# Study of Di-Muon Production Process inpp Collision in CMS Data from Symmetry Scaling Perspective

Authors: Susmita Bhaduri; Anirban Bhaduri; Dipak Ghosh
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

An extensive knowledge of the dynamics of the process ofpp collision serves as input to exhaustive theoretical models of strong interaction. This knowledge is also a baseline for a system to decipher the dynamics of AA collisions at relativistic and ultrarelativistic energies. Recent availability of di-muon data has triggered a spate of interests in revisiting strong interaction process, the study of which in detail is extremely important for enhancement of our understanding of not only the theory of strong interaction but also possible physics scenarios beyond the standard model. Apart from conventional approaches to the study of the dynamics of particle production in high-energy collision the present authors proposed a new approach with successful application in context of symmetry scaling in AA collision data from (ALICE-Collaboration, 2014) in the work (Bhaduri, S. et al., 2019) and pp collision data at 8TeV from (CMS-collaboration, 2017) in the work (Bhaduri, S. et al., 2019) and also in other numerous works with different collision data. This different approach essentially analyses fluctuation pattern from the perspective of symmetry scaling or degree of self-similarity involved in the process. This was done with the help of multifractal scaling analysis and also multifractal cross-correlation analysis using the single variable of pseudorapidity values of di-muon data taken out from the primary dataset of RunA(2011) and RunB(2012) of the pp collision at 7 TeV and 8 TeV, respectively, from (CMS-collaboration, 2016, 2017). High degree of persistent long-range cross-correlations (MF-DXA) exist between pseudorapidity-value and its corresponding azimuthal-value for different rapidity ranges. The different values of scaling exponents (across rapidity ranges and energies) signify that there may be multiple processes other than those conjectured, involved in the underlying dynamics of the production process of oppositely charged di-muons resulting in different kinds of scaling. Otherwise, the scaling exponents at different degrees would have remained the same across the rapidity ranges and also for different energies.


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## Body

## 1. Introduction

In the recent past, fluctuation and correlation have been analyzed widely using novel methods of studying nonstatistical fluctuation which resulted in the better understanding of the dynamics of the pionisation process. The methods including the process of intermittency were introduced by Bialas and Peschanski [1] who have observed association between intermittency indices and anomalous fractal dimension [2, 3]. After that, the parameters of Gq moment and Tq moment [4–8] were introduced which were deduced from various methods based on fractal concepts. Then distinctive approaches of detrended fluctuation analysis (DFA) and multifractal-DFA (MF-DFA) [9, 10] were applied extensively for analyzing nonstationary, nonlinear properties of data series to investigate the long-range correlations inherent in the process of particle production [11–14]. Among various other contemporary works, self-similarity has been analyzed in the areas of particle physics which includes the production process of Jet and Top-quark in the experiments of Tevatron and LHC [15], the procedure of strangeness production in pp collisions at the RHIC [16] experiments, the phenomenon of proton spin and asymmetry inherent in jet production process [17] and the deciphering of the collective phenomena [18], and the process of establishment of the notion of self-similar symmetry of dark energy [19]. The study of long-range cross-correlation between two nonstationary signals detrended cross-correlation analysis (DXA) had been presented by Podobnik and Stanley [20]. Wang et al. [21] introduced multifractal-detrended cross-correlation analysis (MF-DXA) by combining MF-DFA and DXA methods to examine higher degree of multifractal parameters of two cross-correlated series. MF-DXA method has been applied with substantially higher degree of accuracy in the analysis of the unrevealed cross-correlation in the various fields of physics, physiology finance, and power markets [20, 21] and also in the fields of particle physics [22].The main aim of high-energy physics is to prove the existence of the creation of the QGP state, and also the study of the properties of this exotic state, by examining the final state of produced-hadrons, produced in huge numbers. Most of these final state particles may not be produced from the decay of the plasma directly, but rather they are produced or influenced by the hadronic cascade. Hence, studies of resonance states are of great importance because there is a variety of resonances, having very different lifespans, which signifies that these particles decay with differing probabilities in the hadronic stage, and therefore provide valuable information about that stage. We have performed the scaling analysis of the pseudorapidity space taken out from Pb-Pb VSD masterclass data at 2.76 TeV per nucleon pair from ALICE Collaboration [23] using both the method of complex network-based visibility graph and multifractal-DFA (MF-DFA) [9, 10], to study the prospective phase transition and the signature of QGP [24, 25]. We also studied multiplicity fluctuation process in nucleus-nucleus and hadron-nucleus interactions by applying complex network and chaos-based visibility graph methodology in quite a few recent works [22, 26–33]. These techniques have also been successfully applied to identify phase transitions in temperature-driven magnetization properties [34] and also in temperature-driven phase transition from liquid to vapour state [35]. In a recent study [36], different combinations of topological and kinematic input variables from the data of RunA(2011) of the pp collision at 7 TeV at CMS detector have been used, from which several ANNs (artificial neural networks) have been constructed, and then through comparison, the optimally configured ANN is selected .The outcomes of the assessment ofpp and pA systems should be used as a reference to validate the understanding of the processes which are expected to contribute to the dynamics of the process of di-muon production [37]. Moreover, apart from the analysis of AA collisions, an extensive knowledge of pp collisions is required both as an input to comprehensive theoretical models of strong interactions and also as a baseline to decipher the AA collisions at relativistic and ultrarelativistic energy levels. This has been of great interest in the area of theoretical and experimental physics. The process of soft particle generation from ultrarelativistic pp collisions is affected by the flavor distribution among the proton, quark hadronization, and baryon number transport. In the process of AA collisions, the magnitude of the spectrum of transverse momentum of charged particles in pp collisions serves as an important reference. A pp reference spectrum is required for AA collisions to probe for the effects of probable initial states of the collision. The multiplicity distribution of particles generated in pp collisions and the multiplicity dependence of various global event features serve as rudimentary observables which reflect the features of the underlying dynamics of the process of particle production. Therein lies the importance of analyzing the dynamics of di-muon production process in pp collisions. In this work, we have attempted DFA, MF-DFA, and MF-DXA for the scaling analysis of the rapidity and energy dependence of the di-muon production process.The rest of the paper is structured in the following manner. The objective of the study is elaborated in Section2. Section 3 describes the methods of analysis. Section 3.1 presents the algorithm of DFA and MF-DFA, and Section 3.2 presents the method of MF-DXA in detail and the importance of the parameters—the width of multifractal spectrum and the cross-correlation exponent. Section 4.1 describes the data in detail. Section 4.2 describes the details of our study and the deductions from the test results. Section 5 details the physical importance of the proposed parameters and their relevance with regards to the dynamics of the di-muon production process and finally comes the conclusion.


## 2. Goal of the Study

Usingpp collisions at a center-of-mass energy of S=7TeV, the analysis of the production process of exclusive γγ⟶μ+μ− was carried out by the ATLAS and CMS collaborations [38]. A latest review work has been reported about the complexities involved in resonance production process for different high-energy collisions like pp, pA, and AA collisions at LHC(using data from ALICE collaboration), to analyze the complexity and eventually explain the inherent dynamics of the particle production process and the properties of the generated particles for the different collision system [39]. We have elaborated in Section 1 how after few successful ventures by the present authors in the field of analyzing the pionisation process in high-energy interaction using chaos-based procedures and being motivated by the different attempts reported to investigate the dynamics of the generation process of di-muon pairs in [39], we have attempted to revisit the di-muon production process in hadron-hadron interactions. We have proposed to implement the chaos-based methods of DFA, MF-DFA, and MF-DXA to analyze the energy and rapidity dependence of di-muon production process by utilizing a single variable of pseudorapidity values of di-muon data taken out of the primary dataset of RunA(2011) and RunB(2012) of the pp collision at 7 TeV and 8 TeV, respectively, from CMS collaboration [40, 41]. The rapidity and energy dependence of the process are examined by means of fundamental scaling parameter signifying the degree of symmetry scaling or scale-freeness in the di-muon production process, extracted by the proposed method.
(1)
All these rigorous methods provide the information from the deepest level about the particle production process from the emergent di-muons produced from thepp collisions at 7 TeV and 8 TeV from CMS collaboration [40, 41].
(2)
The study reveals that pseudorapidity spaces corresponding to different range of pseudorapidity values are highly scale free and possess multifractal characteristics. They also reveal how the scaling pattern changes from one rapidity range to another and also from one range of energy level to another
(3)
For different rapidity range and at two different levels of energy, differences in the values of scaling exponent signify that there are multiple processes involved in the production process of oppositely charged di-muons which give rise to different kinds of scalingTraditionally invariant-mass/transverse-momentum methods were used to probe various resonance states in high-energy collision. Apart from theJ/Ψ peak observed in their invariant-mass spectrum, there can be existence of other upsilon states and other processes in principle contributing to the di-muon continuum due to open charm and open beauty decays. One may select different ranges of invariant mass by examining the pattern of invariant-mass spectrum of the events producing di-muons. For each such range, the pseudorapidity spaces corresponding to the produced di-muons would be extracted, and then for each of the spaces, the proposed scaling analysis may be done for different ranges of rapidity and energy. Any substantial change in scaling behavior indicated by the width of MF-DFA spectrum and the degree of multifractal cross-correlation may be attributed to the occurrence of different kinds of resonance states from which di-muons are produced in pp collisions. This may act as basic input for studying similar resonance production in pp collisions at higher energy and also for pA and AA collisions.


