Some observations related to intermittency and multifractality in 28Si and 12C-nucleus collisions at

$Page 1: Some observations related to intermittency and multifractality in 28Si and 12C-nucleus collisions at$
Nuclear Physics A 780 (2006) 206–221

Some observations related to intermittency andmultifractality in 28Si and 12C-nucleus collisions

at 4.5A GeV

Shafiq Ahmad ∗, M. Ayaz Ahmad

Physics Department, Aligarh Muslim University, Aligarh 202002, India

Received 16 June 2006; received in revised form 1 September 2006; accepted 19 September 2006

Available online 19 October 2006

Abstract

An attempt is made to study the existence of dynamical fluctuations of relativistic particles using themethods of modified multifractal moments, Gq , and scaled factorial moments, Fq , in terms of new scaledvariable X(η) suggested by Bialas and Gazdzicki. For this purpose analyses of experimental and UrQMDdata involving interactions of 28Si and 12C nuclei at 4.5A GeV/c with nuclear emulsion are used. Thevariation of ln〈Gq 〉 and ln〈Fq 〉 with lnM in pseudorapidity (η) phase space reveals power law behaviour.The values of slopes, τq and φq determined from the analyses of Gq and Fq moments are discussed. Thegeneralized fractal dimensions, Dq , determined from the above methods are found to decrease with theorder of the moments, q, indicating multifractality in multiparticle production. It is also observed that thespectral function f (αq) for heavier projectile is much broader than for lighter beam due to larger numberof participating nucleons present in heavier projectile.© 2006 Elsevier B.V. All rights reserved.

PACS: 25.70.Pq; 13.85.Hd

Keywords: NUCLEAR REACTIONS H, C, N, O, Ag, Br(12C, X), (28Si, X), E at 4.5 GeV per nucleon; measuredfragments multiplicity, pseudorapidity, density distributions; deduced multifractal moments, scaled factorialmoments, spectral functions.

* Corresponding author.E-mail address: [email protected] (S. Ahmad).

0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2006.09.009

S. Ahmad, M.A. Ahmad / Nuclear Physics A 780 (2006) 206–221 207

1. Introduction

Bialas and Peschanski [1] initiated the study of fractal phenomena in high energy particle andnuclear physics with the motivation of searching for a second order phase transition. The exis-tence of large fluctuations in rapidity space was first observed in the cosmic ray experiment bythe JACEE Collaboration [2] and later on in accelerator experiments [3,4]. It has been suggestedthat a quark–gluon phase transition could give rise to large fluctuations. Bialas and Peschanskiintroduced the method of scaled factorial moments (SFMs), which is considered to be the mostsignificant one for extracting the dynamical contribution to the fluctuation in multiplicity dis-tribution in high-energy collisions. They suggested a power law scaling behaviour of the SFMson phase space interval size down to the detector resolution and described the phenomenon as“intermittency”, a term coined from hydrodynamic turbulence [5]. The scaled factorial momentmethod can not only predict the existence of large non-statistical fluctuations but it could alsoinvestigate the pattern of fluctuations and their origin.

Initially, there was a strong speculation that the origin of intermittent type of non-statisticalfluctuation was thought to be the result of the phase transitions from QGP to normal hadronicmatter in relativistic nucleus–nucleus collisions. But no conclusive evidence for the formationof quark–gluon plasma has been found in nucleus–nucleus collisions at ultra-relativistic energies[6]. Therefore the interpretation of intermittency can no longer be related with QGP formation.Only future experiments would clarify the situation. Most of the experimental results on theSFMs support the intermittent behaviour. But the various results presented are not enough foran unambiguous interpretation of the effect. Bialas realized, that Bose–Einstein correlations andintermittency might be deeply connected [7]. Various suggestions such as the jet and minijetformation [8], conventional short-range correlations [9], Bose–Einstein (BE) interference [10]and self-similar random cascade mechanism [11] were considered responsible for the origin ofintermittency. These correlations are frequently referred to as Hanbury–Brown–Twiss (HBT)correlations [12] in the literature of heavy ion physics. However, if intermittency is caused byHBT phenomenon, there is no problem of hadronization and decay of resonances.

There is a strong feeling that the BE interference can play a role in dynamical fluctuations.This correlation arises due to the symmetric wave functions of identical bosons in BE statistics.Increase in the value of the factorial moments, Fq , with decreasing phase-space size could beexplained on the basis of the above correlations between identical charged particles (HBT effect).The phenomenon of intermittency would be stronger for equal charged particles than for allcharged particles. Analysis of some experimental results [13–15] shows that BE effect cannotbe considered as the main source of intermittency, especially in e+e− annihilation [16], lepton–hadron [17] and hadron–hadron collisions [18] at relativistic energies. Most of the experimentalresults on the SFMs observed by NA35 and WA80 Collaborations [19] have been explainedthrough the Bose–Einstein correlations. However, EMU01 data exclude the possibility of BEcorrelations as a dominant source of intermittency in heavy ion collisions [20].

It has been observed that the intermittent behaviour is clearly explained due to short-rangecorrelations [20,21] for lower order of moments. The intermittency has also been observed in avariety of collision processes and therefore it may be considered as a general property of multi-particle production. However, no single mechanism could explain the observed intermittency invarious collision processes. So a detailed study is required to understand the intermittency morerigorously.

