MULTIFRACTAL ANALYSIS OF CHARGED PARTICLE DISTRIBUTION IN 28 Si-Ag/Br INTERACTION AT 14.5A GeV

Transcription

1 Fractals, Vol. 20, No. 3 (2012) 1 13 c World Scientific Publishing Company DOI: /S X Fractals Downloaded from MULTIFRACTAL ANALYSIS OF CHARGED PARTICLE DISTRIBUTION IN 28 Si-Ag/Br INTERACTION AT 14.5A GeV P. MALI and A. MUKHOPADHYAY Department of Physics, University of North Bengal Raja Rammohunpur, Dist. Darjeeling , West Bengal, India amitabha 62@rediffmail.com G. SINGH Department of Computer and Information Science SUNY at Fredonia, Fredonia, New York 14063, USA Received April 13, 2012 Accepted June 26, 2012 Published September 15, 2012 Abstract The multifractal structure of one dimensional charged particle density distribution in 28 Si-Ag/ Br interactions at 14.5 GeV per nucleon is investigated by using two different techniques. The experimental measurements are compared with a microscopic transport model of particle production based on the Ultra-relativistic Quantum Molecular Dynamics (UrQMD). Various parameters related to multifractality, for example the Lévy s index, are obtained. Our analysis shows that multifractal structure is present both in the experiment as well as in the simulation. As far as the self-similar nature of the density fluctuation is concerned, there exists, however, a small but definite quantitative difference between the two. Keywords: Fluctuations; Multifractal Moments; Intermittency; Generalized Fractal Dimension; Multifractal Specific Heat. Corresponding author. 1

2 2 P. Mali, A. Mukhopadhyay & G. Singh 1. INTRODUCTION In high-energy collisions between two sub-atomic particles often many new particles are produced. Most of these particles (typically more than 90%) are pi-mesons, and the phenomenon is known as multiparticle production. The angular distribution of these produced particles in a single interaction (also called an event) exhibits random fluctuations with sharp peaks and deep valleys, which are apparantly devoid of any pattern. Such fluctuations contain trivial statistical noise as well as physically significant dynamical components that are not directly accessible to an experiment. By averaging over many events the noise can be eliminated, but simultaneously the dynamical components are also smoothed out. One can, however, use suitable statistical techniques to disentangle the noise from the dynamical part. One such technique is based on computing the factorial moment (FM) of order q (a positive integer) f q = n(n 1) (n q +1), where n is the number of particles falling within a small phase-space interval (say δx) inanevent.for a Poisson distributed statistical noise an FM of the overall particle distribution is same as the ordinary moment of only the corresponding dynamical part. 1 The FM suitably normalized by the average particle number, for statistical reasons is averaged over many angular intervals and over many events. In one-dimensional analysis, the normalized factorial moment (NFM) F q is found to scale self-similarly with the resolution size δx: F q (δx) φq :asδx 0, (1) down to the experimental limit. In multiparticle production mechanism this type of power law scaling behavior of F q is known as intermittency, a term coined from the turbulence of fluid dynamics, and a phenomenon first observed in ultra-high-energy cosmic-ray events. 2,3 Since the first observation the intermittency phenomenon has been experimentally verified in many different types of high-energy interactions between subatomic particles. 4 The positive valued exponent φ q, known as the intermittency index, provides us with an opportunity to characterize the apparently irregular fluctuations of the particle density function in terms of finite numbered and regularly behaving parameters that are intimately connected to the theory of fractals. This close connection between the intermittency and fractality was first recognized while the intermittent behaviour of turbulent fluids in terms of fractal dimensions was studied. 5 This observation prompted the fractal formalism to be adopted into the multiparticle production in high energy collisions, which subsequently lead to the study of the intermittency from the perspective of (multi)fractality. Several techniques based on the fractal theory are now available to analyze the multiparticle data. 5 7 The most popular of them are developed by Hwa 8 and Takagi. 9 For a self-similar process, the multifractal moment G q (also called the frequency moment) introduced in Hwa 8 exhibits a power law dependence on the phase space resolution size in a way similar to the Eq. (1). The advantage of using the G q moment over the F q moment lies in the fact that, with the former one can study not only the spikes (for q>0) but also the non-empty valleys (for q<0), while the latter is useful only for the spikes of the density distribution. Unlike the FM, the G q moments can be defined for fractional as well as for negative q. However, as the empty bin effect dominates in low multiplicity events, the G q moments saturate with δx 0 and the statistical noise present in the density function cannot be automatically accounted for. In G q technique the noise has to be eliminated by simulating equivalent uncorrelated event sample. It is to be noted that the limiting condition δx 0isonlyamathemat- ical idealization. In practice the reachable limit is only up to the phase space resolution allowed by the detector granularity. Takagi, 9 on the other hand, proposed another set of multifractal moments T q for q>0that are not particularly affected by the finiteness of the event multiplicity. Takagi used his technique to determine several multifractal parameters by analyzing the pp and the e e + annihilation data. However, a definite method of statistical noise elimination in the T q technique, could not be formulated yet. Both the G q and the T q techniques are applied to analyze several high-energy nucleusnucleus (AB) interaction data Recently, we have performed one and two dimensional intermittency analyses of the 28 Si-induced Ag/Br interaction at an incident energy of 14.5GeV per nucleon. 17,18 Correlations among produced particles are observed from both the analyses, and the intermittency strength in two dimensions (2d) is several times larger than that in one dimension (1d). For the same interaction we have also shown that a Monte Carlo event generator

