# Self-Similar Properties of the Proton Structure at Low x within the xFitter
framework

Shahin Atashbar Tehrani1,2 ID<EMAIL_ADDRESS>Fatemeh Taghavi-Shahri3 ID
<EMAIL_ADDRESS>Samira Shoeibi Mohsenabadi3 ID<EMAIL_ADDRESS>(1)School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran.
(2)Department of Physics, Faculty of Nano and Bio Science and Technology,
Persian Gulf University, 75169 Bushehr, Iran.
(3)Department of Physics, Ferdowsi University of Mashhad, P.O.Box 1436,
Mashhad, Iran

(August 28, 2024)

###### Abstract

The structure of the proton exhibits Fractal behavior at low x, where x is the
fraction of the proton’s momentum carried by the interacting partons. This
Fractal behavior is characterized by self-similar properties at different
scales and can be quantified using the concept of Fractal dimension. An
investigation into the Fractal properties of the proton structure at low x is
critical for understanding the fundamental properties of the strong force and
developing a more comprehensive understanding of the hadron structure.
Fractals, characterized by self-similar patterns across scales, demonstrate a
direct correlation between their Fractal dimension and entropy, where higher
Fractal dimensions correspond to increased informational content. Furthermore,
it is essential for designing high-energy physics experiments and developing
more accurate models of subatomic particle interactions. This paper has a
fresh look at the self-similar properties of the proton structure at low x.
Our study involves the use of the xFitter framework to parameterize the proton
structure functions with a Fractal formalism at low x. We also examine how the
inclusion of new data affects the results of our analysis.

Fractal; xFitter.

##  I Introduction

The proton is a subatomic particle that is composed of smaller particles
called quarks and gluons. At high energies, the structure of the proton can be
described using the theory of quantum chromodynamics (QCD), which predicts
that a strong force binds together the quarks and gluons.
At low energies or low momentum transfers, the proton structure becomes more
complex and difficult to describe only using QCD. The proton structure
function, $F_{2}(x,Q^{2})$, goes up with $Q^{2}$ at low Bjorken x (the
fraction of the proton’s momentum carried by the interacting parton), which
indicates rapid growth of the partons at low x. The experiments show that the
number of partons inside a proton goes up at low x, and falls at high x
H1:2001ert ; Abramowicz:2015mha ; H1:2010fzx ; Aaron:2009aa . Proton has three
valence quarks that are more significant at low $Q^{2}$, but at high $Q^{2}$,
the sea quarks, which are quark-antiquark pairs, increase in number and have
an important role at small x. Because the knowledge of these densities at a
much smaller value of x will be needed for any collider predictions, then it
is crucial to know how PDFs behave at low x.
It is shown that the structure of the proton can be characterized by the
concept of Fractals, which are objects that exhibit self-similar properties at
different scales Lastovicka:2002hw ; Lastovicka:2004mq ; Choudhury:2003yy ;
Jahan:2014sqa ; Choudhury:2016fjy ; Mohsenabadi:2021vvj . Fractals possess a
holographic nature too. Intriguingly, Fractals are members of a fundamental
class of statistical objects, with their basis lying at the heart of
information theory. The holographic principle is a concept in theoretical
physics that suggests that the information content of a certain region of
space can be fully encoded on its boundary. Fractals manifest in various
aspects of nature. Scientists were led by Fractals to the realization that
apparent chaos in fact structured. It appears that the complex phenomena have
a hidden order. The holographic principle and Fractals exhibit an intricate
interplay, underpinned by their shared understanding of information
organization in the universe. Fractals, characterized by self-similar patterns
across scales, demonstrate a direct correlation between their Fractal
dimension and entropy, where higher Fractal dimensions correspond to increased
informational content. This connection is rooted in information theory, which
posits that complex, self-similar systems, like Fractals, can be more
efficiently encoded. The holographic principle builds upon this, proposing
that the information within a given volume can be fully described by the
information encoded on its boundary, akin to a hologram. Crucially, the
Shannon entropy of a system, a fundamental information-theoretic measure, is
directly related to its Fractal dimension, suggesting the universe’s
underlying structure may be Fractal-like, enabling the efficient storage and
transmission of information as envisaged by the holographic principle
Mureika:2006tz .
One way to quantify the Fractal properties of the proton structure at low x is
to use the concept of Fractal dimension in the PDFs parameterized by the
Fractal distributions. The Fractal distributions are a type of probability
distribution that is statistically self-similar and look the same at different
scales. Understanding the Fractal patterns of the proton at low x is essential
for learning more about the basic properties of the strong force and the
hadron structure. It also affects high-energy physics experiments that rely on
the proton structure to predict the results of particle collisions.
In this paper, we have a fresh look at the self-similar properties of the
proton structure at low x. We used the xFitter framework Sapronov:2015pba  to
determine the Fractal PDFs and proton’s reduced cross-section at the low x
region by including a selection of data sets (500 data points) from the
combined HERA I+II deep-inelastic scattering data Abramowicz:2015mha . The
various PDF fits are performed to four different data sets: (i) inclusive HERA
I+II cross-section data at 920 (GeV), (ii) inclusive HERA I+II cross-section
data at 920 and 820 (GeV), (iii) inclusive HERA I+II cross-section data at
920, 820, and 575 (GeV) (iv) inclusive HERA I+II cross-section data at 920,
820, 575 and 460 (GeV). Then, the impact of adding each data set on the
proton’s reduced cross-section is investigated.
This paper is organized as follows: In Section 2, we provide a concise
overview of the Fractal dimension and the Fractal distributions that we need
to study the proton structure. Section III describes the parametrization of
the structure-function based on Fractal formalism, which is a key quantity
used in this analysis. Next, in Section IV, we discuss the fitting procedures
used in the QCD analysis of the data. Finally, in Section V, we summarize the
key findings of the study and present our conclusions regarding the
implications of our results. Following this structure, we aim to provide
readers with a clear and comprehensive understanding of the theoretical
concepts, data analysis techniques, and results presented in this paper.

