# Analysis of Prospective Super-Symmetry Inherent in the $pp$ Collision Data
at $7$ TeV from CMS Collaboration Using Novel Two-Dimensional Multifractal-
Detrended Fluctuation Analysis Method with Rectangular Scale

Susmita Bhaduri1, Anirban Bhaduri2
1Moonshot Analytics and Design 3/1/1, Dhakruia Station Lane, Kolkata - 700031,
West Bengal, India
2Moonshot Analytics and Design 3/1/1, Dhakruia Station Lane, Kolkata - 700031,
West Bengal, India
1ORCID:0000-0003-1246-9124, E-mail<EMAIL_ADDRESS>
2ORCID:0000-0002-7787-3550, E-mail<EMAIL_ADDRESS>

###### Abstract

Search for Super-Symmetry in High-Energy-Physics is of enormous interest for
the past few decades. Continuous searches were conducted at LHC regarding
Super-Symmetry for prompt, non-prompt, R-parity conserving and violating
generation and decays. The limits obtained from these analysis to detect the
signatures of Super-Symmetric-particles, revealed greater possibilities of
such experiments in collider. These signatures are usually derived assuming a
bit optimistic conditions of the decaying process of s-particles to final-
states. Moreover, Super-Symmetry might have been in a disguised state in
lower-mass-scales resulting from challenging mass-spectra and mixed-modes of
decays. The proposed chaos-based, novel method of Two-Dimensional-
Multifractal-Detrended-Fluctuation-Analysis((2D)MF-DFA) [1], is extended using
rectangular-scale. The (2D)experimental data-surfaces are constructed using
the component-space(in the $X,Y,Z$ co-ordinates) taken out from the
$4$-momenta of final-state-signatures of the produced di-muons from the
selected events. Two publicly-available datasets are used here. First is the
data from MultiJet primary $pp$ collision-data from RunB(2010) at $7$-TeV [2],
used in analysis of the Super-Symmetry [3, 4] with Razor-variables. Second is
the data from primary-dataset of $pp$ collisions at $7$-TeV from RunA(2011) of
CMS-collaboration [5]. The (2D)Multifractal behaviour of particle-production-
process is studied in terms of symmetry-scaling, the inherent scale-freeness
and multifractality. The analysis outcome for Super-Symmetric-data is compared
with the same for the non-Super-Symmetric-data in terms of the generalized-
Hurst-exponent and (2D)Multifractal-spectrum-width. Unusual and significantly
different scaling-behaviour and long-range-correlation is observed between the
final-state-signatures of the di-lepton production-process of the first and
second datasets. This difference may indicate a possible signature of Super-
Symmetry which may be missed in the conventional method of analysing the
invariant-mass/transverse-momentum-spectrum.

Keywords: Super-Symmetry, Two-Dimensional-Multifractal-Analysis, Symmetry-
based Scaling, CMS collaboration
PCAS Nos.: 10, 11.30.Pb, 24.60.Ky, 24.60.Lz

##  1 Introduction

Over a number of past decades, the Standard Model(SM) of particle physics
gives us an elaboration of the fundamental particles and their respective
interactions [6, 7, 8]. UA1 and UA2 experiments in $1983$ have confirmed that
for short-range weak interactions the the $W$ and $Z$ bosons have huge mass
but bosons in SM have to be massless to keep SM lagrangian invariant under
local gauge transformations. Brout, Englert, Higgs mechanism suggested a
probable solution where gauge bosons can gain mass in gauge invariant process
through spontaneous symmetry breaking [9, 10, 11]. Glashow, Weinberg and Salam
have integrated this concept with the SM by formulating the theory of $W$ and
$Z$ bosons with huge mass and an additional elementary $0$-spin particle which
is the Higgs boson interacting with the gauge bosons and the fermions. The
discovery of the Higgs boson by the ATLAS [12] and CMS [13] collaborations at
the Large Hadron Collider(LHC) [14] gave the final approval of the Standard
Model. The conjectures of the Standard Model(SM) have been established
experimentally with increasing robustness.

However the Standard Model(SM) has few challenges like gravitational
interactions cannot be included in this model, origination of charge parity
violation yielding to matter versus antimatter and also the dark matter is not
found in this model. Also, it does not answer how neutrinos gain mass etc.
There has been search for a framework which unifies all interactions. For
resolution of Standard Model(SM)’s limitations many theories have come up.
Among them, Super-Symmetry(SUSY) has been one of the most reasonable
extensions of the Standard Model(SM) in the field of particle physics [15, 16,
17, 18, 19, 20, 21, 22].It allows to determine coupling unification at the
energy of $10^{16}GeV$. It comes up with a prospective candidate of dark
matter - lightest Super-Symmetric particle(LSP) [23]which is a mandatory
element to decode the concept of quantum gravity based on the framework of
string theory. The signature of Super-Symmetry(SUSY) in the detector, is
driven by the nature of the lightest Super-Symmetric particle(LSP) which is
dependent on the underlying process of Super-Symmetry(SUSY) breaking. The
Super-Symmetry model eliminates the quadratic divergences in radiative
modifications to the Higgs boson mass by default. For each particle in the
Standard Model, Super-Symmetry infuses a super-partner or the sparticle, with
a spin difference from the Standard Model particle by $1/2$ unit. There exist
theoretical propositions which suggest that the masses of sparticles might be
less than $\sim 1TeV$ [21, 22]and hence the experiments at the Large Hadron
Collider(LHC) are best suited to be analyzed for the discovery of Super-
Symmetry particles. For the past decades considerable amount of missing
transverse energy has been considered as the most rigorous observable to
identify the production and decay of Super-Symmetry(SUSY) particles at
colliders and quantity of missing transverse energy depends on the division of
mass among the heavier sparticles. Till date, usual di-lepton SUSY searches in
CMS [24, 25] needed several jets with large transverse momentum($p_{T}$)
correlated with large values of $H_{T}$, the scalar sum of the transverse
momenta of all the jets satisfying the jet selection criteria, and absence of
large transverse energy to distinguish a SUSY signal from the very large SM
backgrounds. In the work [26], comparatively moderate criteria of the absence
of transverse energy($\cancel{E}_{t}>40GeV$) and also $H_{T}(H_{T}>120GeV)$,
has been used. The Standard Model(SM) has been extended with softly broken
Super-Symmetry(SUSY) [27, 28, 29, 30, 23]to predict new elementary particles
which are super-partner of the Standard Model(SM) particles. The Fermilab
Tevatron [31, 32][7,8]and the CERN LHC [eg. [33, 34, 35, 36]]have concentrated
on the events having signatures of high-energy hadronic jets and also of
leptons decayed from squarks and gluinos produced in pairs, considering the
assumption of R-parity [37] conservation. These events usually have high
degree of missing transverse energy resulting from the weakly interacting
super-partners which are stable and one of the super-partners is generated in
each of the two decay chains. In these final states, searches for Super-
Symmetry(SUSY) were conducted by the ATLAS [38] at $13$TeV and CMS [39]
Collaborations at $8$TeV. Chatrchyan et al. [3, 4] have applied the razor
approach to search for Super-Symmetric particles, after considering the
R-parity [37] conservation.In this work they have considered two analyses - a
comprehensive search for new heavy-particle-pairs which decay to final states
with at least two jets and missing transverse energy and an exclusive search
for the final states having at least one jet arising from a bottom quark. For
both the analyses the final states of hadronic, single-lepton and di-lepton
events from the MultiJet primary dataset of the RunB(2010) of $pp$ collision
at $7$TeV from CMS collaboration [2], are considered. Chatrchyan et al. [3, 4]
have analyzed chosen events in the two-dimensional razor plane of $M_{R}$ \-
razor kinematic variable, defined with regards to the momentum of the two
megajets and $R$, a variable without dimensions signifying the missing
transverse energy. These razor variables are established on the generic
procedure of the production of two heavy particles in pair, each one decaying
to an unidentified particle with visible decay products. The two-dimensional
shape analysis in razor-variable plane is validated with simulated events, and
no remarkable excess over the background expectations has been noticed. The
output of the search for Super-Symmetry [3, 4] has been used to drive
simplified Super-Symmetry models [40, 41, 42].

