Intermittency and erraticity of charged particlesproduced in 28Si-Ag/Br interaction at 14.5 A GeV

Provash Mali, Amitabha Mukhopadhyay, and Gurmukh Singh

Abstract: In this paper, we present the intermittency and the erraticity analyses of the distributions of charged particles pro-duced in 28Si-Ag/Br interaction at incident energy 14.5 A GeV. The experimental results are compared with a Monte Carlosimulation using ultra-relativistic quantum molecular dynamics (UrQMD) model. The experimental data show the presenceof a nonstatistical component in the produced charged-particle density. Neither the UrQMD simulation nor the purely statis-tical simulation was found to match the experimental data. The present set of results are compared to those obtained in simi-lar measurements from earlier high-energy nucleus–nucleus experiments.

PACS Nos: 25.75.–q, 25.70.Mn, 24.60.Ky

Résumé : Nous soulignons ici le caractère intermittent et erratique dans l’analyse de la distribution des particules chargéesdans les interactions 28Si-Ag/Br à l’énergie incidente de 14,5 A GeV. Nous comparons les résultats expérimentaux avec unesimulation Monte Carlo utilisant le modèle ultra-relativiste de dynamique moléculaire quantique (UrQMD). Les données ex-périmentales indiquent la présence d’une composante non statistique dans la densité de particules chargées produites. Ni lasimulation UrQMD ni la simulation purement statistique n’arrivent à reproduire les données expérimentales. Nous compa-rons nos résultats avec ceux obtenus dans des mesures similaires lors d’expériences antérieures de collisions noyau–noyau àhaute énergie.

[Traduit par la Rédaction]

1. Introduction

When two nuclei moving with a relative speed close tothat of light collide with each other, a central fireball com-prising very high energy and (or) baryon density can be pro-duced. Depending purely on the initial conditions, thisfireball may achieve the thermal and (or) chemical equili-brium that is necessary for a phase transition from normalnuclear matter to the ever-elusive state of quark–gluonplasma (QGP) to take place [1]. At a later stage, the fireballexpands and cools down, fragmenting into the final-state par-ticles (mostly pions). In many cases, rapidly fluctuating par-ticle densities that are devoid of any apparent regular patternare observed in the final state [2]. The correlations, if thereare any, are often masked under statistical noise and (or) theeffects of trivial kinematic constraints. Using appropriate stat-istical techniques, it is possible to disentangle the dynamicalcontribution to the particle density and to characterize thesame in terms of a finite set of well-behaved parameters.Comparing experimental results with model calculations, itis also possible to extract substantive information regardingthe final freeze out stage of the space–time evolution ofhigh-energy nucleus–nucleus (AB) collisions. Several specu-lative measures, conventional as well as exotic, have beenproposed as probable candidates for the dynamics of corre-lated emission of final-state particles [3]. Each speculationhas its own domain of successes as well as limitations. To

fully understand a phenomenon as complicated as multipar-ticle production, it is first necessary to analyze a wide rangeof experimental data that involve different colliding objectsand varying collision energies. It is further required thatsuch experimental results are compared with model predic-tions so that the validity of any particular mechanism of par-ticle production, or a combination of more than one suchmechanism embedded within the framework of the model,can be subjected to verification. It may be noted that, withthe compressed baryonic matter (CBM) experiment [4] pro-posed to be held within the next 10 years or so, studyinghigh-energy AB interactions from a few to several tens ofgigaelectronvolts have now become more relevant than everbefore.As mentioned earlier, we experimentally observed a com-

binatorial effect of the statistical noise and the contributioncoming from one or the other dynamical process to the par-ticle density. For varying multiplicities the scaled factorialmoment (SFM) of order q, denoted Fq, is capable of filteringout Poisson-type statistical noise. The SFM is the ordinarymoment of the nonstatistical component of the underlyingdistribution, irrespective of its analytical form. A power lawtype of increase in Fq with diminishing phase-space intervalsize (dX) like

Fq � ðdXÞ�fq : dX ! 0 ð1Þis known as intermittency. Had the fluctuations been purely

Received 20 April 2011. Accepted 3 August 2011. Published at www.nrcresearchpress.com/cjp on 7 September 2011.

P. Mali and A. Mukhopadhyay. Department of Physics, University of North Bengal, Darjeeling 734013, West Bengal, India.G. Singh. Department of Computer and Information Science, SUNY at Fredonia, New York, NY 14063, USA.

Corresponding author: Amitabha Mukhopadhyay (e-mail: amitabha_62@rediffmail.com).

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Can. J. Phys. 89: 949–960 (2011) doi:10.1139/P11-078 Published by NRC Research Press

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of statistical origin, the Fq values would have remained inde-pendent of the scale at which they were measured. First intro-duced for a few very high multiplicity cosmic-ray events [5],the intermittency phenomenon has since been investigated formany different types of high-energy interactions [6]. In one-dimensional (1D) analyses, the positive valued exponent fq,known as the intermittency index, is a scale-invariant quan-tity down to experimental resolution, indicating thereby theself-similar nature of the dynamical component of the densityfunction. This kind of self-similarity motivates one to under-stand the intermittency phenomenon in the framework offractal theory, where the density function can be character-ized by a set of (multi)fractal parameters. The first quantita-tive measure of fractality in the particle density in softprocesses was introduced in ref. 7, and it was later developedin ref. 8.The strength of the dynamical fluctuation may vary from

one event to another. All events do not start with the sameinitial condition, and different fluctuation strengths are boundto result in different spatial patterns in their final states.Moreover, an experimental sample usually contains eventswith varying multiplicity, and, consequently, the SFMs willnot have the same value in every event. In intermittency, Fqis averaged over the event sample as well as over many phasespace partitions. This averaged quantity, ⟨Fq⟩, is testedagainst a power law type of scaling behavior. One wouldtherefore loose information regarding the event-space fluctua-tion of Fq, and only a weak dX dependence of ⟨Fq⟩ is ex-pected. To quantify the chaotic nature of the event structure,a new technique called the erraticity was introduced [9]. Thismethod allows one to study the event-to-event fluctuation ofspatial patterns, which the intermittency method cannot. Inerraticity one needs to first define a single event SFM, Fe