##  II The Fractal Dimension and the Fractal Distribution

The concept of the Fractal dimension requires an understanding of the meaning
of the term "dimension." In non-fractional dimensions, the number of
dimensions corresponds to the number of independent directions in a
corresponding coordinate system. For example, as shown in Fig. 1 a line has
dimension 1, a square has dimension 2, and a cube has dimension 3. However,
Fractal objects such as the Sierpinski gasket, shown in Fig. 2, require a more
general definition of dimension than traditional Euclidean geometry. It is
shown in Fig. 1 that the traditional notions of geometry for defining
dimension can be linked to the self-similarity property by this relation:

$D=\frac{log(number\leavevmode\nobreak\ of\leavevmode\nobreak\
self\leavevmode\nobreak\ similar\leavevmode\nobreak\
objects)}{log(magnification\leavevmode\nobreak\
factor)}=\frac{log(N)}{log(r)}$ (1)

here r is the number of each side segment (Magnification factor) and N is the
number of self-similar shapes created by the segmentation. It means when a
line is divided in the middle, two lines of half-length are obtained. By
magnifying one of them by a factor of two, the original line can be
reconstructed. The same principle applies to dividing a square into four
smaller squares or a cube into smaller cubes. Similarly, as another example,
if the magnification factor for a square is 3, then the number of smaller
squares will be $3^{2}=9$. In a similar fashion, a cube magnified by a factor
of 3 will result in $3^{3}=27$ smaller cubes. According to this relation, for
example for 3 segmentation case, we would obtain the values of
D=$\frac{log(3)}{log(3)}$=1 for the dimension of line,
D=$\frac{log(3^{2})}{log(3)}$=2 and D=$\frac{log(3^{3})}{log(3)}$=3 for the
dimensions of square and cube, respectively.
To quantify the dimension of a Fractal object, one cannot simply use the
definition existing for dimension, as this approach fails to capture the
complex and self-similar structure of Fractals. Instead, the concept of the
Fractal dimension provides a more nuanced definition that captures the self-
similarity and irregularity of Fractal objects. The Fractal dimension of an
object can be thought of as a measure of how much space it fills, relative to
its size. The Sierpinski gasket, for example, is a Fractal object that is
formed by repeatedly removing triangles from a larger triangle. As the process
is repeated infinitely many times, the Sierpinski gasket exhibits self-
similarity at different scales. Using Eq. (1), the Fractal dimension of the
Sierpinski gasket is approximately 1.585, which is a non-integer value that
reflects its complex and irregular structure. In iteration No. 1, for example,
we have

${\cal D}=\frac{\log 3}{\log 2}\approx 1.585.$ (2)

Fractal objects, on the other hand, have non-integer dimensions and exhibit
self-similarity at all scales. In contrast to traditional geometric shapes,
real Fractals in nature do not exhibit the same degree of self-similarity for
all magnification factors, and the number of self-similar objects changes in a
more complex manner. The concept of the Fractal dimension provides a way to
quantify the self-similarity and irregularity of the Fractal objects.

