# Detector Requirements and Simulation Results for the EIC Exclusive,
Diffractive and Tagging Physics Program using the ECCE Detector Concept

A. Bylinkin  C. T. Dean  S. Fegan  D. Gangadharan  K. Gates  S. J. D. Kay  I.
Korover  W.B. Li  X. Li  R. Montgomery  D. Nguyen  G. Penman  J. R. Pybus  N.
Santiesteban  S. Shimizu  R. Trotta  A. Usman  M.D. Baker  J. Frantz  D. I.
Glazier  D. W. Higinbotham  T. Horn  J. Huang  G. M. Huber  R. Reed  J. Roche
A. Schmidt  P. Steinberg  J. Stevens  Y. Goto  C. Munoz Camacho  M. Murray  Z.
Papandreou  W. Zha  J. K. Adkins  Y. Akiba  A. Albataineh  M. Amaryan  I. C.
Arsene  C. Ayerbe Gayoso  J. Bae  X. Bai  M. Bashkanov  R. Bellwied  F.
Benmokhtar  V. Berdnikov  J. C. Bernauer  F. Bock  W. Boeglin  M. Borysova  E.
Brash  P. Brindza  W. J. Briscoe  M. Brooks  S. Bueltmann  M. H. S. Bukhari
R. Capobianco  W.-C. Chang  Y. Cheon  K. Chen  K.-F. Chen  K.-Y. Cheng  M.
Chiu  T. Chujo  Z. Citron  E. Cline  E. Cohen  T. Cormier  Y. Corrales Morales
C. Cotton  J. Crafts  C. Crawford  S. Creekmore  C.Cuevas  J. Cunningham  G.
David  M. Demarteau  S. Diehl  N. Doshita  R. Dupré  J. M. Durham  R.
Dzhygadlo  R. Ehlers  L. El Fassi  A. Emmert  R. Ent  C. Fanelli  R. Fatemi
M. Finger  M. Finger Jr  M. Friedman  I. Friscic  S. Gardner  F. Geurts  R.
Gilman  E. Glimos  N. Grau  S. V. Greene  A. Q. Guo  L. Guo  S. K. Ha  J.
Haggerty  T. Hayward  X. He  O. Hen  M. Hoballah  A. Hoghmrtsyan  P.-h. J. Hsu
A. Hutson  K. Y. Hwang  C. E. Hyde  M. Inaba  T. Iwata  H.S. Jo  K. Joo  N.
Kalantarians  G. Kalicy  K. Kawade  A. Kim  B. Kim  C. Kim  M. Kim  Y. Kim  Y.
Kim  E. Kistenev  V. Klimenko  S. H. Ko  W. Korsch  G. Krintiras  S. Kuhn
C.-M. Kuo  T. Kutz  J. Lajoie  D. Lawrence  S. Lebedev  H. Lee  J. S. H. Lee
S. W. Lee  Y.-J. Lee  W. Li  X. Li  X. Li  X. Li  Y. T. Liang  S. Lim  C.-h.
Lin  D. X. Lin  K. Liu  M. X. Liu  K. Livingston  N. Liyanage  W.J. Llope  C.
Loizides  E. Long  R.-S. Lu  Z. Lu  W. Lynch  S. Mantry  D. Marchand  M.
Marcisovsky  C. Markert  P. Markowitz  H. Marukyan  P. McGaughey  M.
Mihovilovic  R. G. Milner  A. Milov  Y. Miyachi  A. Mkrtchyan  P. Monaghan  D.
Morrison  A. Movsisyan  H. Mkrtchyan  A. Mkrtchyan  K. Nagai  J. Nagle  I.
Nakagawa  C. Nattrass  S. Niccolai  R. Nouicer  G. Nukazuka  M. Nycz  V. A.
Okorokov  S. Orešić  J.D. Osborn  C. O’Shaughnessy  S. Paganis  S. F. Pate  M.
Patel  C. Paus  M. G. Perdekamp  D. V. Perepelitsa  H. Periera da Costa  K.
Peters  W. Phelps  E. Piasetzky  C. Pinkenburg  I. Prochazka  T. Protzman  M.
L. Purschke  J. Putschke  R. Rajput-Ghoshal  J. Rasson  B. Raue  K.F. Read  K.
Røed  J. Reinhold  E. L. Renner  J. Richards  C. Riedl  T. Rinn  G. M. Roland
G. Ron  M. Rosati  C. Royon  J. Ryu  S. Salur  R. Santos  M. Sarsour  J.
Schambach  N. Schmidt  C. Schwarz  J. Schwiening  R. Seidl  A. Sickles  P.
Simmerling  S. Sirca  D. Sharma  Z. Shi  T.-A. Shibata  C.-W. Shih  U.
Shrestha  K. Slifer  K. Smith  D. Sokhan  R. Soltz  W. Sondheim  J. Song  J.
Song  I. I. Strakovsky  P. Stepanov  J. Strube  P. Sun  X. Sun  K. Suresh  V.
Tadevosyan  W.-C. Tang  S. Tapia Araya  S. Tarafdar  L. Teodorescu  D. Thomas
A. Timmins  L. Tomasek  N. Trotta  T. S. Tveter  E. Umaka  H. W. van Hecke  C.
Van Hulse  J. Velkovska  E. Voutier  P.K. Wang  Q. Wang  Y. Wang  Y. Wang  D.
P. Watts  N. Wickramaarachchi  L. Weinstein  M. Williams  C.-P. Wong  L. Wood
M. H. Wood  C. Woody  B. Wyslouch  Z. Xiao  Y. Yamazaki  Y. Yang  Z. Ye  H. D.
Yoo  M. Yurov  N. Zachariou  W.A. Zajc  J.-L. Zhang  J.-X. Zhang  Y. Zhang
Y.-X. Zhao  X. Zheng  P. Zhuang

###### Abstract

This article presents a collection of simulation studies using the ECCE
detector concept in the context of the EIC’s exclusive, diffractive, and
tagging physics program, which aims to further explore the rich quark-gluon
structure of nucleons and nuclei. To successfully execute the program, ECCE
proposed to utilize the detector system close to the beamline to ensure
exclusivity and tag ion beam/fragments for a particular reaction of interest.
Preliminary studies confirm the proposed technology and design satisfy the
requirements. The projected physics impact results are based on the projected
detector performance from the simulation at 10 or 100 fb-1 of integrated
luminosity. Additionally, insights related to a potential second EIC detector
are documented, which could serve as a guidepost for future development.

###### keywords:

ECCE , Electron-Ion Collider , Exclusive , Diffractive , Tagging

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###### Contents

  1. 1 Introduction
  2. 2 ECCE central detector and far-farword components
  3. 3 Simulation, reconstruction and analysis framework
  4. 4 Physics impact studies
  5. 5 Insights for EIC detector 2 at IP8
  6. 6 Summary
  7. 7 Acknowledgements
  8. A Detector layout inside B0 magnet
  9. B DVCS off Helium-4 and the TOPEG generator
  10. C XYZ production event generator
  11. D TCS $ep$ and the EpIC generator 2D phase space
  12. E LAGER generator for exclusive J$/\psi$ production

##  1 Introduction

The planned Electron-Ion Collider (EIC), to be constructed at Brookhaven
National Laboratory (BNL) in partnership with the Thomas Jefferson National
Accelerator Facility (JLab), is considered the next generation “dream machine”
to further explore the quark and gluon substructure of hadrons and nuclei, and
provide scientific opportunities for the upcoming decades.

The scientific mission at the EIC was summarized in a 2018 report by the
National Academies of Science (NAS) [1]:

  * 1.

While the longitudinal momenta of quarks and gluons in nucleons and nuclei
have been measured with great precision at previous facilities – most notably
CEBAF at JLab and the HERA collider at DESY – the full three-dimensional
momentum and spatial structure of nucleons are not fully elucidated,
particularly including spin, which requires the separation of the intrinsic
spin of the constituent particles from their orbital motion.

  * 2.

These studies will also provide insight into how the mutual interactions of
quarks and gluons generate the nucleon mass and the masses of other hadrons.
The nucleon mass is one of the single most important scales in all of physics,
as it is the basis for nuclear masses and thus the mass of essentially all of
the visible matter.

  * 3.

The density of gluons and sea quarks which carry the smallest $x_{B}$, the
fraction of the nuclear momentum (or that of its constituent nucleons), can
grow so large that their mutual interactions enter a non-linear regime where
elegant, universal features emerge in what may be a new, distinct state of
matter characterized by a “saturation momentum scale”. Probing this state
requires high energy beams and large nuclear size, and will answer
longstanding questions raised by the heavy ion programs at RHIC and the LHC.

To accomplish the physics program, the EIC requires an accelerator capable of
delivering: 1) Highly polarized electron ($\sim$70%) and proton ($\sim$70%)
beams; 2) Ion beams from deuterons to heavy nuclei such as gold, lead, or
uranium; 3) Variable $ep$ center-of-mass energies from 20-140 GeV at high
collision luminosity of $10^{33}$–$10^{34}$ cm-2 s-1. Additionally, the EIC
requires a comprehensive and hermetic detector to record final-state particles
produced by the $ep$ and $e$A scattering.

The EIC Comprehensive Chromodynamics Experiment (ECCE) is a detector proposal
that was designed to address the full scope of the EIC physics program as
presented in the EIC White Paper [2] and the NAS report. In parallel, the EIC
community developed two additional detector proposals: ATHENA (A Totally
Hermetic Electron Nucleus Apparatus) [3] and CORE (COmpact detectoR for the
EIC) [4]. All three proposals were submitted to the EIC Detector Proposal
Advisory Panel (DPAP) and thoroughly accessed. Following the recommendations
of the DPAP (in March 2022), all three proposals joined efforts to form the
ePIC (Electron Proton-and Ion Collider Experiment)
Collaboration111https://www.jlab.org/conference/EPIC to complete the design of
the first detector of the EIC project.

The specific requirements on each of the ECCE detector systems follow from the
more general detector requirements described in the EIC Yellow Report (YR)
[5]. Through the judicious use of existing equipment, ECCE can be built within
the budget envelope set out by the EIC project while also managing schedule
risks [6].

The YR also identified a set of detector performance requirements that flow
down from the physics requirements of the EIC science program articulated in
the NAS report:

  * 1.

The outgoing electron must be distinguished from other produced particles in
the event, with a pion rejection of $10^{3}$–$10^{4}$ even at large angles, in
order to characterize the kinematic properties of the initial scattering
process. These include $x_{B}$ and the squared momentum transfer ($Q^{2}$).

  * 2.

A large-acceptance magnetic spectrometer is needed to measure the scattered
electron momentum, as well as those of the other charged hadrons and leptons.
The magnet dimensions and field strength should be matched to the scientific
program and the medium-energy scale of the EIC. This requires a nearly 4$\pi$
angular aperture, and the ability to precisely make measurements of the
sagitta of its curved trajectory, to measure its momentum down to low $p_{T}$
(transverse momentum), and to determine its point of origin, in order to
distinguish particles from the charm and bottom hadron decays.

  * 3.

A high-purity hadron particle identification (PID) system, able to provide
continuous $e/\pi$ and $K/\pi$ discrimination to the highest momentum (60
GeV), is important for identifying particles containing different light-quark
flavors.

  * 4.

A hermetic electromagnetic calorimeter system – with matching hadronic
sections – is required to measure neutral particles (particularly photons and
neutrons) and, in tandem with the spectrometer, to reconstruct hadronic jets
that carry kinematic information of the struck quark or gluon, as well as its
radiative properties via its substructure.

  * 5.

Far-Forward detector systems, in the direction of the outgoing hadron beam,
are needed in order to perform measurements of deeply-virtual Compton
scattering (DVCS) through exclusive production as well as diffractive
processes, e.g. by measuring the small deflections of the incoming proton and
suppressing incoherent interactions with nuclei.

  * 6.

Far-Backward detectors, in the direction of the outgoing electron beam, are
needed to reach the very lowest values of $Q^{2}$, and to measure luminosity
to extract the absolute cross-section and spin-dependent asymmetries (with
high precision).

The ECCE concept reuses the BaBar [7, 8] superconducting solenoid (which will
be operated at 1.4 T) as well as the sPHENIX [9] barrel flux return and
hadronic calorimeter. These two pieces of equipment are currently being
installed in RHIC Interaction Point 8 (IR8) as part of the sPHENIX detector.
Engineering studies have confirmed that these critical components can be
relocated to IP6, where the EIC project plans to site the on-project detector.
Additional details concerning ECCE subsystems, performance and selected
physics objectives are provided in separate articles within this same
collection.[6, 10, 11, 12, 13, 14, 15, 16, 17]

Among different types of $ep$ interactions studied by ECCE, in the exclusive
processes: all final state particles are detected and reconstructed, and in
the diffractive processes: no exchange of color-charge between the initial and
final state nucleon. These physics processes share a commonality of requiring
detection (tagging) of the interacted (recoiled) nucleons and electrons close
to the outgoing beamlines. Specialized detector systems are required to
perform such measurements to high precision. The unifying theme of this paper
is to introduce the design and technology used by these specialized detector
systems, and summarize the physics simulation studied based on the expected
detector performance. The conclusion of these studies signifies feasibility
based on the realistic detector acceptance based on the current base
knowledge, however, the process-specific energy-dependent efficiencies and the
resolution of the reconstructed kinematics variables will be further studied
in future publications.

This paper is structured as follows: Sec. 2 provides a short overview of the
ECCE detector and a detailed description of the far-forward region (FFR); Sec.
3 provides a brief description of the structure and workflow of the ECCE
simulation and analysis framework; Sec. 4 presents and discusses the physics
impact related to the Exclusive, Diffractive and Tagging sections; Sec. 5
discusses some improvements and complementary information associated with the
unique second beam focused in IP8; and, finally a summary is presented in Sec.
6.

##  2 ECCE central detector and far-farword components

Figure 1: IP6 interaction region. The far-forward region is encircled by the
blue dashed box, and the far-backward region is encircled by the red dashed
box. It is important to point out that the layout of the ECCE central detector
(green box) reflects the asymmetrical nature of the $ep$ collision laboratory
frame at the EIC. The incoming electron and ion beam form a crossing angle of
25 mrad. Note that the central detector is parallel to the electron beam line.

The ECCE detector consists of three major components: the central detector,
the far-forward, and the far-backward systems. The ECCE central detector has a
cylindrical geometry based on the BaBar/sPHENIX superconducting solenoid
(nominally operated at 1.4 T), and has three primary subdivisions: the barrel
(pseudorapidity coverage $|\eta|<1$), the forward endcap ($-3.5<\eta<-1$), and
the backward endcap ($1<\eta<3.5$). The “forward” region is defined as the
hadron/nuclear beam direction and “backward” refers to the electron beam
direction. These are illustrated in the beam-crossing schematic of Fig. 1. It
is important to note that the electron and ion beams cross at a 25 mrad angle
and that the electron beam passes down the axis of the central detector,
parallel to the magnetic field lines.