q,then study its event-space fluctuation in terms of suitable er-raticity moments, and finally verify the scaling properties thatthese moments may follow. Like intermittency, the erraticityphenomenon has also been tested for several high-energyAB interactions [10–12] (see [6] for more studies).In this paper, we present 1D intermittency and erraticity

analyses of the shower tracks (caused mostly by chargedpions) coming out of 28Si-Ag/Br events at an incident energyof 14.5 GeV per nucleon. Our principal objective is to char-acterize the dynamical part of the apparently random fluctua-tions in the shower track density distribution in terms ofcertain regularly behaving statistical variables; in particular,in terms of a certain set of parameters that are connected tosome kind of scale-invariant self-similar process. It is alsoour objective to compare the experimental results of intermit-tency and erraticity with the ultra-relativistic quantum molec-ular dynamics (UrQMD) prediction [13]. We have alsoperformed the erraticity analysis on a purely statistical samplegenerated by random numbers. On several occasions, resultsof the present analysis are also compared with the results ofother similar experiments on AB interactions [12, 14–17].The paper is organized in the following way. In Sect. 2 the

experimental aspects of the present investigation are dis-cussed. In Sect. 3 the Monte Carlo (MC) methods adoptedfor data analysis and for comparison purposes are summarilydescribed. In Sect. 4 we have discussed the intermittencytechnique and our results on this issue. Similarly, in Sect. 5both the erraticity method of data analysis and our results on

erraticity are elaborated upon. In Sect. 6 we conclude withsome critical observations about the results of the present in-vestigation.

2. ExperimentStacks of Ilford G5 nuclear photo-emulsion pellicles of

size 16 cm × 10 cm × 600 mm, were horizontally irradiatedwith a 28Si beam with an incident flux 3 × 103 ions/cm2 andwith an incident energy 14.5 A GeV from the alternating gra-dient synchrotron (AGS) at the Brookhaven National Labora-tory (BNL) [18]. The nucleon–nucleon (NN) center of massenergy,

ffiffiffiffiffiffiffisNN

p= 5.382 GeV in our case. If the AB collision

is considered a superposition of many incoherent NN colli-sions, then for a central collision where all 28 nucleons ofthe 28Si nucleus participate in the interaction, this amountsto a total center of mass energy

ffiffis

p≈ 151 GeV. On the other

hand, if the AB interaction is considered a coherent collisionbetween an incoming 28Si nucleus and a stationary Ag/Br nu-cleus (for which the weighted average mass number in emul-sion, A ≈ 94), then the center of mass energy comes out tobe

ffiffis

p≈ 275 GeV. It would perhaps be prudent to assume

that the actual situation lies somewhere in between these twoextremes. To observe the primary 28Si-emulsion events, Leitzmicroscopes with a total magnification of 300× were utilized,and the emulsion plates were scanned along individual pro-jectile tracks. Angle measurement and counting of trackswere performed under a total magnification of 1500× withthe help of Koristka microscopes. According to emulsion ter-minology, tracks emitted from an interaction (called an eventor a star) are classified into four categories, namely, showertracks, grey tracks, black tracks, and projectile fragments.Shower tracks are caused by produced charged particles

(mostly pions) moving at relativistic speeds (b > 0.7). Theionization of a shower track I ≤ 1.4I0, where I0 ≈ 20 grains/100 mm, is the minimum ionization due to any track ob-served within a G5 plate. The total number of such tracks inan event is denoted ns.The black and grey tracks predominantly originate from

slowly moving protons and other heavier fragments belong-ing to the target. They fall within a velocity range from 0.7cto <0.3c for protons within a kinetic energy range from 400to <30 MeV, and have ionization I > 1.4I0. The total numberof such heavy fragments (i.e., black and grey tracks taken to-gether) in an event is denoted nh and an event with nh > 8indicates that the interaction is with an Ag/Br nucleus.Projectile fragments are the spectator parts of the incident

projectile nucleus that do not directly participate in an inter-action. They are emitted within a very narrow, extremely for-ward cone whose semivertex angle (qf = 0.21/Pinc) is decidedby the Fermi momenta of the nucleons present in the nu-cleus. Here, Pinc is the incident projectile momentum per nu-cleon in GeV/c. Having almost the same energy andmomentum per nucleon as the incident projectile, these frag-ments exhibit uniform ionization over a long range and suffernegligible scattering. Their number in an event is denoted npf.In an emulsion experiment, the pseudorapidity (h) together

with the azimuthal angle (4) of a track constitutes a conven-ient pair of basic variables in terms of which of the particleemission data can be analyzed. The pseudorapidity, h, is anapproximation of the dimensionless boost parameter rapidity

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of a particle, and it is related to the emission angle (q) of thecorresponding shower track as, h = –ln tan (q/2) An accuracyof dh = 0.1 unit and d4 = 1 mrad could be achieved throughthe reference primary method of angle measurement. Follow-ing the criteria mentioned above, a sample of 331 28Si-Ag/Brevents was considered for further analysis. The averageshower track multiplicity for this sample is ⟨ns⟩ = 52.67 ±1.33, and the present analysis is confined only to the showertracks. To avoid contamination between the singly chargedproduced particles and the spectator projectile protons,shower tracks falling within the Fermi cone were excludedfrom our analysis. The experimental h-distribution of showertracks can be approximated by a Gaussian function with acentroid h0 = 1.90, a width sh = 2.17, and a peak densityr0 = 17.88. The experimental 4-distribution on the otherhand is approximately uniform within its accessible range (0,2p). In spite of their many limitations, nuclear emulsion ex-periments are superior to many other big budget online ex-periments in one respect — they offer very good spatial andangular resolutions between two tracks. When distributionsof particles within narrow phase space regions are to be in-vestigated, this is certainly an important advantage. Thereare some excellent texts where further details of any emulsionexperiment can be found [19].

