels from to-neighborchangesdonotaddmuchtothepatternonce different events. The mixed events preserve the texture theoveralltrend(thelargescalefeature)istakenintoac- associatedwith the detector position offset, the inherent count. dN/dη shape and the dead channels. This is static tex- These two examples illustrate the propertyof scale lo- ture as it produces the same pattern event after event calization, made possible by virtue of the scale orthog- while we are searching for evidence of dynamic texture. onality of the basis. Patterns encountered in multiple Wereducesourcesofthestatictextureinthepowerspec- hadron production involve a variety of scales, and yet tra by empty target subtraction and by subtraction of they are more likely to be of type b), rather than a). An mixed events power spectra, thus obtaining the dynamic important conclusion follows immediately: in this type texturePλ(m) Pλ(m) . Inordertoreproducethe of measurement, large acceptance, like the one used in true mix − electronics cross-talk effects in the mixed event sample, this analysis, rather than fine segmentation, is the way the mixing isdone sector-wise,i.e. the sectorsconstitute to accomplish sensitivity. thesubeventssubjectedtotheeventnumberscrambling. Case c) shows patterns that arise from white noise. We continue with a brief summary of the systematic They produce signals in the power spectrum indepen- errors in the measurements of the DWT dynamic tex- dentofscale,asexpected. Inthefirstapproximation,the ture observable P P . Static texture and dy- whitenoiseexampleprovidesabase-linecaseforcompar- true mix − namicbackgroundtexturepresentthelargestproblemin isons in a search for non-trivial effects. the search for the phase transition-related dynamic tex- Figure6showsthepowerspectrameasuredinPb+Pb turevia powerspectraoflocalfluctuations. The method for one multiplicity range. The unit on the verticalscale (σ2/ dE 2) is chosen so that the power of fluctua- of solving the problem is comparison with the reference MIP tionshwhose viariance σ2 equals the squaredmean energy sample created by event mixing. Thus the Ptrue Pmix − observable was created. For comparison with models, a loss by a minimum ionizing particle traversing the de- Monte Carlo simulation of the Si detector is used. It in- tector, is the unit. The first striking feature is that the cludestheknownstatictextureeffectsandundergoesthe power spectra of physical events are indeed enhanced on same procedure to remove the effects. The “irreducible the coarse scale. The task of the analysis is to quan- remainder” is the residual effect which may tify and, as much as possible, eliminate “trivial” and experiment-specific reasons for this enhancement. 1. survive the elimination procedure 2. emergeas a difference betweenthe data,subjected C. Identification and control of systematic errors totheeliminationprocedure,andtheMCanalyzed in the same manner. The averageevent, formedby summing amplitude images of the measured events in a given multiplicity bin, TableIliststhesourcesofstatictextureandsummarizes and dividing by the number of events, has a much re- themethodsoftheirtreatment. Wegroupthebackground 7 texture sources according to similarity of manifestation the direction angle, with mixing and P P sub- true mix − and treatment, into tractiondonewithinthoseclasses. Neitherreactionplane nordirectionanglewasreconstructedinthepresentanal- statistical fluctuations ysis, and the P P (especially that of the az- • true mix − imuthal and diagonal modes on the coarse scale) retain static texture • theelliptic/directedflowcontribution. Theeffectsofflow backgrounddynamic texture on dynamic texture observables are smaller than other • texture effects, so they can not be singled out and quan- The statisticalfluctuationis the mosttrivialiteminthis tified in this analysis. list. Both event mixing (provided that mixing is done The finite beam cross-section effect belongs to this withinthepropermultiplicityclass)andMCcomparison group, despite the fact that a very similar effect of geo- solve this problem. The statistical fluctuations do not metricaldetector/beamoffsethasbeenclassifiedasstatic result in irreducible systematic errors. texture. An effect must survive mixing with its strength The static texture group includes: unaltered in order to be fully subtracted via event mix- geometrical offset of the detector with respect