Figure 1: Regular self-similarity in Eclidian space Figure 2: Sierpinski
gasket Fractal in iterations No. 1, 3 and 6 (from left). Iteration No. 1
corresponds to the seed image which is arbitrary while the iteration always
converges to the same object.

Figure 3: The prediction for the reduced cross-section in the low x region
below $x<0.001$ for $Q^{2}=3.5GeV^{2}$, $Q^{2}=2GeV^{2}$ , $Q^{2}=20GeV^{2}$
and $Q^{2}=25GeV^{2}$ ($E_{p}=460GeV$). Data points are from NC interactions
in HERA positron- proton DIS processes for fit-4serires data.

Figure 4: . The prediction for the reduced cross-section in the low x region
below $x<0.001$ for $Q^{2}=8.5GeV^{2}$, $Q^{2}=12GeV^{2}$, $Q^{2}=15GeV^{2}$
and $Q^{2}=20GeV^{2}$ ($E_{p}=575GeV$). Data points are from NC interactions
in HERA positron- proton DIS processes for fit-1serires data and fit-2serires
data .

Figure 5: . The prediction for the reduced cross-section in the low x region
below $x<0.0001$ for $E_{p}=820GeV$. Data points are from NC interactions in
HERA positron-proton DIS processes for fit-1serires data, fit-2serires data,
and fit-3serires data.

Figure 6: . The prediction for the reduced cross-section in the low x region
below $x<0.01$. Data points are from NC interactions in HERA positron-proton
DIS processes for fit-1serires data, fit-2serires data, fit-3serires data, and
fit-4serires data. Figure 7: The $xq(x,Q^{2})$ for $Q^{2}=2GeV^{2}$,
$Q^{2}=5GeV^{2}$ , $Q^{2}=10GeV^{2}$ and $Q^{2}=100GeV^{2}$ for fit-4serires
data . Figure 8: The $xq(x,Q^{2})$ form Fractal model in xFitter (Lo
approximation and only qurk PDfs) and those from Mohsenabadi:2021vvj  (NLO
approximation with quark and gluon PDFs).

The concept of the Fractal dimension can also be extended to non-discrete
Fractals, where the magnification factor is a real number $z$ and the number
of self-similar objects is represented by a density function $f(z)$. In this
case, the dimension may vary with the scaling factor and a local dimension can
be defined as the logarithmic derivative of the density function:

${\cal D}(z)=\lim_{z\rightarrow 0}\frac{\log f(z)}{\log z}$ (3)

Here, $f(z)$ represents the density of self-similar objects at scale $z$. We
can rewrite Eq. 3 as:

$\log f(z)={\cal D}\cdot\log z+{\cal D}_{0}$ (4)

where ${\cal D}_{0}$ defines the normalisation of $f(z)$, which thus has a
power law behaviour, $f(z)\propto z^{D}$.
In general, Fractals may have two independent magnification factors, $z$ and
$y$. In this case the density $f(z,y)$ is written in the following way
Lastovicka:2002hw ; Lastovicka:2004mq

$\log f(z,y)={\cal D}_{zy}\cdot\log z\cdot\log y+{\cal D}_{z}\cdot\log z+{\cal
D}_{y}\cdot\log y+{\cal D}_{0}.$ (5)

where $D_{zy}$ demonstrates the dimensional correlation related to two
magnification factors, $y$ and $z$.
It is important to mention that there is a certain freedom in selecting
magnification factors without changing the shape of the function $f(z,y)$. It
is possible to use any non-zero power of a factor multiplied by a constant:
$z\rightarrow az^{\lambda}$ , without affecting the underlying self-similarity
of the Fractal distributions. The only effect of such a change is a
redefinition of the dimensional parameters ${\cal D}_{{z,y,zy}}$ and of the
normalization ${\cal D}_{0}$, respectively.

