A Study of Multifractal Analysis in 16O-AgBr Collisions at 60A and 200A GeV

A multifractal analysis to study the multiparticle dynamics in 60A and 200A GeV, O-AgBr collisions has been performed in the pseudorapidity phase space. Multifractal moments, Gq, 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, Dq, and the multifractal spectral function, ( ) q f α , with order of the moments, q, have been studied thoroughly. Dq 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.


Introduction
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 [1] [2] which is known as intermittency. The concept of self-similarity is closely related to the fractal theory which is a natural consequence of the cascading mechanism prevailing in the multiparticle production process. The power law behaviour, scaled factorial moments, indeed implies the existence of some kind of fractal pattern [2] in the dynamics of particles produced in their final state of reaction.
In order to investigate the cascading mechanism [3] of multiparticle production in the framework of the multifractal technique, a formalism has been de-veloped by R.C. Hwa [4] [5] for a systematic study of the fractal properties and it also provides an effective means of defining non-uniform rapidity distribution of produced particles in nuclear collisions. The aim of this technique is to verify the authenticity of basic scaling properties, which occur in multifractal theories, when applied to particle production for the dynamical mechanism responsible for the hadronisation process in nucleus-nucleus (AA) collisions [6] [7] [8]. The explanation of cascading as self-similar process in analogy with geometrical objects such as fractals has been allowed by scaling law [9].
It is worth-mentioning that the issues of possible signals of Quark Gluon Plasma (QGP) formation in relativistic heavy-ion collisions are still on discus- Some workers [10] [11] have proposed entropy based tools which are commonly used to describe the dynamics of complex systems. In the last few decades, it has been observed that Tsallis entropy and multifractal technique are found to be useful to characterize on large scale scaling behaviour. Furthermore, the Shannon, Reyni and Tsallis entropies [12] related formalisms are used as multifractal measures. However, the concept of divergence is being employed to quantify the departure between two distributions. It is one of the most used divergences related to the Tsallis entropy.
It is also interesting to mention that A. Deppman [13] describes a system which has a fractal structure in its thermodynamical functions called as thermofractals. The author has mentioned that its thermodynamics is more naturally described by the Tsallis statistics and a relation between fractal dimension and entropy index has been established. Furthermore, according to the Deppman such complex hadronic system exhibits thermofractal structure.
Although many attempts have been made to study the fractal properties, using p µ , pp and e e + − data [14] [15], using the method of multifractal moments, G 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

Theory
In order to examine the dependence of multifractal moments, G q , on pseudorapidity, η, which is defined as; ( ) . 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; 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; where, τ q stands for mass exponents. The linear dependence of ln q G on ln δη over all the windows is related as; The multifractal spectrum ( ) q f α is related to the mass exponents τ q and calculated from Legendre transform as: The properties of ( ) q f α for multifractal behaviour are defined [4] [5] [16] [17] by; This downward concave form of curve of ( ) q f α has the following characteristics for multifractals;

Experimental Details
The data analyzed in the present paper were collected using two emulsion stacks

Results and Discussions
The variation of q G as a function of 1 δη for 16 O-AgBr at 60A and 200A GeV are plotted in Figure 2 and   saturate as δη decreases whereas for positive q values, this shows linearity over a wide range of δη . This saturation could be due to decrease in number of particles as bin size is reduced. The variation of ln q G as a function of ln δη − also plotted in Figure 4. The data have been fitted using method of least squares for ln δη − 1. The linear rise of the multifractal moments with decreasing bin size of pseudorapidity shows a power-law behaviour in experimental data which is the indication of the self-similarity in the production mechanism of investigated reactions.
The mass exponents ( q τ ) have been calculated for the linear region of plots ln q G verses ln δη − and plotted as a function of q in Figure 5. It is observed from the figure that the values of q τ increases linearly with increasing order of moments and independent of the collision energy.
The generalized dimensions, D q , have been calculated using Equation (8) and plotted as a function of q in Figure 6 for both data sets. At both the energies D 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.     increases with increasing beam energy for the same projectile.

Conclusion
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, q τ , 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, q D , 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 ( ) q f α in pseudorapidity space manifests self similarity in the mechanism of multiparticle production