The purpose of the far-forward and far-backward detectors is to measure the
reaction kinematics of the colliding systems. This information is vital for
the interpretation of the data from the central detectors. The goal of the
far-backward system is to determine the luminosity, and measure the momentum
of the scattered electron, while the far-forward detectors are designed around
detecting the forward (close to the hadron beamline) charged hadrons,
neutrons, photons, and light nuclei or nuclear fragment photons over the
maximum possible acceptance with high position and momentum resolution.

ECCE’s barrel, far-forward, and far-backward detector systems were implemented
and studied using a Geant4 simulation [18] within the Fun4all framework [19]
(see Sec. 3 for further detail).

###  2.1 A brief description of central detector

The layout of the ECCE central detector is intended to be asymmetrical. In the
laboratory frame of the EIC, the collisions are asymmetric as the ion beam
will carry higher momentum and interact will the electron beam at 25 mrad
angle (the crossing angle). The incoming electron beam is defined as
$\eta\rightarrow-\infty$. Note that the central detector is parallel to the
electron beam line, therefore, the spectating or recoiled nucleon could be
tagged by the integrated detector systems along the beam momentum.

The ECCE central barrel detector features a hybrid-tracking detector design
using three state-of-the-art technologies to determine vertex positions (for
both primary and decay vertices), track momenta, and distance of closest
approach with high precision over the $|\eta|\leq 3.5$ region with full
azimuthal coverage. This tracking detector consists of the Monolithic Active
Pixel Sensor (MAPS) based silicon vertex/tracking subsystem, the $\mu$RWELL
tracking subsystem, and the AC-LGAD outer tracker, which also serves as the
ToF detector.

The PID system in the barrel, forward, and backward endcaps consists of high-
performance DIRC (hpDIRC), dual-radiator Ring Imaging Cherenkov (dRICH), and
modular RICH (mRICH), respectively. Their key features are:

hpDIRC

with coverage of $-1<\eta<1$, provides PID separation with 3$\sigma$ (standard
deviations) or more for $\pi/K$ up to 6 GeV/c, $e/\pi$ up to 1.2 GeV/c, and
$K/p$ up to 12 GeV/c.

dRICH

with coverage of $1<\eta<3.5$ (hadron direction), is designed to provide
hadron identification in the forward endcap with 3$\sigma$ or more for $\pi/K$
from 0.7 GeV to 50 GeV, and for $e/\pi$ from $\sim$100 MeV up to 15 GeV/c.

mRICH

with coverage of $-3.5<\eta<-1$ (electron direction), is to achieve 3$\sigma$
$K/p$ separation in the momentum range from 3 to 10 GeV/c, within the physical
constraints of the ECCE detector. It also provides excellent $e/p$ separation
for momenta below 2 GeV. In addition, the RICH detectors contribute to $e/\pi$
identification. e.g., when combined with an EM calorimeter, the mRICH and
hpDIRC will provide excellent suppression of the low-momentum $\pi^{\pm}$
backgrounds, which can limit the ability to measure the scattered electron in
kinematics where it loses most of its energy.

The ECCE electromagnetic calorimeter system consists of three major
components, it allows high-precision electron/hadron detection and suppression
in the backward, barrel, and forward directions. Hadronic calorimetry is
essential for the barrel and forward endcap regions for hadron and jet
reconstruction. Jet yields in the backward region were found to be
sufficiently infrequent that hadronic calorimetry would provide little to no
scientific benefit.

EEMC

The Electron Endcap EM Calorimeter is a high-resolution electromagnetic
calorimeter designed for precise measurement of scattered electrons and final-
state photons towards the electron endcap. The design of the EEMC is based on
an array of 3000 lead tungstate (PbWO4) crystals of size 2 cm $\times$ 2 cm
$\times$ 20 cm and readout by SiPMs yielding an expected energy resolution of
$2\%/\sqrt{E}\oplus 1\%$.

oHCAL and iHCAL

The energy resolution of reconstructed jets in the central barrel will be
dominated by the track momentum resolution, as the jets in this region have
relatively low momentum, and the measurement of the energy in the hadronic
calorimeter does not improve knowledge of the track momentum. The primary use
for a hadronic calorimeter in the central barrel will be to collect neutral
hadronic energy. The sPHENIX Outer Hadronic Calorimeter (oHCAL) will be
reused, which instruments the barrel flux return steel of the BaBar solenoid
to provide hadronic calorimetry with an energy resolution of
$75\%/\sqrt{E}\oplus 14.5\%$. There is also a plan to instrument the support
for the barrel electromagnetic calorimeter to provide an additional
longitudinal segment of hadronic calorimetry. This will provide an Inner
Hadronic Calorimeter (iHCAL) layer very similar in design to the sPHENIX inner
HCAL. The primary inner HCAL is useful to monitor shower leakage from the
barrel electromagnetic calorimeter as well as improve the calibration of the
combined calorimeter system.

BEMC

The barrel electromagnetic calorimeter (BEMC) is a projective homogeneous
calorimeter based on an inorganic scintillator material that produces shower
due to high $Z$ components. Scintillating Glass (SciGlass) blocks of size 4 cm
$\times$ 4 cm $\times$ 45.5 cm, plus an additional 10cm of radial readout
space. SciGlass has an expected energy resolution of $2.5\%/\sqrt{E}\oplus
1.6\%$ [20], comparable to PbWO4 for a significantly lower cost. The BEMC’s
optimal acceptance region is ($-1.4<\eta<1.1$).

FEMC and LFHCAL

The forward ECal (FEMC) will be a Pb-Scintillator shashlik calorimeter (the
scintillator layers consists of polystyrene panels). It is placed after the
tracking and PID detectors and made up of two half disks with a radius of
$\approx$1.83 m. It employs modern techniques for the readout as well as
scintillation tile separation. The towers were designed to be smaller than the
Molière radius in order to allow for further shower separation at high
rapidity. The longitudinally segmented forward HCal (LFHCAL) is a Steel-
Tungsten-Scintillator calorimeter. It is made up of two half disks with a
radius of $\approx$2.6 m. The LFHCAL towers have an active depth of 1.4 m with
additional space for the readout of $\approx$20–30 cm depending on their
radial position. Each tower consists of 70 layers of 1.6 cm absorber and 0.4
cm scintillator material. For the first 60 layers, the absorber material is
steel, while the last 10 layers serve as the tail catcher and are thus made
out of tungsten to maximize the interaction length within the available space.
The front face of the tower is 5 $\times$ 5 cm2.

Further details of the central barrel detector stack are described in Ref.
[6].

Figure 2: The layout of the EIC far-forward region. Table 1: Summary of far-
forward detector locations and angular acceptances for charged hadrons,
neutrons, photons, and light nuclei or nuclear fragments. In some cases, the
angular acceptance is not uniform in $\phi$, as noted. For the three silicon
detectors (Roman Pots, Off-Momentum Detectors, and B0 spectrometer), the 2D
size of the silicon planes are given and the thickness is not specified. For
the Roman Pots and Off-Momentum Detectors, the simulations have two silicon
planes spaced 2 m apart, while the B0 detectors have four silicon planes
evenly spaced along the first 1 m length of the B0pf dipole magnet bore. The
planes have a hole for the passage of the hadron beam pipe that has a radius
of 3.2 cm.

Detector | (x,z) Position [m] | Dimensions |  $\theta$ [mrad] | Notes
---|---|---|---|---
ZDC | (-0.96, 37.5) | (60 cm, 60 cm, 1.62 m) |  $\theta<$ 5.5 |  $\sim$4.0 mrad at $\phi=\pi$
Roman Pots (2 stations) | (-0.83, 26.0), (-0.92, 28.0) | (30 cm, 10 cm) |  $0.0<\theta$ $<5.5$ | 10$\,\sigma$ cut.
Off-Momentum Detector | (-1.62, 34.5), (-1.71, 36.5) | (50 cm, 35 cm) | $0.0<\theta<5.0$ | $0.4<x_{L}<0.6$
B0 Trackers and Calorimeter | (x = -0.15, $5.8<$ z $<7.0$) | (32 cm, 38 m) | $6.0<\theta<22.5$ |  $\sim$20 mrad at $\phi$=0

###  2.2 Schematics of the far-forward

Operating forward detectors at colliders will be a challenge since space is
very limited and radiation loads and backgrounds are high. To simplify the
operation of such a complex system of detectors, a uniform, and common
technology (such as the central barrel) for electromagnetic calorimetry
(PbWO4) and tracking (AC-LGAD) is explored and proposed. Such uniformity also
allows for the implementation of common monitoring and calibration systems.
The luminosity will be determined using complementary approaches following
what was learned from HERA, as described in the YR.

A schematic of the far-forward detectors is shown in Fig. 2. They include the
B0 spectrometer, off-momentum trackers, Roman Pots, and ZDC (see Table 1 for
position and dimensions).

###  2.3 Zero Degree Calorimeter (ZDC)

The Zero-Degree Calorimeter (ZDC) plays an important role in many physics
topics. The production of exclusive vector mesons in diffraction processes
from electron-nucleus collisions is one of the important measurements. For the
coherent processes, where the nucleus remains intact, the momentum-transfer
($t$) dependent cross section can be related to the transverse spatial
distribution of gluons in the nucleus, which is sensitive to gluon saturation.
In this case, however, the coherence of the reaction needs to be determined
precisely. Incoherent events can be isolated by identifying the break-up of
the excited nucleus. The evaporated neutrons produced by the break-up in the
diffraction process can be used in most cases (about 90%) to separate coherent
processes [21]. In addition, photons from the de-excitation of the excited
nuclei can help identify incoherent processes even in the absence of
evaporated neutrons. Therefore, in order to identify coherent events over a
wide $t$ range, neutrons and photons must be accurately measured near zero
degrees.

The geometry of the collision is important to understand the characteristics
of each event in electron-nucleus collisions. It has been proposed that
collision geometry can be studied by tagging it with the multiplicity of
forward neutrons emitted near zero degrees (see for instance [22]).
Determining the geometry of the collision, such as the “travel length” of the
struck partons in the nucleus, which correlates with the impact parameters of
the collision, is very useful in the study of nuclear matter effects.
Determining the geometry of the collision will allow us to understand the
nuclear structure with greater accuracy.

The physics requirements of the ZDC are summarised in Table 2.

Table 2: Physics requirement for ZDC. |  Energy range |  Energy |  Position |  Others
---|---|---|---|---
|  |  resolution |  resolution |
Neutrons |  up to the beam energy |  $\dfrac{50\%}{\sqrt{E}}\oplus 5\%$, ideally $\dfrac{35\%}{\sqrt{E}}\oplus 2\%$ |  $\dfrac{3\mathrm{mrad}}{\sqrt{E}}$ |  Acceptance: 60 cm $\times$ 60 cm
|  Note:
|  The acceptance is required for meson structure measurements.
|  Pion structure measurements may require a position resolution of 1 mm.
Photons |  $0.1-1$ GeV |  $20-30\%$ |  |  Efficiency: $90-99\%$
|  Note:
|  Used as a veto in $e$Pb exclusive J/$\psi$ production
$20-40$ GeV |  $\dfrac{35\%}{\sqrt{E}}$ |  0.5–1 mm |
|  Note:
|  u-channel exclusive electromagnetic $\pi^{0}$ production has a milder
requirement of $\frac{45\%}{\sqrt{E}}\oplus 7\%$ and 2 cm, respectively.
Events will have two photons, but single-photon tagging is also useful.
|  Kaon structure measurement requires tagging a neutron and 2 or 3 photons,
as decay products of $\Lambda$ or $\Sigma$.
Figure 3: Simulated energy deposits in each layer of active materials of the
ZDC, namely a silicon, crystal, or scintillator layer, shown for 40 GeV single
photons and single neutrons. The first silicon layer has the layer ID = 0 and
the next crystal layer has ID = 1. Other detector parts are indicated in the
figure. The shown energy deposits are averaged values for an event, where a
single photon or a neutron hits the ZDC by a particle gun.

####  2.3.1 ZDC design

The ZDC design is shown in Fig. 2 (bottom right). It consists of four
different calorimeters. Particles come in from the left side of the figure.
The detector consists of a 7 cm crystal layer (yellow) with a silicon pixel
layer attached (magenta), 22 layers of Tungsten/Silicon planes (light purple)
with an additional silicon pixel layer attached in front, 12 layers of
Lead/Silicon planes (gray), and 30 layers of Lead/Scintillator planes (green),
corresponding to the thickness of $8X_{0}$, $22X_{0}$, $2\lambda_{I}$, and
$5\lambda_{I}$, respectively. The energy deposition in each layer of active
material in shown in Fig. 3. The total size is 60 cm$\times$60 cm$\times$162
cm and the weight is greater than 6 tons.

Crystal calorimeter:  For good measurement of low-energy photons, the first
part of ZDC is designed to use a layer of crystal calorimeter towers which is
7 cm in thickness. The layer consists of $3\times 3$ cm2 crystals in an array
of $20\times 20$. $\mathrm{PbWO}_{4}$ is considered as the material choice for
the crystal, but LYSO is another candidate as the radiation hardness of
$\mathrm{PbWO}_{4}$ could be an issue. In front of the crystal layer, a
silicon pixel layer, which has the same design as in the W/SI calorimeter, is
attached.

Figure 4: Estimation of the energy resolution for single photons (left) and
single neutrons (right). The photon or neutron energy is reconstructed from
the deposited energy in each active material. The readout system was not
included in the simulation. The energy of the crystal layer is smeared by
$2.5\%/\sqrt{E}\oplus 1\%$ and is compared to $5\%/\sqrt{E}\oplus 1\%$
smearing for the photon case. The estimated resolution is fitted as a function
of the induced energy and compared to the physics requirements,
$35\%/\sqrt{E}$ for photons and $50\%/\sqrt{E}\oplus 5\%$ for neutrons.

W/SI sampling calorimeter:  This is an ALICE FoCal [23] style calorimeter and
consists of tungsten plates and silicon sensor planes placed one after the
other. It will measure the rest of the photon energy and extract the shower
development of photons and neutrons. The tungsten plates have 3.5 mm thickness
($\sim 1X_{0}$) and the silicon sensor planes have a thickness of
$300-320~{}\mu\mathrm{m}$. Two types of silicon sensors are considered. Pad
sensors have $1\times 1$ cm2 segmentation, while pixel sensors have $3\times
3$ mm2. There are 22 tungsten layers and each of these layers is followed by a
silicon pad layer except for the $11^{th}$ and $22^{nd}$ tungsten layers. For
those tungsten layers, a silicon pixel layer is inserted instead of a pad
layer. Another silicon pixel layer is attached in front of the first tungsten
layer, for the photon position measurement. The W/SI calorimeter has 22
tungsten layers, 20 silicon pad layers, and 3 silicon pixel layers in total.