A significant feature of multiparticle production process is the inherent
fluctuation. Correlation analysis may yield valuable information about the
dynamics of this process. In the past, for quite a few years, fluctuation and
correlation had been analyzed extensively using novel perspective of studying
non-statistical fluctuation which had resulted in rigorous interpretation of
the pionisation process. To begin with, Bialas and Peschanski [43] introduced
a novel concept to analyze multiplicity fluctuations with respect to scaled
factorial moments to identify and analyze the pattern of large non-statistical
fluctuation, and finally heading towards the physical explanation of the
origin of the fluctuations. They suggested that the character of the factorial
moments is similar to the process of intermittency observed in the
hydrodynamics of turbulent fluid flow. Intermittency is a process which
manifests prominent local fluctuations in consistent and large statistical
systems. Moreover, it was noticed that multipion production process manifests
a power-law trend of the factorial moments in regards to the magnitude of
phase-space intervals in decreasing mode [43]. Bialas and Peschanski [43] also
suggested a connection between intermittency and fractal behaviour. Initially
fractal pattern of multi-pion production process was analyzed from the context
of intermittent fluctuations by utilizing the technique of factorial moment.
Then a simple relationship was found between the intermittency indices and
anomalous fractal dimension [44, 45]. The inherent cascading process in the
multipion production process naturally generates a fractal structure. Also the
scale invariance existing in the process of hadronization was obvious from the
spectrum of fractal dimensions. After that, innumerable methods based on the
fractals had been applied to study the multipion production process using the
parameters of Gq moment and Tq moment [46, 47, 48, 49, 50]. Then different
approaches like the Detrended Fluctuation Analysis(DFA), multifractal-DFA(MF-
DFA) method [51, 52] were implemented exhaustively for studying non-
stationary, nonlinear data series to detect their long-range correlations. In
the recent years, various studies based on the multi-fractal properties of the
process of particle production had been reported [53, 54, 55, 56]. In a number
of recent works, self-similarity had been probed in the field of particle
physics, like - in the Jet and Top-quark generation process at the experiments
of Tevatron and LHC [57], in the process of strangeness production in $pp$
collisions at the experiments of RHIC [58], in the proton spin phenomena and
asymmetry of jet production process [59] to explain the collective phenomena
[60] and in the implementation of the concept of self-similar symmetry to dark
energy [61].

A fundamental change in approach occurred with the latest advancements in the
area of research using complex-network based methodologies. Albert and Barabá
made remarkable contributions in this area applying the analytic models and
tools for small-world, random and also scale-free graphs [62, 63]. Lacasa et
al. developed the Visibility Graph methodology [64, 65] which attained
prominence because of its completely novel and accurate perspective to
estimate the fractal behaviours. Studies had been conducted in the area of the
multiplicity fluctuation process in nucleus-nucleus and hadron-nucleus
interactions, applying this Visibility Graph methodology in few recent works
[66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. Also, Pb-Pb VSD masterclass data at
$2.76TeV$ per nucleon pair from ALICE Collaboration [76] was used for scaling
analysis of the pseudorapidity space, using both the method of complex network
based Visibility Graph and multifractal-DFA(MF-DFA) [51, 52], to study the
prospective phase transition and the signature of QGP in some latest works
[77, 74]. In a latest work [78], the fluctuation pattern inherent in the
dynamics of particle production process in high energy collision has been
analyzed using the multifractal-DFA(MF-DFA) method and also multifractal-
detrended-cross-correlation(MF-DXA) analysis using the pseudorapidity values
of di-muon data taken out from the $pp$ collision at $7$TeV and $8$TeV
respectively from CMS collaboration [5, 79]. A recent review work had been
reported about the complexities involved in resonance production for various
high energy collisions like $pp$, $pA$ and $AA$ collisions at LHC (using data
from ALICE collaboration), to understand the complexity and origin of
different resonance states and eventually understand the inherent dynamics of
the particle production process and the properties of the produced particles
for the various collision systems [80].

Chatrchyan et al. [26] have presented a search for Super-Symmetry by applying
an ANN (Artificial Neural Network) model to differentiate prospective Super-
Symmetry signals from the background of Standard Model. They implemented the
search for Super-Symmetry in the events having pairs of oppositely signed
leptons in the final state, hadronic jets and missing transverse energy, from
the data of Run-A(2011) of the $pp$ collision at $7$TeV at CMS detector. They
have confirmed accordance between the expectations and observations of the
Standard Model which gives rise to more optimized Super-Symmetric Standard
models. Such successful attempts in this area of analysis of pionisation
process in high energy interaction using chaos-based fractal, multi-fractal
and complex-network procedure including attempts to probe the origin of the
resonance states [80] as well as the use of new computational techniques such
as neural networks in the area of high energy collisions have encouraged and
inspired us to extend the methods to analyze another significant area of high
energy interaction using a novel multi-fractal method.

In this work we have attempted to analyze the inherent symmetry scaling in the
di-lepton production process using a novel method of extended two-
dimensional(2D) MF-DFA method using rectangular scales. The resultant dataset
(in totality) analyzed for the search of Super-Symmetry using razor filter [3,
4] generated from the MultiJet primary $pp$ collision data from RunB of 2010
at $7$TeV from the CMS collaboration [2] has been used for this analysis. The
symmetry scaling for di-muon production process in case of Super-Symmetry is
compared with the same for the non-Super-Symmetric di-muon data produced in
$pp$ collision in RunA of 2011 at $7$ TeV from CMS collaboration [5], using
the same method of (2D) MF-DFA. The scaling analysis is based on the
component-space(in the $X,Y,Z$ co-ordinates) taken out from the $4$-momenta of
final state signatures of the produced di-muons, obtained from the data from
CMS collaboration [2, 5]. The signature of Super-Symmetry may be identified
from the remarkable differences in the scaling behaviour of the di-muon
production process between the two kinds of data from CMS collaboration,
indicated by the Hurst exponent, generalized Hurst exponent and the width of
(2D) MF-DFA spectrum, without using conventional invariant mass or transverse
momentum techniques.

The rest of the paper is structured as follows. Section 2 describes the
methods of analysis in detail and the importance of various properties of
scale-freeness. Inside the Section 3, the Section 3.1 elaborates the details
of the data and Section 3.2 describes the detailed steps of our analysis and
the relevant inferences. The test results are further analyzed with regards to
the the physical relevance of the proposed parameters with respect to the
conventional ones for analysing Suer-Symmetry in the high energy interactions
in Section 4.

##  2 Method of analysis

Fractals are geometric pattern which is reiterated over various scales to
generate self-similar patterns [81]. Natural artifacts like coastal lines,
profile of trees, leaves, clouds or a mountain etc. are proved to be fractals.
Magnifying a small part of a fern leave, generates similar geometrical pattern
like the whole fern leave, which shows that they are self-similar. Most
significant property of fractals is the reiteration of the geometric pattern
over different scales, which is known as the self-similarity. Fractal
dimension is a ratio signifying a non-statistical index of complexity,
comparing detailed change of a fractal pattern along with the scale. Fractals
are categorized into two types: mono and multi-fractals. For mono-fractal
systems, the scaling properties are consistent across different scales but for
multi-fractals these properties are different for different regions and are
more complex in nature. Multi-fractal objects have a range of different non-
integer dimensions. Various methods that have been suggested to measure
Fractal Dimension - Wavelet Transform Modulus Maxima (WTMM), Fluctuation
Analysis (FA), Detrended Fluctuation Analysis (DFA), Detrended Moving Average
(DMA), Multi-fractal Detrended Fluctuation Analysis (MF-DFA) and Hurst
exponent, Power of Scale-freeness of a Visibility Graph (PSVG) etc.

In the present paper we have extended two-dimensional (2D) Multifractal-
Detrended Fluctuation Analysis (MF-DFA) proposed Yeh et al. [1], by selecting
rectangular scale for doing Multifractal analysis for experimental data
surfaces. To elaborate the methodology we have briefly explained the Detrended
Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis
(MF-DFA) method for single dimensional data series first [51, 82, 52] and then
elaborated the extended (2D) (MF-DFA) method in detail in the Section 2.1.