## 3. Method of Analysis

We have elaborated the multifractal-detrended fluctuation analysis (MF-DFA) method [9, 10, 42] used to calculate the Hurst exponent and the width of the multifractal spectrum. We have extracted these parameters for analyzing the fluctuation of data series extracted from the experimental data as elaborated in Section 4.1.


### 3.1. MF-DFA Method

(1)
Here, we denote the experimental data series asxi for i=1,2,⋯,N, where N=numberofpoints. The average of this series is computed as x¯=1/N∑i=1Nxi. Then, the collective deviation series for xi is calculated as
(1)Xi≡∑k=1ixk−x¯,i=1,2,⋯,N.
This deduction of the average (x¯) from the input data series is a conventional method of eliminating noise from the input data series. The result of this subtraction would be removed by the detrending process in the fourth step.
(2)
Xi is then divided into Ns nonoverlapping segments, with Ns≡intN/s and s as the length of the segment. In this experiment, s ranges from 16 (minimum) to 1024 (maximum) value in log scale
(3)
For eachs, a particular segment is denoted by vv=1,2,⋯,Ns. Least-square fitting is performed for each segment to derive the local trend for that specific segment [9]. xvi denotes the least-square fitted polynomial for the segment v in series Xi. xvi is computed according to the equation xvi=∑k=0mCkim−k, with Ck as the kth coefficients of the fitted polynomial of degree m. Different kinds of fitting—linear, quadratic, cubic, or higher m-order polynomial—may be used [10, 42]. In this experiment, linear least-square fitting is applied with m=1
(4)
Now, to detrend the data series, the least-square fitted polynomial is subtracted from the data series. There is existence of slow-varying trends in natural data series. In order to extract the scale invariant structure of the dissimilarity around the trend, detrending is necessary. For each value ofs and segment v∈1,2,⋯,Ns, detrending is executed by deducting the least-square fit xvi from the specific portion of the data series Xi, for the segment v to calculate the variance which is denoted by F2s,v computed as
(2)F2s,v≡1s∑i=1sXv−1s+i−xvi2,with s∈16,32,⋯,1024 and v∈1,2,⋯,Ns.
(5)
Next, theqth order function of fluctuation, denoted by Fqs, is computed by averaging the values of F2s,v over all the segments (v) produced for each s∈16,32,⋯,1024 and for a specific q, as
(3)Fqs≡1Ns∑v=1NsF2s,vq/21/q.
Here,q≠0 as in that case 1/q would blow up. In this experiment q varies from (−5) to (+5). For q=2, computation of Fqs would sum up to conventional method of detrended fluctuation analysis (DFA) [9].
(6)
The above steps are repeated for various values ofs∈16,32,⋯,1024, and it is observed that for a particular q, Fqs rises in value with increasing s. If the data series is long-range power correlated, then Fqs vs s for a specific q will display power-law behavior
(4)Fqs∝shq.
If this type of scaling exists, thenlog2Fqs depends on log2s in a linear fashion, where hq is the slope which is dependent on q. h2 is alike to the so-called Hurst exponent [42]. So hq is defined as the generalized Hurst exponent.
(7)
The scaling pattern of the varianceF2s,v is the same for all segments in case of a monofractal series. In other words, the averaging of F2s,v would show uniform scaling behavior for various values of q, and hence, hq becomes independent of q for monofractals.
But if large and small fluctuations in the series have varying scaling pattern, thenhq becomes substantially dependent on q. In these cases, for positive values of q, hq delineates the scaling pattern of the segments with large fluctuations, and for negative values of q, hq describes scaling behavior of the segments with smaller fluctuations. The generalized Hurst exponent hq for a multifractal data series is associated with the classical multifractal scaling exponent τq according to
(5)τq=qhq−1.
(8)
As multifractal series have numerous Hurst exponents, soτq depends nonlinearly upon q [43]. The singularity spectrum, here denoted by fα, is associated with hq as
(6)α=hq+qh′q,fα=qα−hq+1.
Here, the singularity strength is denoted byα, and fα describes the dimension of the subset series denoted by α. Different values of fα for different α results into multifractal spectrum of fα which is an arc, and for this spectrum, the gap between the maximum and minimum values of α is the width of the multifractal spectrum or the measurement of the multifractality of the input data series.
(9)
Forq=2, if hq or h2=0.5, no correlation exists in the data series. There is persistent long-range cross-correlations in the data series, which means a large value in the series is presumably to be followed by another large value in the series, if h2>0.5, whereas for h2<0.5, there would be antipersistent long-range correlations which implies that a large value would probably be followed by a small value in the series and vice versa.


### 3.2. MF-DXA Method

Wang et al. [21] have introduced MF-DXA method based on the MF-DFA method [10, 42] and analyzed the cross-correlation between two nonstationary series quantitatively. The broad steps for the MF-DXA method are as follows.
(1)
Letxi and yi be two data series for i=1,2,⋯,N, of length N. The mean of these series is calculated as x¯=1/N∑i=1Nxi and y¯=1/N∑i=1Nyi, respectively. Then, accumulated deviation series for xi and yi are calculated as per equation (1) and denoted by Xi and Yi, respectively. Both Xi and Yi are divided into Ns nonoverlapping segments, where Ns=intN/s, s is the length of the segment. In our experiment, s varies from a minimum of 16 to a maximum of 512 value in log scale
(2)
For eachs, we denote a particular segment by vv=1,2,⋯,Ns. Here, xvi and yvi denote the least-square fitted polynomials for the segment v in Xi and Yi, respectively. xvi and yvi are calculated as per the equations xvi=∑k=0mCxkim−k and yvi=∑k=0mCykim−k, where Cxk and Cyk are the kth coefficients of the fit polynomials with degree m. For this experiment, m is taken as 1 [21].
For eachs and segment v, v=1,2,⋯,Ns, detrending is done by subtracting the least-square fits xvi and yvi from the part of the data series Xi and Yi, respectively, for the segment v. The covariance of the these residuals, denoted by fxy2s,v for a particular s and v, is then calculated as follows.
(7)fxy2s,v=1s∑i=1sXv−1s+i−xvi×Yv−1s+i−yvi,for each segment v, v=1,2,⋯,Ns.
(3)
Then, theqth order detrended covariance, denoted by Fxyq,s, is calculated by averaging fxy2s,v over all the segments (v) generated for a particular s and q, as per the equation below [10, 21, 42].
(8)Fxyq,s=1Ns∑v=1Nsfxy2s,vq/21/q.
Here,q≠0 because in that case, 1/q would blow up.
(4)
The above process is repeated for different values ofs∈16,32,⋯,512, and it can be seen that for a specific q, Fxyq,s increases with increasing s. If the series are long-range power correlated, the relation between Fxyq,s versus s for a particular q will show power-law behavior as below [21].
(9)Fxyq,s∝shxyq.
If this kind of scaling exists,log2Fxyq,s would depend linearly on log2s, where hxyq is the slope and represents the degree of the cross-correlation between the data series xi and yi.
In general,hxyq depends on q. q ranges from negative to positive values. For q=2, the method is referred as the so-called method of DXA [21].
(5)
As confirmed from several experiments done by Wang et al. [21], if hxyq=0.5, there is no cross-correlation. Further, if hxyq>0.5, there are persistent long-range cross-correlations, where the large value of one variable, which is in this study the η values, is likely to be followed by a large value of another variable, which is the corresponding ϕ values, in the series, whereas in case of hxyq<0.5, there are antipersistent long-range cross-correlations, where a large value of one variable is most likely to be followed by a small value and vice versa in the series
(6)
hxyq for q=2, i.e., hxy2, is the DXA exponent. As per Podobnik and Stanley, the cross-correlation exponent between two nonstationary series, denoted by γi, is calculated as per the equation γi=2−2hxy2 [20]. For uncorrelated data series, γi=1, the lower the value of γi, the more correlated the data series are