Pb/SI sampling calorimeter:  This is a calorimeter with 3 cm-thick lead planes
as absorbers and silicon pad layers as active material, where the pad-layer
design is as in the W/SI calorimeter. The silicon layers (with good radiation
hardness) are used for the measurement of the neutron shower development. It
consists of 12 lead layers and 12 silicon pad layers.

Pb/Sci sampling calorimeter:  This is to measure hadron shower energy and uses
3-cm-thick lead planes as absorbers with 2-mm-thick scintillator planes as
active material. The calorimeter is segmented as $10\times 10$ cm2 on a plane
and 15 layers of scintillator planes will be read out together, comprising a
tower. The length of a tower is 48 cm. The Pb/Sci calorimeter has 6$\times$6
towers in the transverse and two towers in the longitudinal direction. In
total, it consists of 30 layers of lead planes and 30 layers of scintillator
planes.

####  2.3.2 Simulated performance study

The performance of the designed ZDC was studied using the Geant4 simulation
[18]. In the simulation, a single photon or a neutron is shot at the center of
the ZDC plane. The readout system is not implemented in the simulation but the
deposited energy in the active materials is studied. The materials for the
readout system were not fully implemented for the crystals and the
scintillator layers.222For each layer of silicon plane, a readout board with
chips is inserted. Empty spaces were used to represent the readout planes,
thus, the study provides an optimistic estimation.

Fig. 3 shows the deposited energy in each layer of ZDC active materials for
photons and neutrons with an energy of 40 GeV. It shows a clear difference in
the ZDC response against photons and neutrons. Photons deposit more energy in
the crystal layer and early layers in the W/Si calorimeter, while neutrons
continuously deposit their energy to the scintillator layers, owing to the
difference in their shower development.

The photon energy is reconstructed from the crystal layer and the W/SI
calorimeter. In the crystal, a tower with $E>15$ MeV is taken as a seed and
$3\times 3$ towers build a cluster. The crystal energy is smeared by
$2.5\%/\sqrt{E}\oplus 1\%$ (note that $5\%/\sqrt{E}\oplus 1\%$ was also
studied). In the resolution that follows (and throughout this paper),
$\sqrt{E}$ is taken to be in units of GeV. The energy in the W/SI calorimeter
is reconstructed from a $9\times 9$ cm2 region of interest (RoI), with a scale
factor corresponding to the sampling fraction. The neutron energy is
reconstructed from all the crystal, W/SI, Pb/SI, and Pb/Sci calorimeters. The
W/SI, Pb/SI, and Pb/Sci calorimeters need scale factors in order to convert
the energy deposits in the active material to the reconstructed energy, as
corrections for the sampling fraction and the $e/h$ compensation. For
extraction of the factors, the crystal calorimeter is taken out from the
simulation, and neutrons are shot directly on the sampling calorimeters. In
this setup, the factors are determined by fitting the following function:

$E_{N}=a\cdot E_{\text{SI in W/SI}}+b\cdot E_{\text{SI in Pb/SI}}+c\cdot
E_{\text{Sci. in Pb/Sci}},$

where $a$, $b$, and $c$ are the scale factors, performed for $E_{N}=$20, 40,
60, 80, and 100 GeV.

The estimated energy resolution is shown in Fig. 4. For high-energy photons,
the resolution is well below the requirement stated in the YR. For the low
energy photons, the estimated resolution for 100 MeV photons using 5% smearing
reaches 20% but is still acceptable. The neutron energy resolution is larger
than the ideal value of $35\%/\sqrt{E}\oplus 2\%$, but is smaller than the
required value of $50\%/\sqrt{E}\oplus 5\%$.

Position reconstruction is accomplished using the first silicon pixel layer
after the crystal calorimeter. For 40 GeV and 20 GeV photons, the position
resolution is estimated as 1.1 mm and 1.5 mm, respectively. On the crystal
layer, the cluster finding efficiency is $>95\%$ for both 20 GeV photons and
100 MeV photons, with a seed energy requirement of 15 MeV for the clustering.

Although the simulation results are optimistic without the readout system’s
geometry and materials, the results show a reasonable performance of the ZDC,
which practically fulfills the physics requirements listed in Table 2.

###  2.4 Roman Pots

The LHC forward-proton detectors have shown the capability of thin silicon
detectors to deliver both excellent precisions in position and timing with
pixelated detectors [24, 25].

The Roman Pots (RP) envisioned for ECCE largely follow the concept outlined in
the YR, namely the use of AC-LGADs to provide both precise timing and
excellent position resolution. The sensor will be laid out in a grid pattern.
Fig. 5 shows an example of such a layout from CMS.

Figure 5: Layout of the CMS Roman Pot silicon sensors [26]. Parameter | Interaction Point/Region
---|---
| IP6 | IP8
Beam crossing angle | 25 mrad | 35 mrad
Outer radius of B0 detector | 19 cm | 23.5 cm
Spanning angle Packman | 240 deg | 240 deg
Detector cut off for hadron beam pipe, tracker | 3.5 $\times$ 9.5 cm | 3.5 $\times$ 10.5 cm
Detector cut off for hadron beam pipe, calorimeter | 3.5 $\times$ 10.0 cm | 3.5 $\times$ 11.2 cm
Pipe hole offset in x-axis w.r.t. the center of the B0 magnet | -1.0 cm | -1.4 cm
‘PAC-man’ cut off for electron beam pipe, radius difference | 7 cm | 7 cm
Si layer thickness | 0.1 cm | 0.1 cm
Dead material (Cu) thickness | 0.2 cm | 0.2 cm
B0 EM section ($\rm PbW0_{4}$) thickness | 10 cm | 10 cm
B0 EM section z-position (relative to the B0-magnet) | 48 cm | 48 cm
Table 3: Shape parameters of the B0 detector. These are derived using
specifications provided in Ref. [27].

It is essential that such detectors be temperature stabilized. This can be
accomplished by using a cooled heat sink to pull heat off the detector via a
copper bus. We propose using a foam metal heat sink that will be cooled via
compressed air. Such systems have already been deployed at the LHC by a group
from the Technical University of Prague. The timing and resolution of the RP
layers are similar to the expected values of the B0 tracker (identical in
technology).

###  2.5 B0 magnet detector stack

Figure 6: B0 tracking resolution. Difference between reconstructed and true
momentum scaled by the true momentum as a function of $\eta$ (top) and
generated momentum (bottom).

Figure 7: Reconstructed momentum and its resolution for $\mu^{-}$ tracks found
in the B0 tracker (top). Acceptance of the B0 tracker (bottom). A particle gun
was used for this study.

The tracker and calorimeter stack inside of the B0 magnet provide detection
capability for far-forward charged tracks and photons. Such capability is
important for forward ($\eta>3$) particle measurements as well as event
characterization and separation.

The B0 spectrometer is located inside the B0pf dipole magnet. Its main use is
to measure forward-going hadrons and photons to identify exclusive reactions.
The B0 acceptance is defined by the B0pf magnet. Its design is challenging due
to the two beam pipes (electron and hadron) that must be accommodated and the
fact that these pipes are not parallel to each other, due to the 25 mrad IP6
crossing angle. Moreover, service access to the detectors inside of the dipole
is only possible from the IP side, where the distance between beam pipes is
the narrowest. To satisfy these constraints, the B0 detector design requires
the use of compact and efficient detection technologies.

The B0 detector stack design uses four AC-LGAD tracker layers with 30 cm
spacing between each layer (top left Fig. 2 in yellow). These will provide
charged particle detection for $6<\theta<22.5$ mrad. The use of such sensors
will provide good position and timing resolutions. The AC-LGAD sensors will
have a $3.2\times 3.2$ cm2 area, with four dedicated ASIC units on each
sensor. In addition, a PbWO4 calorimeter (Fig. 2 top left in magenta) will be
positioned behind the fourth tracking layer 683 cm away from the IP. The
calorimeter is constructed from 10 cm long $3\times 2$ cm2 PbWO4 crystals
positioned to leave 7 cm for the detector and readout system (before the B0
magnet exit). In order to consume less space inside the magnet, the processing
of the signals from the detector will be performed outside the magnet volume.
Both trackers and the calorimeter have oval holes in the center to accommodate
the hadron beam pipe, and a cutaway on the side to accommodate the electron
beam and allow installation and service of the detector system. An additional
circular cutoff (with 2 cm radius) on the side opposite the electron beam pipe
is assumed for cabling in each detector plane.

The parameters of the B0 detector are summarized in Table 3 for the two IPs.
To help visualize the trackers and calorimeter layout within the compact B0
magnet, CAD drawings (in realistic dimensions) are documented in A.

Figure 8: The left plot shows the photon detection acceptance in the B0 EM
calorimeter; the right plot is the ratio between the reconstructed and
generated photon energies.

####  2.5.1 Track Reconstruction in the B0 Calorimeter

Reconstructing tracks requires an accurate understanding of the magnetic field
in the B0 magnet. The field map implemented in the simulation combined the
field map of the central detector (1.4 T) with that of the B0 dipole magnet
(1.18 T). A Kalman filter was used to reconstruct the track momentum of
generated $\mu^{-}$ in the momentum range $1<p<100$ GeV, using the
reconstructed hits in the tracking layers and this field. Fig. 7 shows the
difference between the reconstructed and true momentum of the track, scaled by
its true momentum as a function of $\eta$ (top) and generated momentum
(bottom). This difference was found to be uniform as a function of
pseudorapidity and increasing slightly with the momentum of the generated
particle, and staying below 2% for the studied kinematic region.

The simulated momentum and its resolution $\sigma[\Delta p/p]$ are shown in
Fig. 7 (top), as a function of the truth momentum; the momentum resolution is
less than 5% for the studied kinematic region. The effect of the presence of
dead material (2 mm of Cu after each Si plane) on the momentum resolution is
also shown and estimated to degrade the resolution by 2% uniformly as a
function of $p$. Fig. 7 (bottom) also shows the acceptance of the B0 tracker
in the pseudorapidity-momentum plane.

####  2.5.2 Photon Reconstruction in the B0 Calorimeter

The studies of the efficiency of photon detection with the B0 electromagnetic
calorimeter have been performed for photons going from the interaction vertex
in the forward direction in the pseudorapidity range $4<\eta<6$ and having
energy $0<E_{\gamma}<60$ GeV. The granularity of the crystals of the B0 EM
section was assumed to be $2\times 2$ cm2.

The photon reconstruction algorithm search is based on a matrix of $2\times 2$
crystals. Other algorithms, for example, based on a Swiss-cross pattern, are
being considered and require further study.

The acceptance of the calorimeter in the $\eta-E_{\gamma}$ plane and the
average ratio of the reconstructed to generated energy are shown in Fig. 8
(left) and Fig. 8 (right), respectively. In general, about 60% of the energy
is reconstructed within a 2x2 crystal grid.

Figure 9: (Top) Reconstructed energy of photons and its resolution in the B0
calorimeter; (bottom) effect of the presence of dead material layers in the B0
tracker on the efficiency of photon reconstruction with the B0 calorimeter.

A scatter plot of the reconstructed versus generated photon energy together
with the energy resolution is shown in Fig. 9 (top). The resolution is found
to be below 7% for the studied kinematic region. The fraction of photon energy
that is reconstructed within the B0 calorimeter as a function of photon energy
$E_{\gamma}$ is portrayed in Fig. 9 (bottom). The effect of dead material
layers (the 2 mm of Cu after each silicon tracking plane) on the efficiency of
photon reconstruction with the B0 calorimeter is also shown and does not
exceed 10%.

##  3 Simulation, reconstruction and analysis framework

Figure 10: Simulation workflow, from the generated input from a variety of
physics event generators to the data output in the format of root trees and
histograms.

The ECCE proto-collaboration made a conservative decision to utilize
developed, supported, and established software tools to support the proposal
writing process in 2021. The primary consideration was the condensed proposal
writing timeline, as several data production campaigns would be necessary to
allow the physics and detector working groups to analyze data as well as
exercise the full simulation production system. Under such context, the
Fun4All software framework was chosen to perform Geant4 simulations [19].

Fun4All is an integrated simulation, reconstruction, and analysis framework.
Fun4All is an actively developed event processing framework that was
originally written for the PHENIX experiment [28]. In 2015, the framework was
moved to an open-source project and is now used by the sPHENIX and SpinQuest
[29] experiments. As the EIC-related activities increased towards the
proposal, a significant amount of software infrastructure was created to
support EIC-related studies prior to proto-collaboration formation, such as
the various Fun4All related repositories in Ref. [30]. This, and ongoing
Fun4All software development, was the basis for the studies that were
performed to develop the ECCE proposal.

A workflow diagram for using the Fun4All is shown in Fig. 10. As the input to
the simulation framework, the users need to generate physics event samples
with the generators (a few example generators are shown in the top grey
boxes). The fast simulation tool: eic-smear, is used to convert the generated
event data into ROOT trees or HepMC2 format, without modifying the underlying
event data. The users are also required to write their individual analysis
modules to interpret the simulation output, which takes the form of analysis
plugins within the Fun4All framework. The beam effects are handled within the
Fun4All framework and are further explained in Sec. 3.2.

The Fun4All framework (enclosed in the blue rectangle) is based upon the
Fun4AllServer, which can handle a variety of inputs, reconstruction modules,
and outputs. The modularity of the framework allowed users in the detector and
physics working groups to develop the relevant code asynchronously, while the
computing and simulation teams were then responsible for quality assurance and
code integration for deployment in large-scale productions. In this design,
various calibration and analysis modules were developed as part of the
coresoftware111https://github.com/eic/fun4all_coresoftware, fun4all
eicdetectors222https://github.com/eic/fun4all_eicdetectors, ecce-
detectors333https://github.com/ECCE-EIC/ecce-detectors, and
calibration444https://github.com/ECCE-EIC/calibrations repositories. These
modules were then aggregated in a series of ROOT macros that were steered by
one top macro. The top-most macro defined the event generation, the geometry
of the detector, input or output, and anything else that might be relevant for
the job. This ran as a standalone ROOT macro to produce the data summary tapes
(DSTs) and eventual micro DST data that the physics and detector working
groups analyzed as a part of the larger simulation campaigns.

###  3.1 Beam parameters

Figure 11: Material scans of the July-concept detector with build tag prop.4.
(Top) The radiation length, (bottom) the hadronic interaction length. Note:
this setup/study is made prior to the final ECCE detector configuration in the
proposal.