###  2.1 Two-dimensional (2D) MF-DFA algorithm

The DFA algorithm implemented on an one dimensional data series, measures
self-similar or fractal-like correlations by calculating the scaling pattern
of the Root-Mean-Square fluctuation of both the integrated and detrended time
series [51, 82, 52]. Let us denote time series with $N$ samples as $x(i)$ for
$i=1,2,\ldots N$ and average of the series is denoted by
$\bar{x}=\frac{1}{N}\sum_{i=1}^{N}x(i)$. Then the integrated random walk
series $y(i)$ is calculated as per the equation 1.

$\displaystyle y(i)=\sum_{k=1}^{i}[x(k)-\bar{x}],i=1,2,\ldots,N$ (1)

Then, series $y(i)$ is detrended for a particular scale of $s$ by dividing
$y(i)$ with $N_{s}$ number of non-overlapping segments of length
$s(N_{s}\equiv int(N/s))$ and then subtracting the local trend of the each
segment derived from least-square fitting for that segment, denoted by
$x_{v}(i)$ for segment sequence $v$ corresponding to a specific value of $s$.
For each $s$, the detrending function $F^{2}(s,v)$ of the integrated series
$y(i)$ for a particular segment $v=1,2,\ldots N_{s}$ is calculated as per the
equation 2.

$F^{2}(s,v)=\frac{1}{s}\sum_{i=1}^{s}\\{y[(v-1)s+i]-x_{v}(i)\\}^{2}$ (2)

with different $s$ and corresponding $v\in 1,2,\ldots,N_{s}$.

Then, the $q^{th}$-order function of fluctuation, $F_{q}(s)$, is calculated by
averaging the values of $F^{2}(s,v)$ computed for all the set of
segments-$v=1,2,\ldots N_{s}$ produced for each $s$, as per the equation 3.

$\displaystyle
F_{q}(s)=\left\\{\frac{1}{N_{s}}\sum_{v=1}^{N_{s}}[F^{2}(s,v)]^{\frac{q}{2}}\right\\}^{\frac{1}{q}}$
(3)

For $q=2$, computation of $F_{2}(s)$ for different values of $s$ would
correspond to conventional method of Detrended Fluctuation Analysis(DFA) [51,
82, 52]. If the data series $x(i)$ is long range power correlated, then
$F_{q}(s)$ vs $s$ for a specific $q$, will display power-law behaviour and
$\log_{2}[F_{q}(s)]$ would depend on $\log_{2}s$ in a linear fashion, where
the slope for $q=2$ corresponds to the so-called Hurst exponent [51, 82, 52]
and the slope for other non-zero values of $q$ is denoted as the generalized
Hurst exponent. After extending Detrended Fluctuation Analysis (DFA) for
monofractal data series, Multifractal Detrended Fluctuation Analysis (MF-DFA)
method has been proposed for multifractal data series and the width of the
multifractal spectrum denotes the magnitude of the multifractality for self
similar data series [51, 82, 52].

DFA and MF-DFA [51, 82, 52] methods were proposed for one-dimensional self-
similar data series but in this experiment scaling analysis is based on the
components of the momemtum($GeV$) in the $X,Y$ and $Z$ co-ordinate of the
produced di-leptons. So, for this analysis, a two dimensional (2D) MF-DFA
algorithm is applied [1], where the data is represented as a two dimensional
matrix and the scale is chosen as square matrix segment. However, in most
real-life and natural data surfaces, their two-dimensional matrix
representations are not perfect square. Hence here we propose a (2D) MF-DFA
analysis for self-similar surfaces using a rectangular scale.

The proposed method is elaborated in the below steps.

  1. 1.

Consider $X$ is a self-similar surface of dimension $M\times N$ where $X(i,j)$
denotes the value of $i^{th}$ row and $j^{th}$ column. So, like equation. 1,
the integrated random walk surface or the profile surface $Y$ is calculated
as:

$Y(i,j)=\sum_{m=1}^{i}[X(m,j)-{\bar{X}}_{jc}]+\sum_{m=1}^{j}[X(i,m)-{\bar{X}}_{ir}]$
(4)

for $1\leq i\leq M$, $1\leq j\leq N$. Where, $Y(i,j)$ is the value $(i,j)$-th
element on the integrated matrix, $X(m,j)$ is the $m^{th}$ element on the
$j^{th}$ column, $X(i,m)$ is the $m^{th}$ element on the $i^{th}$ row.
${\bar{X}}_{jc}$ is the average of the values in $j^{th}$ column and
${\bar{X}}_{ir}$ is the average of the values of $i^{th}$ row.

  2. 2.

The integrated surface represented in matrix $Y(i,j)$ is partitioned into
$M_{ms}\times N_{ns}$ disjoint rectangular surfaces of the size $ms\times ns$.
Here, $M_{ms}=[\frac{M}{ms}]$, $N_{ns}=[\frac{N}{ns}]$ and $ms_{min}\leq
ms\leq ms_{max}$, $ns_{min}\leq ns\leq ns_{max}$. $ms\times ns$ can be
regarded as the scale, denoted by $S$, for which the fluctuation is measured.

  3. 3.

There are a number of factors to choose the maximum and minimum size of the
scale $S$ i.e. $ms\times ns$ as we need to repeat the below steps varying the
values of $S$ and this is important for the computation of local fluctuations
for each segments. Different arguments, statistical and phenomenological exist
for choosing the maximum and minimum segment size [52]. The statistical
argument is to choose minimum and maximum segment sizes that provide a
numerical stable estimation of variance function and different orders of
fluctuation function for each surface segment. Further, it is desirable to
have a equal spacing between scales when they are arranged in a log-log scale
i.e. $log(S)$ versus the $q^{th}$ order fluctuation function for an optimal
performance of the linear regression to estimate the $q^{th}$ order Hurst
exponent $H_{q}$. Considering the above we choose the below

     * •

$ms_{max}$ as mentioned in step 2 as $\frac{M}{4}$ is optimal to provide
sufficient number of segments in the computation.

     * •

$ns_{max}$ as mentioned in step 2 as $\frac{N}{4}$ is optimal to provide
sufficient number of segments in the computation.

     * •

$ms_{min}$ as mentioned in step 2 as $16$ is optimal to prevent over-fitting.

     * •

$ns_{min}$ as mentioned in step 2 as $16$ is optimal to prevent over-fitting.

     * •

Scale resolutions $S_{res}$ i.e. the number of different scale sizes for which
we repeat the above steps is chosen as $S_{res}=19$.

     * •

To have equal spacing between scale sizes following is done

       * –

We calculate $\hat{exp_{ms}}=linspace\\{log_{2}(ms_{min}),\\\
log_{2}(ms_{max}),S_{res}\\}$ where $\hat{exp_{ms}}$ is a vector with
$S_{res}$ elements placed at equal interval in log scale between
$log_{2}(ms_{min})$ and $log_{2}(ms_{max})$.

       * –

Similarly $\hat{exp_{ns}}=linspace\\{log_{2}(ns_{min}),\\\
log_{2}(ns_{max}),S_{res}\\}$ where $\hat{exp_{ns}}$ is a vector with
$S_{res}$ elements placed at equal interval in log scale between
$log_{2}(ns_{min})$ and $log_{2}(ns_{max})$.

       * –

We thereby get a vector $\hat{ms}$ of size $S_{res}$ according to the equation
$\hat{ms}=\lfloor{2^{\hat{exp_{ms}}}}\rfloor$ for each element in
$\hat{exp_{ms}}$.

       * –

Similarly a vector $\hat{ns}$ of size $S_{res}$ is obtained as per the
equation $\hat{ns}=\lfloor{2^{\hat{exp_{ns}}}}\rfloor$ for each element in
$\hat{exp_{ns}}$

       * –

Thus a new vector of scales $\hat{S}$ is calculated from $\hat{ms}$ and
$\hat{ns}$ where $S(i)=ms(i)\times ns(i)$ and $i\in[1,2\ldots S_{res}]$.

  4. 4.