The self-similarity observed in the power law dependence of scaled factorial moments revealsa connection between intermittency and fractality. The importance of the multiplicity fluctua-

208 S. Ahmad, M.A. Ahmad / Nuclear Physics A 780 (2006) 206–221

tions in terms of multifractal behaviour has been carried out to focus the production process ofthe high energy relativistic hadrons. In multifractal approach, it has been suggested that the nu-clear interactions can be treated as geometrical objects with non-integer dimensions. The study ofchaotic system [22–24] and intermittent behaviour in turbulent fluids has been performed usingthe fractal dimension. Various methods have been proposed to investigate the fractal structures.Hwa [25] was the first to provide the idea of the multifractal moments Gq to study the multifrac-tality and self-similarity in multiparticle production. If the particle production process exhibitsself-similar behaviour, then a modified form of Gq moment in terms of step function [26] showsthe remarkable power law dependence on phase space bin size.

In the present work, an attempt has been made to investigate the non-statistical fluctuationin pseudorapidity phase space using the methods of modified Gq moments and Fq moments interms of new scaled variable X(η) for 28Si–AgBr and 12C–AgBr interactions at 4.5A GeV/c.The values of Gq and Fq moments calculated for the experimental data as well as for datagenerated using the ultra-relativistic quantum molecular dynamics (UrQMD) model [27], arecompared. Some interesting results have also been discussed on generalized fractal dimensions,Dq , and the multifractal spectral function, f (αq). To see the presence of statistical fluctuations,uncorrelated Monte Carlo generated events have also been compared with the experimental andUrQMD data. In the present investigation, the interactions with Ns � 8 are only considered, asthe events with low multiplicity (Ns < 8) would have large statistical fluctuations. The numberof interactions with Ns � 8 for silicon and carbon beams with emulsion nuclei are 481 and 332,respectively. Out of these events only 323 and 258 events, respectively, were finally selected asgenuine AgBr interactions with Nh � 8. Only AgBr events were considered to maximize thecontribution of dynamical fluctuations [26].

2. Experimental details

In the present work, two stacks of BR-2 emulsion exposed to 4.5A GeV/c silicon and carbonbeams at Synchrophasotron of Joint Institute of Nuclear Research (JINR), Dubna, Russia, havebeen utilized. The method of line scanning has been adopted to scan the stacks, which was carriedout using Japan made NIKON (LABOPHOT and Tc-BIOPHOT) high resolution microscopeswith 8 cm movable stage using 40X objectives and 10X eyepieces. The interactions due to beamtracks with an angles < 2◦ to the mean direction and lying in emulsion at depths > 35 µm fromeither surface of the pellicles were included in the final statistics. The sensitivity of nuclearemulsion used for singly charged particles was about 30 grains per 100 µm. Some other relevantdetails about the present experiment are reported elsewhere [28]. All charged secondary particlesin the interactions were classified in accordance with the emulsion terminology on the basis oftheir ionization in nuclear emulsion [29].

2.1. Grouping of targets

The black and grey tracks together in an event are known as heavily ionizing tracks (Nh =Nb + Ng). The exact identification of target in emulsion experiment is not possible since themedium is composed of H, C, N, O, Ag and Br nuclei. The events produced due to the collisionswith different targets in nuclear emulsion are usually classified into three main categories onthe basis of the multiplicity of heavily ionizing tracks in it. Thus, the events with Nh � 1, 2 �Nh � 7 and Nh � 8 are classified as collisions with hydrogen (H,AT = 1), group of light nuclei(CNO, 〈AT 〉 = 14) and group of heavy nuclei (AgBr, 〈AT 〉 = 94) respectively. However, the


grouping of events only on the basis of Nh values does not lead to the right percentage of eventsof interactions due to light and heavy group of nuclei. In fact, a considerable fraction of stars withNh � 7 are produced in the interactions with heavy group of nuclei. Therefore we have used thefollowing criteria.

AgBr events: (i) Nh > 7, or(ii) Nh � 7 and at least one track with range, R � 10 µm

and no track with 10 < R < 50 µmCNO events: (i) 2 � Nh � 7 and no tracks with R � 10 µmH events: (i) Nh = 0, or

Nh = 1 and no track with R � 50 µm.

In the present investigations, the pseudorapidity (η) of relativistic shower particles has beenfound using the relation, η = − ln tan(θ/2), where θ is the space angle of the shower particleswith respect to the beam direction in the laboratory system. At very high energies, η is foundto be a good approximation to rapidity, Y , because experimentally it is not always possible tomeasure energy and momentum of a particle and hence different distributions in rapidity spaceare generally studied in terms of pseudorapidity variable, η, instead of the rapidity variable, Y .Moreover, pseudorapidity, η, is conventionally used to obtain the phase space distribution of theparticles. The fluctuations are studied in the range of pseudorapidity, �η = −0.5 to 5.5, leavingout the fragmentation tails where the statistics are low.

