Furthermore, at low $y$ the reconstruction of the relevant kinematics in the
Breit-frame suffers from low resolution. These issues have been shown to be
significantly improved using the hadronic final state as input to ML/AI
methods Pecar and Vossen (2022) or translating the kinematics into the lab-
frame Gao  _et al._ (2022). However, even with these improvements, larger $y$
still offers advantages in the resolution that can be reached.

  * •

To map out the structure of the nucleon encoded in TMDs and twist3 PDFs, high
precision, multi-dimensional measurements are needed, which requires very high
statistics. For our understanding of the evolution and proper domain of these
objects, it is essential to cover an extended kinematic phase space region
connecting the future collider to the ongoing fixed-target precision
measurements, e.g. by the JLab experiments. Figure 19 shows the estimated
phase space covered by the existing JLab12 program compared to the lowest and
highest EIC energy options.

  * •

Finally, intermediate energies have an advantage for a SIDIS program, as its
foremost detector requirements are excellent tracking and particle
identification. The most significant signals are expected for particles that
carry a large momentum fraction $z$ of the fragmenting quark, as these
particles are most closely connected to the original quark properties. As
illustrated in Fig. 20, at intermediate EIC energies, all particles that are
detected at mid-rapidity are within the momentum acceptance range of the
reference detectors. This is not necessarily true for the highest energies,
when particle identification within the typical EIC detector dimensions
becomes challenging.

Figure 14: Relative kinematic factors entering beam and target spin
asymmetries and polarization independent cross-section for $5\times 41$ beam
energy. These so-called depolarization factors are dependent on $y$ and
$\epsilon=\nicefrac{{(1-y-1/4\gamma^{2}y^{2})}}{{(1-y+1/2y^{2}+1/4\gamma^{2}y^{2})}}$
where $\gamma=\nicefrac{{2Mx}}{{Q}}$ Bacchetta  _et al._ (2007). The
nomenclature using $A,B,C,V,W$ is taken from Gliske  _et al._ (2014). They can
be approximated by $A\approx(1-y+\frac{1}{2}y^{2})$, $B\approx(1-y)$,
$C\approx y(1-\frac{1}{2}y)$, $V\approx(2-y)\sqrt{1-y}$ and $W\approx
y\sqrt{1-y}$. The rows indicate the different beam and target polarization
combinations while the first two columns relate to twist-2 quantities and the
third column to twist-3 quantities. For the spin independent cross-section,
the factor $A$ impacts the transverse momentum independent part, $B$ the
asymmetry relating to the Boer-Mulders $h_{1}^{\perp}$ function and $V$ the
asymmetry relating to the twist-3 FF $D_{1T}$. For the target-spin asymmetries
$UL$ and $UT$, the factor $B/A$ impacts the extraction of the transversity,
pretzelosity $h_{1T}^{\perp}$ and worm-gear $h_{1L}^{\perp}$ asymmetries,
whereas $V/A$ impacts the extraction of $h_{L}$ from $UL$ asymmetries. While
the factors described so-far become small only for large $y$, the factors
entering asymmetries with respect to the beam helicity, $LU$, $LL$ and $LT$
shown in the third row become small for medium and small $y$. Here the $C/A$
factor enters the extraction of the wormgear (LT) and helicity dependent FFs,
whereas $W/A$ enters the extraction of the twist-3 PDFs $g_{T}$ and $e$. As
illustrated by the figures, beam helicity dependent asymmetries are
significantly suppressed at low values of $y$. This restricts the minimal
$Q^{2}$ value that can be accessed and limits the statistical precision of the
measurement.  Figure 15: Like Fig. 14 but for $18\times 275$ beam energies.
Due to the higher $\sqrt{s}$, the accessible $Q^{2}$ range for TMDs extracted
from beam-helicity dependent asymmetries is higher. At large $\sqrt{s}$ a
large fraction of the data is at low $y$, making these measurements even more
challenging. Figure 16: Quantity $(\sqrt{N_{i}}/d)^{-1}$ for the $5\times 41$
configuration, where $N_{i}$ is the normalized count rate in a bin and $d$ the
depolarization factor. The quantity is proportional to the relative
statistical uncertainty in the respective bin with a proportionality factor of
$N^{-1}_{\textrm{total}}$. This illustrates the relative statistical
uncertainties one can reach for TMDs dependent on different polarization
factors.  Figure 17: Same as Fig. 14, but for the $18\times 275$ beam energy
configuration. The relative impact of the depolarization factor on asymmetries
dependent on the electron beam helicity is increased due to the phase space
distribution of the data.

The remainder of this section is organized as follows. Section III.2 will
discuss the physics case for twist-3 observables, Sec. III will give a short
overview of the TMD framework and impact studies for unpolarized and Sivers
TMD, which were identified as golden channels in the Yellow Report. This
section will also briefly discuss TMDs in medium. Finally, Sec. III.4 will
introduce the case for jet physics at intermediate energies and high
luminosity. Radiative corrections might complicate the picture, as the impact
on cross-sections and asymmetries can be sizable, depending on the kinematic
regime. The interplay between radiative corrections and TMD extraction is
still very much under investigation with recent studies Akushevich and
Ilyichev (2019a); Liu  _et al._ (2021) showing potential significant effects
on the angular reconstruction for TMDs in certain parts of the phase space.
However, as those studies are still in their initial stages, these effects are
not considered for the studies shown in this section.

Figure 18: Left: Projected Sivers asymmetry for various EIC run settings.
(Example for ATHENA pseudodata), 2% point-to-point systematic uncertainties
assumed. Right: projected Sivers asymmetries for 100 days of data taking at
each CM setting with the baseline luminosity vs. $Q^{2}$ for $0.25<x<0.35$ and
$0.4<z<0.6$ at the luminosity optimized EIC, JLab12 and the proposed JLab24.
For the JLab projections, the acceptance of the CLAS detector is used. The
proposed SoLID experiment will be able to run at higher luminosity values and
is expected to improve on these projections Chen  _et al._ (2014); Sol . The
drop of the amplitude with $Q^{2}$ is evident. At the same time the projected
uncertainties rise, as the valence quark region is harder to access at high
$Q^{2}$. A constraint of $y>0.05$ is used for this figure. Figure 19:
Estimated coverage of JLab12, HERMES and EIC data for different energy
configurations. The need to deliver high luminosity for the low and medium
energy configurations to fill in the phase space between fixed target
experiments and the higher EIC options is obvious. The data are constraint to
$y>0.05$. Figure 20: Acceptance of an exemplary EIC detector (here: ATHENA) in
laboratory frame $\eta/p$ for various energy configurations and $x,Q^{2}$
regions. PID limits exemplary for the ATHENA proposal are indicated with red
lines. At the highest energies a significant fraction of high $z$ particles is
outside the PID range. The horizontal axes are momenta from 0.1 to 100 GeV,
and the vertical axes are pseudo-rapidity from -4 to +4.

###  III.2 Accessing Quark-Gluon Correlations at sub-leading Twist

The interest for contributions that are suppressed by factors of $(M/Q)^{t-2}$
has recently grown with the possibility to access them in low-energy
experiments, such as HERMES and CLAS. Moderate $Q^{2}$ values at EIC will
offer unique opportunities for precision analyses of higher-twist distribution
functions. Such PDFs are often associated to multi-parton correlations as, to
some extent, the operator that defines such objects is made of quarks and
gluon fields. Such operators are almost unexplored by phenomenology Efremov
_et al._ (2003); Accardi  _et al._ (2009); Sato  _et al._ (2016); Aschenauer
_et al._ (2016); Boglione and Prokudin (2016); Anselmino  _et al._ (2020). As
argued below, the physics of twist-$3$ distributions is broader than the
already important quark-gluon-quark interaction, whose third Mellin moments
receive an interpretation in terms of forces Burkardt (2013).

A well-known example of higher-twist objects is the twist-$3$ contribution to
the axial-vector matrix element, $g_{T}$. The latter can be expressed in terms
of a leading-twist distribution through the Wandzura-Wilczek relation, and a
genuine twist-$3$ contribution. Data have shown that the genuine term is not
necessarily small Accardi  _et al._ (2009); Sato  _et al._ (2016). In the
Yellow Report for the EIC, the access to $g_{T}$ through double-spin asymmetry
$A_{LT}$ in inclusive DIS has been proposed as the golden channel towards the
study of multi-parton correlations. It was shown that the impact on the
uncertainty, based on the previous JAM analyses, is expected to be
significant. Figure 21 shows the impact of the EIC data with high luminosity
at low and medium energies on $g_{T}$ extraction.

Figure 21: Impact of EIC data with high luminosity at low/medium energies on
$g_{T}$ extraction. The improvement at high x is moderate (but not zero) due
to pre-existing data. This extraction uses data at $18\times 275$, $10\times
100$, $5\times 100$ and $5\times 41$, assuming an integrated luminosity of
$10$fb-1 at 18x275 and the other energies scaled according to their relative
instantaneous luminosities.

The scalar PDF, $e(x)$, is preeminent in that it relates to diverse aspects of
non-perturbative dynamics, such as the scalar charge of the nucleons and an
explicit quark-mass term, in addition to the quark-gluon correlations. The
scalar charge is particularly interesting in view of the mass decomposition of
the proton as it constitutes a unique avenue towards the phenomenological
extraction of the scalar condensate Efremov and Schweitzer (2003). While there
exist semi-phenomenological approaches to the determination of the pion-
nucleon sigma-term, e.g. Alarcon  _et al._ (2012); Hoferichter  _et al._
(2016), the twist-$3$ $e(x)$ can provide a determination that is minimally
biased by the underlying theoretical assumptions. Some model dependence is,
based on our current understanding, inevitable, since the extraction of the
sigma requires knowledge of $e(x)$ in particular down to $x=0$, which is not
experimentally accessible. The access to the scalar PDF through longitudinal
beam-spin asymmetries in (dihadron) SIDIS Bacchetta and Radici (2004) was
proposed as a silver channel in the Yellow Report. Up to date, the scalar PDF
has been accessed at JLab, in CLAS Mirazita  _et al._ (2021) and CLAS12
Hayward  _et al._ (2021), for low values of $Q^{2}$ and $x$ ranging from
$0.1-0.5$, leading to the first point-by-point phenomenological extraction
Courtoy  _et al._ (2022). While the parameterization of $e(x)$ is still a work
in progress, the impact from the EIC was shown to be significant thanks to the
broad kinematical reach. The $x$ range will be extended towards small-$x$
values, in the region relevant for the evaluation of the sum rules – such as
the relation to the scalar charge. The $Q^{2}$ range, spanning a broad window
of mid-$Q^{2}$ values, will allow analyses that account for QCD evolution
effects on each contribution. EIC thus represents a unique opportunity to
expand the curent exploratory studies towards global QCD analyses of the rich
phenomenology of higher-twist distribution functions.

In Fig. 22 the theoretical predictions are shown for the contribution of
$e^{a}(x)$ to the beam spin asymmetry in semi-inclusive di-hadron production
in the collinear framework for two different center of mass energies, showing
larger projected asymmetries for lower energies as expected. This asymmetry
receives a contribution not only from $e^{a}(x)$ but also from a term
involving a twist-3 di-hadron fragmentation function together with
$f_{1}^{a}(x)$ Bacchetta and Radici (2004). The latter has not been considered
here Courtoy  _et al._ (2022). The uncertainties in Fig. 22 come from the
envelope of the uncertainties on the interference fragmentation function
Radici  _et al._ (2015) and two models for $e^{a}(x)$, the light-front
constituent quark model Pasquini and Rodini (2019) and model of the mass-term
contribution to $e^{a}(x)$ with an assumed constituent quark mass of $~{}300$
MeV and the unpolarized PDF from MSTW08LO. All PDFs and fragmentation
functions are taken at $Q^{2}=1$ GeV2 and the projected uncertainties for the
EIC are shown only for $Q^{2}$ values smaller than $10$ GeV2.

Figure 22: Beam Spin Asymmetry in semi-inclusive di-hadron production.
Predictions corresponding to $Q^{2}=1$ GeV2 based on the di-hadron
fragmentation functions of Ref. Radici  _et al._ (2015), low-energy models for
the twist-$3$ PDF $e(x)$ (see text) and MSTW08 for the unpolarized PDF at LO.
Figure is taken from the Yellow Report Abdul Khalek  _et al._ (2021). The
twist-3 fragmentation is neglected. The upper and lower panel show two
different energy configuration ; the left (blue) and right (green) plots
correspond, respectively, to the fragmentation kinematics of ($0.2<z<0.3$,
$0.7<M_{h}<0.8$ GeV) and ($0.6<z<0.7$, $0.9<M_{h}<1.2$ GeV). The bands give
the envelope of the model projections discussed in the text folded with the
uncertainty of the interference fragmentation function. The projected
statistical uncertainties are plotted at zero and correspond to 10 fb-1 at
each CM setting. This illustrates that the data at lower $\sqrt{s}$ will have
a larger impact on constraining $e(x)$. Furthermore, the $Q^{2}<10$ GeV2 data,
where the signal is still expected sizable, is restricted to low $x$ for large
$\sqrt{s}$, where in turn $e(x)$ is expected to be small.

As the leading twist analysis addressed further below, all higher-twist
analyses will rely on the possibility to separate the contributions of the
various flavors from different observables, and mostly from different targets.
In particular, deuteron and 3He nuclei will provide effective neutron targets
to complement the proton data.

The phenomenological efforts can be paired with the progress made from the
lattice Lin  _et al._ (2018a); Constantinou  _et al._ (2021). Moments of
higher-twist distributions have been determined on the lattice Bhattacharya
_et al._ (2021a), frameworks for quasi-PDFs are being studied as well Braun
_et al._ (2021).