to ing. Preservingtheeffectoftherandomvariationsinthe • the beam’s“centerofgravity”inthe verticalplane Pb+Pbvertexonthe powerspectrainthe mixedevents requires classification of events according to the vertex dead pads position and mixing only within such classes. This re- • quires knowledge of the vertex for each event, which is dN/dη shape – a genuine large scale multiparticle • not available in this experiment. Therefore, MC simula- correlationsensitivetothephysicsoftheearlystage tionofthebeamprofileremainstheonlywaytoquantify of the collision false texture arising from vertex variations. MC studies with event generators show that the beam spatial ex- Cleanliness of the static texture elimination via event tent and the resulting vertex variation is the source of mixing has been checked by simulating the contribut- the growth of the coarse scale azimuthal texture corre- ing effects separately. First, by running the detector re- lation with multiplicity (see Fig. 7). Uncertainty in our sponse simulation on MC-generated events without the knowledge of the beam’s geometrical cross-section must beam/detector offset and with a beam of 0 thickness it be propagated into a systematic error on P P . was ascertained that the remaining dynamic texture is true mix − very small compared with the systematic errors due to The other two effects in this groupare difficult to sep- the background Si hits and the beam geometrical cross- arate and simulate and the error estimate reflects the section, for all scales and all directional modes λ. Due combined effect. The systematic errors were evaluated tothefinite sizeofthe multiplicitybin,the mixedevents by removing the Pb target and switching magnetic field consistofsubeventscomingfromeventsofdifferenttotal polarity to expose the given side of the detector to δ- multiplicity. With the sector-wisemixing, this causes an electrons (from the air and T0), while minimizing nu- additional sector-to-sector variation of amplitude in the clear interactions. This gives an “analog” generator of mixed events, thus resulting in an enhancement of Pφ uncorrelated noise. All correlations (i.e. deviations of mix primarily on the finest scale, with respect to Pφ . On Pλ(m)true from Pλ(m)mix) in this noise generator are true Fig. 7, this effect can be seen as the Pφ Pφ values treated as systematic uncertainties. Thus this compo- true− mix nent of the systematic error gets a sign, and the sys- progressivelygrownegativewithmultiplicityinthefinest tematic errors are asymmetric. The effect of increasing scale plot. However, as can be seen on the same figure, texture correlation (for diagonal and azimuthal modes) the effect is small compared with the total systematic withmultiplicityonthecoarsescale,attributedtothege- error bars shown as boxes. ometricaloffsetof the detector with respectto the beam The backgrounddynamic texture group includes: (the leading one in the static group), is present in the elliptic and directed flow switched polarity empty target runs as well. For this • reason, it was impossible to disentangle the background finiteness of the beam cross-section dynamic contribution on the coarsest scale. In Table I, • the “irreducible remainder estimate” for the diagonal, backgroundhits in the Si • coarse scale is bracketed with two numbers, which form thelowerandupperestimates. Thelowerestimateisob- channel-to-channelcross-talk • tained by taking the scale one unit finer and quoting its Elliptic and directed flow, observed at SPS [16], are number. This, indeed, sets the lower limit because the large scale dynamic texture phenomena of primarily az- deviations of Pλ(m) from Pλ(m) generally grow true mix imuthal (elliptic) and diagonal (directed flow) modes. withscalecoarseness. Theupperlimitissetbyascribing Because both reaction plane and direction angle vary the entire texture correlation,observed in the δ-electron event by event, the respective dynamic textures can not data,to the backgroundhits and channelcross-talk,and besubtractedbyeventmixing,unlesstheeventsareclas- ignoring the fact that significant portion of it must be sified according to their reaction plane orientation and due to the vertex fluctuation (finite beam profile). This 8 1 a . f f fh f h e fhfh ff hh o -1 c 10 22)) 44 ee ce ææE MIP E MIP 22 coarscoars den 10 -2 dd 00 fi 22ÆÆ// 11 (cid:222)(cid:222) on -3 ssm) (m) (mixmix --0000....005555 s. err. c 