3 based on the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) 19 produces almost no intermittency. Even inclusion of the Bose-Einstein correlation (BEC) effect (known to be the most dominant factor of such correlated emission of particles) into the simulation, could not fully reproduce the experiment. 18 Hence, at this stage it is difficult to single out any definite and satisfactory reason behind the observed behaviour of the density fluctuation. However, a clear signature of self-similarity in the 1d density fluctuation, and of self-affinity in the 2d fluctuation indicate that fractal characteristics are involved in the particle emission process. In this paper we therefore, present a 1d multifractal analysis of the charged meson distribution in the 28 Si-Ag/ Br interaction at 14.5 GeV per nucleon. Following the trend of our previous works 17,18 the experimental measurement is compared with the corresponding UrQMD simulation. The paper is organized according to the following sequence: in Sec. 2, we summarily discuss the experimental aspects like the event selection criteria, the track classification and the gross data characteristics etc. In Sec. 3, the simulation technique is briefly outlined. In Sec. 4, the statistical methods employed for data analysis and the results obtained thereof are discussed. And finally, in Sec. 5, we conclude with a summary of our observation and a few critical remarks. 2. EXPERIMENT The nuclear photo-emulsion technique is used to collect the data. Ilford G5 nuclear emulsion pellicles of size 16 cm 10 cm 600 µm were irradiated by the 28 Si beam obtained from the Alternating Gradient Synchrotron at the Brookhaven National Laboratory (USA). The incident beam energy was 14.5 GeV per nucleon. Each emulsion plate was scanned along individual projectile tracks until either the track leads to an event, or it leaves the plate. The event scanning was performed with Leitz microscopes under a total magnification of 300. Angle measurement, track counting and track classification were performed under a total magnification of 1500 by using a pair of Koristka microscopes. Minute details of the nuclear emulsion technique are discussed in Refs. 20 and 21. In emulsion terminology tracks emerging from an interaction (also called a star) are classified in the following ways: (1) Shower tracks: The shower tracks are caused by the singly charged particles produced in an Multifractal Analysis of Charged Particle Distribution 3 interaction, and moving with a speed v>0.7c. The ionization of this class of particles I 1.4I 0, I 0 being the minimum ionization due to any particle in the emulsion plate. In the present case I 0 20 grains/100 µm. Mostly charged pion tracks, contaminated by a small percentage of other meson tracks and e + /e tracks, belong to the shower track category. The total number of such tracks in a single event is denoted by n s. (2) Grey tracks: The grey tracks are caused mainly by the target recoil fast protons with ionization 1.4I 0 I 10I 0. Their velocity values range between 0.3 c and 0.7 c, and their kinetic energies can move up to 400 MeV. The total number of grey tracks in an event is denoted by n g. (3) Black tracks: The black tracks are due to the heavy and slow moving fragments evaporating out of the remnants of target nuclei, having ionization I > 10I 0, velocity less than 0.3 c, and energy less than 30 MeV. The total numberoftracksofthiskindinaneventisdenoted by n b. (4) Projectile fragments: The projectile fragments are caused by the spectator parts of the incident nucleus that do not directly participate in an interaction. They are emitted within a very narrow and extremely forward cone of semi-vertex angle θ f 0.21/p inc,wherep inc is the incident projectile momentum per nucleons in GeV/c. The projectile fragments have a uniform ionization over a very long range and they possess almost the same energy per nucleon as the incident projectile. Their number in an event is denoted by n pf. To ensure that either an Ag or a Br nucleus is the target in an interaction, only the stars with the number of heavy fragments n h (= n g + n b ) > 8are considered. The present analysis however, is confined only to the light charged mesons (pions) producing the shower tracks. The number of 28 Si-Ag/ Br events present in the present data sample is N ev = 331 and the average shower track multiplicity for this sample is, n s =52.67 ± In the multiparticle dynamics the phase space is usually constituted by any one or a suitable combination of the following variables: the rapidity variable y, the azimuthal angle ϕ, and the transverse momentum p t of the emitted particles. The rapidity variable is a dimensionless boost parameter that can be used to locate a particle in one-dimensional phase space.