Table 1:  The list of all data sets: DIS HERA I+II used in the present analysis. For each data set, we indicate process, measurement, reference and the ranges of their kinematic cuts such as $x$, $Q^{2}$ (GeV2).  Data set |  Experiment | Ref. |  Kinematic ranges and details |
---|---|---|---|---
HERA I+II |  |  |  |
$\sigma_{r,NC}$ |  HERA I+II NC $e^{-}p$ 460 | Abramowicz:2015mha  |  $1\leq Q^{2}\leq 100$ |  $1\times 10^{-6}\leq x\leq$0.01
|  HERA I+II NC $e^{-}p$ 575 | Abramowicz:2015mha  |  $1\leq Q^{2}\leq 100$, |  $1\times 10^{-6}\leq x\leq$0.01
|  HERA I+II NC $e^{+}p$ 820 | Abramowicz:2015mha  |  $1\leq Q^{2}\leq 100$, |  $1\times 10^{-6}\leq x\leq$0.01
|  HERA I+II NC $e^{+}p$ 920 | Abramowicz:2015mha  |  $1\leq Q^{2}\leq 100$, |  $1\times 10^{-6}\leq x\leq$0.01

It should be noted that the Fractal behavior is seen on the shape of the quark
distribution functions or PDFs( parton distribution functions) at low Bjorken-
x values, not in the shape of the proton. The parton distribution functions at
low x has a linear behavior in log-log space and a power-law behavior in
ordinary phase space of x and $Q^{2}$. Then, we can describe these PDFs with
Fractal distributions. In this work, we want to show that the quark
distribution functions have a monofractal behavior at low x (and $Q^{2}$)
which means it has self-similar properties with fixed exponents (Fixed Fractal
dimension). In the study of proton structure at small-x values, researchers
have observed an increased rate of interactions between gluons, leading to
enhanced densities of both gluons and sea quarks. This phenomenon exhibits
fractal characteristics, as discussed here, which has sparked interest in
applying fractal formalism to Parton Distribution Functions (PDFs). The key
concept behind this application lies in the recognition of scaling behaviors
following a power law. A notable reference Abdulov:2021ivr  demonstrates that,
when plotted on a log-log scale, the density functions of sea quarks, which
depend on momentum transfer $Q^{2}$ and the Bjorken variable x specifically
for $x<0.01$ (outside the valence quark region), exhibit a linear trend.
Consequently, the kinematical variables x and $Q^{2}$ emerge as suitable
magnification factors. It is important to note that these magnification
factors must be positive, non-zero, and dimensionless. Thus, in addition to x
and $Q^{2}$, Reference Lastovicka:2002hw  introduces 1/x and 1 +
$Q^{2}$/$Q_{0}^{2}$ as magnification factors. The former confirms that as the
probing goes deeper into the proton’s structure and x approaches zero, the
magnification factor increases accordingly. The latter normalizes the physical
dimension of $Q^{2}$ by dividing it with a constant $Q_{0}^{2}$, preventing
the issue of $Q^{2}=0$ by adding 1 to the ratio $Q^{2}/Q_{0}^{2}$. With these
preparations in place, all the necessary components are ready to begin
depicting the proton’s structure function at low x values using the framework
of Fractal formalism. The following section is dedicated to outlining the
parameterization of quark Parton Distribution Functions (PDFs) within the
context of the Fractal framework.

##  III Parametrization of the Structure Function

The concept of self-similarity, when applied to the structure of the proton’s
confinement, leads to a simple parametrization of quark densities within the
proton. This parametrization is based on Eq. (5), and can be constructed using
magnification factors $1/x$ and $1+Q^{2}/Q_{0}^{2}$ Lastovicka:2002hw . The
distributions described by Eq. 5 have linear behavior in log-log space. This
linearity is also found for unintegrated sea quark densities at low x and
$k_{t}^{2}$ Abdulov:2021ivr . An unintegrated quark density may be written in
the following general form:

$\displaystyle\log f_{i}(x,Q^{2})$ $\displaystyle=$ $\displaystyle{\cal
D}_{1}\cdot\log\frac{1}{x}\cdot\log(1+\frac{Q^{2}}{Q_{0}^{2}})+{\cal
D}_{2}\cdot\log\frac{1}{x}$ (6) $\displaystyle+{\cal
D}_{3}\cdot\log(1+\frac{Q^{2}}{Q_{0}^{2}})+{\cal D}^{i}_{0}$

where $i$ denotes a quark flavor. Following Eq. (5), ${\cal D}_{1}$,${\cal
D}_{2}$ and, ${\cal D}_{3}$ are the Fractal dimensions related to $x$ and
$Q^{2}$ parameters respectively. The normalization parameter ${\cal
D}^{i}_{0}$ is related to each kind of partons. Conventional, integrated quark
densities $q_{i}(x,Q^{2})$ are defined as a sum over all contributions with
quark virtualities smaller than that of the photon probe, $Q^{2}$. Thus,
$f_{i}(x,Q^{2})$ has to be integrated over $Q^{2}$, yielding the following
relationship between the integrated and unintegrated quark densities:

$q_{i}(x,Q^{2})=\int^{Q^{2}}_{0}f_{i}(x,q^{2})\ dq^{2}.$ (7)

By Integrating on $f_{i}(x,q^{2})$ in equation (7), one can obtain the
following analytical expression for the parametrization of a quark density:

$\displaystyle q_{i}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{e}}^{{\cal D}^{i}_{0}}\leavevmode\nobreak\
Q_{0}^{2}\leavevmode\nobreak\ x^{-{\cal D}_{2}}}{1+{\cal D}_{3}-{\cal
D}_{1}\log{x}}$ (8) $\displaystyle\left(x^{-{\cal
D}_{1}\log(1+\frac{Q^{2}}{Q_{0}^{2}})}(1+\frac{Q^{2}}{Q_{0}^{2}})^{{\cal
D}_{3}+1}-1\right).$

It is important to note that in this parametrization, only the normalization
parameter ${\cal D}_{0}^{i}$ depends on the quark flavor, while the other
parameters are independent of flavor. This assumption implies that all quarks
follow the same Fractal structure, i.e., the dimensions $D_{i}$ and
magnification factors are common for all quarks, and they differ only in
normalization.

The proton structure function $F_{2}$ is directly related to the quark
densities, given by $F_{2}=x\sum_{i}e_{i}^{2}(q_{i}+\bar{q}_{i})$. Therefore,
assuming the flavour symmetry of Eq. (8), we can express $F_{2}$ directly in
terms of the parameters in Eq. (8) by replacing $x^{-{\cal D}_{2}}$ with
$x^{-{\cal D}_{2}+1}$ and introducing a common normalisation factor
$\mathbf{e}^{{\cal D}_{0}}$, as follows:

$\displaystyle F_{2}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{e}}^{{\cal D}_{0}}\leavevmode\nobreak\
Q_{0}^{2}\leavevmode\nobreak\ x^{-{\cal D}_{2}+1}}{1+{\cal D}_{3}-{\cal
D}_{1}\log{x}}$ $\displaystyle\left(x^{-{\cal
D}_{1}\log(1+\frac{Q^{2}}{Q_{0}^{2}})}(1+\frac{Q^{2}}{Q_{0}^{2}})^{{\cal
D}_{3}+1}-1\right).$ $\displaystyle F_{L}(x,Q^{2})$ $\displaystyle=$
$\displaystyle F_{2}(x,Q^{2})\times\frac{R}{1+R}$ (9)

A free parameter R will be set by global analysis. To investigate the internal
structure of the proton at low values of $x$, one must analyze the reduced
cross-section data obtained from $e^{\pm}p$ Deep Inelastic Scattering (DIS)
processes at low $x$. For unpolarized $e^{\pm}p$ scattering, the reduced
cross-section at low $Q^{2}$ values, i.e. $Q^{2}\ll M_{Z}^{2}$, can be
expressed as H1:2010fzx :

$\sigma_{r,NC}=F_{2}(x,Q^{2})-\frac{y^{2}}{Y_{+}}F_{\rm
L}(x,Q^{2})\leavevmode\nobreak\ ,$ (10)

where the kinematic variables $x$, $Q^{2}$, and $y$ are defined as:

$Q^{2}=-q^{2},:x=\frac{Q^{2}}{2(P.q)},:y=\frac{(P.q)}{P.k},$ (11)

Here, $P$, $k$, and $q$ denote the four-momentum of the incoming proton,
incoming lepton, and exchanged boson, respectively, and $Y_{+}=1+(1-y)^{2}$.
It should be noted that for $y$ values larger than approximately 0.5, the
contribution of the longitudinal structure function $F_{\rm L}$ becomes
significant. The detailed information for each data set is summarized in Table
1.