10 0 2 4[P6l (18) 0 -2Pl (41)6 8]/R0MS2 4 6 8 (( --11 sy true mix mix llPP --ee --11..55 (m)(m)trutru 00..0022 FstIrGen.g8t:hC. onfiRdMenScmeicxoedffiecnioetnetsasahfPuλn(c1ti)o2mnixof−thhPeλfl(u1c)tmuiaxtii2oin. llPP The multiplicity bin is 326 < dN/dη < 398 (6-10% central- e e --00..22 p rr ity),as in Fig.6. uu extext 00..22 mic tmic t 00 (cid:220)(cid:220)fine fine can be used to characterize the relative strength of local aa nn --00..22 fluctuations in an event. The distribution for different λ yy DD (or directions) is plotted on Figure 8 in an integral way, 00 225500 00 225500 00 225500 ddNN//ddhh cchhaarrggeedd i.e. as anα(x) graphwhere for everyx, α is the fraction of the distribution above x. ∞ dN +∞ dN FIG. 7: Multiplicity dependence of the texture correlation. α(x)= dξ dξ, (8) (cid:13)–theNA44data,•–RQMD.Theboxesshowthesystem- dξ dξ atic errors vertically and the boundaries of the multiplicity Zx .Z−∞ bins horizontally; the statistical errors are indicated by the where ξ denotes the fluctuation strength verticalbarson thepoints. Therows correspond tothescale Pλ(1) Pλ(1) fineness m, the columns – to the directional mode λ (which true mix ξ = − , (9) can bediagonal φη,azimuthal φ, and pseudorapidity η). RMS(Pλ(1) ) mix and dN/dξ is the statistical distribution of ξ, obtained fromtheexperimentallyknowndistributionsofPλ(1) upper limit is likely to be a gross overestimation, and in true and Pλ(1) . Expression 9 is constructed to be sensi- Fig. 7 we show systematic errors,obtained by adding in mix tive to the difference between Pλ(1) and Pλ(1) , quadraturethefinitebeamerrorwiththebackgroundhit true mix whileminimizing detectorspecificsto enablecomparison error. between different experiments in future. The latter is accomplished by normalizing to RMS . This normal- mix izationalsoeliminatesthetrivialmultiplicitydependence IV. RESULTS of the observable. The fluctuation strength observable provides a limit Fig. 7presentsacomparisonoftheDWTdynamictex- on the frequency and strength of the fluctuations and ture in the measured and RQMD-simulated[25] Pb+Pb expresses the result in a model-independent way. The collision events. The three directional sensitivity modes confidence level with which local fluctuations of a given (diagonal φη, azimuthal φ, and pseudorapidity η) have strength (expressed through the event by event observ- four scales each, so that there are 12 sets of points in ables via Eq. 9) can be excluded is then 1 α. Fluc- − the DWT dynamic texture as a function of the charged tuations greater than 3 RMS are excluded in the mix × multiplicity dNch/dη bin. The systematic errors on the azimuthal and pseudorapidity modes with 90%and 95% points (shown by vertical bars) have been evaluated fol- confidence, respectively. The monotonic fall of the curve lowing the procedure described in detail in Section III. isconsistentwiththeabsenceofabnormalsubsamplesin Fig.6 demonstrated that the major fraction of the ob- the data. served texture exists also in mixed events. A detailed RQMD events were fed into the GEANT detector re- accountofthe causeswasdiscussedinthe precedingsec- sponse simulation and analyzed using the same off-line tion, including known physics as well as instrumental ef- procedure as used for the experimental data. The de- fects. It is therefore clear that the observable most di- tector offset with respect to the beam center of gravity rectly related to the dynamical correlations/fluctuations andthe beam profile were included in the simulation. In is not Pλ(m), but Pλ(m) Pλ(m) . This quan- a separate simulation run, the beam profile was identi- true mix tity, normalized to the RMS −fluctuation of Pλ(m) , fied as the cause of the rise of the azimuthal dynamic mix 9 400 texture with the multiplicity on the coarse scale. In our experiment,thispurelyinstrumentaleffectdominatesthe 200 azimuthal component of the DWT dynamic texture. The most apparent conclusion from Fig. 7 is that a 0 large fraction of the texture (seen on Fig. 6) is not dy- d namic i.e. not different between true and mixed events. e 400 g Being monotonic (or absent), the change of the data r 200 points with multiplicity does not reveal any evidence of a h 0 a region of impact parameters/baryochemicalpotentials c with qualitatively different properties, such as those of y 300 a critical point neighborhood. The RQMD comparison d 200 / confirms that particle production via hadronic multiple N 100 d scattering, following string decays (without critical phe- 0 nomenaorphasetransition)canexplaintheobservedre- 300 sultswhendetectorimperfectionsaretakenintoaccount. 