Beyond the collinear twist-$3$ mentioned above, there is a plethora of higher-
twist TMDs that could be studied at the EIC. Moreover, the second IR will
grant us the opportunity to explore the relations between twist-$3$ collinear
PDFs and twist-$2$ TMDs, the understanding of which is key for the
interpretation of low-energy dynamics.

###  III.3 Measurements of TMDs

The lepton-hadron semi-inclusive deep inelastic scattering (SIDIS) at the EIC
will provide excellent opportunities to probe the confined motion of quarks
and gluons inside the colliding hadron, which are encoded in the transverse
momentum dependent parton distribution functions (TMD PDFs, or simply, TMDs).
With the scattered lepton and an observed hadron (or jet) with sensitivity to
transverse momentum in the final-state, SIDIS provides not only a hard scale
$Q\gg\Lambda_{\rm QCD}$ from the virtuality of the exchanged virtual photon to
localize an active quark or gluon inside the colliding hadron, but also a
natural “soft” scale from the momentum imbalance between the observed lepton
and hadron in the final-state, which is sensitive to the transverse momentum
of the active quark or gluon.

With the one-photon approximation, the “soft” scale is the transverse momentum
of the observed hadron in the photon-hadron (or the Breit) frame, ${\bf
P}_{h_{T}}\gtrsim\Lambda_{\rm QCD}$. When $Q\gg|{\bf P}_{h_{T}}|$, the
unpolarized SIDIS cross section can be factorized as Bacchetta  _et al._
(2007),

$\frac{d\sigma^{\rm SIDIS}}{dx_{B}dQ^{2}d^{2}{\bf P}_{h_{T}}}\propto
x\sum_{i}e_{i}^{2}\int d^{2}{\bf p}_{T}\,d^{2}{\bf k}_{T}\,\delta^{(2)}({\bf
p}_{T}-{\bf k}_{T}-{\bf P}_{h_{T}}/z)\,\omega_{i}({\bf p}_{T},{\bf
k}_{T})f_{i}(x,p_{T}^{2})D_{h/i}(z,k_{T}^{2})\equiv{\cal C}\left[\omega
fD\right]\,,$ (39)

which provides the direct access to the TMD PDFs, $f_{i}(x,p_{T}^{2})$ of
flavor $i$ and transverse momentum $p_{T}^{2}\equiv{\bf p}_{T}^{2}$, and TMD
fragmentation functions (FFs), $D_{h/i}(x,k_{T}^{2})$ for a parton of flavor
$i$ and transverse momentum $k_{T}^{2}\equiv{\bf k}_{T}^{2}$, to evolve into
the observed hadron $h$ of transverse momentum $P_{h_{T}}$ in this photon-
hadron frame. In Eq. (39), the $\omega_{i}({\bf p}_{T},{\bf k}_{T})$ is a
known function depending on the kinematics, the type of TMDs and corresponding
angles between the parton transverse momenta.

With many more TMDs than PDFs, it will be possible to learn much more on QCD
dynamics that holds the quarks and gluons together to form the bound hadron,
despite being harder to extract and separate these TMDs from experimental
data. On the other hand, with a good detector able to cover the angle
distribution between two well-defined planes, the leptonic plane determined by
the colliding and scattered leptons, and the hadronic plane defined by the
colliding and observed hadrons, SIDIS measurements at the EIC will allow the
extraction of various TMDs by evaluating independent angular modulations of
the angle distribution between the two planes as well as the distribution
between the hadron spin vector and one of the planes.

####  III.3.1 Impact on the understanding of TMD factorization and
applicability to fixed target data

The TMD factorization formula Eq 39 receives corrections which enter in terms
of powers of $\delta\sim P_{hT}/z/Q$. Identifying the domain of applicability
of TMD factorization is not trivial Boglione  _et al._ (2017). In recent
analyses, usually the choice $\delta<0.25$ is adopted, at least for high $Q$
Bacchetta  _et al._ (2017); Scimemi and Vladimirov (2018, 2020); Bacchetta
_et al._ (2020a). These restrictions reduce the significance of a large amount
of existing measurements, in particular a majority of data from existing fixed
target experiments. Figure 23 illustrates this issue by showing the results of
Ref. Boglione  _et al._ (2022) where the regions of pion production in SIDIS
at the EIC are studied using results of Ref. Boglione  _et al._ (2019). The
so-called affinity to TMD factorization region (i.e. the probability that the
data can be described by TMD factorization) is calculated for each bin of the
EIC measurements. The affinity represents the probability of the bin to belong
to TMD factorization region and spans from 0% to 100%, indicated by color and
symbol size in the figure. One can see from Fig. 23 that only at relatively
high $z$ and $P_{hT}$ (and relatively large $x$ and $Q^{2}$) corrections to
the TMD factorization description are expected to be negligible. The reach of
the EIC data into other regions, will be important for the study the
connections to other types of factorization, for instance the collinear
factorization or the region accessed by fixed target experiments, where
sizable corrections to the current TMD formalism are expected. Comparing this
figure with the reach of the different energy option shown in Fig. 20, it can
be seen that intermediate beam energy option such as $10\times 100$ GeV2
operate largely in a region where TMD factorization holds, but also contain
phase space in the transition region towards other QCD regimes. The
flexibility to go from one regime of factorization to the other will be a
crucial ingredient in our understanding of QCD, and in the interpretation of
the vast amount of fixed target data, which has a low TMD affinity.

Figure 23:  TMD affinity for EIC kinematics. Bin centers are located in the
points corresponding to the bin averaged values of $x_{b}$ and $Q^{2}$, and in
each of these bins various values of $z_{h}$ and $q_{T}/Q$ can be measured. In
each bin of fixed $z_{h}$ and $q_{T}/Q$, the affinity is indicated by a dot
with size proportional to the corresponding affinity value. The affinity is
color coded according to the scheme on the right of the panels: red (and
smaller) symbols correspond to low TMD affinity, while dark blue (and larger)
symbols correspond to high TMD affinity. The plot is from Ref. Boglione  _et
al._ (2022)

Figure 24: Left: Impact on unpolarized TMD measurements integrated within
$0<q_{T}/Q<1.0,z>0.2$ , figure from Athena Proposal. Fit based on P17. The
color-code shows the datasets with the highest impact at a given $x,Q^{2}$
point. The assumed systematic uncertainty of 2% point-to-point is dominating.
However, the extraction of a specific point in $b$ is sensitive to the
collected statistics as shown in the right plot. Right: Impact of the EIC data
on the extraction of the CS kernel as function of $b$ (GeV-1) at $\mu=2$ GeV
using SV19 as a baseline compared to several other global extractions not
using EIC data. Figure from the Yellow Report Abdul Khalek  _et al._ (2021).

####  III.3.2 Impact on TMD PDF extraction

The theoretical description of TMDs has been extensively studied in coordinate
space labeled by $b$ as the conjugate variable of transverse momentum. In the
large $b$ region (small $q_{T}\approx p_{T}/z$), TMDs are non-perturbative and
encode intrinsic properties of hadrons while in the small $b$, TMDs are
dominated by QCD radiation which is calculable in perturbative QCD. In the
latter, TMDs can be connected with their corresponding collinear counterparts
such as PDFs and fragmentation functions offering a new venue to constrain
collinear distributions using TMD observables. While the experimental data is
sensitive to all regions in coordinate space, as discussed above, the relative
contribution of each region to the physical observables depend on the
kinematics of the final state particles accessible at a given collision
energy. Because of this, different collision energies from low to high at high
luminosity is needed at the EIC in order to systemically probe TMDs at
different regions of coordinate space. In the sections below, we concentrate
on the impact on the unpolarized TMD PDFs as well as the Sivers TMD PDF as
exemplary cases that would profit from increased precision at moderate
energies.

####  III.3.3 The impact study on the unpolarized TMDs

The unpolarized TMD distributions and fragmentation functions have been
extracted in Refs. Scimemi and Vladimirov (2018); Bacchetta  _et al._ (2017);
Scimemi and Vladimirov (2020); Bacchetta  _et al._ (2020a, 2022) (SV17, PV17,
SV19, PV19, MAPTMD22) with high perturbative accuracy up to NNLO and up to
N3LL of TMD logarithmic resummation. The data used in these global analyses
includes Drell-Yan and SIDIS processes measured at fixed target experiments
Derrick  _et al._ (1996); Adloff  _et al._ (1997); Asaturyan  _et al._ (2012);
Airapetian  _et al._ (2013); Adolph  _et al._ (2013); Aghasyan  _et al._
(2018a); Ito  _et al._ (1981); Moreno  _et al._ (1991); McGaughey  _et al._
(1994) at relatively low energies, and the collider measurements at higher
energy scales Aidala  _et al._ (2019); Affolder  _et al._ (2000); Aaltonen
_et al._ (2012); Abbott  _et al._ (2000); Abazov  _et al._ (2008, 2010); Aad
_et al._ (2014, 2016); Chatrchyan  _et al._ (2012); Khachatryan  _et al._
(2017); Aaij  _et al._ (2015, 2016a, 2016b). The span in the resolution scale
$Q$ and in observed transverse momentum $q_{T}$ allows for an extraction of
the non-perturbative Collins-Soper kernel (CS-kernel) and the unpolarized
TMDs. These extractions demonstrate an agreement between the theory and the
experimental measurements.

The extremely precise LHC measurements at $Q\simeq M_{Z}$ provide very
stringent constraints on the CS-kernel and TMDs in the region of small values
of $b$. However, the uncertainty of extractions grows in the region of $b>1$
GeV-1 due to the lack of the precise low-$q_{T}$ data. The large $b$ region is
important for the understanding of the non-perturbative nature of TMDs and the
primordial shapes TMDs and CS-kernel. In particular for the $Q$ range accessed
by intermediate energies, $Q\geq 5-10$ GeV, TMDs are only very poorly
constrained. Low and intermediate energies at the EIC will naturally provide
precision data in this kinematic regime as shown below. Predictions from
various groups are different in this region, see Ref. Vladimirov (2020), and
also disagree with the lattice measurements Zhang  _et al._ (2020); Shanahan
_et al._ (2020); Schlemmer  _et al._ (2021). This disagreement is problematic
since it points to a limited understanding of the TMD evolution encoded in the
CS-kernel, which dictates the evolution properties of all TMDs and describes
properties of the QCD vacuum Vladimirov (2020). The measurements from the EIC
will fill in the gap between the low-energy and high-energy experiments, and
will pin down these functions at higher values of $b$ corresponding to lower
values of $k_{T}$. Ultimately, it will help to unravel the 3D nucleon
structure in a very wide kinematic region.

Figure 25:  Comparison of relative uncertainty bands for unpolarized u-quark
TMD PDFs at different values of $b$ as a function of $x$. Lighter blue band is
the impact of $18\times 275$ data, light brown band is the impact of $5\times
41$ EIC pseudo data. The dataset used for these projections is the same as
used for the Yellow Report Abdul Khalek  _et al._ (2021). In particular all
energy options use the same integrated luminosity.

The unpolarized structure function is the leading contribution to the
differential SIDIS cross-section and also serves as the weight for polarized
asymmetries. As discussed above, mapping the unpolarized TMD over the full
phase space is a also necessary to probe TMD evolution effects which partially
cancel in the extraction of spin asymmetries. Therefore, the knowledge of
unpolarized TMDs is of paramount importance for the whole momentum tomography
program.

To demonstrate the impact, in particular of medium- and low energy data, we
consider the PV17 and SV19-fits. Figure 24, left shows the relative impact of
the different energy options on the extraction of the PV17 based TMD fit. It
is evident, that low and medium energies dominate over a wide range of phase
space, in particular at intermediate $x-Q^{2}$. This is even more impressive
considering that the impact plot is based on the baseline luminosities.

The estimation of the impact on the nonperturbative parts of the CS-kernel and
unpolarized TMDs has been done using the SV19-fit as the baseline. The
analysis was performed with the inclusion of EIC pseudo-data (in $5\times 41$,
$5\times 100$, $10\times 100$, $18\times 100$ and $18\times 275$ beam-energy
configurations). The pseudo-data, generated by pythia Sjostrand  _et al._
(2006), includes expected statistical and estimated systematic uncertainties,
for a hand-book detector design with moderate particle identification
capability. The estimate for the improvement in the uncertainties for the
extraction of the unpolarized TMDs is shown in the right panel in Fig. 24
exemplary for $f_{1T}^{u}$. In general, the main impact in the unpolarized
sector occurs for the CS-kernel, whose uncertainty reduces by a factor of
$\sim 10$. This is only possible with precise and homogeneous coverage of the
$(Q,x,z)$ domain, which can efficiently de-correlate the effects of soft gluon
evolution and internal transverse motion.

Fig. 25 shows the impact of the same integrated luminosity with the highest,
$18\times 275$, energy configuration and the lowest, $5\times 45$ energy
configuration on the extraction of the unpolarized u-quark TMD PDFs at
different values of $b$ as a function of $x$. As expected, the lower energy
data has a signifcant impact to constrain the PDF at in the valence quark
region for all $b$ and over the majority of the $x$ range at higher values of
$b$. This is thanks to the sensitivity to smaller values of $p_{T}$. Notice
that the high energy option has little impact in the valence region, as large
$x$ values can only be accessed at large $Q^{2}$. The combination of low and
high energy measurements will have the most homogeneous coverage of the
kinematics required for the studies of TMDs.

