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An attempt has been made to apply the wavelet methodology for the study of the results of the chaotic behavior of multiparticle production in relativistic heavy ion collisions. We reviewed the data that describes the collisions of relativistic heavy ion for the case
*η*
-space in 1-D phase space of variable. We compared the experimental data and UrQMD data using wavelet coherency. We discussed the results of the comparison.

Particle physics is the science of the fundamental structure of matter. One of the directions of particle physics is the study of high-energy nuclear matter. The interest in the study of high-energy nuclear matter has increased many folds due to the possibility of studying unstable states of nuclear matter under extreme condition of high energy density and high temperature. The study of non-statistical fluctuations in relativistic nuclear collisions has recently also attracted a great deal of attention due to the possibility of extracting important information about the mechanism of multiparticle production in such collisions [

But it is appropriate to study the field in terms of theoretical and the experimental points of view. This will help understand the relationship between theory and experiment, understand the complicated moments which may arise. Theory predicts phenomenon, which can be verified by experiments, and experiments very often provide new insight through unexpected results, which in turn lead to the improvement in theoretical description. In the present paper, an attempt has been made to find the relationship between the stability of a theoretical model and experiment to study the collisions of relativistic heavy ion. This exercise has been made to perform the study of (E-by-E) spatial fluctuations of relativistic shower particles produced in the collisions of ^{28}Si + Em at energy 14.6A GeV in 1-D phase space of

To solve this problem we use wavelets ideology [

To obtain the data, FUJI nuclear emulsion pellicles were irradiated horizontally with a beam of 28Si nuclei at 14.6A GeV at Alternating Gradient Synchrophasotron (AGS) of Brookhaven National Laboratory (BNL), New York, USA have been used. The nuclear emulsion experiment is a versatile detector for the study of nuclear reactions in high energy heavy ion collisions. It has the ability to detect and identify the secondary charged particles in the outlet channel of nuclear reactions. The method of line scanning has been adopted to scan the stacks, which was carried out carefully using Japan made NIKON (LABOPHOT and Tc-BIOPHOT) high-resolution microscopes with 8 cm movable stage using 40× objectives and 10× eyepieces by two independent observers, so that the bias in the detection, counting and measurements can be minimized. The interactions due to beam tracks making an angle < 2˚ to the mean direction and lying in emulsion at depths > 35 mm from either surface of the pellicles were included in the final statistics [

In order to perform a meaningful analysis of chaoticity, normalized cumulative variable (

where,

Various experimental efforts have established the existence of the empirical phenomenon of “intermittency” in multiparticle production using normalized scaled factorial moments. On the basis of bin averaging the normalized scaled factorial moments of the order of q is defined in vertical form as [

and its horizontal form is defined as [

where, ^{th} bin, m can take values from 1 to M and N represents the total multiplicity of charged shower particles in a particular event in the pseudo-rapidity interval

Recently, Cao and Hwa first introduced to measure the spatial pattern of particles in an event using normalized factorial moments associated with it. In contrast to the horizontally averaged vertical moments, ^{th} order, they define event factorial moments as [

where, M is the partition number in phase space, ^{th} bin and

The event factorial moments,

where,

In order to quantify the degree of the fluctuations, a new normalized moment related to the chaotic nature of the system is defined as [

where, p is any positive real number, it should not be negative,

The other relevant details may be seen in works [

Thus, we examine the relationship between,

We can see (see

The basis of formal generalization of continuous wavelet transformation on the time interval

where

In this case, feasibility of wavelet analysis in time series study is determined by the fact that the method of wavelet analysis allows to discover the local features of the studied time series due to the decomposition of the input data [

In addition, the empowerment of study time series using wavelet analysis methodology promotes the use of various procedures of wavelet transformation: even-scaled analysis, cross wavelet transformation, wavelet coherence. Then wavelet analysis methodology has been widely used in the disclosure of dynamics of time series that define various data [

One of the main wavelet transformation methods used for generalized cross-reference analysis between different time series is wavelet coherence. It allows calculating local correlation of two time series in a region of time-frequency. It uses the following formalized model: wavelet coherence as the squared absolute value of the smoothed cross wavelet spectra

where

We use Morlet wavelet that is a complex wavelet with a good time-frequency localization, as a parent one [

We consider the wavelet coherency for each pair of time series of the natural log

In Figures 2-6 you can check the results of wavelet coherence between selected time series. Each of the following figures indicated a sepa-rate group of time series that match each other. This comparison is displayed in the time-frequency plane. In this case the time scale is equal to the consistent change of values

On the vertical axis are the weighted characteristics of the analyzed data series in frequency space. Along each of the figures importance scale is presented as separate columns for reflections. Maximum reflection is speaks about full coherence between the experimental data and UrQMD data. Defined lines are a manifestation of localization for individual irregularities within studied time series according to importance of irregularities. In general, each point of wavelet reflects shown in Figures 2-6 is their value in the time-frequency space, which is calculated through wavelet transformation.

The phase difference, indicated by arrows, gives us details about delays of oscillation of the two examined time series. Arrows pointing to the right (left) when the time series are in-phase (anti-phase) or are positively (negatively) correlated. Arrow pointing up means that the first time series leads the second one, arrow pointing down indicates that the second time series leads the first one. In our case, these data suggest discrepancy of the experimental data according to the model.

We see that with increasing values q of unbalance occurs between the experimental data and UrQMD data. This is observed for all values p. With increasing the values of p and when is increased the values q imbalance between the experimental data and UrQMD data increasing. Especially it is typical for q = 4. Such

Sequence number | Value of | Sequence number | Value of |
---|---|---|---|

1 | 0.693 | 15 | 2.773 |

2 | 1.099 | 16 | 2.833 |

3 | 1.386 | 17 | 2.890 |

4 | 1.609 | 18 | 2.944 |

5 | 1.792 | 19 | 2.996 |

6 | 1.946 | 20 | 3.045 |

7 | 2.079 | 21 | 3.091 |

8 | 2.197 | 22 | 3.135 |

9 | 2.303 | 23 | 3.178 |

10 | 2.398 | 24 | 3.219 |

11 | 2.485 | 25 | 3.258 |

12 | 2.565 | 26 | 3.296 |

13 | 2.639 | 27 | 3.332 |

14 | 2.708 |

an increase is observed for the mean values

The results of the study can determine the appropriate application of wavelet analysis methodology as disclosure of wavelet coherency between the studied data series as a tool for the study collisions of relativistic heavy ion. Some comparative results between the experimental data and UrQMD data have been obtained from the analysis of event-by-event fluctuations of produced charged particles in heavy ion collisions at 14.6A GeV. We showed overall consistency between the experimental data and UrQMD data. But we also note that coherence decreases with increasing values of q. This result was obtained for the case

The authors would like to acknowledge the keen support for this work of the Department of Physics, Faculty of Science, University of Tabuk, Saudi Arabia and also the Department of Informatics, Kharkiv National University of Radio-Electronics, Kharkiv, Ukraine [

Lyashenko, V.V., Ahmad, M.A., Deineko, Z.V. and Rasool, M.H. (2017) Methodology of Wavelets in Relativistic Heavy Ion Collisions in One Dimensional Phase Space ( -Space). Journal of High Energy Physics, Gravitation and Cosmology, 3, 254-266. https://doi.org/10.4236/jhepgc.2017.32021