3. Results and discussion

It should be pointed out that the single charged particle pseudorapidity distribution is non-uniform. This non-uniformity of the distribution influences the scaling behaviour of factorialand multifractal moments. Bialas and Gazdzicki [30] proposed a new scaled variable X(η)

which drastically reduces the distortion produced in the study of multiplicity fluctuations dueto non-uniformity of single charged particle distribution. We have used this new-scaled variableX related to the single particle density distribution ρ(η) as:

X(η) =∫ η

η1ρ(η′) d(η′)∫ η2

η1ρ(η′) d(η′)

, (1)

where ρ(η) = (1/N)dn/dη is the single particle pseudorapidity density of the shower particlesand η1 and η2 are the minimum and maximum values of the pseudorapidity distribution for agiven value of the pseudorapidity η falling in the interval �η = η2 − η1 of an individual showertracks in an events. The variable X(η) corresponding to single particle density distribution isuniformly distributed from 0 to 1 in X-space. Fig. 1 shows the single particle density distributionin X space for 28Si and 12C projectiles. Such uniformity in density is an implicit assumption inthe derivation of the power law scaling behaviour of the SFMs and phase space interval size (seeEq. (13)). In terms of cumulative variable X(η), the single particle density distribution is alwaysuniform and both horizontal averaging and vertical averaging should produce the same result.

3.1. Multifractality in terms of modified G-moments

In order to minimize the contribution of the statistical fluctuations, a modified multifractalmoment [26] has been used. In this method N single charged shower particles in X(η) intervals


Fig. 1. Single particle density distributions of scaled variable X(η) for 28Si and 12C projectiles.

�X = Xmax − Xmin are distributed into M non-empty bins of width δX = �X/M . The powerdependence of a modified Gq moment of order q on phase space partition number M is expressedas

〈Gq〉 =Nev∑

1

M∑j=1

(nj /N)qθ(nj − q) ∝ M−τq , (2)

where q is a positive integer, nj denotes the number of charged particles in j th bin and N =∑Mj=1 nj is the total number of particles in the interval X = 0–1 in an event. The step function,

θ(nj − q) is equal to unity if nj � q , otherwise it is zero. Nev stands for the total number ofevents in a given ensemble and τq is the fractal index and can be found from a linear dependenceof ln〈Gq〉 on lnM over all windows.

In order to examine the dependence of ln〈Gq〉 on lnM , the values of the multifractal moments〈Gq〉 for q = 2–6 have been calculated using Eq. (2) for our data at 4.5A GeV/c. The variationsof ln〈Gq〉 with lnM of the emitted relativistic shower particles in X-phase space are shown inFig. 2(a) and (b) for 28Si–AgBr and 12C–AgBr collisions with Ns � 8. It is clear from the fig-ure that ln〈Gq〉 now exhibits a linear dependence on lnM for q > 1 without saturation. Thislinear behaviour is found to satisfy the power law dependence as described in Eq. (2). The ob-served linear dependence of the multifractal moments gives an evidence of self-similarity for themechanism of particle production and an initial indication of fractal structure in multiparticle pro-duction. A similar trend of power law dependence of ln〈Gq〉 on lnM has also been obtained froman inspection of results in μp, pp and e+e− data [31,32]. In order to compare the experimentalresults with the prediction of standard generators of particle production in nucleus–nucleus colli-sions, we have generated 5630 28Si–AgBr and 12C–AgBr collisions events at 4.5A GeV/c usingstring hadronic transport model, UrQMD. The UrQMD results shown by dotted lines are alsopresented in Fig. 2, which also exhibit a linear dependence on lnM similar to the experimentaldata. A comparison of real and UrQMD results from the figures shows that the experimental val-ues of ln〈Gq〉 are somewhat smaller in comparison to those obtained for UrQMD data for q > 4.Furthermore, the results from the figures reflect that there is an agreement between the experi-mental data and the corresponding results for events generated using the string hadron model,


Fig. 2. Variation of ln〈Gq 〉 as a function of lnM in the interactions of (a) 28Si–AgBr and (b) 12C–AgBr collisionsat 4.5A GeV. Open stars represent the experimental results, dotted lines the UrQMD events and closed circles theuncorrelated MC events in (a) and (b). Solid lines represent the best linear fits of the experimental points.

Table 1Values of φq , τq , �τq and (q − 1 − τ

dynq ) for our data. The errors are statistical

Energy4.5A GeV/c

q φq τexpq τ stat

q τdynq q − 1 − τ

dynq �τq = τ stat

q − τq

28Si–AgBr 2 0.12 ± 0.03 0.70 ± 0.01 0.79 ± 0.04 0.91 ± 0.11 0.09 ± 0.01 0.09 ± 0.013 0.24 ± 0.05 1.35 ± 0.01 1.67 ± 0.01 1.68 ± 0.16 0.32 ± 0.01 0.32 ± 0.024 0.58 ± 0.05 1.92 ± 0.02 2.63 ± 0.02 2.29 ± 0.03 0.75 ± 0.03 0.72 ± 0.035 0.95 ± 0.06 2.41 ± 0.04 3.59 ± 0.03 2.83 ± 0.05 1.17 ± 0.05 1.17 ± 0.056 1.17 ± 0.23 2.84 ± 0.06 4.37 ± 0.05 3.48 ± 0.06 1.50 ± 0.07 1.53 ± 0.07