####  III.3.4 The impact study on the Sivers functions

Figure 26: Expected impact on the u-quark Sivers functions as a function $x$
as obtained from semi-inclusive pion and kaon EIC pseudo-data for $10\times
100$, $18\times 275$) beam-energy configurations and the combined impact. Fit
uses pseudodata from the EIC reference detector described in the Yellow Report
Abdul Khalek  _et al._ (2021) and SV19 fit. Left: impact of equal time data
taking with the base configuration, right: impact of proposed luminosity
increase at low and mid energies.

The non-vanishing Sivers asymmetry triggered a lot of interest in the physics
community and many groups have performed extractions of the Sivers functions
from the available experimental data Anselmino  _et al._ (2005); Collins  _et
al._ (2006); Vogelsang and Yuan (2005); Anselmino  _et al._ (2009); Bacchetta
and Radici (2011); Sun and Yuan (2013); Echevarria  _et al._ (2014); Boglione
_et al._ (2018); Luo and Sun (2020); Cammarota  _et al._ (2020); Bacchetta
_et al._ (2020b); Echevarria  _et al._ (2021); Bury  _et al._ (2021a, b).
However, currently the global pool of Sivers asymmetry measurements offers a
relatively small number of data points that could be consistently analysed
using the TMD factorization approach. The future measurements by the EIC will
provide a significant amount of new data in a wide and unexplored kinematic
region, and thus have a decisive impact in the determination of the Sivers
functions.

To determine the impact of EIC measurements on the Sivers function, the
pseudo-data generated by Pythia-6 Sjostrand  _et al._ (2006) was used with a
successive reweighing by a phenomenological model for the Sivers and
unpolarized structure functions from Ref. Anselmino  _et al._ (2009). The
pseudo-data for $\pi^{\pm}$ and $K^{\pm}$ production in $e+p$ and $e+^{3}He$
collisions at the highest ($18\times 275$) and the lowest ($5\times 41$) beam-
energy configurations were analyzed. The resulting pseudo-data set is about
two orders of magnitude larger in comparison with the current data. Performing
the fit of the new pseudo-data with the initial set of Sivers functions taken
from the global analysis made in Refs. Bury  _et al._ (2021a, b) and based on
the current SIDIS Airapetian  _et al._ (2009); Adolph  _et al._ (2012, 2017);
Alekseev  _et al._ (2009); Qian  _et al._ (2011) and Drell-Yan Aghasyan  _et
al._ (2017); Adamczyk  _et al._ (2016) measurements, a substantial reduction
of uncertainties is obtained. The uncertainty bands are reduced by an order of
magnitudes for all flavors.

Fig. 26 shows the impact on the uncertainty of the u-quark Sivers function at
$b=0$ GeV-1 as a function of $x$. The distribution of impact between $5\times
41$ and $18\times 275$ beam-energy configurations is similar to the
unpolarized case. Namely, $5\times 41$ configuration constrains mainly the
large-$x$ region, while $18\times 275$ configuration constrains the low-$x$
region. The combined set of the data gives the most homogeneous error
reduction. In turn, it reduces significantly uncertainties of the integral
characteristics. For example, the integral over Qiu-Sterman function has about
$3\%$ uncertainty (in the combined case) versus $6\%$ (for $18\times 275$
case) or $12\%$ (for $5\times 41$ case). Figure 18 shows the projected
experimental uncertainties compared to projections based on the extraction in
Ref. Bacchetta  _et al._ (2020b) for more energy options and vs $Q^{2}$.
Intermediate energies are most advantageous, since the expected asymmetries
are large while still enough statistics for a multi-dimensional analysis are
collected. This is in particular evident when plotting the asymmetries vs
$Q^{2}$ where the drop of the expected asymmetries at high $Q^{2}$ can be
observed as well as the drop of statistics expected from the EIC in the
valence region at high $Q^{2}$.

####  III.3.5 TMDs in nuclei

QCD multiple scattering in the nuclear medium has been demonstrated to be
responsible for the difference between TMDs in bound and free nucleons within
a generalized high-twist factorization formalism Liang  _et al._ (2008) and
the dipole model Mueller  _et al._ (2016, 2017). In these models, the scale of
the power corrections which modify the relevant distribution for the process
is proportional at leading order to $\alpha_{s}(Q)$, which becomes small at
large $Q$, see for instance Qiu and Vitev (2004, 2006). Thus while the EIC
will be capable of performing $e-A$ collisions for a wide range of nuclear
targets, a low center of mass energy is optimal for probing nuclear medium
modifications to TMDs.

From a phenomenological standpoint, nuclear modifications to collinear PDFs
have been performed in Refs. Eskola  _et al._ (1999); de Florian and Sassot
(2004); Hirai  _et al._ (2007); Eskola  _et al._ (2007); Schienbein  _et al._
(2009); Atashbar Tehrani (2012); Khanpour and Atashbar Tehrani (2016); Eskola
_et al._ (2017); Walt  _et al._ (2019); Kovarik  _et al._ (2016); Abdul Khalek
_et al._ (2019a, 2020) and for the collinear fragmentation function in Ref.
Sassot  _et al._ (2010); Zurita (2021). In these global analyses, the medium
modifications to the distributions enter into the non-perturbative
parameterizations. In the TMD description, the QCD multiple scattering
naturally leads to a broadening of the transverse momentum distributions.
Recently, the first extraction of the unpolarized nuclear modified TMDs have
been performed in Ref. Alrashed  _et al._ (2021). The authors of this paper
performed a global analysis at NLO+NNLL to the world set of experimental data
from hadron multiplicity production ratio at HERMES Airapetian  _et al._
(2007), Drell-Yan reactions at Fermilab Alde  _et al._ (1990); Vasilev  _et
al._ (1999) and RHIC Leung (2018), as well as $\gamma^{*}/Z$ production at the
LHC Khachatryan  _et al._ (2016); Aad  _et al._ (2015). In analogy to the work
that has been done in the past, this analysis took the medium modifications to
enter into the non-perturbative parameterization of the collinear
distributions as well as the parameterization for the non-perturbative Sudakov
factor, which controls the broadening of the transverse momentum distribution.
Despite the success of work in Alrashed  _et al._ (2021) in describing the
world set of experimental data, there are currently few data points which can
be used in order to constrain the TMD FFs. While the HERMES measurement of the
hadron multiplicity ratio probed a relatively wide kinematic region, the
stringent kinematic cuts applied to ensure the data are within the proper TMD
region vastly reduces the total number of useful experimental points. Since
Semi-Inclusive DIS is sensitive to both the TMD PDFs as well as the TMD FFs,
experimental measurements within the brod kinematical reach of EIC at small
and medium $Q$ represents the optimal process for probing nuclear
modifications to TMDs.

###  III.4 Jet Hadronization Studies

Jets are collimated sprays of particles, which are observed in collider
experiments. They exhibit a close connection to energetic quarks and gluons
that can be produced in hard-scattering processes at the EIC Hinderer  _et
al._ (2015); Abelof  _et al._ (2016); Boughezal  _et al._ (2018); Borsa  _et
al._ (2020); Arratia  _et al._ (2020a); Page  _et al._ (2020). Besides event-
wide jet measurements, significant progress has been made in recent years to
better understand jet substructure observables, see Refs. Larkoski  _et al._
(2020); Kogler  _et al._ (2019); Marzani  _et al._ (2019) for recent reviews.
Jet substructure observables can be constructed to be Infrared and Collinear
Safe making them less sensitive to experimental resolution effects.
Nevertheless, hadronization corrections can be sizable for these observables.
For several jet substructure observables it is possible to connect the
relevant hadronization correction to universal functions. The scaling of these
functions can be predicted from first principles which can be tested
experimentally by studying jets at different energies and by varying
parameters of specific observables. EIC jets at different center of mass
energies have different quark/gluon fractions and a different quark flavor
decomposition. Therefore, the measurement of jets at high luminosity and low
center of mass energies can provide important complementary information to
better disentangle the flavor decomposition of the hadronization corrections
of jets and also to study their correlation with different initial state PDFs.
Several jet observables in the literature have been studied which are
particularly sensitive to the quark flavor and quark/gluon differences.
Examples include jet angularities Lee and Sterman (2007); Ellis  _et al._
(2010); Aschenauer  _et al._ (2020); Caletti  _et al._ (2021), the jet charge
Waalewijn (2012); Kang  _et al._ (2020), angles between jet axes Cal  _et al._
(2020), groomed jet substructure Hoang  _et al._ (2019), flavor correlations
Chien  _et al._ (2021), energy-energy correlators Dixon  _et al._ (2019); Chen
_et al._ (2021); Li  _et al._ (2021a), jets at threshold Dasgupta  _et al._
(2008); Arratia  _et al._ (2021), and T-odd jets Liu and Xing (2021); Lai  _et
al._ (2022a). The EIC provides a clean environment with a minimal background
contamination from the underlying event/multi-parton interactions making it an
ideal place to study low-energy aspects of jets. In addition, the measurements
of jets for multiple jet radii at different energies may help to explore in
detail the connection of hadron and jet cross sections. Recently, it was
demonstrated that inclusive hadron cross sections can be obtained from
inclusive jet calculations by taking the limit of a vanishing jet radius Neill
and Ringer (2020); Neill (2021).

An important aspect of jet observables is their sensitivity to TMD PDFs and
FFs. For example, lepton-jet cross sections in the laboratory frame Liu  _et
al._ (2019a); Kang  _et al._ (2021); Hatta  _et al._ (2021) and the Breit
frame Gutierrez-Reyes  _et al._ (2018, 2019a, 2019b) give access to (spin-
dependent) quark TMD PDFs where the final state radiation can be calculated
perturbatively. Similarly, di-jet production can be used to study gluon TMD
PDFs Kang  _et al._ (2019); del Castillo  _et al._ (2020). Moreover, the
transverse momentum of hadrons inside the jet relative to the jet axis can
provide access to TMD FFs, which is independent of initial state TMD PDFs
Arratia  _et al._ (2020b). Here the choice of the jet axis is important and
different physics can be probed Neill  _et al._ (2017). Especially, due to the
separation of initial and final state TMD PDFs and FFs, jet observables can
provide important complementary information to semi-inclusive deep inelastic
scattering. All of these observables and the information content they provide
benefit greatly from measurements over a wide kinematic range. In particular,
high luminosity at the EIC will allow for a unique quark flavor decomposition.

A measurement that is in particular luminosity hungry, is the detection of
diffractive di-jet events. This observable is sensitive to the elusive
Generalized TMDs (GTMDs) Hatta  _et al._ (2016, 2020) of gluons. Lower
collision energies provide constraints for the moderate $x$-range of the gluon
distribution, while higher energies are sensitive to the small-$x$ gluon
distribution. If, as typically assumed, the gluon spin (helicity and orbital
angular momentum) is sizable at moderate $x$, it is critical to have very high
luminosity at lower/intermediate collision energies at the EIC.

##  IV Exotic meson spectroscopy

###  IV.1 Motivations for an exotic spectroscopy program at the EIC

Modern electro/photoproduction facilities, such as those operating in
Jefferson Lab, have demonstrated the effectiveness of photons as probes of the
hadron spectrum. However the energy ranges of these facilities are such that
most states with open or hidden heavy flavor are out of reach. This is
unfortunate as there remains significant discovery potential for
photoproduction in this sector. Already electron scattering experiments at
HERA Chekanov  _et al._ (2002); Aktas  _et al._ (2006) observed low-lying
charmonia, demonstrating the viability of charmonium spectroscopy in
electroproduction at high-energies but were limited by luminosity. Now the
proposed EIC, with high luminosity, will provide a suitable facility for a
dedicated photoproduction spectroscopy program extended to the heavy flavor
sectors. In particular, the study of heavy-quarkonia and quarkonium-like
states in photon-induced reactions will not only be complementary to the
spectroscopy programs employing other production modes but may give unique
clues to the underlying non-perturbative QCD dynamics.

One of the most striking features of quarkonium spectra is the wealth of
observed experimental signals which seem to indicate an exotic QCD structure
beyond conventional $Q\bar{Q}$ mesons. Starting with the observation of the
narrow $\chi_{c1}(3872)$ in the $J/\Psi\,\pi^{+}\pi^{-}$ invariant mass
spectrum by the BELLE Collaboration in 2003 Choi  _et al._ (2003), these
states, collectively denoted the $XYZ$’s, now number in the dozens. The
dramatic change in landscape from 2003 up to 2021 is illustrated in figure 27
where new states beyond quark model charmonium are highlighted. These states
exhibit properties which are not consistent with expectations of conventional
QCD bound states, for example : large isospin violation in the case of the
$\chi_{c1}(3872)$; iso-vector quarkonium-like character for the $Z$’s;
supernumeracy of the vector $Y$ states. We refer to reviews such as Brambilla
_et al._ (2020); Olsen  _et al._ (2018) for more detailed discussion. The
underlying dynamics governing their nature is not unambiguously known. The
experimental signals of these states, usually in the form of sharp peaks in
invariant mass spectra or broader enhancements that are required to describe
distributions in a more complex amplitude analysis, allow multiple
interpretations of their structure, e.g. multi-quark states, hadron-hadron
molecules, kinematic cusps or triangle singularities. Disentangling these
possibilities is one of the foremost missions of exotic spectroscopy and would
further our understanding of the non-pertubative nature of QCD in heavy
sectors.