4 4 P. Mali, A. Mukhopadhyay & G. Singh It is defined as y = 1 2 ln E + p l, E p l where E is the energy and p l is the longitudinal momentum of the particle. The phase space in the present analysis is constituted by the pseudorapidity variables η of all the shower particles. Knowing the emerging angle θ of a particle/track η can be defined as η = ln tan(θ/2). For light mass particles like mesons moving with relativistic speed, the pseudorapidity is a convenient approximation of the rapidity variable. Moreover, the energy and momentum measurements are difficult for emulsion experiments. An accuracy of δη =0.1unit in η could be achieved through the reference primary method of angle measurement. Taking all 331 events together the η-distribution can be approximated by a Gaussian function for which the fit parameters are: the centroid η 0 =1.90 ± 0.01, the width σ η =2.17 ± 0.03, and the peak density ρ 0 =17.88 ± It should be noted that the single particle density distribution is not uniform and that nonuniformity influences the multifractal results. The shape dependence problem can be resolved by replacing the phase space variable η with a cumulative variable X η defined as 22 η / ηmax X η = ρ(η )dη ρ(η )dη, η min η min where ρ(η) = (1/N )dn/dη is the single particle pseudorapidity density of the shower tracks and η max (η min ) is the maximum(minimum) value of η. Irrespective of the nature of phase space variable, the density distribution in terms of the cumulative variable is always uniform between X η = 0 and X η = 1. Though the analysis has been performed taking X η as the basic variable, we shall continue to call it η-variable, and the phase space will be called the η-space. 3. SIMULATION The UrQMD model 19 is a nonequilibrium, microscopic transport theory combined with the stochastic binary scattering, resonance decay and color string fragmentation. The model considers covariant propagation of hadrons along their respective classical trajectories. In the mathematical framework, a relativistic Boltzmann equation is to be solved to obtain the hadrons in the final stage of the collision. A nucleon-nucleon interaction would occur if the impact parameter b< σ tot /π, where the total cross-section σ tot depends on the isospin of the interacting hadrons, their flavors and the center of mass energy involved. The Fermi gas model is utilized to describe the projectile and the target nuclei. The initial momentum of each nucleon belonging to the target (or to the projectile) is distributed at random between zero and the local Thomas-Fermi momentum. Each nucleon is described by a Gaussian shaped density distribution, and the wave function for each nucleus is taken as a product of such single nucleon Gaussian functions. The Slater determinant necessary for antisymmetrization is not taken into account. In configuration space the centroids of the Gaussian functions are distributed at random within a sphere, and finite widths of these Gaussians result in a diffused surface region. At low and intermediate energies ( s 5 GeV) like the present case, the phenomenology of hadronic interactions is described in terms of interactions between known hadrons and their resonances. At higher energies, s > 5 GeV, the excitation of color strings and their subsequent fragmentation into hadrons dominate the multiple production of particles. For AB collisions the soft binary and ternary interactions between nucleons are described by a non-relativistic density dependent Skyrme potential. In addition, Yukawa, Coulomb and Pauli (optional) potentials are also implemented in the model. The model allows for subsequent rescattering. The framework allows to bridge with one concise model, the entire available range of energies from the SIS ( s NN 5GeV) to the RHIC ( s NN = 200 GeV). Using the UrQMD model code (version 3.1) we have simulated a sample of 28 Si-Ag/Br events at 14.5 GeV per nucleon that is five times as large as the experimental sample. Events with Ag and Br as target nuclei are first generated separately, and they are thereafter mixed with each other according to the proportional abundance of Ag and Br nuclei in Ilford G5 emulsion. All newly produced charged mesons in the simulated events are retained for subsequent analyses. The simulated event sample possesses identical multiplicity distribution, and similar η-distribution as the experiment. Obviously, the average multiplicity n ch of the UrQMD generated sample is same as the experimental n s quoted above. The parameters of the best Gaussian fitted

5 to the η-distribution of the UrQMD, are also very close to the respective experimental values. We have generated another set of event sample based on the random numbers. An inverse of integral method have been used to generate the random numbers with Gaussian distribution. 23 The corresponding η distribution has the same set of parameters as the experiment. The random sample size is also five times as large and possesses the identical multiplicity distribution as the experiment. While generating the random numbers no correlation has been incorporated. It corresponds therefore, to an independent emission of particles, which can be utilized to discriminate between the statistical component of the fluctuation from the so desired dynamical component. 