# Probing gluon distributions with $D^{0}$ production at the EicC

Daniele Paolo Anderle<EMAIL_ADDRESS>Key Laboratory of Atomic and
Subatomic Structure and Quantum Control (MOE), Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China  Guangdong Provincial
Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China
Normal University, Guangzhou 510006, China  Guangdong-Hong Kong Joint
Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South
China Normal University, Guangzhou 510006, China  Aiqiang Guo
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, Gansu Province 730000, China  University of Chinese Academy of
Sciences, Beijing 100049, China  Felix Hekhorn<EMAIL_ADDRESS>Tif
Lab, Dipartimento di Fisica, Università di Milano and
INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy  Yutie Liang
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu
Province 730000, China  University of Chinese Academy of Sciences, Beijing
100049, China  Guangdong Provincial Key Laboratory of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China  Yuming Ma  Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, Gansu Province 730000, China  Lei Xia  University of Science and
Technology of China, Hefei, Anhui Province 230026, China  Hongxi Xing  Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE),
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China  Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China  Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou, Guangdong Province 516000,
China  Yuxiang Zhao  Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, Gansu Province 730000, China  Southern Center for Nuclear-Science
Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences,
Huizhou, Guangdong Province 516000, China  University of Chinese Academy of
Sciences, Beijing 100049, China  Key Laboratory of Quark and Lepton Physics
(MOE) and Institute of Particle Physics, Central China Normal University,
Wuhan 430079, China

###### Abstract

The Electron-Ion Collider in China (EicC) has been proposed to study the inner
structure of matter and fundamental laws of strong interactions. In this
paper, we will present a conceptual design of the tracking system based on the
state-of-art silicon detector and Micro-Pattern Gaseous Detector at the EicC
and demonstrate that it will enable us to reconstruct charm hadron with good
significance, hence study gluonic parton distribution functions in nucleons
and nuclei, as well as gluon helicity distributions. The impact study using
reweighting techniques shows that the impact of the EicC will be mainly in the
large $x$ region. It complements similar physics programs at the Electron-Ion
Collider at Brookhaven National Laboratory.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II A tracking detector conceptual design at the EicC
    1. II.1 Requirements
    2. II.2 The magnetic field
    3. II.3 Choice of technology
    4. II.4 Geometry
    5. II.5 Performance
      1. II.5.1 Momentum resolution
      2. II.5.2 Spatial resolution and efficiency
  3. III Impact study on gluon distributions
    1. III.1 Unpolarized proton gluon distribution function
    2. III.2 Unpolarized nuclear gluon PDF
    3. III.3 Proton gluon helicity PDF
  4. IV Summary

##  I Introduction

Understanding the visible world we are living in is of fundamental importance
to modern sciences. The building blocks of visible matter, nuclei, and
nucleons, are in turn composed of quarks and gluons bound together by a strong
force. The underlying theory describing the strong interactions between quarks
and gluons is Quantum Chromodynamics (QCD). To gain a deeper insight into the
inner structure of nucleons, an Electron-ion collider in China (EicC) has been
proposed Anderle  _et al._ (2021a). It will provide highly polarized electrons
and protons with a variable center-of-mass (c.m.) energies ranging from 15 to
20 GeV and with a luminosity of $(2-3)\times
10^{33}\,\textrm{cm}^{-2}\cdot\textrm{s}^{-1}$. A series of unpolarized ion
beams ranging from Carbon to Uranium will also be available at the EicC.

The EicC will explore a wide range of physics topics: the spin structure of
nucleons, the partonic structure of nuclei, the parton interaction with the
nuclear environment, and exotic states, especially those with heavy flavor
quark contents. In order to achieve such broad physics goals, a hermetical
detector system will be constructed with cutting-edge technologies. In this
paper, we discuss the requirements for charged-particle tracking and describe
a conceptual design of a tracking system based on state-of-the-art silicon
pixel detectors coupled with Micro-Pattern Gaseous Detectors (MPGD). Based on
the high precision of the vertexing and momentum resolution for the tracking
system, a feasibility study with quantitative estimations on the measurements
of $D^{0}$/$\bar{D}^{0}$ production at the EicC is presented. Measurements of
charm hadron production in deep-inelastic scattering (DIS) can provide
constraints on the gluon distributions as the relevant unpolarized/polarized
charm structure functions provide direct access to the respective gluon
distributions from the leading order, through the photon-gluon fusion (PGF)
process.

These constraints can complement the extraction of gluon distributions via QCD
evolution or higher-order corrections in fully inclusive DIS. Similar studies
have already been performed for the Electron-Ion Collider (EIC) at Brookhaven
National Laboratory Kelsey  _et al._ (2021); Anderle  _et al._ (2021b); Dong
_et al._ (2022); Abdul Khalek  _et al._ (2022). Our study here shows that the
EicC can make a significant impact on the unpolarized or polarized gluon
distributions of nucleons and nuclei through charm hadron production in the
large-$x$ region ($x>0.1$), which is complementary to the studies at the EIC.

The rest of the paper is organized as follows: Section II describes the
proposed tracking system as well as its performance and Section III presents
the physics goals and simulation studies on the determination of gluon
distributions with $D^{0}$/$\bar{D}^{0}$ production in the DIS process. It
contains both unpolarized gluon distribution studies with unpolarized e+p and
e+Au collisions and polarized gluon distribution study with longitudinally
polarized e+p collisions. In Section IV we summarize our conclusions.

##  II A tracking detector conceptual design at the EicC

###  II.1 Requirements

In the EicC baseline design, an electron beam energy is 3.5 GeV, and proton
beam energy of 20 GeV, resulting in a c.m. energy of 16.7 GeV and a cross-
section of 20.8 $\mu b$. The luminosity is expected to be $L\rm=2\times
10^{33}\,cm^{-2}s^{-1}$ with an interaction rate of 83.2 kHz. The PYTHIA
Sjostrand  _et al._ (2006) simulation shows the final state particles are
concentrated near pseudo-rapidity $\eta$ = 1, with a particle density rate of
$dN/d\eta dt=\rm 8\times 10^{4}/s$.

Among the final state particles, the scattered electrons provide crucial
information to most of the physics programs, in particular those focusing on
the processes of DIS, semi-inclusive deep inelastic scatterings (SIDIS),
deeply virtual Compton scattering (DVCS), and so on. It is noticed that with
increasing $Q^{2}$, the scattered electrons are less boosted to negative
pseudo-rapidity. For physics requiring $Q^{2}$ larger than 1 $\rm{GeV^{2}}$
(such as SIDIS or DVCS), a detector coverage of $\eta>-2$ is sufficient for
the scattered electron. In addition to scattered electrons, the other final
state particles are also important and have been studied. The number of final
state pions is about 1–2 orders of magnitude larger than those of kaons and
anti-protons. The momenta of hadrons in other $\eta$ regions are also
investigated. At a pseudo-rapidity smaller than 1, the momenta of final state
hadrons are expected to be smaller than 6 GeV, while at large pseudo-rapidity
region ($\eta>2$), the final state hadrons have momenta up to 15 GeV. Thus,
the detection and particle identification (PID) of the final state hadrons
needs to be considered at various pseudo-rapidity regions.

