Comparative Multi Fractal De-trended Fluctuation Analysis of heavy ion interactions at a few GeV to a few hundred GeV

We have studied the multifractality of pion emission process in 16O-AgBr interactions at 2.1AGeV&60AGeV, 12CAgBr&24Mg-AgBr interactions at 4.5AGeV and 32S-AgBr interactions at 200AGeV using Multifractal Detrended Fluctuation Analysis (MFDFA) method which is capable of extracting the actual multifractal property filtering out the average trend of fluctuation. The analysis revels that the pseudo rapidity distribution of the shower particles is multifractal in nature for all the interactions i.e. pion production mechanism has in built multi-scale self-similarity property. We have employed MFDFA method for randomly generated events for 32S-AgBr interactions at 200 AGeV. Comparison of expt. results with those obtained from randomly generated data set reveals that the source of multifractality in our data is the presence of long range correlation. Comparing the results obtained from different interactions, it may be concluded that strength of multifractality decreases with projectile mass for same projectile energy and for a particular projectile it increases with energy. The values of ordinary Hurst exponent suggest that there is long range correlation present in our data for all the interactions.


Introduction
The study of correlation and fractality is an active area of research in many fields including heavy ion collisions [1-provide information about the characteristics of system evolution. Moreover, multifractal analysis is effective for understanding the underlying dynamics of any complex system such as pionisation in high energy nucleus-nucleus interactions. To get both qualitative and quantitative idea concerning the multiparticle production mechanism [7] multifractal analysis is expected to be very fruitful. Such a behavior has been observed for vast majority of high energy multiparticle production experiments [8][9][10].
The investigation of fractal dimension in hadronic multiparticle production was carried out probably first time by Carruthers and Minh [11]. But there was no formalism developed for a systematic fractal study. A systematic approach for the fractal study was suggested by Hwa (G q moment) [12]. But that G q moments are found to be influenced by statistical fluctuations especially for the low multiplicity events. Later in order to avoid large statistical fluctuations and exclude low multiplicity events Hwa and Pan [13] proposed a modified G q moment method introducing a step function, which acts as a filter to the low multiplicity events [14].Also this modified G q moments suffer from the demerit that they are defined only for positive orders (q) and hence it is unable to explore the whole multifractal spectrum. Afterwards a number of techniques [15][16][17][18] were developed for the fractal study of multiparticle data. The techniques developed by Hwa [12] and Takagi [15] are the most popular and have been used in many cases [2,[19][20][21] to analyze multi-pion production process. However, none of these methods can disentangle the dynamical "signal" from the "background". Trending behaviors usually give rise to spurious multifractal effects for the analyzed series. Therefore it is essential to study the intrinsic fluctuations characterizing the dynamical process after filtering out the average trending behavior.
Sophisticated methods have been invented to characterize the actual fluctuations extracted from the average behavior, and the fractal nature of non-stationary time series. These include-detrended fluctuation analysis (DFA) and its variance [22,23], the wavelet transform [24,25] based multi-resolution analysis [26,27], multifractal detrended fluctuation analysis (MFDFA) [28] etc.. DFA technique [22] was developed in order to determine minutely the presence of any long range correlation [22,29] in a non-stationary series. However, despite a multitude of real-data analyses, a proper detection of the multifractality in the experimental data still presents much difficulty and is not always reliable [30].MFDFA technique [28] is actually a generalization of standard DFA technique for the characterization of multifractal nature of a series. One main reason to employ MFDFA method is to avoid fallacious detection of correlations leading to multi-fractality which are artifacts evolving due to the non-stationarity of the signal. Thus MFDFA is a powerful technique which has been applied successfully to characterize fluctuation in a variety of fields like finance [31][32][33][34],medicine [35,36], natural science [37,38],solid state physics [39,40] etc..