4. METHODOLOGY AND RESULTS 4.1. Hwa s Moments The frequency moment or the multifractal moment of order q (positive or negative, integer or fraction), averaged over many events as well as over many equal size phase-space intervals, is defined as 8,24 G q = 1 N ev N ev e=1 m=1 M [ ] n e q m. Here M represents the total number of equal size intervals into which the entire phase-space accessible to the experiment is partitioned, n e m is the number of shower tracks falling within the mth n e s Multifractal Analysis of Charged Particle Distribution 5 such interval of the eth event, and n e s is the total number of shower tracks in the eth event (i.e., n e s = M m=1 ne m ). For finite ne s thesingleeventg q moments are subjected to large statistical fluctuations, that can be minimized through event averaging. To get rid of the saturation problem mentioned in the introduction section, Hwa and Pan 25 introduced a step function (Θ) in the definition of G q, which acts like a filter for the empty bin effect that particularly affects the low multiplicity events, G q = 1 N ev N ev e=1 m=1 M [ n e m n e s ] q Θ(n e m q), (2) where Θ is the step function. According to the theory of fractals, if the single particle density distribution possesses multifractal structure then G q should exhibit a scaling relation like G q (δx η ) τ(q) :asδx η 0, (3) where τ(q) is called the mass exponent or the fractal exponent. The phase space interval size can be reduced to a considerable smallness which is sufficient to extract significant results. We have calculated the G q moments as a function of phase space partition number M over a wide range of q values for all three data sets used in this analysis. Figure 1 shows the corresponding results where ln G q have been plotted against ln M for (a) the experiment, (b) the UrQMD, and (c) the random numbers. The diagrams show that the phase space dependence of G q is more or less similar for all three event samples, e.g., (i) ln G q increases for q<0 and decreases for (a) Experiment (b) UrQMD (c) Random Fig. 1 ln G q as a function of ln M for 28 Si-Ag/Br interaction at 14.5GeV/n.(a) Experiment, (b) UrQMD, and (c) Random numbers. In all diagrams, line joining points are drawn to guide the eye.

6 6 P. Mali, A. Mukhopadhyay & G. Singh q>1, (ii) ln G q tends to saturate in the large M region, and (iii) the saturation effect is more prominent for the higher positive q( 4) values. The saturation effects as mentioned before, are simply an outcome of the finiteness of n s. Equation (3) suggests a linear dependence of ln G q on ln δx η, that can be used to determine τ(q), ln G q τ(q) = lim. δx η 0 lnδx η For each q the mass exponent τ(q) is extracted from the best linear fit of the ln G q versus ln M data. While obtaining these straight lines we did not take into account the points falling in the saturation region. From the knowledge of τ(q) onecannow establish a connection between intermittency and multifractality, can evaluate the fractal dimensions, and can also construct the most important multifractal spectral function f(α q ): f(α q )=qα q τ(q), (4) where the Lipschitz-Holder exponent α q is given by α q = τ(q)/ q. In Fig. 2a we have shown a plot of the mass exponent τ(q) against q both for the experiment and for the UrQMD. Corresponding α q values are also graphically presented in the same diagram. It can be seen that (i) there exists a small but definite nonlinearity in the variation of τ q, and (ii) the experimental and the simulated results are not significantly different from each other. However, the decreasing trend of α q with increasing q, assuggested by the phenomenology of multifractality, 8 is well reproduced. The multifractal spectral functions f(α q ) are plotted in Fig. 2b. Both the experimental and the simulated spectra are found to be very stable, smooth and concave downwards having peaks at α 0. The straight line f(α q )=α q, tangentially touches both the spectra around α q 1.0. The maxima of f(α q ) is very close to unity, indicating that the empty bin effect, specially within the region of analysis, has properly been taken care of. According to the theory of fractals, the width of the spectral function indicates the degree of multifractality. Awidef(α q ) spectrum for both the data sample is, therefore, an indication of the multifractal nature of the respective density function. However, the experimental spectrum is wider than the UrQMD. The spectral functions obtained in our analysis exhibit similar features that are observed in the pp collision of the UA1, 26 or in the AB collision of the EMU01 11 experiment. It is to be noted that the UA1 results were compared with the GENCL and the PYTHIA prediction, whereas a stochastic model is used to compare the EMU01 results. As mentioned above, the G q moment calculated from Eq. (2) contains a statistical component, which is simply due to the finiteness of n s. While the dynamical component in the particle density fluctuation can be automatically filtered in the (a) (b) Fig. 2 (a) Event averaged mass exponents τ(q) and Lipschitz-Holder exponents α q versus q plots. Data points represent the experimental values and the lines represent the corresponding UrQMD predictions. Here the statistical errors cannot be seen as they are of the same size as the data points. (b) Multifractal spectral functions for experiment (solid circles) and the corresponding UrQMD prediction (open circles). The straight line represents f(α q)=α q.