200 Moredetaileddiscussionoftheimplicationsofthesedata 100 on various phase transition models will be given in Sec- 0 tion VI. 0 1 2 3 4 5 6 y V. SENSITIVITY FIG.9: dN/dydistributionofchargedparticlesinthemulti- fireballeventgeneratorinfourindividualevents withdifferent Interesting physics can manifest itself in the ensemble numberoffireballs: △–2fireballs,2–4fireballs,3–8fire- probability density distributions as well as in the event- balls, (cid:13)–16 fireballs. Onecan seehowthetexturebecomes by-event (EbyE for short) observables. To illustrate the smootherasthenumberoffireballsincreases. Weremindthe power of the EbyE observable we used, we should con- readerthatthedetector’sactiveareacoversπazimuthallyand struct final states of charged particles indistinguishable pseudorapidity 1.5 to 3.3. In general, acceptance limitations fromthe point ofview of“traditional”,orensemble-wise makeit more difficult to detect dynamic textures. observables, such as 1. dN/dy distribution fireball is 0; its total p is chosen to reproduce the ob- Z 2. dN/dp , 1/mTdN/dm distribution etc. T T served dN/dy of charged particles by Lorentz-boosting 3. multiplicity distribution the fireballs along the Z direction, keeping the total p~ of an event at 0 in the rest frame of the colliding nuclei. and compare the sensitivity of the above-mentioned ob- The fireballs hadronize independently into charged and servables with that of the EbyE one. neutral pions and kaons mixed in a realistic proportion. Asensitivitystudywasperformedusingamultifireball ByvaryingnumberofparticlesN perfireball,onevaries p event generator created specially for this purpose. The “grain coarseness”of the event texture in η. generatorproducestexturesofknownmagnitudebysim- Toillustratethe discussion,Fig.9presentsexamplesof ulating the observed multiplicity as arising from an ar- dN/dydistributionsinfoureventswithdifferentnumber bitrary number of fireballs. Correlations among groups of fireballs. The dynamic textures seen on the figures of particles arise when the particles come from the same arepeculiarto these particulareventsandaregone after fireball. We do not suggest that the physics of Pb+Pb dN/dy of many events are added. collisionsisproperlydescribedbyasuperpositionoffire- ballsofafixedsize. Rather,weusethe fireballsasaway We simulated average fireball multiplicities of 10, 50, to generate controlled multiparticle correlations. 90(withRMSfluctuationof3)andlarger. Fig. 10shows This picture is inspired by Van Hove’s scenario [4] comparisonof our data with the simulated pseudorapid- of a first order phase transition via droplet fragmenta- ity texture. With 104 events, the detector+software ∼ tion of a QGP fluid. We measure texture in two direc- can differentiate between the cases of 50 and 90 parti- tions, spanned by polar and azimuthal angles, and are cle fireballs. The signal grows with the charged particle also sensitive to the spatial texture of longitudinal flow. multiplicity and with N . Fig. 10 provides quantitative p Forboost-invariantexpansion[26]twodroplets,separated information on the sensitivity of the texture measure- along the longitudinal coordinate, will be separated in y ments by relating the expected strength of response to and η. As long as there is longitudinal expansion,a spa- the strength of texture via Monte Carlosimulation. The tial texture will be manifested as (pseudo)rapidity tex- sensitivity is limited by systematic errors of the mea- ture. In the multifireball event generator, we generate surement, discussed in Section III. Nevertheless, it is the pseudorapidity texture explicitly, omitting the spa- instructive to compare sensitivity of this method with tial formulation of the problem. The total p of each other methods; in particular with two point correlators. T 10 2) 5 pectedthatafirstorderphasetransitionwouldbeeasier æ to observe. Our dynamic texture measurement tests the P NA44 data I hypothesis of the first order phase transition via QGP M 4 MC fireballs : droplet hadronization[4] in a way more direct than in- E of 90 particles (+,-,0) terpretation of p spectra involving latent heat. Our re- d T of 50 sultcanbeusedtoconstrainphenomenologicalquantities