12C–AgBr 2 0.15 ± 0.03 0.62 ± 0.01 0.79 ± 0.01 0.83 ± 0.02 0.17 ± 0.02 0.17 ± 0.023 0.37 ± 0.05 1.21 ± 0.02 1.68 ± 0.03 1.53 ± 0.05 0.47 ± 0.03 0.47 ± 0.044 0.72 ± 0.04 1.74 ± 0.03 2.59 ± 0.03 2.15 ± 0.03 0.85 ± 0.04 0.85 ± 0.045 1.09 ± 0.02 2.23 ± 0.04 3.50 ± 0.04 2.73 ± 0.05 1.27 ± 0.05 1.27 ± 0.066 1.51 ± 0.03 2.68 ± 0.06 4.39 ± 0.05 3.30 ± 0.06 1.71 ± 0.06 1.70 ± 0.07

UrQMD. The experimental values of τq obtained using the least-square fitting of the experimen-tal points in the above figures are shown in Table 1. The values of τq for each data set are plottedas a function of the order of moments, q , in Fig. 3 for different projectiles and energies [33,34].From the figure, one can readily observed that the values of τq are nearly independent of collisionenergy and mass of the projectiles.

In order to check the presence of the statistical fluctuations to 〈Fq〉 and 〈Gq〉, uncorrelatedMonte Carlo events, (MC-RAND) were generated randomly in η-space based on the assumptionof independent emission of particles according to the following criteria [35]:

(i) N such particles in each event are distributed randomly in the give �η-interval;(ii) the multiplicity distribution of generated events should be similar to those of the experimen-

tal data and


Fig. 3. Variation of τq versus q at different energies.

(iii) the single particle η-distribution of generated event in η-space reproduces Gaussian shapewith its mean value, 〈η〉, and dispersion, σ , comparable to the corresponding experimentalvalues obtained for the entire experimental sample.

The variations of ln〈Gstatq 〉 with lnM for the uncorrelated MC events (conventionally also

called mixed events) are also included in Fig. 2(a) and (b) by full circles with the correspondingexperimental and UrQMD data in X-phase space. The slopes of the plots, τ stat

q , thus obtained arealso listed in Table 1 along with the experimental data. A clear difference of the pseudorapidityfluctuations is observed for statistically generated events. The pattern of variation of ln〈Gq〉 onlnM for mixed events is more or less similar to that of the experimental data, but the magnitudesof Gq moments are always significantly less in comparison to the experimental values. Thisfeature can be attributed to the existence of the dynamical fluctuations. There is a good agreementbetween our data and the results reported by other workers [33,34,36]. The variation of 〈Gstat

q 〉on M like 〈Gq〉 also exhibits a power law dependence on M as⟨

Gststq

⟩ ∝ M−τ statq . (3)

The dynamical contributions to the Gq moment suggested by Chiu et al. [37] is expressed as

〈Gq〉dyn = [〈Gq〉/⟨Gstatq

⟩]M1−q . (4)

If 〈Gq〉dyn also exhibits a power law dependence on M as

〈Gq〉dyn ∝ M−τdynq , (5)

then the dynamical contribution to τq can be extracted from the experimental τq and the τ statq

using the following formula [26,34,37]

τdynq = τq − τ stat

q + q − 1. (6)

Expression (4) reduces to Gdynq = M1−q provided 〈Gq〉 is purely statistical, which implies from

Eq. (6) that τdynq = q −1 for trivial dynamics. The dynamical values of τ

dynq obtained by inserting

the values of τq and τ statq in Eq. (6) for q = 2–6 are also listed in Table 1. Therefore, it is evident


from this table that the values of τdynq deviate significantly from (q − 1). This indicates the

presence of the dynamical fluctuations in our data.

3.2. Relationship between Fq and Gq moments

There is an increasing evidence for self-similar properties of multiparticle production at highenergy. This behaviour is explained by the power law dependence of normalized factorial mo-ments Fq , whereas Gq moments are described as a means of studying the multifractal propertyof self-similar process. It has been proposed [26] that to establish a relation between the fractalbehaviour of Fq and Gq , the normalized factorial moments Fq and Gq moments have been ex-plained in terms of basic functions Bq,k(M), which is used to characterize the fractal behaviour.We have also used this method to find a connection between Fq and Gq .

It is well known that the Fq and Gq moments are defined for integer values of q � 1 and realvalues of q respectively. In order to establish a relation between them, q values for Gq momentsare also restricted to integer values of q � 1. In order to suppress the statistical fluctuations, theGq moments can be defined in terms of the basic functions Bq,k(M) as

⟨Gq(M)

⟩ =∞∑

k=0

Bq,k(M)(q + k)q . (7)

The functions Bq,k(M) express the basic fractal structure of the data, which is given as

Bq,k(M) = ⟨N−qQq+k(M,N)

⟩, (8)

where Qq+k(M,N) are defined as the number of bins containing (q + k) particles in an event ofmultiplicity N in the pseudorapidity range �η, k = 0,1,2, . . . and 〈〉 indicates an average overall events.

