Multifractal structure of multiplicity distributions
and negative binomials
S. MALIK
INFN, Sezione di Pisa - Via Livornese 1291, S. Piero a Grado, I-56010 Pisa, Italy Department of Physics and Astrophysics, University of Delhi - Delhi 110007, India (ricevuto il 18 Novembre 1996; revisionato il 16 Luglio 1997; approvato il 7 Ottobre 1997)
Summary. — We present experimental results of the multifractal structure analysis in proton-emulsion interactions at 800 GeV. The multiplicity moments have a power law dependence on the mean multiplicity in varying bin sizes of pseudorapidity. The values of generalised dimensions are calculated from the slope values. The multifractal characteristics are also examined in the light of negative binomials. The observed multiplicity moments and those derived from the negative-binomial fits agree well with each other. Also the values of Dq, both observed and derived from the
negative-binomial fits not only decrease with q typifying multifractality but also agree well with each other showing consistency with the negative-binomial form. PACS 24.60.Ky – Fluctuation phenomena.
PACS 13.85.Hd – Inelastic scattering: many-particle final states. PACS 29.40.Rg – Nuclear emulsions.
1. – Introduction
The observation [1] of non-statistical multiplicity fluctuations in rapidity phase space, called intermittency, has aroused considerable interest in the possible mechanism of multiparticle production in high-energy interactions. This chaotic behaviour has been studied extensively with the availability of precise and very high multiplicity data in terms of bin-averaged normalised factorial moments introduces by Bialas and Peschanski [ 2 , 3 ]. To investigate this phenomenon of intermittency, whether it concerns particle or nuclear physics of hydrodynamic turbulence [4], the scaling behaviour of the moments of the relevant distributions is studied as a function of the bin size in phase space which might unravel the fundamental properties of an interaction process. It has naturally brought fractal dynamics [5] into play and the approaches based on the concept of (multi-)fractal [ 3 , 6 , 7 ] are exciting as they may be related to self-similar cascades [ 2 , 3 ], phase transition [8], etc. In fact, to probe the multifractal characteristics in high-energy interactions several methods [ 6 , 7 , 9 , 10 ] have been suggested and some interesting results have already been obtained from e1e annihilations [11], lepton-hadron [12], hadron-hadron [13], hadron-nucleus [14] and nucleus-nucleus [15] interactions. A second feature of multiparticle production that has
come into focus is the widespread occurrence of negative-binomial distribution over a large energy range [16-21]. The results of UA5 Collaboration [ 22 , 23 ] on the comprehensive analysis of the charged multiplicity distribution of pp collisions at ks 4540 GeV in different rapidity windows has given a new impetus to the study of negative-binomial distributions (NBD) in high-energy interactions. Analysis of charged multiplicity distribution shows that NBD describes it well not only for full phase space domains but for different intervals [ 16 , 19 , 21 -30 ] as well is pseudorapidi-ty (h) space. It is found to hold in hadron-hadron [ 16 , 31 ], hadron-nucleus [ 21 , 28 , 29 ], nucleus-nucleus [30], semileptonic [32] and leptonic [ 19 , 26 , 27 ] interactions. NBD has been physically interpreted in terms of clan structure [ 33 , 34 ] typifying a cascade mechanism. However the point to be emphasized is that the occurrence of relative fluctuations and negative binomials are both related to the study of multiplicity distributions. We exploit this common connection here to understand the process underlying multiparticle dynamics. We investigate the bin size dependence of relative fluctuations in particle density distribution in h space in terms of multiplicity moments. The multifractal characteristics are analysed in the light of Takagi’s formalism [6] in proton-emulsion nuclei interactions at a primary beam energy of 800 GeV, the highest so far for fixed-target experiments. Takagi’s method overcomes the non-linear behaviour observed in the log-log plots. The behaviour of generalized dimensions favours multifractality and cascading in multiparticle production. The multiplicity moments of proton-emulsion nuclei interactions have been further obtained on the basis of NBD fits to the data. We also obtain Dq values derived from NBD fits. We
further compare NBD fits with values observed experimentally to check the consistency between multifractal behaviour and NBD.
