# Biomimetic Evaluation of an Underwater Soft Hand Through
Deep Learning-based 3D Pose Reconstruction

Haihang Wang, He Xu, and Yihan Meng This work was supported by the Natural
Science Foundation of China under Grant 51875113, Natural Science Joint
Guidance Foundation of the Heilongjiang Province of China under Grant
LH2019E027. The first author thanks China Scholarship Council (CSC) for
financial support through No. 201906680045. (Corresponding author: He Xu)The
authors are with College of Mechanical and Electrical Engineering, Harbin
Engineering University, China. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Soft robotic hand shows considerable promise for various grasping
applications. However, the sensing and reconstruction of the robot pose will
cause limitation during the design and fabrication. In this work, we present a
novel 3D pose reconstruction approach to analyze the grasping motion of a
bidirectional soft robotic hand using experiment videos. The images from top,
front, back, left, right view were collected using an one-camera-multiple-
mirror imaging device. The coordinate and orientation information of soft
fingers are detected based on deep learning methods. Faster RCNN model is used
to detect the position of fingertips, while U-Net model is applied to
calculate the side boundary of the fingers. Based on the kinematics, the
corresponding coordinate and orientation databases are established. The 3D
pose reconstructed result presents a satisfactory performance and good
accuracy. Using efficacy coefficient method, the finger contribution of the
bending angle and distance between fingers of soft robot hand is analyzed by
compared with that of human hand. The results show that the soft robot hand
perform a human-like motion in both single-direction and bidirectional
grasping.

##  I Introduction

Soft robots have advantages over rigid robots due to their flexibility,
compliance, and adaptability to the object and environment [1]. Dexterity of
the human hand is of tremendous importance for exploring and interacting with
the world and a top design objective for robotic hands [2]. Fluidic elastomer
actuators provide particular advantages over rigid underwater actuators, owing
to their flexibility, light weight, low cost, and waterproofing features
inherently endowed by soft materials [3, 4]. Bidirectional actuators that are
capable of both inward and outward bending show potential on improving the
grasping capability of grippers. Soft robotic hands with anthropomorphic
design are emerging and being developed by researchers. Compared to joint-like
bending motion with a small bend radius and a large bend range of human
fingers, soft bending actuators lack the capability of controlling the angle
of the fingertip. An opposable and flexible palm enables fingers with more
grasping dexterity.

Soft robotic hands with anthropomorphic design are emerging and being of
interest of many scientists. Examples of the soft biomimetic hands include RBO
Hand 2 [5] and 3 [6], the soft neuroprosthetic hand [7], BCL-26
Anthropomorphic hand [7], human-inspired soft hand with active palm [8] and so
on. Although various SRHs have been developed for safe, adaptable grasping,
most of them still lack sensory feedback [9]. For achieving the proprioception
in soft robots, a large variety of flexible tactile sensors provides a much
convenient way for the soft bending actuators. Most of these sensors are based
on capacitive, resistive and piezoresistive, or optical principles [10, 11,
12]. However, the capacitance of capacitive sensors is affected by the change
of the gap between the conductive plates on the imposition of external force.
Resistive sensors generate error in electrical resistance when mechanical
deformation occurs during working. Optical fiber sensors are immune to
magnetic fields and are inherently safe for interacting with human beings
[13]. Besides, the addition of the sensors into the soft material will greatly
increase the risk of rupture under higher pressure. Moreover, integrating the
sensor into soft actuator is usually cumbersome during fabrication.

Compared to these sensing modalities, directly acquiring the motion
information through experiment video has no influence on the design and
fabrication. Ref. [14] reconstructed numerically soft deformable morphologies
based on experiment images. Inoue et al. [15] skeletonized the dynamics of
soft robot body from video. Using the image from three cameras, Ref. [16]
reconstructed the swimming motion of the underwater soft robotic octopus.
However, these vision-based reconstruction method is aiming at process arm-
like soft actuators, which is unsuitable for the complex motion of soft
robotic hands, especially when the robotic hand grasps with object.

Figure 1: The 3D imaging system for bidirectional grasp.

In this paper, we present a novel 3D reconstruction approach to analyze the
complex grasping motions of soft robots in underwater environment through
experiment videos. The videos were captured by the 3D imaging system, as shown
in Fig. 1. The proposed reconstruction method perform the reconstructed hand
pose when the hand is both with and without grasp objects.

##  II 3D Imaging System

###  II-A Experimental Apparatus

The soft robotic hand used in this study was based on our previous work [17],
which works in underwater environment and is able to achieve bidirectional
grasping. During bidirectional grasping, the image from single view can not
fully present and collect comprehensive hand grasp information. Thus, we
developed a 3D imaging system to providing all-sided images about this novel
bidirectional grasp. As shown in Fig. 1, the 3D imaging system consists of
three parts. One is the water hydraulic experiment platform, which powers,
sense and control the water pressure into the soft hand. The second includes
the soft robotic hand, water tank and metal support. The third is the one-
camera-multiple-mirror imaging device.

###  II-B One-Camera-Multiple-Mirror Imaging Device

The one-camera-multiple-mirror imaging device consists of 1 camera and 7
mirrors placed around the observed water tank. Fig. 2 describes its imaging
principle. The image information of the soft hand will be reflected by the
mirrors and captured by one camera. As shown in Fig. 2(a), the arrangement of
the camera and mirrors has been optimized. To increase the resolution of the
experimental images, and reduce the imaging distance, the light path was
designed as close to the tank as possible. To decrease the effect of camera
distortion, the camera was arranged up to balance the top and horizontal
directions.

Figure 2: The principle drawing of the light paths from the top, front, back,
left and right directions.

##  III 3D Reconstruction Approach

###  III-A Deep Learning Based Feature Detection

Deep learning methods, particularly artificial neural networks, provide a
powerful tool to learn the motion feature of soft robots [18]. With the 5
experiment videos from top/front/back/left/right views, a finger feature
detection framework has been developed for achieving the position and
orientation of the 6 soft fingers. In this study, we always define F1
represents the index finger, F2 represents the middle finger, F3 represents
the ring finger, F4 represents the little finger and F5/F6 represents the
main/vice thumb. As shown in Fig. 3, two deep learning algorithm, Faster RCNN
and U-Net, are applied to acquire the 3D pose data of the soft hand. The key
data of the 3D reconstruction model generated in the figure includes the
current frame number, the x, y, and z coordinates of the 6 orange fingertips,
the finger bending angle, the finger deflection angle and the palm bending
angle. These 7 key data are recorded.