Table 2:  The numerical values and their uncertainties extracted for parameters related to fit-4serires data, fit-3serires data, fit-2serires data, and fit-1serires data . Parameter | fit-4ser. data | fit-3ser. data | fit-2ser. data | fit-1ser. data
---|---|---|---|---
’${\cal D}_{0}$’ | $1.60\pm 0.24$ | $1.83\pm 0.31$ | $1.95\pm 0.36$ | $1.69\pm 0.31$
’${\cal D}_{1}$’ | $0.0807\pm 0.0016$ | $0.0793\pm 0.0016$ | $0.0793\pm 0.0017$ | $0.0798\pm 0.0019$
’${\cal D}_{2}$’ | $0.992\pm 0.010$ | $0.986\pm 0.012$ | $0.982\pm 0.013$ | $0.991\pm 0.013$
’${\cal D}_{3}$’ | $-1.339\pm 0.013$ | $-1.321\pm 0.014$ | $-1.318\pm 0.014$ | $-1.330\pm 0.015$
’$Q_{0}^{2}$’ | $0.067\pm 0.010$ | $0.0566\pm 0.0099$ | $0.0524\pm 0.0099$ | $0.062\pm 0.012$
’$R$’ | $0.347\pm 0.020$ | $0.366\pm 0.022$ | $0.383\pm 0.024$ | $0.393\pm 0.027$
Fit status | converged | converged | converged | converged
Table 3:  The numerical values and their uncertainties extracted for parameters related to Ref.Lastovicka:2002hw  and Ref.Mohsenabadi:2021vvj  . Parameter | Ref.Lastovicka:2002hw  | Ref.Mohsenabadi:2021vvj
---|---|---
’${\cal D}_{0}$’ | $0.330\pm 0.195$ | $4.81\pm 0.01$
’${\cal D}_{1}$’ | $0.073\pm 0.001$ | $-0.0051\pm 0.00009$
’${\cal D}_{2}$’ | $1.013\pm 0.01$ | $1.138\pm 0.002$
’${\cal D}_{3}$’ | $-1.287\pm 0.01$ | $-1.285\pm 0.005$
’$Q_{0}^{2}$’ | $0.062\pm 0.010$ | $0.0187\pm 0.0001$
Fit status | converged | converged
Table 4: The numerical results for the correlated $\chi^{2}$, log penalty $\chi^{2}$, total $\chi^{2}$ and the total $\chi^{2}$/ degree of freedom (dof) of each data sets for different fit-4serires data, fit-3serires data, fit-2serires data and fit-1serires data. Dataset | fit-4ser. data | fit-3ser. data | fit-2ser. data | fit-1ser. data
---|---|---|---|---
HERA1+2 NCep 820 | 67 / 41 | 67 / 41 | 67 / 41 | -
HERA1+2 NCep 460 | 144 / 111 | - | - | -
HERA1+2 NCep 575 | 155 / 143 | 161 / 143 | - | -
HERA1+2 NCep 920 | 241 / 211 | 241 / 211 | 244 / 211 | 240 / 211
Correlated $\chi^{2}$ | 73 | 56 | 39 | 33
Log penalty $\chi^{2}$ | +0.69 | +1.6 | +2.2 | +3.2
Total $\chi^{2}$ / dof | 680 / 500 | 527 / 389 | 353 / 246 | 276 / 205

##  IV Fitting Contents and results

In this section, we utilize experimental data obtained from HERA to study
electron(positron)-proton scattering in DIS processes within the xFitter
framework. The H1 and ZEUS Collaborations investigated a vast kinematical
phase space in ($x,Q^{2}$), where the experimental data covered $0.005<x<0.65$
and $0.045<Q^{2}<50000GeV^{2}$ for neutral current (NC) interactions and
$0.01<x<0.4$ and $200<Q^{2}<50000GeV^{2}$ for charged current (CC)
interactions, respectively. Since we focus on the low-$x$ region, we select a
subset of the combined HERA data related to NC interactions in the $x<0.01$
region. The total number of experimental data used in our analysis is
$N_{data}=500$, and we summarize them in Table 1.
The following step involves a brief explanation of error estimation and the
results we obtained by xFitter Abdolmaleki:2019tbb ; Tooran:2019cfz ;
Vafaee:2017nze . xFitter is a powerful tool that provides a flexible platform
for fitting various types of parton distribution functions (PDFs) to
experimental data. It includes a variety of PDF sets, including those based on
different parameterizations and fitting methods, as well as a set of commonly
used models for the strong coupling constant and heavy quark masses.
Additionally, xFitter supports the implementation of different types of
systematic uncertainties in the fit, such as experimental and theoretical
uncertainties. To estimate the uncertainties associated with the experimental
data used in the fit, xFitter provides various methods, such as the Hessian
method and Monte Carlo methods. The Hessian method involves calculating the
second derivatives of the $\chi^{2}$ function with respect to the fit
parameters at the minimum $\chi^{2}$ point, which can be used to estimate the
PDF uncertainties. On the other hand, Monte Carlo methods involve generating a
large number of pseudo-data sets by adding Gaussian-distributed random errors
to the experimental data and then fitting each of them to obtain a set of PDF
replicas. The spread of the resulting PDF replicas can be used to estimate the
PDF uncertainties. It is important to carefully evaluate the uncertainties
associated with the experimental data to ensure that they are properly taken
into account in the fit. This can help to obtain reliable PDFs and associated
uncertainties, which are essential for making predictions for a wide range of
high-energy physics observable. The $\chi^{2}$-function is a tool to determine
how well a particular QCD model fits experimental measurements. In our
analysis, we aim to obtain the best values of 6 independent free parameters by
minimizing the $\chi^{2}$-function. When all of the correlated uncertainties
associated with experimental measurements are known, the $\chi^{2}$-function
in the xFitter framework is given by Aaron:2009aa :