2. – Experimental details
A stack of 40 Ilford G5 emulsion pellicle of dimension ( 10 3830.06) cm was exposed to a proton beam of energy 800 GeV at Fermilab. The beam flux was 8 .7 3 104particlesOcm2 and the dispersion in energy of the primary was E 0.05%. The emulsion plates were carefully area scanned by each observer and average efficiency for detecting events was found to be 96%. Scanning was done starting from a distance of 1 cm from the leading edge of the emulsion plate, so as to ensure that all interactions considered are only due to primary protons and not to any otherOsecondary particle. In order to select events due to primary protons, the incident particle should make an angle E 27 with the mean beam direction. All the secondary tracks were back-followed to their starting point in the emulsion, in order to eliminate events due to secondary tracks. Events lying up to 25 mm from the surface or the glass side of the emulsion pellicle were not considered. Using the above criteria, a total of 3390 events were obtained. Following the usual emulsion terminology [35], secondary particles with b(vOc) F0.7 and bE0.7 were designated as shower particles and heavy tracks, respectively. It was convenient to further subdivide the heavily ionizing particles into grey and black tracks whose multiplicities are designated as Ng and Nb, respectively. The black tracks result from secondary particles with specific ionization D10I0, while grey tracks have ionization between 1 .4 I0 and 10 I0, where I0 is the ionization of the primary beam tracks. Shower tracks have ionization E1.4I0[35]. The pseudorapidity (h) of all shower particles was obtained from the measured space angle (u) with reference to the beam direction by the relation h 42ln tan(uO2).
3. – Formalism
Consider a multiparticle production process at a given incident energy and distributed over a single bin of varying Dh in h space around h 40. Let n be the number of shower particles in one event in Dh . To measure fluctuations, we consider the multifractality function
Aq4 [ 1 O(q 2 1 ) ] ln [anqb Oanb] ,
(1)
where q is a positive real number. The angular brackets denote the average taken over all the events of a particular target type. To look for evidence of multifractality we consider the quantity Aq regarding its linear behaviour with the logarithm of anb,
i.e.
Aq4 Bq1 Dqln anb ,
(2)
where Bq and Dqare independent of the bin size Dh . If such a behaviour exists over a
considerable range of anb [6], then this points towards the fractal structure in multiparticle production. The generalized dimensions Dqdefine such a structure.
For q F2, eq. (2) implies a relation of the form [6] ln anq
b 4 (q21) Bq1 [ (q 2 1 ) Dq1 1 ] ln anb .
(3)
The case q 41 is obtained by taking an appropriate limit of eq. (1) A14 an ln nb Oanb ,
(4)
whose linear behaviour with ln anb gives the information dimension D1, as [6] A14 B11 D1ln anb .
(5)
The negative-binomial distribution of charged shower particles is given [33] by
P(n , n, k) 4 k(k 11) R (k1n21) n!
nnkk
(n 1k)n 1k ,
(6)
where n refers to the multiplicity of charged shower particles in a window and k is a parameter related to the dispersion by D2
O n24 1 O n 1 1 Ok . The values of the NBD parameters n and k were obtained by fitting the experimental data with the CERN MINUIT program.
To check the consistency of multifractal behaviour with the negative binomial distribution fits, we calculate the values of normalised multiplicity moments or Cq
moments from the NB fitted parameters n and k. The Cq moments are defined as the
ratio of the q-th moment divided by the mean multiplicity as Cq4 anqb Oanbq,
(7)
where angular brackets stand for the event average. The predictions of NBD for Cq moments (q 42, 3, 4, 5) can be expressed in terms of the parameters n and k
as follows:
.
`
/
`
´
C24111O n11Ok , C34(111Ok)(112Ok)13O n(111Ok)11O n2,C44(111Ok)(112Ok)(113Ok)16O n(111Ok)(112Ok)17O n2(111Ok)11O n3, C54(111Ok)(112Ok)(113Ok)(114Ok)110O n(111Ok)(112Ok)(113Ok)1
125 O n2( 1 11Ok)(112Ok)115O n3( 1 11Ok)11O n4. (8)
Using eqs. (7) and (8) we find the values of anqb for q 42, 3, 4, 5. Similarly to Takagi’s method one can obtain the Dqvalues from plots of ln anqb vs. lnanb for the NBD analysis.