## 3.1. MF-DFA Method

(1)
Here, we denote the experimental data series asxi for i=1,2,⋯,N, where N=numberofpoints. The average of this series is computed as x¯=1/N∑i=1Nxi. Then, the collective deviation series for xi is calculated as
(1)Xi≡∑k=1ixk−x¯,i=1,2,⋯,N.
This deduction of the average (x¯) from the input data series is a conventional method of eliminating noise from the input data series. The result of this subtraction would be removed by the detrending process in the fourth step.
(2)
Xi is then divided into Ns nonoverlapping segments, with Ns≡intN/s and s as the length of the segment. In this experiment, s ranges from 16 (minimum) to 1024 (maximum) value in log scale
(3)
For eachs, a particular segment is denoted by vv=1,2,⋯,Ns. Least-square fitting is performed for each segment to derive the local trend for that specific segment [9]. xvi denotes the least-square fitted polynomial for the segment v in series Xi. xvi is computed according to the equation xvi=∑k=0mCkim−k, with Ck as the kth coefficients of the fitted polynomial of degree m. Different kinds of fitting—linear, quadratic, cubic, or higher m-order polynomial—may be used [10, 42]. In this experiment, linear least-square fitting is applied with m=1
(4)
Now, to detrend the data series, the least-square fitted polynomial is subtracted from the data series. There is existence of slow-varying trends in natural data series. In order to extract the scale invariant structure of the dissimilarity around the trend, detrending is necessary. For each value ofs and segment v∈1,2,⋯,Ns, detrending is executed by deducting the least-square fit xvi from the specific portion of the data series Xi, for the segment v to calculate the variance which is denoted by F2s,v computed as
(2)F2s,v≡1s∑i=1sXv−1s+i−xvi2,with s∈16,32,⋯,1024 and v∈1,2,⋯,Ns.
(5)
Next, theqth order function of fluctuation, denoted by Fqs, is computed by averaging the values of F2s,v over all the segments (v) produced for each s∈16,32,⋯,1024 and for a specific q, as
(3)Fqs≡1Ns∑v=1NsF2s,vq/21/q.
Here,q≠0 as in that case 1/q would blow up. In this experiment q varies from (−5) to (+5). For q=2, computation of Fqs would sum up to conventional method of detrended fluctuation analysis (DFA) [9].
(6)
The above steps are repeated for various values ofs∈16,32,⋯,1024, and it is observed that for a particular q, Fqs rises in value with increasing s. If the data series is long-range power correlated, then Fqs vs s for a specific q will display power-law behavior
(4)Fqs∝shq.
If this type of scaling exists, thenlog2Fqs depends on log2s in a linear fashion, where hq is the slope which is dependent on q. h2 is alike to the so-called Hurst exponent [42]. So hq is defined as the generalized Hurst exponent.
(7)
The scaling pattern of the varianceF2s,v is the same for all segments in case of a monofractal series. In other words, the averaging of F2s,v would show uniform scaling behavior for various values of q, and hence, hq becomes independent of q for monofractals.
But if large and small fluctuations in the series have varying scaling pattern, thenhq becomes substantially dependent on q. In these cases, for positive values of q, hq delineates the scaling pattern of the segments with large fluctuations, and for negative values of q, hq describes scaling behavior of the segments with smaller fluctuations. The generalized Hurst exponent hq for a multifractal data series is associated with the classical multifractal scaling exponent τq according to
(5)τq=qhq−1.
(8)
As multifractal series have numerous Hurst exponents, soτq depends nonlinearly upon q [43]. The singularity spectrum, here denoted by fα, is associated with hq as
(6)α=hq+qh′q,fα=qα−hq+1.
Here, the singularity strength is denoted byα, and fα describes the dimension of the subset series denoted by α. Different values of fα for different α results into multifractal spectrum of fα which is an arc, and for this spectrum, the gap between the maximum and minimum values of α is the width of the multifractal spectrum or the measurement of the multifractality of the input data series.
(9)
Forq=2, if hq or h2=0.5, no correlation exists in the data series. There is persistent long-range cross-correlations in the data series, which means a large value in the series is presumably to be followed by another large value in the series, if h2>0.5, whereas for h2<0.5, there would be antipersistent long-range correlations which implies that a large value would probably be followed by a small value in the series and vice versa.


## 3.2. MF-DXA Method

Wang et al. [21] have introduced MF-DXA method based on the MF-DFA method [10, 42] and analyzed the cross-correlation between two nonstationary series quantitatively. The broad steps for the MF-DXA method are as follows.
(1)
Letxi and yi be two data series for i=1,2,⋯,N, of length N. The mean of these series is calculated as x¯=1/N∑i=1Nxi and y¯=1/N∑i=1Nyi, respectively. Then, accumulated deviation series for xi and yi are calculated as per equation (1) and denoted by Xi and Yi, respectively. Both Xi and Yi are divided into Ns nonoverlapping segments, where Ns=intN/s, s is the length of the segment. In our experiment, s varies from a minimum of 16 to a maximum of 512 value in log scale
(2)
For eachs, we denote a particular segment by vv=1,2,⋯,Ns. Here, xvi and yvi denote the least-square fitted polynomials for the segment v in Xi and Yi, respectively. xvi and yvi are calculated as per the equations xvi=∑k=0mCxkim−k and yvi=∑k=0mCykim−k, where Cxk and Cyk are the kth coefficients of the fit polynomials with degree m. For this experiment, m is taken as 1 [21].
For eachs and segment v, v=1,2,⋯,Ns, detrending is done by subtracting the least-square fits xvi and yvi from the part of the data series Xi and Yi, respectively, for the segment v. The covariance of the these residuals, denoted by fxy2s,v for a particular s and v, is then calculated as follows.
(7)fxy2s,v=1s∑i=1sXv−1s+i−xvi×Yv−1s+i−yvi,for each segment v, v=1,2,⋯,Ns.
(3)
Then, theqth order detrended covariance, denoted by Fxyq,s, is calculated by averaging fxy2s,v over all the segments (v) generated for a particular s and q, as per the equation below [10, 21, 42].
(8)Fxyq,s=1Ns∑v=1Nsfxy2s,vq/21/q.
Here,q≠0 because in that case, 1/q would blow up.
(4)
The above process is repeated for different values ofs∈16,32,⋯,512, and it can be seen that for a specific q, Fxyq,s increases with increasing s. If the series are long-range power correlated, the relation between Fxyq,s versus s for a particular q will show power-law behavior as below [21].
(9)Fxyq,s∝shxyq.
If this kind of scaling exists,log2Fxyq,s would depend linearly on log2s, where hxyq is the slope and represents the degree of the cross-correlation between the data series xi and yi.
In general,hxyq depends on q. q ranges from negative to positive values. For q=2, the method is referred as the so-called method of DXA [21].
(5)
As confirmed from several experiments done by Wang et al. [21], if hxyq=0.5, there is no cross-correlation. Further, if hxyq>0.5, there are persistent long-range cross-correlations, where the large value of one variable, which is in this study the η values, is likely to be followed by a large value of another variable, which is the corresponding ϕ values, in the series, whereas in case of hxyq<0.5, there are antipersistent long-range cross-correlations, where a large value of one variable is most likely to be followed by a small value and vice versa in the series
(6)
hxyq for q=2, i.e., hxy2, is the DXA exponent. As per Podobnik and Stanley, the cross-correlation exponent between two nonstationary series, denoted by γi, is calculated as per the equation γi=2−2hxy2 [20]. For uncorrelated data series, γi=1, the lower the value of γi, the more correlated the data series are


## 4. Experimental Details

The datasets for the proposed analysis are taken out from two publicly available experimental primary datasets from CMS collaboration. The details of the data is given in Section4.1, and the complete method of the experiment is explained in step by step in Section 4.2.