The identification of fingertip and grasp object was based on the Faster R-CNN
method developed by Ren et al. . [19], which is a mainstream deep learning
method in object detection. The average number of the training samples for the
fingertips in the top, front and left view and the bottle is 93. We got a
stable and satisfactory accuracy after finishing an average 1000 iterations.
The finger side-boundary was based on the U-Net method developed by
Ronneberger et al. [20], which provides an effective network and training
strategy for image segmentation. We used 109 training samples and implemented
10 epochs.

Using the Faster RCNN and U-Net models, the $(x,y,z)$ coordinate information
of the index, middle, ring and little fingers are calculated and collected.
Note that the pixel position data transmitted between different view images
has been calibrated by pixel/length scale. For the main and vice thumbs, we
get the $(y,z)$ coordinate information and their orientation in the left view.

Figure 3: Finger feature detection framework based on deep learning. (A):
Calculate the coordinate of the center point in the top image; (B): Locate the
center point of fingertip according to $x_{\rm{F}i}$ in the left image; (C):
Calculate the slope of thumb in the left/right images based on the
skeletonization image processing; (F1: Index finger; F2: Middle finger; F3:
Ring finger; F4: Little finger; F5: Main thumb; F6: Vice thumb.).

###  III-B Kinematic Modeling

Figure 4: Kinematics modeling of the soft actuator. (a) The soft actuator at
linear state. (b) The soft actuator at bending state.

The robotic hand kinematics is described using homogeneous transformation
matrix, which maps the bending angles of the soft actuators to the major
parameters of the robot configuration space. As shown in Fig. 4, the position
and pose of the soft finger can be quantified by the points $O_{\rm{FB}}$ and
$O_{\rm{FT}}$ on the centerline. As illustrated by the red dashed line, the
length of the strain limiting layer $l_{SL}$ is constant. The curvature and
place can be calculated by the trigonometric relationship [21]. The origin is
at the base part of the soft hand. The geometrical relations of the soft hand
provides a means of determining the relative position of key points on the
outlines.

For the bending segment of soft finger, the orientation angle
${\phi_{{\rm{F}}i}}$ represents the rotation around the $z$-axis, curvature
angle ${\theta_{{\rm{F}}i}}$ represents the rotation around the $y$-axis,
where $i$ indicates the $i$-th finger. The transformation matrix for the
flexible segment of soft finger is expressed as Eq. 1. That of the soft palm
can be acquired by a similar, simplified transmission.

$\displaystyle{{\bf{T}}_{{\rm{Fi}}}}=$
$\displaystyle\left[{\begin{array}[]{*{20}{c}}{{{\bf{R}}_{z}}({\phi_{{\rm{F}}i}})}&0\\\
0&1\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}1&{{{\bf{p}}_{{\rm{F}}i}}}\\\
0&1\end{array}}\right]$ (1) $\displaystyle\cdot$
$\displaystyle\left[{\begin{array}[]{*{20}{c}}{{{\bf{R}}_{y}}({\theta_{{\rm{F}}i}})}&0\\\
0&1\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{{\bf{R}}_{z}}(-{\phi_{{\rm{F}}i}})}&0\\\
0&1\end{array}}\right]$

${{\bf{p}}_{{\rm{F}}i}}={[\begin{array}[]{*{20}{c}}{{r_{{\rm{F}}i}}(1-\cos{\theta_{{\rm{F}}i}})}&0&{{r_{{\rm{F}}i}}\sin{\theta_{{\rm{F}}i}}}\end{array}]^{\rm{T}}}$

The soft hand configuration and pose can be simulated by importing the angle
variables ${\phi_{{\rm{P}}j}}$, ${\phi_{{\rm{F}}i}}$ and
${\theta_{{\rm{F}}i}}$. ${\phi_{{\rm{P}}j}}$ represents the bending angle of
the $j$-th soft palm actuator. Workspace sampling was performed in terms of
finger level, by selecting values of configuration parameters as
${\phi_{{\rm{P}}j}}$, ${\phi_{{\rm{F}}i}}$, ${\theta_{{\rm{F}}i}}$ within
certain range, as shown in Fig. 5(a). The sampled data points for each
parameter were spaced by the same value. It is intuitive that the denser the
sampling points is, the higher the accuracy should be. However, larger the
database size is, slower the mapping process would be. Considering the error
caused by the feature detection accuracy and to shorten the further process of
data mapping, the workspace database including 4000 points was established.

$\alpha_{{\rm{F}}i}=\arctan k_{{\rm{F}}i}$ (2)

The slope angle $\alpha_{{\rm{F}}i}$ database was created to help map the side
orientation of the thumbs . As shown in Fig. 5(b), we separately simulate the
configuration of the main (F5) and vice (F6) thumbs. Following the same image
processing method (C), we calculate the slope angle of all the points in the
workspace database from the left/right views.

Figure 5: The establishment of (a) the workspace database and (b) the slope
angle database.

###  III-C Workspace Mapping Strategies

Different strategies are designed for the workspace mapping of the F1-F4 and
F5/F6. As illustrated in Fig. 6(a), after achieving the feature detection and
coordinate transmission, the searching strategy for F1-F4 is quite
straightforward by sequentially comparing closeness in space between each data
point and the target position. With respect to that of F5/F6, we first search
the data points within a distance range in $y$-$z$ plane. Then, we select the
data point that has closet slope angle difference. Finally, the
${\phi_{{\rm{P}}j}}$, ${\phi_{{\rm{F}}i}}$, ${\theta_{{\rm{F}}i}}$ of the
mapped data points will be imported in the kinematic model. The 3D pose
reconstruction method of the soft hand is integrated in the dedicated program
with a GUI, as shown in Fig. 6(b). It is built on a laptop with MATLAB 2016a
and Python 3.7 or higher version on the Windows operating system.

Figure 6: (a) Workspace mapping approach. (b) The Graphical User Interface
(GUI) consisting of image presentation and processing, feature detection,
workspace mapping and 3D pose reconstruction.

Fig. 6 presents the reconstruction results of the key frames during the
bidirectional grasp, which are origin pose single-direction grasping pose,
bidirectional grasping pose and its grasping pose when with one or two objects
(bottle). Comparing the reconstructed pose with the experiment images from the
top, front and left view, it is validated that the effectiveness and accuracy
of our proposed 3D reconstruction approach is convincing and satisfactory.