As a high luminosity machine, EicC could reduce the statistical uncertainty
down to a few percent for many measurements. To cope with the small
statistical uncertainty, we need a matching systematic uncertainty, which
requires a good detector. For example, a tracking detector with a tracking
resolution of a few percent is necessary. A momentum resolution of 1% for the
central coverage and 2% for small angles are marked according to the
experiences from similar experiments. For the first version of the conceptual
design, we divide the EicC detector into the central detector and the forward
(backward) detector. The central detector consists of the barrel part and two
endcaps, and it will be constructed inside a solenoid. In this paper, we focus
on the central detector.

###  II.2 The magnetic field

The magnetic field, which is provided by a superconducting solenoid, plays an
essential role in momentum measurement for the charged tracks.

Figure 1: The calculated Bz of EicC solenoid (in Tesla). The black lines show
the detector schematic.

The superconducting solenoid has a maximum field strength of 1.5 T with an
inner bore radius of 1.5 m and a coil length of 3.0 m. A stronger magnetic
field certainly is beneficial for the momentum resolution, meanwhile, it also
decreases the tracking efficiency for the low momentum track due to the larger
bending capability. From the PYTHIA simulation, the transverse momentum of the
majority of the track is less than 2 GeV, therefore, a field strength of 1.5 T
is at the balance point between the momentum resolution and tracking
efficiency. The 2D map of the field is also calculated as shown in Fig. 1. And
it is sufficient to provide the barrel tracking with a uniform high-field
region, while also containing the PID devices and the Electromagnetic
Calorimeter (ECal).

###  II.3 Choice of technology

To satisfy the requirements of EicC physics programs, we investigated various
tracking technologies and detector designs, including all silicon tracking
systems based on the Monolithic Active Pixel Sensors (MAPS) and a hybrid pixel
detector and MPGD tracking system. The latter one is proven to have better
performance and is utilized in this paper. We will explain why these
technologies are chosen for the first conceptual design.

MAPS provides high granularity, low power consumption, and consequently low
material budget, as well as the required readout speed in one device.
Therefore, it is considered the best detector technology to satisfy the
requirements of the EicC vertex and tracking detector. In addition, The
integration of charge collection and readout capabilities into one silicon
substrate is well-suited for the required level of integration and acceptance
coverage of the EicC. The MAPS detector has been developed for three
generations. The first generation was deployed in the STAR Heavy Flavour
TrackerContin  _et al._ (2018) and in the ALICE ITS2 Abelev  _et al._ (2014).
The latter used the ALPIDE sensorAglieri Rinella (2017), which is fabricated
in a commercial 180 nm Complementary Metal Oxide Semicondutor (CMOS) imaging
process provided by Tower Jazz. It provides better charge collection
properties, radiation hardness, and signal processing capabilities compared to
the traditional MAPS. The second generation of MAPS was designed to reach the
rate and radiation tolerance capability typically required by high luminosity
particle physics experiments. Recently, a large community is gathering to
develop a third-generation MAPS in a 65 nm CMOS imaging technology for future
experiments through the ALICE ITS3 project Musa (2019) and the CERN EP R&D
program. The ITS3 project is aiming to develop a three-layer vertex detector
with an extremely low material budget. It will apply low power design
techniques, large area, 2D stitched sensors thinned below 50 $\rm\upmu m$,
thus it can be bent around the beam pipe to minimize cooling, support
structure, and services in the active area, therefore, enabling a material
budget of only 0.05% $X_{0}$. Such a detector concept is a very attractive
solution for the EicC vertex layers where an extremely low material budget
coupled with the sensor’s high granularity will deliver the required vertex
resolution. For the tracking layers and disks, a reticule-size version of the
ITS3 sensor will be developed with a more conventional design of support
structures.

MPGD technologies such as Gas Electron Multiplier (GEM) Sauli (1997), Micro
Mesh Gaseous Structures (Micromegas) Giomataris  _et al._ (1996), Resistive
Micro Well (µRWELL) Bencivenni  _et al._ (2015) are widely used for tracking
in various particle physics experiment across the world. These technologies
typically combine a gaseous device for electron amplification with high
granularity strips or pads anode readout PCB to provide a combined excellent
2D space point resolution ($\sim 50\,\rm\upmu m$), fast signal per layer
($\sim$5 ns), radiation hardness and large area capabilities at a
significantly lower cost compared to silicon trackers.

EicC will adopt silicon MAPS near the interaction point and MPGDs farther out.
This configuration allows for low-material budget tracking with sufficient
redundancy over a large lever arm, which is critical to achieving the required
momentum resolution.

###  II.4 Geometry

The layout of the vertex and tracking system is illustrated in Fig. 2. The
tracker coverage for low values of pseudorapidity ($|\eta|<1.1$) is utilized
by a barrel hybrid tracking system including the inner silicon and outer MPDG
layers. The inner silicon barrel consists of three vertex layers and two
tracking layers. They occupy a region that has a maximum radius of 15 cm and a
total length of 38 cm. The vertex layers deploy the wafer-scale, stitched
sensors that are bent around the beam pipe, which is made of a beryllium
cylinder with a radius of 3.17 cm. The tracking layers comprise the same
stitched sensors but have different support structures. Two closely-spaced 2-D
layers of Micromegas are chosen to cover the outermost barrel region. Their
mean radii are approximately 48 cm and 77 cm, and their maximum total length
of approximately 200 cm. The radii of each layer and their corresponding
lengths along z are summarized in Table 1.