### 4.1. Data Description

The primary dataset of thepp collision at 8 TeV in AOD format from RunB of 2012 [40] and another dataset of pp collision at 7 TeV in the same AOD format from RunA of 2011 [41] of the CMS collaboration are taken as the source datasets for this experiment. The run numbers which are selected and qualified by CMS to be processed along with the appropriate parameters for generation of the collision datasets are provided in the links—link1 and link2 for 8 TeV and 7 TeV, respectively. These datasets are made available for experiment. We have extracted the pseudorapidity-η space and corresponding azimuthal-ϕ space for the generated di-muons from these runs qualified by CMS from the primary datasets in the following formats—text (.txt) and .root format. In this analysis, we have utilized these pseudorapidity space and the corresponding azimuthal space from the text (.txt) file.


### 4.2. Data Analysis and Results

(1)
The pseudorapidity-η space for each of the datasets for 8 and 7 TeV extracted from the primary datasets of the CMS collaboration as described in Section 4.1 is divided into the following 5 ranges of η values
(a)
−2.5 to −1.5
(b)
−1.5 to −0.5
(c)
−0.5 to 0.5
(d)
0.5 to 1.5
(e)
1.5 to 2.5
For all the 5 ranges, theη values are extracted from the full-phase space of the two source datasets and mapped to data series. The data series is plotted with the X-axis denoting the sequence number of η values and the Y corresponds to the η values corresponding to the sequence number as in the X-axis.
For each of these data series, the following values are calculated.(i)
The width of the multifractal spectrum
(ii)
Degree of cross-correlation between theη space and their corresponding ϕ space
(2)
For each of the 10 datasets (5 for 8 TeV and 5 for 7 TeV datasets) created for the 5 ranges of pseudorapidity values, as specified in Step 1, the multifractal analysis is done and the width of multifractal spectrum is calculated as per the method elaborated in Section3.1
Theqth order detrended variance Fqs is calculated as per equation (3) in the Step 5 of the MF-DFA methodology as described in Section 3.1. Figures 1(a) and 1(b) show the log2Fqs vs log2ss trend for q=−5,0,5, extracted for a particular range of η values for 8 and 7 TeV datasets, respectively.
Their linear trend confirms the power-law behavior ofFqs versus s for all the values of q. Similar calculation is done for all the η ranges for both 8 and 7 TeV datasets, and similar trend is observed.
(3)
For each of theη-data series corresponding to the ranges specified in Step 1, a randomized version of data is produced and widths of the multifractal spectrum are calculated as per the same methodology elaborated in Section 3.1. The calculated values of the parameters are compared to those for the experimental data. In Figures 2(a) and 2(b), the widths of the multifractal spectrum of the original datasets and their randomized versions calculated for one of the ranges η values are shown for 8 and 7 TeV datasets, respectively. The below points must be noted for the shape and widths of the multifractal spectrum of the original datasets and their randomized versions
(i)
The shape of the multifractal spectrum does not necessarily have to be symmetric. The spectrum might have either a right or a left truncation that arises from the consistent/(almost constant) trend of Hurst exponents for positive or negative values ofqs. The consistent trend of qth order Hurst exponent signifies that the qth order RMS (calculated as per the equations in Step 5 and Step 6 of Section 3.1) is not much sensitive to the extent of the local fluctuations
(ii)
The width and trend of multifractal spectrum between the original and the randomized version of the particularη-space for 8 TeV dataset shown in Figure 2(a) shows that the spectrum for the randomized version has a long right tail which signifies that the series have a multifractal structure not much affected by the local fluctuations with large magnitudes
(iii)
Whereas the multifractal spectrum calculated for the original and the randomized version of theη-space for the same range of η values for 7 TeV dataset shown in Figure 2(b) shows that the spectrum for the randomized version has a long left tail, which means that the randomized series is not much sensitive to the local fluctuations with small magnitudes.
(iv)
If the source data is long-range correlated, that is eradicated by the randomization process, and the data should start to become uncorrelated. This makes the multifractal spectrum or the scaling pattern of randomized series insensitive to the local fluctuations with large or small magnitudes, which is not the case with the actual experimental data. Hence, it results in the different widths of multifractal spectrum calculated the randomized version from those for the original version. Moreover, for two completely different experimental datasets the randomized data may be different with regards to peak, shape, and trend. The width of the multifractal spectrum has normally been less for the randomized data than the one for the experimental data
(v)
The main conclusion comes from the fact that the values of the width of the multifractal spectrum calculated experimental data being significantly different from shuffled ensembles essentially confirms that thedegree of scale-freeness is indicative of the dynamics inherent in the di-muon production process, and this degree of scale-freeness may be indicative of different processed responsible for di-muon production even beyond standard model (SM).
(vi)
It is evident from the figures that the widths of the multifractal spectrum of the original datasets and their randomized versions differ substantially for both energies. Similar trend is observed from the comparison of the original and the randomized version of the 5 ranges ofη values for both 8 and 7 TeV datasets
The comparison of the widths of the multifractal spectrum generated for theη spaces for all the 5 ranges of η values for 7 and 8 TeV datasets with respect to their rapidity and energy dependence is shown in Figure 3. It should be noted that:
(i)
The comparison of the width of the multifractal spectrum offα, denoted by the difference between the maximum and minimum values of α, between the original and the randomized version of the η space for both energy ranges confirm the mutifractality of the original η spaces
(ii)
For the 2nd, 3rd, and 5th range of η values, the widths of multifractal spectrum is substantially different between the energy ranges
(iii)
For both 7 and 8 TeV the 1st and 4th range of η-space display minimum or no difference with respect to multifractality
(iv)
The degree of multifractality is found to be the least for 2nd and 3rd range for 8 and 7 TeV data, respectively
Table1 details the widths of the multifractal spectrum of the original datasets and their randomized versions for all the 10 datasets (5 for 8 and 5 for 7 TeV) corresponding to the η values. The values of the width of MF-DFA spectrum, essentially is an indicator of inherent symmetry and scale-freeness (different at different energy and rapidity ranges) with which the produced di-muons create the signatures in terms of η values. The values of these parameters in experimental data being significantly different from shuffled ensembles confirm that this inherent symmetry and degree of scale-freeness is never the outcome of randomization process but is indicative of the dynamics involved in the di-muon production process.
(4)
For each of the 10 datasets (5 for 8 and 5 for 7 TeV) ofη values extracted for the ranges specified in Step 1, the corresponding azimuthal-ϕ space is also extracted. The 10 corresponding ϕ space is sorted in the ascending order and then mapped into data series. They in turn are mapped into a two-dimensional space with their sequence along the X-axis and the corresponding values of ϕ along the Y-axis
(5)
For the 10ϕ spaces (5 for 8 and 5 for 7 TeV), the qth order detrended variance Fqs is analyzed as per equation (3) in Step 5 of the MF-DFA methodology as described in Section 3.1. Figures 4(a) and 4(b) show the log2Fqs vs log2ss trend for q=−5,0,5, extracted for corresponding ϕ values for the same range of η values for which the same trend is shown in Figures 1(a) and 1(b) for 8 and 7 TeV datasets, respectively
It is to be noted that the linear trend confirms the power-law behavior ofFqs versus s for all the values of q for the ϕ spaces. The same analysis is done for all the ϕ spaces corresponding to the η ranges for both 8 and 7 TeV datasets, and a similar trend is observed.
(6)
Multifractal cross-correlation analysis is done as per the method described in Section3.2 between the 10 pairs of datasets (5 for 8 TeV and 5 for 7 TeV), one being the sorted ϕ values and the other being the corresponding η values. The trend of generalized Hurst exponent hq for different order (q) is analyzed for all the 10 pairs of η and ϕ datasets as per the process described in Section 3.1. Along with that, for the same pairs of datasets the degree of cross-correlationhx,yq for different order (q) is analyzed as per the methodology described in Section 3.2. The trend of hq and hx,yq versus q for the particular sample pair of η and ϕ space for which trends of log2Fqs vs log2ss are shown in Figure 1(a) (η space) and Figure 4(a) (ϕ space) for 8 TeV dataset and Figure 1(b) (η space) and Figure 4(b) (ϕ space) for 7 TeV dataset is shown in Figures 5(a) and 5(b) for 8 and 7 TeV datasets, respectively. The values are shown in the figures for q=−5,−4,⋯,5. It should be noted that:
(i)
As shown in Figures5(a) and 5(b), the trend of dependence of hq on q for individual η and ϕ spaces confirm their multifractality and the same for hx,yq on q for the same pair of η and ϕ spaces confirm their cross-correlation for both 8 and 7 TeV datasets
(ii)
Figures5(a) and 5(b) show that hq decreases at a lower rate for the increasing values of negative qs for both φ and η data ranges in both energies of 8 and 7 TeV. This signifies that for negative values of q, the qth order RMS (calculated as per the equations in Step 4 and Step 5 of Section 3.1) is not much sensitive to the local fluctuations with small magnitudes for the datasets (ϕ and η) for both energies