3. The simulationHigh-energy AB collisions have been phenomenologically

investigated using the UrQMD approach [13]. Basically, theUrQMD model is a microscopic transport theory of covariantpropagation of all hadrons along their classical trajectories,combined with stochastic binary scattering, resonance decay,and color string formation. In the mathematical framework ofthis scheme, a relativistic Boltzmann equation has to besolved for the hadrons in the final stage of the collision. Thebasic input to such hadronic transport models is that ahadron–hadron interaction would occur if the impact parame-ter b <

ffiffiffiffiffiffiffiffiffiffiffistot=p

p, where total cross section, stot, depends on

the isospin of interacting hadrons, their flavors, and the cen-ter of mass–energy involved. Partial cross sections are alsoused to compute the relative weights of different channels.The Fermi gas model is utilized to describe the projectileand the target nuclei, the initial momentum of each nucleonbeing randomly distributed between zero and the localThomas–Fermi momentum. Each nucleon is described by aGaussian density distribution, and the wave function for eachnucleus is taken as a product of single-nucleon Gaussianfunctions without invoking the Slater determinant necessaryfor antisymmetrization. In configuration space, the centroidsof the Gaussian are randomly distributed within a sphere,and finite widths of these Gaussians result in a diffused sur-face region. At low and intermediate energies (

ffiffis

p< 5 GeV),

the phenomenology of hadronic interactions is described interms of interactions between known hadrons and their reso-nances. At higher energies (

ffiffis

p> 5 GeV), the excitation of

color strings and their subsequent fragmentation into hadronsdominate the multiple production of particles. For AB colli-sions, the soft binary and ternary interactions between nucle-ons are described by a nonrelativistic density-dependentSkyrme potential. In addition, Yukawa, Coulomb, and Pauli(optional) potentials are also implemented in the model.

These potentials allow us to calculate the equation of state ofthe interacting many-body system, as long as it is dominatedby nucleons. Note that the potential interactions are onlyused for baryons and (or) nucleons with relative momentaDp < 2 GeV/c. The model allows for subsequent rescatter-ing(s). The collision term in the UrQMD model includesmore than 50 baryon species and five meson nonets (45 mes-ons). The framework bridges, with one concise model, theentire available range of energies from the Bevalac region(

ffiffiffiffiffiffiffisNN

p∼ a few GeV) to the relativistic heavy ion collider

(RHIC) (ffiffiffiffiffiffiffisNN

p= 200 GeV).

Using the UrQMD code, we have simulated a sample of28Si-Ag/Br interactions at 14.5 A GeV that is five times aslarge as the experimental sample. Events with Ag and Br nu-clei are first generated separately, and they are thereaftermixed with each other according to their proportional abun-dance in nuclear emulsion. All newly produced charged mes-ons in the simulated events have been retained for subsequentanalyses. The simulated event samples possess identical mul-tiplicity distributions, and similar h- and 4-distributions tothe experimental ones. The average charged meson multiplic-ity ⟨nch⟩ of the UrQMD-generated sample is the same as theexperimental ⟨ns⟩. The centroid of the best Gaussian fitted h-distribution (h0 = 1.75), and the width of the h-distribution(sh = 2.15) are also close to the respective experimental val-ues. For error calculation, we have generated event samplesbased on random numbers where a pair of random numbershas been associated with each track. There are 10 such gen-erated event samples in the present case, each is of the samesize as the experimental sample, and they all possess identi-cal multiplicity distributions and the same h-distribution (ac-tually the best Gaussian fitted distribution) and 4-distribution(uniform) as the experimental sample. A linear congruentialiterative method and a derivative method have been used togenerate the random numbers [20]. While generating the ran-dom numbers, no correlation has been assumed, and hencethese data sets correspond to independent emission of par-ticles. While the UrQMD-generated intermittency results arecompared with the experiment to check the statistical contri-bution in erraticity, both UrQMD and random number gener-ated results are used for comparison with the experiment.

4. The intermittencyThe factorial moment of order q for multiplicity n within a

particular phase space interval is defined as

fq ¼ nðn� 1Þ . . . ðn� qþ 1Þwhere fq is actually equal to the qth order ordinary momentof the dynamical component of the underlying multiplicitydistribution, irrespective of its exact analytic form. When theentire phase space of width DX is divided into M equal inter-vals of width dX = DX/M, and the number of particles fallingwithin the mth interval of the eth event is denoted by nðeÞm , theSFM for an event normalized with respect to the average par-ticle number in an interval nðeÞ = ⟨ns⟩/M, is defined as

Feq ¼

1

M

XMm¼1

nðeÞm

�nðeÞm � 1

�. . .

�nðeÞm � qþ 1

��nðeÞ

�q ð2Þ

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Feq can now be averaged over an entire sample of events,

hFqi ¼ 1

Nev

XNev

e¼1

Feq ð3Þ

to result in the ultimately required statistical variable ⟨Fq⟩ forintermittency. One can now vary dX and verify the power lawtype of scaling as prescribed in (1). It may be noted that thereare other ways of averaging and normalizing fq [6, 15]. How-ever, these differences are only quantitative in nature andmarginal in amount. In terms of the cumulant variables [21],different ways of averaging do not produce significantly dif-ferent results. The intermittency phenomenon, and issues re-lated to it, can be investigated by choosing any particularway of defining ⟨Fq⟩. We have replaced our phase space vari-ables h (4) by the respective cumulant variable Xh (X4). Asany cumulant variable is always uniformly distributed within[0,1], we can therefore ensure that the intermittency resultsare not dependent on the shape of the underlying distribution.As DX = 1, the power law behavior now reduces to a simplelinear relation