One challenge in this endeavor is that, with few exceptions, the $XYZ$ signals
have only been observed in single production modes, usually $e^{+}e^{-}$
annihilation or $B$ meson decays. Observation of any of these states at the
EIC through photoproduction would thus provide independent and complementary
verification of their existence. Further, an ubiquitous feature of $XYZ$
signals is their proximity to open thresholds and the presence of additional
particles in the reconstructed final state. This complicates the
interpretation of experimental peaks as complicated kinematic topologies
involving nearby open channels may modify or mimic a resonant signal. Here
photoproduction provides a unique opportunity to produce $XYZ$ in isolated
final states, thus alleviating the role of kinematic singularities. In this
way a null result may be equally important towards uncovering the spectrum of
genuine bound-states. Additionally the polarized electron and proton beam
setups enable the determination of spin-parity assignments of states for which
these are not yet known. The EIC would also have real discovery potential for
exotic heavy flavor mesons.

A dedicated spectroscopy effort can make meaningful contributions to several
aspects of non-exotic quarkonium physics. Theoretical understanding of
photoproduction processes conventionally rely on Regge theory and exchange
phenomenology which have been tested extensively in the light sector Nys  _et
al._ (2018). Measurement of quarkonium photoproduction cross-sections serves
as a testing ground of scattering phenomenology in heavy sectors where
perturbative QCD inputs may also be used. In particular the microscopic
structure of $\gamma Q\bar{Q}$ interaction and assumptions such as Vector
Meson Dominance (VMD) may be tested Mamo and Zahed (2021b); Xu  _et al._
(2021).

Beyond the charmonium sector, the energy reach of the EIC will also allow the
study of near-threshold bottomonium photoproduction which may be sensitive to
the trace anomaly contribution to the nucleon mass and would be complementary
to ongoing studies of $J/\Psi$ photoproduction studies currently underway at
Jefferson Lab Ali  _et al._ (2019); Meziani  _et al._ (2016). Further, this
mass range is predicted to also exhibit a rich landscape of pentaquark-like
structures Cao  _et al._ (2020); Paryev (2020) the as yet unobserved hidden-
bottom partners of the $P_{c}$ signals observed in the $J/\Psi p$ mass spectra
in $\Lambda_{c}$ decays.

Figure 27: Experimentally measured charmonia, XYZ and pentaquark spectra from
Collaboration  _et al._ (2021). A ’?’ refers to unknown spin or parity.

####  IV.1.1 Photoproduction with the EIC

Given the many physics opportunities around photoproduction of heavy
quarkonia, new measurements at the EIC will be essential for understanding
both exotic and conventional quarkonium spectra. Photoproduction provides a
flexible production mode, able to produce the full spectrum of hadrons of any
quantum number. This gives such measurements significant discovery potential
and allows mapping out of patterns within the observed spectrum. The trade-off
however is that the cross sections for photoproducing heavy mesons are small,
only up to $\mathcal{O}(1$ nb), meaning a dedicated spectroscopy program will
require high luminosity at sufficiently large centre-of-mass energies to make
a meaningful contribution. The proposed EIC, maintaining high luminosity at
its lower centre-of-mass energies, would be well placed to meet these
conditions. In particular, even with the lower centre-of-mass settings of 29
and 45 $(GeV/c^{2})$ there is sufficient energy to directly produce many
exotic states of interest in the charmonium sector without the constraints in
bounds from parent masses which occur in decay processes. Kinematic generation
of peaks through final state interactions, such as triangle diagrams, will
also be suppressed over the entire $W$ range.

When combined with complete measurement of the final state, the polarized
electron and proton beams offer means for detailed partial wave analysis to
disentangle overlapping states, deduce the quantum numbers of resonant states
and study production mechanisms. This is of particular importance for many of
the excited $XYZ$ states which have intrinsically greater decay widths and
contribute to more complicated final states. The use of partial-wave analysis
through polarized photoproduction set-ups for exotic searches is currently
being pursued in the light-quark sector at the GlueX experiment and much of
the expertise will be readily applicable to the EIC setup. This includes the
possibility to measure polarized cross-sections, spin density matrix elements,
and asymmetries.

The variable beam setups of the EIC allow exploration of Primakoff production
of axial vector charmonium Albaladejo  _et al._ (2020) and simultaneous
measurement of charged charmonium-like isospin multiplets with deuteron beams.
Additionally, the electroproduction mode of the EIC allows measurement of
$Q^{2}$ dependence and photocouplings, a detailed study of which may be a
reliable probe of the microscopic nature of exotic hadrons Kawamura and Kumano
(2014); Anikin  _et al._ (2004). Electroproduction studies are of particular
importance for the $\chi_{c1}(3872)$ and the closely related $\tilde{X}(3872)$
candidate claimed in muoproduction by the COMPASS experiment in the
$J/\Psi\pi^{+}\pi^{-}$ mass spectrum Aghasyan  _et al._ (2018b). Although this
new state closely resembles the $\chi_{c1}(3873)$ in mass and width, its
dipion mass distribution was suggestive of a scalar wave instead of the usual
$\rho J/\Psi$ decay mode of the $\chi_{c1}(3872)$, implying a different
C-parity. Further this state was observed in production with an additional
pion in the final state but not in exclusive production, raising further
questions as to the nature of the muoproduced peak. Detailed study of the
$J/\Psi\pi\pi$ mass spectra in virtual photoproduction would help to
understand the COMPASS result.

####  IV.1.2 States of interest

The first goal of an exotic spectroscopy program will be to identify the
production of the most established states, $\chi_{c1}(3872)$, Y(4260) and
$Z_{c}(3900)$. The decay of these states to a $J/\Psi$ and pions will provide
a clean and well studied final state and we discuss in Section (IV.2.5) the
prospects for measuring this with the EIC. After that there are many open
questions in XYZ physics, particularly with respect to the nature of peaks in
invariant mass distributions which we hope to address. Here we consider a few
examples with decays which should be readily measurable and make rate
estimates for these in Section (IV.2.4).

A recent publication from LHCb show structure in the $J/\Psi K^{+}$ mass
spectra which they can reproduce with the addition of two new resonances with
strangeness and hidden charm, $Z_{cs}(4000)$ and $Z_{cs}(4220)$ Aaij  _et al._
(2021) with widths around 100-200 MeV. A similar, narrower state, the
$Z_{cs}(3985)$, has also been seen in $K^{+}D\bar{D}^{*}$ by BESIII Ablikim
_et al._ (2021).

The X(6900) or $T_{c\bar{c}c\bar{c}}(6900)$ tetraquark candidate has been seen
from its decay to 2$J/\Psi$ Aaij  _et al._ (2020). Analogue Z states have been
seen in the b-quark sector by Belle, with the $\Upsilon$ or $h_{b}$ mesons in
combination with a charged pion Bondar  _et al._ (2012). Production of these
states are also well within EIC centre-of-mass energies. In addition,
spectroscopy at the EIC will be able to search in a variety of other final
states replacing pions for other mesons such as vectors. We can also look for
charm quarks via reconstructing D mesons the most accessible decay mode of
which will be $K^{-}\pi^{+}$ with a branching ratio of around 4%, while the
decay of XYZ into final states with D mesons is likely to be quite high. As
seen later XYZ decay products populate the detector region relatively
uniformly giving good potential for reconstructing events including pairs of D
mesons. This would be particularly useful for investigating the molecular
picture of these states.

###  IV.2 Estimates for the EIC

####  IV.2.1 JPAC Photoproduction Amplitudes

In order to estimate the feasibility of quasi-real photoproduction for states
of interest at EIC energies we followed the approach of a recent JPAC
Collaboration study in Albaladejo  _et al._ (2020). Here, general principles
are used to construct exclusive photoproduction amplitudes of the charmonium
states of interest on the per-helicity-amplitude basis. In this way, full
kinematic dependence is retained and the production may be propagated along
decay chains to reconstructed final states.

In general the amplitude of producing a meson, $\mathcal{Q}$ via the exchange
of a particle, $\mathcal{E}$ with spin $j$ take the form:

$\langle\lambda_{\gamma}\,\lambda_{N}|T_{\mathcal{E}}|\lambda_{\mathcal{Q}},\lambda_{N^{\prime}}\rangle=\mathcal{T}^{\mu}_{\lambda_{\gamma}\,\lambda_{\mathcal{Q}}}\;\mathcal{P}_{\mu\nu}^{(\mathcal{E})}\;\mathcal{B}^{\mu}_{\lambda_{N}\,\lambda_{N^{\prime}}}$
(40)

where $\mathcal{T}$ and $\mathcal{B}$ are Lorentz tensors of rank-$j$ and
given by effective interaction Lagrangians which provide an economical way to
satisfy kinematic dependencies and discrete symmetries of the reaction. Such
methods have been widely used to motivate searches for exotic hadrons through
photoproduction Wang  _et al._ (2015a); Cao  _et al._ (2021); Winney  _et al._
(2019); Wang  _et al._ (2019); Karliner and Rosner (2017); Hiller Blin  _et
al._ (2016); Wang  _et al._ (2015b); Galata (2011); Lin  _et al._ (2013) The
form of the exchange propagator, $\mathcal{P}$, provides means to consider
production in two kinematic regions of interest: near-threshold and at high-
energies, where production is expected to proceed through exchanges of
definite-spin and Reggeized particles respectively. The center-of-mass range
available at the EIC provides wide coverage in energy, thus for first
estimates we used a simple linear interpolation between the low- and high-
energy models provided in Albaladejo  _et al._ (2020).

####  IV.2.2 Electroproduction

We generalized the aforementioned (real) photoproduction to consider exclusive
electroproduction with low-$Q^{2}$ quasi-real virtual photons via a factorized
model whereby the amplitude for producing a virtual photon beam is followed by
the t-channel photoproduction of the meson. The produced meson subsequently
decays to specific final states which can be measured in the EIC detector:

$\frac{d^{4}\sigma}{ds\,dQ^{2\,}dt\,d\phi}=\Gamma(s,Q^{2},E_{e})\;\frac{d^{2}\sigma_{\gamma*+p\rightarrow
V+p}(s,Q^{2})}{dt\,d\phi}$ (41)

$\Gamma(s,Q^{2},E_{e})$ is the virtual photon flux and
$\frac{d^{2}\sigma_{\gamma*+p\rightarrow V+p}(s,Q^{2})}{dtd\phi}$ is the two-
body photoproduction cross section calculated from the model of Albaladejo
_et al._ (2020), modified by an additional $Q^{2}$ dependence taken from
Adloff  _et al._ (2000). Eqn. (41) was integrated numerically to give the
total cross section for determining event rates. Note, the virtual photon flux
integration leads to a factor of around 0.2 for the case of $\chi_{c1}(3872)$
production relative to real photoproduction for the 5x41 GeV beams.

####  IV.2.3 Other Models

To estimate how reliable our production rates may be we compared to other
approaches that have been published recently.

In Yang and Guo (2021) a semi-inclusive production mechanism for hadron
molecules was investigated. Here the molecular constituents were first
photoproduced via Pythia, and then allowed to interact together in given $X$
and $Z$ states. Cross sections for semi-inclusive production were given for
the highest proposed EIC centre-of-mass energy for $\chi_{c1}(3872)$, $Z_{c}$
and $Z_{cs}$ and are compared to our estimates for exclusive production in
Table 1. While the estimates for $\chi_{c1}(3872)$ are an order of magnitude
lower than this work, the $Z_{c}$ cross section is an order of magnitude
higher. We note that the calculations of Yang and Guo (2021) should be valid
for larger $Q^{2}$, in the central region (large $p_{T}$), those from
Albaladejo  _et al._ (2020) should be valid at $Q^{2}<1(GeV/c^{2})^{2}$, and
peak in the peripheral region (small $p_{T}$), where we expect the bulk of
events to be produced.

Using the same method, Ref. Shi  _et al._ (2022a) estimates the semi-inclusive
production rates of more exotic hadrons, and finds that copious $P_{cs}$
pentaquarks and $\Lambda_{c}\bar{\Lambda}_{c}$ dibaryons can be produced at
EIC. It is also promising to search for double-charm tetraquarks at EIC. In
addition, Ref.[2208.02639] also suggests that the possible 24 GeV upgrade of
CEBAF [proper ref.] can play an important role in the search of hidden-charm
tetraquarks and pentaquarks.

A very similar approach to the current work is taken in Xie  _et al._ (2021),
where the models of Albaladejo  _et al._ (2020) were coupled to a virtual
photon produced from electron-proton scattering interactions. Their results
are compared to ours in table 1, where our estimates are just over a factor 2
lower for the low energy setting and more comparable for the high energy
setting. The differences are likely due to our interpolation of low and high
models, or handling of phase space and virtual photon flux factors when
performing the integration. The threshold at $Q^{2}>0.01~{}(GeV/c^{2})^{2}$ is
applied in this comparison but not in our later results where we integrate the
full allowable $Q^{2}$ range.

In general we can expect integrated cross sections for electroproduction of up
to order 1 nb for production of mesons with charm quarks.

Table 1: Model Comparisons. Note, in the Lanzhou calculations cuts are applied to $Q^{2}$ and W, as indicated in the column title with units in GeV. The same cuts are applied to our calculation when comparing to Lanzhou, but not to the comparisons with Yang. The cut on $W>20~{}GeV/c^{2}$ has a very large effect on the calculated electroproduction cross sections as the photoproduction cross section for X and Z of Albaladejo  _et al._ (2020) falls rapidly. |  3.5x20 $Q^{2}>0.01$;$W<16$ |  18x275 $Q^{2}>0.01$; $20<W<60$ |  18x275 $Q^{2}>0$
---|---|---|---
| JPAC |  LanzhouXie  _et al._ (2021) | JPAC |  LanzhouXie  _et al._ (2021) | JPAC |  YangYang and Guo (2021)
$\chi_{c1}(3872)$ | 0.47 nb | 1.2 nb | 0.00014 nb | 0.00021 nb | 3.5 nb | 0.216-0.914 nb
$Y(4260)$ | 0.06 nb | 0.2 nb | 1.5 nb | 2.0 nb | 14 nb | -
$Z_{c}^{+}(3900)$ | 0.06 nb | 0.16 nb | 0.00018 nb | 0.00048 nb | 0.41 nb | 3.8-14 nb

####  IV.2.4 Estimates

In table 2 we give estimates for the production of a variety of exotic states
with the EIC. These are based on the models and parameters detailed in
Albaladejo  _et al._ (2020), with the addition of the $Z_{cs}(4000)$
production using kaon exchange; and the modification of the X(6900) model to
use a higher branching ratio to $\Psi\omega$ of 3%, which was previously taken
as 1%. These estimates assume a luminosity of $10^{34}$ cm-2s-1. The
additional branching ratios, used to calculate events per day, of
$J/\Psi\rightarrow e^{+}e^{-}$ was taken as 6% and $\Upsilon(2S)\rightarrow
e^{+}e^{-}$ as 1.98%.