Figure 2: The conceptual design of EicC tracking system. The silicon detctor, MPGD, and support structures are shown in the red, dark-yellow, and light-yellow components, respectively. The beryllium section of the beam pipe is shown in green, and the rest of the beam pipe is shown in cyan. The solenoid is shown in gray. Table 1: The radii (R), lengths, pitch(pixel) size, material budget ($X/X_{0}(\%)$), and technology (Tech.) for the barrel tracking system R(cm) | Length(cm) | Pitch Size ($\rm\upmu m$) | $X/X_{0}(\%)$ | Tech.
---|---|---|---|---
3.3 | 28.0 | 10 | 0.08 | Silicon
4.4 | 28.0 | 10 | 0.08 | Silicon
5.4 | 28.0 | 10 | 0.08 | Silicon
8.0 | 28.0 | 10 | 0.08 | Silicon
15.0 | 38.7 | 10 | 0.08 | Silicon
47.7 | 127.5 | 150($\rm r\upphi$)$\times$150(z) | 0.40 | MPGD
49.6 | 127.5 | 150($\rm r\upphi$)$\times$150(z) | 0.40 | MPGD
75.6 | 202.0 | 150($\rm r\upphi$)$\times$150(z) | 0.40 | MPGD
77.5 | 202.0 | 150($\rm r\upphi$)$\times$150(z) | 0.40 | MPGD

To cover the larger values of pseudorapidity ($|\eta|>1.1$), the tracking
system consists of silicon disks followed by large-area Micromegas in the
forward (proton/nucleus-going) directions and all silicon disks in backward
(electron-going) direction. The silicon disks will use the same sensor
technology as the vertex and tracking layers in the barrel region. There are
five silicon disks in the both backward direction and the forward direction.
They start from 25 cm along the z direction on either side of the interaction
point and extend to 145 cm in the backward direction and 134 cm in the forward
direction. The minimum radii are determined by the divergence of the beam pipe
while the maximum outer radius of the disks is approximately 77 cm. One
Micromegas detector is implemented at 165 cm in the forward direction from the
IP. Its inner and outer radii are about 8 cm and 150 cm, respectively. In
addition, it is located behind the dual ring imaging Cerenkov (dRICH) detector
in the forward direction. This detector helps not only the momentum resolution
in the forward direction but also the seed finding of the dRICH ring. All the
geometry parameters and position as well as the material budget are listed in
Table 2 and Table 3.

Table 2: The radii ($R_{in}$ and $R_{out}$), position (Z), pitch(pixel) size, material budget ($X/X_{0}(\%)$), and technology (Tech.) for the end-cap region tracking system (proton going direction) $R_{in}$ (cm) |  $R_{out}$ (cm) | Z (cm) | Pitch Size ($\rm\upmu m$) | $X/X_{0}\%$ | Tech.
---|---|---|---|---|---
3.2 | 18.6 | 25.0 | 10 | 0.08 | Silicon
3.2 | 36.5 | 49.0 | 10 | 0.08 | Silicon
3.5 | 54.7 | 73.0 | 10 | 0.08 | Silicon
5.1 | 77.5 | 103.7 | 10 | 0.08 | Silicon
6.6 | 77.5 | 134.3 | 10 | 0.08 | Silicon
8.2 | 150.0 | 165.0 | 50($\rm r\upphi$)x250(r) | 0.40 | MPGD
Table 3: The radii ($R_{in}$ and $R_{out}$), position (Z), pitch(pixel) size, material budget ($X/X_{0}(\%)$), and technology (Tech.) for the end-cap region tracking system (electron going direction) $R_{in}$ (cm) |  $R_{out}$ (cm) | Z (cm) | Pitch Size ($\rm\upmu m$) | $X/X_{0}\%$ | Tech.
---|---|---|---|---|---
3.2 | 18.6 | -25.0 | 10 | 0.08 | Silicon
3.2 | 36.5 | -49.0 | 10 | 0.08 | Silicon
3.2 | 54.7 | -73.0 | 10 | 0.08 | Silicon
4.0 | 77.5 | 109.0 | 10 | 0.08 | Silicon
5.6 | 77.5 | 145.0 | 10 | 0.08 | Silicon
Figure 3: The material budget as a function of pseudorapidity of EicC tracking
system. The different colors show the contribution from the beam pipe (light
green), vertex detector (dark brown), tracking doctor in the barrel part
(blue) and end-cap part (cyan), the support structure (light brown), and the
rest part (dark green).

Both the barrel layers and the disks are made up of large-area, wafer-scale,
stitched sensors based on ALICE-ITS3 design. Besides the active silicon
volume, each layer includes components such as support structures between the
layers, which combined correspond to an average material budget of 0.08%
$X_{0}$ per layer. Each Micromegas layer in the barrel and each of the forward
disks has a material thickness well below 0.5% $X_{0}$. The total amount of
material budget and each component’s contribution to the tracker geometry are
shown in Fig. 3. With the current configuration, the material budget
contributed by the barrel and disk staves is $<5\%\,X_{0}$. A potential future
optimization to improve the performances of the barrel and end-cap tracker are
ongoing.

In the simulation, the sensor silicon pixels have a pitch of $10\times
10\,\rm\upmu m^{2}$ and silicon thickness of 50 $\rm\upmu m$. The pitch sizes
of the readout of Micromegas are $150\,\rm\upmu m$ in both z and $r-\phi$
directions for the barrel region. Meanwhile, the Micromegas disk in the
forward direction consists of many trapezoidal models, so the pitch size in
$r-\phi$ direction ($50\,\rm\upmu m$) is much smaller than that in the $r$
direction (250 $\rm\upmu m$).

The detector is mounted on a conical aluminum and carbon-fiber support
structure with a thickness of 4.5 mm, which is tapered for $z>58$ cm. Although
this support structure adds a huge amount of material to the detector as shown
in Fig. 3, the projective design concentrates this material into a narrow
pseudorapidity range at $|\eta|$ = 1.1. More realistic support structures and
services are needed to be implemented in the future.

###  II.5 Performance

####  II.5.1 Momentum resolution

The performance studies were carried out within the EicCRoot software
framework, which is being developed based on the FairRoot package Al-Turany
_et al._ (2012). It is an object-oriented simulation, reconstruction, and data
analysis framework. The geometry was implemented in GEANT4 Agostinelli  _et
al._ (2003) and the charged particles (e.g. pions, electrons, protons, and
muons) are generated from nominal IP and cover the entire detector acceptance
by multiple generators provided by the EicCRoot. The momentum is set from 0 to
15 GeV/$c$ and distributed uniformly. The magnetic field is produced by a
superconducting solenoid introduced by SectionII.2. The interaction between
the generated particles and the detector is handled by GEANT4, in which the
multiple-scattering effect and energy loss of the track are taken into
account. A hit is defined as the position where the particle enters the active
area of the detector. Then, these hit positions are smeared according to the
detector resolution. All the hits belonging to one track are selected
according to the truth-track information and used for track fitting algorithm
with the Kalman filtering method Rauch and Schlüter (2015).