To fulfill the physics requirements (see Sec. 4), the EIC accelerator and
detector design must enable the detection of scattered protons with a minimum
transverse momentum of $p_{T}=200$ MeV, which at a hadron beam energy of 275
GeV corresponds to a scattering angle of 730 $\mu$rad in the horizontal plane.
The RMS divergence of the proton beam at the IP must not exceed one-tenth of
this minimum scattering angle: $\sigma_{0}\leq 73$ $\mu$rad. This requirement
may be violated in the vertical plane, provided the beam divergence in the
horizontal plane meets the requirement. A smaller horizontal RMS beam
divergence of 56 $\mu$rad allows the detection of 50% of all scattered protons
with a transverse momentum of 200 MeV. For $\sim 10\%$ of the operation time,
the EIC will run with a large horizontal beta function at IP, $\beta^{*}_{x}$
(related to the transverse beam size at the IP), that results in this low
divergence and thus provides high acceptance at the expense of reduced
luminosity; this beam configuration is referred to as the high-acceptance
configuration. Because of the large cross-section for small $p_{T}$, a large
amount of data can be collected in a short amount of time. For about 90% of
the time, the EIC will operate at small $\beta^{*}_{x}$ for high-luminosity
but with a divergence angle exceeding 73 $\mu$rad and this is referred to as
the high-divergence configuration. Combining the high-acceptance configuration
running (with higher cross-section at lower $p_{T}$) for a shorter time, with
the high-acceptance configuration running (smaller cross-section at higher
$p_{T}$) for a longer run time, a comparable amount of data (between both
settings) can be collected at all $p_{T}$ values from 200 MeV to 1.3 GeV. This
scenario substantially increases the effective luminosity of the facility
[31].

The beam parameters for electron-proton collisions (including the resulting
luminosities) at different center-of-mass energies ($\sqrt{s}$) for high-
divergence and high-acceptance are listed in Tables 3.3 and 3.4 of Ref. [31];
the beam parameters for $e$Au collisions (fully stripped gold ions with A=197)
are listed in Table 3.5 of Ref. [31]. All three sets of beam
parameters/configurations were implemented and used for the full simulation
during physics studies.

###  3.2 Applying beam effects on physics data

The beam effects are introduced via a generator-agnostic after-burner, which
has been integrated into the ECCE software setup since early 2021. Being the
standard procedure to take beam effects into account in the ECCE software, the
after-burner implements the beam effects on final-state particles on an event-
by-event basis based on the choice of beam configuration, such as high-
divergence, high-acceptance, or $e$A scattering.

The beam-parameter after-burner first boosts the generated physics events
horizontally, from the head-on frame, towards the beam crossing direction. The
amplitude of the boost is $\tan(\theta_{CA}/2)$, ignoring the beam divergence
and crab-cavity kick. Here, $\theta_{CA}=25$ mrad, which is the crossing angle
at IP6. In the presence of these variations, the final boost direction and
amplitude are chosen according to the final angle between the two beams in the
lab frame. In the last step, a simple rotation of $\theta_{CA}/2$ around the
vertical axis in the lab coordinate system aligns the electron beam back to
the $-z$ axis, which leaves the proton beam with the intended crossing angle
of $\theta_{CA}$. A more detailed discussion on the beam effects in EIC
simulation is summarized in a technical note [32].

Table 4: Summary ECCE Exclusive, Diffractive and Tagging physics studies with full simulations. Associated physics event generators and physics objectives of individual topics are given. Physics Impact Study Topics | Subsection | Physics Objective | Event Generator
---|---|---|---
Pion Form Factor | Sec. 4.1 | #4 | DEMPGen [33]
$\pi$ Structure Function | Sec. 4.2 | #4 | EIC_mesonMC [34]
Double Tagged e-He3 | Sec. 4.3 | #1 | DJANGOH [35]
ep DVCS | Sec. 4.4 | #2 | MILOU3D [36, 37]
eA DVCS via e-He4 | Sec. 4.5 | #3 | TOPEG [38]
ep DEMP $J/\psi$ | Sec. 4.6 | #4 | LAGER [39]
TCS | Sec. 4.7 | #2 | EpIC [40]
XYZ Spectroscopy | Sec. 4.8 | #5 | elSpectro [41]

###  3.3 Simulation campaign status

Four detector concepts were assembled in the ECCE simulation, one for each
simulation campaign. The information and overall simulation status are
documented in the wiki
database333https://wiki.bnl.gov/eicug/index.php/ECCE_Simulations_Working_Group.
The corresponding software branch name for the simulation campaigns are given
below.

  1. 1.

First simulation campaign: June-Concept (2021), which is tagged with proposal
software build prop.2.

  2. 2.

Second simulation campaign: July-Concept (2021), which is tagged with proposal
software build prop.4.

  3. 3.

Third simulation campaign: October-Concept (2021) and a variation with an AI-
optimized inner tracker, which is tagged with proposal software build prop.5.

  4. 4.

Fourth simulation campaign: January-Concept (2022) with the full beam
configuration set, which is tagged with proposal software build prop.7.1.

Each software build is developed under a branch at the GitHub
repository444https://github.com/ECCE-EIC/macros. The prop.4 simulation is the
baseline for the ECCE detector proposal; the material profile as a function of
$\eta$ is shown in Fig. 11.

##  4 Physics impact studies

The physics objectives derived from the National Academy of Sciences (NAS)
questions (Sec. 1) to the EIC project can be expressed as follows:

  1. 1.

Origin of nucleon spin.

  2. 2.

Three-Dimensional structure of nucleons and nuclei.

  3. 3.

Gluon structure of nuclei.

  4. 4.

Origin of hadron mass.

  5. 5.

Science beyond the NAS Report.

See Table 4 for the full list of physics topics covered in this paper.

To achieve these objectives, ECCE conducted a variety of studies with
Exclusive, Diffractive and Tagging processes utilizing the Fun4All simulation
(Sec. 3). The said processes were categorized as exclusive electro- and
photoproduction of mesons and photons, as well as $ep$ and $e$A vector meson
production through a diffractive process. One commonality among these
processes is the requirement that a nucleon (or nucleus) be tagged by the far-
forward instrumentation (see Sec. 2). It is important to note that fully
reconstructing all final-state particles is experimentally challenging.
Detailed background studies are required in the future to better gauge the
sensitivity required to complete the relevant studies under realistic
experimental conditions.

###  4.1 Pion form factor - $F_{\pi}$

The elastic electromagnetic form factor of the charged pion, $F_{\pi}(Q^{2})$,
is a rich source of insights into basic features of hadron structure, such as
the roles played by confinement and Dynamical Chiral Symmetry Breaking (DCSB)
in determining the size and mass of hadrons and defining the transition from
the strong- to perturbative-QCD domains. Studies during the last decade, based
on JLab 6-GeV measurements, have generated confidence in the reliability of
$\pi^{+}$ electroproduction as a tool for pion form factor extractions.
Forthcoming measurements at the 12-GeV JLab will deliver pion form factor data
that are anticipated to bridge the region where QCD transitions from the
strong (color confinement, long-distance) to perturbative (asymptotic freedom,
short-distance) domains.

The experimental determination of $F_{\pi}$ is challenging. The theoretically
ideal method for determining $F_{\pi}$ would be electron-pion elastic
scattering. However, the lifetime of the $\pi^{+}$ is only 26.0 ns. Since
$\pi^{+}$ targets are not possible, and $\pi^{+}$ beams with the required
properties are not yet available, one must employ high-energy exclusive
electroproduction, $p(e,e^{\prime}\pi^{+}n)$. This is best described as quasi-
elastic ($t$-channel) scattering of the electron from the virtual $\pi^{+}$
cloud of the proton, where $t$ is the Mandelstam momentum transfer
$t=(p_{p}-p_{n})^{2}$ to the target nucleon. Scattering from the $\pi^{+}$
cloud dominates the longitudinal photon cross section ($d\sigma_{L}/dt$), when
$|t|\ll m_{p}^{2}$ [42]. To reduce background contributions, normally one
separates the components of the cross-section due to longitudinal (L) and
transverse (T) virtual photons (and the LT, TT interference contributions),
via a Rosenbluth separation. The value of $F_{\pi}(Q^{2})$ is determined by
comparing the measured $d\sigma_{L}/dt$ values at small $-t$ to the best
available electroproduction model. The obtained $F_{\pi}$ values are in
principle dependent upon the model used, but one anticipates this dependence
to be reduced at sufficiently small $-t$. JLab 6 GeV experiments were
instrumental in establishing the reliability of this technique up to
$Q^{2}=2.45$ GeV2 [43], and extensive further tests are planned as part of
JLab E12-19-006 [44].

At the EIC, pion form factor measurements can be extended to still larger
$Q^{2}$, by measuring the unseparated electroproduction cross section
($\sigma_{uns}$) of the Deep Exclusive Meson Production (DEMP) reaction
$p(e,e^{\prime}\pi^{+}n)$. The value of $F_{\pi}(Q^{2})$ can be determined
from these measurements by comparing the measured $\sigma_{uns}$ at low $-t$
to the best available electroproduction model, incorporating pion pole and
non-pole contributions. The form factor extraction model would be validated by
$\pi^{-}/\pi^{+}$ ratios from deuterium data ($D(e,e^{\prime}\pi^{-}p)p_{sp}$
and $D(e,e^{\prime}\pi^{+}n)n_{sp}$) in the same kinematics as the
$p(e,e^{\prime}\pi^{+}n)$ measurements on the proton. The measurements would
be made over a range of small values of $-t=-(p_{p}-p_{n})^{2}$, and gauged
with theoretical and phenomenological expectations to verify the reliability
of the pion form factor extraction in EIC kinematics.

Figure 12: [Top] Detection efficiency for $e^{\prime}\pi^{+}n$ triple
coincidences in ECCE versus $Q^{2}$ and $-t$. [Bottom] Predicted distribution
of neutron hits from the DEMP process in the ZDC.

####  4.1.1 Kinematics, acceptance and reconstruction resolution

A DEMP $p(e,e^{\prime}\pi^{+}n)$ event generator [33] was written and used to
perform simulations demonstrating the feasibility of pion electric form factor
measurements at the EIC. A sample of 0.3M simulated events from the DEMP
generator in EIC-Smear format was passed through Fun4All including the ZDC
plug-in. The neutrons from the DEMP reactions of interest take 80-98% of the
proton beam momentum and are detected at very forward angles (0–2∘) in the
ZDC. The scattered electrons and pions have similar momenta, except that the
electrons are distributed over a wider range of angles. For $5\times 100$ beam
energies, the 5–6 GeV/c electrons are primarily scattered 25–45∘ from the
electron beam into the lepton end cap and the central barrel detector. The
5–12 GeV/c $\pi^{+}$ are 7–30∘ from the proton beam and enter the hadron end
cap and central barrel detector.

$e^{\prime}\pi^{+}n$ triple coincidence events were identified in the
simulated data by utilizing a series of conditional selection cuts:

  * 1.

at least one hit in the ZDC, with an associated energy deposit above $40$ GeV.

  * 2.

exactly two charged tracks: a positively charged track going in the $+z$
direction ($\pi^{+}$) and a negatively charged track going in the $-z$
direction ($e^{\prime}$).

Both conditions had to be satisfied for a given event for it to be considered
a $e^{\prime}\pi^{+}n$ triple coincidence event.

Figure 13: Deviation of the reconstructed neutron track momentum from the
neutron “truth” track, expressed as a percentage, $\Delta
p_{n}=(p_{ntrack}-p_{ntruth})/p_{ntruth}$ for $e^{\prime}\pi^{+}n$ triple
coincidence events.

The ECCE detection efficiency for these triple coincidence events is
fortunately quite high, $\sim 80\%$, and nearly independent of $Q^{2}$. A
density plot of detection efficiency versus $-t$ ($y$-axis) and $Q^{2}$
($x$-axis) is shown in the top panel of Fig. 12. The detection efficiency is
highest for the small $-t<0.15$ GeV2 events needed for the pion form factor
measurement, decreasing rapidly with $-t$ thereafter. The $t$ range of optimal
acceptance is dictated by the size of the ZDC, as the energetic neutrons from
high $-t$ events are emitted at an angle larger than the ZDC acceptance. The
distribution of neutron hits on the ZDC for $5\times 100$ beam energy up to
$-t=0.4$ GeV2 is given in the bottom panel of Fig. 12.

Figure 14: Reconstructed $t$ versus true $t$ for simulated
$e^{\prime}\pi^{+}n$ triple coincidence events with $15<Q^{2}<20$ GeV2, where
$t$ is reconstructed as $t=(p_{e}-p_{e^{\prime}}-p_{\pi})^{2}$ (top) and as
$t_{alt}=(p_{p}-p_{n})^{2}$ (bottom). $p_{n}$ here is the reconstructed
neutron track that combines the missing momentum with the ZDC position
information. $t$ reconstruction using the lepton and meson information alone
shows little correlation with the true value (top), while the reconstruction
from the charged tracks and the ZDC position information is more reliable.
Note the vastly different horizontal scales of the two plots.

The simulation successfully detected and reconstructed the $\pi^{+}$ and
$e^{\prime}$ tracks. The momentum of the detected tracks was reconstructed to
within a few percent of the “true” momentum for these particles. The two
charged tracks were utilized to determine the missing momentum from the
reaction,
$\vec{p}_{miss}=\vec{p}_{e}+\vec{p}_{p}-\vec{p}_{e^{\prime}}-\vec{p}_{\pi^{+}}.$
As there is already a requirement for a high-energy hit in the ZDC as a veto,
this missing momentum track is treated as being the exclusive neutron track.
As discussed in Section 4.1.2, additional cuts were utilized to remove
potential contamination from SIDIS or other background reactions. However,
since the hit positions of the neutron track in the ZDC were known to have a
high degree of accuracy, they were utilized to “correct” the missing momentum
track and form a new “reconstructed neutron track”. The angles,
$\theta_{Miss}$ and $\phi_{Miss}$ from the missing momentum track were
switched to the values determined from the ZDC hit position $\theta_{ZDC}$ and
$\phi_{ZDC}$. The mass of the particle for this track was also fixed to be
that of the neutron. Following these adjustments, the subsequent
reconstruction of the neutron track proved to be sufficiently accurate. The
resulting reconstructed neutron track momentum was within $1\%$ of the “true”
momentum, as seen in Fig. 13.

Figure 15: The reconstructed neutron track momentum for DEMP
$e^{\prime}\pi^{+}n$ triple coincidence events compared to $\vec{p}_{miss}$
for simulated SIDIS background events ($y$-axis scaled arbitrarily,
$\vec{p}_{miss}=\vec{p}_{e}+\vec{p}_{p}-\vec{p}_{e^{\prime}}-\vec{p}_{\pi^{+}}$).
The SIDIS events can be cleanly separated from the DEMP events of interest.
Note that both plots display events with $15<Q^{2}<20$ GeV2. Figure 16: The
difference between the reconstructed ($\theta_{pMiss}$, $\phi_{pMiss}$) and
detected ($\theta_{ZDC}$, $\phi_{ZDC}$) simulated angles for the neutron in
$e^{\prime}\pi^{+}n$ triple coincidence events. The indicative cut range is
shown by the area enclosed within the four red lines,
$-0.6\degree<\theta_{pMiss}-\theta_{ZDC}<0.6\degree$ and
$-3\degree<\phi_{pMiss}-\phi_{ZDC}<3\degree$.