The expansion of Eq. (7) can be written as

⟨Gq(M)

⟩ = Bq,0(M)qq

[1 + Bq,1(M)

Bq,0(M)

(q + 1

q

)q

+ · · ·], (9)

where only first few terms in above expression make a significant contribution at large M andthe values of τq are found by the sum of those terms. The power law dependence of 〈Gq(M)〉are given as⟨

Gq(M)⟩ ∝ M−τq for large M. (10)

To establish a link between the scaled factorial moments [1] and the self similarity discussedabove on the basis of the multifractals, the usual definition of the scaled factorial moments arerepresented by the following expression:

〈Fq〉 = 1

M

M∑j=1

〈nj (nj − 1) · · · (nj − q + 1)〉(〈N〉/M)q

, (11)

which reduces to

〈Fq〉 = Mq−1M∑ 〈nj (nj − 1) · · · (nj − q + 1)〉

〈N〉q . (12)
j=1


The above form of Fq is in close agreement to the original form proposed by Bialas andPeschanski [1]. Thus one can write the following dependence:

〈Fq〉 ∝ Mφq for large M. (13)

In the above equation, φq are the intermittency exponent. The expression of scaled factorialmoments Fq in terms of Bq,k(M) can also be written as

〈Fq〉 = Mq−1∞∑

k=0

Bq,k(M)(q + k)!

k! . (14)

There is clear similarity between Eqs. (7) and (14) for 〈Gq〉 and 〈Fq〉. The values of 〈Fq〉 and〈Gq〉 show a minor difference in terms of Bq,k(M). The first few terms of Eq. (14) are given as

〈Fq〉 = Mq−1Bq,k(M)q![

1 + Bq,1(M)

Bq,0(M)(q + 1) + · · ·

]. (15)

It can be concluded from Eqs. (9) and (15) that M-dependence of 〈Gq〉 and 〈Fq〉 are not identical.The experimental data should show a power-law behaviour for large M provided the evidenceof fractal structure exists in the data. However, the following expression between φq and τq arerelated by considering only the leading terms as

φq ≈ q − 1 − τq . (16)

In terms of modified Gq moments, the above relation reduces to

φq ≈ q − 1 − τdynq . (17)

3.3. Intermittency in pseudorapidity phase space

It has been shown [1] that the values of Fq for purely statistical fluctuation saturate withdecreasing phase space size, whereas in dynamical fluctuation, Fq moments are supposed toincrease with decreasing phase space size and exhibit a power law behaviour of normalized fac-torial moments, Fq . In order to apply above expression for our data, the variation of ln〈Fq〉 asa function of lnM is presented in Fig. 4(a) and (b) for 28Si–AgBr and 12C–AgBr interactionsrespectively in X-phase space for each order of the moments. It has already been shown [38] thatthe horizontally and vertically averaged moments give essentially the same results. It is interest-ing to note that these results from the figures once again suggest that a clear dependence of ln〈Fq〉on lnM shows a power law dependence, which confirms the presence of intermittent behaviour.The nature of linear dependence of ln〈Fq〉 on lnM is reflected in the values of intermittencyexponents, φq , listed in Table 1. The values of φq and τq are determined by using the method ofleast squares. The error bars in Figs. 2 and 4 are estimated by considering them as independentstatistical errors only and the solid lines drawn indicate the least squares fit to the respectiveexperimental data points. Though the effect of point-to-point correlations of the statistical errorsfor different bin sizes has not been taken into consideration in the present study, it is expectedthat the exclusion of the correlation of the statistical errors will not change the main result appre-ciably [13,39]. In order to get a quantitative comparison of ln〈Fq〉 versus lnM with the mixedMC results and UrQMD events, we have also plotted these results in Fig. 4 with the correspond-ing experimental results. It is observed from the figure that UrQMD data also exhibit a lineardependence on lnM similar to the experimental data. It may be seen from the figure that the ex-perimental values of ln〈Fq〉 are somewhat larger than the events simulated using UrQMD model.


Fig. 4. Variation of ln〈Fq 〉 as a function of lnM in the interactions of (a) 28Si–AgBr and (b) 12C–AgBr collisionsat 4.5A GeV. Open stars represent the experimental results, dotted lines the UrQMD events and closed circles theuncorrelated MC events in (a) and (b). Solid lines represent the best linear fits of the experimental points.

Further, it is shown that the experimental data on intermittency exhibit a remarkable closeness toanalogous data obtained from the UrQMD model. However, the mixed generated events exhibitno such dependence on M . This gives an indication for the absence of statistical contribution inthe experimental data. The flat behaviour in mixed events is expected for independent emissionof particles. Thus, the experimental results are comparable to the results obtained in nucleus–nucleus collisions at different energies [34,40]. Akesson et al. [41] also analyzed their data in16O–Em and 32S–Em interactions to investigate the intermittent behaviour observed in heavy-ion induced collisions. Similar analysis was also carried out on MC generated events taking intoaccount the Dalitz decay and gamma conversion by the same collaborator [41]. EMU01 Collabo-ration [42] also carried out comparison between the data and MC events for 32S–Au interactionsat 200A GeV and found an identical behaviour.