The spectrum of references of application of MFDFA technique is not a complete one. Recently, MFDFA method has been applied to analyze the pseudo rapidity and azimuthal angle distribution of the pions produced in Au + Au interactions at 200 GeV/nucleon by Zhang et al. [41]. They studied a sample of only 10 events. Another group, Wang et al. [42], studied the same interactions for UrQMD generated data using the same method. The DFA and MF-DFA methods are also used by Mali et al. [43] to characterize the particle density fluctuation for 28 Si + Ag/Br interactions at 14.5 GeV/nucleon and 32 S + Ag/Br interactions at 200GeV/nucleon. These analyses [41][42][43] suggest that the MF-DFA approach is a reasonably good technique for the multifractal analysis of multiparticle production process in high-energy nucleus-nucleus (A-A) interactions and hence should be applied for understanding the dynamics of the process.
In this paper we have studied the pseudo rapidity distribution of the pions produced in 16  Two observers scanned each plate independently so that the biases in detection, counting and measurement could be minimized and consequently the scanning efficiency could be increased. After finding a primary interaction induced by the incoming projectile, the number of secondary tracks in an event belonging to each category was counted using oil immersion objectives. Measurements were carried out with the help of an oil-immersion objective of 100× magnification. The measuring system fitted with both the microscopes has 1μm resolution along the X-and Y-axes and 0.5μm resolution along the Z-axis.
Events were chosen according to the criteria given below. According to nuclear emulsion terminology [48], particles emitted after interactions can be classified as the shower, gray and black particles. Shower particles are mostly (about more than 90%) due to pions with a small admixture of K-mesons and hyperons having ionization ≤ 1.4 and velocity greater than 0.7c where c.I 0 is the minimum ionization of a singly charged particle produced in the emulsion medium. Grey particles are mainly fast target recoil protons with energies up to 400 MeV. They have ionization 1.4I 0 I 10I 0 . They have velocities lying between 0.3c and 0.7c.Black particles consist of both singly and multiply charged fragments of the target nucleus with ionization and velocity less than 0.3c. Along with the above stated three kinds of particle, there could also be a few projectile fragments. These projectile fragments are the spectator parts of the incident projectile nuclei that do not directly participate in an interaction.
In the experiments with the nuclear emulsion track detectors, interactions may be with three different type of targets e.g., hydrogen (H), light nuclei (CNO) and heavy nuclei (AgBr) present in the emulsion medium. Events,with , occur because of the collision between hydrogen and the projectile beam. Events with are due to collisions of projectile with light nuclei and events with are due to collisions with heavy nuclei.
Here , the number of heavy tracks, is the total number of black and gray tracks. In our study, only events having a number of heavy tracks greater than 8 ( ) have been selected to exclude the H and CNO events.
According to the selection procedure mentioned above, we have chosen 730 events of 16 O-AgBrinteractions at 2.1AGeV [44], 800 events of 12 C-AgBr [49] and 24 Mg-AgBr [50] interactions at 4.5AGeV, 250 events of 16 O-AgBr interactions at 60AGeV [51] and 140 events of 32 S-AgBr interactions at 200AGeV [52]. The present analysis has been performed on the pion tracks only. The emission angle (θ) was measured for each pion track with respect to the beam direction by taking readings of the coordinates of the interaction point (X 0 , Y 0 , Z 0 ), coordinates (X 1 , Y 1 , Z 1 ) of a point at some distance away from the interaction point on each secondary track and coordinates (X i , Y i , Z i ) of a point on the incident beam. From 'θ' the pseudo-rapidity variable ( ), which may be treated as a convenient substitute of the rapidity variable of a particle when the rest mass of the particle can be neglected in comparison to its energy or momentum, was calculated for each pion track.