Reconstruction of $t=(p_{e}-p_{e^{\prime}}-p_{\pi})^{2}$ from the detected
$\pi^{+}$ and $e^{\prime}$ tracks proved to be highly unreliable, as can be
seen in the top panel of Fig. 14. Fortuitously, due to the exclusive nature of
the reaction, $t$ can also be calculated from the proton beam and the
reconstructed neutron via $t=(p_{p}-p_{n})^{2}$. With this information, $t$
could be reconstructed from the neutron track in a manner that reproduced the
“true” value closely (see bottom panel of Fig. 14). This also demonstrates the
importance of combining the ZDC hit information with the charged particle
tracks to determine the neutron four-momentum. Reliable reconstruction of $t$
is essential for the extraction of the pion form factor from the
$p(e,e^{\prime}\pi^{+})n$ data. The cross-section falls rapidly with $-t$ as
the distance from the pion pole $(t-m_{\pi}^{2})$ is increased. This steep
decrease in the cross-section needs to be measured to confirm the dominance of
the Sullivan mechanism.

Figure 17: Predicted $e\pi^{+}n$ triple coincidence rates for different
$Q^{2}$ bins after application of the $p_{miss}$ and $\theta_{n}$ cuts
described in the text. Each $-t$ bin is $0.04$ GeV2 wide. The luminosity
assumed in these rate calculations: L$=10^{34}$ cm-2s-1. Figure 18:  Existing
data (blue, black, yellow, green) and projected uncertainties for future data
on the pion form factor from JLab (cyan, red) and EIC (black), in comparison
to a variety of hadronic structure models. The ECCE projections clearly cover
a much larger $Q^{2}$ range than the JLab measurements, providing access to
the emergent mass scale in QCD.

Our finding that $t$ reconstructed from the baryon information is
significantly more reliable than the version reconstructed from the lepton and
meson is similar to the studies of $t$ resolution reported as part of
exclusive vector meson production studies in the YR (Sec. 8.4.6) and as
observed in the TCS study detailed in Sec. 4.7. The high-quality ZDC proposed
by ECCE is clearly of paramount importance to the feasibility of this
measurement.

####  4.1.2 Other event selection cuts

Guided by previous work [45], cuts are applied on the detected neutron angle
($\pm 0.5^{\circ}$ from the outgoing proton beam) and on the missing momentum,
computed as
$\vec{p}_{miss}=\vec{p}_{e}+\vec{p}_{p}-\vec{p}_{e^{\prime}}-\vec{p}_{\pi^{+}}$.
The missing momentum cut corresponds to the momentum of the tagged forward-
going neutron and is $Q^{2}$-bin dependent, varying from $p_{miss}<$96 GeV/c
at $Q^{2}$=6.25 GeV2 to $<$77.5 GeV/c at $Q^{2}$=32.5 GeV2. In earlier
studies, these cuts were highly effective in separating DEMP events from
background SIDIS ($p(e,e^{\prime}\pi^{+})X$) events, as can be seen in Fig.
15. After the application of these cuts, the exclusive
$p(e,e^{\prime}\pi^{+}n)$ events were found to be cleanly separated from the
simulated SIDIS events. Due to the compressed ECCE proposal timeline, we did
not have time to repeat this study and used the same cuts as our earlier study
shown in the YR.

Due to the method used to reconstruct the neutron four-momentum, an additional
set of cuts was also implemented. A cut was applied on the difference between
the angle reconstructed from the missing momentum of the charged track pair
($\theta_{pMiss}$) and the angle of the neutral particle detected in the ZDC
($\theta_{ZDC}$). A cut was also applied based on the difference in $\phi$.
This pair of cuts is likely to be needed to distinguish DEMP events from SIDIS
events and will need further study. For now, a conservative, but indicative,
cut a range of $-0.6^{\circ}<\theta_{pMiss}-\theta_{ZDC}<0.6^{\circ}$ and
$-3^{\circ}<\phi_{pMiss}-\phi_{ZDC}<3^{\circ}$ was applied, as shown in Fig.
16.

After application of these cuts, the predicted $p(e,e^{\prime}\pi^{+}n)$ event
rates at an instantaneous luminosity of $10^{34}$ cm-2s-1 for $Q^{2}$ bins
over the full simulated kinematic range are shown in Fig. 17.

####  4.1.3 Results

After the exclusive $\pi^{+}n$ event sample is identified, the next step is to
separate the longitudinal cross-section $d\sigma_{L}/dt$ from
$d\sigma_{T}/dt$, needed for the extraction of the pion form factor. However,
a conventional Rosenbluth separation is impractical at the EIC. Fortunately,
at the high $Q^{2}$ and $W$ accessible at the EIC, phenomenological models
predict $\sigma_{L}\gg\sigma_{T}$ at small $-t$. This is expected since in the
hard scattering regime, QCD scaling predicts $\sigma_{L}\propto Q^{-6}$ and
$\sigma_{T}\propto Q^{-8}$, hence $\sigma_{L}$ is expected to dominate at
sufficiently high $Q^{2}$. For example, the Vrancx and Ryckebusch Regge-based
model [46] predicts $R=\sigma_{L}/\sigma_{T}>10$ for $Q^{2}>10$ GeV2 and
$-t<0.06$ GeV2, and $R>25$ for $Q^{2}>25$ GeV2 and $-t<0.10$ GeV2. Thus, the
transverse cross-section contributions are expected in these cases to be only
4-10%. The most practical choice appears to be to use a model to isolate the
dominant $d\sigma_{L}/dt$ from the measured $d\sigma_{uns}/dt$.

The value of $F_{\pi}(Q^{2})$ is then determined by comparing the measured
$d\sigma_{uns}/dt$ at small $-t$ to the best available electroproduction
model, incorporating pion pole and non-pole contributions and validated with
$\pi^{-}/\pi^{+}$ data. The model should have the pion form factor as an
adjustable parameter, so that the best-fit value and its uncertainty at fixed
$(Q^{2},W)$ are obtained by comparison of the magnitude and $t$-dependencies
of the model and data. If several models are available, the form factor values
obtained with each one can be compared to better understand the model
dependence. The importance of additional $p(e,e^{\prime}\pi^{+}n)$ model
development to improve knowledge of pion form factors cannot be overestimated,
and additional activity in this area is strongly encouraged.

Using this technique, ECCE can enable a pion form factor measurement up to
$Q^{2}=32.5\,$GeV2, as shown in Fig. 18. Note that the $y$-axis positions of
the projected data points in the figure are arbitrary. However, the error bars
represent the real projected errors for these points. The errors in the yields
are based on the following assumptions: cross sections parameterized from the
Regge model in [47], integrated luminosity of 10 fb-1 for 5$\times$100
measurement, clean identification of exclusive $p(e,e^{\prime}\pi^{+}n)$
events by tagging the forward neutron, and a cross-section systematic
uncertainty of 2.5% point-to-point and 12% scale (similar to the HERA-H1 pion
structure function measurement [48]). One should then apply an additional
uncertainty, since the form factor will be determined from unseparated, rather
than L/T-separated data: $\delta R=R$ systematic uncertainty in the model
subtraction to isolate $\sigma_{L}$, where
$R=\sigma_{L}/\sigma_{T}=0.013-0.14$ at $-t_{\rm min}$. Finally, the model
fitting procedure is used to extract $F_{\pi}(Q^{2})$ from the $\sigma_{uns}$
data, where one assumes the applied model is validated at small $-t$ by
comparison to data. Additional model uncertainties in the form factor
extraction are not estimated here, but the EIC should provide data over a
sufficiently large kinematic range to allow the model dependence to be
quantified in a detailed analysis.

Regarding the projected uncertainties in Fig. 18, for the lowest $Q^{2}$ bins
($Q^{2}<10$ GeV2) the uncertainty in $R$ is among the largest systematic
uncertainties, arising from the inability to perform an L/T-separation, and
the relatively less favorable T/L ratio. At intermediate $Q^{2}$
($10<Q^{2}<25$ GeV2), the T/L ratio is more favorable and the experimental
systematic uncertainties dominate. The statistical uncertainties dominate the
highest $Q^{2}$ bins ($Q^{2}>$25 GeV2), as the rates in these regions are very
low (see Fig. 17).

To conclude, the extraction of the pion form factor to high $Q^{2}$ with ECCE
depends on very good ZDC angular resolution for two reasons: 1) the necessity
to separate the small exclusive $\pi^{+}$ cross-section from dominant
inclusive backgrounds via $p_{miss}$ and $\theta_{n}$ cuts, 2) the need to
reconstruct $t$ to better than $\sim$0.02 GeV2, such resolution is only
possible when reconstructed from the initial proton and final neutron momenta.
The ZDC is thus of crucial importance to the feasibility of a pion form factor
measurement.

###  4.2 $\pi$ structure function

Studies of the meson structure functions were identified as a key science
topic in the YR. The far-forward detection region is particularly important,
as the recoiling baryon and its decay products have to be detected with
sufficient precision to achieve the desired resolution for meson structure
studies. This region provides a broad acceptance for charged and neutral
particles for a variety of interactions. For meson structure experiments, it
maximizes the kinematic coverage for a range of beam energies.

Figure 19: Top plots: B0 occupancy of the simulated leading neutron for
$p(e,e^{\prime}\pi^{+}n)$ meson structure study at 5$\times$41 (left) and
10$\times$100 (right). Bottom plots: ZDC acceptance of the simulated leading
neutron for a range of energies 5$\times$41 (left) and 10$\times$100 (right).

Similar to the inclusive $ep$ structure-function, the neutron-tagged structure
function rises at low $x_{B}$. As shown by HERA, by determining the neutron-
tagged cross-section relative to the inclusive $ep$ cross-section it is
possible to precisely determine the leading neutron production [48]. Tagged
deep-inelastic scattering (TDIS) can then be used to probe the meson content
of the neutron structure function, thus extracting the pion structure function
using the Sullivan process.

There are limited data on the pion structure function with (HERA TDIS data
[49]) which looked at the low $x_{B}$ region using the Sullivan process, and
the pionic Drell-Yan data [50] from nucleons at large $x_{B}$. The one-pion
exchange seems to be the dominant mechanism that makes it possible to extract
the pion structure function through the use of an in-depth model and kinematic
studies, which include effects such as re-scattering and absorption. These
projected capabilities of EIC (104 higher luminosity compares to HERA) will
directly balance the ratio of Sullivan processed tagged pion structure
function measurements in various bins of $t$ to the proton structure
measurements. At high $x_{\pi}$, it is possible for a direct comparison to
fixed target experiments and Drell-Yan. Upcoming experiments, like
COMPASS++/AMBER Drell-Yan [51] and the JLab 12 GeV TDIS experiment [52], will
be vital consistency checks of pion structure information obtained at the EIC.

####  4.2.1 Kinematics, acceptance, and reconstruction resolution

The $\pi$ structure studies were conducted over a range of beam energies with
the EIC_mesonMC generator [34]. The highest energy of 18$\times$275 was used
to maximize the kinematics coverage. However, to improve access to the high
$x_{\pi}$ region, alternate lower beam energies 10$\times$100, 5$\times$100,
and 5$\times$41 were also utilized. These lower beam energies allow access to
this high $x_{\pi}$ regime over a wider range of $Q^{2}$.

The leading neutrons for these two energy settings are at a very small forward
angle while carrying nearly all of the proton beam momentum. These leading
neutrons will be detected by the ZDC.

The ZDC must reconstruct the energy and position well enough to constrain both
the scattering kinematics and the four-momentum of the pion. Constraining the
neutron energy around 35$\%$/$\sqrt{E}$ will assure an achievable resolution
in $x_{B}$. Fig. 19 bottom row shows the acceptance plots for neutrons in the
ZDC for the two energy settings. As one can see, the spatial resolution of the
ZDC plays an important role in the highest energy setting, since it is
directly related to the measurements of $p_{T}$ or $t$. The $t$-reconstruction
was produced from the proton beam and the reconstructed neutron via
$t=(p_{p}-p_{n})^{2}$ as outlined in Sec. 4.1. For the lowest energy setting,
the total acceptance coverage of the ZDC is important. This sets a requirement
for the total size of the ZDC to be a minimum of 60$\times$60 cm2.

The B0 occupancy in Fig. 19 top row plots show a significant amount of leading
neutrons hitting the detector for the lowest energy settings (i.e.
5$\times$41). The ZDC acceptance in Fig. 19 bottom row plots for the leading
neutron also show a significant drop in neutron detection for the lowest
energy setting (i.e. 5$\times$41). This corresponds to the increased occupancy
in the B0.

As mentioned earlier, the spatial resolution of the ZDC plays a crucial role
in determining measurements of $t$. Fig. 20 breaks down the $t$-distribution
for the two energies for a range of $Q^{2}$ bins. The drop in events at the
higher $Q^{2}$ bins is expected for the lower energies. Fig. 21 shows the
deviation of $t$ from its detected value. The deviation,
$\Delta{t}=t-t_{truth}$, is clearly much greater for the lowest energy
(5$\times$41), providing a consistent picture of the energy ranges.

Figure 20: The $-t$ distributions of $p(e,e^{\prime}\pi^{+}n)$ meson structure
study at 5$\times$41 (left) and 10$\times$100 (right). There are four $Q^{2}$
bins presented (7, 15, 30, 60 GeV2) of bin width $\pm$5 GeV2.

Figure 21:  The deviation of generated $-t$ from the detected $t_{truth}$
value for $p(e,e^{\prime}\pi^{+}n)$ meson structure study,
$\Delta{t}=t-t_{truth}$, for a range of energies (5$\times$41, 10$\times$100)
at IP6. There are four $Q^{2}$ bins presented (7, 15, 30, 60 GeV2) of bin
width $\pm$5 GeV2. The lowest energy (5$\times$41) sees a strong deviation.
5$\times$41 is the same energy that sees the drop in ZDC acceptance.

####  4.2.2 Results

Figure 22:  _Left:_ Comparison of uncertainties on the pion’s valence, sea
quark, and gluon PDFs before (yellow bands) and after (red bands) inclusion of
EIC data. _Right:_ Ratio of uncertainties with EIC data to without,
$\delta^{\rm EIC}/\delta$, for the valence (green line), sea quark (blue), and
gluon (red) PDFs, assuming 1.2% experimental systematic uncertainty but no
model systematic uncertainty, and (inset) the corresponding ratios of the
momentum fraction uncertainties, $\delta\langle x\rangle^{\rm
EIC}/\delta\langle x\rangle$, for valence, sea, total quark and gluon PDFs
[53], at a scale $Q^{2}=10$ GeV2.