Thus one can compare the multiplicity moments and generalized dimensions observed experimentally and derived from NBD fits.
4. – Data analysis
To analyse the data in h space we have taken a central pseudorapidity bin centered around h 40 and subsequently reduced it in steps of 0.25. The h values were determined in proton-nucleon system. Our aim was to calculate an ln nb and lnanqb over
Fig. 1. – Behaviour of ln anq
b (q 42, 3, 4, 5) and an ln nbOanb vs. anb in proton-emulsion nuclei interactions. The dots and circles correspond, respectively, to the observed and NBD derived values in the various pseudorapidity intervals. The solid and dashed lines are the best fits to the observed and NBD derived values, respectively. The errors shown are statistical.
TABLE I. – The observed and NBD derived values of the generalized dimensions Dq for the
proton-emulsion nuclei interactions. The errors are statistical. Interaction proton-emulsion D2 D3 D4 D5 observed NBD derived 0.854 6 0.003 0.843 6 0.004 0.832 6 0.005 0.830 6 0.008 0.816 6 0.012 0.814 6 0.007 0.812 6 0.008 0.800 6 0.015
a wide range of anb for h space with all the emulsion nuclei. We investigated the multifractal structure for shower particles in proton-emulsion nuclei interactions by looking at the behaviour of ln anqb vs. lnanb for integral values of q from 2 to 5 in a single bin of varying size for h space. This is shown in fig. 1 as full dots. The solid line gives the best fit to the data points. The moments vary smoothly with anb according to the relation
ln anq
b 4 (q21) Bq1 Kqln anb .
(9)
The variation in an ln nb Oanb with anb for h space is also shown in fig. 1. We parametrize it by eq. (5). The slope values for this is 0 .885 60.003. The presence of linear behaviour of ln anq
b vs. lnanb and an ln nb Oanb vs. lnanb is indicative of the fractality in the data.
The generalized dimensions Dq characterizing the fractal behaviour are determined
from the slope values Kqusing eqs. (2) and (3), which yield
Dq4 (Kq2 1 ) O(q 2 1 ) .
(10)
The information dimension D1equals the slope parameter in eq. (5), i.e. 0 .885 60.003. Table I shows the generalized dimensions Dq for q 42–5 for all emulsion nuclei. The
value of Dq decreases with q, typifying the multifractal characteristics of the data and
supports a cascading mechanism for the underlying process.
To analyse the multiplicity distribution in terms of negative-binomial law, the values of the NBD parameters n and k were obtained by fitting the experimental data with the CERN MINUIT program with x2
Od.o.f. C 1 for each h interval. We have checked that the values of n and k agree within a few percent with values calculated from the mean multiplicity and dispersion, respectively. All the multiplicity distributions in pseudorapidity windows 0 .25 GDhG3.50 are found to be fitted well by NB distributions. The values of n and k are found to increase from 2 .78 60.05 and 3.106 0 .24 to 18 .18 60.20 and 5.7560.14 respectively with Dh increasing from 0.25 to 3.50. This behaviour of n and k is shown in figs. 2 and 3.
It can be seen from fig. 2 that n increases linearly with Dh up to Dh 41.5, thereafter saturation sets in which could be due to energy and momentum conservation. We have not applied any correction to the pseudorapidity distribution, in order to make it flat around h 40, which could have led to the linear behaviour of n up to higher values of Dh . Also the behaviour of k with h is linear only after Dh D0.5. The
values of k for Dh G0.5 are off the linear trend. This kind of behaviour is unlike the one found e.g. in pp interactions by the UA5 Collaboration [23]. This discrepancy could be due to the fact that our interactions do not involve a single target but are of emulsion type, where the effect of more than one type of target is present and this target effect is more important near Dh 40. Various investigations [28, 29] on proton-nucleus interactions show that, when the k parameter is studied separately in forward and backward hemisphere of h variable, its behaviour is quite different in the two cases. So it could be that, if one studies k with Dh and without forward-backward distinction, an average effect from different types of target could lead to this non-linearity.