Current measurements of X and Y states contain up to order 10 thousand and 1
thousand events respectively. This is similar to the daily production rate of
our estimates. So with an overall detector acceptance of order 10 % the EIC
would be able to make significant contributions to our understanding of these
states.

We note that a previous investigation of charged final states in
electroproduction at an electron-ion collider Klein and Xie (2019) through a
Regge exchange mechanism found similar production rates for the $Z_{c}(4430)$,
approximately a factor 2 lower than our estimates for the $Z_{c}(3900)$. They
also conclude that the final state rapidity depends on the beam energy, at
lower center of mass energies production shifts toward mid-rapidity, where the
final state may be reconstructed in a central detector.

Table 2: Summary of results for production of some states of interest at the EIC electron and proton beam momentum $5\times 100(GeV/c)$ (for electron x proton). Columns show : the meson name; our estimate of the total cross section; production rate per day, assuming a luminosity of $6.1\times 10^{33}$ cm-2s-1; the decay branch to a particular measurable final state; its ratio; the rate per day of the meson decaying to the given final state.  Meson | Cross Section (nb) | Production rate (per day) | Decay Branch | Branch Ratio (%) | Events (per day)
---|---|---|---|---|---
$\chi_{c1}(3872)$ | 2.3 | 2.0 M | $J/\Psi\;\pi^{+}\pi^{-}$ | 5 | 6.1 k
$Y(4260)$ | 2.3 | 2.0 M | $J/\Psi\;\pi^{+}\pi^{-}$ | 1 | 1.2 k
$Z_{c}(3900)$ | 0.3 | 0.26 M | $J/\Psi\;\pi^{+}$ | 10 | 1.6 k
$X(6900)$ | 0.015 | 0.013 M | $J/\Psi\;J/\Psi$ | 100 | 46
$Z_{cs}(4000)$ | 0.23 | 0.20 M | $J/\Psi\;K^{+}$ | 10 | 1.2 k
$Z_{b}(10610)$ | 0.04 | 0.034 M | $\Upsilon(2S)\;\pi^{+}$ | 3.6 | 24

####  IV.2.5 Detection of final states

Meson photoproduction at the EIC will require a detector with full hermicity.
Quasi-real photoproduction results in the scattered electron being very close
to the incident beam line. t-channel production provides very little
transverse momentum for the recoiling baryon, which will likewise be scattered
within a degree or so of the beam. On the other hand the meson itself will be
produced relatively centrally at the lower centre-of-mass settings making for
excellent detection of its decay products.

The individual particle momentum distributions for the 5x100 centre-of-mass
setting are shown in Fig. 28. Also shown are the distributions expected when
reconstructed with the EIC Yellow Report matrix detector via the eic-smear
package Abdul Khalek  _et al._ (2021). It is clear the meson decay products
are almost entirely directed at the high acceptance central detector region.
Protons pass to the far-forward detector region, while there is some electron
detection in the backward electron region.

For final states including a $J/\Psi$, which are mostly under consideration
here, excellent electron/pion separation will allow a clean tag of $J/\Psi$
events through its narrow width in the $e^{+}e^{-}$ invariant mass. Coupled
with a very high detection efficiency this should allow for full
identification of the meson decay products and provide a means for peak
hunting in many final states including a final $J/\Psi$.

Supplementing the meson detection with far-forward and far-backward detector
systems will enhance the spectroscopy program by allowing measurements of the
full production process, that is measurement of the reaction variables W, from
the $e^{-}$ and $t$ from the recoil baryon. Detecting the scattered electron
also allows determination of the longitudinal and transverse polarisation
components of the virtual photon, providing further information on the
production processes through access to the meson spin density matrix elements.
This can be done with the backward detector around 5-10% of the time when the
electron beam momentum is lowest (5 GeV), due to the transverse kick to the
electron from the Lorentz boost due to the more energetic proton. A dedicated
far-backward electron detector such as the proposed low-$Q^{2}$ tagger could
increase the electron detection rate significantly. Detection of both the
electron and baryon can also allow for superior background rejection for
exclusive event reconstruction.

Figure 28: Momentum and angle distributions for X production. Left(right)
columns are for beam configuration 5x41(5x100). Rows, from top to bottom, show
$J/\Psi$ decay $e^{+}$; X decay $\pi^{+}$; the scattered proton; and the
scattered electron. Red lines show the true generated distributions while blue
are the detected particles as expected with the EIC Yellow Report matrix
detector.

###  IV.3 Outlook

We have briefly examined the case for producing exotic mesons through quasi-
real photoproduction at the EIC. Although it is difficult to make strong
statements on what we might expect this is exactly due to the uncertainty
around the nature and structure of the new states seen at other labs. We have
shown that if real exotic states exist then many of these should have
sufficiently high cross sections to be measurable. The low centre-of-mass
configurations are particularly suited to mesons produced through fixed spin
exchanges of light mesons, which have a high cross section close to threshold.
Coupled with a high luminosity this would provide a very high production rate,
while the kinematics and hermetic detector systems are ideal for
reconstructing the mesons we wish to study and allow us to exploit the EIC’s
discovery potential in exotic heavy flavor spectroscopy.

##  V Science highlights of light and heavy nuclei

###  V.1 Introduction

Lepton-induced high-energy scattering with nuclei will be measured at fixed
target facilities such as Jefferson Lab 12 GeV. These facilities have a rich
experimental program that will yield interesting results for years to come. To
complement these programs, the EIC will be the first high-energy facility that
has the ability to collide electrons and nuclei, which means it comes with
unique capabilities:

  * •

The EIC has a wide kinematic range in $Q^{2}$ and Bjorken $x$, enabling high-
energy nuclear measurements in unexplored kinematics.

  * •

The EIC can have beams of polarized light ions (3He, deuteron, etc. Bai  _et
al._ (2011)), enabling studies of the polarized nuclear (neutron) structure,
the polarized EMC effect, and nuclear spin-orbit phenomena. The deuteron,
being spin-1, offers possibilities of spin studies beyond that of the nucleon.

  * •

Measurements on nuclei inherently have to deal with nuclear effects such as
the Fermi motion, nuclear binding and correlation effects, and possible non-
nucleonic components of the nuclear wave function Frankfurt and Strikman
(1988). In inclusive measurements these nuclear effects form one of the
dominant sources of systematic uncertainties. With its extensive far-forward
detection apparatus in both interactions regions, detecting particles
originating from the breakup of the nuclear target (nuclear target
fragmentation region) is possible and can help to eliminate or control these
nuclear effects. (See Fig. 29 for a schematic diagram.) As a consequence,
these more exclusive measurements will push the capabilities of the EIC as a
precision machine for high-energy nuclear physics.

Figure 29: Schematic diagram of a nuclear breakup process. The virtual photon
$q$ interacts with a constituent of the nucleus $A$ and particles originating
from the breakup of the nucleus can be detected in the far forward region of
the EIC detector.

Measuring nuclear breakup reactions at the EIC has several advantages. In
collider kinematics nuclear fragments are still moving forward with a certain
fraction of the initial beam momentum and in non-coherent scattering they have
a different rigidity from the beam particles. This makes their detection more
straightforward than in fixed target experiments where they typically have low
momenta in the laboratory frame (10s of MeV/$c$). The detection of these
fragments enables additional control over the initial nuclear state in the
high-energy scattering event. It can be used to probe effective targets, for
instance, free neutron structure in tagged spectator DIS Frankfurt and
Strikman (1981); Sargsian and Strikman (2006), and pion and kaon structure in
the Sullivan process Arrington  _et al._ (2021). Nuclear breakup measurements
also determine which nuclear configurations (densities, virtualities, initial
nucleon momentum) play a role in the process, important for instance in a
multivariate disentanglement of nuclear medium modification effects such as
the EMC effect. A special case of detecting fragments is coherent nuclear
scattering in hard exclusive reactions, where the initial nucleus receives a
momentum kick but stays intact (no breakup). Measurements of these coherent
reactions allow us to perform tomography of light nuclei in quark and gluon
degrees of freedom as for the nucleon (Sec. I) and to study coherent nuclear
effects in these systems.

For all these reactions, having high event rates is of high importance
(multidimensional cross sections measured with sufficient precision, probing
rare nuclear configurations). To obtain these high event rates one needs both
high luminosity for a wide kinematic range and high acceptance for the
detection of final-state particles. In both interaction regions, the EIC will
have a dedicated set of far-forward detectors that enable the detection of
nuclear fragments with high acceptance. Due to the intricate engineering
challenges (magnets, beam pipe, crossing angle of the beam), each interaction
region will have some holes in the acceptance. Having these holes in different
regions of the kinematic phase space would enforce the complementarity between
the two interaction regions. Having a secondary focus would also increase
acceptance of detected fragments down to lower $p_{T}$ values. This is
especially important for coherent scattering of light nuclei, where the
$p_{T}$ values are much lower than for the free proton. (see Section VII.)

In the remainder of the section we offer a brief overview of nuclear reactions
that can be studied at the EIC and the physics motivation behind them. These
can all benefit from the complementarity offered by having a second IR. We
discuss these according to the nature of the measurements, starting with
inclusive measurement, then semi-inclusive and tagged reactions, and we
conclude with a discussion on exclusive nuclear channels and charm-flavored
hypernuclei.

###  V.2 Inclusive measurements

EIC can measure inclusive DIS on a wide range of nuclei, from the lightest to
heaviest nuclei, and in a wide range of Bjorken $x$ and $Q^{2}$. This can shed
light on the dynamics of nuclear modifications of partonic distributions
functions: shadowing and anti-shadowing at low values of $x$ and the so-called
EMC effect at high $x$. These high-$x$ measurements benefit from lower center
of mass energies and, with the $Q^{2}$ range that can be explored at the EIC,
the $Q^{2}$ dependence of the EMC effect could be further explored. This would
enable the disentanglement of leading and higher-twist effects in the medium
modifications. QCD evolution applied to the wide $Q^{2}$-range offers a way of
getting access to the gluon EMC effect at high $x$. In addition, for polarized
light nuclei the polarized EMC effect Cloet  _et al._ (2005) could be further
explored, which is so far an unknown quantity that will be explored in an
upcoming JLab experiment Brooks and Kuhn (2014).

###  V.3 Semi-inclusive and tagged spectator measurements

The use of semi-inclusive reactions on nuclei for nuclear TMD studies was
highlighted earlier in Section III.3.5. Here, we focus on so-called tagged
spectator measurements, where one or more nuclear fragments from the nuclear
breakup are detected. This helps, as previously outlined, to control the
nuclear configurations playing a role in the hard scattering processes. One
example is the use of deuteron or 3He as effective neutron targets by tagging
one (resp. two) spectator protons Frankfurt and Strikman (1981); Sargsian and
Strikman (2006); Kondratyuk and Strikman (1984); Del Dotto  _et al._ (2017);
Cosyn and Weiss (2019, 2020); Friscic  _et al._ (2021); Jentsch  _et al._
(2021). These neutron data are an essential ingredient in the quark flavor
separation of the partonic distribution functions. In the tagged spectator
reactions, an effective free neutron target can be probed by performing a so-
called on-shell extrapolation of the measured cross sections or asymmetries
Sargsian and Strikman (2006); Jentsch  _et al._ (2021). The presence of
polarized light ion beams enables the extraction of polarized neutron
structure in this manner Kondratyuk and Strikman (1984); Cosyn and Weiss
(2019); Del Dotto  _et al._ (2017); Friscic  _et al._ (2021).

Measuring tagged spectator reactions at larger nucleon momenta (several 100
MeV relative to the ion rest frame) is of interest to several outstanding
questions in nuclear physics and how these are interconnected. What is the QCD
nature of the short-range, hard core part of the nucleon-nucleon force Boeglin
and Sargsian (2015); Miller  _et al._ (2016); Tu  _et al._ (2020)? How do
nuclear medium modifications of partonic properties manifest themselves and
what nuclear configurations play a role in these Hen  _et al._ (2017)? In
these kinematics, however, the influence of final-state interactions between
products of the hard scattering and the spectator(s) and between the
spectators has to be accounted for Cosyn and Sargsian (2017); Strikman and
Weiss (2018) in order to disentangle them from the QCD phenomenon of interest.
These final-state interactions are moreover little explored in high-energy
scattering and are an interesting topic that can teach us about the space-time
evolution of hadronization dynamics.