Statistical uncertainties, with the addition of the leading neutron detection
fraction, were incorporated into the overall uncertainty for an integrated
luminosity of $\mathcal{L}~{}=100\,\mathrm{fb}^{-1}$. For 10$\times$100
energy, the coverage in $x_{B}$ extends down to 10-2, with reasonable
uncertainties in the mid-to-high $x_{B}$ region, increasing rapidly as
$x\rightarrow 1$. Even with these restrictions, the coverage in mid-to-high
$x_{B}$ is unprecedented.

In Fig. 22, we show the impact of EIC data on the pion PDFs themselves and
their uncertainties, folding in the estimated systematic uncertainty and the
projected statistical uncertainties from the simulations. The resulting access
to a significant range of $Q^{2}$ and $x_{B}$, for appropriately small $-t$,
will allow for much-improved insights in the gluonic content of the pion.

The ratio of the uncertainty of the $F_{2}^{\pi}(x_{\pi},Q^{2})$ structure
function resulting from a global fit with EIC projected data to that without
it is displayed in Fig. 23. We show various $Q^{2}$ values over a wide range
between a few GeV2 and a few hundred GeV2, over the range $10^{-3}<x_{\pi}<1$,
to investigate the $Q^{2}$ dependence of the impact. Strikingly, the
$F_{2}^{\pi}$ structure function’s uncertainties reduce by 80-90% in the range
of $x_{\pi}$ between $3\times 10^{-3}$ and $0.4$ in the presence of EIC data,
independent of the values of $Q^{2}$. Within the whole range, the
uncertainties reduce by 65% or more. Below $x_{\pi}$ of 0.1, the error in the
$F_{2}^{\pi}$ structure function reduces by a factor of 10 for the case when
$Q^{2}=2$ GeV2. The EIC provides a unique opportunity to improve our knowledge
of the $F_{2}^{\pi}$ structure function over a large range in $Q^{2}$ and
$x_{B}$.

Figure 23:  The ratio of the uncertainty of the $F_{2}^{\pi}$ structure
function from the global fit with and without including EIC projected data to
the uncertainty of the $F_{2}^{\pi}$ as a function of $x_{\pi}$ for various
$Q^{2}$ values.

###  4.3 Neutron spin structure

Polarized 3He plays an important role as an effective neutron target in many
neutron spin structure experiments. For inclusive measurements, as often done
with fixed targets, the two protons not only dilute the signal, but they also
have a small net polarization which is not known, leading to rather large
systematic uncertainties on the extracted quantities. The EIC has a unique
capability to measure the two protons in the far forward region; this allows
for the extraction of neutron information with reduced systematic, as well as
an enhanced asymmetry, as compared to inclusive measurements, as will be shown
in this section.

Figure 24: A diagram of Deep Inelastic $e\,^{3}$He scattering with double
spectator tagging. The channel shown here is electron scattering off a neutron
in 3He; the two spectator nucleons are the protons in the process
3He$(e,e^{\prime}p_{s1}p_{s2})X$.

####  4.3.1 Event generation

This study used the output of the DJANGOH 4.6.10 [54, 55] event generator to
produce neutral-current DIS events from 3He, with a fully-calculated hadronic
final-state from the leading nucleon. The event generation was performed using
the CTEQ6.1 PDF set [56]. As DJANGOH events already include the effects of QED
radiation and final-state hadronization, it is only necessary to add the
kinematics of the spectator system.

The method used to determine the distributions of the spectators comes from
the convolution approximation for nuclear structure functions in the Bjorken
limit [57]:

$F_{1A}(x_{B},Q^{2})\approx\int\frac{d\alpha
d\Gamma_{s}}{\alpha}\frac{A}{\alpha}\sum_{N=p,n}F_{1N}\left(\frac{x_{B}}{\alpha},Q^{2}\right)\rho_{N}(\alpha,\Gamma_{s})$
(1) $F_{2A}(x_{B},Q^{2})\approx\int\frac{d\alpha
d\Gamma_{s}}{\alpha}\sum_{N=p,n}F_{2N}\left(\frac{x_{B}}{\alpha},Q^{2}\right)\rho_{N}(\alpha,\Gamma_{s})$
(2)

Here, $\alpha=\frac{A}{m_{A}}p^{+}$ is the light-cone fraction of the struck
nucleon, $\Gamma_{s}$ are the remaining kinematic degrees of freedom of
spectator system, and $\rho_{N}(\alpha_{s},\Gamma_{s})$ is the light-front
decay function of the 3He nucleus which gives the distribution of these
kinematic variables (described in Ref. [58]). Inserting these formulae into
the DIS cross-section formula and removing the convolution, we arrive at the
following cross-section differential in the spectator kinematics:

$\frac{d\sigma}{dx_{B}dy\frac{d\alpha
d\Gamma_{s}}{\alpha}}=\sum_{N=p,n}\frac{4\pi\alpha_{em}^{2}}{x_{B}yQ^{2}}\rho_{N}(\alpha,\Gamma_{s})\times\\\
\left[\left(1-y-\frac{x_{B}^{2}y^{2}m_{N}^{2}}{Q^{2}}\right)F_{2N}+y^{2}\frac{x_{B}}{\alpha}F_{1N}\right]$
(3)

Figure 25: Distribution of the momentum vector sum of two spectator protons,
$\vec{p}_{s1}$ and $\vec{p}_{s2}$, in the ion rest frame, for $5\times 41$
(top) and $18\times 166$ (bottom).

Here, $F_{iN}=F_{iN}\left(\frac{x_{B}}{\alpha},Q^{2}\right)$ , $i=1$, $2$.

For a given nucleon, the inclusive differential cross-section in terms of
variables $x_{B}$ and $y$ gives a distribution

$P(\alpha,\Gamma_{s}|x_{B},y)=\\\
\frac{\left[\left(1-y-\frac{x_{B}^{2}y^{2}m_{N}^{2}}{Q^{2}}\right)F_{2N}+y^{2}\frac{x_{B}}{\alpha}F_{1N}\right]\rho_{N}(\alpha,\Gamma_{s})}{\int\frac{d\alpha
d\Gamma_{s}}{\alpha}\left[\left(1-y-\frac{x_{B}^{2}y^{2}m_{N}^{2}}{Q^{2}}\right)F_{2N}+y^{2}\frac{x_{B}}{\alpha}F_{1N}\right]\rho_{N}(\alpha,\Gamma_{s})}$
(4)

This distribution was applied by sampling the light-front decay function and
is included in the event-by-event weighting factors:

$w=\frac{\left[\left(1-y-\frac{x_{B}^{2}y^{2}m_{N}^{2}}{Q^{2}}\right)F_{2N}+y^{2}\frac{x_{B}}{\alpha}F_{1N}\right]}{\int\frac{d\alpha
d\Gamma_{s}}{\alpha}\left[\left(1-y-\frac{x_{B}^{2}y^{2}m_{N}^{2}}{Q^{2}}\right)F_{2N}+y^{2}\frac{x_{B}}{\alpha}F_{1N}\right]\rho_{N}(\alpha,\Gamma_{s})}$
(5)

The $F_{1N}$ and $F_{2N}$ models used in calculating these weights were
provided by Ref. [59, 60].

####  4.3.2 Event selection

The full simulation framework, Fun4All, was used to process the generated
event samples to account for the detector acceptance effects. For each EIC
energy setting (eN: $5\times 41$ and $18\times 166$), a sample of 1M events
was generated for each Bjorken-x region ($x_{B}<0.2$, $x_{B}>0.2$ and
$x_{B}>0.5$). These samples are scaled by their corresponding normalization
factors and combined to provide 3M events for the full $x_{B}$ range. The
output of Fun4All is used as pseudo data for analysis. In this study we will
select two different event samples, inclusive and tagging using selection cuts
as below:

#### Inclusive sample e3He(e,e’)

The event selection cuts were applied to the scattering electron leptonic
reconstructed variables:

  * 1.

$E_{e}^{{}^{\prime}}>2$ GeV, $\eta_{e}>-3.5$

  * 2.

$Q^{2}>2$ GeV2

  * 3.

$W^{2}>10$ GeV2

  * 4.

$0.05<y<0.95$

#### Double tagging sample e3He(e,e’$p_{1s},p_{2s}$)

In addition to inclusive cuts, the tagging sample requires two spectator
protons to be detected. In order to identify the double tagging event, we use
the hit information from the Roman Pot. Only the first layer was considered in
the selection cuts. The occupancy plots for each spectator proton for two
energy settings were shown in Fig. 26. First, we require both spectator
protons to have a hit on the first layer and the hit’s local position to
satisfy the condition: $-12.5<x<12.5$ cm and $-5<y<5$ cm. In addition, the
beam contribution is excluded using the cut $-5<x<5$ cm and $-1<y<1$ cm.

After the double tagging events are identified in the collider frame, the
4-vectors of two spectator protons are boosted to the ion rest frame, and
their total momentum ($|\vec{p}_{s1}+\vec{p}_{s2}|$) as shown in the Fig. 25.
A cut of $|\vec{p}_{s1}+\vec{p}_{s2}|<0.1$ GeV was placed to ensure minimal
nuclear effects, where $\vec{p}_{si}$ is the 3-momentum of spectator proton
$i$. Due to the state of the far-forward reconstruction at the time of this
study, we used only the truth information (directly from Geant4 simulation
without smearing) of the far-forward protons.

Figure 26: The Roman Pot occupancy layer 1 for spectator proton 1 (left) and
spectator proton 2 (right) for the double tagging events

####  4.3.3 Extracted $A_{1}^{n}$ vs double tagged $A^{n}_{1}$

Uncertainties were calculated on $A^{n}$ given both extraction from
$A_{1}^{{}^{3}\text{He}}$ and measurement via double-spectator tagging. The
uncertainties were calculated given the estimated yields from the DJANGOH
event samples, as well as systematic uncertainties in the case of inclusive
extraction. Events were binned in $x_{B}$ and $Q^{2}$ and unfolded to Born-
level from reconstructed values. The unfolding procedure was completed using a
4-iteration Bayesian unfolding algorithm using the RooUnfold [61] framework,
trained with the Born-level and reconstructed values in the data; this means
that the impact of unfolding was to increase the uncertainty, but to perfectly
reconstruct the Born-level values. The effects of unfolding are reflected in
the yield uncertainties in each kinematic bin.

We compare the uncertainty of extracted $A^{n}_{1}$ from
$A_{1}^{{}^{3}\text{He}}$, and directly measured $A^{n}_{1}$, using the double
spectator tagging measurements. In a simple approximation, the relation
between $A_{1}^{n}$ and $A_{1}^{{}^{3}\text{He}}$ can be expressed as:

$A_{1}^{{}^{3}\text{He}}=P_{n}\frac{F_{2}^{n}}{F_{2}^{{}^{3}\text{He}}}A_{1}^{n}+2P_{p}\frac{F_{2}^{p}}{F_{2}^{{}^{3}\text{He}}}A_{1}^{p},$
(6)

Eq. 6 is used to calculate the prediction value for inclusive
$A_{1}^{{}^{3}\text{He}}$ where:

  * 1.

The values of $A_{1}^{n}$ and $A_{1}^{p}$ are taken from the parameterization
provided in [62]. The uncertainties and the correlation matrix associated with
$A_{1}^{n}$ and $A_{1}^{p}$ parameterization have also been obtained from
[62].

  * 2.

The structure functions $F_{2}^{p}$ and $F_{2}^{\text{D}}$ are taken from the
world data fit NMC E155 [63]. The larger of the asymmetric uncertainties is
chosen as the symmetric uncertainty for these structure functions.

  * 3.

Assuming no off-shell or nuclear-motion corrections, the value of $F_{2}^{n}$
is obtained using $F_{2}^{n}=F_{2}^{\text{D}}-F_{2}^{p}$. Similarly,
$F_{2}^{{}^{3}\text{He}}$ is obtained by using
$F_{2}^{{}^{3}\text{He}}=F_{2}^{\text{D}}+F_{2}^{p}$. The uncertainties of
$F_{2}^{n}$ and $F_{2}^{{}^{3}\text{He}}$ are propagated from the
uncertainties of $F_{2}^{\text{D}}$ and $F_{2}^{p}$.

  * 4.

The effective polarization of neutron and proton are $P_{n}=0.86\pm 0.02$ and
$P_{p}=-0.028\pm 0.004$ taken from [64].

Experimentally, the virtual photon asymmetry $A_{1}$ can be extracted from the
measured longitudinal electron asymmetry $A_{||}$ and transverse electron
asymmetry $A_{\perp}$, where

$A_{||}=\frac{\sigma_{\downarrow\Uparrow}-\sigma_{\uparrow\Uparrow}}{\sigma_{\downarrow\Uparrow}+\sigma_{\uparrow\Uparrow}}\quad\text{and}\quad
A_{\perp}=\frac{\sigma_{\downarrow\Rightarrow}-\sigma_{\uparrow\Rightarrow}}{\sigma_{\downarrow\Rightarrow}+\sigma_{\uparrow\Rightarrow}}.$

Considering electromagnetic interaction only,
$\sigma_{\downarrow\Uparrow}(\sigma_{\uparrow\Uparrow})$ is the cross-section
of the electron spin anti-parallel (parallel) to beam direction scatter off
the longitudinally polarized target.
$\sigma_{\downarrow\Rightarrow}(\sigma_{\uparrow\Rightarrow})$ is the cross-
section of the electron spin anti-parallel (parallel) scatter off the
transversely polarized target. The relation between $A_{1}$, $A_{||}$ and
$A_{\perp}$ is

$A_{1}=\frac{A_{\|}}{D(1+\eta\xi)}-\frac{\eta A_{\perp}}{d(1+\eta\xi)},$ (7)

where
$D={y(2-y)(2+\gamma^{2}y)}/(2(1+\gamma^{2})y^{2}+(4(1-y)-\gamma^{2}y^{2})(1+R))$,$\gamma=4M^{2}x^{2}/Q^{2}$,
$d=\sqrt{4(1-y)-\gamma^{2}y^{2}}D/(2-y)$,
$\eta=\gamma(4(1-y)-\gamma^{2}y^{2})/(2-y)/(2+\gamma^{2}y)$,
$\xi=\gamma(2-y)/(2+\gamma^{2}y)$, [65, 62] and $R$ is the ratio of the
longitudinal and transverse virtual photon absorption cross sections
$\sigma_{L}/\sigma_{T}$ [66]. The world fit parameters in Ref.[67] are used to
calculate the value of $R$.

Figure 27: A direct comparison of extracted $A^{n}_{1}$ from inclusive
measurements (blue band) and double tagging measurements (black square) which
are superimposed on the blue band. The left plot is for beam energy setting
5$\times$41 and the right plot is for 18$\times$166\. The blue points are the
$A_{1}^{{}^{3}\text{He}}$ measured values from inclusive measurements from
which the blue band is extracted. The uncertainties for both techniques are
compared in the bottom box where the blue (black inverted) triangles are the
absolute uncertainties of inclusive (tagged) measurements. The data points
were located at the average value for each $x_{B}$ bin. The asymmetry
calculation for each data point corresponds to the average value of $Q^{2}$
for each $x_{B}$ bin.