After the tracking fitting, the standard deviation of the
$dp/p=(|\overrightarrow{p}_{truth}|-|\overrightarrow{p}_{reco}|)/|\overrightarrow{p}_{truth}|$
can be measured and defined as the momentum resolution, where
$|\overrightarrow{p}_{truth}|$ and $|\overrightarrow{p}_{reco}|$ are the
generated and reconstructed absolute value of the particle momentum,
respectively. Figure 4 (top) shows the momentum resolution as a function of
momentum for charged pions, electrons, muons, and protons in the
pseudorapidity range $0.0<|\eta|<0.5$. The multiple-scattering effect is more
pronounced for protons and kaons below 2 GeV/$c$, therefore, worsening the
resolution significantly. With the increase of the moment, the $dp/p$ rises
almost linearly. This phenomenon is understandable because the measured
sagitta will be decreased for the stiffer tracks (due to the higher momentum).
Figure 4 (bottom) shows the momentum-resolution results as a function of
pseudorapidity in the momentum range $5.0<p<7.5$ GeV/$c$. The momentum
resolution is approximately constant when $\eta<2$, and then it rises quickly.
In addition, the electron momentum resolution is systematically worse than
other particles in most of the studied range because of its smaller mass.
Overall, the performance is very similar for all kinds of particles.

Figure 4: Momentum resolution for different particles in the 1.5 T magnetic
field. Left: $dp/p$ as a function of momentum in the $0<\eta<0.5$ range.
Right: $dp/p$ as a function of pseudorapidity in the $5.0<p<7.5$ GeV/$c$
range. Figure 5: Momentum resolution as a function of momentum in different
pseudorapidity regions. The markers represent the resolutions obtained from
the simulations, and the lines correspond to the best fit to the momentum-
dependent resolution. The functional form used in the fits is $dp/p=Ap\oplus
B$, and the parameters $A$ [%/(GeV/$c$)] and $B$ [%] are given in the plots.

We also studied the momentum resolution in different $\eta$ regions for pions
as shown in Fig. 5. Momentum resolutions are typically parameterized by the
function $dp/p=A\cdot p\oplus B$, where A and B are fit parameters and
$\oplus$ indicates sum in quadrature. The fit results are shown in the figure.
In the case of the current magnetic field, the conceptual tracker design
satisfies the physics requirements in the momentum range of $0<p<15$ GeV/$c$
for $|\eta|<2.5$.

####  II.5.2 Spatial resolution and efficiency

The tracking system is designed not only to measure the momentum of the
charged particles but also to determine the primary vertex for an event as
well as the secondary vertex for the long-life particle decay. This capability
is crucial for background suppression and therefore improves the sensitivity
significantly for the process with long-life particles. The spatial resolution
can be studied by measuring the Distance of the Closest Approach (DCA), which
is defined as the spatial separation between the primary vertex and the
reconstructed track projected to the z-axis ($DCA_{z}$) or to the x-y plane
($DCA_{r\phi}$). The DCA resolutions are determined as the standard deviation
of normal functions that is used to fit to the $DCA_{z}$ and $DCA_{r\phi}$
distributions. DCA-resolution could be parameterized by the function
$\sigma(DCA)=A/p_{T}\oplus B$, where $p_{T}$ stands for the transverse
momentum. The results, including the fits and fitted parameters, for pions are
shown in Figs. 6 and 7.

Figure 6: Longitudinal DCA resolution as a function of transverse momentum in
different pseudorapidity regions. The markers represent the resolutions
obtained from the simulations, and the lines correspond to the best fits to
the momentum-dependent resolution. The functional form used in the fits is
$\sigma(DCA)=A/p_{T}\oplus B$, and the parameters $A$ [$\rm\upmu m\cdot$
GeV/$c$] and $B$ [$\rm\upmu m$] are given in the plots.

The primary vertex represents the average vertex position of all the
reconstructed charged tracks in an event assuming they originate from IP. To
study the resolution of the primary vertex, millions of $e+p$ events at 3.5
GeV $\times$ 20 GeV collisions are generated by the PYTHIA generator in the
EicCRoot framework. In the simulation, we require $Q^{2}>1\,\rm{GeV^{2}}$, and
all the hadrons and scattering electrons are used to reconstruct the primary
vertex. The primary vertex is obtained by fitting all the tracks with the
Kalman filtering method Waltenberger (2011). The vertex residual distribution
with respect to the generated truth vertex for track multiplicity of three is
shown in Fig. 8. Since these distributions contain non-Gaussian tails, it is
difficult to be parameterized as a normal distribution. Therefore, three
coherent Gaussian functions are used to parameterize these residuals. To
investigate the resolution change with respect to the track multiplicity, we
fit the peak of these distributions with a single Gaussian function and define
the standard deviation as the primary vertex resolution. The bottom plot of
Fig. 9 shows the resulting primary vertex resolution as a function of track
multiplicity in events. The primary vertex resolution gets improved with the
increase of the tracking number and the values vary from $\rm 35\,\upmu m$ to
$\rm 25\,\upmu m$ in the studied range. In the fast simulation, the primary
vertex is smeared by sampling the fitted three Gaussian functions.

Figure 7: Transverse DCA resolution as a function of transverse momentum in
different pseudorapidity regions. The markers represent the resolutions
obtained from the simulations, and the lines correspond to the best fits to
the momentum-dependent resolution. The functional form used in the fits is
$\sigma(DCA)=A/p_{T}\oplus B$, and the parameters $A$ [$\rm\upmu m\cdot$
GeV/$c$] and $B$ [$\rm\upmu m$] are given in the plots.  Figure 8: Fit to the
residuals of the primary vertex with three Gaussian functions. The markers
correspond to the residual distribution from the simulation. The solid lines
show the best fits to the residuals and the dashed lines are the three sub-
components of the fit model.

Figure 9 top plot shows the charged pion tracking efficiency in different
$\eta$ regions. It should be noticed that pattern recognition efficiency is
not included in this study. The efficiency loss is mainly due to the detector
acceptance and the track fitting quality. To reconstruct a track, we require
that at least 3 hits should be found. Therefore, in the central part of the
detector ($|\eta|<$1), the efficiency in low $p_{T}$ is worse than in the
forward/backward region. It is related to the minimum $p_{T}$ threshold needed
for a track to reach the outer layer. For the $|\eta|>1$ region, the loss of
efficiency at higher $p_{T}$ because of the lower acceptance at the edge of
the barrel-to-endcap transition region and also at the small radii where the
beam pipe openings in the silicon disk. These tracking efficiencies obtained
from this simulation were applied in the following performance studies through
fast simulation.

Figure 9: Top: The tracking efficiency as a function of transverse momentum in
several pseudo-rapidity bins determined in the full simulation with PYTHIA
$e+p$ collision events at 18 GeV $\times$ 275 GeV with an event level
selection of $Q^{2}>1\,\rm{GeV^{2}}$. Bottom: the vertex resolution versus
track multiplicity

##  III Impact study on gluon distributions

Figure 10: The leading-order diagram for charm and anticharm pair production
in $e+p/A$ deep inelastic scatterings.