However, for theϕ data series corresponding to the particular range of η at 8 TeV, hq decreases at a higher rate for positive values of q than the corresponding ones calculated for the same range of η at 7 TeV. This means, in this case, hq is a bit more affected by the local fluctuations with large magnitudes, resulting in higher rate of decrease for hq with increasing order of positive q for the ϕ data series of 8 TeV than the corresponding one for 7 TeV data. This has happened because there may exist fluctuations with comparatively large magnitude for the φ data series for higher energy—8 TeV than the one for 7 TeV.
(i)
It is further observed in Figures5(a) and 5(b) that hx,yq (calculated as per the equation in Step 4 of Section 3.2) for multifractal cross-correlation changes almost at the same rate for both 8 and 7 TeV energies. It should be noted that the maximum and minimum values of hx,yq for 8 and 7 TeV are different; however, the change of hx,yq with the increasing values of q is almost the same. Although for 8 TeV, there is higher rate of decrease for hq with increasing order of positive qs for the ϕ data series than the corresponding one for 7 TeV data, resulting from possible higher magnitude of fluctuation for ϕ data series, that decreasing trend gets nullified while calculating the covariance of ϕ data series and corresponding η data series at 8 TeV energy. Hence, the trend of hx,yq with the increasing values of q is almost the same for both energies. However, this has resulted in the lower ranges of the values of degree of cross-correlation hx,yq for 8 TeV, making them less cross-correlated
(ii)
Forq=2, both hq and hx,yq are >0.5 and for ϕ space hq is much higher than that for the corresponding η space
(iii)
Also,hx,yq is much higher than 0.5 for the pair of datasets for q=2. This suggests the presence of long-range correlation and persistence in both spaces
(iv)
Moreover, there is a drop in the value ofhx,yq around q=−1. In most of the previous works, it has been shown that there exists similar trend of hx,yq or the degree of cross-correlation with increasing values of order or q, as seen in the present analysis
(v)
Similar analysis has been done for all the 10 pairs of datasets, and similar trend is observed for all of them
(7)
Figures6(a) and 6(b) show the comparison of the trend of different values of fα versus α for the same η, ϕ spaces and the same trend calculated for their cross-correlation, for 8 and 7 TeV datasets, respectively
(i)
For both energy ranges, width of the cross-correlation curve is the maximum, followed by the width of the multifractal spectrum of theϕ space and then that of the η space
(ii)
Again, similar trend is observed for all the pairs of datasets in this experiment. The more wide the spectrum is, the more degree of multifractality is inherent in the data series
(8)
Theqth order detrended covariance Fxyq,s is calculated for a particular range of η values and their corresponding ϕ values as per the Step 4 of the MF-DXA methodology described in Section 3.2, and the trend of log2Fxyq,s vs log2s for q=−5,0,5 is shown in Figures 7(a) and 7(b) for 8 and 7 TeV datasets, respectively
(i)
Their linear trend (more prominent for the values ofq>0) confirms the power-law behavior of Fxyq,s versus s for all the values of q
(ii)
Similar calculation is done for all theη ranges and their corresponding ϕ spaces, for both 8 and 7 TeV datasets, and similar trend is observed
As explained in Step 5 of the MF-DXA methodology described in section3.2, here, hxyq>0.5 implies that there are persistent long-range cross-correlations, where a large value in ϕ space is likely to have an equally large corresponding η value. The higher the value of hxyq, the higher the cross-correlation. Figures 7(a) and 7(b) show how the qth order detrended covariance or Fxy(q,s) varies with increasing values of scale (s) in log scale. The straight line fitting of Fxyq,s versus s in log scale is shown in the figures and the relevant power-law coefficients or degree of cross-correlations (hxyq) are also shown in the fitting equations in the diagram. It is evident from the figures that for negative values of q, degree of cross-correlation is higher than that for positive values of q. Covariance of small magnitudes are reflected more prominently for negative values of q, which signifies that the average cross-correlation between φ spaces and their corresponding η spaces fluctuates more for the values of q<0 and gives rise higher value of hxyq. This fluctuation becomes lesser as q increases from q=−5,−4,⋯,−1. For positive values of q (q≥0), the qth order detrended covariance starts to fluctuate very less and gives rise to lesser values of hxyq. Hence, the value of hxyq also changes more for q<0 than for q>0. This different degree of detrended covariance between φ values and their corresponding η values for different values of q gives a range of hx,yq forming the multifractal cross-correlation spectrum.
Figures7(a) and 7(b) show that the qth order detrended covariance or Fxyq,s calculated for large values of a particular η space and its corresponding ϕ space having large values is fluctuating more with increasing scale (s) for negative values of q yielding to comparatively higher values of hxyq—power-law coefficient. Also, for q>0, Fxyq,s would flatten or change in almost constant rate with s as evident in the figures and yielding to comparatively lesser values of hxyq for both 8 and 7 TeV energies. For calculating the degree of multifractal cross-correlation denoted by γis, the trend of Fxyq,s∝shxyq for q=2 is analyzed [20], as elaborated in Section 3.2. Further, in most of the previous works, it had been shown that Fxyq,s varies with increasing values of scale or s in log scale for different order or values of q, in similar manner as it is varying in this experiment.
(9)
Two sets of multifractal cross-correlation coefficient, denoted byγi for i=1,2,⋯,5 for each of the 8 and 7 TeV datasets, are computed as per the method described in Section 3.2. This way, the degree of cross-correlation between ϕ and η spaces for all the 5 ranges of η values as specified in Step 1 for both 8 and 7 TeV datasets are calculated
(10)
Then, each of the azimuthalϕ spaces extracted in Step 4 is randomized, and the multifractal cross-correlation coefficients between the randomized ϕ spaces and the corresponding η spaces are extracted for both 8 and 7 TeV datasets are calculated as per the method described in Section 3.2
(11)
Figure8 shows the comparison of multifractal cross-correlation coefficients (γi) between ϕ and η spaces for all the 5 ranges of η values for 7 and 8 TeV datasets with respect to their rapidity as well as energy dependence. Here, we notice that:
(i)
For both 7 and 8 TeV data, all the 5η spaces are highly cross-correlated with their corresponding ϕ spaces
(ii)
It should be noted thatγi=1 for uncorrelated data series. The more correlated the data series are, the lower the value of γi. For 8 TeV data, the first range of η values is most cross-correlated with the corresponding ϕ space, and the most cross-correlated range for 7 TeV data is the third one
The comparison ofγis for all the 5 η ranges for 7 and 8 TeV datasets between the original and the randomized version is shown in Table 2. The values of the parameter, γi, calculated for experimental data being significantly different from the shuffled ensembles, confirm that this inherent degree of multifractal cross-correlation between each η space and its corresponding ϕ space is never the outcome of randomization process but is indicative of the inherent multifractal cross-correlation among η and corresponding ϕ values in the di-muon production process.
(12)
Hence, the values ofγis calculated for the original and the randomized version differ substantially, clearly establishing the statistical significance of the results obtained from the actual data.Figure 1
Trend oflog2Fqs vs log2ss for q=−5,0,5, extracted for a particular range of η (a) for 8 TeV dataset and (b) for 7 TeV dataset.
(a)
(b)Figure 2
Comparison of the trend of different values offα versus α between the original and the randomized version of the η space for a particular range of η values for (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 3
Comparison of the widths of the multifractal spectrum generatedη spaces for all the 5 ranges of η values for 7 and 8 TeV datasets.Table 1
Comparison of the widths of the multifractal spectrum generatedη spaces for all the 5 ranges of η values for 7 and 8 TeV datasets, between the original and the randomized version.
η ranges
MF-DFA spectrum width
8 TeV
7 TeV
Original
Random
Original
Random
−2.5 to −1.5
0.09
0.06
0.09
0.04
−1.5 to −0.5
0.02
0.01
0.07
0.02
−0.5 to 0.5
0.09
0.04
0.02
0.04
0.5 to 1.5
0.03
0.02
0.03
0.04
1.5 to 2.5
0.05
0.04
0.07
0.06Figure 4
Trend oflog2Fqs vs log2ss for q=−5,0,5, analyzed for the ϕ space corresponding to the η space for (a) 8 TeV dataset, as shown in Figure 1(a), and (b) 7 TeV dataset, as shown in Figure 1(b).
(a)
(b)Figure 5
Trend ofhq and hx,yq versus q for q=−5,−4,⋯,5, calculated for a particular range of η values and their corresponding ϕ values for (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 6
Comparison of the trend of different values offα versus α among the same η, ϕ spaces and the same trend calculated for their cross-correlation, for the (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 7
Trend oflog2Fxyq,s vs log2s for q=−5,0,5, calculated for a particular range of η values and their corresponding φ values for (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 8
Comparison of multifractal cross-correlation coefficient (γi) between φ and η spaces for all the 5 ranges of η values for 7 and 8 TeV datasets.Table 2
Comparison of the experimental values of multifractal cross-correlation coefficients (γi) between ϕ and η spaces for all the 5 ranges of η values for 7 and 8 TeV datasets, between the original and the randomized version.
η ranges
MFD-XA coefficients (γi)
8 TeV
7 TeV
Orig
Rand
Orig
Rand
−2.5 to −1.5
−0.57
1.01
−0.49
0.94
−1.5 to −0.5
−0.48
1.00
−0.53
0.99
−0.5 to 0.5
−0.48
1.02
−0.55
1.00
0.5 to 1.5
−0.54
0.98
−0.50
1.01
1.5 to 2.5
−0.45
0.96
−0.53
0.94