Hwa and Pan [26] have also given a similar relation between intermittency index and fractalindex, which are approximately related as

φq ≈ q − 1 − τdynq = τ stat

q − τq . (18)

The above equation is not an exact relation because Fq and Gq moments are different andapproaches each other only in the limiting case of infinite N . The deviation of φq from zero givesa measure of non-statistical fluctuations of dynamical origin, which is equivalent to the deviationof τq from q −1. Fig. 5(a) and (b) gives a comparison of φq , �τq (= τ stat

q −τq) and (q −1−τdynq )

as a function of q obtained from Table 1 for our data in η-phase space. It is evident from the figurethat the values of φq are not exactly equal to �τq and (q − 1 − τ

dynq ) for 28Si and 12C projectiles,

respectively. They show a similar trend for all q except at q = 6, while these values are closeto one another only up to q � 3, and beyond q > 3 they are different. This difference can beattributed to the difference in the definitions of Fq and Gq moments, since the former is closelyrelated to the correlation function while the latter gives a measure of the fractal structures. Thecontributions of the statistical fluctuations in Fq moments are automatically suppressed, whereasin Gq moments, actually they are not eliminated completely. Further, it can be concluded that


Fig. 5. Comparison of φq , �τq and (q − 1 − τdynq ) with the order of moments, q , in the interactions of (a) 28Si–AgBr

and (b) 12C–AgBr collisions at 4.5A GeV.

the deviation of τq from the statistical τ statq is a measure of the real dynamics involved. This is in

agreement with the observation reported by other workers [3,30,43].

3.4. Generalized fractal dimensions

The power law behaviour of 〈Fq〉 and 〈Gq〉 on M reveals self-similarity and in general,it indicates the existence of fractal properties which are called multifractals. According to thefractal theory, self-similar system are characterized by infinite spectrum of non-integer general-ized dimensions, Dq . Therefore, the generalized dimensions, Dq , that characterize multiparticleproduction process can be determined from Gq moments and Fq moments analyses using thefollowing relations:

Ddynq = τ

dynq /(q − 1) (19)

and

Dq = 1 − φq

(q − 1). (20)

We now compare the values of Dq obtained from the intermittency and multifractal for-malisms. The values of, Dq , are plotted in Fig. 6 as a function of q for our data in η-space.It is obvious from the figures that the decreasing trend of the generalized fractal dimensions,Dq , with increasing order of moment clearly indicates the presence of multifractality using themethods of intermittency and multifractality. The observed behaviour of Dq with q is in agree-ment with the multifractal cascade model [44]. However, the generalized dimensions determinedby the intermittency analysis are consistently larger and do not match with those obtained fromGq moments. This difference may be due to the different definitions of two moments, whosedifferences become more prominent when N is low. The analysis of Fq moments may eliminatethe statistical fluctuations completely, whereas Gq moments are unable to eliminate them com-pletely and the values of Dq obtained from Gq moments might be influenced due to the statistical


Fig. 6. Dependence of the generalized dimension, Dq on q for (a) 28Si–AgBr and (b) 12C–AgBr collisions using Gq

and Fq moments methods at 4.5A GeV.

fluctuations. Therefore, the analysis in terms of Fq , may reveal a self-similarity characteristic inmultiparticle production. Sengupta et al. [45] have reported no systematic trend in their data ei-ther with increasing energy or projectile size. Some other workers [33–35] have also observedsimilar behaviour in their data. On the basis of above discussion, it may be observed that thestudy of dynamical fluctuations using Fq moment method is preferred over Gq moment method.

3.5. The spectral function f (αq)

The knowledge of the spectral function f (αq) can also give an idea about the presence ofmultifractality. The theory of multifractals [46] can be employed to find a relation for f (αq) func-tions from the known values of τq . The relation between f (αq) and τq is obtained by Legendretransformation as follows:

f (αq) = qαq − τq, (21)

with

αq = dτq/dq and df (αq)/dαq = q. (22)

The above relations are interpreted in terms of thermodynamical equations, which shows thatthe f (αq), αq and q play the role of the entropy, the energy density and inverse of tempera-ture [46].

The variation of the spectrum f (αq) as a function of the Lipschitz–Holder exponent, αq , isshown in Fig. 7 for our data in η-space. This figure also includes the results of 28Si, 32S and197Au at 14.5, 200 and 10.6A GeV/c respectively [40] for comparison. The values of f (αq)

and αq are determined using relations (21) and (22) for Si–AgBr and C–AgBr data. The resultsof the simulated events obtained using the string hadronic model, UrQMD are also shown inFig. 7 for the sake of comparison. The spectrum f (αq) in each case is reproduced with almostidentical shape like the experimental one. It is evident from the figure that the multifractal spectra


Fig. 7. Plots of spectrum f (αq) as a function of αq at different energies. Open square and open circle represent theUrQMD result for 28Si–AgBr and 12C–AgBr data. Solid line represents a 45◦ line, tangent to f (αq) at αq = α0.

Table 2Values of D0, D1 and D2 obtained from f (αq) for our data

Projectiles Fractal dimensionD0 = f (α0)

Information dimensionD1 = f (α1)

Correlation dimensionD2 = 2α2 − f (α2)

28Si 0.853 ± 0.005 0.795 ± 0.004 0.735 ± 0.00412C 0.804 ± 0.003 0.716 ± 0.004 0.639 ± 0.003