While technically not a nuclear process, the Sullivan process $e+p\to
e^{\prime}+X+(N\;\text{or}\;Y)$ share characteristics with the previously
discussed processes. The physics interest of the Sullivan process lies in the
extraction of pion and kaon structure Aguilar  _et al._ (2019); Arrington  _et
al._ (2021). The pion being the pseudo-Goldstone boson of dynamical chiral
symmetry breaking, this can shed light on the mechanism of emergent hadronic
mass (EHM) within QCD. For the kaon, the presence of the heavier strange quark
opens up the study of the interplay between EHM and the Higgs mechanism. In
the Sullivan process, a nucleon or hyperon is tagged in the far-forward region
at low four momentum transfer squared $-t$. In this manner, the process is
dominated by meson exchange in the $t$-channel and, by extrapolating the
observables to the on-shell pole of the exchanged meson, one can extract pion
(nucleon tagging) or kaon (hyperon tagging) structure. Compared with the
earlier HERA extractions, the high luminosity and wide kinematic range of the
EIC would result in an order of magnitude decrease of statistical errors on
the extracted pion pdfs. These measurements require high luminosity
($>10^{33}$ cm-2 sec-1) in order to compensate for the few times $10^{-3}$
fraction of the proton wave function related to the pion (kaon) pole.
Additionally, for kaon structure lower center of mass energies are preferable
so that sufficient $\Lambda$ decays happen in the far forward region, see Sec.
VII.

Nuclear properties beyond that of the mean-field shell model can be studied
using $A(e,e^{\prime}NN)$ two-nucleon knockout reactions. These can especially
shed light on the nature of the nuclear short-range correlations (SRCs) and
their potential relation to nucleon medium modifications Hen  _et al._ (2017).
The EIC will enable measurements of these processes up to $Q^{2}$ values a
factor of 3-4 higher than has been achieved so far in fixed target setups
Hauenstein  _et al._ (2022). In these two-nucleon knockout reactions in
selected kinematics, one leading nucleon originates from the interaction with
the photon, while the other is the recoil partner that originated from the
SRC-pair. As with the previous discussed processes, the detection of recoil
nucleons happens in the far forward detector apparatus, due to the boost in
the collider lab frame relative to the ion rest frame. Additionally, detection
of nuclear fragments ($A-2$), and/or veto its breakup, could be possible
improving control over the reaction mechanism in these reactions Patsyuk  _et
al._ (2021).

Measurements of single-nucleon knockout reactions in mean-field kinematics are
possible at EIC up to $Q^{2}\approx 20~{}\text{GeV}^{2}$ Hauenstein  _et al._
(2022). These would help to constrain the onset of the nuclear color
transparency phenomenon Dutta  _et al._ (2013), which has not been observed
for proton knockout up to $Q^{2}=14~{}\text{GeV}^{2}$ Bhetuwal  _et al._
(2021). Color transparency could also be explored in other kinematics and
reaction mechanisms. One example that was recently explored is meson
electroproduction on nuclei in backward kinematics Huber  _et al._ (2022), see
also Sec. I.4.

Concerning the detection capabilities of the EIC for these 2N knockout
reactions, for the leading nucleon the detection region depends on the ion
beam energy. With 41 GeV/A beams, the majority of the leading nucleons is
detected in the central detector, while for 110 GeV/A it is detected in the
far-forward region, see Fig. 3 of Ref. Hauenstein  _et al._ (2022). Moreover,
at 110 GeV/A higher acceptance for recoil nucleons is also achieved. For
leading neutrons, however, with 110 GeV the neutrons are outside the angular
coverage of the ZDC, and these channels have to be measured at the lower ion
beam energy.

###  V.4 Exclusive measurements

Hard exclusive reactions on light nuclei can be measured in both the coherent
and incoherent (nuclear breakup) channels Dupré and Scopetta (2016). The
coherent channel, similarly to the case of the nucleon discussed in Secs. I
and II, would give access to 3D tomography of light nuclei in quark and gluon
degrees of freedom and the extraction of mechanical properties of light
nuclei. It could also potentially shed light on the size of non-nucleonic
components of the nuclear wave function. The incoherent channel, on the other
hand, can be used to study medium modifications of nucleon tomography Fucini
_et al._ (2020a, b) and to probe neutron 3D structure Rinaldi and Scopetta
(2013). Three of the lightest nuclei (d,3He,4He) have the interesting feature
that they have different spin and binding energies Cano and Pire (2004); Guzey
and Strikman (2003); Scopetta (2004); Cosyn and Pire (2018); Fucini  _et al._
(2018). 4He being spin-0 has the advantage that it has only one leading twist
GPD in the chiral even sector. 3He is a spin-1/2 nucleus, meaning that hard
exclusive observables can be similarly defined to those of the free nucleon.
Lastly, the spin-1 deuteron has a richer structure of GPDs beyond that of the
nucleon (associated with its tensor polarization modes), meaning that new
spin-orbit phenomena can be studied. In terms of binding energy, the deuteron
is very loosely bound, while 4He is very tightly bound and 3He falling
somewhat in between. This gives us access to different degrees of nuclear
effects that can be studied in these systems. Additionally, the availability
of high-precision ab initio nuclear wave functions for these light nuclei
results in a high degree of theoretical control in calculations. The
challenges of detecting these exclusive reactions are covered in more detail
in Sec. VII. There, the influence of a secondary focus on the lower limit of
the measurable $t$-range for the exclusive channel especially deserves
highlighting.

###  V.5 Charm-flavored hypernuclei

Hypernuclear physics has been one of the crucial tools for studying the
interactions between nucleons and strange hyperons. Most experimental studies
on hypernuclei have been focused on $\Lambda$ hypernuclei and many precise
measurements have been performed as reviewed in Ref. Feliciello and Nagae
(2015). Recently, these efforts are extended to hypernuclei with multi-
strangeness such as $\Xi$ hypernuclei.

Recently, there have been interests in charm hypernuclei of which the
existence was predicted almost 45 years ago Tyapkin (1975); Dover and Kahana
(1977) right after the discovery of the charm quark. As the strange
hypernuclei structure heavily depends on the $\Lambda$-nucleon interactions,
the stability of charm hypernuclei depends on the $\Lambda_{c}$-nucleon
interactions. Following the seminal works of 1980s, there have been many
theoretical model calculations on various states of $\Lambda_{c}$ hypernuclei.
The calculated spectra of charm hypernuclei are found to be sensitive to the
$\Lambda_{c}$-nucleon interactions. (See, for example, Refs. Krein  _et al._
(2018); Wu  _et al._ (2020) for a review.) As there is no empirical
information on the $\Lambda_{c}N$ interactions, various ideas were adopted for
modeling the potential between $\Lambda_{c}$ and the nucleon. In recent
calculations, lattice simulation results were used to model this potential.
However, depending on the approach to the physics point from the unphysical
quark masses used in lattice calculations, the extrapolated potentials lead to
very different results for the $\Lambda N$ interactions Haidenbauer and Krein
(2021). Figure 30 shows different predictions for the $\Lambda_{c}N$
${}^{3}D_{1}$ phase shift extrapolated from the same lattice calculations but
with different extrapolation methods. It shows that the results are completely
different depending on the extrapolation approaches. Therefore, experimental
measurements on charm hypernuclei are strongly required to shed light on our
understanding of the $\Lambda_{c}N$ interactions.

Figure 30: Predictions for the $\Lambda_{c}N$ ${}^{3}D_{1}$ phase shift. (a)
Results from covariant $\chi$EFT taken from Ref. Song  _et al._ (2020). (b)
Results based on the $\Lambda_{c}N$ potential from Ref. Haidenbauer and Krein
(2018). Red (black), green (dark grey), and blue (light grey) bands correspond
to $m_{\pi}=138$, 410, and 570 MeV, respectively. The width of the bands
represent cutoff variations/uncertainties. Lattice results of the HAL QCD
Collaboration corresponding to $m_{\pi}=410$ MeV (filled circles) and 570 MeV
(open circles) are taken from Ref. Miyamoto (2019). The figure is from Ref.
Haidenbauer and Krein (2021).

Experimentally, earlier efforts to find charm hypernuclei started right after
the seminal work of Ref. Tyapkin (1975) and a few positive report on the
existence of charm hypernuclei (called supernuclei at that time) were claimed
Lyukov (1989). However, no serious follow-up research was reported and, in
practice, there is no experimental information on charm hypernuclei. The
experimental investigations in this topic would be possible at future hadron
beam facilities Barucca  _et al._ (2021). The experimental instrumentation of
the EIC allows for precise measurements and would offer a chance to study
charm hypernuclei. So far $\Lambda$ hypernuclei have been studied extensively
with high intensity meson beams as well as electron beams. Electro-production
of Lambda hypernuclei was studied with the
${}^{A}Z(e,e^{\prime}K^{+})^{A}_{\Lambda}(Z-1)$ reaction and similar reaction,
${}^{A}Z(e,e^{\prime}D^{-})^{A}_{\Lambda_{c}^{+}}Z$ will produce charm
hypernuclei by converting a neutron to $\Lambda_{c}^{+}$ and $D^{-}$. Through
observation of produced $D^{-}$ and scattered electron, the missing mass
technique can be applied to the spectroscopic study of charm hypernuclei.
Therefore, studying charm hypernuclei with electron-ion collider would open a
new way to study heavy-flavored nuclei with the future hadron beam facilities.
This investigation can also be extended to the bottom sector Feliciello
(2012), which is simpler than the charm sector as there is no Coulomb
interaction between $\Lambda_{b}$ and nucleons. Therefore, comparing the
properties of bottom hypernuclei and strange hypernuclei would give a clear
clue on the mass dependence of the strong interactions. The designed energy
range of EIC would allow further investigations.

##  VI Precision studies of Lattice QCD in the EIC era

Lattice QCD enables the first-principles solution of QCD in the strong-
coupling regime, and thereby facilitates calculations that can both guide the
analysis of key physics quantities to be determined at the EIC, and provide
complementary calculations that can further the physics potential of the EIC.
The calculation of the internal structure of the nucleon, pion and other
hadrons in terms of the fundamental quarks and gluons of QCD has been a key
effort of lattice calculations since the inception of lattice QCD. Notably,
there have been the first-principles calculation of the electromagnetic form
factors, and of the low moments of the unpolarized and polarized parton
distribution functions and of the generalized form factors. Similarly, the
low-lying spectrum of QCD has been a benchmark calculation that now including
the electroweak splittings. Nevertheless, the formulation of lattice QCD in
Euclidean space imposes important restrictions. Firstly, time-dependent
quantities, and in particular those related to matrix elements of operators
separated along the light cone, could not be calculated, thereby precluding
the computation of quantities, such as the $x$-dependent parton distribution
functions. Further, scattering amplitudes, and thereby information about
resonances in QCD, eluded direct computation. In both the fields of three-
dimensional imaging and spectroscopy key theoretical advances have
circumvented these restrictions and transformed our ability to address key
questions of QCD in the strong-coupling regime.

###  VI.1 Three-dimensional Imaging of the Nucleon

The electromagnetic form factors, and the generalized form factors
corresponding to the moments with respect to $x$ of the GPDs, can be expressed
as the matrix elements of time-independent, local operators amenable to
computation in lattice QCD on a Euclidean grid. In particular, there has been
a progression of calculations of the lowest moments of the isovector
generalized form factors Hagler  _et al._ (2003, 2008); Bratt  _et al._ (2010)
that have already provided important insight into three-dimensional imaging of
the nucleon, notably in discerning the role of orbital angular momentum.

The realization that $x$-dependent distributions including the one-dimensional
parton distribution functions and the quark distribution amplitudes, and the
three-dimensional GPDs could be computed from the matrix elements of operators
at Euclidean separations, with its genesis in Large-Momentum Effective Theory
(LaMET) Ji (2013), or quasi-PDF approach, has spurred a renewal in the first-
principles calculation of hadronic and nuclear structure. For the isovector
distributions, the basic matrix elements are those of spatially separated
quark and anti-quark fields, joined by a Wilson line so as to ensure gauge
invariance; an alternative approach to relating the resulting lattice matrix
elements to the familiar PDFs is the pseudo-PDF framework Radyushkin (2017a).
While both the quasi- and pseudo-PDFs methods share the same matrix elements,
the former matches the lattice data to the light-cone PDFs using a large
momentum expansion, while the latter is based on a short distance expansion. A
further framework that encompasses both the quasi-PDF and pseudo-PDF
approaches is that of the so-called “Good Lattice Cross Sections” method that
admits spatially separated gauge-invariant operators thereby simplifying the
lattice renormalization at the expense of computational cost Ma and Qiu
(2018). Characteristic of any of these approaches is the need to attain high
spatial momentum on the lattice in order to obtain a controlled description of
the $x$-dependent PDF. For the most easily accessible isovector nucleon PDFs,
there are now several calculations at the physical light- and strange-quark
masses. Recent reviews can be found in Refs. Lin  _et al._ (2018a); Cichy and
Constantinou (2019); Detmold  _et al._ (2019); Ji  _et al._ (2021c); Lin
(2020); Constantinou (2021); Constantinou  _et al._ (2021); Cichy (2021);
Constantinou  _et al._ (2022).

Each of the approaches introduced above admits the calculation of the GPDs,
and both the incoming and outgoing hadrons now have to be boosted to high but
distinct spatial momenta to introduce a non-zero momentum transfer $-t$.