For a particular $S(i)\in\hat{S}$ and corresponding $ms(i)\in\hat{ms}$,
$ns(i)\in\hat{ns}$, the full integrated surface $Y(i,j)$ can be partitioned
into smaller surface segments denoted by $Y_{k,l}$ of size $ms(i)\times ns(i)$
such that each element in surface $Y_{k,l}$ can be denoted by $Y_{k,l}(o,p)$
mapped to the original surface represented by $Y(i,j)$ such that —

     * •

$0\leq k<M_{ms}$ and $0\leq l<N_{ns}$, where $M_{ms}$ and $N_{ns}$ are
calculated as per the particular $ms(i)$ and $ns(i)$.

     * •

$1+(k\times ms(i))\leq o\leq(k+1)\times ms(i)$, $1+(l\times ns(i))\leq
p\leq(l+1)\times ns(i)$ i.e. here the generic sequence numbers denoted by $o$
and $p$, starts from $1+(k\times ms(i))$ and $1+(l\times ns(i))$ respectively
for particular values of $k$ and $l$ and are incremented by $1$ until $o$ and
$p$ reach the values of $(k+1)\times ms(i)$ and $(l+1)\times ns(i)$
respectively. Eg. for $k=0$ and $l=1$, $o=1,2,\dots ms(i)$ and
$p=1+ns(i),(1+ns(i))+1,\dots(2\times ns(i))$. This way set of $Y_{k,l}$ for
different values of $k$ and $l$ make up the original surface $Y(i,j)$ for the
particular scale $S$.

     * •

Each surface segment $Y_{k,l}$ in the full surface $Y(i,j)$, thereby becomes a
separate matrix of size $ms(i)\times ns(i)$ for each scale $S\in\hat{S}$ and
corresponding $ms(i)\in\hat{ms}$ and $ns(i)\in\hat{ns}$. Also each element of
the independent surface segment denoted by $Y_{kI,lI}$, can be denoted by
$Y_{kI,lI}(x,y)$ with $x=1,2\dots ms(i)$ and $y=1,2\dots ns(i)$, for $1\leq
kI\leq M_{ms}$ and $1\leq lI\leq N_{ns}$, where $M_{ms}$ and $N_{ns}$ are
calculated as per the particular $ms(i)$ and $ns(i)$ corresponding a specific
scale $S(i)\in\hat{S}$ as defined in the step 3.

  5. 5.

For each individual surface segment $Y_{kI,lI}$ as defined in the step 4,
corresponding to each scale $S(i)\in\hat{S}$, $ms(i)\in\hat{ms}$,
$ns(i)\in\hat{ns}$ and $i\in[1,2\ldots S_{res}]$ as defined in the step 3, we
can obtain a local trend $\tilde{Y}_{kI,lI}$ by fitting it with a pre-chosen
bi-variate polynomial function. The simplest function could be a plane in
equation 5. In this work, we have adopted this trending function.

$\tilde{Y}_{kI,lI}(x,y)=Ax+By+C$ (5)

where $x=1,2\dots ms(i)$ and $y=1,2\dots ns(i)$ and $A,B$ and $C$ are
parameters to be determined and can be estimated easily through simple matrix
operations, derived from the least-square method.

  6. 6.

For each $Y_{kI,lI}$ the residual matrix $R_{kI,lI}$ obtained by subtracting
the least-square fitted bi-variate polynomial function for the particular
surface segment, can be determined as per equation 6, for a particular scale
$S(i)\in\hat{S}$.

$R_{kI,lI}(x,y)=[Y_{kI,lI}(x,y)-\tilde{Y}_{kI,lI}(x,y)]$ (6)

  7. 7.

For each scale $S(i)\in\hat{S}$, and corresponding $ms(i)\in\hat{ms}$,
$ns(i)\in\hat{ns}$ for $i\in[1,2\ldots S_{res}]$, de-trending is performed as
per equation 7, by subtracting the least-square fitted surface segment
$\tilde{Y}_{kI,lI}(x,y)$ from the corresponding actual surface segment
$Y_{kI,lI}(x,y)$ to deduce the variance.

$F^{2}_{kI,lI}(S(i))=\frac{1}{s_{n}}\sum_{x=1}^{ms(i)}\sum_{y=1}^{ns(i)}{[R_{kI,lI}(x,y)}]^{2}$
(7)

where count of elements in $Y_{kI,lI}(x,y)$ is denoted by $s_{n}=ms(i)*ns(i)$.
This variance $F^{2}_{kI,lI}(S(i))$ is calculated for all the surface segments
$Y_{kI,lI}$, where $kI=1,2,\ldots M_{ms}$ and $lI=1,2,\ldots M_{ns}$ and
$M_{ms}$ and $N_{ns}$ are calculated for each of the $ms(i)$ and $ns(i)$
corresponding to a particular $S(i)\in\hat{S}$. Hence for a particular $S(i)$,
there would be $M_{ns}*N_{ns}$ number of $F^{2}_{kI,lI}(S(i))$-s for the same
number of surface segments $Y_{kI,lI}$ for the full integrated surface
$Y(i,j)$.

The steps 5, 6, 7 are repeated each $S(i)\in\hat{S}$ where $i=1,2\ldots
S_{res}$ and for each scale a set of $M_{ns}*N_{ns}$ number of
$F^{2}_{kI,lI}(S(i))$-values, is calculated.

  8. 8.

Then the $q^{th}$ order fluctuation of fluctuation, denoted by $F_{q}(S(i))$
is calculated by averaging the values of $[F^{2}_{kI,lI}(S(i))]^{\frac{q}{2}}$
corresponding to $F^{2}_{kI,lI}(S(i))$-values calculated for all surface
segments $Y_{kI,lI}$ as per step 7, where $i=1,2\ldots S_{res}$. This $q^{th}$
order fluctuation of fluctuation function is defined as per the equation 8.

$F_{q}(S(i))=\left[\frac{1}{M_{ms}*N_{ns}}\sum_{kI=1}^{M_{ms}}\sum_{lI=1}^{N_{ns}}[F^{2}_{kI,lI}(S(i))]^{\frac{q}{2}}\right]^{\frac{1}{q}}$
(8)

where

     * •

$M_{ms}$ and $N_{ns}$ are as calculated for each of the $ms(i)\in\hat{ms}$ and
$ns(i)\in\hat{ns}$ corresponding to each scale $S(i)\in\hat{S}$ with
$i=1,2\ldots S_{res}$, as per the method specified in step 2 and 3.

     * •

For $q=0$ the values of $q^{th}$ order fluctuation of fluctuation -
$F_{q}(S(i))$ are not affected by the values of segments with small and large
values of variance function $F^{2}_{kI,lI}(S(i))$. Also for $q=0$ the
$\frac{1}{q}$ would blow up to infinity. Hence, for $q=0$, $F_{0}(S(i))$ is
calculated as per the equation9

$F_{0}(S(i))=e^{\left[0.5*\frac{1}{M_{ms}*N_{ns}}*\log(F^{2}_{kI,lI}(S(i)))\right]}$
(9)

for each scale $S(i)\in\hat{S}$.

     * •

For $q=2$, computation of $F_{2}(S(i))$ for all the values of $S(i)\in\hat{S}$
would correspond to conventional method of Detrended Fluctuation Analysis
(DFA) [51].

     * •

In this experiment $q$ varies from $(-5)$ to $(+5)$.

  9. 9.

After executing the above steps, it is observed that for a particular $q$,
$F_{q}(S(i))$ increases with increasing $S(i)$. It must be noted here that
each scale $S(i)$ is actually the area of corresponding rectangular scale with
dimensions of $ms(i)$ and $ns(i)$, which amounts to $ms(i)*ns(i)$. If the data
surface is long range power correlated, then $F_{q}(S(i))$ vs $S(i)$ for a
specific $q$, will display power-law behaviour as per the equation[ 10].

$\displaystyle F_{q}(S(i))\propto S(i)^{h(q)}$ (10)

where $S(i)\in\hat{S}$ with $i=1,2\ldots S_{res}$. If this type of scaling
exists then $\log_{2}[F_{q}(S(i))]$ would depend on $\log_{2}S(i)$ in a linear
fashion, where $h(q)$ is the slope which is dependent on $q$. Here $h(2)$ is
analogous to the so-called Hurst exponent [82] and $h(q)$ is termed as the
generalized Hurst exponent.

  10. 10.