Figure 7: Selected 3D pose reconstruction results of the soft hand during
bidirectional grasping.

##  IV Biomimetic Evaluation and Analysis

###  IV-A Grasp Evaluation

Using the 3D reconstructed algorithm, the experimental video about
bidirectional grasping was calculated frame by frame. We skipped the image in
the frames when the researcher is operating the bottle to be grasped, or the
bottle is greatly shelter the detection features. To eliminate the data
fluctuation, the data in Fig. 8 was smoothed using Savizky-Golay method with 1
polynomial order. The number of experimental frames is the number of frames in
the experimental video. I captured a total of 3948 frames of the experimental
video, of which 1 to 3200 frames are the entire capture process, and the
experiments from 3200 to 3948 frames are useless and static. Among the 3200
frames, 2758 frames were reconstructed in 3D through Matlab. After manually
removing abnormal and obviously incorrect data, a total of 2560 frames were
the final valid results, and the algorithm recognition and reconstruction rate
was about 80%.

Figure 8: The three key angles of the reconstructed result based on the
bidirectional grasping video. (a) The bending angle of soft fingers in the
main grasping axis. (b) The deflection angle of soft fingers in the side-
deflection axis. (c) The bending angle of soft palm actuators. (P1: Palm
actuator 1; P2: Palm actuator 2; P3: Palm actuator 3.)

Human hand dexterity has provided numerous insights on development of robotic
hands. Min et al. evaluated the robot hand with human hand with respect to
human-like grasping features [22]. Since we aim to using the comprehensive
efficacy coefficient to analyze the contribution of each fingers of
human/robot hand for successful grasp, we perform experiments on the same
grasping motion as single-direction grasping to acquire the motion data of
human hand. The 3D hand pose data of human hand was estimated using the
pretrained network model developed in Ref. [23]. Each finger has a total of 4
feature points, plus a common base point of the palm and wrist. A total of 21
feature points. The bending angle of each finger can be obtained from the
included angle of the line segment composed of 4 feature points.

Figure 9: The grasping data of human hand was collected according to the same
grasping motion of single-direction grasp. (a) The 3D pose was estimated by
human hand grasp image. (b) The bending angle of human fingers by frame.

$\displaystyle
Q_{{\rm{S}}i}={}^{(\sum\limits_{m=1}^{M}{w_{{\rm{F}}m}}+\sum\limits_{n=1}^{N}{w_{{\rm{R}}n}})}\sqrt{\prod\limits_{m=1}^{M}{q_{{\rm{S}}i,{\rm{F}}m}^{w_{{\rm{F}}m}}}\times\prod\limits_{n=1}^{N}{{{\left({q_{{\rm{S}}i,{\rm{R}}n}-s(q_{{\rm{S}}i,{\rm{R}}n}-1)}\right)}^{w_{{\rm{R}}n}}}}}-1$
(3) $\displaystyle
Q_{{\rm{B}}i}={}^{(\sum\limits_{m=1}^{M}{w_{{\rm{F}}m}}+\sum\limits_{n=1}^{N}{w_{{\rm{R}}n}})}\sqrt{\prod\limits_{m=1}^{M}{{{\left({q_{{\rm{B}}i,{\rm{F}}m}-s(q_{{\rm{B}}i,{\rm{F}}m}-1)}\right)}^{w_{{\rm{F}}m}}}}\times\prod\limits_{n=1}^{N}{q_{{\rm{B}}i,{\rm{R}}n}^{w_{{\rm{R}}n}}}}-1$
(4)

Based on the collected data of the fingers of the soft robot and human hands,
the proportion of the bending angle were calculated to analyze the
contribution of each fingers. As shown in Figure 10, for the single-direction
grasp, the thumb, index and middle fingers act as the main roles for the grasp
pose. Beside, the relative high weight of the human ring finger is caused by
the coupling motion relationship between middle and ring fingers. With respect
to the bidirectional grasp, the soft hand performs a prominent advantage
compared with human hand. It indicates that the bidirectional grasp fully
releases the potential abilities of the ring and little fingers, while
promises the quality of the single-direction grasp.

Figure 10: The finger contribution comparison of the bending angle for human
and robot hands. (a) Single-direction grasp; (b) Bidirectional grasp.

In the grasping work of software manipulators or human hands, it is necessary
to conduct coupled evaluation of multi-finger and complex data from multiple
data sources. Considering the data of one finger alone, it is impossible to
evaluate its contribution to the final grasping result as a whole. The
efficacy coefficient method can reflect the multi-factor connection of the
bionic efficacy index in the multi-finger coupled bionic grasping process, and
can quantify the comprehensive system, which is convenient for comparison.
Using Pearson correlation analysis and sliding window dynamic correlation
coefficient analysis, the bionic correlation between human hand and soft hand
was compared and analyzed. In view of the characteristic bionic two-way
grasping of the underwater soft hand, based on the deep learning software for
3D reconstruction of the soft hand, the dynamic data of the one-way grasping
by the human hand is collected at the same time, and a bionic grasping
evaluation system based on the efficacy coefficient method is established, and
the dynamic bionic correlation is carried out. Analysis, the correlation
analysis of the bending angles of the fingers of the soft hand and the fingers
of the human hand when completing the one-way or two-way grasping task.
Considering the contribution of the bending angle of each finger and the
grasping space of each fingertip to the grasping success, and converting it
into grasping efficacy, the dynamic bionic correlation analysis is carried out
on the process of grasping and releasing the target object.

Figure 11: The biomimetic correlation analysis of the fingers of human/soft
hands. (SF: Soft finger; HF: Human finger).

###  IV-B Comprehensive Efficacy Evaluation

The efficacy coefficient method is a method mostly used for comprehensive
evaluation of economic benefits. In the grasping work of software manipulators
or human hands, it is necessary to conduct coupled evaluation of multi-finger
and complex multi-data source data. Considering the data of a finger alone, it
is not possible to evaluate it as a whole. Contribution to the final crawl
result. The efficacy coefficient method can reflect the multi-factor
relationship of the bionic efficacy index in the process of multi-finger
coupled bionic grasping, and can calculate the comprehensive system
quantitatively, which is convenient for comparison.

A successful grasp is the collaboration result of multiple fingers. During the
grasping motion of soft/human hands, the complex motion parameters generated
by the fingers needs to be comprehensively analyzed. Efficacy coefficient
method, which evaluates the comprehensive efficacy of the studied object, is
used to evaluate the grasping efficacy.