ln hFqi ¼ fq lnM þ bq ð4Þwhere fq is the intermittency index, characterizing thestrength of the effect. In Fig. 1 we have plotted ln ⟨Fq⟩against lnM in h-space for q = 2, ..., 6. The experimental re-sults and the UrQMD simulated results have been plottedside by side. A similar set of plots in 4-space is shown inFig. 2. One can easily see that for each q there is a definitelinear rise in the experimental ln ⟨Fq⟩ values with lnM,which confirms the power law behavior suggested in (1). Onthe other hand, the simulated ln ⟨Fq⟩ values remain practi-cally uniform with varying lnM over its entire range, indicat-ing almost no intermittency in the UrQMD-generated events.This is true in both h- and 4-spaces. At this point we maynote that the string fragmentation model FRITIOF [22] alsodid not show any intermittency for AB interactions at200 A GeV/c [16, 17].The values of the intermittency index along with the Pear-

son’s R2 coefficients, which indicate the goodness of fit [23],are shown in Table 1. For each q, a linear fit of ln ⟨Fq⟩against lnM data has been performed by excluding the verysmall M (or large dX) region. The errors associated with⟨Fq⟩ (as shown in Figs. 1 and 2) and those associated withfq (as listed in Table 1) are only of statistical origin. EachFeq is assumed to be an error-free quantity, and the standard

error of the mean ⟨Fq⟩ over event space is shown in the dia-grams. On the other hand, as the data points in the ln ⟨Fq⟩ vs.lnM plot are highly correlated, the errors in fq are nontri-vially estimated [24] by generating several event samples us-ing random numbers [20]. Following (4), the fq values foreach individual simulated event sample are determined, andthen the statistical spread about the mean, ⟨fq⟩, over ten dif-ferent samples are quoted as errors in Table 1.One can also see that the fq values are consistently larger

in the 4-space than in the h-space. To conserve transversemomentum, the particles probably experience extra correla-tion in the azimuthal plane. However, such differences in thefq values are more prominent in interactions induced by nu-clei with higher mass numbers (e.g., 28Si or 32S [16]) and are

smaller when the interaction is induced by a comparativelylighter (16O) nucleus [17]. Results of the present investigationdiffer significantly from those obtained from the analyses ofanother set of 28Si-Ag/Br data at the same incident energy(14.5 A GeV) as in the present case [14, 15]. This discrep-ancy is probably due to the nonconversion of the phase space

Fig. 1. Intermittency of shower track emission in the h-space of28Si-Ag/Br interaction at 14.5 A GeV. The error bars represent stan-dard error. Best linear fits are drawn.

Fig. 2. Intermittency of shower track emission in the 4-space of28Si-Ag/Br interaction at 14.5 A GeV. The error bars represent stan-dard error. Best linear fits are drawn.

Table 1. Intermittency exponents of 28Si-Ag/Br interaction at14.5 A GeV.

h-space 4-space

Order fq R2 fq R2

Experimentq = 2 0.011±0.001 0.989 0.016±0.001 0.955q = 3 0.039±0.003 0.972 0.065±0.002 0.940q = 4 0.112±0.007 0.949 0.214±0.005 0.911q = 5 0.255±0.013 0.941 0.600±0.011 0.908q = 6 0.477±0.023 0.938 1.941±0.029 0.934UrQMDq = 2 –0.0001±0.0006 0.971 0.002±0.001 0.982q = 3 0.0001±0.0018 0.963 0.004±0.002 0.985q = 4 0.006±0.005 0.960 0.005±0.002 0.951q = 5 0.028±0.011 0.951 0.008±0.003 0.943q = 6 0.075±0.048 0.943 0.018±0.011 0.927

Note: The errors are only of statistical origin.

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h (4) variables to their respective cumulant variables inrefs. 14 and 15. Hence, the intermittency phenomenon de-pends more on the colliding system and less on the collisionenergy. It is also dependent on the choice of the phase spacevariable, which, to a certain extent is in contradiction to theobservation of ref. 24, but is consistent with our previous ob-servations on AB interactions [16, 17].One can now put the fq values to further tests, and explore

the underlying physical processes (e.g., phase transition or nophase transition) that have resulted in the observed pattern.To check to what extent the particle correlation embeddedwithin a higher-order SFM is influenced by the contributionfrom lower order (two or three particle) correlation(s), onecan introduce normalized exponents and can then study theirdependence on q. These exponents are defined as [24]

zq ¼ fq=qC2 ð5Þ

and

zð3Þq ¼ ðq� 2Þz3 � ðq� 3Þz2 ð6Þ

In Fig. 3, the normalized exponents for the experimentaland the UrQMD simulated data have been plotted against qin h- and 4-spaces. As prescribed in (6), the result confirmsa linear relationship between zð3Þq and q. Expectedly, the ex-perimental and the simulated results behave in quite differentways, and within statistical error the simulated values areconsistent with no intermittency. Thus, our results on normal-ized exponents suggest that higher-order (q ≥ 4) correlationscan be understood in terms of two- and three-particle correla-tions. As the shower tracks are caused by all kinds ofcharged mesons, these correlations are, however, not entirelyof the usual Bose–Einstein type arising from the symmetriza-tion of the wave function of a system of identical bosons.The power law behavior of 1D SFMs characterizes some

kind of scale invariance of the dynamics of multiparticle pro-duction. This can be realized either in the self-similar randomcascading models [25], or in the statistical systems at criticaltemperature for a second-order phase transition [26] (seeref. 6 for a review). The dependence of fq on q would bedifferent in these two cases, which can be examined by estab-lishing a connection between intermittency and (multi)fractal-ity [8]. The generalized Rényi dimensions, Dq, are directlyrelated to fq as,

Dq ¼ D� fq

q� 1ð7Þ

where D is the topological dimension of the supporting space(e.g., D = 1 for h- or 4-space). On the other hand, the anom-alous dimensions are defined as

dq ¼ D� Dq ð8ÞFor a system at critical temperature of a second-order

phase transition, the multifractal behavior reduces to a mono-fractal behavior like dq/d2 = 1. A plot of dq/d2 against q, pre-sented in Fig. 4a, shows that our results are certainly notindicating a monofractal structure of the density function,and hence do not indicate a second-order phase transition in28Si-Ag/Br interaction at 14.5 A GeV/c. For multiplicative

cascade mechanisms like the a-model [5], where the final-state particle density is given as a product of random num-bers, the density function can be approximated by a long-tailed log–Lévy distribution. Under this approximation, thefollowing relation holds:

ðq� 1Þ dqd2

¼ qm � q

2m � 2ð9Þ

where m (0 ≤ m ≤ 2) is called the Lévy stable index [25].While m < 1 is indicative of a second-order (thermal) phasetransition, m > 1 indicates a nonthermal phase transition. Afit of our experimental data in terms of (9), however, resultsin m = 3.15 ± 0.003 in h-space, and m = 3.70 ± 0.03 in 4-space. Both of these values far exceed the allowed limit ofthe stability index, thereby showing that the (dynamical) den-sity function in the present case cannot exactly be representedby a log–Lévy distribution. A similar observation was alsomade in the 1D intermittency analysis in 12C-(Cu, Ne) inter-actions at 4.5 A GeV/c [27], where a second-order phasetransition was ruled out as a possible mechanism of hadroni-zation. Note that the present energy per nucleon, 14.5 GeV,is not very much different either. The reason may be due tothe “projection” effect, as it is usually found inlower-dimensional intermittency analysis. The stability indexcan, however, be obtained through other approaches, for ex-ample, by using the multifractal spectrum [28]. Though thepresent m-values (>1) indicate a nonthermal phase transitionduring the particle emission process, the issue needs furtherscrutiny to reach a definite conclusion.Intermittency can also be studied in the framework of the

Ginzburg–Landau (GL) model. According to GL theory, theratio dq/d2 should obey the relation

dq

d2¼ ðq� 1Þn�1 ð10Þ

where n (= 1.304) is a dimension-independent universal para-meter [29]. For our 28Si-Ag/Br experimental data, we foundn = 2.30 ± 0.02 in h-space and n = 2.65 ± 0.03 in 4-space.A graphical representation of the GL prediction has beenshown in Fig. 4b. Thus, our intermittency results cannot beexplained in terms of the GL theory either.The phase transition, if there is any, may not necessarily

always be a thermal one, as the new phase is not essentially

Fig. 3. Variation of the normalized intermittency indices against q in28Si-Ag/Br interaction at 14.5 A GeV. Curves are drawn to guide theeye.

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defined by a (set of) thermodynamic parameter(s). Simultane-ous existence of two nonthermal phases (like those in a spin-glass system) is a possibility that can be investigated by theintermittency parameter [30]

lq ¼fq þ 1

qð11Þ

In a self-similar cascade mechanism, lq should exhibit aminimum at some critical point q = qc, where the regionsq < qc and q > qc are dominated by a large number of smallfluctuations (liquid phase), and a small number of large fluc-tuations (dust phase), respectively. The variation of lq hasbeen schematically presented in Fig. 5a for the experimentand simulation. In h-space there is a hint of a probable mini-mum beyond our region of investigation (q > 6), whereas, in4-space a clear minimum at qc = 4 can be seen. TheUrQMD-simulated values exhibit no intermittency. Thepresent experimental results are similar to what was observedin C-Cu interaction at 4.5 A GeV/c [31] (see ref. 6 for laterexperimental confirmation of this issue).A more direct measure of the intermittency strength can be

obtained from its connection with (multi)fractality; at first inthe framework of a random cascading model like the a-model, and subsequently in a model-independent way, irre-spective of any particular hypothesized mechanism of particleproduction [32]. According to the a-model the strength pa-rameter, aq, is related to Dq by a simple relation

aq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 ln 2

qðD� DqÞ

sð12Þ

The aq values in 1D have been calculated and their varia-tion with q has been schematically presented in Fig. 5b. Onecan see that, for the experiment, the strength parameter line-arly increases with increasing order, whereas for the simula-tion it hovers around a very small value (≈ 0.05). From thepreceding discussion, it cannot be ascertained in clear termsas to which process (i.e., second-order phase transition orrandom cascading) is actually responsible for the observedintermittency. For arbitrary underlying dynamics it is possibleto define an effective fluctuation strength

aeff ¼ffiffiffiffiffiffiffiffi2f2

pð13Þ

We found that aeff = 0.15 ± 0.003 in h-space and aeff =0.18 ± 003 in 4-space for the present set of 28Si-Ag/Br ex-perimental data. These values are about 1/6 times the maxi-mum fluctuation strength (a = 1.0) allowed in the a model.Comparing with our previous results, we find that the presentvalues are smaller than those obtained in 16O-Ag/Br interac-tion at 200 A GeV/c (about 1/4 of the maximum value), butwithin error are of the same magnitude as the 32S-Ag/Br in-teraction at 200 A GeV/c [16, 17]. Thus, it appears that, forthe same target, the fluctuation strength depends more on themass number than on the energy of the incident projectile.A thermodynamic interpretation of multifractality has also

been given in terms of a constant specific heat, C, that is re-lated to the Rényi dimensions as [33]

Dq ¼ D1 þ C ln q

q� 1ð14Þ

While deriving (14), it has been assumed that only Ber-noulli-type fluctuations are responsible for a transition frommonofractality to multifractality. A monofractal to multifrac-tal transition corresponds to a jump in the value of C fromzero to a nonzero finite value. By examining the variation ofDq with q one can obtain the value of C. A plot of Dq can befound in Fig. 5c for the experimental as well as for the simu-lated data. The simulated Dq values are always very close tothe dimension of the supporting space (D = 1), whereas theexperimental values are consistently different from unity.Over the full range (q = 2, ..., 6), the experimental Dq variesnonlinearly with ln q/(q – 1), and the nonlinearity is moreprominent in 4-space than in h-space. The C value will obvi-ously depend on the range of q over which the data are fitted.In Table 2, the fit results in different q ranges are given,which show nonzero multifractal specific heat for the experi-mental data. One can also see that, contrary to what wefound in the 32S-Ag/Br interaction at 200 A GeV/c [34], inthe present case the C value in 4-space is always higher thanthat in h-space. The C values are, however, not consistentwith any universal value, as claimed in ref. 33.