To check the consistency of multifractal behaviour with negative binomial fits we have calculated the values of multiplicity moments Cqfrom the fitted valued of n and k
using eq. (8). The plots of these NB derived values of C-moments are shown in fig. 1 as open circles. The dashed lines are the best fits to these points. The close proximity of dots and open circles point to the consistency of multifractal behaviour with NBD. Further table I also shows the Dq values based on NB fits. These values of Dq also
decrease with q, pointing to multifractality in the present interactions. A good agreement between the observed and NB derived values of Dqfurther adds up to the
above consistency. 5. – Conclusions
In this work we have analysed the multifractal characteristics of proton-emulsion nuclei interactions and as well studied the multiplicity distributions of charged shower particles in terms of negative-binomial fits in various pseudorapidity intervals of decreasing size. The behaviour of low-order multiplicity moments as a function of bin size is suggestive of fractality in the data. The decrease of Dqvalues with q points to the
multifractal geometry in the data and hence cascading. Furthermore the values of multiplicity moments and Dq, when calculated from NB fits, agree well with the
corresponding observed values. It is also known that NB has been successfully interpreted in terms of the clan model which represents a particular type of cascading. This close agreement points to a deeper common connection between these two features to understand the mechanism of multiparticle production through multiplicity distribution.
* * *
We are grateful to the Fermi National Accelerator Laboratory of exposure facilities at the Tevatron and to Prof. R. STEFANSKI for help during exposure of the emulsion
stack. We would like to thank Prof. R. WILKESfor processing facilities.
R E F E R E N C E S
[1] BURNETTT. H. et al., Phys. Rev. Lett., 50 (1983) 2062. [2] BIALASA. and PESCHANSKIR., Nucl. Phys. B, 273 (1986) 703.
[3] BIALASA. and PESCHANSKIR., Nucl. Phys. B, 308 (1988) 857; Phys. Lett. B, 207 (1988) 59. [4] PALADING. and VULPIANIA., Phys. Rep. C, 56 (1987) 147.
[5] BOHRT. and JENSENM., Phys. Rev. A, 36 (1987) 4904; SZEPFALUSYP. P. et al., Phys. Rev. A, 36 (1987) 3525; BIALASA. and HWAR. C., Phys. Lett. B, 253 (1991) 436.
[6] TAKAGIF., Phys. Rev. Lett., 72 (1994) 32.
[7] HWAR. C., Phys. Rev. D, 41 (1990) 1456; HWAR. C. and PANJ., Phys. Rev. D, 45 (1992) 1476; DERADOI. et al., Phys. Lett. B, 283 (1992) 151.
[8] PESCHANSKIR., Nucl. Phys. B, 327 (1989) 144.
[9] CHIUC. B. and HWAR. C., Phys. Rev. D, 43 (1991) 100. [10] HWAR. C., Phys. Rev. D, 31 (1995) 3323.
[11] BUSCHBECK B., LIPA P. and PESCHANSKI R., Phys. Lett. B, 215 (1988) 788; TASSO COLLABORATION (W. BRAUNSCHWEIG et al.), Phys. Lett. B, 231 (1989) 548; DELPHI COLLABORATION(P. ABREUet al.), Phys. Lett. B, 247 (1990) 137; Nucl. Phys. B, 386 (1992) 471; OPAL COLLABORATION (M. Z. AKRAWAY et al.), Phys. Lett. B, 262 (1991) 35; ALEPH COLLABORATION(D. DECAMPet al.), Z. Phys. C, 53 (1992) 21.
[12] DERADO I. et al., Phys. Lett. B, 283 (1992) 151; Z. Phys. C, 54 (1992) 357; WA25O59 COLLABORATION(L. VERTUYTENet al.), Phys. Lett. B, 260 (1991) 456.