Let ${x_{ki}}$ ($k=1,2,\cdots,K$) present the values of motion indexes at
$i$-th frame. $k$ is the number of evaluation indexes. In general, the
coefficient value of $q_{ki}$ is between the objective and the worst values,
that is, $x_{ki}^{{\rm{(s)}}}\leq{x_{ki}}\leq x_{ki}^{{\rm{(h)}}}$. A single
efficacy coefficient, namely ${q_{ki}}$, can be described by the following
function:

${q_{ki}}={f_{i}}({x_{ki}})=\frac{{{x_{ki}}-x_{k}^{{\rm{(s)}}}}}{{x_{k}^{{\rm{(h)}}}-x_{k}^{{\rm{(s)}}}}}+1$
(5)

where $x_{ki}^{{\rm{(h)}}}$ denotes the satisfactory value, which usually the
optimal limit of the grasping task, and $x_{ki}^{{\rm{(s)}}}$ denotes the
impermissible value, which is usually the most unfavorable limit for the task.
Then, we establish the total efficacy coefficient:

${Q_{i}}={}^{2\times({w_{1}}+{w_{2}}+\cdots+{w_{K}})}\sqrt{q_{1i}^{{w_{1}}}\times
q_{2i}^{{w_{2}}}\times\cdots\times q_{Ki}^{{w_{K}}}}$ (6)

where $w_{k}$ represents the weight of $k$-th evaluation index. $w_{k}$ can be
acquired by Eq. 7.

$w_{k}=\frac{{x_{ki}^{{\rm{(h)}}}-x_{k}^{{\rm{(s)}}}}}{{\sum\limits_{k=1}^{K}{\left({x_{k}^{{\rm{(h)}}}-x_{k}^{{\rm{(s)}}}}\right)}}}$
(7)

For the bidirectional grasping motion in this study, the total efficacy
coefficient $Q_{i}$ are the combined index that expresses one or two
successful object grasping. Thus, $Q_{i}$ can be expressed as the sum of two
parts:

${Q_{i}}={Q_{{\rm{S}}i}}+{Q_{{\rm{B}}i}}$ (8)

where $Q_{{\rm{S}}i}$ denotes the efficacy coefficient for single-direction
grasping and $Q_{{\rm{B}}i}$ denotes the extra efficacy coefficient for the
bidirectional grasping.

${q_{{\rm{S}}i,{\rm{F}}m}}$ and ${q_{{\rm{B}}i,{\rm{F}}m}}$ are the variation
indexes that contributing to the front-directional grasp during the single-
direction and bidirectional grasping status. $M$ is the number of front-
directional grasping indexes. ${q_{{\rm{S}}i,{\rm{R}}n}}$ and
${q_{{\rm{B}}i,{\rm{R}}n}}$ are those of the reversed-directional grasp. And
$N$ is the number of reverse-directional grasping indexes. When in single-
direction grasping status, $s=0$. When in bidirectional grasping status,
$s=1$.

###  IV-C Biomimetic Analysis

The efficacy coefficients of both human and robot hands $Q_{{1}i}$,
$Q_{{1\rm{F}}i}$, $Q_{{1\rm{R}}i}$, $Q_{{2\rm{F}}i}$ were calculated. We
selected the bending angle of fingers and the distance between the thumb and
other fingers as the two evaluation index, which are the key indexes for hand
grasping. The results are respectively presented presented in Fig. 12(a) and
(b). The time scale defined by frame of the human and robot hands were
normalized.

At Stage 2, the efficacy coefficients of both human and robot hand generally
increase. At Stage 4, the soft robotic hand implements reverse-direction
grasping. Thus, the $Q_{{2}i}$ of the human hand during this stage is
constant. Meanwhile, the efficacy coefficients $Q_{{1\rm{R}}i}$ starts to
increase, which also leads the total efficacy coefficients $Q_{{1}i}$
continues to increase. The trade at Stage 6 is inverse compared with Stage 2.

For single-direction grasping, the efficacy coefficient of human hand (black
line) presents a quite similar variation with that of robot hand through all
the 3 stages. This testify that the soft hand performs a human-like motion of
hand grasp. When considering bidirectional grasping, the variation of
$Q_{{1\rm{F}}i}$ at Stage 2 is quite similar with that of $Q_{{1\rm{R}}i}$.
This indicates that the two successful grasps of bidirectional grasping is
mutually independent.

Figure 12: The biomimetic evaluation by comparing the soft robotic hand with
human hand during the single-direction grasp, the bidirectional grasp and
grasp release. (a) The grasp contribution of fingers about bending angle. (b)
The grasp contribution of fingers about fingertip distance.

##  V Conclusion

In this paper, we present a novel 3D reconstruction approach to analyze the
complex grasping motions of soft robots in underwater environment through
experiment videos. Using the videos captured by the 3D imaging system, we
reconstruct the configuration of the hand. The proposed reconstruction method
perform the reconstructed hand pose when the hand is both with and without
grasp objects. The 3D reconstructed pose result presents a satisfactory
performance and good accuracy. Using efficacy coefficient method, the finger
contribution of the bending angle and distance between fingers of soft robot
hand is analyzed by compared with that of human hand. The results show that
the soft robot hand perform a human-like motion in both single-direction and
bidirectional grasping.

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# Extracting the QCD ${\Lambda_{\overline{MS}}}$ parameter in Drell-Yan
Process using Collins-Sopper-Sterman approach

R. Taghavi (a), A. Mirjalili (a)
r.taghavi<EMAIL_ADDRESS>
(a) Physics Department, Yazd University, Yazd, Iran

###### Abstract

In this work we directly fit the QCD dimensional transmutation parameter,
${\Lambda_{\overline{MS}}}$, to experimental data of Drell-Yan (DY)
observables. For this purpose we first obtain the evolution of transverse
momentum dependent parton distribution functions (TMDPDFs) up to the NNLO
approximation based on Collins-Sopper-Sterman formalism. As is expecting the
TMDPDFs are appearing at larger values of transvers momentum by increasing the
energy scales and also the order of approximation. Then we calculate the cross
section related to the TMDPDFs in the DY process. As a consequence of fitting
to the E288 experimental data at center of mass energy $\sqrt{s}=23.8$ $GeV$,
we obtain ${\Lambda_{\overline{MS}}}=249\pm 7MeV$ corresponding to the
renormalized coupling constant ${\alpha_{s}}(M_{z}^{2})=0.119\pm 0.001$ which
is within the acceptable range for this quantity. The results for DY cross
section at different energy scales are in good agreement within the available
data.