The scaling trend of the variance function $F^{2}_{kI,lI}(S(i))$ is similar
for all surface segments in case of a monofractal surface. To put differently,
the averaging of $F^{2}_{kI,lI}(S(i))$ would exhibit consistent scaling
pattern for various values of $q$ and hence for monofractal surfaces $h(q)$
becomes independent of $q$.

But, if small and large fluctuations in the surface have differing scaling
pattern, then $h(q)$ becomes significantly dependent on $q$. In these
scenarios, for values of $q>0$, $h(q)$ depicts the scaling behaviour of the
surface segments with large fluctuations and for values of $q<0$, $h(q)$
depicts scaling pattern of the surface segments with smaller fluctuations. The
generalized Hurst exponent $h(q)$ for a multifractal surfaces is related with
the classical multifractal scaling exponent $\tau(q)$ as per the equation 11.

$\tau(q)=qh(q)-1$ (11)

  11. 11.

As multifractal surface has a range of Hurst exponents, $\tau(q)$ depends upon
$q$ [83] in a nonlinear fashion. The singularity spectrum, here denoted
by$f(\alpha)$, is related to $h(q)$ as per the equation[ 12].

$\displaystyle\alpha=h(q)+qh^{\prime}(q),f(\alpha)=q[\alpha-h(q)]+1$ (12)

Here the singularity strength is represented by $\alpha$ and $f(\alpha)$
represents the dimension of the subset of surface segments corresponding to
$\alpha$. Different values of $f(\alpha)$ corresponding to different values of
$\alpha$ result into multifractal spectrum of $f(\alpha)$ which forms an arc
and the difference between the maximum and minimum values of $\alpha$ for this
spectrum, is the width of the multifractal spectrum or the magnitude of the
multifractality of the input self-similar surface, here denoted by $X$.

  12. 12.

For $q=2$, if $h(q)$ or $h(2)=0.5$ there is no correlation in the data
surface. $h(2)>0.5$ infers that a high value of $X(i,j)\in X$ is followed by
another large value in the surface or there is existence of persistent long-
range cross-correlations in the surface. However if $h(2)<0.5$, there must be
anti-persistent long-range correlations existing in the surface suggesting a
high value of $X(i,j)\in X$ would possibly be followed by a small value in the
surface and vice verse.

##  3 Experimental details

Two experimental primary datasets which are made publicly available by CMS
collaboration are used for the the proposed analysis. The detailed
specification of the data is given in Section 3.1 and the detailed method of
the experiment is elaborated step by step in Section 3.2.

###  3.1 Data description

From the CMS collaboration, the MultiJet primary dataset of $pp$ collisions in
AOD format from RunB of 2010 at $7$ TeV [2], is extracted and considered as
the source data-set of first level, for this experiment. During data selection
procedure all the runs recorded by CMS collaboration, are qualified as good
for physics analysis if all sub-detectors, trigger (an algorithm to select
particle collisions to be stored in the primary dataset and rest are
discarded), lumi and physics objects like electron, muon, photon, jet etc.
show the expected performance. First level of validation is done based on the
evaluations of off-line shifters and then the feedback is taken from the
detector and experts from Physics Object Group(POG). All these information is
stored in a specific database named Run Registry. The CMS Data Quality
Monitoring group validate the integrity of the certification and creates a
(.json) file of qualified runs to be utilized for physics analysis. The
(.json) file link1, describe luminosity sections containing the good runs
which have been selected from the primary dataset of events for further
processing. This is the second level of selection. Then the filter is run over
the MultiJet primary dataset from the CMS open data, after the second level of
selection, and the events with jet candidates with certain parameters having
the required threshold values are selected and resultant files in (.root) and
(.csv) formats are created. The jets are reconstructed and grouped into two
mega-jets - one leading mega-jet with largest transverse momentum($p_{T}$) and
another subleading mega-jet with largest transverse momentum($p_{T}$). The
mega-jets are derived as sum of the $4$-momenta of their constituent
particles. From the reconstructed jets from the events with large degree of
missing transverse energy, the razor variables $M_{R}$ and $R^{2}$ are
calculated to be used in Super-Symmetric particle searches [4]. This is the
final level of selection. In this analysis we have used this set of $X,Y$ and
$Z$ components taken out from the $4$-momenta of final state signatures of the
constituent di-muons from the pair of mega-jets. This is the data space for
Super-Symmetric data.

Similarly, the primary dataset of $pp$ collisions at $7$ TeV from RunA of 2011
[5] of the CMS collaboration, in AOD format is taken as another source dataset
in this experiment. Events stored in this dataset were selected due of the
presence of precisely two global muons in the event with specific range of
invariant mass. This is the first level of selection. Similar to the case of
MultiJet primary dataset from CMS collaboration, here also the (.json) file is
created after validating the certification by the CMS Data Quality Monitoring
group, is gives in the link - link2 for $7$ TeV. This (.json) file describes
the luminosity sections containing the good runs which have been selected from
the primary dataset for further processing. This is the second level of
selection. Here also, the output containing di-muon event information is
extracted in both (.root) and (.csv) format from the collision datasets. Then
the component-space(in the $X,Y,Z$ co-ordinates) taken out from the
$4$-momenta of final state signatures of the produced di-muons, is extracted
from output dataset generated from the $pp$ collisions data at $7$ TeV from
CMS collaboration [5]. This is the data space for non-Super-Symmetric data.

In this analysis we have used these two set of component-spaces(in the $X,Y,Z$
co-ordinates) taken out from the $4$-momenta of final state signatures of the
produced multi particle data - one Super-Symmetric and another non-Super-
Symmetric data.

###  3.2 Data analysis and results

####  3.2.1 Data Preparation

The two sets of component-space(in the $X,Y,Z$ co-ordinates) taken out from
the $4$-momenta of final state signatures of the produced multi particle data,
are extracted from the datasets created out of the MultiJet primary dataset
from RunB of 2010 at $7$ TeV [2] for supersymmetry data and the primary
dataset of $pp$ collisions at $7$ TeV from RunA of 2011 [5] for $pp$ collision
data from the CMS collaboration for non-Super-Symmetric data.

For each of the $2$ data surfaces(one Super-Symmetric and another non-Super-
Symmetric) two-dimensional matrix is constructed as per the definition of $X$,
elaborated in the step 1 in the Section 2.1. Here, each element of the matrix
defined as $X(i,j)$, in the step 1 in the Section 2.1, where $i$ is the value
of the $X$-component of the $4$-momenta of final-state-signature of the first
muon of the produced di-muons of a single extracted event, $j$ is the value of
the $Y$-component of the $4$-momenta of final-state-signature of the same muon
and the value of the element $X(i,j)$ is the value of the $Z$-component of the
$4$-momenta of final-state-signature of the same muon. For each element of the
matrix $X(i,j)\in X$, $i$ denotes the sequence of the row and $j$ denotes the
sequence of column and $X(i,j)$ is the value of the element. The element next
to $X(i,j)$ in the matrix would be defined by the $X,Y$ and $Z$ components of
the $4$-momenta of final-state-signature of the second muon of the produced
di-muons of the same extracted event.

This way, the matrix $X$ would be defined with the $X,Y$ and $Z$ components of
the $4$-momenta of final-state-signature of the produced di-muons from each
extracted event for each of the $2$ datasets. Each of the $2$ matrices
corresponding to $2$ datasets, is of dimension $M\times N$ where

  * •

$M=($Maximum of the values of $X$-component of the $4$-momenta of final-state-
signature of the produced di-muons for a dataset$)-($Minimum of the values of
$X$-component$)+1$.

  * •

$N=($Maximum of the values of $Y$-component of the $4$-momenta of final-state-
signature of the produced di-muons for a dataset$)-($Minimum of the values of
$Y$-component$)+1$.

For each of the $2$ data surfaces (one Super-Symmetric and another non-Super-
Symmetric) of $X,Y$ and $Z$ components of the $4$-momenta of final-state-
signature of the produced di-muon data extracted from the MultiJet primary
dataset from RunB of 2010 at $7$ TeV [2] and the primary dataset of $pp$
collisions at $7$ TeV from RunA of 2011 [5] for $pp$ collision data from the
CMS collaboration, $3$ sets of $2$ two-dimensional matrices are defined
following the above instructions, as described below.

  1. 1.