[13] UA5 COLLABORATION(G. J. ALNERet al.), Phys. Rep., 154 (1987) 247; UA1 COLLABORATION (C. ALBAJARet al.), Nucl. Phys. B, 345 (1990) 1; NA22 COLLABORATION(F. BOTTERWECKet al.), Z. Phys. C, 51 (1991) 37; EHSONA22 COLLABORATION (N. M. AGABABYAN et al.), Phys. Lett. B, 261 (1991) 165.
[14] KLM COLLABORATION(R. HOLYNSKIet al.), Phys. Rev. Lett., 62 (1989) 733; SHIVPUTIR. K. and PARASHARNEETI, Phys. Rev. D, 49 (1994) 219; MALIKS. and SONIS. K., Phys. Rev. C, 53 (1996) 1876; MAITYD. K. et al., Mod. Phys. Lett. A, 8 (1993) 3853.
[15] NA35 COLLABORATION (I. DERADO), in Proceedings of the Ringberg Workshop on Multiparticle Production, Fluctuations and Fractal Structure, Ringberg, Germany, June, 1991, edited by R. C. HWA, W. OCHSand N. SCHMITZ (World Scientific, Singapore) 1992; NA35 COLLABORATION(A. BAMBERGERet al.), Phys. Lett. B, 203 (1989) 320; LAHANASM. et al., Nucl. Phys. A, 525 (1991) 327c; WA80 COLLABORATION(R. ALBRECHTet al.), Phys. Lett. A, 221 (1989) 427; ADAMOVICHM. I. et al., Phys. Lett. B, 263 (1991) 539; ALBRECHTR. et al., Phys. Rev. C, 50 (1994) 1048; SENGUPTAK. et al., Phys. Rev. D, 43 (1993) 3174.
[16] ADAMUSM. et al., Phys. Lett. B, 177 (1986) 239. [17] ARNEODOM. et al., Z. Phys. C, 35 (1987) 335. [18] AMMARR. et al., Phys. Lett., 178 (1986) 124. [19] DERRICKM., et al., Phys. Lett. B, 168 (1986) 299.
[20] ADAMUSM. et al., Z. Phys. C, 32 (1986) 475; 37 (1988) 215. [21] MALIKS., Z. Phys. C, 70 (1996) 621; 67 (1995) 239.
[22] UA4 COLLABORATION(G. J. ALNERet al.), Phys. Lett. B, 160 (1985) 199. [23] UA4 COLLABORATION(G. J. ALNERet al.), Phys. Lett. B, 160 (1985) 193. [24] THOMEW. et al., Nucl. Phys. B, 219 (1977) 365.
[25] BREAKSTONEA. et al., Nuovo Cimento A, 102 (1989) 1199.
[26] TASSO COLLABORATION(W. BRAUNSCHWEIGet al.), Z. Phys. C, 45 (1989) 193. [27] DELPHI COLLABORATION(P. ABREUet al.), Z. Phys. C, 52 (1991) 271. [28] DENGLERF. et al., Z. Phys. C, 33 (1986) 187.
[29] MAITYD. K. et al., Z. Phys. C, 65 (1995) 75; EHSONA22 COLLABORATION(I. V. AJINENKOet al.), Z. Phys. C, 46 (1990) 569; AGGARWALM. M., Int. J. Mod. Phys. A, 6 (1991) 865. [30] JAIN P. L. et al., Proceedings of Second International Conference on Physics and
Astro-physics of QGP, 1983, Calcutta, p. 478.
[31] ALNERG. J. et al., Phys. Lett. B, 167 (1986) 476.
[32] AMMARR. et al., Phys. Lett., 178 (1986) 124; BAILLYJ. L., et al., Z. Phys. C, 40 (1988) 215. [33] GIOVANNINI A. and VAN HOVEL., Z. Phys. C, 30 (1986) 391; Acta Phys. Pol. B, 19 (1988)
495.
[34] LUPIAS., GIOVANNINIA. and UGOCCIONIR., Z. Phys. C, 59 (1995) 427; 66 (1995) 195. [35] SHIVPURR. K., MALIKS. and SONIS. K., Z. Phys. C, 62 (1994) 609.