7 intermittency technique, 17,18 the same is not true for the multifractality. Therefore, it would be a useful exercise to estimate, at least qualitatively, the nontrivial nonstatistical component of the G q moments, and to see whether it can match the same obtained from our intermittency analysis. 17 To find out the statistical contribution, we have calculated the G q moments over an uncorrelated event sample (call it G sta q ) generated by (pseudo)random numbers as discussed in Sec. 3. Variation of ln G sta q with ln M is shown in Fig. 1c. For each q corresponding mass exponents τ sta (q) are also calculated from the best fitted straight lines to the data points. It has been shown in Ref. 25 that the dynamical component of τ(q), denoted by τ dyn (q) is related to the statistical component τ sta (q) by the following relation τ dyn (q) =τ(q) τ sta (q)+q 1. (5) In deriving the above relation, it has been assumed that G dyn q obeys the same scaling law as G q [Eq. (3)]. It then follows from Eq. (5) thatfora trivial dynamics, τ dyn (q) should be equal to (q 1). Therefore, any deviation in τ dyn (q) from (q 1) may be considered as an outcome of the nontrivial dynamics. A phenomenological relation between the intermittency exponent φ q and the dynamical component of mass exponent τ dyn (q), as developed in Hwa and Pan, 25 is given by τ dyn (q) q +1 φ q. (6) The effect of eliminating the statistical contribution to the G q moments can be readily seen from Multifractal Analysis of Charged Particle Distribution 7 Fig. 3, where the q 1 τ(q), the q 1 τ dyn (q), along with the φ q values taken from Mali et al. 17 are plotted against q. The experiment and the UrQMD results are shown separately. Both in the experiment and in the simulation, the q 1 τ(q) values are far above the corresponding φ q values, but as soon as the statistical contribution is subtracted, the respective q 1 τ dyn (q) values come down very close to the intermittency index. It is to be noted that very little evidence of dynamical fluctuation has been observed in the intermittency analysis (φ q 0) of the UrQMD data. 17 Consistent with this, we find that for the UrQMD, q 1 τ dyn (q) 0. In spite of the fact that the model produces only statistical fluctuations, the UrQMD shows multifractality. As is seen from Fig. 1c, the multifractal nature is also observed for the random number generated G q moments. Hence, characterization of the dynamical fluctuation (if there is any) can not be done simply by looking at the scaling behavior of the G q moments. It has to be done at the level of scaling exponents (parameters). A fractal system can also be characterized by a parameter called the Lévy stable index (µ) within the range of stability 0 µ 2. 27,28 The upper limit µ = 2 corresponds to minimum fluctuation for a self-similar branching process, whereas the lower limit µ = 0 corresponds to maximum fluctuation, i.e. monofractals, which may be a signal of a second order phase transition. In Hegyi, 29 it has been argued though that the µ value can even be outside its stability range. For a multiplicative cascade mechanism of particle production like the (a) Experiment (b) UrQMD Fig. 3 Variation of φ q,(q 1 τ dyn (q)) and (q 1 τ(q)) against q for 28 Si-Ag/Br interaction at 14.5GeV/n: (a) experimental and (b) UrQMD. In both diagrams, lines joining points are drawn to guide the eye.