5. The erraticityAs mentioned in the Introduction, the erraticity technique

measures an event-space fluctuation of Feq. In Fig. 6 we have

graphically shown the experimental distributions of Feq in h-

space corresponding to two different M (equal to 10, 20) val-ues. Respective distributions generated by the UrQMD arealso incorporated in these diagrams. Though most of the Fe

q

values are concentrated within a small initial range, one cansee that long tails corresponding to unusually large Fe

q valuesare also present in each distribution. It is our objective toquantify these large fluctuations of Fe

q in terms of the erratic-ity moments and the associated scaling parameters. Themethod of analysis [9] as well as the erraticity results on28Si-Ag/Br interaction at 14.5 A GeV are presented below.The technique starts with a normalized moment, Fq, definedfor an individual event as

Fq ¼ Feq=hFqi

Fig. 4. Plots of dq/d2 against q for the experimental data of 28Si-Ag/Br interaction at 14.5 A GeV: (a) the curves represent (9); (b) thecurves are drawn following (10).

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and then by defining a pair of erraticity moments in terms ofFq. On one hand, we have the Cpq moments defined as thevertically averaged pth-order moment of Fq

Cpq ¼ hFpqi ð15Þ

and on the other we have an entropy-like quantity, Sq, de-fined as

Sq ¼ hFq lnFqi ð16ÞNote that, unlike q, here p is not necessarily an integer,

and we are particularly interested in studying the erraticitybehavior in the region p ≈ 1. These erraticity moments are

capable of characterizing the event-space fluctuation of Feq,

whereas the phenomenon actually refers to the followingscaling law:

Cpq � Mjðp;qÞ ð17Þwhere j(p, q) is called the erraticity index. If the spatial pat-tern of the particle density function does not change fromone event to another, one would expect the distributionPðFe

qÞ to behave like a delta function. Correspondingly, bothFq and Cpq would reduce to unity, and j(p, q) to zero.Hence, any deviation of j(p, q) from zero can be considereda measure of erraticity. In high-energy AB interactions, how-ever, Cpq may not exhibit as strict a scaling law as prescribedin (17), but would rather follow a more generalized form as

Cpq / gðMÞejðp;qÞ ð18Þwhere g(M) is some well-behaved function of M. Similar to(18), one would expect a generalized scaling law also for theSq moments as

Sq / emq ln gðMÞ ð19Þ

From the definitions of Cpq and Sq it follows that

emq ¼d

dpejðp; qÞjp¼1 ð20Þ

Termed as the entropy index, emq is actually the slope ofthe ejðp; qÞ vs. p curve at p = 1, and is an efficient parameterto characterize erraticity. It has been found [9] that the en-tropy index signifies chaotic behavior in the quantum chro-modynamics (QCD) branching processes. Small emq

corresponds to large Sq and hence to less chaotic systems.On the other hand, a large emq corresponds to a small entropyand hence to a chaotic system. The entropy index, emq, islarger if the QCD branching process is initiated by a quarkthan if it is initiated by a gluonic jet [35].In Fig. 7, the experimentally obtained Cpq moments for dif-

ferent q and p values have been plotted as functions of thephase space partition number, M. For an easy comparison,beside each experimental plot corresponding UrQMD-generated results are also similarly presented. From these

Fig. 5. (a) Variation of the intermittency parameter, lq, with q. (b) Variation of the intermittency strength withq. (c) Variation of the general-ized Rényi dimensions, Dq, with ln q/(q – 1) in 28Si-Ag/Br interaction at 14.5A GeV. Curves are drawn to guide the eye.

Table 2. The multifractal specific heat in 28Si-Ag/Br interaction at 14.5 A GeV.

Fit region h-space 4-space

Experiment2 ≤ q ≤ 4 0.103±0.035 0.224±0.0892 ≤ q ≤ 5 0.158±0.049 0.047±0.1563 ≤ q ≤ 5 0.273±0.061 0.758±0.229UrQMD2 ≤ q ≤ 4 0.009±0.006 –0.001±0.00102 ≤ q ≤ 5 0.022±0.011 0.0001±0.00133 ≤ q ≤ 5 0.046±0.017 0.0006±0.0004

Fig. 6. F2 distribution for two different h-space partitions in 28Si-Ag/Br interaction at 14.5 A GeV.

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graphs one can see that the experimental Cpq values varyover a wider range than the corresponding UrQMD-generated values. A smooth but nonlinear variation of lnCp2 with lnM can be seen over its entire range, which indi-cates the necessity of invoking a generalized scaling lawlike (18). For q = 3, 4, and 5 the variation patterns aremore or less similar to those for q = 2. However, severalkinks (i.e., discontinuities) in the experimental distributionsat large M are observed in these cases. As the 28Si-Ag/Brevent sample does not possess a very high ⟨ns⟩ value, withincreasing q more events become susceptible to the emptybin effect, and a kind of saturating trend beyond M = 10can be seen with growing fluctuation in the data points. Toestablish the generalized scaling law, as suggested in ref. 9,we have assumed ln g(M) = (lnM)b where b is a free pa-rameter that has to be adjusted from the linear fit of lnC22values against ln g(M). Such plots can be found in Fig. 8,both for the experiment and for the UrQMD with the re-spective best fitted values being b = 2.95 and 2.45. A sim-ilar fit for the purely statistical sample (not showngraphically) results in b = 2.70. Here, the Pearson’s R2 co-efficients (>0.98 in our case) have been used to decide thegoodness of the fit. Compare these values with b = 3.23and 2.08, respectively, for the 16O-Ag/Br and the 32S-Ag/Brinteractions at 200 A GeV/c [11]. The slopes of thesestraight lines give us another erraticity parameter, ej(2, 2),whose values are ej(2, 2) = 0.0138 ± 0.0006 (experiment),ej(2, 2) = 0.0068 ± 0.0003 (UrQMD), and ej(2, 2) =0.0082 ± 0.0004 (random number). Corresponding entropyindices, em2, can be obtained from similar linear relation-ships between S2 and ln g(M). These graphs are also in-cluded in the same diagrams as the lnC22 vs. ln g(M)diagrams, and em2 comes out as the slope of the respectivebest linear fit as em2 = 0.00493 ± 0.00009 (experiment),em2 = 0.00377 ± 0.00008 (UrQMD), and em2 = 0.00375 ±0.0008 (random number).Noting that all the Cp,q moments depend similarly on lnM,

one can use lnC22 in place of ln g(M). The scaling relationcan thus be converted to