Æ/ 3 2 of 10 whichrepresentbasic QCDproperties and affecttexture s ( formation in this class of hadronization models [4, 6, 8]. x mi 2 Such quantities are the energy flux, or rate at which the 1) QGPtransmitsitsenergytohadrons[29,30],criticalsize ( of the QGP droplet[6], and initial upper energy density hP 1 of the transition ǫ′. - 0 ue The specific experimental signature of second order r )t phase transition (known since the discovery of critical 1 0 ( opalescence[31]) is the emergence ofcriticalfluctuations hP of the orderparameter with an enormous increase of the 0 100 200 300 400 correlation lengths. However, for physical quark masses dN/dh charged RajagopalandWilczek [12,32]arguedthatdue toclose- ness of the pion mass to the critical temperature, it would be unlikely for the correlation volumes to include FIG. 10: Coarse scale η texture correlation in the NA44 large numbers of pions, if the cooling of the plasma and data, shown by (cid:13) (from the top right plot of Figure 7), is hadronization proceeds in an equilibrated manner. If, compared with that from the multifireball event generator onthe contrary,the hightemperatureconfigurationsud- for three different fireball sizes. Detector response is simu- denly finds itself at a low temperature, a self-organized lated. The boxes represent systematic errorbars (see caption criticality regime settles in, and the critical local fluctu- to Fig. 7). ations develop fully[12, 32]. The NA44 data reported here signifies absence of dynamical fluctuations on the scales probed, within the The sensitivity of the method is remarkable indeed if limitofsensitivitydiscussedinSectionV. Convincingev- one takes into account that statistics in the fifth multi- idenceofthermalequilibrationcanbeprovidedbyevent- plicity bin for each of the three event generatorpoints is below 3 104 events – too scarce, e.g., to extract three by-event observables. Our data is consistent with local × thermal equilibrium, understood as an absence of phys- source radius parameters via HBT analysis even with a ically distinguished scales between the scale of a hadron well optimized spectrometer! andthe scale of the system,or scale invarianceof fluctu- The use of two particle correlation in rapidity R2(y) ations [33] (“white noise”). However to probe equilibra- to search for droplets was discussed for pp¯ collisions at √s = 1.8 TeV (at FNAL)[27]. R2 was reported to de- tiondirectly withthismethod,texturesensitivityatleast crease withmultiplicity,sothatitwouldnotbeexpected down to the typical fireball (cluster) sizes observed in pN collisions in cosmic raysandacceleratorexperiments to be visible for dN/dy above 20; the signal would be weaker in a scenario with corr≈elated droplets. In con- [34,35]wouldbenecessary. Intheabsenceofsuchdirect evidence, the non-observationof critical fluctuations can trast, the wavelet transformation retains sensitivity at implyeitherabsenceofthesecondorderphasetransition high multiplicity, as we see in Fig.10. In the fifth mul- orpresenceofthermalequilibration–thelattervoidsthe tiplicity bin, with total number of hadrons at freeze-out around 1.5 102, a typical fraction of particles coming criticalitysignature,accordingtoRajagopalandWilczek × [12]. from the same fireball for the clustering parameters of 50 (90) would be 3% (6%) [40]. In either case there is little hope of seeing any trace of suchdynamics either in ensemble-averaged dN/dy orin dN/dy ofasingleevent, VII. CONCLUSION but the systematic difference between the power spectra of the real and mixed events, integrated over multiple Wehavedevelopedamethodofmeasuringthedynamic events, nevertheless reveals the difference. The data are component of local fluctuations in charged particle den- consistent with clustering among 3% of the particles. sity in pseudorapidity and azimuthal angle, and applied ≤ the analysisto Pb+Pbcollisionsmeasuredbythe NA44 experiment. Comparison of the data to a simple Monte VI. DISCUSSION Carlotextureeventgeneratorindicatesthatsensitivityto pseudorapidity density clusters of 3% is accomplished ≥ The order of the expected QCD phase transitions is in this experiment. The probability of encountering a known to be a complex issue for realistic current masses real event whose dynamic azimuthal texture exceeds in ofquarksinthe systemofafinite size. Itisgenerallyex- strength that of a randommixed event by 3 RMS, is be-