## 4.1. Data Description

The primary dataset of thepp collision at 8 TeV in AOD format from RunB of 2012 [40] and another dataset of pp collision at 7 TeV in the same AOD format from RunA of 2011 [41] of the CMS collaboration are taken as the source datasets for this experiment. The run numbers which are selected and qualified by CMS to be processed along with the appropriate parameters for generation of the collision datasets are provided in the links—link1 and link2 for 8 TeV and 7 TeV, respectively. These datasets are made available for experiment. We have extracted the pseudorapidity-η space and corresponding azimuthal-ϕ space for the generated di-muons from these runs qualified by CMS from the primary datasets in the following formats—text (.txt) and .root format. In this analysis, we have utilized these pseudorapidity space and the corresponding azimuthal space from the text (.txt) file.


## 4.2. Data Analysis and Results

(1)
The pseudorapidity-η space for each of the datasets for 8 and 7 TeV extracted from the primary datasets of the CMS collaboration as described in Section 4.1 is divided into the following 5 ranges of η values
(a)
−2.5 to −1.5
(b)
−1.5 to −0.5
(c)
−0.5 to 0.5
(d)
0.5 to 1.5
(e)
1.5 to 2.5
For all the 5 ranges, theη values are extracted from the full-phase space of the two source datasets and mapped to data series. The data series is plotted with the X-axis denoting the sequence number of η values and the Y corresponds to the η values corresponding to the sequence number as in the X-axis.
For each of these data series, the following values are calculated.(i)
The width of the multifractal spectrum
(ii)
Degree of cross-correlation between theη space and their corresponding ϕ space
(2)
For each of the 10 datasets (5 for 8 TeV and 5 for 7 TeV datasets) created for the 5 ranges of pseudorapidity values, as specified in Step 1, the multifractal analysis is done and the width of multifractal spectrum is calculated as per the method elaborated in Section3.1
Theqth order detrended variance Fqs is calculated as per equation (3) in the Step 5 of the MF-DFA methodology as described in Section 3.1. Figures 1(a) and 1(b) show the log2Fqs vs log2ss trend for q=−5,0,5, extracted for a particular range of η values for 8 and 7 TeV datasets, respectively.
Their linear trend confirms the power-law behavior ofFqs versus s for all the values of q. Similar calculation is done for all the η ranges for both 8 and 7 TeV datasets, and similar trend is observed.
(3)
For each of theη-data series corresponding to the ranges specified in Step 1, a randomized version of data is produced and widths of the multifractal spectrum are calculated as per the same methodology elaborated in Section 3.1. The calculated values of the parameters are compared to those for the experimental data. In Figures 2(a) and 2(b), the widths of the multifractal spectrum of the original datasets and their randomized versions calculated for one of the ranges η values are shown for 8 and 7 TeV datasets, respectively. The below points must be noted for the shape and widths of the multifractal spectrum of the original datasets and their randomized versions
(i)
The shape of the multifractal spectrum does not necessarily have to be symmetric. The spectrum might have either a right or a left truncation that arises from the consistent/(almost constant) trend of Hurst exponents for positive or negative values ofqs. The consistent trend of qth order Hurst exponent signifies that the qth order RMS (calculated as per the equations in Step 5 and Step 6 of Section 3.1) is not much sensitive to the extent of the local fluctuations
(ii)
The width and trend of multifractal spectrum between the original and the randomized version of the particularη-space for 8 TeV dataset shown in Figure 2(a) shows that the spectrum for the randomized version has a long right tail which signifies that the series have a multifractal structure not much affected by the local fluctuations with large magnitudes
(iii)
Whereas the multifractal spectrum calculated for the original and the randomized version of theη-space for the same range of η values for 7 TeV dataset shown in Figure 2(b) shows that the spectrum for the randomized version has a long left tail, which means that the randomized series is not much sensitive to the local fluctuations with small magnitudes.
(iv)
If the source data is long-range correlated, that is eradicated by the randomization process, and the data should start to become uncorrelated. This makes the multifractal spectrum or the scaling pattern of randomized series insensitive to the local fluctuations with large or small magnitudes, which is not the case with the actual experimental data. Hence, it results in the different widths of multifractal spectrum calculated the randomized version from those for the original version. Moreover, for two completely different experimental datasets the randomized data may be different with regards to peak, shape, and trend. The width of the multifractal spectrum has normally been less for the randomized data than the one for the experimental data
(v)
The main conclusion comes from the fact that the values of the width of the multifractal spectrum calculated experimental data being significantly different from shuffled ensembles essentially confirms that thedegree of scale-freeness is indicative of the dynamics inherent in the di-muon production process, and this degree of scale-freeness may be indicative of different processed responsible for di-muon production even beyond standard model (SM).
(vi)
It is evident from the figures that the widths of the multifractal spectrum of the original datasets and their randomized versions differ substantially for both energies. Similar trend is observed from the comparison of the original and the randomized version of the 5 ranges ofη values for both 8 and 7 TeV datasets
The comparison of the widths of the multifractal spectrum generated for theη spaces for all the 5 ranges of η values for 7 and 8 TeV datasets with respect to their rapidity and energy dependence is shown in Figure 3. It should be noted that:
(i)
The comparison of the width of the multifractal spectrum offα, denoted by the difference between the maximum and minimum values of α, between the original and the randomized version of the η space for both energy ranges confirm the mutifractality of the original η spaces
(ii)
For the 2nd, 3rd, and 5th range of η values, the widths of multifractal spectrum is substantially different between the energy ranges
(iii)
For both 7 and 8 TeV the 1st and 4th range of η-space display minimum or no difference with respect to multifractality
(iv)
The degree of multifractality is found to be the least for 2nd and 3rd range for 8 and 7 TeV data, respectively
Table1 details the widths of the multifractal spectrum of the original datasets and their randomized versions for all the 10 datasets (5 for 8 and 5 for 7 TeV) corresponding to the η values. The values of the width of MF-DFA spectrum, essentially is an indicator of inherent symmetry and scale-freeness (different at different energy and rapidity ranges) with which the produced di-muons create the signatures in terms of η values. The values of these parameters in experimental data being significantly different from shuffled ensembles confirm that this inherent symmetry and degree of scale-freeness is never the outcome of randomization process but is indicative of the dynamics involved in the di-muon production process.
(4)
For each of the 10 datasets (5 for 8 and 5 for 7 TeV) ofη values extracted for the ranges specified in Step 1, the corresponding azimuthal-ϕ space is also extracted. The 10 corresponding ϕ space is sorted in the ascending order and then mapped into data series. They in turn are mapped into a two-dimensional space with their sequence along the X-axis and the corresponding values of ϕ along the Y-axis
(5)
For the 10ϕ spaces (5 for 8 and 5 for 7 TeV), the qth order detrended variance Fqs is analyzed as per equation (3) in Step 5 of the MF-DFA methodology as described in Section 3.1. Figures 4(a) and 4(b) show the log2Fqs vs log2ss trend for q=−5,0,5, extracted for corresponding ϕ values for the same range of η values for which the same trend is shown in Figures 1(a) and 1(b) for 8 and 7 TeV datasets, respectively
It is to be noted that the linear trend confirms the power-law behavior ofFqs versus s for all the values of q for the ϕ spaces. The same analysis is done for all the ϕ spaces corresponding to the η ranges for both 8 and 7 TeV datasets, and a similar trend is observed.
(6)
Multifractal cross-correlation analysis is done as per the method described in Section3.2 between the 10 pairs of datasets (5 for 8 TeV and 5 for 7 TeV), one being the sorted ϕ values and the other being the corresponding η values. The trend of generalized Hurst exponent hq for different order (q) is analyzed for all the 10 pairs of η and ϕ datasets as per the process described in Section 3.1. Along with that, for the same pairs of datasets the degree of cross-correlationhx,yq for different order (q) is analyzed as per the methodology described in Section 3.2. The trend of hq and hx,yq versus q for the particular sample pair of η and ϕ space for which trends of log2Fqs vs log2ss are shown in Figure 1(a) (η space) and Figure 4(a) (ϕ space) for 8 TeV dataset and Figure 1(b) (η space) and Figure 4(b) (ϕ space) for 7 TeV dataset is shown in Figures 5(a) and 5(b) for 8 and 7 TeV datasets, respectively. The values are shown in the figures for q=−5,−4,⋯,5. It should be noted that:
(i)
As shown in Figures5(a) and 5(b), the trend of dependence of hq on q for individual η and ϕ spaces confirm their multifractality and the same for hx,yq on q for the same pair of η and ϕ spaces confirm their cross-correlation for both 8 and 7 TeV datasets
(ii)
Figures5(a) and 5(b) show that hq decreases at a lower rate for the increasing values of negative qs for both φ and η data ranges in both energies of 8 and 7 TeV. This signifies that for negative values of q, the qth order RMS (calculated as per the equations in Step 4 and Step 5 of Section 3.1) is not much sensitive to the local fluctuations with small magnitudes for the datasets (ϕ and η) for both energies