Two-dimensional matrix is constructed as per the definition of $X$, defined in
the step 1 the Section 2.1, where for each element $X(i,j)$, $i=$ the value of
the $X$-component, $j=$ the value of the $Y$-component and $X(i,j)=$ value of
the corresponding $Z$-component. These are denoted by $[x,y,z]_{ss}$ and
$[x,y,z]_{pp}$.

  2. 2.

Two-dimensional matrix is constructed as per the definition of $X$, defined in
the step 1 the Section 2.1, where for each element $X(i,j)$, $i=$ the value of
the $X$-component, $j=$ the value of the $Z$-component and $X(i,j)=$ value of
the corresponding $Y$-component. These are denoted by $[x,z,y]_{ss}$ and
$[x,z,y]_{pp}$.

  3. 3.

Two-dimensional matrix is constructed as per the definition of $X$, defined in
the step 1 the Section 2.1, where for each element $X(i,j)$, $i=$ the value of
the $Y$-component, $j=$ the value of the $Z$-component and $X(i,j)=$ value of
the corresponding $X$-component. These are denoted by $[y,z,x]_{ss}$ and
$[y,z,x]_{pp}$.

Once the all the two-dimensional, rectangular matrices are constructed for
each of the experimental dataset, the (2D) MF-DFA analysis is done and the
width of (2D) MF-DFA spectrum is deduced as per the method elaborated in
Section 2.1, for each of the matrix data. The findings and inferences are
listed as follows.

####  3.2.2 Observations and inferences

  1. 1.

The $q^{th}$ order fluctuation of fluctuation $F_{q}(S(i))$ of each of the
surfaces (here represented by matrices) is calculated as per the equation 8 of
the step 8 of the (2D) MF-DFA methodology as elaborated in Section 2.1
corresponding to each scale $S(i)\in\hat{S}$ with $i=1,2\ldots S_{res}$. The
Figure 3-(a) and 3-(b) show the trend of $log_{2}[F_{q}(S(i))]$ vs
$log_{2}[S(i)]s$ for $q=-5,0,5$, computed for $[x,y,z]_{ss}$ and
$[x,y,z]_{pp}$ for all the scales $S(i)\in\hat{S}$ with $i=1,2\ldots S_{res}$.
Similarly, the Figure 3-(a) and 3-(b) show the same for $[x,z,y]_{ss}$ and
$[x,z,y]_{pp}$ and the Figure 3-(a) and 3-(b) show the same for $[y,z,x]_{ss}$
and $[y,z,x]_{pp}$.

Their linear trend of $log_{2}[F_{q}(S(i))]$ vs $log_{2}[S(i)]s$ for all the
values of $q$ establishes the power law trend of $F_{q}(S(i))$ versus $S(i)$
for the $q^{th}$ orders. This trend in turn confirms the self-similarity and
long range power correlation of different orders of the experimental data
surfaces. This is true for both Super-Symmetric and non-Super-Symmetric $pp$
collision data at $7$ TeV, which is evident from the Figures.

(a) (b)

Figure 1: Trend of $log_{2}[F_{q}(S(i))]$ vs $log_{2}[S(i)]s$ for $q=-5,0,5$,
computed for (a) $[x,y,z]_{ss}$ (b) $[x,y,z]_{pp}$.

(a) (b)

Figure 2: Trend of $log_{2}[F_{q}(S(i))]$ vs $log_{2}[S(i)]s$ for $q=-5,0,5$,
computed for (a) $[x,z,y]_{ss}$ (b) $[x,z,y]_{pp}$.

(a) (b)

Figure 3: Trend of $log_{2}[F_{q}(S(i))]$ vs $log_{2}[S(i)]s$ for $q=-5,0,5$,
computed for (a) $[y,z,x]_{ss}$ (b) $[y,z,x]_{pp}$.

  2. 2.

As already elaborated in step 9 of the (2D) MF-DFA methodology in Section 2.1,
that for $q=2$, computation of $F_{2}(S(i))$ for all the values of
$S(i)\in\hat{S}$ would correspond to conventional method of Detrended
Fluctuation Analysis(DFA) [51] and if the trend of $F_{q}(S(i))$ versus $S(i)$
is in accordance with the power law, then the slope of $\log_{2}[F_{2}(S(i))]$
versus $\log_{2}S(i)$ with straight-line fitting, denoted by $h(2)$ is the
Hurst exponent [82] and the slope of $\log_{2}[F_{q}(S(i))]$ versus
$\log_{2}S(i)$ with straight-line fitting, denoted by $h(q)$ is the
generalized Hurst exponent.

Table 1 Hurst exponent calculated for $6$ experimental data
surfaces(matrices)[$3$ for Super-Symmetric data and $3$ for non-Super-
Symmetric $pp$ collision data at $7$ TeV] and their randomized versions.
Hurst Exponent $[x,y,z]_{ss}$ $[x,y,z]_{pp}$ Original $0.320\pm 0.020$
$0.160\pm 0.020$ Randomized $0.011\pm 0.002$ $0.009\pm 0.001$ $[x,z,y]_{ss}$
$[x,z,y]_{pp}$ Original $0.250\pm 0.020$ $0.130\pm 0.020$ Randomized $0.013\pm
0.002$ $0.008\pm 0.001$ $[y,z,x]_{ss}$ $[y,z,x]_{pp}$ Original $0.250\pm
0.020$ $0.130\pm 0.020$ Randomized $0.013\pm 0.002$ $0.008\pm 0.001$

For each of the matrices corresponding to $6$ data surfaces ($3$ Super-
Symmetric and $3$ non-Super-Symmetric $pp$ collision data at $7$ TeV)
constructed as per the method described in step 3.2.1 in the current section,
a randomized or shuffled version of data surface or matrix is generated and
for all of them. The randomization is done by keeping the $Z$ values the same
and randomizing the matrix co-ordinates they are placed into. Then the same
(2D) MF-DFA analysis is done and the width of (2D) MF-DFA spectrum is
calculated. Hurst exponent calculated for each of the data surfaces and their
randomized version is listed in the Table 2.

From the Table 2, it’s evident that for the data surfaces constructed out of
$X$ from $3$ perspectives - $[x,y,z]$, $[x,z,y]$, $[y,z,x]$, as mentioned in
the step 3.2.1, that Hurst exponents or $h(2)$-s are $<0.5$. This signifies
that for both data surfaces(Super-Symmetric and non-Super-Symmetric) contain
anti-persistent long-range correlations. This means, high value of $X(i,j)\in
X$ would possibly be followed by a small value in the surface and vice verse
for all the data surfaces constructed for the $3$ perspectives.

Further all Hurst exponents calculated for experimental surfaces are
substantially different from those calculated for the randomized ones. This
signifies that the inherent anti-persistent long-range correlation and also
self-similarity of the data surfaces indicated by the power law behaviour of
the data surfaces with respect to the scale, in the experimental surface is
not the consequence of shuffling or randomization. Rather these are
consequence of the inherent dynamics of the data surfaces. Hence this
noticeably different scaling behaviour between experimental and randomized
data establishes the statistical significance of the result of the experiment.

Lastly, it’s also noted in the Table 2, that the values of the Hurst exponent
calculated for the Super-Symmetric surfaces is significantly more(around
$50\%$) than those of non-Super-Symmetric surfaces. This indicates that the
anti-persistent long-range correlation and self-similarity inherent in the
data surfaces changes significantly from non-Super-Symmetric surfaces to the
Super-Symmetric ones and this is an unusual observation arising out of the
dynamics of Super-Symmetric data.

  3. 3.

For each of the $3$-pairs data surfaces (one Super-Symmetric and another non-
Super-Symmetric for each pair) and their corresponding randomized versions,
the widths of the (2D) multifractal spectrum are calculated as per the steps
9, 10 and 11 following the (2D) MF-DFA method described in Section 2.1. The
values of the the widths of the (2D) multifractal spectrum of the experimental
and their randomized version for each pair of the data surfaces are compared.

(a) (b)

Figure 4: Comparison of the trend of different values of $f(\alpha)$ versus
$\alpha$ between the experimental and the randomized version for the data
surfaces (a) $[x,y,z]_{ss}$ (b) $[x,y,z]_{pp}$.