8 8 P. Mali, A. Mukhopadhyay & G. Singh α-model, 2,3 where the final state particle density is given by a product of random numbers, the density function can be approximated by a log-lévy distribution. Under this approximation the Lévy index µ is read from the relation 28,30 D D q = 1 q µ q D D 2 q 1 2 µ 2, (7) where D is the topological dimension of the supporting space, which is 1 in the present case. D q are the generalized dimensions of integer order q defined in terms of the intermittency exponents φ q as, D q = D φ q q 1. (8) Following Eq. (6) one can also set D q τ dyn (q) q 1. (9) We shall discuss more about D q in the next subsection (4.2). In practice only a few D q (q>2) values are obtained from the intermittency and/or multifractal analysis. Therefore, the µ value extracted by fitting Eq. (7) toasmallsetofdata points is likely to be unreliable. Hence we follow an alternative method of calculating the µ value from the multifractal spectral function f(α q ), as describedinochs. 31 AccordingtoOchs 31 f(α q )is related to µ by the following relation 1 f(α q ) (B α q ) µ/(µ 1), for α q <B, (10) with B =1+(1 D 2 )/(2 µ 2). Note that for any q positive or negative, f(α q )is a smooth and continuous function of α q. Therefore, the Lévy index µ can be extracted from the slope C(= µ/µ 1) of the ln(1 f(α q )) versus ln(b α q ) straight line in the α q < B region. As it follows from Eq. (10) the only criterion that has to be satisfied here is f(b) = 1. Figure 4 shows the results of such calculations, where (a) the experimental, and (b) the UrQMD simulated plots are shown side by side. In either of these plots, it is clear that a single straight line cannot reproduce the data well, and hence the µ value depends on the region of fitting the straight lines. In the positive and low q-value region (straight line fits are shown by the dotted lines), we obtained µ = ± for the experiment. For the UrQMD C 1whichleads to a very large (diverging) µ value. On the other hand, µ = ± for the experiment and ± for the UrQMD in the positive and high q region (straight line fits are shown by the solid lines). All µ values are beyond the range of stability. From the 1d intermittency analysis 17 we obtained µ = 3.15 ± 0.03 for experiment. All these observations therefore, show that the (dynamical) density fluctuation is not compatible to a log-lévy stable distribution Takagi s Moments In order to study the fractal structure of multiparticle density distributions, Takagi 9 proposed a new set of multifractal moments (T q ) based on the following two assumptions: (i) the density distribution (a) Experiment (b) UrQMD Fig. 4 ln[1 f(α q)] versus ln(b α q)plotsfor 28 Si-Ag/Br interaction at 14.5GeV/n: (a) experiment, and (b) UrQMD. In both diagrams the dotted (solid) lines represent straight line fits to data points in the low (high) q region.

9 ρ is uniform all over the phase space interval, and (ii) the multiplicity distribution P n does not depend on the location of the phase space interval. Both these criteria are found to be valid in the present case, as the cumulative variable X η has been used as the basic phase space variable. Takagi s moments are defined for positive integer order as N ev M T q (δx η )=ln (p ij ) q, (11) i=1 j=1 where p ij = n ij /K, K being the total number of tracks in all N ev events, and n ij is the number of tracks in the jth bin of the ith event. Unlike the G q moments the T q moments so defined [Eq. (11)] are not affected by the finiteness of the event multiplicity n s. According to Takagi, 9 T q (δx η ) should be a linear function of ln δx η, i.e., T q (δx η )=A q + B q ln(δx η ), (12) where A q and B q are constants independent of δx η. When a linear relation like Eq. (12) is observed over a considerable range of δx η, once again the generalized dimension D q can be obtained in terms of the fit parameter B q as, D q = B q, for q 2. (13) q 1 For a sufficiently large N ev one can set N ev i=1 j=1 M (p ij ) q = nq K q 1 n, (14) where n represents the average bin multiplicity. From Eqs. (11) (14) and by replacing δx η with n the following relation is derived: ln n q = A q + {(q 1)D q +1} ln n. (15) The generalized dimensions D q can be calculated from the above relation for q 2. For q = 1 it is known as the information dimension, and is obtained by taking an appropriate limit of Eq. (13). 32 This is equivalent to considering an entropy like function defined as N ev M S(δX η )= p ij ln p ij, (16) i=1 j=1 and to looking for a linear dependence of S(δX η ) such as S(δX η )= D 1 ln(δx η )+constant. (17) Multifractal Analysis of Charged Particle Distribution 9 Using Eq. (14) one can easily obtain an expression for D 1 as n ln n / n = C 1 + D 1 ln n. (18) Following the prescription of Takagi, 9 we have calculated n ln n and ln n q (for q =1 7) with increasing phase space width δx η in a symmetric interval about the centroid of the η-distribution (η 0 =1.90 ± 0.01). The results are shown graphically in Fig. 5 for all three data sets employed in this analysis. The information dimension (D 1 ), as well as the generalized dimensions (D q )areevaluated from the slopes of best fitted straight lines. It turns out that D 1 =0.933 ± for the experiment, D 1 = ± for the UrQMD, and D 1 =0.934 ± for the random number generated event sample. The generalized dimensions D q obtained from Takagi s moments [Eq. (15)], and those from Hwa s moments [Eq. (9)] are plotted against q in Fig. 6. On the same figure we have incorporated the D q values obtained from the intermittency analysis [see Eq. (8)]. 