The $A_{1}^{n}$ extraction from $A_{1}^{{}^{3}\text{He}}$ follows the below
procedure:

#### Inclusive $A_{1}^{{}^{3}\text{He}}$

The number of DIS e3He$(e,e^{\prime})$ events passing the selection cuts were
binned in $x_{B}$ and normalized up to the EIC total luminosity. Assuming that
we will measure $A_{||}$ and $A_{\perp}$ using the same luminosity, 100 fb-1,
the statistical uncertainty can be defined as:

$\delta A_{||,\perp}^{{}^{3}\text{He}}=\frac{\delta N}{NP_{e}P_{N}},$ (8)

where $N$ is the number of events for a given bin after normalization, and
$\delta N$ is the uncertainty on the number of counts. and $P_{e}$ and $P_{N}$
are the polarization of the electron and ion beam respectively, both taken to
be $70\pm 1$% as stated in the YR. $\delta N$ reflects the inflation of
uncertainty related to the unfolding (during the reconstruction to the Born-
level values). The $\delta A_{1}^{{}^{3}\text{He}}$ is the propagation
uncertainty of $\delta A_{||,\perp}^{{}^{3}\text{He}}$ through Eq. 7. The
prediction values for $A_{1}^{{}^{3}\text{He}}$ for each $x_{B}$ bin are
calculated using Eq. 7 at the average values of $x_{B}$ and $Q^{2}$ for that
given bin.

#### Inclusive extracted $A_{1}^{n}$

For each $x_{B}$ bin, using the obtained value of inclusive
$A_{1}^{{}^{3}\text{He}}$ from previous step, $A_{1}^{p}$ from [62],
$F_{2}^{p}$ and $F_{2}^{\text{D}}$ from fit NMC E155 [63], and $P_{p(n)}$ from
[64], we extract $A_{1}^{n}$ using Eq. 7. The total uncertainty of extracted
$A_{1}^{n}$ is propagated from statistical uncertainty of
$A_{1}^{{}^{3}\text{He}}$ and systematic uncertainty from $A_{1}^{p}$,
$F_{2}^{n}$, $F_{2}^{\text{D}}$ and $P_{p(n)}$.

#### Double tagging $A_{1}^{n}$

The double-tagging sample was binned in the same way as the inclusive sample
and normalized to the same total luminosity. Also assuming $A_{||}^{n}$ and
$A_{\perp}^{n}$ are measured with the same total luminosity, the statistical
uncertainties $\delta A_{||,\perp}^{n}$ can be calculated similarly as Eq. 8.
The total uncertainty of double tagging $A_{1}^{n}$ is propagated from the
$\delta A_{||,\perp}^{n}$ using Eq. 7.

####  4.3.4 Projections and impacts

We show the direct comparison of uncertainty from double tagging to the
extracted $A_{1}^{n}$ for two energy settings (5$\times$41 and 18$\times$166)
in Fig. 27. The study shows that the double tagging method results in reduced
uncertainties by a factor of $2$ on the extracted neutron spin asymmetries for
overall kinematics, and by a factor of $10$ in the low-$x_{B}$ region for
energy setting 5x41.

The EIC coverage of $A_{1}^{n}$ as a function of $x_{B}$ and $Q^{2}$ is shown
in Fig. 28. These new data points cover a previously unreachable kinematic
region, especially for neutron spin structure function study. This provides
valuable input for the polarized parton distribution global fit and the flavor
separation. In addition, the overlap in the moderated $x_{B}$ region with much
higher $Q^{2}$ compared to existing fixed target data will be the perfect
place to test the nuclear correction that has been used to extract the neutron
information.

Figure 28: The EIC kinematic coverage of the neutron asymmetry $A_{1}^{n}$ as
a functions of $x_{B}$ and $Q^{2}$ for two electron-nucleon energy settings
5$\times$41 and 18$\times$166.

###  4.4 DVCS

Deeply Virtual Compton Scattering (DVCS), $(ep,e^{\prime}p\gamma)$, provides
an excellent tool to study the Generalized Parton Distributions (GPDs) of the
proton, Fig. 30, and the three-dimensional structure of the nucleon. These
non-perturbative quantities encode the correlated momentum and spatial
distributions of the quark and gluons within the proton. In addition, these
important quantities offer a unique opportunity to probe the energy-momentum
tensor and thus open the door to deepening our understanding of the nucleon
mass.

Current knowledge of GPDs from DVCS is mainly based on data from fixed target
experiments from JLab at high $x_{B}$, and the HERA collider at low $x$. EIC
offers a unique opportunity in kinematics coverage which will create a linkage
between the JLAB and HERA data, $ep$-DVCS was labeled as one of the future
flagship measurements and was described extensively in the YR.

In this work, we estimated the feasibility of the ECCE detector for measuring
$ep$ DVCS without addressing the separation between DVCS and pure Bethe-
Heitler (BH) scattering. Both DVCS and BH have the same final states (see Fig.
30), and both processes contribute to the total cross-section. This separation
will be reported elsewhere, in the future. Additional background can originate
from deeply virtual $\pi^{\circ}$ production. However, due to the high photon
acceptance and resolution of the barrel and forward endcap calorimeters, this
background is expected to play a minor role.

Figure 29:  Feynman diagrams for the $ep$-DVCS process. (a) Quark and (b)
Gluon contributions to GPDs.

Figure 30:  Feynman diagrams for Bethe-Heitler process, where the final state
particles are identical to $ep$-DVCS.

The $ep$-DVCS study used the MILOU3D generator [36] [37] at three beam energy
configuration: 5$\times$41, 10$\times$100 and 18$\times$275\. This generator
was selected based on the existing YR studies and was selected in order to
assess a comparison of the ECCE detector versus the original YR conceptual
detector performance. However, this generator is not suitable for beam spin
asymmetry studies, because it integrates the angular dependence between the
leptonic and hadronic planes. Therefore, we have not reported on those studies
here, but we are currently looking at other event generators for these
purposes and will also report on that in the future.

Parameters input to the MILOU3D code for event generation were based on the
settings used for the YR studies. We performed simulations for three beam
energies 5x41, 10x100, and 18x275. For all three setups, we limited the
kinematical range to:

  * 1.

$1<Q^{2}<1000.0$ (GeV)2,

  * 2.

$10^{-5}<x_{B}<0.7$,

  * 3.

$-2.0<t<-0.01$ (GeV)2.

For each setup, 500 000 events were generated and used as an input to the ECCE
Fun4All detector simulation. The particle kinematics generated by the MILOU3D
software are described in the head-on (i.e. center of mass) frame. The Fun4All
software took the head-on kinematics from the MILOU3D input files, applied the
beam crossing effects (as described elsewhere), and then propagated each
particle through the realistic ECCE detector.

In exclusive $ep$-DVCS measurements, the goal is to detect all three emerging
particles, the electron, proton, and real photon. In all three kinematics
studied, electrons and real photons are measured in the central detector
(pseudorapidity $-3.5<\eta<3.5$ ), while the scattered protons escape through
the beam pipe opening in the hadronic endcap, and thus their detection is
required in the far-forward region ($\eta>3.5$) described in Sec. 2.

####  4.4.1 Event Selection

The starting point of this study was to reconstruct the detected electrons.
This was done using the track reconstruction algorithm that was used in
Fun4All software (SvtxTrackEval). To identify the scattered electron, we
required a track multiplicity equal to 1 (almost 100% of events) and the
particle charge to be negative ($\sim$ 99.5% of events). For most of the
kinematical region, this selection technique yielded excellent performance,
except at the geometric edges ($\sim$15% of the events were lost). Most of
this inefficiency can be recovered by the use of calorimeters for electron
reconstruction (this was outside the scope of the work reported here, and
again will be studied in the future).

In the second step of the DVCS event selection, the real photons were
reconstructed. Their reconstruction was based on the identification of
clusters in the electromagnetic calorimeters (EEMC, BECAL, and FEMC). The
reconstructed photon energy was based on the energy deposition in the cluster.
For this, the total momentum of the reconstructed photon was calibrated using
the energy of the “truth” photon information, where the truth momentum is the
known momentum of the particle from the MILOU3D generator output. The photon
direction was reconstructed based on the electron vertex and the position of
the cluster.

Finally, the scattered protons that elude the central barrel were detected
using either the B0 detector or the Roman Pots in the far-forward region. As
described in Sec. 2, a realistic geometry of the B0 detector was encoded in
the simulation, allowing for accurate modeling of the geometric acceptance
directly in the Fun4All simulation. However, the Roman Pot beam pipe cutout
was not included in the Fun4All software. Hits in the B0 were therefore
selected directly, based on which layers were hit first per event. Geometric
cuts of $\pm$5 cm in $x$ (detector horizontal plane) and $\pm$1 cm in $y$
(detector horizontal plane) were applied to the center of both Roman Pots in
the analysis of the Fun4All output, to remove events which would have
otherwise been lost down the beam pipe into the beam dump. For the results
shown here, the analysis used “truth” momentum values, as currently there was
no reconstruction of momentum from the far forward detectors. For each hit in
the B0 detector planes or Roman Pots, the Geant4 particle ID was used to
select the detected protons. To simulate the expected level of response of the
detector, the “truth” momentum of the detected protons was smeared by 1%. For
these studies, position resolution effects were not studied and the proton
directions were kept intact. This smearing level was selected as it is
consistent with the proposed detector technology, AC-LGAD, and its expected
segmentation.

####  4.4.2 Results

The results shown here present the acceptances of $ep$-DVCS photons and
protons, which enabled us to assess the accessible $-t$ range with the ECCE
detector, required for nucleon imaging purposes. The uncertainties shown in
this study are only statistical for 10 fb-1 integrated luminosity. The
resulting projected differential cross-section measurements are also given. In
the case of $ep$-DVCS, the $-t$ variable can be calculated using two different
methods. The first one is based solely on reconstruction from
$e^{\prime}+\gamma$, while the second corresponds to the more standard
definition, which is $t=(p-p^{\prime})^{2}$. During the study, both methods
gave comparable results. We chose to complete the study with the latter method
because the former is subject to significant radiation correction which is
poorly understood at the current stage (larger uncertainty at certain
kinematics regions).

Simulation of the current detector configuration exhibits good performance for
photon detection. Fig. 31 presents the acceptance as a function of $\eta$ of
the real photon for the highest beam setup of 18x275. The acceptance is
defined as the ratio of reconstructed photons in the calorimeters to the
number of generated photons in the MILOU3D generator.

Figure 31:  DVCS photon acceptance in the backward (green), barrel (blue), and
forward (grey) ECAL’s, as a function of pseudorapidity. The red dotted line
shows the distribution of (generated) DVCS photons

Contrary to the photon acceptance, which exhibits similar behavior from the
lowest to the highest beam configurations (the minimum energy of DVCS photons
must be much higher than the detection limit of the calorimeters), in the
proton case the acceptance is very sensitive to the beam energies. The recoil
proton acceptances of the B0 spectrometer and Roman Pots for different energy
configurations as a function of the momentum transfer to the proton
$t=(p-p^{\prime})^{2}$, for each energy configuration studied, are shown in
Fig. 32. The resulting $-t$ acceptance is shown to be very wide, continuous,
and extends to low-$t$. Such a wide coverage is essential for the precision
extraction of the transverse position distributions of quarks and gluons
inside the nucleon. It is also worth noting, that for the highest beam setup,
the minimal $-t$ value is limited by the beam size and the mandatory gap
between Roman Pots and the beam.

Figure 32:  Acceptance for DVCS protons as a function of $-t$ in the far-
forward detectors for different beam energy configurations. The inserts show
the $-t$ distributions of generated events.

The full exploration of nucleon GPDs will require multi-dimensional
measurements of the $ep$-DVCS differential cross-section in $Q^{2}$, $x_{B}$,
$t$ and the azimuthal angle $\phi$ between the lepton and hadron planes in the
initial hadron rest frame. Fig. 33 shows the projected precision and coverage
of $ep$-DVCS differential cross-section measurements for several beam energy
configurations and in multi-dimensional bins of $Q^{2}$, $x_{B}$ and $t$,
whilst due to the aforementioned MILOU3D limitation the $\phi$ dependence is
integrated. The uncertainties of the differential cross-section are based on
the expected integrated luminosity of $\mathcal{L}=$10 fb-1.

Figure 33:  Projected DVCS differential cross-section measurements as a
function of the momentum transfer $-t$ for different bins in $Q^{2}$ and
$x_{B}$. The assumed integrated luminosity is 10 fb -1 for each beam energy
configuration.

####  4.4.3 Summary

To summarize, our study shows that the ECCE detector is suitable to deliver a
wide $Q^{2}$ and $x_{B}$ coverage for the $ep$-DVCS process with reasonable
statistical uncertainties. Additional studies must be performed with a fully
realistic implementation of the far-forward region of the detector, due to the
large proton acceptance sensitivity in this reaction. The Roman Pots must be
sufficiently separated from the beam ($\sim 10\sigma$ according to YR), in
order to avoid saturation and radiation damage. This issue will be addressed
in future work, through the study of different beam configurations, such as
high-acceptance or high-divergence, and will be reported in future studies.

###  4.5 eA DVCS

This study aimed to investigate the feasibility of measuring coherent
exclusive DVCS off 4He, i.e. $(e\,^{4}$He,
$e^{\prime}\,{}^{4}$He${}^{\prime}\,\gamma)$ with the ECCE detector.

Including measurements of exclusive DVCS off light nuclei at the EIC, in
addition to DVCS off the proton discussed in Sec. 4.4, would provide access to
several physics topics of interest. These topics are only named here, as
further details may be found in Sec. 7.2.5 of the YR and the references
therein. This reaction is thought to allow one to look in detail at the
European Muon Collaboration effect in the transverse plane. As with the
proton, coherent DVCS on light nuclei also allows one to extract GPDs which
encode the spatial distributions of partons in the nucleon. Furthermore, with
4He, in particular, the separation of coherent and incoherent channels in DVCS
is a recent theoretical milestone [68]. The EIC will be able to probe the
required low xB values in wide ranges of momentum transfer $t$, offers a
unique opportunity to make measurements of these topics.

Figure 34: Reconstructed (black) and generated (red) for $-t$ distributions
for $(e\,^{4}$He, $e^{\prime}\,{}^{4}$He${}^{\prime}\,\gamma)$, using
different methods, as described in the body of text, and normalized to the EIC
luminosity.