##  1  Introduction

To discuss the details of probing hadronic structure in high energy
collisions, the intrinsic transverse momentum carried by partons plays an
essential role. Allowing the parton distribution functions (PDFs) and
fragmentaion function (FFs) to depend additionaly on intrinsic transverse
momentum rather than the usual momentum fraction variables will cause the
teransverse momentum dependent (TMD) factorization formalism go beyond the
framework of standard factorization [1, 2]. To describe the processes which
are sensitive to intrinsic parton transverse, TMD factorization will be
needed. There are large variety of situations which can be considered as an
eveidnent to appear the usefulness of the TMD concept. Some examples are
Drell-Yan (DY) process, hadron production in electron-positron annihilation at
small transverse momentum and semi inclusive deep inelastic scattering
(SIDIS). In this article, we are focusing on the DY processes and try to do
some phenomenological investigation to extract an improtant parameter in
perturbative QCD analysis.

The renormalized strong coupling constant ${\alpha_{s}}(\mu)$ is a crucial
quantity in high energy collisions . Equivalently the Renormalization Group
(RG) invariant scale parameter ${\Lambda_{\overline{MS}}}$ is the fundamental
QCD scale which is depend on the number of active quark flavour, ${n_{f}}$,
and can be extracted using fit to the different experimental data confronted
with theoretical predictions. Although lots of works has been done to
determine this parameter, it is still of interest to be estimated by other
methods and using different available experimental data. Since the Drell-Yan
process is one of the good context to test QCD theory, we choose the available
low energy experimental data related to this process to find the
${\Lambda_{\overline{MS}}}$ parameter by makeing use of Collins, Sopper and
Sterman(CSS) resummation formalism [3] which is one of the theoretical
framework designed to account for QCD effects. The fitting is performed at
first four mass bins of 5 $GeV$ to 9 $GeV$ of the E288 experimental data .
Since the used energy scale, $Q$, is more than the mass of bottom quark, five
active flavour is considered in our analysis. Totally 28 data points are taken
into account to extract ${\Lambda_{\overline{MS}}}$ in the fit.

The organization of paper is as following. In Sec.2 we consider the evolution
of TMD PDF which is employed to calculate the cross section in DY process by
making use of the Collins-Sopper-Sterman (CSS) approach. We extend the current
results in [4] and obtain the TMD PDF up to the NNLO approximation. In Sec.3
we review the theoretical assumption to calculate the cross section of DY
processes in which the Brock-Landry-Nadolsky-Yuan (BLNY) parametrization is
used for the non-perturbative part of the calculation. Using fitting process
the unknown parameter ${\Lambda_{\overline{MS}}}$ is obtained in Sec.4.
Finally we give our conclusion in Sec.5.

##  2 TMD evolution

The evolution of quark-TMD PDF in the CSS formalism is given by [4, 5]:

$\displaystyle{{\tilde{F}}_{f/P}}(x,{b_{T}};\mu,{\zeta_{F}})=\overbrace{\sum\limits_{j}{\int_{x}^{1}{\frac{{d\hat{x}}}{{\hat{x}}}}}{{\tilde{C}}_{f/j}}(x/\hat{x},{b_{*}};{\mu_{b}}^{2},{\mu_{b}},g({\mu_{b}})){f_{j/P}}(\hat{x},{\mu_{b}})}^{(a)}$
$\displaystyle\times\overbrace{\exp\\{\ln\frac{{\sqrt{{\zeta_{F}}}}}{{{\mu_{b}}}}\tilde{K}({b_{*}};{\mu_{b}})+\int_{{\mu_{b}}}^{\mu}{\frac{{d{\mu^{{}^{\prime}}}}}{{{\mu^{{}^{\prime}}}}}}[{\gamma_{F}}(g({\mu^{{}^{\prime}}});1)-\ln\frac{{\sqrt{{\zeta_{F}}}}}{{{\mu^{{}^{\prime}}}}}{\gamma_{K}}(g({\mu^{{}^{\prime}}}))]\\}}^{(b)}$
$\displaystyle\times\overbrace{\exp\\{{g_{j/P}}(x,{b_{T}})+{g_{K}}({b_{T}})\ln\frac{{\sqrt{{\zeta_{F}}}}}{{\sqrt{{\zeta_{F,0}}}}}\\}}^{(c)}.$
(1)

In this equation, the ${f_{j/P}}(\hat{x},{\mu_{b}})$ is the ordinary PDF and
${\tilde{C}_{f/j}}(x/\hat{x},{b_{*}};{\mu_{b}}^{2},{\mu_{b}},g({\mu_{b}}))$,
$\tilde{K}({b_{*}};{\mu_{b}})$, ${\gamma_{K}}(g({\mu^{{}^{\prime}}}))$ and
${\gamma_{F}}(g({\mu^{{}^{\prime}}});1)$ are functions which for all ${b_{T}}$
are perturbatively calculable. The non-perturbative ${b_{T}}$ behavior of
${\tilde{F}_{f/P}}(x,{b_{T}};\mu,{\zeta_{F}})$ and $K(b;{\mu_{b}})$ are
governed respectively by the functions ${g_{j/P}}(x,{b_{T}})$ and
${g_{K}}({b_{T}})$ . These functions are scale-independent and universal. The
anomalous dimensions of the ${{\tilde{F}}_{f/P}}(x,{b_{T}};\mu,{\zeta_{F}})$
and $\tilde{K}({b_{*}};{\mu_{b}})$ are represented by the ${\gamma_{F}}$ and
${\gamma_{K}}$ respectively. In the small
${b_{T}}\ll\frac{1}{{{\Lambda_{QCD}}}}$ limit the first phrase, $(a)$, in
Eq.(1) matches the TMD PDF to a collinear treatment . The hard part of this
phrase is given by Wilson coefficient function, ${\tilde{C}_{f/j}}$, where it
also contains the standard integrated PDF which is denoting a collinear
factor. The second phrase, $(b)$, of Eq.(1) includes exponential functions
which are different and all can perturbatively be calculated . The matching
between the small and large ${b_{T}}$-dependence is implementing by the third
phrase, $(c)$, of this equation. The non-perturbative feature, intrinsic to
the proton, for large-${b_{T}}$ values is given by ${g_{j/P}}(x,{b_{T}})$
function. On the other hand the non-perturbative behavior of
$\tilde{K}({b_{*}};{\mu_{b}})$ at large values of ${b_{T}}$ is presented by
${g_{K}}({b_{T}})$. However the function ${g_{j/P}}(x,{b_{T}})$ depends on the
external hadron but it can be considered universal. For ${g_{K}}({b_{T}})$
function there is different situation in which it is universal and also do not
depend on the external types of the hadrons.