In the deep inelastic scattering, charm or anti-charm quarks can be produced
through PGF process at the leading order: $\gamma^{*}+g\rightarrow c+\bar{c}$.
The charm or anti-charm quarks are further fragmented into the D mesons. The
Feynman diagram of PGF is shown in Fig. 10. By tagging a D meson in the final
state for which tracking is critical, one can access unpolarized/polarized
gluon distributions in the nucleon, depending on different experimental
observables. In this section, we will discuss the impact studies on
unpolarized gluon distribution via $F_{2}^{c\bar{c}}$ measurements and the
$\Delta g$ distribution via $A_{1}^{c}$ measurements at the EicC.

Without conducting entirely new Parton Distribution Function (PDF) fits, we
employ well-established re-weighting techniques Giele and Keller (1998); Ball
_et al._ (2011, 2012) to facilitate our studies. The central values and
uncertainty bands of the distributions can be determined by statistically
averaging and calculating the variance of a replica set. This set can be
generated through random smearing of the data points within their error bands,
followed by re-fitting the data. Re-weighting procedures leverage Bayesian
interference to incorporate the information from a new data set into the
probability distribution of the initial replica set. This is achieved by
assigning appropriate weights to different replicas. The resulting re-weighted
replica set, which incorporates the new data information, is then utilized to
compute the central values and uncertainties of the updated distributions
while preserving the statistical rigor of the original set. However, it should
be noted that if a large number of replicas are suppressed due to vanishing
weights, the statistical relevance of the new replica set is compromised,
necessitating a new fit.

Figure 11: Illustration of the $D^{0}$ to $K\pi$ two-body decay topology.

Figure 12: The figure-of-merit, FOM = $S/\sqrt{S+B}$, distributions in the
criteria optimization for $K\pi$ pair-DCA (top), the cos$\theta_{r\phi}$
(middle), and $D^{0}$ Decay-Lengthrϕ in the transverse plane (bottom). The
blue lines show the position at the maximum FOM.

Figure 13: Fits to the $K\pi$ invariant-mass distributions in a few different
Bjorken-$x$ bins for 3.5 GeV $\times$ 20 GeV $e$+$p$ collisions. The red and
green dashed curves are the signal (Gaussian) and background (linear) fits,
and the blue curve is the sum.

The EicC pseudo data for reweighting analysis was produced as follows: The
geometry of the tracking system has been implemented in the GEANT4 and studied
within the full Monte-Carlo framework for detector simulation, as discussed in
Section II. The full simulation yields the detector response for momentum
resolution, single-track DCA resolution, primary vertex resolution, as well as
tracking efficiency. Afterward, the implementation of resolution
parametrizations in a fast-smearing simulation framework enables the
generation of ample statistics, facilitating comprehensive studies for physics
projections.

In our study, two beam energy configurations have been used for electron-
proton collisions: 3.5 GeV $\times$ 20 GeV and 5 GeV $\times$ 25 GeV. For the
nuclear gluon distribution study, we use electron-gold collisons with 3.5 GeV
$\times$ 10.35 GeV/u and 3.5 GeV $\times$ 12.93 GeV/u energy configurations.
The data was generated by pythiaeRHIC pyt  (PYTHIA V6.4) and then fed into the
fast-smearing framework to accommodate detector response. We take advantage of
the $K\pi$ two-body-decay to identify the $D^{0}$($\bar{D^{0}}$). The decay-
topology is illustrated in Fig. 11. To obtain a data sample with a good
signal-to-background ratio, three decay topological distributions were
investigated: the $K\pi$ pair-DCA, the $D^{0}$ Decay-Length ${r\phi}$ in the
transverse plane, and the angle $\cos(\theta_{r\phi})$, where $\theta$
represents the angle between the $D^{0}$ direction with respect to the primary
vertex and the momentum vector of the $K\pi$ pair. In practice, taking 3.5 GeV
$\times$ 20 GeV electron-proton collision as an example, the criteria of these
topological requirements are optimized by maximizing the Figure-Of-Merit, FOM
= $S/\sqrt{S+B}$, where $S$ is the number of $D^{0}$ and B is the estimated
background from the simulation. The results of optimization are shown in Fig.
12.

In addition to the $D^{0}$-decay topology requirements, the following
kinematic constraints were applied during the analysis: $Q^{2}>$ 2 GeV2, 0.05
$<$ y $<$ 0.8 and $W^{2}>$ 4 GeV2. The pion/kaon separation was assumed to be
feasible up to the momentum limits: 4, 6, 15 GeV/$c$ in pseudo-rapidity
regions (-3,-1), (-1,1) and (1,3) respectively. Here, the pseudo-rapidity is
positive in the ion-going direction. Following the selection requirements, the
data were binned in Bjorken-$x$ and $Q^{2}$. Within each bin, the $K\pi$
invariant mass spectrum was fitted to a Gaussian function for the signal,
along with a linear background, as shown in Fig. 13, which depicts electron-
proton collisions with an energy of 3.5 GeV $\times$ 20 GeV for various
Bjorken-$x$ bins.

###  III.1 Unpolarized proton gluon distribution function

Structure functions provide valuable insights into the partonic structure of
hadrons. The structure function $F_{2}(x,Q^{2})$ specifically characterizes
the interaction between transversely polarized photons and the proton. The
open-charm contribution, $F_{2}^{c\bar{c}}$ can be defined in terms of the
inclusive double differential $c\bar{c}$ cross-section in $x$ and $Q^{2}$ by

$\frac{d^{2}\sigma^{c\bar{c}}}{dx\,dQ^{2}}=\frac{2\pi\alpha^{2}}{xQ^{4}}\left\\{[1+(1-y)^{2}]F_{2}^{c\bar{c}}(x,Q^{2})-y^{2}F_{L}^{c\bar{c}}(x,Q^{2})\right\\}.$
(1)

The equation can be rewritten to form a reduced cross-section as

$\displaystyle\sigma_{r}^{c\bar{c}}(x,Q^{2})$
$\displaystyle=\frac{d^{2}\sigma^{c\bar{c}}}{dx\,dQ^{2}}\times\frac{xQ^{4}}{2\pi\alpha^{2}[1+(1-y)^{2}]}$
(2)
$\displaystyle=F_{2}^{c\bar{c}}(x,Q^{2})-\frac{y^{2}}{1+(1-y)^{2}}F_{L}^{c\bar{c}}(x,Q^{2})$

To extract $F_{2}^{c\bar{c}}$ at a fixed $Q^{2}$ and $x$, the Rosenbluth
technique Rosenbluth (1950) requires a minimum of two collision energies. In
this study, beam configurations of 3.5 GeV $\times$ 20 GeV and 5 GeV
$\times$25 GeV were chosen, each with an integrated luminosity of 10 fb-1. The
differential cross section can be determined experimentally