$\chi^{2}=\sum_{i}\frac{[d_{i}-t_{i}(1-\sum_{j}\beta_{j}^{i}s_{j})]^{2}}{\delta^{2}_{i,unc}t^{2}_{i}+\delta^{2}_{i,stat}d_{i}t_{i}}+\sum_{j}s^{2}_{j}\;,$
(12)

The $\chi^{2}$-function presented above calculates the goodness of fit between
theoretical predictions $t_{i}$ and experimental measurements $d_{i}$ for each
data point $i$, taking into account statistical and systematic uncertainties.
The systematic uncertainties are categorized as correlated and uncorrelated,
with $\beta_{j}^{i}$ and $s_{j}$ representing the corresponding uncertainties
and nuisance parameters, respectively. The $\delta^{2}_{i,stat}$ and
$\delta^{2}_{i,unc}$ terms describe the relative statistical and uncorrelated
systematic uncertainties. In addition to obtaining the central values of the 6
free parameters, it is essential to determine their uncertainties. The
uncertainties of parton distribution functions have been estimated in several
studies by the Hessian method, such as Refs. Hirai:2007cx ; Pumplin:2001ct ;
Pumplin:2002vw ; Martin:2002aw ; Blumlein:2002qeu ; Hirai:2006sr ;
Leader:2005ci ; deFlorian:2005mw ; Hirai:2004wq . One of the most commonly
used approaches for estimating uncertainties is the Hessian minimum iteration
method. We utilize this method in our QCD analysis and details of this method
are available in these references. Four different fits with the name
fit-4serires data, fit-3serires data, fit-2serires data, and fit-1serires data
are introduced in Table 2. The first data set,fit-1serires data, only contains
HERA I+II-920 data to prepare a fine base for investigating the impact of
other data sets on PDFs. In the second data set,fit-2serires data, the HERA
I+II-820 data are added to the first data set. The third data set,
fit-3serires data, and the HERA I+II-575 data are added to the second data
set. Finally, the third data set,fit-4serires data, and the HERA I+II-460 data
are added to contain all previously mentioned data. According to Table 4 the
extracted values of $\chi^{2}$/dof for (fit-1serires data) are 1.34 ,
(fit-1serires data) 1.43 ,(fit-3serires data) 1.35 and (fit-4serires data)
1.36 , respectively.
We have also a comparison of our results with those from Lastovicka:2002hw
and Mohsenabadi:2021vvj  in Table 3. The results show an agreement with those
from Lastovicka:2002hw  because we did the same but in the frame of xFitter
and also including new data from HERA2015 Abramowicz:2015mha . The results
show a negative value for Fractal dimension of $\mathcal{D}_{3}$. To see why,
we revisit Equation 6 in this paper and also refer to Abdulov:2021ivr . In
Abdulov:2021ivr , the authors demonstrate that the up quark parton
distribution function exhibits a linear behavior with a negative slope in log-
log space at the low x region and also for $Q^{2}$ greater than 1 ${GeV}^{2}$
(see Figures 1 and 2 in Abdulov:2021ivr ). This implies that the quark
distribution functions in this region can be described by lines with negative
slopes. A line in log-log space corresponds to a power-law function in
ordinary space. We then parameterized the quark distribution functions in this
region in Eq. (6) with three parameters, $\mathcal{D}_{2}$,$\mathcal{D}_{3}$
and, $\mathcal{D}_{1}$ , which represent the Fractal dimensions related to x
and $Q^{2}$ and, their correlations respectively. Our results also shows this
behavior for quark distribution functions at the low x region, exhibiting
linearity with a negative slope. Note that our PDFs in Eq. (6) are
parameterized as a function of $\frac{1}{x}$, not x , and thus we obtain a
positive value for $\mathcal{D}_{2}$ . These PDFs are parameterized as a
function of $Q^{2}$ as $\mathcal{D}_{3}\cdot\log(1+\frac{Q^{2}}{Q_{0}^{2}})$ ,
from which we obtain the negative value for $\mathcal{D}_{3}$.
In Fig. 3, the predictions for the reduced cross-section in the low x region
below $x<0.001$ for $Q^{2}=3.5GeV^{2}$, $Q^{2}=2GeV^{2}$, and
$Q^{2}=25GeV^{2}$ at $E_{p}=460GeV$ are presented. Data points are from NC
interactions in HERA positron-proton DIS processes for fit-4 series data. We
also have a comparison between the theoretical model and experimental data. In
Fig. 4 To evaluate the reduced cross-section in the low x region, specifically
below $x<0.001$, for three different values of $Q^{2}$ ($Q^{2}=8.5GeV^{2}$,
$Q^{2}=12GeV^{2}$, $Q^{2}=15GeV^{2}$ and $Q^{2}=20GeV^{2}$) at a proton energy
of $E_{p}=575GeV$, data from neutral current (NC) interactions in positron-
proton DIS processes at HERA are utilized. The experimental data are available
as two sets, fit-3series and fit-4series, and are used to compare theoretical
models with the measured values.
In Fig. 5, to predict the reduced cross-section in the low x region,
specifically below $x<0.0001$, at a proton energy of $E_{p}=820GeV$, data from
neutral current (NC) interactions in positron-proton DIS processes at HERA are
utilized. The experimental data are available as three sets: fit-2serires
data, fit-3serires data, and fit-4serires data.
In Fig. 6 To make predictions for the reduced cross-section in the low x
region, specifically below $x<0.01$, data from neutral current (NC)
interactions in positron-proton DIS processes at HERA are utilized. The
experimental data points are available as four sets: fit-1serires data,
fit-2serires data, fit-3serires data, and fit-4serires data.
In Fig 7 we show the behavior of the Fractal quark density for fit-4serires
data at $Q^{2}=2GeV^{2}$, $Q^{2}=5GeV^{2}$, $Q^{2}=10GeV^{2}$ and
$Q^{2}=100GeV^{2}$. Finally in Fig 8, we compare our results with those with
did at NLO approximation with including the gluon distribution function at low
x Mohsenabadi:2021vvj .
This work demonstrates that the Fractal approach is a promising method to
investigate the hadron structure at the low x region. We used the reduced
cross-section at the LO approximation and only considered the contribution of
sea quarks at the low x region, neglecting the contribution of gluons. We also
used the xFitter framework to study the effect of new data on the results.