To do the calculations, as it is customary, the choose $\sqrt{{\zeta_{F}}}=Q$
and $\sqrt{{\zeta_{F,0}}}={Q_{0}}$ are applied. In this work, we first obtain
the TMD PDF of up quark at the NNLO approximation with x = 0.09 for the small
$Q=\sqrt{2.4}$, medium $Q=5$ and $Q=91.9$ $GeV$ values. To do this we use the
package of $PDFCollinear[parton,x,{b_{T}}]$ [6] which gives us the collinear
part of the TMD PDF for a parton of a specified quark. Since up-quark is
concerned in our calculations, the $PDFCollinear[up,x,{b_{T}}]$ is used for
first phrase, $(a)$, of Eq.(1). Considering the second phrase, $(b)$, we are
able to increase the accuracy of calculation, taking into account the NNLO
contributions of $\tilde{K}({b_{*}};{\mu_{b}})$ ,${\gamma_{F}}$ and
${\gamma_{K}}$ functions.

The one-loop values for $\tilde{K}({b_{*}};{\mu_{b}})$ ,${\gamma_{F}}$ and
${\gamma_{K}}$ functions are as following [5]:

$\displaystyle\tilde{K}(\mu,{b_{T}})=-\frac{{{\alpha_{s}}{C_{F}}{L_{\bot}}}}{\pi},$
(2)
$\displaystyle{\gamma_{F}}(\mu)=\frac{{{\alpha_{s}}{C_{F}}}}{\pi}(\frac{3}{2}-\ln(\frac{{{\zeta_{F}}}}{{{\mu^{2}}}})),$
(3) $\displaystyle{\gamma_{K}}(\mu)=2\frac{{{\alpha_{s}}{C_{F}}}}{\pi}\;,$ (4)

where:

$\displaystyle{L_{\bot}}(\mu,b_{T})=Log(\frac{{{\mu^{2}}{b_{T}}^{2}}}{{4{e^{-2{\gamma_{E}}}}}}).$
(5)

Considering Eqs.(2-5) the TMD PDF for up quark at three different scales can
be calculated in the NLO approximation. To reach our main aim to obtain the
NNLO approximation in evolving the TMD PDF, we need to add the NNLO
contributions to the Eqs.(2-4) which are available in [5] as:

$\displaystyle\tilde{K}(\mu,{b_{T}})=-\frac{{{\alpha_{s}}{C_{F}}{L_{\bot}}}}{\pi}+{(\frac{{{\alpha_{s}}}}{\pi})^{2}}(\frac{1}{{32}}{\Gamma_{0}}{\beta_{0}}{L_{\bot}}{(\mu,{b_{T}})^{2}}-\frac{1}{8}(\frac{{{\Gamma_{1}}}}{2}{L_{\bot}}(\mu,{b_{T}}))-\frac{1}{8}{d_{20}}),$
(6)
$\displaystyle{\gamma_{F}}(\mu;{\zeta_{F}}/{\mu^{2}})={\alpha_{s}}\frac{{{C_{F}}}}{\pi}(\frac{3}{2}-\ln(\frac{{{\zeta_{F}}}}{{{\mu^{2}}}}))+{(\frac{{{\alpha_{s}}(\mu)}}{\pi})^{2}}(-4{C_{F}}((\frac{{67}}{9}-\frac{{{\pi^{2}}}}{3}){C_{A}}-\frac{{20}}{9}{T_{F}}{N_{f}})(Log[\frac{\zeta}{{{\mu^{2}}}}])$
$\displaystyle-{C_{F}}^{2}(-3+4{\pi^{2}}-48{\varsigma_{3}})-{C_{F}}{C_{A}}(-\frac{{961}}{{27}}-\frac{{11{\pi^{2}}}}{3}+52{\varsigma_{3}})-{C_{F}}{T_{F}}{N_{f}}(\frac{{260}}{{27}}+\frac{{4{\pi^{2}}}}{3})),$
(7)
$\displaystyle{\gamma_{K}}(\mu)=2\frac{{{\alpha_{s}}{C_{F}}}}{\pi}+2{(\frac{{{\alpha_{s}}}}{\pi})^{2}}({C_{F}}{C_{A}}(\frac{{67}}{{36}}-\frac{{{\pi^{2}}}}{{12}})-\frac{5}{{18}}{C_{F}}{N_{f}})\;,$
(8)

where:

$\displaystyle{\Gamma_{0}}=4{C_{F}},$ (9)
$\displaystyle{\beta_{0}}=\frac{{11}}{3}{C_{A}}-\frac{4}{3}{T_{F}}{N_{f}},$
(10)
$\displaystyle{\Gamma_{1}}=4{C_{F}}((\frac{{67}}{9}-\frac{{{\pi^{2}}}}{3}){C_{A}}-\frac{{20}}{9}{T_{F}}{N_{f}}),$
(11)
$\displaystyle{d_{20}}={C_{F}}{C_{A}}(\frac{{404}}{{27}}-14{\varsigma_{3}})-(\frac{{112}}{{27}}){C_{F}}{T_{F}}{N_{f}}.$
(12)

By inserting Eqs.(6-8) in Eq.(1) and doing the required numerical calculations
the TMD PDF of up quark can be obtained for the small, medium and large values
of $Q=\sqrt{2.4}$ , 5, and 91.19 GeV with x = 0.09 in ${Q_{0}}=1.6$. We depict
the related plots run over a range typical for studies of TMD-functions from
${P_{T}}=0$ to 6 GeV in Fig.1.

Figure 1:  The TMDPDF for up quark at $Q=\sqrt{2.4}$ , 5.0 and 91.19 GeV with
x = 0.09. The plot shows the result at the NLO (dashed and dash-dotted curves)
and NNLO (solid curves) approximations. Solid curves are resulting from our
results with ${b_{\max}}=0.5\;Ge{V^{-1}}$.