f (αq) is represented by continuous curves. The figure shows a distinct peak at αq = α0 and thesolid line in each figure represents a common tangent at an angle of 45◦ at α1 = f (α1). Thisis the consequence of a general property of the multifractals [47]. An upward convex shapeof the spectral function f (αq) in Fig. 7 gives an evidence for self-similar cascade mechanism[48,49], which might be responsible for producing relativistic particles in nuclear collisions. Thisbehaviour is in excellent agreement with the prediction of gluon model [47]. The left-hand sides(q > 0) of the spectra f (αq) are sensitive to peaks and the right-hand sides (q < 0) describe thevalleys of the single particle η-distribution [47]. The spectrum f (αq) indicates the existence offluctuations of the experimental pseudorapidity distribution and depends on the projectile mass,energy and the impact parameter. It is also inferred from the figure that the spectrum f (αq) forheavier beam is much broader than that for lighter beam. This characteristic is observed dueto larger number of participating nucleons present in heavier projectile. One of the most basiccharacteristic of the fractal theory is its conventional dimensions for q = 0, 1 and 2, which couldbe used to find the fractal dimension, D0 = f (α0), the information dimension, D1 = f (α1) andcorrelation dimension, D2 = 2α2 −f (α2). The values of three dimensions are recorded in Table 2for our data. From the table it is seen that the values of D0, D1 and D2 are found to be somewhathigher for 28Si–AgBr collisions than for 12C–AgBr collisions. These dimensions are assumedto be sensitive for the production mechanism of multiparticle production process [50]. Fromthe discussion of the spectral function f (αq), it can be said that no phase transition is taking


Fig. 8. Plots of rescaled spectrum f (αq ) as a function of αq at different energies.

place. Jain et al. have also reported similar result in heavy ion collisions at BNL and CERN SPSenergies [40]. Very similar results were also found in p–Em interaction at 800 GeV [51].

For the purpose of understanding a connection between the spectra for different projectilesand energy, a rescaled spectrum f (αq) is given by the following relations:

f (αq) = f (αq)/f (α1) = f (αq)/α1, (23)

αq = αq/α1. (24)

The rescaled spectrum f (αq) as a function of αq is shown in Fig. 8 for our data as well asother data [40]. It is realized from the figure that all the curves coincide for αq < 1 (q > 0). Thedistribution indicates a kind of universality of the multifractal structure in this region.

4. Conclusions

On the basis of the results presented, it should be emphasized that the existence of dynami-cal fluctuations of the relativistic particles in the pseudorapidity phase space using the factorialmoments and the multifractal moments reveals power law behaviour. The power dependenceobtained in two methods gives an evidence of self-similar structure in multiparticle production.The decrease of the generalized fractal dimensions, Dq , with q clearly indicates the presenceof multifractality using the intermittency and multifractality approaches. These results, there-fore, suggest that the existence of the fractal structure observed in 28Si–AgBr and 12C–AgBrcollisions at 4.5A GeV/c are consistent with the predictions of other results. The existence ofdynamical fluctuations is expected to be an intrinsic characteristic prevailing in the dynamics ofmultiparticle production in relativistic high energy nuclear collisions. The multifractal spectra ofthe generalized dimensions, f (αq), of the produced shower particles from our data represents asmooth and continuous function with universal features that characterizes the fluctuations of theexperimental pseudorapidity distribution. It is also observed that the spectrum f (αq) for heavierprojectiles at the same incident energy is broader than for light projectiles due to a large numberof participating nucleons in heavier beam.


Finally, we suggest a similar analysis may be carried out in heavy-ion collisions at higher en-ergies, e.g., at RHIC to provide a better understanding of multiplicity fluctuation in multiparticleproduction.

Acknowledgements

We would like to express our thanks to Prof. K.D. Tolostov, JINR, Dubna, Russia for providingthe exposed and developed nuclear emulsion plates. We also gratefully acknowledge Dr. ShakilAhmad for his help in generating events using UrQMD model.

References

[1] A. Bialas, R. Peschanski, Nucl. Phys. B 273 (1986) 703;A. Bialas, R. Peschanski, Nucl. Phys. B 308 (1988) 857.

[2] JACEE Collaboration, T.H. Burnett, et al., Phys. Rev. Lett. 50 (1983) 2062.[3] UA5 Collaboration, G.J. Alner, et al., Phys. Lett. B 138 (1984) 304.[4] NA22 Collaboration, M. Adamus, et al., Phys. Lett. B 185 (1987) 2002;

E. Stenlund, et al., Nucl. Phys. A 498 (1989) 541.[5] B.L. Ha, Chaos, World Scientific, Singapore, 1984.[6] BRAHMS Collaboration, D. Rohrich, et al., J. Phys. G: Nucl. Part. Phys. 31 (2005) S659.[7] A. Bialas, Acta Phys. Pol. B 23 (1992) 561.[8] W. Och, J. Wosiek, Phys. Lett. B 214 (1988) 617.[9] P. Carruthers, I. Sarsevic, Phys. Rev. Lett. 63 (1989) 1562.

[10] P. Carruthers, et al., Phys. Lett. B 222 (1989) 487.[11] A. Bialas, R.C. Hwa, Phys. Lett. B 253 (1991) 436.[12] R. Hanbury Brown, R.Q. Twiss, Philos. Mag. 45 (1976) 663;

R. Hanbury Brown, R.Q. Twiss, Nature 178 (1956) 1046.[13] NA22 Collaboration, I.V. Ajinenko, et al., Phys. Lett. B 222 (1989) 306.[14] TASSO Collaboration, W. Braunscheweig, et al., Phys. Lett. B 231 (1989) 548.[15] I. Derado, G. Jancso, N. Schmitz, P. Stopa, Z. Phys. C 54 (1992) 357.[16] B. Buschbeck, R. Lipa, R. Peschanski, Phys. Lett. B 215 (1988) 788.[17] W. Shaoshun, Z. Jie, Y. Yunxiu, X. Chingua, Z. Yu, Phys. Rev. D 49 (1994) 5787;