$\frac{d^{2}\sigma^{c\bar{c}}(x,Q^{2})}{dxdQ^{2}}=\frac{dN(D^{0}+\bar{D}^{0})/2}{L\cdot\varepsilon\cdot
B(D^{0}\rightarrow K\pi)\cdot f(c\rightarrow D^{0})\cdot dxdQ^{2}}.$ (3)

Here, $L$ is the integrated luminosity, $\varepsilon$ is the detection
efficiency, $B(D^{0}\rightarrow K\pi)$ is the branching fraction of $D^{0}$ to
$K\pi$ from PDG Group  _et al._ (2022), and $f(c\rightarrow D^{0})$ is the
$D^{0}$ fragmentation fraction in PYTHIA Sjöstrand  _et al._ (2006). The
$dN(D^{0}+\bar{D}^{0})$ in the numerator is the number of detected D mesons in
the specific $(x,Q^{2})$ bin which is determined from a fit to the $K\pi$
invariant mass spectrum.

Figure 14: The reduced cross sections (left panel) and the projected
$F_{2}^{c\bar{c}}$ (right panel) for the proton beams as a function of the
photon virtuality. We show two beam configurations for the reduced cross
sections from which we then extract the structure function.

In Fig. 14, we present the projected measurements based on one year’s worth of
data collected at the EicC. The left plot shows the reduced cross sections for
each $Q^{2}$ and $x$ bins at the two energies. The red boxes indicate the low
energy pseudo-data, while the black triangles are the high energy counterpart.
Using the reduced cross sections at these two energies, the $F_{2}^{c\bar{c}}$
for the proton is extracted and shown in the right plot of Fig. 14.

By utilizing these measurements, we can enhance our understanding of the
unpolarized proton gluon PDF, denoted as $g(x,Q^{2})$, through reweighting
techniques. For this initial estimate, our focus is on evaluating the impact
on the most recent CTEQ PDF set Hou  _et al._ (2021). We employ the mcgen
program Gao and Nadolsky (2014); Hou  _et al._ (2017) to convert the Hessian
PDF representation into a Monte Carlo replica representation. Furthermore, we
provide theoretical predictions, accurate to next-to-leading order (NLO)
precision, for all the required observables using yadism Candido  _et al._
(2022).

In Fig. 15, we present a comparison between the projected experimental
uncertainty for each bin and the corresponding PDF uncertainty before and
after reweighting. The subplots are organized based on a fixed value of
virtuality $Q^{2}$, while the horizontal axis represents the variation of
Bjorken-$x$. Here, our kinematics are constrained within the region
$0.01<x<0.2$ due to the specific beam configurations. Moreover, it should be
noted that PDF uncertainties are influenced by assumptions inherited from the
fitting procedure.

Figure 15 allows us to extract the influence of the measurement on the
underlying PDF, as demonstrated in Fig.17. The upper plots depict the singlet
and gluon PDF of the CTEQ PDF set Hou  _et al._ (2021), while the middle and
lower plots focus on the corresponding uncertainties. Through the reweighting
procedure, the number of replicas contributing to the uncertainties are
reduced from 100 to approximately effective 25. Consequently, we observe a
modest impact on the singlet PDF, resulting in an uncertainty reduction of
around 50%. In contrast, the gluon PDF shows a significant impact, with a
maximum uncertainty reduction by a factor of 3. It is important to note that
despite the use of a general-mass variable flavor number scheme (GM-VFNS)
Forte  _et al._ (2010) in computing the theory predictions, the relatively
small virtuality $Q^{2}$ still leads to the dominance of the PGF mechanism,
directly influencing the gluon PDF.

Figure 15: The predictions for the projected measurement of the structure-
function: in red are the projected experimental uncertainties, in light grey
(blue) is the theoretical predictions using the CT18ANLO Hou  _et al._ (2021)
PDF set before (after) re-weighting.

###  III.2 Unpolarized nuclear gluon PDF

Likewise, by performing measurements on reduced cross sections in electron-ion
collisions, it is possible to extract the nuclear $F_{2}^{c\bar{c}}$, thereby
gaining insights into the unpolarized nuclear gluon distributions. In this
context, we demonstrate the potential of the EicC through electron-gold
collisions. For this purpose, we generate two sets of pseudo-data for e+Au
collisions: 3.5 GeV$\times$10.35 GeV/u and 5 GeV$\times$12.93 GeV/u, each with
an integrated luminosity of 10 fb-1. The resulting projected ratios of
$F_{2}^{c\bar{c}}$ in e+Au collisions relative to e+p collisions are displayed
in Fig. 16.

Figure 16: Ratio of $F_{2}^{c\bar{c}}$ for the nuclear case (gold) towards the
proton case. Figure 17: Impact on CTEQ Hou  _et al._ (2021) singlet $\Sigma$
and gluon $g$ PDFs using a NLO computation. The hashed bands show the impact
of the pseudo data on the distributions’ uncertainties, whereas the solid
bands show the original uncertainty. In the middle plot, we show the absolute
uncertainty $\sigma$. The bottom plots show the ratios between the absolute
uncertainties before and after reweighting.

For our impact study, we focus on the latest PDF set of the EPPS family Eskola
_et al._ (2022). Through the reweighting procedure, the effective set of
replicas is reduced from 1000 to approximately 108. Figure 18 illustrates the
impact on the singlet and gluon PDF for a gold target. Similar to the proton
case, we observe a mild impact on the singlet PDF, while the gluon PDF
exhibits a significant influence.

Figure 18: Same as Fig. 17 but for the EPPS Eskola  _et al._ (2022) nuclear
PDF set with the bottom plot showing the ratio of the nuclear gluon
distribution $g^{Au}(x)$ from EPPS Eskola  _et al._ (2022) for gold to the
respective proton gluon distribution $g(x)$ from CTEQ Hou  _et al._ (2021) as
a function of the parton momentum $x$. For small $x\sim 10^{-2}$ the shadowing
region is visible, while for $x\sim 10^{-1}$ the anti-shadowing region is
visible.

In the bottom plot of Fig. 18, we directly compare the impact on the proton
and nuclear gluon PDF by plotting the respective ratio. Notably, we observe a
substantial reduction in uncertainties in the anti-shadowing region around
$x\sim 0.1$. This finding highlights the pronounced effect of the measurements
on both the proton and nuclear gluon PDFs.