####  VI.1.1 Parton distribution functions

The direct calculation of distribution functions is not possible in lattice
QCD as the latter is formulated with a Euclidean metric, while the former have
a light-cone nature. The last decade has been instrumental in attaining the
$x$-dependence of PDFs through a number of approaches, such as the hadronic
tensor Liu and Dong (1994), auxiliary quark field Detmold and Lin (2006);
Braun and Mueller (2008), the quasi-PDFs Ji (2013), pseudo-PDFs Radyushkin
(2017b), current-current correlators Ma and Qiu (2018), and with an OPE
Chambers  _et al._ (2017). The most intensively-studied methods are the quasi-
and pseudo-PDFs, which rely on calculation of matrix elements of non-local
operators that are coupled to hadronic states that carry non-zero momentum.
The non-local operators contain a straight Wilson line with a varying length
in the same spatial direction as the momentum boost. Naturally, the
corresponding matrix elements are defined in coordinate space, and can be
transformed to the desired momentum space, $x$, with a Fourier transform. A
factorization process relates the quasi and pseudo distributions to the light-
cone PDFs, with the matching kernel calculated in perturbation theory. Both
methods have been used for lattice calculations using ensembles of gauge
configurations at physical quark masses Lin  _et al._ (2018b); Alexandrou  _et
al._ (2018a); Chen  _et al._ (2018); Alexandrou  _et al._ (2018b); Liu  _et
al._ (2018); Lin  _et al._ (2018c); Joó  _et al._ (2020); Bhat  _et al._
(2021); Lin  _et al._ (2020). Such studies correspond to different lattice
discretizations (actions) and parameters and a comparison may reveal
systematic effects related to the employed methodology, discretization and
volume effects.

Figure 31: Upper left: A selection of lattice-QCD results on the unpolarized
PDF using the quasi-PDFs method Alexandrou  _et al._ (2019a) (red band) and
pseudo-ITDs from Ref. Joó  _et al._ (2020) (gray band) and Ref. Bhat  _et al._
(2021) (blue band). A comparison of unpolarized isovector nucleon PDFs from
lattice QCD (upper right), helicity (lower left) and transversity (lower
right) at or near the physical pion mass Joó  _et al._ (2020); Chen  _et al._
(2018); Lin  _et al._ (2018c); Liu  _et al._ (2018); Alexandrou  _et al._
(2018a, b, 2019a); Bhat  _et al._ (2021) with global fits. Plots taken from
Ref. Constantinou  _et al._ (2022). All results are given in the
$\overline{\text{MS}}$ scheme at a renormalization scale of 2 GeV.

In Fig. 31 we show results for the unpolarized isovector valence PDF for the
proton. The results indicated by HadStruc’20 Joó  _et al._ (2020) and ETMC ’20
Bhat  _et al._ (2021) have been obtained using the pseudo-PDFs method, while
ETMC’18 Alexandrou  _et al._ (2018a) uses the quasi-PDFs approach. The results
are very encouraging, exhibiting agreement for a wide range of values for $x$.
The small tension at large $x$ is due to systematic effects such as higher-
twist contamination and the ill-defined inverse problem in the reconstruction
of the $x$ dependence of the PDFs. In fact, Refs. Alexandrou  _et al._
(2018a); Bhat  _et al._ (2021) analyze the same raw data, and they differ in
the analysis (quasi-PDFs versus pseudo-PDFs). This corroborates that the
large-$x$ region has contamination from the aforementioned systematic effects.
A similar tension is also present in the comparison of the lattice data, e.g,
of Ref. Joó  _et al._ (2020) with the global analyses of experimental data
sets shown in the right panel of Fig. 31. When predicting spin-dependent PDFs,
lattice calculations may already provide comparable predictions to
phenomenological global analyses. The lower panel of Fig. 31 summarizes the
lattice predictions for helicity and transversity nucleon isovector PDFs at
physical pion mass Alexandrou  _et al._ (2018a); Lin  _et al._ (2018c);
Alexandrou  _et al._ (2018b); Liu  _et al._ (2018). The helicity lattice
results are compared to two phenomenological fits, NNPDFpol1.1 Nocera  _et
al._ (2014) and JAM17 Ethier  _et al._ (2017), exhibiting nice agreement. The
lattice results for the transversity PDFs have better nominal precision than
the global analyses by PV18 and LMPSS17 Lin  _et al._ (2018d). The success in
extracting the $x$ dependence of PDFs is a significant achievement for lattice
QCD, and has the potential to help constrain PDFs in kinematic regions where
experimental data are not available. The synergy of lattice QCD results and
global analysis is currently under study and some results can be found in
Refs. Cichy  _et al._ (2019); Bringewatt  _et al._ (2021); Del Debbio  _et
al._ (2021).

####  VI.1.2 Generalized parton distributions

Information on GPDs from lattice QCD is mostly extracted from their Mellin
moments, that is the form factors (FFs) and generalized form factors (GFFs).
This line of research has been very successful within lattice QCD, and several
results for the form factors using ensembles with physical quark masses
appeared in the last five years. Furthermore, the flavor decomposition for
both the vector and axial form has been performed, giving the individual up,
down, strange and charm contributions to these quantities Sufian  _et al._
(2017a, b); Alexandrou  _et al._ (2018c, 2019b, 2020b, 2021a). A summary of
state-of-the-art calculations can be found in Ref. Constantinou  _et al._
(2021). In the left panel of Fig. 2 we show results on the axial form factor
at physical quark masses from various lattice groups employing different
lattice discretization and analysis methods. Its forward limit is the axial
charge, $g_{A}\equiv G_{A}(0)$, which is a benchmark quantity for lattice QCD,
and is related to the intrinsic spin carried by the quarks in the proton. As
can be seen, the results are in very good agreement, despite the fact that not
all sources of systematic uncertainties have been fully quantified. The level
of agreement indicates that remaining systematic effects are small. Further,
$g_{A}$ is found to be in agreement with the world average of experimental
data Märkisch  _et al._ (2019). This is a breakthrough for lattice QCD
calculations, as they demonstrate that agreement with experiment is achieved
once systematic uncertainties are eliminated.

Figure 32: Summary of lattice calculations of $G_{A}(-t)$ (left) and
$A_{20}(-t)$ (right) using ensembles at or near physical quark masses. The
label of $G_{A}$ results correspond to: ETMC ’20 Alexandrou  _et al._ (2021b),
RQCD ’20 Bali  _et al._ (2020), PNDME ’19 Jang  _et al._ (2020), PACS ’18
Shintani  _et al._ (2019), RQCD ’18 Bali  _et al._ (2019a), ETMC ’17
Alexandrou  _et al._ (2017b), LHPC ’17 Hasan  _et al._ (2018) and MSULat’21
Lin (2022) . The corresponding results for $A_{20}$ are: ETMC ’19 Alexandrou
_et al._ (2020c)222Larger-volume results are plotted for the ETMC’19 2f
calculation., RQCD ’18 Bali  _et al._ (2019b), and MSULat’20 Lin (2021).

More recently, lattice results on the GFFs associated with the sub-leading
Mellin moments of GPDs (one-derivative operators) became available at the
physical pion mass. In the right panel of Fig. 2 we show results on $A_{20}$,
which appears in the decomposition of the energy momentum tensor. Its forward
limit is the quark momentum fraction, $\langle x\rangle\equiv A_{20}$, which
enters the spin decomposition Ji (1997a). Extracting GFFs is more challenging
for a number of reasons. First, the introduction of covariant derivatives
increases the gauge noise, as well as the uncertainties due to cutoff effects.
Second, in general the number of GFFs increases, requiring independent matrix
elements to disentangle the GFFs. Third, beyond the NNNLO Mellin moments,
there is unavoidable mixing under renormalization. The introduction of matrix
elements with greater than three covariant derivatives introduces power-
divergent mixing with matrix elements with few derivatives, thereby precluding
the calculation of the higher Mellin moments. Consequently, there are
limitations in mapping the three-dimensional structure of the nucleon from the
FFs and GFFs.

Methods such as large momentum factorization (quasi-distributions) and short
distance factorization (pseudo-distributions) are very promising in extracting
the $x$-dependence of GPDs Ji  _et al._ (2015a); Liu  _et al._ (2019b);
Radyushkin (2019); Hou  _et al._ (2022) avoiding the challenges associated
with renormalization that are present in the calculation of GFFs mentioned
above. However, the calculations are very taxing because, unlike FFs and GFFs,
GPDs are frame dependent objects and are defined in a symmetric (Breit) frame.
This increases significantly the computational cost, as a separate calculation
is needed for each value of $t$. The $x$-dependence of nucleon GPDs has
already been explored, in the Breit frame, for the unpolarized ($H,\,E$),
helicity ($\widetilde{H},\,\widetilde{E}$) and transversity
($H_{T},\,E_{T},\,\widetilde{H}_{T},\,\widetilde{E}_{T}$) GPDs Alexandrou  _et
al._ (2020d, 2021c). Such calculations are very timely, since the EIC will
measure the DVCS process with polarized electrons and longitudinal and
transverse polarized protons to extract the CFFs of $H,\,E$ and
$\widetilde{H}$. It should be noted that, to date, lattice calculations of
GPDs are exploratory and are available for only a few values of $t$ for zero
and nonzero skewness, $\xi$. Nevertheless, lattice results are useful for a
qualitative understanding of GPDs. For instance, one can find characteristics
for the $t$ dependence for each operator under study. For instance, the
lattice results of Fig. 33 indicate that the decay of the GPD with $t$ is
fastest in $H$, followed by $H_{T}$, and then $\widetilde{H}$. Also, one can
compare the hierarchy of GPDs at each value of $t$. On this aspect, it is
found that at $t=0$, $f_{1}\equiv H(t=0)$ is dominant, followed by
$h_{1}\equiv H_{T}(t=0)$ and $g_{1}\equiv\widetilde{H}(t=0)$. As $-t$
increases, $H$ remains dominant, while the hierarchy of $H_{T}$, and then
$\widetilde{H}$ interchanges. Finally, lattice results can be used to check
sum rules. For more details we refer the reader to Refs. Bhattacharya  _et
al._ (2020); Alexandrou  _et al._ (2021c). We emphasize that lattice
calculations on GPDs are at the proof-of-concept stage, but results are
promising. Once the lattice data can access a wide range of $t$, their
$t$-dependence can be parameterized. This is very useful because the
parameterizations can be used to extract the GPDs in the impact-parameter
space as done in Refs. Lin (2021, 2022) at physical pion mass. The green bands
in Fig. 2 show the moments of lattice $x$-dependent GPD results at zero
skewness; they are in nice agreement with the traditional local-operator
methods, which shows there will be a promising future for lattice QCD
contributions in GPD tomography. Figure 34 shows the first LQCD results of
impact-parameter–dependent 2D distributions at $x=0.3$, 0.5 and 0.7 Lin
(2021). Similar tomography results for helicity GPD,
$\tilde{H}(x,\xi=0,Q^{2})$ can be found in Ref. Lin (2022).

The progress in the field of $x$-dependent GPDs from lattice QCD is being also
extended to twist-3 GPDs Bhattacharya  _et al._ (2021b). We anticipate that,
in the near future, lattice results will be incorporated in phenomenological
analysis of GPDs at both the twist-2 and twist-3 level. Lattice-computed
twist-3 GPDs can have advantages with regards to extracting twist-2 GPDs at
kinematics where twist-3 contributions aren’t negligible. In fact, this may
even be a required step before one attempts to extract twist-2 GPDs from DVEP
data.

Figure 33: The non-polarized $H$ and $E$, helicity $\widetilde{H}$ and
transversity $H_{T}$ GPDs at $\\{t,|\xi|\\}=\\{0,0\\}$, $\\{-0.69\text{
GeV}^{2},0\\}$, $\\{-1.02\text{ GeV}^{2},1/3\\}$ extracted from the 260-MeV
pion mass lattice calculations of Ref. Alexandrou  _et al._ (2020d, 2021c).

Figure 34:  (left) Nucleon tomography: three-dimensional impact
parameter–dependent parton distribution as a function of $x$ and $b$ using
lattice $H$ at physical pion mass. (right) Two-dimensional impact-
parameter–dependent isovector nucleon GPDs for $x=0.3$, 0.5 and 0.7 from the
lattice at physical pion mass. Source: Ref. Lin (2021).

####  VI.1.3 Transverse momentum dependent distributions

In contrast to GPDs, TMDs describe the three-dimensional structure in terms of
the longitudinal momentum-fraction $x$, and the transverse momentum of the
partons. One of the additional challenges that arise in TMD calculations is
the presence of the rapidity divergences that need an additional regulator.
Such divergences can be factorized into the so-called soft function, which can
be separated into a rapidity-independent and a rapidity-dependent part. The
latter defines the Collins-Soper (CS) kernel, which depicts the rapidity
evolution. One of the challenges is that the soft function is non-perturbative
for small transverse momenta.

The TMDs involve the matrix elements of staple-like Wilson lines that extend
along the light cone, imposing analogous restrictions on their calculation
within lattice QCD as encountered for the case of PDFs and GPDs described
above. The first efforts at overcoming these restrictions employed space-like-
separated staples that approached the light-cone as the length of the staple
increased Musch  _et al._ (2011), in particular focusing on the time-odd Boer-
Mulders and Sivers functions Musch  _et al._ (2012); Yoon  _et al._ (2017) and
their relation to the corresponding processes in Drell-Yan and SIDIS,
including calculations for the pion Engelhardt  _et al._ (2016).

More recently, there has been extensive work on exploring TMDs within the
quasi-PDF approach Ji  _et al._ (2015b, 2019); Ebert  _et al._ (2019), as well
as the soft function Ji  _et al._ (2020a, b). The Collins-Soper kernel has
been studied by a few collaborations Shanahan  _et al._ (2020); Zhang  _et
al._ (2020); Schlemmer  _et al._ (2021); Li  _et al._ (2021b); Shanahan  _et
al._ (2021) and a comparison is shown in Fig. 35. Presently, such a comparison
is qualitative, as systematic uncertainties are not fully quantified.
Nevertheless, the agreement is very good and encouraging.