MFDFA Method
The MFDFA technique was developed by Kantelhardt et al. [28] as a generalization of standard DFA method to analyze non-stationary time series. If {x k , k=1, 2, 3…, N} be the signal of length N, MFDFA consists of the following steps [28] among which the first three steps includes the ordinary DFA technique.
Calculation of the signal profile, which is the cumulative sum of the signal to be analyzed, according to the equation (1) , where j runs from 1 to N …………(1) Step 2. Division of the profile into numbers of non-overlapping segments of length s. Since N is not an integer multiple of s, a small part of the signal will be left at the end. In order to include that part the same process is repeated starting from the other end. Thus one obtains altogether number of segments.
Step 3. Determination of the local trend associated with each of such segments by a least square polynomial fit of the series in a particular segment. And calculation of the variance of the series. The variance of the series relative to the local trend in a given segment ν of length s can be calculated as …………. (2) where is the fitting polynomial of order l for the ν th segment.
To investigate the scaling behavior one has to calculate for various s values. It should be remembered that the sample size of the smallest segment (or scale) should be much larger than the polynomial order l in order to prevent an over fitted trend.
Scaling behavior and multifractality: Presence of any long range correlation in the system will manifest in the multifractal behavior of the system and The multifractal exponent can be derived from [28,53] using equation (6) ………… (5) The singularity spectrum can be obtained from through a Legendre transform [18,53,54]: where ………… (6) The width of the multifractal spectrum is the difference between the maximum and minimum value of i.e. ………… (7) It has been proposed by Y. Ashkenazy et al. [55] and Shimizu et al. [56] that the width of a multifractal spectrum is a measure of the degree of multifractality. Broader the spectrum richer the multifractality [57].
Though originally MFDFA technique was developed for non-stationary time series analysis but it can also be used equally for non-uniform distribution like rapidity distribution of high energy multiparticle production process [41][42][43]. In principle multifractality of a natural system may originate either (a) due to broad probability density function (non-Gaussian distribution) of the concerned parameter, or (b) due to the simultaneous presence of dissimilar characteristics of long range correlations for large as well as small fluctuations, or sometimes (c) due to both of reason (a) and (b) [58]. MFDFA technique allows one to identify the source of multifractality when it is applied to the randomly shuffled distribution of the same. For shuffled distribution though the probability distribution function remains unchanged but all possible origin of correlation is wiped out from the distribution. Thus if multifractality is originated from type b) then randomly generated distribution will show an arbitrary behavior with h(q) random =0.5. On the other hand if the source of multifractality is type a) then for the generated distribution generalized Hurst exponent will be identical to that of the original one (h(q) random = h(q) experimental ). Furthermore for the reason c) the generated series will show a weaker multifractality than that of the experimental one.

Results and Discussions
In the current analysis we have focused on 16  We have also calculated for different q values according to equation (5). The plot of against q is shown in figure 3. The non-linear variation of with q also confirms the fact that the pion density fluctuation in the rapidity space is multifractal in nature i.e. pion production mechanism has an inbuilt multi-scale self-similarity property.
In order to shed light on the possible origin of the demonstrated multifractal behavior we have randomly shuffled the rapidity distribution of each selected event for all interactions and have employed MFDFA method to the shuffled data using the same approach as before. Results found to be identical for all the interactions. Here we present for 32 S-AgBr interactions only.  To compare the strength of multifractality we have calculated the width of the multifractal spectrum for each of the interactions. These values have been presented in Table 2. The spectrum width is found to vary for different interactions suggesting that the strength of multifractality depends upon the mass and energy of the projectile beam.
The spectrum width values reveal the following notable features of the heavy ion interaction process  Comparison of 16   Finally if we go from relativistic energy regime to the ultra relativistic one, influence of energy is much more prominent than mass. This is noticed comparing 16 O-AgBr at 2.1 AGeV, 24 Mg-AgBr at 4.5 AGeV, 12 C-AgBrat 4.5 AGeV interactions with the 32 S-AgBr interactions at 200 AGeV.

Conclusions
We have presented a systematic study on pion density fluctuation in pseudo rapidity spectra in the framework of sophisticated MFDFA technique using various heavy ion projectiles covering a wide range of energy starting from a few GeV to a few hundred GeV. The main observations of the analysis may be summarized as follows:  Both the h(q) and τ(q) spectra speaks in favor of multifractal pion density fluctuation in pseudo rapidity space for all the considered A-A interactions.
 Multifractal parameter h q and  q and the multifractal spectrum for the shuffled data clearly demonstrate that the observed multifractality of the experimental data does not originate from the trivial broad probability density distribution but occurs due to dynamics.
 The study suggests multifractal pion production dynamics in heavy ion interactions.
 The strength of multifractality is influenced both by mass and energy of the projectile beam.
 For the same projectile beam multifractality increases with projectile energy. On the other hand it decreases with mass of the projectile beams having same energy.
 When mass and energy both changes, effect of mass is dominant if their energy does not differ much. But multifractality is much more influenced by energy of the projectile beam if mass difference is small.
 From the value of ordinary Hurst exponent we can conclude that long range correlation is present among the pions for all of the considered heavy ion interactions irrespective of projectile mass and energy.