####  4.5.1 Simulated Settings

For this study, 1 M events were generated with the TOPEG generator [38]. More
information on our use of TOPEG is available in B. The nominal EIC beam
configuration 5$\times$41 was used. For 4He, with four nucleons, this gives an
ionic energy configuration of 5$\times$164.

The Fun4All [19] simulation software was used to simulate the physics events
in Geant4, using the generator data as an input. Initial results presented in
the exclusive physics note of the ECCE detector proposal [69] were obtained
using the prop.4 simulation build. However, the Fun4All software has developed
since, as detailed in Sec. 3, and further results have been obtained.
Specifically, using software build prop.7.1, this study was repeated for the
high-acceptance and high-divergence beam parameterizations detailed in Sec.
3.1. The most important differences in each of these simulation builds, and
indeed in the feasibility of this measurement, manifests itself in the
acceptance of scattered 4He ions in the far forward Roman Pot detectors. This
is detailed in Table 5.

Simulation |  $x$ [cm] |  $y$ [cm] | RP Acceptance
---|---|---|---
prop.4.0 | 5.0 | 1.0 | 14.4%
prop.7.1 hi.-div. | 4.25 | 0.80 | 8.0%
prop.7.1 hi.-acc. | 2.08 | 0.34 | 60.0%
Table 5: 10$\sigma$ cuts in x and y on RP hits (to reject the ion beam), and
scattered 4He acceptance in the RP in each beam configuration (high divergence
and high acceptance). The $x$ and $y$ boundary cuts are based on standard
deviations of simulation beam spot widths and heights at different beam
configurations at the maximum energy (18x275).

Due to the unique charge and momentum considerations with an 4He beam,
specific magnetic steering of 82 GeV2 was employed in the simulation.

####  4.5.2 Event Selection and Analysis

Final event selection is exclusive with three particles: the scattered
electron $e^{\prime}$, the scattered 4He, $A^{\prime}$, and the real photon
produced by the DVCS process, $\gamma^{\prime}$. Due to obvious similarities
in the channels, much of the final analysis methodology and selection is
identical to the DVCS-$ep$ case described in Sec. 4.4.

The electron selection is as in Sec. 4.4, solely using the SvtxTrackEval
Fun4All container and choosing events with explicitly 1 track in this silicon
tracker. All momentum information about the electron is available in the
container, so reliance on truth information is the lowest for this particle.

The photon selection is also very similar to the method described in Sec. 4.4.
The highest energy photon in any of the three calorimeters is selected using
its PID and energy in the container and assumed to match the DVCS
$\gamma^{\prime}$. It is important to subsequently calibrate the response of
the calorimeters. For this, the energy of the photons in each calorimeter is
plotted against the true energy of the photons, and a straight line was fitted
to the data. This is done separately for each calorimeter in the central
detector (FEMC, EEMC, BECAL). The energy of the selected photon is then
corrected using the coefficients extracted from the calibration fit. The
momentum components of the photon are reconstructed using the calibrated
energy and the available angular information of the track provided in the
container.

Finally, the scattered 4He is selected using hits in the B0 detector or Roman
Pot detectors. For each, all hits in the container with correct Geant4
tracking IDs are selected. When the study was performed, an accurate B0
geometry had already been implemented in the simulation. However, the RP
geometry was still preliminary. Hits in the B0 are therefore selected directly
based on which layers are hit first per event. A geometric cut is applied to
the center of both Roman Pots, to remove events that would have otherwise been
lost down the beam pipe. The size of the cut is proportional to the size of
the beam spot and as such is different in each simulation build. These are
detailed in Table 5.

Currently, the analysis uses truth momentum values for the 4He ions, as the
reconstruction of momentum in the far forward detectors is limited. As in Sec.
4.4, a 1% momentum smearing was applied to the scattered 4He ion to account
for detector effects. The IP8 detector configuration may offer further
improved resolution in the far forward region, as well as a higher acceptance
for the forward going 4He ions due to the secondary focus region. Because of
this, it is planned to repeat this study for the IP8 setup in the near future.

Figure 35: Projected differential cross-sections in ECCE as functions of
physics variables $Q^{2}$, $x_{B}$, $-t$ and $\phi$ for DVCS-e4He. Each plot
is integrated over the phase space denoted in the legend.

The momentum transfer $t$ can be calculated using the reconstructed 4He or
using only the reconstructed photon and electron as described in Eq. 9. Both
reconstruction methods are shown in Fig. 34 for comparison’s sake only. In the
final analysis for this reaction, $t$ was reconstructed using only the
scattered helium method, to avoid potential contamination of the channels
cross-section with non-exclusive or incoherent events.

$t=-\frac{MQ^{2}+2M\nu(\nu-\sqrt{\nu^{2}+Q^{2}}cos(\theta_{\gamma^{*}\gamma}))}{M+\nu-\sqrt{\nu^{2}+Q^{2}}cos(\theta_{\gamma^{*}\gamma})}$
(9)

####  4.5.3 Results

For analysis, the acceptance in each phase space bin is calculated using the
reconstructed events in the bin divided by the number of events generated by
TOPEG in the bin. The fourth differential of the cross-section is then given
by the formula:

$\frac{d^{4}\sigma}{dQ^{2}dx_{B}dtd\phi}=\frac{1}{\mathcal{L}\cdot
Acc_{bin}\cdot\Delta\Omega}(N\pm\sqrt{N}),$ (10)

where $\mathcal{L}$ is the integrated luminosity and in this case is equal to
$\mathcal{L}$eic/u; Accbin is the acceptance in the bin; $\Delta\Omega$ is the
multidimensional bin width given as $\Delta\Omega=\Delta Q^{2}\Delta\
x_{B}\Delta t\Delta\phi_{h}$; and $N$ is the number of counts in the bin.

We then integrated over three-dimensional phase space, and the projected
differential cross-sections are given as a function of $Q$2, $t$ (using 4He
and 4He′ ions for $t$ reconstruction only), $x_{B}$ and $\phi$ (the angle
between the leptonic and hadronic scattering planes). The results are shown in
Fig. 35.

####  4.5.4 Analysis and Summary

The simulation build which offered the best ion detection in the far forward
region was chosen for the result presented here. From Table 5 it is clear that
this is the high acceptance parameterization of the most recent prop.7.1
version. From a generated data sample of 107 events, we find no hits in the B0
detector layers. A realistic beam pipe cut is implemented in the center of
this detector. As such we conclude that due to the proximity to the
interaction point and high pseudorapidity of deflected ions, almost all will
pass through this central hole. Most simulated events can be reconstructed in
the Roman Pot detectors, however, after photon and electron exclusivity cuts
and the beam pipe cut on the Roman Pot geometry, we measure 607703 events.
This yields a final acceptance for the IP6 high acceptance of 60.8%.

$Q^{2}$ and $x_{B}$ acceptances are fairly high in the probed region of phase
space and $t$ acceptance is non-zero across the entire region of generated
space. Un-physical ($>1$) acceptances in $t$ are within the statistical
errors, and acceptance drops in the region of $<0.03$ are likely still due to
forward ion acceptance

Overall, we can make the statement that these results are promising for
realizing DVCS measurements on light ions at the EIC with ECCE if the design
of the far forward region is fully considered. Initially, we were able to
confirm the YR findings that the study would be heavily dependent on ion
acceptance. Our results with the latest Fun4All builds demonstrate that if
this beam setting is realized at the EIC, large $\eta$ ions from a highly
rigid beam will be detectable to an acceptable degree for the measurement. The
kinematic region on which cross-sections are measured is most sensitive to the
$Q^{2}$ versus $x_{B}$ space. However, this is purely kinematics, and as such,
probing a larger phase space becomes a task of generating data in a given
region, rather than refining analysis or simulation. To conclude, these
results lend much confidence that the ECCE detector matches extremely well
with any stated possibilities and reaches of this measurement that were
previously outlined in the YR.

###  4.6 ep DVMP

Hard exclusive meson electroproduction processes, also known as deeply virtual
meson production (DVMP), complement the DVCS reaction. In the DVMP case, the
scattering reaction produces a meson instead of a photon. Heavy vector mesons,
such as J/$\psi$, probe the gluon GPDs and ultimately provide information
about saturation when studying the change of gluon spatial distribution from
low to high (see [5] page 114).

Following the lead of the YR (page 334), the detector performance and
efficiency in the context of J$/\psi\rightarrow e^{+}e^{-}$ events from $ep$
collisions were evaluated with ECCE. The main goal of this study was to
quantify the detector acceptance for this reaction in one of the kinematic
regions. The final results are estimated for 10 fb-1 luminosity.

####  4.6.1 Electroproduction of J$/\psi$ decaying in an electron-positron
pair

This section summarizes the event selection and the simulation details in the
analysis of the J$/\psi\rightarrow e^{+}e^{-}$ reaction with the ECCE
detector. The event generator is summarized in E, and the kinematics studied
in this analysis correspond to electron and proton beam energies of 18 GeV and
275 GeV, respectively.

The generated events were selected with the requirements summarized in Table
6.

The sample of reconstructed events was chosen such that only three tracks were
detected. Two of the tracks were positive and the third one was negative. The
J$/\psi$ selection had the negative track and the two possible combinations
with the positive tracks. If the J$/\psi$ reconstructed mass was in the 2 to 5
GeV window for a single combination of tracks (1 negative and 1 positive), the
event was processed, otherwise, the event was discarded. The proton was
detected in the far forward region with the Roman Pots since B0 was out of the
acceptance for this kinematic sample.

Variable | Definition | Range
---|---|---
$Q^{2}$ [GeV] | $Q^{2}=-q^{2}=-(k_{e}-k{e^{\prime}})$ | 0 - 50 GeV2
$x_{B}$ | $x_{B}=\frac{Q^{2}}{2\cdot k_{p}\cdot q}$ | 0 - 0.15
Table 6:  Kinematic limits in the J$/\psi$ study.

Fig. 36 shows that the scattered electron is detected mostly in the backward
region, but also in the barrel. Fig. 37 shows that the lepton pair daughter of
the J$/\psi$ is detected in three regions (backward, central, and forward). In
addition, Fig. 38 shows the distribution of the protons detected in the Roman
Pots, where the majority of the generated events that are not reconstructed
are lost to the Far Forward region in an exclusive measurement. These studies
have shown that, given the current IP6 design, a large number of protons go
through the beam pipe and cannot be detected.

Figure 36: Scattered electron detection in the calorimeters. Most of the
electrons go to the far backward region. Figure 37:  Electron (left) and
positron (right) from J$/\psi$ detection in the calorimeters.

Figure 38:  Proton detection in Roman Pot 1 (left) and Roman Pot 2 (right) for
the kinematic setting studied in this work.

The acceptance of electrons and positrons is 80 $\%$ and appears to be
independent of the kinematic setting. In the case of the protons, they are
limited by the far forward region; protons with $\eta<6$ in the head-on frame
are not detected, and there is an average of 50$\%$ efficiency for the other
$\eta$ regions.

The acceptance of electrons and positrons does not seem to depend on the beam
setting. The dips in those acceptances correspond to the transitions in the
tracking system and/or calorimeters.

The $e^{+}e^{-}$ invariant mass, and the missing mass reconstruction for the
whole process, will be essential to check the exclusivity of the measurement.
Fig. 39 shows the $e^{+}e^{-}$ reconstructed mass from this simulation, even
if they are difficult to interpret in the absence of a background study.

Figure 39: Reconstructed J$/\psi$ mass, for the 18x275 GeV kinematic setting.

Figure 40:  Physics kinematicas variables and resolutions for $ep$ scattering
of 18x275 GeV2.

####  4.6.2 Physics Variables: distribution and resolutions

The various quantities that are relevant to the physics at hand are $\phi$,
$Q^{2}$, $-t$, $x_{b}$, $x_{v}$ and $x_{L}$. Fig. 40 shows the distributions
of these variables. As expected, the effective range of these physical
variables is limited by the acceptance of the protons. This directly affects
the range of the variable $-t$: events with $-t>1$ GeV2 are not reconstructed
due to the outer acceptance of the Roman Pots and the lack of statistics. For
the case of 18$\times$275 in this study, events with small $-t$ ($<$ 0.2 GeV2)
are not detected because of the inside edge of the Roman Pots.

Figure 41: Differential cross-section vs Momentum transfer t for the
18$\times$275 beam setting studied in xv slices, 0.0016 $<$ xv $<$ 0.0025
(black), 0.016 $<$ xv $<$ 0.025 (blue) and 0.16 $<$ xv $<$ 0.25 (red).

####  4.6.3 Cross-section

The cross-section, assuming a luminosity of 10 fb-1, was extracted as a
function of $-t$ and is displayed in Fig. 41. The acceptance of the ECCE
detector was fully considered for the events generated, but as expected it is
not the limiting factor in the measurement of these processes. The Far Forward
detectors are the main limitation of this measurement.

The statistical precision shown is for an integrated luminosity of 10 fb-1,
while the YR study (page 342) was performed for an integrated luminosity of
100 fb-1. The physics interest resides in the evolution of the $-t$ dependence
of the cross-section. The $Q^{2}$ dependence is also important to allow for
multi-dimensional binning. To a large extent, the $Q^{2}$ accepted range is
independent of the $-t$ range, and we have shown the evolution with $-t$ only
here.

###  4.7 Time-like Compton scattering

The following study investigated the feasibility of measuring Time-like
Compton Scattering (TCS) off the proton with the ECCE detector. The YR
description on this topic (Sec. 8.4.4) was conducted with a toy Monte Carlo
generator, eic-pi0-toy-MC [70]. The main aim for ECCE TCS activities was to
conduct the same study performed for the YR, however, this time taking
detector effects into account via the Geant4 simulation of the ECCE detector,
available with the Fun4All software [19]. Detector acceptance and
reconstruction of the final state were key areas of study for these
activities.

Figure 42:  Representation of TCS kinematics in the hadronic plane (yellow)
and leptonic plane (blue). The planes are separated by an angle $\phi$.
Initial four momenta of the beam proton and the real photon are represented by
convention as $p,q$, and the final state four momenta (the scattered proton
and produced virtual photon) are represented as $p^{\prime},q^{\prime}$. The
momenta of the decay lepton pair are represented as $k,k^{\prime}$. The angle
between the decay lepton $k$ and the scattering axis of the proton is
represented as $\theta$ [71]. For the study in this note, the decay lepton
pair was $e^{+}e^{-}$.

A sketch of the TCS process may be found in Fig. 42. TCS is an inverse process
to DVCS. Both measurements provide access to the same Generalised Parton
Distributions (GPDs), yet each have different experimental advantages over the
other. Complimentary TCS and DVCS measurements at the EIC will be crucial for
testing factorization, the transition between the space-like and time-like
regimes, and the universality of GPDs. The physics accessible via TCS is
further described in sections 7.2.2 and 8.4.4 of the YR. The dominating
background channel for TCS is the BH process, in which an incoming or
scattered electron radiates a photon, and scatters elastically off the proton,