UA1 Collaboration, C. Albajar, et al., Nucl. Phys. B 345 (1990) 1.[18] NA22 Collaboration, I.V. Ajinenko, et al., Phys. Lett. B 222 (1989) 306.[19] NA35 Collaboration, J. Bachler, et al., Z. Phys. C 61 (1994) 551;

WA80 Collaboration, R. Albrecht, et al., Phys. Rev. 50 (1994) 1048.[20] A. Capella, K. Fialkowski, A. Krzywicki, Phys. Lett. B 230 (1989) 149;

P. Carruthers, I. Sarcevie, Phys. Rev. Lett. 63 (1999) 1562;R.K. Shivpuri, V.K. Verma, Phys. Rev. D 47 (1993) 123.

[21] P. Carruthers, H.C. Eggers, I. Sarsevic, Phys. Lett. B 254 (1991) 258;P. Carruthers, Int. J. Mod. Phys. A 4 (1989) 5587.

[22] G. Paladin, A. Valpiani, Phys. Rep. 156 (1987) 147.[23] H. Hiramoto, M. Kohmoto, Int. J. Mod. Phys. B 6 (1992) 281.[24] M. Janssen, Int. J. Mod. Phys. B 8 (1994) 943.[25] R.C. Hwa, Phys. Rev. D 41 (1990) 1456.[26] R.C. Hwa, J. Pan, Phys. Rev. D 45 (1992) 1476.[27] S.A. Bass, et al., Prog. Part. Nucl. Phys. 42 (1998) 255;

M. Bleicher, et al., J. Phys. G: Nucl. Part. Phys. 25 (1999) 1859.[28] S. Ahmad, M.A. Ahmad, M. Irfan, M. Zafar, J. Phys. Soc. Jpn. 75 (2006) 64604.[29] C.F. Powell, P.H. Fowler, D.H. Perkins, The study of Elementary Particles by Photographic Methods, Pergamon,

Oxford, p. 450.[30] A. Bailas, M. Gazdzicki, Phys. Lett. B 252 (1990) 483.[31] I. Derado, R.C. Hwa, G. Jansco, N. Schnitz, Phys. Lett. B 283 (1992) 151.[32] UA1 Collaboration, C. Albajar, et al., Z. Phys. C 56 (1992) 37.


[33] R. Hasan, M. Mohib-ul Haq, S. Islam, Int. J. Mod. Phys. 13 (2004) 479;R. Hasan, S. Islam, M. Mohib-ul Haq, Fractals 25 (2005) 1029.

[34] D. Ghosh, et al., Phys. Rev. C 58 (1998) 3553;D. Ghosh, et al., Fractals 11 (2003) 331;D. Ghosh, et al., Nucl. Phys. A 707 (2002) 213.

[35] D. Ghosh, et al., Phys. Rev. D 47 (1993) 1235.[36] P.L. Jain, A. Mukhopadhyay, Phys. Rev. C 47 (1993) 342.[37] C.B. Chiu, K. Fialkowaski, R.C. Hwa, Mod. Phys. Lett. A 5 (1990) 2651.[38] S. Ahmad, M.A. Ahmad, J. Phys. G: Nucl. Part. Phys. 32 (2006) 1279.[39] N.M. Agababyyan, et al., Phys. Lett. B 382 (1996) 305;

D. Ghosh, et al., Int. J. Phys. A 78 (2004) 353.[40] P.L. Jain, G. Singh, A. Mukhopadhyay, Nucl. Phys. A 561 (1993) 651;

P.L. Jain, K. Sen Gupta, G. Singh, Phys. Lett. B 241 (1990) 273;P.L. Jain, G. Singh, Nucl. Phys. A 596 (1996) 700.

[41] HELIOS Collaboration, T. Akesson, et al., Phys. Lett. B 252 (1990) 303.[42] EMU01 Collaboration, M.I. Adamovich, et al., Z. Phys. C 49 (1991) 395.[43] EMU 01 Collaboration, M.A. Adamovich, et al., Europhys. 44 (1998) 571.[44] C. Meneveau, K.R. Sreenivasan, Phys. Rev. Lett. 59 (1987) 1474.[45] K. Sengupta, P.L. Jain, G. Singh, S.N. Kim, Phys. Lett. B 236 (1990) 219;

M.Kr. Ghosh, A. Mukhopadhyay, Phys. Rev. C 68 (2003) 034907;B. Bhattacharjee, Nucl. Phys. A 748 (2005) 641.

[46] L.D. Landau, E.M. Lifshitz, Phys. Sz. Sowjet 6 (1934) 244;L.D. Landau, E.M. Lifshitz, Statistical Physics, Part I, Pergamon, Oxford, 1980.

[47] C.B. Chiu, R.C. Hwa, Phys. Rev. D 43 (1991) 100.[48] A. Bialas, R. Peschanski, Phys. Rev. Lett. B 207 (1991) 59.[49] W. Ochs, J. Wosiek, Phys. Rev. Lett. 214 (1988) 617.[50] R.C. Hwa, World Scientific, Singapore, 1991.[51] N. Parashar, J. Phys. G: Nucl. Part. Phys. 22 (1996) 59.

Documents

Some observations related to intermittency and multifractality in 28Si and 12C-nucleus collisions at