Cpq / ðC22Þcðp;qÞ ð21ÞWe have found that, for q = 2, the expected linear depend-

ence of lnCpq on lnC22 is almost exact, and for different pthe results are graphically presented in Fig. 9. The experi-mental result and the UrQMD simulated result are plotted onthe same set of axes. For q > 2, the dependence is approxi-mately linear only in the low M region, and once again finitemultiplicity saturation effects are seen at high M. For q > 2we have, however, obtained c(p, q) through linear fit of theapproximate linear dependence of ln Cpq on lnC22 within alimited region (M ≤ 12), where lnCpq is found to behave sys-tematically. In Fig. 10, the c(p, q) values so obtained for theexperiment and for the UrQMD are plotted against p for dif-ferent q values, and for each q the nonlinear variation hasbeen fitted with a quadratic function like

cðp; qÞ ¼ a2p2 þ a1pþ a0 ð22Þ

The first-order derivatives cq′ = @c(p, q)/@p|p=1 can nowbe obtained by using this diagram and the corresponding

quadratic fit results, and can further be utilized to determinethe entropy index from the following relation:

emq ¼ ej2;2c0q ð23Þ

The entropy-like moments, Sq, for different q are alsoplotted against lnM in Fig. 11. As expected, one can seethat these moments are also not linearly varying with lnMover the entire range of the latter. However, for all q the var-iation pattern looks similar. Hence, in place of g(M) one canuse S2 and make a plot of Sq against S2, which is given inFig. 12. In both of these diagrams, the UrQMD predictionhas been incorporated along with the experimental plot. Theslope parameters, uq = @Sq/@S2, of these curves have beenobtained by making a linear fit of the data points within alimited M (≤ 12) region. Subsequently, following a differentrelationship

emq ¼ uqem2 ð24Þone can obtain the entropy index in addition to (23). All thescaling parameters pertaining to the erraticity analysis of our28Si-Ag/Br data, namely cq′, uq, and the two sets of emq,using both (23) and (24), have been presented in Table 3 forq = 2–5. The experimental results, the UrQMD-generated re-sults, and the random number generated pure statistical re-sults are presented together. Several observations can now bemade regarding the erraticity behavior of the 28Si-Ag/Br in-teraction at 14.5 A GeV. First of all, in general, the erraticityof a particle distribution is observed for all three data sets.However, with increasing q, while the experimental cq′ para-meter increases at a slower rate, the uq parameter increases ata faster rate than the corresponding UrQMD values. Bothparameters for the purely statistical sample change at aslower rate than both the experiment and the UrQMD. For28Si-Ag/Br interaction at 14.5 A GeV both parameters in-crease at a much slower rate than what was previously ob-served in other AB interactions at 200 A GeV/c [11]. Theentropy index obtained from two different methods (i.e., (23)and (24)) are very close to each other within statistical uncer-tainties. The variation of emq with q is plotted in Fig. 13,which for the experimental data shows a rapid increase withincreasing q. The experimental values are always larger thanthe simulated values, and the difference increases with q. Theexperimental emq values in 28Si-Ag/Br interaction at 14.5 AGeV are slightly larger than the corresponding emq values ob-tained in 16O-Ag/Br interaction at 200 A GeV/c, but are al-most of the same order of magnitude as the emq valuesobtained in 32S-Ag/Br interaction at 200 A GeV/c. Onceagain, dependence on the projectile mass number seems todominate the erraticity behavior over the collision energy.We note that the emq parameter characterizes the event-spaceentropy and hence the chaoticity of the interacting system.An increase in the entropy index signals a possiblequark–hadron phase transition [36]. This feature has indeedbeen obtained for a possible nonthermal phase transition[37], where it is shown that the mq index increases fast withq in the case of a “critical” M interval. We can find a similarincreasing trend of our emq parameter for the present set of28Si-Ag/Br data, which is consistent with the nonthermalphase transition indicated in Fig. 5a.

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Fig. 7. The erraticity moments Cp,q are plotted against the h-space partition number M in 28Si-Ag/Br interaction at 14.5 A GeV. The curvesare drawn to guide the eye.

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However, the em2 values of the present analysis are signifi-cantly (by two orders of magnitude) smaller than what wasobtained in a similar experiment on the 28Si-Ag/Br interac-tion at 14.6 A GeV [12]. We note that in ref. 12 the averageshower track multiplicities are always (i.e., for all event sub-samples) lower than the present set of data, and with increas-ing ⟨ns⟩ the entropy index, m2, is found to decreaseconsistently. We also note that in ref. 12 the erraticity mo-

ments were plotted against lnM, and not against ln g(M).The entropy index, m2, obtained thereby is altogether a differ-ent parameter than em2 measured in the present investigation.We have checked that a similar linear fit of the lnCpq vs.ln M data in the region 11 ≤ M ≤ 25 for the present eventsample results in m2 = 0.092 ± 0.009, which for the Ag/Brevents is close to the m2 values obtained in ref. 12 within asimilar M-range.We observe that the UrQMD fails to replicate any of the

intermittency predictions of our 28Si-Ag/Br data, but both theUrQMD-generated sample and the random number generatedsample exhibit reasonable amounts of erraticity, though lowervalues than experiment. Similarly, we earlier found that FRI-TIOF also reproduced erraticity behavior to a lesser extentthan experiment [11]. Neither FRITIOF nor UrQMD has any-thing to do with the dynamics of erraticity of particle produc-tion, as no particle correlation is included in these models.The erraticity behavior comes merely as a statistical effect,as is also confirmed from the purely statistical results. Simi-lar observations regarding the statistical contribution to erra-ticity behavior were highlighted in other experiments [6]. As,for example, in ref. 38 it was demonstrated that the erraticitybehavior of a set of low multiplicity data of the NA27 experi-ment, could be reproduced by purely statistical fluctuations.In ref. 39 it was found that the FRITIOF and Venus MC sim-ulations on AB interactions fit very well with the expectedfrom of the 3D pure statistical fluctuation model. The resultswere independent of the event generator, colliding nuclei, in-cident energy, particle type, and phase space region used in

Fig. 8. Plot of lnC2,2 against ln g(M) and S2 against ln g(M) =(ln M)b in 28Si-Ag/Br interaction at 14.5 A GeV; b = 2.95 for ex-periment and b = 2.45 for the UrQMD. The best linear fits areshown.