As is expected by increasing the energy scale the effect of TMD PDF at larger
values of $P_{T}$ is growing. By raising the accuracy of calculations and
increasing the energy scales, the amount of of TMDPDF at small $P_{T}$ value
is deceasing. In this case the effect of this function at large values of
$P_{T}$ is slightly extending.

##  3 Hadronic production and DY processes in CSS formalism

The differential cross section for the Drell-Yan process in the CSS
resummation formalism can be expressed as [7, 8]:

$\displaystyle\frac{{d\sigma}}{{d{Q^{2}}d{Q_{T}}^{2}dy}}=\frac{1}{{{{(2\pi)}^{2}}}}\delta({Q^{2}}-{M_{V}}^{2})\int{{d^{2}}b{e^{i{{\vec{Q}}_{T}}.{{\vec{b}}_{T}}}}{{\tilde{W}}_{j\bar{k}}}(b_{T},Q,{x_{1}},{x_{2}})+Y(Q,{Q_{T}},}{x_{1}},{x_{2}}).$
(13)

In this equation $Q,{Q_{T}}$ and $y$ are the invariant mass, transverse
momentum and rapidity of vector boson V respectively in which
${x_{1}},{x_{2}}$ are presented by ${x_{1}}={e^{y}}Q/s$, ${x_{2}}={e^{-y}}Q/s$
where s is the overall center-of-mass energy of colliding hadrons. In Eq.(13)
the two dimensional fourier transform can be converted to one dimensional
integral, using the first type of Bessel function in which Eq.(13) can be
appeared as [5]:

$\displaystyle\int{{d^{2}}{b_{T}}{e^{i{P_{T}}.{{\vec{b}}_{T}}}}{{\tilde{W}}_{j\bar{k}}}({b_{T}},Q,{x_{1}},{x_{2}})}=2\pi\int{{b_{T}}d{b_{T}}{J_{0}}({b_{T}}{P_{T}}){{\tilde{W}}_{j\bar{k}}}({b_{T}},Q,{x_{1}},{x_{2}})}.$
(14)

The ${{\tilde{W}}_{j\bar{k}}}$ term is defined by [9]:

$\displaystyle{\widetilde{W}_{j\overline{k}}}(b_{T},Q,{x_{1}},{x_{2}})={e^{-S(Q,b_{T},{C_{1}},{C_{2}})}}\sum\limits_{j,\overline{k}}{\frac{{{\sigma_{0}}}}{s}{{\overline{P}}_{j,{h_{1}}}}}({x_{1}},b_{T},\frac{{{C_{1}}}}{{{C_{2}}b_{T}}}){\overline{P}_{\overline{k},{h_{2}}}}({x_{2}},b_{T},\frac{{{C_{1}}}}{{{C_{2}}b_{T}}})+(j\leftrightarrow\overline{k})\;.$
(15)

In Eq.(15) the indices ${h_{1}}$ ,${h_{2}}$ denotes to the parton number 1 and
2. The
$\sum\limits_{j}{{{\overline{P}}_{j{h_{1}}}}}({x_{1}},b,\frac{{{C_{1}}}}{{{C_{2}}b}})$
corresponds to the part $(a)$ of Eq.(1) where we calculate it, using the
package of $PDFCollinear[up,{x_{1}},{b_{T}}]$ as we described in Sec.2. The
${C_{1}}$ and ${C_{2}}$ are renormalization constants. To remove some of the
logarithmic divergences in
${\widetilde{W}_{j\overline{k}}}(b_{T},Q,{x_{1}},{x_{2}})$, the best choice
for these constants are ${C_{1}}=2{e^{{\gamma_{E}}}}\equiv{b_{0}}$ and
${C_{2}}=\frac{{{C_{1}}}}{{{b_{0}}}}=1$ [10] . The Sudakov factor S in Eq.(15)
which is equivalent to the square of the $(b)$ factor in Eq.(1) is given by
[9]:

$\displaystyle
S(Q,b_{T},{C_{1}},{C_{2}})=\int_{{C_{1}}^{2}/{b_{T}^{2}}}^{{C_{2}}^{2}{Q^{2}}}{\frac{{d{{\overline{\mu}}^{2}}}}{{\overline{\mu}}}}[{\rm
A}({\alpha_{s}}(\overline{\mu}),{C_{1}})\ln(\frac{{{C_{2}}^{2}{Q^{2}}}}{{{{\overline{\mu}}^{2}}}})+B({\alpha_{s}}(\overline{\mu}),{C_{1}},{C_{2}})]\;.$
(16)

The A and B functions in Eq.(16) can be found up to NNLO approximation in [7]
as:

$\displaystyle
A({\alpha_{s}}(\bar{\mu}),{C_{1}})=\sum_{N=1}^{2}{A^{(N)}}({C_{1}}){[\frac{{{\alpha_{s}}(\bar{\mu})}}{\pi}]^{N}},$
(17) $\displaystyle
B({\alpha_{s}}(\bar{\mu}),{C_{1}},{C_{2}})=\sum_{N=1}^{2}{B^{(N)}}({C_{1}},{C_{2}}){[\frac{{{\alpha_{s}}(\bar{\mu})}}{\pi}]^{N}}\;,$
(18)

where

$\displaystyle{A^{(1)}}({C_{1}})=\frac{4}{3},$ (19)
$\displaystyle{A^{(2)}}({C_{1}})=\frac{{69}}{9}-\frac{1}{3}{\pi^{2}}-\frac{{10}}{{27}}{N_{f}}+\frac{8}{3}{\beta_{1}}\ln({C_{1}}\frac{1}{2}{e^{\gamma}}),$
(20)
$\displaystyle{B^{(1)}}({C_{1}},{C_{2}})=\frac{8}{3}\ln(\frac{{{C_{1}}}}{{2{C_{2}}}}{e^{\gamma-3/4}}),$
(21)
$\displaystyle{B^{(2)}}({C_{1}},{C_{2}})=2[\frac{{67}}{9}-\frac{1}{3}{\pi^{2}}-\frac{{10}}{{27}}{N_{f}}]\ln(\frac{{{C_{1}}}}{{2{C_{2}}}}{e^{\gamma-3/4}})+\frac{8}{3}{\beta_{1}}\\{{\ln^{2}}({C_{1}}\frac{1}{2}{e^{\gamma}})-$
$\displaystyle{\ln^{2}}({C_{2}}{e^{3/4}})\\}-\frac{9}{8}+\frac{7}{9}{\pi^{2}}+\frac{2}{3}\varsigma(3)+(\frac{5}{{36}}-\frac{2}{{27}}{\pi^{2}}){N_{f}}.$
(22)

To achieve more precise results for DY cross section, one can use the NNLO
contributions for TMD PDF as was obtained in Sec.2.