(a) (b)

Figure 5: Comparison of the trend of different values of $f(\alpha)$ versus
$\alpha$ between the experimental and the randomized version for the data
surfaces (a) $[x,z,y]_{ss}$ (b) $[x,z,y]_{pp}$.

(a) (b)

Figure 6: Comparison of the trend of different values of $f(\alpha)$ versus
$\alpha$ between the experimental and the randomized version for the data
surfaces (a) $[y,z,x]_{ss}$ (b) $[y,z,x]_{pp}$.

In the Figures 6-(a) and 6-(b), the trends of different values of $f(\alpha)$
versus $\alpha$ extracted for the (2D) multifractal spectrum of the
experimental data surfaces and their randomized versions, are shown for Super-
Symmetric data surface $[x,y,z]_{ss}$ and non-Super-Symmetric data surface
$[x,y,z]_{pp}$ respectively. The same trend for Super-Symmetric data surface
$[x,z,y]_{ss}$ and non-Super-Symmetric data surface $[x,z,y]_{pp}$ are shown
in the Figures 6-(a) and 6-(b). And for Super-Symmetric data surface
$[y,z,x]_{ss}$ and non-Super-Symmetric data surface $[y,z,x]_{pp}$, the same
the trends of different values of $f(\alpha)$ versus $\alpha$ are shown in
Figures 6-(a) and 6-(b). The widths of (2D) multifractal spectrum for all the
experimental data surfaces and their corresponding randomized versions are
deduced as per step 11 in Section 2.1 and listed in Table 3.

Table 2 Widths of (2D) multifractal spectrum calculated for $6$ experimental
data surfaces(matrices)[$3$ for supersymmetry data and $3$ for $pp$ collision
data at $7$ TeV] and their randomized versions.

Width of (2D) multifractal spectrum
---
| $[x,y,z]_{ss}$ | $[x,y,z]_{pp}$
Original | $0.760$ | $0.640$
Randomized | $0.095$ | $0.151$
| $[x,z,y]_{ss}$ | $[x,z,y]_{pp}$
Original | $0.730$ | $0.710$
Randomized | $0.156$ | $0.255$
| $[y,z,x]_{ss}$ | $[y,z,x]_{pp}$
Original | $0.750$ | $0.710$
Randomized | $0.135$ | $0.135$

Comparison of the trend of different values of generalized Hurst exponent
$h(q)$ versus $q$ for $q$ for $q=-5,\ldots 5$ between the Super-Symmetric and
non-Super-Symmetric data surfaces constructed out from the $3$ perspectives -
$[x,y,z]$, $[x,z,y]$, $[y,z,x]$ are shown in Figure 7-(a), (b) and (c)
respectively. The values of generalized Hurst exponent $h(q)$ for $q=-5,\ldots
5$, are calculated as per the steps 7, 8, 9 in Section 2.1.

The observations and inferences from the Figures and the Table are listed
below

  * •

It must be noted that the shape of all the (2D) multifractal spectrum (figures
6,6 and 6), denoted by the trends of different values of $f(\alpha)$ versus
$\alpha$, is non-symmetric. All the spectrum are truncated from the left which
signifies that the trend of the generalized Hurst exponents-$h(q)$s calculated
for positive values of $q(q>0)$-s for all the data surfaces, is almost
consistent or uniform in nature. The almost uniform trend of generalized Hurst
exponents for $q(q>0)$-s for the Super-Symmetric and non-Super-Symmetric data
surfaces constructed out from the $3$ perspectives - $[x,y,z]$, $[x,z,y]$,
$[y,z,x]$, is evident from the Figure 7. This consistent trend of generalized
Hurst exponents-$h(q)$s for $q(q>0)$-s indicate that the $q^{th}$ order
fluctuation of fluctuation function $F_{q}(S(i))$ is not much affected by the
extent of the local fluctuations for positive values of $q$. Also the (2D)
multifractal spectrum have long right tails indicating the (2D) multifractal
structures of the data surfaces are not affected by the higher magnitudes of
the local fluctuations.

(a) (b)

(c)

Figure 7: Comparison of the trend of different values of generalized Hurst
exponent $h(q)$ versus $q$ for $q$ for $q=-5,\ldots 5$ between the Super-
Symmetric and non-Super-Symmetric data surfaces (a) $[x,y,z]_{ss}$ and
$[x,y,z]_{pp}$ (b) $[x,z,y]_{ss}$ and $[x,z,y]_{pp}$ (c) $[y,z,x]_{ss}$ and
$[y,z,x]_{pp}$

  * •

The long range correlation inherent in the experimental data, is eliminated by
the process of randomization making the experimental data uncorrelated. This
makes the width of (2D) multifractal spectrum signifying the trend of scaling
pattern for different orders($q$-s), substantially less for the of randomized
version of the data surfaces than the experimental ones.

In this experiment the shape of the (2D) multifractal spectrum for randomized
version is similar to that of experimental ones. This implies for both the
experimental data surfaces and their randomized versions signifying the (2D)
multifractal structures of the data surfaces and their randomized versions are
not affected by the large values of the local fluctuations, as explained in
the previous point. However, it’s evident from the Table 3 that the widths of
the (2D) multifractal spectrum of the experimental data is significantly more
than that of their randomized versions. This establishes the statistical
significance of the experimental result.

Hence, the widths of the (2D) multifractal spectrum computed for experimental
data surfaces being noticeably different from their randomized ensembles,
fundamentally establishes that the degree of scale-freeness is manifestation
of the inherent dynamics in the di-muon production process for both Super-
Symmetric and non-Super-Symmetric data surfaces and not the characteristic of
the dynamics of the randomized version and the values of the degree of scale-
freeness indicate various and unusual processes responsible for di-muon
production

  * •

It’s evident from the Figure 7 that there exists a consistently changing trend
of $h(q)$ with $q$, though for $q<0$, $h(q)$ changes in a higher rate than for
$q>0$. But this consistently changing behaviour of $h(q)$ with $q$ indicates
the multifractal behaviour of the experimental data surfaces. Comparison of
the trend of different values of generalized Hurst exponent $h(q)$ versus $q$
for $q$ for $q=-5,\ldots 5$ between the Super-Symmetric and non-Super-
Symmetric data surfaces show that $h(q)$s for Super-Symmetric surfaces are
around $50-100\%$ more than those calculated for the non-Super-Symmetric data
surfaces for all $q$-s.

  * •

Finally, if we compare the widths of the (2D) multifractal spectrum computed
from the $3$ perspectives - $[x,y,z]$, $[x,z,y]$, $[y,z,x]$ for the range of
$q=-5,\ldots 5$ for the Super-Symmetric data surfaces with those calculated
for corresponding non-Super-Symmetric data surfaces in Table 3, we find that
the widths for Super-Symmetric data surfaces is around $10\%$(average) more
than those of corresponding non-Super-Symmetric data surfaces.

##  4 Conclusion

For the past $60$ years Super-Symmetry has been been area of immense interest
as it is the most encouraging theory in the are of high energy physics. The
LHC has been enabling ATLAS and CMS collaborations to prove for it in various
high energy interactions of $pp$ collisions at $7$, $8$, $13$ TeV and so on.
Several hundreds of searches have been conducted to enrich our understanding
of Nature. For the past few decades, substantial amount of missing transverse
momentum($p_{T}$) has been studied as the most potential observable for
detecting the generation and decay of Super-Symmetric particles at different
experiments at the colliders. At the LHC, the standard models of Super-
Symmetry obeying the $R$-parity conservation predict signatures with jets
originating from decaying pairs of squarks or gluinos, Super-Symmetric
partners of quarks and gluons, and missing transverse momentum($p_{T}$)
originating from undetected weakly communicating lightest Super-Symmetric
particles or LSP. However, searches for such events have not been quite
encouraging in displaying the signature of Super-Symmetry and also have not
yielded strongest bounds as per the considered details of the Super-Symmetric
models. The majority these bounds or limits obtained are considered as lower
limit on the mass parameter. It has been concluded that Super-Symmetric
partners must have masses higher than the limits deduced by the analysis.
However, searches for such events have not been quite encouraging in
displaying the signature of Super-Symmetry and also have not yielded strongest
bounds as per the considered details of the Super-Symmetric models. In
summary, most of these decay processes share a distinct feature among
themselves - the existence of many high energy jets originating from the
expected high degree of difference between gluinos and the so-called daughter
particles, and obviously, significant amount of missing transverse
momentum($p_{T}$) yielding from the couple of high momentum LSP which escapes
detection. At the LHC, there have been non-stop searches for Super-Symmetry
for prompt as well as non-prompt, for R-parity conserving as well as R-parity
violating generation and decays. From the results of these searches, it is
inferred that in these searches, LHC excludes the presence of gluinos below
$2$ TeV, and existence of stops and gauginos below $1$ TeV. These checks or
limits showed greater possibilities of the experiments in the collider.
However, these signatures of Super-Symmetric particles are commonly derived
under the assumption of a bit optimistic scenario where sparticles decay in to
the final states under the study, with a $100\%$ branching fraction. The fact
is that Super-Symmetry might have been in a disguised state at lower mass-
scales as a results of difficult and challenging mass spectra and mixed modes
of decays.