However, for theϕ data series corresponding to the particular range of η at 8 TeV, hq decreases at a higher rate for positive values of q than the corresponding ones calculated for the same range of η at 7 TeV. This means, in this case, hq is a bit more affected by the local fluctuations with large magnitudes, resulting in higher rate of decrease for hq with increasing order of positive q for the ϕ data series of 8 TeV than the corresponding one for 7 TeV data. This has happened because there may exist fluctuations with comparatively large magnitude for the φ data series for higher energy—8 TeV than the one for 7 TeV.
(i)
It is further observed in Figures5(a) and 5(b) that hx,yq (calculated as per the equation in Step 4 of Section 3.2) for multifractal cross-correlation changes almost at the same rate for both 8 and 7 TeV energies. It should be noted that the maximum and minimum values of hx,yq for 8 and 7 TeV are different; however, the change of hx,yq with the increasing values of q is almost the same. Although for 8 TeV, there is higher rate of decrease for hq with increasing order of positive qs for the ϕ data series than the corresponding one for 7 TeV data, resulting from possible higher magnitude of fluctuation for ϕ data series, that decreasing trend gets nullified while calculating the covariance of ϕ data series and corresponding η data series at 8 TeV energy. Hence, the trend of hx,yq with the increasing values of q is almost the same for both energies. However, this has resulted in the lower ranges of the values of degree of cross-correlation hx,yq for 8 TeV, making them less cross-correlated
(ii)
Forq=2, both hq and hx,yq are >0.5 and for ϕ space hq is much higher than that for the corresponding η space
(iii)
Also,hx,yq is much higher than 0.5 for the pair of datasets for q=2. This suggests the presence of long-range correlation and persistence in both spaces
(iv)
Moreover, there is a drop in the value ofhx,yq around q=−1. In most of the previous works, it has been shown that there exists similar trend of hx,yq or the degree of cross-correlation with increasing values of order or q, as seen in the present analysis
(v)
Similar analysis has been done for all the 10 pairs of datasets, and similar trend is observed for all of them
(7)
Figures6(a) and 6(b) show the comparison of the trend of different values of fα versus α for the same η, ϕ spaces and the same trend calculated for their cross-correlation, for 8 and 7 TeV datasets, respectively
(i)
For both energy ranges, width of the cross-correlation curve is the maximum, followed by the width of the multifractal spectrum of theϕ space and then that of the η space
(ii)
Again, similar trend is observed for all the pairs of datasets in this experiment. The more wide the spectrum is, the more degree of multifractality is inherent in the data series
(8)
Theqth order detrended covariance Fxyq,s is calculated for a particular range of η values and their corresponding ϕ values as per the Step 4 of the MF-DXA methodology described in Section 3.2, and the trend of log2Fxyq,s vs log2s for q=−5,0,5 is shown in Figures 7(a) and 7(b) for 8 and 7 TeV datasets, respectively
(i)
Their linear trend (more prominent for the values ofq>0) confirms the power-law behavior of Fxyq,s versus s for all the values of q
(ii)
Similar calculation is done for all theη ranges and their corresponding ϕ spaces, for both 8 and 7 TeV datasets, and similar trend is observed
As explained in Step 5 of the MF-DXA methodology described in section3.2, here, hxyq>0.5 implies that there are persistent long-range cross-correlations, where a large value in ϕ space is likely to have an equally large corresponding η value. The higher the value of hxyq, the higher the cross-correlation. Figures 7(a) and 7(b) show how the qth order detrended covariance or Fxy(q,s) varies with increasing values of scale (s) in log scale. The straight line fitting of Fxyq,s versus s in log scale is shown in the figures and the relevant power-law coefficients or degree of cross-correlations (hxyq) are also shown in the fitting equations in the diagram. It is evident from the figures that for negative values of q, degree of cross-correlation is higher than that for positive values of q. Covariance of small magnitudes are reflected more prominently for negative values of q, which signifies that the average cross-correlation between φ spaces and their corresponding η spaces fluctuates more for the values of q<0 and gives rise higher value of hxyq. This fluctuation becomes lesser as q increases from q=−5,−4,⋯,−1. For positive values of q (q≥0), the qth order detrended covariance starts to fluctuate very less and gives rise to lesser values of hxyq. Hence, the value of hxyq also changes more for q<0 than for q>0. This different degree of detrended covariance between φ values and their corresponding η values for different values of q gives a range of hx,yq forming the multifractal cross-correlation spectrum.
Figures7(a) and 7(b) show that the qth order detrended covariance or Fxyq,s calculated for large values of a particular η space and its corresponding ϕ space having large values is fluctuating more with increasing scale (s) for negative values of q yielding to comparatively higher values of hxyq—power-law coefficient. Also, for q>0, Fxyq,s would flatten or change in almost constant rate with s as evident in the figures and yielding to comparatively lesser values of hxyq for both 8 and 7 TeV energies. For calculating the degree of multifractal cross-correlation denoted by γis, the trend of Fxyq,s∝shxyq for q=2 is analyzed [20], as elaborated in Section 3.2. Further, in most of the previous works, it had been shown that Fxyq,s varies with increasing values of scale or s in log scale for different order or values of q, in similar manner as it is varying in this experiment.
(9)
Two sets of multifractal cross-correlation coefficient, denoted byγi for i=1,2,⋯,5 for each of the 8 and 7 TeV datasets, are computed as per the method described in Section 3.2. This way, the degree of cross-correlation between ϕ and η spaces for all the 5 ranges of η values as specified in Step 1 for both 8 and 7 TeV datasets are calculated
(10)
Then, each of the azimuthalϕ spaces extracted in Step 4 is randomized, and the multifractal cross-correlation coefficients between the randomized ϕ spaces and the corresponding η spaces are extracted for both 8 and 7 TeV datasets are calculated as per the method described in Section 3.2
(11)
Figure8 shows the comparison of multifractal cross-correlation coefficients (γi) between ϕ and η spaces for all the 5 ranges of η values for 7 and 8 TeV datasets with respect to their rapidity as well as energy dependence. Here, we notice that:
(i)
For both 7 and 8 TeV data, all the 5η spaces are highly cross-correlated with their corresponding ϕ spaces
(ii)
It should be noted thatγi=1 for uncorrelated data series. The more correlated the data series are, the lower the value of γi. For 8 TeV data, the first range of η values is most cross-correlated with the corresponding ϕ space, and the most cross-correlated range for 7 TeV data is the third one
The comparison ofγis for all the 5 η ranges for 7 and 8 TeV datasets between the original and the randomized version is shown in Table 2. The values of the parameter, γi, calculated for experimental data being significantly different from the shuffled ensembles, confirm that this inherent degree of multifractal cross-correlation between each η space and its corresponding ϕ space is never the outcome of randomization process but is indicative of the inherent multifractal cross-correlation among η and corresponding ϕ values in the di-muon production process.
(12)
Hence, the values ofγis calculated for the original and the randomized version differ substantially, clearly establishing the statistical significance of the results obtained from the actual data.Figure 1
Trend oflog2Fqs vs log2ss for q=−5,0,5, extracted for a particular range of η (a) for 8 TeV dataset and (b) for 7 TeV dataset.
(a)
(b)Figure 2
Comparison of the trend of different values offα versus α between the original and the randomized version of the η space for a particular range of η values for (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 3
Comparison of the widths of the multifractal spectrum generatedη spaces for all the 5 ranges of η values for 7 and 8 TeV datasets.Table 1
Comparison of the widths of the multifractal spectrum generatedη spaces for all the 5 ranges of η values for 7 and 8 TeV datasets, between the original and the randomized version.
η ranges
MF-DFA spectrum width
8 TeV
7 TeV
Original
Random
Original
Random
−2.5 to −1.5
0.09
0.06
0.09
0.04
−1.5 to −0.5
0.02
0.01
0.07
0.02
−0.5 to 0.5
0.09
0.04
0.02
0.04
0.5 to 1.5
0.03
0.02
0.03
0.04
1.5 to 2.5
0.05
0.04
0.07
0.06Figure 4
Trend oflog2Fqs vs log2ss for q=−5,0,5, analyzed for the ϕ space corresponding to the η space for (a) 8 TeV dataset, as shown in Figure 1(a), and (b) 7 TeV dataset, as shown in Figure 1(b).
(a)
(b)Figure 5
Trend ofhq and hx,yq versus q for q=−5,−4,⋯,5, calculated for a particular range of η values and their corresponding ϕ values for (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 6
Comparison of the trend of different values offα versus α among the same η, ϕ spaces and the same trend calculated for their cross-correlation, for the (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 7
Trend oflog2Fxyq,s vs log2s for q=−5,0,5, calculated for a particular range of η values and their corresponding φ values for (a) 8 TeV dataset and (b) 7 TeV dataset.
(a)
(b)Figure 8
Comparison of multifractal cross-correlation coefficient (γi) between φ and η spaces for all the 5 ranges of η values for 7 and 8 TeV datasets.Table 2
Comparison of the experimental values of multifractal cross-correlation coefficients (γi) between ϕ and η spaces for all the 5 ranges of η values for 7 and 8 TeV datasets, between the original and the randomized version.
η ranges
MFD-XA coefficients (γi)
8 TeV
7 TeV
Orig
Rand
Orig
Rand
−2.5 to −1.5
−0.57
1.01
−0.49
0.94
−1.5 to −0.5
−0.48
1.00
−0.53
0.99
−0.5 to 0.5
−0.48
1.02
−0.55
1.00
0.5 to 1.5
−0.54
0.98
−0.50
1.01
1.5 to 2.5
−0.45
0.96
−0.53
0.94