The form factor $\widetilde{W}_{j\bar{k}}(b_{T},Q,{x_{1}},{x_{2}})$ in Eq.(15)
can be expressed in terms of its perturbative part
$\widetilde{W}_{j\bar{k}}^{pert}$ and nonperturbative function
$\widetilde{W}_{j\bar{k}}^{NP}$ as [10]:

$\displaystyle\widetilde{W}_{j\bar{k}}(b_{T})=\widetilde{W}_{j\bar{k}}^{pert}({b_{*}})\widetilde{W}_{j\bar{k}}^{NP}(b_{T}).$
(23)

with:

$\displaystyle{b_{*}}=\frac{{{b_{T}}}}{{\sqrt{1+{b_{T}}^{2}/{b_{\max}}^{2}}}}.$
(24)

In our calculations, the cross section of DY process is considered and
compared with the available experimental data. Indeed we take the
${\Lambda_{\overline{MS}}}$ as an unknown parameter that souled be determined
by fitting process. We analyze the DY process with BLNY fitting
parametrization in the non-perturbative part of the cross section using [8]:

$\displaystyle\exp\\{[-{g_{1}}-{g_{2}}\ln(\frac{Q}{{2{Q_{0}}}})-{g_{1}}{g_{3}}\ln(100{x_{1}}{x_{2}})]{b^{2}}\\},$
(25)

where ${g_{1}},{g_{2}},{g_{3}}$ are the unknown parameters that should be
determined by fitting to the expression in Eq.(25) which is corresponding to
the square of $(c)$ factor in Eq.(1). Using the E288 experimental data (for Q
= 5 to 9) [12] we can obtain the values of these parameters as well as
${\Lambda_{\overline{MS}}}$ , using CERN subroutine MINUIT [13]. We calculate
the theoretical cross section using b-star method [11], choosing
${b_{\max}}=0.5\;Ge{V^{-1}}=0.1$ $fm$ in all part of fitting.

To get better consistency between the theoretical results and the related
experimental data, the data at each mass bin including 5 $<$ Q $<$ 6, 6 $<$ Q
$<$ 7, 7 $<$ Q $<$ 8 and 8 $<$ Q $<$ 9, is multiplied by individual
normalization factors. Then we have totally four normalization factors in our
calculations.

##  4 Results and discussions

As mentioned before our purpose is to determine the
${\Lambda_{\overline{MS}}}$ in DY process together with the unknown parameters
$g_{1}$, $g_{2}$ and $g_{3}$ at low energy scales, considering transverse
momentum distributions by CSS resummation formalism. Therefor we should
include the experimental data for which the transverse momentum distributions
have more effect in non-perturbatve part of our calculations. Hence we use the
low energy DY data in the region where the transverse momentum of the lepton
pair is much smaller than its invariant mass Q. The reason is that the CSS
formalism describes the production of DY pairs better in the central rapidity
region [10]. The E288 data from the $p+Cu\to{\mu^{+}}{\mu^{-}}+X$ process at
$\sqrt{s}=23.8$ $GeV$ with $x=0.03$ is a good case with those properties. By
accessing to the TMDPDFs we are able to calculate the related cross section in
DY process. Using the E288 experimental data, we do the fitting and obtain the
values of the non-perturbative parameters $g_{1}$, $g_{2}$ and $g_{3}$, using
the BLNY parametrization [8] and finally ${\Lambda_{\overline{MS}}}$ which is
placed in the perturbative part of calculations. The fitting is done by MINUIT
programming and the results are:

$\displaystyle{g_{1}}=0.43\pm 0.09\ ,\;{g_{2}}=0.67\pm 0.04\
,\;{g_{3}}=-0.63\pm 0.07,$ (26)
$\displaystyle{\Lambda_{\overline{MS}}}=249MeV\pm 7.$ (27)

which is corresponding to ${\alpha_{s}}(M_{z}^{2})=0.119\pm 0.001$ for energy
scale of Z-mass boson at the NLO approximation. The goodness fitting with
$\chi^{2}/{d.o.f.}=1.12$ and finding for ${\Lambda_{\overline{MS}}}$ a value
within the acceptable range indicates that the BLNY parametrization is
suitable to describe low energy events.

We use the obtained numerical results for the unknown parameters and plot the
related DY cross section. We then compare the result with the available
experimental data and depict them in Fig.2 for $p+Cu\to{\mu^{+}}{\mu^{-}}+X$
reaction at $\sqrt{s}=23.8$ $GeV$ with $x=0.03$. As can be seen a good
consistency is existing between them.

Figure 2:  The theoretical calculation of DY cross section which is compared
to E288 experimental data [12] for the $p+Cu\to{\mu^{+}}{\mu^{-}}+X$ reaction
at $\sqrt{s}$=23.8 GeV.

##  5 Conclusion

We considered the evolution of transverse momentum dependent parton
distribution function for up quark in CSS formalism up to NNLO approximation
which can be easily extended to obtain the TMD PDF for other quarks by similar
calculations. In particular, we have used the TMD evolution kernel at NNLO
approximation which to our knowledge, has not been done before. We also
indicated that one can directly relate QCD observables to the underlying
dimensional transmutation parameter of the theory,
${\Lambda_{\overline{MS}}}$. We extracted the ${\Lambda_{\overline{MS}}}$
within the appropriate range of perturbative QCD and showed that the TMD
formalism is a powerful tool to analyze perturbative and non-perturbative
effects in cross section spectra of DY process. The result of global fit
showed that the BLNY parametrization is a good one for the non-perturbative
part of cross section and the CSS formalism can formulate and describe
properly the E288 experimental data (for low energy Drell-Yan process).

The calculations for DY cross section can be extended to the NNLO
approximation which we hope to report them in our further research activity.
Employing the experimental data for DY cross section at high energy scales is
also a valuable task to increase the precision of our fitting which can be
done in future.

## Acknowledgements

R.T and A.M acknowledge Andria Signore which provided us the package of PDF
collinear and thanks S.Atashbar Tehrani for useful discussions.

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