In the proposed investigation, we have extended (2D)(MF-DFA) proposed in [1],
by selecting rectangular scale for doing Multifractal analysis for
experimental data surfaces made up of the component-space(in the $X,Y,Z$ co-
ordinates) taken out from the $4$-momenta of final state signatures of the
produced di-muons for - Super-Symmetric data from the MultiJet primary $pp$
collisions dataset from RunB of 2010 at $7$ TeV and non-Super-Symmetric data
from the primary dataset of $pp$ collisions at $7$ TeV from RunA of 2011 of
the CMS collaboration, the details of which is written in Section 3.1. The
fundamental dynamics obtained from the signatures of final state particles of
the di-lepton production process is reflected through the nonlinear and non-
stationary parameters of symmetry based scaling analysis done using the novel
method of extended (2D)(MF-DFA) method. The scaling exponent and other
parameters differ substantially from non-Super-Symmetric to Super-Symmetric
data which signifies that there is significant change in scaling behaviour
from non-Super-Symmetric to Super-Symmetric data, as evident from the figures
and table. This may be attributed to the occurrence of some unusual phenomena
like Super-Symmetry.

We have analyzed how the inherent pattern of scaling obtained from the
signatures of final state particles has developed in the multi particle
production process from non-Super-Symmetric to Super-Symmetric $pp$ collisions
at $7$ TeV from the CMS collaboration and the inferences are summarised below.

  1. 1.

The Table 2 shows that the Hurst exponents denoted by $h(2)$ calculated from
the slope of $log_{2}[F_{q}(S(i))]$ vs $log_{2}[S(i)]s$ for $q=2$ obtained
from the straight-line fitting for all the data surfaces (experimental and
their randomised versions) from the $3$ perspectives - $[x,y,z]$, $[x,z,y]$,
$[y,z,x]$, corresponding to both Super-Symmetric and non-Super-Symmetric $pp$
collision data at $7$ TeV, are all $<0.5$. This indicates that all the Super-
Symmetric and non-Super-Symmetric data surfaces have inherent anti-persistent
long-range correlations. Table 2 shows that the values of the Hurst exponent
calculated for the Super-Symmetric surfaces is significantly more (around
$50\%$) than those of non-Super-Symmetric surfaces from the $3$ perspectives.
This indicates that the anti-persistent long-range correlation and self-
similarity inherent in the data surfaces changes significantly from non-Super-
Symmetric surfaces to the Super-Symmetric ones and this is an unusual
observation arising out of the dynamics of Super-Symmetric data.

Also, significantly different values of Hurst exponent between all the
experimental data surfaces and their randomized versions establishes the
statistical significance of the result of the experiment.

  2. 2.

The linear trend of $log_{2}[F_{q}(S(i))]$ vs $log_{2}[S(i)]s$ for $q=-5,0,5$
calculated for the data surfaces from the $3$ perspectives - $[x,y,z]$,
$[x,z,y]$, $[y,z,x]$, corresponding to both Super-Symmetric and non-Super-
Symmetric $pp$ collision data at $7$ TeV, shown in the Figures 3, 3 and 3,
establish both fractal and multifractal nature of the data surfaces. This
linear trend also confirms the self-similarity and long range power
correlation of different orders of the experimental data surfaces.

  3. 3.

It must be noted that all the (2D) multifractal spectrums generated from
different values of $f(\alpha)$ versus $\alpha$ for all the experimental and
their randomized version of both Super-Symmetric and non-Super-Symmetric data
surfaces, are of similar non-symmetric shape. The (2D) multifractal spectrum
have long right tails indicating the (2D) multifractal structures of the data
surfaces are not affected by the higher magnitudes of the local fluctuations.
However, for Super-Symmetric and non-Super-Symmetric data, it’s evident from
the Table 3 and Figures 6-(a) and 6-(b) for $[x,y,z]$ perspective, Figures
6-(a) and 6-(b) for $[x,z,y]$ perspective and Figures 6-(a) and 6-(b) for
$[y,z,x]$ perspective, that the widths of the (2D) multifractal spectrum of
the experimental data is significantly different than that of their randomized
versions. It indicates that long-range correlation and the degree of scale-
freeness inherent in the experimental data is completely different from the
inherent dynamics of its randomised version. This establishes the statistical
significance of the experimental result.

Also comparison of the widths of the (2D) multifractal spectrum computed for
the Super-Symmetric data surfaces with those calculated for corresponding non-
Super-Symmetric data surfaces, from the $3$ perspectives, listed in Table 3,
reveals that the structural difference between sections of small and large
fluctuations of the Super-Symmetric data surfaces is more than that of
corresponding non-Super-Symmetric data surfaces. This structural difference
yields to around $10\%$(average) higher value of the widths of the (2D)
multifractal spectrum computed for the Super-Symmetric data surfaces than the
same computed for the non-Super-Symmetric data surfaces.

  4. 4.

Interestingly, it must also be noted from the comparison of the trend of
different values of generalized Hurst exponent $h(q)$ calculated for different
$q$-orders for $q=-5,\ldots 5$ between the Super-Symmetric and non-Super-
Symmetric data surfaces constructed out from the $3$ perspectives - $[x,y,z]$,
$[x,z,y]$, $[y,z,x]$, that $h(q)$-s for Super-Symmetric data are consistently
$50-100\%$ more than those of non-Super-Symmetric data. This difference is
evident from the Figures 7-(a), (b) and (c).

As we know that for multifractal data surfaces $q^{th}$ order $h(q)$ for $q>0$
reflects the scaling behaviour of the surface segments with large fluctuations
and for values of $q<0$, $h(q)$ depicts scaling pattern of the surface
segments with smaller fluctuations, hence we can confirm that the self-
similarity and the scaling pattern reflected by the Hurst exponent or $h(2)$
and also generalized Hurst exponents or $h(q)$-s inherent in the non-Super-
Symmetric data are substantially different than those of Super-Symmetric data.
We may infer that this difference is yielded from the unusual dynamics of
Super-Symmetric interactions for $pp$ collision at $7$ TeV from CMS
collaboration, which is manifested in the component-space(in the $X,Y,Z$ co-
ordinates) taken out from the $4$-momenta of final state signatures of the
produced di-muons.

We can conclude that the noticeable changes in scaling behaviour, as displayed
in the figures and table, may be designated to the occurrence of various types
of resonance-like states or other unusual phenomena. So, the proposed novel
method of two-dimensional Multifractal analysis using rectangular scales is a
robust one and is rigorous enough to identify different types of resonance-
like states or other unusual phenomena like Super-Symmetry which cannot be
identified by the conventional method of analysing the spectrum of invariant
mass/transverse momentum. Moreover, in regards to the advantages of proposed
method, we may emphasize that the proposed method can identify prospective
Super-Symmetry which the conventional methods of analysing invariant mass
spectrum/transverse momentum may miss, let alone the simplicity and novelty of
the analysis methodology using the component-space(in the $X,Y,Z$ co-
ordinates) taken out in totality, from the $4$-momenta of final state
signatures of the di-muons produced in the $pp$ collision at $7$ TeV from CMS
collaboration.

## Acknowledgements

We thank CERN, the CMS collaboration, and the CMS Data Preservation and Open
Access(DPOA) team for making research-grade collider data available to the
public.

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