Fig. 9. Plot of lnCp,2 against ln C2,2 in 28Si-Ag/Br interaction at14.5 A GeV. The best linear fits are shown.

Fig. 10. Plot of c(p, q) against p for different q in 28Si-Ag/Br inter-action at 14.5 A GeV. The best quadratic fits are shown.

Fig. 11. Plot of Sq against lnM. Curves are drawn to guide the eye.

Fig. 12. Plot of Sq against S2. Best linear fits for M ≤ 12 areshown.

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the calculation. Thus, in the framework of the FRITIOF and(or) Venus event generators, even in the central collision ofheavy nuclei at energies up to 200 A GeV, the statistical fluc-tuations dominated the erraticity behavior. The erraticityanalysis of hadronic interaction at 250 GeV/c in the NA22experiment [40] confirms the dominance of statistical fluctu-ations in the experimental measurement. In emulsion experi-ments [12, 41], the experimental data on AB interactionswere found to be far apart from the MC-generated resultsand (or) results generated by purely statistical considerations,an observation that is similar to the present analysis.

6. ConclusionIn this paper, a 1D analysis of charged particles (pions)

produced in 28Si-Ag/Br interactions at an incident energy of14.5 A GeV is presented. The features of short-range correla-tion, namely, the intermittency and the erraticity phenomena,are studied. The experimental results are compared to theprediction from a microscopic transport model, UrQMD. Tosummarize, we obtained a nonstatistical ingredient in themeasured 1D density fluctuation of the charged particle dis-tribution, and these fluctuations are shown to be ofself-similar nature. The UrQMD predictions show almost nointermittency effect. The event-by-event fluctuations are

found to be of a chaotic nature. The following critical obser-vation can be made from the analysis:The present analysis shows that intermittency is dependent

more on the colliding system than on the collision energy in-volved in AB interactions. The 1D intermittency also de-pends on the choice of the underlying phase space variable.Intermittency in the 4-space is found to be slightly strongerthan in the h-space. The intermittency exponents can be ex-plained in terms of two- and (or) three-particle correlations.The shower tracks are caused by nonidentical mesons (bothpositive and negative charge). Hence, these correlations arenot entirely of the usual Bose–Einstein type.In the present study, the order dependence of the intermit-

tency indexes is neither in conformity with a second-orderthermal phase transition, nor with a multiplicative cascademechanism. Instead, our intermittency results indicate thatthere may be a nonthermal phase transition and (or) simulta-neous coexistence of two nonthermal states of hadronic mat-ter. Using the generalized Rényi dimensions, the multifractalspecific heat, C, has also been obtained for the 28Si-Ag/Br in-teraction at 14.5 A GeV. However, no universality is found tobe associated with this parameter.The event-by-event fluctuation of small phase-space struc-

ture in a single event has been addressed in the erraticityanalysis of our 28Si-Ag/Br data. The erraticity moments arefound to abide by a less stringent generalized scaling lawthan the factorial moments. Quantitatively this is true for theexperimental as well as for the UrQMD simulated data. Thepresent sample of 28Si-Ag/Br events at 14.5 A GeV seems tobe almost as chaotic as the 16O-Ag/Br and 32S-Ag/Br sys-tems at 200 A GeV/c. Unlike the intermittency, theUrQMD-generated results on 28Si-Ag/Br interaction also ex-hibit an erratic nature, and the simulated system appears tobe almost as chaotic as the experimental one. It may benoted that, as was found earlier, the string fragmentationmodel, FRITIOF, does not fully reproduce either the inter-mittency results or the erraticity measurements at similar orhigher energies in nucleus–nucleus interactions as well as inhadronic interactions. However, the erratic behavior ispresent in the MC models as well as in the pure statisticalfluctuations, signaling the statistical fluctuations to be domi-

Table 3. The erraticity parameters in 28Si-Ag/Br interaction at 14.5 A GeV.

Order cq′ uq emq (23) emq (24)Experimentq = 2 0.368±0.111 — 0.0051±0.0015 0.0049±0.0001q = 3 1.671±0.904 5.727±0.132 0.0230±0.0124 0.0282±0.0008q = 4 5.995±0.922 16.554±0.536 0.0825±0.0132 0.0816±0.0031q = 5 11.918±1.672 27.729±0.588 0.1641±0.0242 0.1367±0.0039UrQMDq = 2 0.546±0.033 — 0.0037±0.0003 0.0037±0.0001q = 3 2.929±0.463 4.452±0.227 0.0199±0.0032 0.0168±0.0009q = 4 8.138±1.808 12.487±0.803 0.0553±0.0124 0.0471±0.0032q = 5 17.908±1.537 24.638±1.536 0.1217±0.0115 0.0929±0.0061Random numberq = 2 0.441±0.094 — 0.0037±0.0001 0.0032±0.0001q = 3 1.367±0.077 3.431±0.104 0.0116±0.0001 0.0111±0.0005q = 4 3.451±0.127 8.709±0.249 0.0294±0.0016 0.0283±0.0011q = 5 6.308±0.226 16.529±0.599 0.0537±0.0030 0.0537±0.0024

Fig. 13. Plot of emq against q: (a) using (23); and (b) using (24).Lines joining points are drawn to guide the eye.

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nant in the erraticity measurements and hence need to betaken into account.

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