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A multifractal analysis to study the multiparticle dynamics in 60A and 200A GeV,
^{16}O-AgBr collisions has been performed in the pseudorapidity phase space. Multifractal moments,
*G*
_{q}, as the function of pseudorapidity bin size for different order of the moments, q, have been calculated. The power-law behaviour has been observed in the considered data sets. The variations of multifractal dimension,
*D _{q}*, and the multifractal spectral function,

*f(a*

_{q}

*)*, with order of the moments,

*q*, have been studied thoroughly.

*D*

_{q}_{ }is found to decrease with increasing order of moments, q, indicating thereby a self-similar behaviour in the multiparticle production for the considered collisions. We have also found a concave downward curve of multifractal spectral function with maximum at

*q*= 0.

The study of event-by-event fluctuations in the pseudorapidity windows of decreasing bin width in multiparticle production at high energy has divulged self-similar properties as speculated by Bialas and Peschanski [

In order to investigate the cascading mechanism [

It is worth-mentioning that the issues of possible signals of Quark Gluon Plasma (QGP) formation in relativistic heavy-ion collisions are still on discussion stage. Fluctuations with different features are visualized to occur in different types of phase transitions. A study of multiplicity fluctuations in the final state of collisions might enable one to find out new observables linked to the characteristics of different types of phase transitions.

Some workers [

It is also interesting to mention that A. Deppman [

Although many attempts have been made to study the fractal properties, using μ p , p p ¯ and e + e − data [_{q}, however AA collisions are less studied using this method. In the present paper, an attempt is made to investigate various interesting features of multifractality in ^{16}O-AgBr collisions at 60A and 200A GeV. The Hwa’s proposal which is based on the general theory of multifractals [_{q}, generalized dimension, D_{q} and multifractal spectral function f ( α q ) for different order of the moments. These parameters serve as the measure of the multifractal structure in the considered data sets.

In order to examine the dependence of multifractal moments, G_{q}, on pseudorapidity, h, which is defined as; η = − ln tan ( θ s / 2 ) , θ s is the space angle of a secondary particle with mean direction of primary, a given pseudorapidity range Δ η = η max − η min is divided into M_{0} bins of width δ η = Δ η / M 0 . A multifractal moment is defined as;

G q = ∑ j = 1 M p j q (1)

where, p j = n j / n , such that n ( = n 1 + n 2 + ⋯ + n M ) . M denotes the number of non-empty bins. q is a real number and may has both positive or negative values. Once G_{q} is calculated, its average over the entire sample is determined;

〈 G q 〉 = 1 N e v ∑ 1 N e v G q (2)

where, N_{ev} stands for the total number of events. If there is self-similarity in the production of particles, G_{q} moments can be written in the form of a power law;

〈 G q 〉 = ( δ η ) τ q for δ η → 0 (3)

where, t_{q} stands for mass exponents. The linear dependence of l n 〈 G q 〉 on l n δ η over all the windows is related as;

τ q = lim δ η → ∞ ( Δ ln 〈 G q 〉 / Δ ln δ η ) (4)

The multifractal spectrum f ( α q ) is related to the mass exponents t_{q} and calculated from Legendre transform as:

α q = d τ q / d q , (5)

f ( α q ) = q α q − τ q . (6)

The properties of f ( α q ) for multifractal behaviour are defined [

d f ( α q ) d α q = q , d 2 f ( α q ) d α q 2 < 0 (7)

This downward concave form of curve of f ( α q ) has the following characteristics for multifractals;

1) f ( α q ) is downward concave

2) f ( α q = α 0 ) is maximal

3) f ( α q ) < f ( α 0 ) , for q ≠ 0

The behaviour of f ( α q ) shows that f ( α q ) = α q = 1 for all q as a special case if there are absolutely no fluctuations. The width of f ( α q ) is a measure of the size of the fluctuations and the value f ( α 0 ) < 1 is a measure of the number of empty bins.

The generalized dimensions may be defined as;

D q = τ q q − 1 (8)

Different types of dimensions are given as under:

1) D 0 = f ( α 0 ) ; Capacity dimension: This shows how the data points of multifractal pattern fill the phase space domain

2) D 1 = f ( α 1 ) = α 1 ; Entropy dimension: This is a measure of order-disorder of the data points in the phase space domain under study. Larger the value higher the disorder.

3) D 2 = 2 α 2 − f ( α 2 ) ; Correlation dimension: This quantifies the degree of clustering. Larger the value corresponds to higher level clustering.

The data analyzed in the present paper were collected using two emulsion stacks exposed to Oxygen beams at 60A and 200A GeV at CERN, SPS (EMU01 Collaboration) [

Two data samples of 391 and 212 interactions of ^{16}O with AgBr at 60A and 200A GeV respectively having N s ≥ 10 were used for analysis using standard emulsion criteria [^{16}O-AgBr interaction are plotted in

The variation of 〈 G q 〉 as a function of 1 / δ η for ^{16}O-AgBr at 60A and 200A GeV are plotted in _{q} moments.

From the figures, it may be noted that the moments with negative q values

saturate as δ η decreases whereas for positive q values, this shows linearity over a wide range of δ η . This saturation could be due to decrease in number of particles as bin size is reduced. The variation of l n 〈 G q 〉 as a function of − l n δ η also plotted in

The mass exponents ( τ q ) have been calculated for the linear region of plots l n 〈 G q 〉 verses − l n δ η and plotted as a function of q in

The generalized dimensions, D_{q}, have been calculated using Equation (8) and plotted as a function of q in _{q} decreases with increasing q, which shows the multifractal behaviour in multipartcle production. It may also be mentioned that for positive q values D_{q} increases with increasing beam energy for same projectile, whereas for negative q values it seems to be independent of the projectile beam energy.

The multifractal spectrum, f ( α q ) , can also give an idea about the presence of multifractal behaviour in multipartcle production. We have calculated the multifractal spectrum f ( α q ) and plotted in

These observations are in accordance with those reported earlier [^{16}O-AgBr collisions. It is also observed from

increases with increasing beam energy for the same projectile.

On the basis of the results discussed in the present paper, it is observed that, the moments having positive q values saturate with decrease in the bin size. Further, the variation of mass exponent, τ q , with the order of the moment, q, shows a power law behavior. As far as multifractality is concerned, a decreasing trend in the variation of multifractal dimension, D q , with q is observed. This indicates the presence of multifractality in the particle production process for the considered nuclear reactions. The observed behavior of multifractal spectral function f ( α q ) in pseudorapidity space manifests self similarity in the mechanism of multiparticle production

We are grateful to Professor Anju Bhasin, Jammu University, Jammu, INDIA for providing the experimental data used in the present analysis.

Ahmad, N., Ahmad, T., Singh, O. and Ahmad, S. (2018) A Study of Multifractal Analysis in ^{16}O-AgBr Collisions at 60A and 200A GeV. Journal of Modern Physics, 9, 1029-1036. https://doi.org/10.4236/jmp.2018.95064