###  III.3 Proton gluon helicity PDF

With longitudinally polarized electron and proton beams, one can perform
double spin asymmetry $A_{\rm LL}$ measurements in the $\vec{e}+\vec{p}\to
e^{\prime}+D^{0}+X$ process to access the gluon helicity distribution:

$\displaystyle A_{LL}^{\vec{e}+\vec{p}\to e^{\prime}+D^{0}+X}$
$\displaystyle=\frac{1}{P_{e}P_{p}}\frac{N^{++}-N^{+-}}{N^{++}+N^{+-}}$ (4)
$\displaystyle\approx\frac{y(2-y)}{y^{2}+2(1-y)}\frac{g_{1}^{c}(x,Q^{2})}{F_{1}^{c}(x,Q^{2})}$
$\displaystyle=\frac{y(2-y)}{y^{2}+2(1-y)}A_{1}^{c},$

where $N^{++}$ and $N^{+-}$ are the counts normalized by luminosity for
different beam helicity states respectively and $P_{e}$ ($P_{p}$) is the
electron (proton) beam polarization. The $A_{1}^{c}$ observable provides
direct access to the $\Delta g/g$ ratio at leading order. The COMPASS
collaboration conducted a pioneering measurement in polarized $\mu$-proton
collisions Adolph  _et al._ (2013). In the era of the EIC, this measurement
can be significantly enhanced. First, the presence of a high-quality vertex
detector enables thorough studies of $D^{0}$ decay topology, leading to
improved significance in $D^{0}$ reconstruction. Second, the combination of
high luminosity and large acceptance allows for precise measurements across a
wide range of kinematics. This has recently been investigated using an all-
silicon tracking system at the EIC Anderle  _et al._ (2021b). With a lower
center-of-mass energy, the EicC will extend the measurement into the
relatively high $x$ region. It is expected that in this region, the ratio of
polarized and unpolarized gluons, $\Delta g/g$, will exhibit significantly
larger values compared to the small $x$ region Khan  _et al._ (2022); Xu  _et
al._ (2022).

Figure 19: Projections of the A${}_{1}^{c}$ (formula 4) in the
$\vec{e}+\vec{p}\to e+D^{0}+X$ process in bins of Bjorken-$x$ for two beam-
energy configurations. The position of each data point in the plot is defined
by the weighted center of Bjorken-$x$ and Q2 for each particular bin. The
uncertainty indicated for each data point should be interpreted using the
scale shown on the right-side vertical axis of the plot.

In practice, the uncertainty of $A_{1}^{c}$ is determined within each
Bjorken-$x$ bin based on the number of $D^{0}$ particles extracted through a
fit to the $K\pi$ invariant mass spectrum. The projected uncertainties are
displayed in Fig. 19 for two collision energies: 3.5 GeV $\times$ 20 GeV and 5
GeV $\times$ 25 GeV. The mean Bjorken-$x$ and $Q^{2}$ values for each data
point correspond to the positions on the respective axes, while the magnitude
of the error is represented by the scale on the right-side vertical axis. The
integrated luminosity for each collision energy is set to 100 fb-1. The
assumed beam polarizations are 80% for the electron beam and 70% for the
proton beam.

Figure 20: Same as Fig. 17 but for the NNPDFpol1.1 polarized PDF Nocera  _et
al._ (2014) set for the collision energies 5 GeV $\times$ 25 GeV (top) and 3.5
GeV $\times$ 20 GeV (bottom). The bottom plots show the impact on the ratio of
$\Delta g(x)/g(x)$ plotted as a function of the momentum fraction $x$. To
produce the ratio we use the central value of NNPDF2.3 Ball  _et al._ (2013)
for the unpolarized gluon distributions.

The reweighting study is based on the NNPDFpol1.1 Nocera  _et al._ (2014)
helicity distribution set. Fig. 20 illustrates the impact on the singlet and
gluon helicity distributions for the two different energy configurations. The
reweighting procedure yields effective replicas of approximately 63 and 49 for
the high and low energy configurations, respectively. In both cases, we
observe a moderate reduction in the uncertainties of the singlet distributions
and a substantial reduction in the uncertainties of the gluon helicity
distributions. Notably, the lower energy configuration enhances the precision
of the gluon helicity distribution in the relatively high $x$ region, where
the signal significance is greater, as depicted in the bottom panel of the
plot.

To demonstrate the dependence of the constraining power on the energy
configuration, we compare the impact of the EIC Anderle  _et al._ (2021b) and
the present study in Fig. 21. The plot showcases the potential uncertainty
reduction as a function of momentum fraction for the two energy configurations
at the EIC and the EicC configuration of a 3.5 GeV $\times$ 20 GeV collision.

Figure 21: Comparison of ratios between the absolute uncertainties for the
gulon helicity distribution before and after reweighting for three
configuration energies: two from the EIC as studied in Anderle  _et al._
(2021b), namely for the collision energies 5 GeV $\times$ 41 GeV (green) and
18 GeV $\times$ 275 GeV (blue), and one from the EicC energy configuration
studied in this paper, 3.5 GeV $\times$ 20 GeV (red).

##  IV Summary

In summary, this study presented a conceptual design of a tracking system at
the EicC, consisting of silicon and MPGD components. The performance of this
system was evaluated through Geant4 simulations. A feasibility study focusing
on the reconstruction of $D^{0}/\bar{D}^{0}$ mesons was conducted, leveraging
the high resolution and excellent vertex reconstruction capabilities of the
tracking system. The results indicate that the EicC, as a lower center of mass
energy electron-ion collider with high luminosity, offers promising prospects
for probing gluon distributions in the relatively high $x$ region by utilizing
the abundant $D^{0}$ production. Specifically, in cases like the determination
of $\Delta g/g$, the EicC has an advantage in terms of its extended reach in
$x$ where the signal is more prominent. It is important to note that the EicC
encompasses a wide range of physics programs, and ongoing efforts involve full
simulations and optimizations of the detector system to meet the requirements
of various physics channels across diverse kinematic ranges. Although the
tracking system design may undergo future revisions, the stringent
requirements for heavy flavor measurements, including charm hadron
reconstructions, will be upheld at the EicC.

## Acknowledgements

The authors are grateful for the support from Professor Nu Xu. We also thank
Professor Feng Yuan for his helpful suggestions on this work. This work is
supported in part by the Strategic Priority Research Program of Chinese
Academy of Sciences, under grant number XDB34030301; the National Natural
Science Foundation of China (U203220, 12222512) and 100 Talents Program of CAS
(E129642Y). D. A. and H. X. is supported by the Guangdong Major Project of
Basic and Applied Basic Research No. 2020B0301030008, No. 2022A1515010683, by
the National Natural Science Foundation of China under Grant No. 12022512, No.
12035007. F. H. is supported by the European Research Council under the
European Unions Horizon 2020 research and innovation Programme (grant
agreement number 740006).