17 We observe a general decreasing trend of D q with increasing q. In the case of experiment, the D q values obtained from the F q moments fall at a much faster rate than those obtained from the G q moments, whereas for the UrQMD the D q values calculated either form the F q moments or from the G q moments, within the errors are very close to unity (the dimension of the supporting space). Their rate of fall against q is also very slow. On the other hand, the D q values obtained from the T q moments for both the data sets are significantly and consistently lower than those obtained from the F q and the G q moments. Such a large and systematic deviation might reflect the fact that the Takagi s moments are not corrected with respect to the statistical noise, the technique of which is still unknown to us. Moreover, the moments F q, G q and T q are defined in different ways, which might be another source of such inconsistencies. Note that for a simple Poissonian type of multiplicity distribution within a given interval δx η, the generalized dimensions would all be equal to unity. Any deviation from unity, as it is observed in our data, should therefore be considered as a signature of nonstatistical elements in the particle production dynamics. The thermodynamic interpretation of fractals, in terms of the multifractal specific heat, has been given by Bershadski. 33 If the monofractal to the multifractal transition is governed by a Bernoulli

10 10 P. Mali, A. Mukhopadhyay & G. Singh (a) Experiment (b) UrQMD (c) Random Fractals Downloaded from (d) Experiment (e) UrQMD (f) Random Fig. 5 Multifractal Takagi s moments for 28 Si-Ag/Br interaction at 14.5 GeV/n. In all diagrams the lines represents best linear fits to data points, and the errors are statistical only. type of fluctuation only, then the multifractal specific heat C can be derived from the relation, 33 D q = D + C ln q q 1. (19) A monofractal to multifractal transition corresponds to a jump from C = 0 to a nonzero finite value. In Fig. 7 we have plotted D q against ln q/(q 1) both for (a) the experiment and (b) for the UrQMD data, as well as for both the techniques (a) Experiment (b) UrQMD Fig. 6 Generalized dimensions D q calculated from F q moments (solid circled), G q moments (open circles) and T q moments (solid squares) against q for 28 Si-Ag/Br interaction at 14.5GeV/n: (a) experiment and (b) UrQMD. In both diagram lines joining points are shown to guide the eye.

11 Multifractal Analysis of Charged Particle Distribution 11 (a) Experiment (b) UrQMD Fractals Downloaded from Fig. 7 Generalized dimensions D q calculated from G q moments (solid circles) and T q moments (open circles) against ln q/ (q 1) for 28 Si-Ag/Br interaction at 14.5GeV/n: (a) experiment and (b) UrQMD. In both diagrams the lines represents best linear fits to data points. of analysis employed here. An almost linear dependence of D q on ln q/(q 1) is observed in the experimental data as well as in the UrQMD, indicating the relevance of Bershadski s interpretation of multifractality 33 in the context of a phase transition in the present case. The C values are extracted from the best linear fits to the data points. For experiment the values are: C =0.049 ± from the G q analysis and C =0.115 ± from T q analysis, whereas, the corresponding UrQMD values are, respectively, C 0 and C = ± The effect of eliminating the statistical noise using a random number generated event sample is manifested once again in the C value obtained from Hwa s method of analysis, while in Takagi s method no such distinction is possible between the experiment and the UrQMD. The present set of C values are not consistent with the universality as claimed in Bershadski SUMMARY We have presented a set of results on multifractal analysis of the shower track distribution in the pseudorapidity (η) spacefor 28 Si-Ag/Br interaction at an incident energy of 14.5GeV per nucleon. Systematic comparison between the experiment and the UrQMD simulation has been made. The data behave expectedly and the results are consistent with those obtained from similar other experiments on nucleus-nucleus interaction. Observations of the present analysis can be summarized in the following way. The multifractal moments introduced by Hwa follow a power law and scale with the diminishing phase-space resolution size almost identically for the experiment, for the UrQMD simulation, and even for the random number generated events. The intermittency results on the same sets of data, however, behave differently for the experiment and the simulation. Whereas the self-similar nature of the 1d intermittency of the density fluctuation observed previously 17 is the primary motivation of the present work, we observe that the difference between experiment and simulation lies not in the scaling pattern of the multifractal moments, but in the quantitative aspects of the scaling parameters and in the derivatives thereof. The multifractal mass exponent or the Lipschitz- Holder exponent can not discriminate between the experiment and the UrQMD either. The parameters themselves are probably not sensitive enough to the nature of fluctuation (noise or dynamical) present in the data. However, when the statistical contribution is properly taken care of, we observe that within experimental uncertainties, the intermittency exponents overlap with the corresponding multifractal parameter. A small but definite departure from the simulation can be traced into the experiment. The multifractal spectrum, consistent in all aspects with