## 5. Conclusion

Oppositely charged di-muon production is the outcome of several hypothetical processes and investigation for such processes have been the goal of experiment ofpp collision at 7 and 8 TeV and also at other energies at the CMS collaboration. However, the present investigation, as we have pointed out earlier, is based on a deep-rooted dynamics of di-lepton production process in hadron-hadron interaction from the basic perspective of symmetry-based scaling in different pseudorapidity ranges in two different energies. We have used two rigorous and robust methodologies, namely, MF-DFA and MF-DXA, for the scaling analysis of the dynamics of the di-muon production process using di-muon data taken from the primary dataset of RunA(2011) and RunB(2012) of the pp collision at 7 TeV and 8 TeV, respectively, from CMS collaboration. We have analyzed how this scaling pattern has evolved from one rapidity range to the next one and how this change evolved from lower energy range of 7 TeV to the higher one 8 TeV, and the findings are listed below.
(1)
The linear trend ofFqs vs s for all the values of q for all the 5 ranges of η values for 8 and 7 TeV datasets confirms the fractality as well as the multifractality of all the pseudorapidity spaces. Figures 1(a) and 1(b) show similar trend for a particular range of η values for both energy ranges. Similar linear trend is observed for the ϕ spaces corresponding to the η spaces, which again confirm the fractality and the multifractality of the ϕ spaces as well. Figures 4(a) and 4(b) show the linear trend calculated for the ϕ spaces corresponding to the particular η range for both energy ranges
(2)
Table1 and Figure 3 show how the widths of the multifractal spectrum differ from one η space to the other and how they in turn differ from one energy range to another. It is interesting to note that for both 7 and 8 TeV energies, the η space corresponding to the first range of η has the maximum width of multifractal spectrum/degree of complexity, or in other words, they are most multifractal in nature among the other five ranges. Moreover, they have exactly the same value for the parameter. As for the minimum width of multifractal spectrum, the second η range for 8 TeV data and third η range for 7 TeV data is 0.02 which is again the same for both energy ranges
(3)
The linear trend oflog2Fxyq,s vs log2s for q=−5,0,5 which is shown in Figures 7(a) and 7(b) for 8 and 7 TeV datasets, respectively, for the same specific η range and its corresponding ϕ space confirms the self-similar cross-correlation between the spaces which is evident from the goodness-of-fit for all the values of q. Similar trend is observed for the rest of the η ranges
(4)
Table2 and the Figure 8 show that for both 7 and 8 TeV data, all the 5 η spaces are highly cross-correlated with their corresponding ϕ spaces and how the degree of cross-correlation changes from one η space to the other and from one energy range to another. It should be noted that the degree of multifractal cross-correlation, γi, is maximum for the first η range for 8 TeV data, and the same is maximum for the third range of 7 TeV data. γi is minimum for the fifth η range for 8 TeV data and for the first range of 7 TeV dataThis analysis manifests different degree of symmetry scaling or scale-freeness in different pseudorapidity domains and at the same time different degree of cross-correlation between pseudorapidity and azimuthal space at both energy. The differences in the values of scaling and cross-correlation exponents representing thedegree of symmetry scaling and degree of cross-correlation, respectively, calculated for different ranges of rapidity and at two different energy values, indicate the existence of several processes involved in the production process of oppositely charged di-muons giving rise to varying degree of scaling. The observed difference of degree of symmetry scaling in different rapidity domains at two different energy values may provide a clue for exploring other processes in regards to di-muon production even from the perspective beyond the standard model. This novel method has the prospect for applications in different high-energy interactions to detect not only different possible resonance states but also for identification of exotic resonance states proposed by theories.



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