## References

  * Anderle  _et al._ (2021a) D. P. Anderle _et al._ , Front. Phys. (Beijing) 16, 64701 (2021a), arXiv:2102.09222 [nucl-ex] .
  * Kelsey  _et al._ (2021) M. Kelsey, R. Cruz-Torres, X. Dong, Y. Ji, S. Radhakrishnan, and E. Sichtermann, Phys. Rev. D 104, 054002 (2021), arXiv:2107.05632 [hep-ph] .
  * Anderle  _et al._ (2021b) D. P. Anderle, X. Dong, F. Hekhorn, M. Kelsey, S. Radhakrishnan, E. Sichtermann, L. Xia, H. Xing, F. Yuan, and Y. Zhao, Phys. Rev. D 104, 114039 (2021b), arXiv:2110.04489 [hep-ex] .
  * Dong  _et al._ (2022) X. Dong, Y. Ji, M. Kelsey, S. Radhakrishnan, E. Sichtermann, and Y. Zhao, (2022), arXiv:2210.08609 [hep-ph] .
  * Abdul Khalek  _et al._ (2022) R. Abdul Khalek _et al._ , Nucl. Phys. A 1026, 122447 (2022), arXiv:2103.05419 [physics.ins-det] .
  * Sjostrand  _et al._ (2006) T. Sjostrand, S. Mrenna, and P. Z. Skands, JHEP 05, 026 (2006), arXiv:hep-ph/0603175 .
  * Contin  _et al._ (2018) G. Contin _et al._ , Nucl. Instrum. Meth. A 907, 60 (2018), arXiv:1710.02176 [physics.ins-det] .
  * Abelev  _et al._ (2014) B. Abelev _et al._ (ALICE), J. Phys. G 41, 087002 (2014).
  * Aglieri Rinella (2017) G. Aglieri Rinella (ALICE), Nucl. Instrum. Meth. A 845, 583 (2017).
  * Musa (2019) L. Musa,  _Letter of Intent for an ALICE ITS Upgrade in LS3_, Tech. Rep. (CERN, Geneva, 2019).
  * Sauli (1997) F. Sauli, Nucl. Instrum. Meth. A 386, 531 (1997).
  * Giomataris  _et al._ (1996) Y. Giomataris, P. Rebourgeard, J. P. Robert, and G. Charpak, Nucl. Instrum. Meth. A 376, 29 (1996).
  * Bencivenni  _et al._ (2015) G. Bencivenni, R. De Oliveira, G. Morello, and M. P. Lener, JINST 10, P02008 (2015), arXiv:1411.2466 [physics.ins-det] .
  * Al-Turany  _et al._ (2012) M. Al-Turany, D. Bertini, R. Karabowicz, D. Kresan, P. Malzacher, T. Stockmanns, and F. Uhlig, J. Phys. Conf. Ser. 396, 022001 (2012).
  * Agostinelli  _et al._ (2003) S. Agostinelli _et al._ (GEANT4), Nucl. Instrum. Meth. A 506, 250 (2003).
  * Rauch and Schlüter (2015) J. Rauch and T. Schlüter, J. Phys. Conf. Ser. 608, 012042 (2015), arXiv:1410.3698 [physics.ins-det] .
  * Waltenberger (2011) W. Waltenberger, IEEE Trans. Nucl. Sci. 58, 434 (2011).
  * Giele and Keller (1998) W. T. Giele and S. Keller, Phys. Rev. D 58, 094023 (1998), arXiv:hep-ph/9803393 .
  * Ball  _et al._ (2011) R. D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J. I. Latorre, J. Rojo, and M. Ubiali (NNPDF), Nucl. Phys. B 849, 112 (2011), [Erratum: Nucl.Phys.B 854, 926–927 (2012), Erratum: Nucl.Phys.B 855, 927–928 (2012)], arXiv:1012.0836 [hep-ph] .
  * Ball  _et al._ (2012) R. D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, N. P. Hartland, J. I. Latorre, J. Rojo, and M. Ubiali, Nucl. Phys. B 855, 608 (2012), arXiv:1108.1758 [hep-ph] .
  * (21) “PYTHIA6 with Radiative Corrections,” https://eic.github.io/software/pythia6.html.
  * Rosenbluth (1950) M. N. Rosenbluth, Phys. Rev. 79, 615 (1950).
  * Group  _et al._ (2022) P. D. Group, R. L. Workman, V. D. Burkert, _et al._ , Progress of Theoretical and Experimental Physics 2022 (2022), 10.1093/ptep/ptac097, 083C01, https://academic.oup.com/ptep/article-pdf/2022/8/083C01/45434166/ptac097.pdf .
  * Sjöstrand  _et al._ (2006) T. Sjöstrand, S. Mrenna, and P. Skands, Journal of High Energy Physics 2006, 026 (2006).
  * Hou  _et al._ (2021) T.-J. Hou _et al._ , Phys. Rev. D 103, 014013 (2021), arXiv:1912.10053 [hep-ph] .
  * Gao and Nadolsky (2014) J. Gao and P. Nadolsky, JHEP 07, 035 (2014), arXiv:1401.0013 [hep-ph] .
  * Hou  _et al._ (2017) T.-J. Hou _et al._ , JHEP 03, 099 (2017), arXiv:1607.06066 [hep-ph] .
  * Candido  _et al._ (2022) A. Candido, F. Hekhorn, and G. Magni, “N3pdf/yadism: Fonll-b,”  (2022).
  * Forte  _et al._ (2010) S. Forte, E. Laenen, P. Nason, and J. Rojo, Nucl. Phys. B 834, 116 (2010), arXiv:1001.2312 [hep-ph] .
  * Eskola  _et al._ (2022) K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, Eur. Phys. J. C 82, 413 (2022), arXiv:2112.12462 [hep-ph] .
  * Adolph  _et al._ (2013) C. Adolph _et al._ (COMPASS), Phys. Rev. D 87, 052018 (2013), arXiv:1211.6849 [hep-ex] .
  * Khan  _et al._ (2022) T. Khan, T. Liu, and R. S. Sufian, (2022), arXiv:2211.15587 [hep-lat] .
  * Xu  _et al._ (2022) S. Xu, C. Mondal, X. Zhao, Y. Li, and J. P. Vary, (2022), arXiv:2209.08584 [hep-ph] .
  * Nocera  _et al._ (2014) E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo (NNPDF), Nucl. Phys. B 887, 276 (2014), arXiv:1406.5539 [hep-ph] .
  * Ball  _et al._ (2013) R. D. Ball _et al._ , Nucl. Phys. B 867, 244 (2013), arXiv:1207.1303 [hep-ph] .