12 12 P. Mali, A. Mukhopadhyay & G. Singh its expected behavior, has a slightly smaller width in the UrQMD generated curve than that in the experiment. Therefore, the multifractal spectrum is considered as a sensitive observable that can distinguish the dynamical contribution from the statistical noise in the density fluctuation. The stability index (µ) associated with the log- Lévy distribution is an important parameter that needs to be mentioned separately. Our intermittency analysis of the same data on 28 Si-Ag/Br interaction resulted in a µ value that is way beyond its stability limit. We have adopted a method based on the multifractal spectral function, and have obtained different µ values in different q-region. Our results from Hwa s multifractal analysis on this issue are consistent with the observation of intermittency analysis. Once again the µ values are beyond its allowed limit. However, in the high q-region its value within errors is very close to the upper limit µ = 2. Hence in the present case, a log-lévy distribution can not appropriately describe the multiplicity fluctuation. Takagi s multifractal moments also exhibit expected power law type scaling behaviour. Though Takagi s technique has a few advantages over Hwa s technique of analysis, the T q moments are contaminated with statistical noise. This limitation is reflected in the generalized fractal dimensions derived from Takagi s moments. The experimental D q values obtained from T q moments are not significantly different from the corresponding UrQMD simulated values. This is not true either for the intermittency (F q ) or for Hwa s method (G q ), where the statistical noise has been taken care of. The variation of D q with q is consistent with the thermodynamic interpretation of monofractal to multifractal transition. However, the corresponding specific heat significantly deviates from any kind of universality as claimed in Bershadski. 33 REFERENCES 1. R. C. Hwa, Factorial moments of continuous order, Phys. Rev. D 51 (1995) A. Bialas and R. Peschanski, Moments of rapidity distributions as a measure of short-range fluctuations in high-energy collisions, Nucl. Phys. B 273 (1986) A. Bialas and R. Peschanski, Intermittency in multiparticle production at high energy, Nucl. Phys. B 308 (1988) W. Kittel and E. A. De Wolf, Soft Multihadron Dynamics (World Scientific, Singapore, 2005). 5. G. Paladin and A. Vulpiani, Anomalous scaling laws in multifractal objects, Phys. Rep. 156 (1987) P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Physica D 13 (1984) T. C. Halsey et al., Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A 33 (1986) ; T. C. Halsey and M. H. Jensen, Spectra of scaling indices for fractal measures: theory and experiment, Physica D 23 (1986) R. C. Hwa, Fractal measure in multiparticle production, Phys. Rev. D 41 (1990) F. Takagi, Multifractal structure of multiplicity distribution in particle collisions at high energies, Phys. Rev. Lett. 72 (1994) N. Parashar, Multifractal structure in two dimensions in proton-nucleus interactions at high energy, J. Phys. G 22 (1996) M. I. Adamovich et al. (EMU01 Collaboration), Multifractal analysis of particles produced in 197 Au, 32 Sand 16 O induced interactions at high energies, Europhys. Lett. 44 (1998) D. Ghosh et al., Multifractal behavior of nuclear fragments in high-energy leptonic interactions, Phys. Rev. C 70 (2004) M. K. Ghosh, A. Mukhopadhyay and G. Singh, Multifractal moments of particles produced in 32 S- Ag/Br interaction at 200 A GeV/c, J. Phys. G 32 (2006) S. Ahmad and M. A. Ahmed, A comparative study of multifractal moments in relativistic heavy-ion collisions, J. Phys. G 32 (2006) ; Some observations related to intermittency and multifractality in 28 Si and 12 C-nucleus collisions at 4.5 A GeV, Nucl. Phys. A 780 (2006) M. K. Ghosh et al., Centrality dependence of nonstatistical fluctuation in single particle density distribution in 32 S-Ag/Br interaction at 200 GeV/c, Int. J. Mod. Phys. E 19 (2010) S. Bhattacharyya et al., Multifractality in charged pion production at a few GeV/n, Physica A 390 (2011) P. Mali, A. Mukhopadhyay and G. Singh, Intermittency and erraticity of charged particles produced in 28 Si-Ag/Br interaction at 14.5A GeV, Can. J. Phys. 89 (2011) P. Mali, A. Mukhopadhyay and G. Singh, Selfaffine two dimensional intermittency in 28 S-Ag/Br interaction at 14.5A GeV, Acta Phy. Pol. B 43 (2012)

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