##  V Summary and Conclusions

In this paper, we have presented the Fractal parton distribution functions,
including HERA I+II DIS experimental data as a base data set to investigate
the proton structure at the low x region in the xFitter framework. Our
analysis incorporates the HERA I+II Deep Inelastic Scattering (DIS)
experimental data as a foundational dataset to scrutinize the proton structure
within the low x ($x<0.01$) and $Q^{2}>1GeV^{2}$ domain, employing the xFitter
framework. By juxtaposing the proton’s reduced cross-section with empirical
data, we find that the Fractal approach provides a plausible description of
the physics at low x. This comparison not only bolsters the validity of the
Fractal model but also enhances our understanding of the proton’s behavior
under these conditions. A comparison of the proton’s reduced cross-section
with experimental data shows that the Fractal approach may describe the low x
physics well. Investigating PDFs at low x helps in understanding the non-
perturbative aspects of QCD and the dynamics of parton interactions in this
regime. The behavior of PDFs at low x also influences the parton-parton
collision cross sections, which are fundamental for calculating various
scattering processes in particle physics. The results indicate that the
Fractal approach is a promising method, offering novel insights into the
proton structure at low x regions and exploring new frontiers in physics.

## Acknowledgments

Fatemeh Taghavi-Shahri and S. Shoeibi acknowledge the Ferdowsi University of
Mashhad for the provided facilities to do this project. S. Shoeibi and Shahin
Atashbar Tehrani are grateful to the School of Particles and Accelerators,
Institute for Research in Fundamental Sciences (IPM) to make the required
support to do this project. Authors also appreciate Hamed Abdolmaleki for
useful discussion about xFitter.

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