Figure 35: The Collins-Soper kernel as a function of $b_{T}$ as extracted from
various lattice QCD calculations. We show results from SWZ Shanahan  _et al._
(2020, 2021), LPC Zhang  _et al._ (2020), Regensburg/NMSU Schlemmer  _et al._
(2021), and ETMC/PKU Li  _et al._ (2021b). Open and filled symbols of the same
shape and color correspond to results from the same lattice group. Source:
Ref. Shanahan  _et al._ (2021).

####  VI.1.4 Gluon and flavor-singlet structure

The calculation of the flavor-singlet structure of hadrons is considerably
more challenging than those for the flavor-non-singlet quantities that have
been the focus of the most precise studies. The challenges are primarily
related to the degrading signal-to-noise ratio that impacts calculations both
of the gluon distributions, and of the flavor-singlet quark distributions with
which they mix. Recently, the first calculations of the unpolarized
$x$-dependent gluon distributions in the nucleon have been performed using
quasi-PDF Fan  _et al._ (2018) and pseudo-PDF Fan  _et al._ (2021); Khan  _et
al._ (2021); Fan and Lin (2021) methods, as well as the first lattice gluon
helicity study Egerer  _et al._ (2022). Within the present statistical
precision and through a qualitative comparison with global analyses of the
gluon helicity distribution, the lattice calculation hinted at a positive
gluon polarization contribution to the nucleon spin budget.

Figure 36: Left: lattice results on the unpolarized nucleon gluon PDF using a
two-parameter parametrization $xg(x)=Nx^{\alpha}(1-x)^{\beta}$ by MSULat’20
Fan  _et al._ (2021) and HadStruc’21 Khan  _et al._ (2021) Also shown are the
unpolarized gluon PDFs extracted from global fits to experimental data: CT18
Hou  _et al._ (2021), NNPDF3.1 Ball  _et al._ (2017), and JAM20 Moffat  _et
al._ (2021). Right: the gluon nucleon GFF in a lattice calculation
corresponding to $M_{\pi}=450(5)~{}{\rm MeV}$; the bands show a multipole fit
with $n=3$ (green), and a model-independent $z$ expansion (blue). Source: Ref.
Pefkou  _et al._ (2021).

A comparison of the calculation with phenomenological parametrizations is
shown as the left-hand panel in Fig. 36. While this calculation is at
unphysically large pion masses, with limited understanding of the systematic
uncertainties, it demonstrates the potential of lattice QCD to complement and
augment insights into hadron structure from experiment, notably at large $x$.

The calculation of the gluon contributions to three-dimensional structure of
hadrons proceeds as in the case of that of the valence quarks described above.
In particular, the gluonic contribution to the GFF has been computed Shanahan
and Detmold (2019a, b); Pefkou  _et al._ (2021) thereby enabling, when
combined with the corresponding quark contributions, the pressure and shear
forces within a nucleon to be computed, shown as the right-hand panel of Fig.
36.

###  VI.2 LQCD and Spectroscopy

The ability to study multi-hadron states and resonances from lattice QCD
calculations was transformed by the realization that, for the case of two-body
elastic scattering, infinite-volume, momentum-dependent phase shifts could be
related to energy shifts at finite volume on a Euclidean lattice Luscher
(1986a, b, 1991). The formalism for elastic scattering has now been extended
to coupled-channel scattering, and to multi-hadron final states facilitating a
range of calculations that impact our understanding of the spectroscopy of
QCD. Notably, there are now calculations of coupled-channel scattering
describing the nature of the isoscalar $a_{0},f_{0}~{}{\rm and}~{}f_{2}$
resonances Briceno  _et al._ (2018), and the first calculation of the decays
of the exotic $1^{-+}$ meson Woss  _et al._ (2021).

Figure 37: The upper panel shows the $\pi^{+}\gamma\rightarrow\pi^{+}\pi^{0}$
cross section as a function of the $\pi\pi$ center-of-mass energy in a
calculation with a pion mass $m_{\pi}\simeq 400~{}{\rm MeV}$. The lower panel
shows the $l=1$ elastic $\pi\pi$ scattering cross section, with the $\rho$
resonance visible in both cases. Source: Ref. Briceño  _et al._ (2016).

Beyond the challenge of computing the spectrum of resonances and their decays,
an important development has been that of a formalism for the photo- and
electro-production of two-hadron final states, an example of the so-called
$1+{\cal J}\rightarrow 2$ processes Briceño  _et al._ (2015); Briceño and
Hansen (2015). The formalism has been applied to the case of
$\pi^{+}\gamma\longrightarrow\pi^{+}\pi^{0}$, shown in Figure 37. Recently,
this has been extended to the case of coupled-channel, multi-hadron final
states Briceño  _et al._ (2021a) thereby providing an essential framework
underpinning the spectroscopy opportunities through photoproduction at the
EIC.

The calculation of the spectrum of the exotic charmonium and bottomonium
states anticipated at the EIC poses several additional challenges beyond those
encountered in the light-quark sector. Firstly, a precise understanding of
light- and strange-quark spectroscopy is a precursor to precision calculations
in the heavy-quark sector since the $c\bar{c}$ can mix with such states in
many of the most interesting channels. Secondly, with increasing mass of the
quark constituents, the splitting between the different energies on the
lattice is compressed, with many $J^{P}$ states at similar energies requiring
additional constraints to identify the states from the lattice data. Finally,
there are the many open channels that must be included. The work so far is
largely exploratory Chen  _et al._ (2019); Prelovsek  _et al._ (2021), with
the inclusion of only a limited number of coupled channels. However,
controlled calculations of many of the exotic states anticipated at the EIC
are now computationally feasible, with studies both of the $\chi_{c1}(3872)$
and the $X(6900)$ most easily attainable.

###  VI.3 Outlook

Many of the ”no-go” theorems that until recently have imposed limitations on
the range of quantities accessible to first-principles calculation in lattice
QCD have now been circumvented through a progression of theoretical advances,
with demonstrations of the ability of lattice QCD calculations to add to our
understanding of the internal structure and spectroscopy of hadrons. The
advent of the era of exascale computing will enable the precision calculations
needed to exploit the opportunities afforded by the EIC Detmold  _et al._
(2019); Joó  _et al._ (2019). Notably, in addition to the emerging precision
computations of the isovector quantities, such calculations will be extended
to the isoscalar sector. Precise computations within lattice QCD of the three-
dimensional measures of hadron structure, combined with the two-dimensional
Generalized Form Factors accessible through exclusive processes at the EIC,
will constrain the model dependence in global analysis of experimental data,
and will facilitate a more precise three-dimensional imaging of hadrons that
either experiment or first-principles calculation can achieve alone.

Despite these advances, there remain physical processes that elude current
lattice QCD calculations, notably the direct calculation of real-time
scattering cross sections, fragmentation functions, and nuclear response
functions. The rapid advance of Quantum Information Science, and its role as a
high-priority research area, will play an increasingly important role in
addressing many of these key problems, recognised in the report of the NSAC
subcommmittee NSA (2019). Thus far, the investigation of quantum field theory
on quantum computers has been restricted to far simpler systems than that of
QCD, but the role of QIS both in advancing lattice gauge theory is reviewed in
ref. Bañuls  _et al._ (2020). Further, strategies for exploiting quantum
computing to directly address processes relevant to the EIC, such as Compton
Scattering, are now being formulated Briceño  _et al._ (2021b).

##  VII Science of far forward particle detection

###  VII.1 Far-forward detection overview

In contrast to colliders that are mainly built to study particles produced at
central rapidities, much of the EIC physics critically relies on excellent
detection of the target and target fragments moving along, and often within,
the outgoing ion beam. Consequently, EIC detectors are from the outset
designed with an elaborate far-forward detection system that is closely
integrated with the interaction region of the accelerator. The forward
detection has several stages: the endcap of the central detector, trackers
within a large-bore dipole magnet in front of the accelerator quadrupole
(quad) magnets, two sets of Roman pots (one for charged particles at lower
rigidities, so-called “off-momentum detectors”; the other for tagging protons
or light ions near the beam momentum) after a larger dipole behind the quads
as seen in Fig. 38 which shows the layout of IP6 during the time of the Yellow
Report, which is largely unchanged. Additionally, a zero-degree calorimeter is
employed for tagging neutrons and photons at very small ($<$5 mrad) polar
angles.

This arrangement allows for high-$p_{T}$ cutoffs to be determined by the
magnet apertures, such as is the case for the neutron/photon cone going toward
the zero-degree calorimeter (which must traverse the full hadron lattice), and
for charged particles and photons being tagged in the first, large-bore dipole
magnet after the IP, which contains a detector for far-forward particles at
polar angles roughly between 5.5 and 20 mrad. The bore of the first dipole
(called B0pf in IP6) has a radius of 20 cm (while the pre-conceptual design
for IP8 has an equivalent dipole magnet with a slightly larger radius), which
in principle allows for larger acceptance than 20 mrad, but support structure
and services for the detectors will limit how much of the bore can be filled
with active detector material. As designs progress, it may be possible to
achieve a larger acceptance in the dipole spectrometer at both IP6 and IP8.

On the other hand, for lower-energy proton beams, unavoidable inefficiencies
will occur in the transition regions. There is a low-$p_{T}$ cutoff due to the
beam itself, which is most severe for the detection of recoil protons from
mid- to high-energy beams (which provide the highest luminosity), for light
ions at all energies, and for heavy ion fragments with A/Z close to that of
the original beam. For ions, where the $p_{T}$ per nucleon is usually small,
acceptance at very low-$p_{T}$ is extremely important. With a traditional IR
layout, low-$p_{T}$ acceptance can be improved by reducing the angular spread
of the beam via reduced beam focusing. However, this has the drawback that it
also reduces luminosity and still does not make it possible to reach
$p_{T}$=0.

Figure 38: Layout of the IP6 Far-Forward region generated with the EICROOT
simulation package EIC (6 16) including the dipole magnets (rectangular
boxes), quadrupole magnets (cylinders), and the four detector subsystems
currently proposed to cover the geometric acceptance.

The kinematics of the EIC are uniquely suited to a more sophisticated forward
detection concept than previous colliders. In DIS, the typical longitudinal
momentum loss $dp/p\sim x$. At the same time, the intrinsic momentum spread of
the particles in the beam is a few $\times 10^{-4}$. With a 10$\sigma$ margin,
all recoil protons with $x>0.01$ will thus separate out from the beam even at
$p_{T}$=0, and at much lower $x$ for non-zero $p_{T}$. Since this method only
relies on a fractional longitudinal momentum loss (magnetic rigidity), it is
independent of the beam energy. For heavy ions, which typically only
experience small changes in momentum, rigidity ($\sim A/Z$) can change through
emission of nucleons. In particular, emission of a single neutron from an
$A\sim 100$ nucleus corresponds to a change in rigidity at the 1% level, which
in principle also allows the EIC to detect most nuclear fragments.

To take full advantage of the EIC kinematics, the forward detection requires
two elements: dispersion and focusing. The former is generated by dipole
magnets and translates a momentum (rigidity) change into a transverse position
offset: $dr=Ddp/p$ (e.g., with $D=40cm$, the transverse displacement for a
particle with $dp/p=0.01$ and $p_{T}=0$ will be 4 mm). This value has to be
compared with the (10$\sigma$) beam size at the detection point (Roman pot).
Without focusing, this is typically a few cm, but with a secondary focus it
can be reduced to 2-3 mm (depending on the beam momentum spread). The beam
size on the Roman pot does in principle not depend on the focusing of the beam
at the collision point ($\beta^{*}$), but in a practical implementation the
same magnets are used to generate both the primary and secondary focus.
However, in contrast to the unfocused case, this means that with a secondary
focus the best low-$p_{T}$ acceptance is achieved at the highest luminosity. A
secondary focus could in principle be used at either IP6 or IP8 of the EIC.
However, while the current IP6 layout has some dispersion (17 cm), it does not
have a secondary focus. In contrast, IP8 is designed for a much larger
dispersion and incorporates a secondary focus – making it complementary to IP6
and opening up unique physics capabilities, as can be seen in Fig. 39.

Figure 39: Two-dimensional plots of proton acceptance in transverse momentum,
$p_{T}$ the nucleon momentum fraction. The acceptance is shown for three
configurations: accepted protons in the IP6 Roman pots with the CDR high
divergence optics (left), for accepted protons in the IP6 Roman pots with the
CDR high acceptance optics (middle), and for accepted protons in the IP8 Roman
pots at the secondary focus with the pre-conceptual optics configuration
(right). All samples were generated for 18 GeV on 275 GeV protons with an
$x_{L}>0.8$ and with $0<\theta<5$ mrad; the cutoff at the top of the plots is
due to the event generation region, while the acceptance in the bottom right
varies with different configurations.

###  VII.2 Detection of recoil baryons and light ions

As discussed in sections 7.2.2 and 7.3.8 of the Yellow Report Abdul Khalek
_et al._ (2021) and earlier in this paper, exclusive reactions on the proton
and light nuclei form an essential part of the EIC physics program. The wide
kinematic reach of the EIC makes it ideal for probing different parts of the
nuclear wave function, revealing how the internal landscape of nucleons and
nuclei changes with $x$. Measurements of exclusive processes require high
luminosity, a range of collision energies, and excellent far-forward
detection. Key issues are detector acceptance for the recoil proton or light
ion and optimized reconstruction resolution of the momentum transfer, $t$.

##### Proton detection:

Detecting the recoiling nucleons is important to cleanly establish the
exclusivity of the reaction. It also makes it possible to reconstruct $t$
directly from the proton. Since the EIC reaches its highest luminosity with
the most asymmetric beam energies (i.e., 5-10 GeV electrons colliding with
hadrons at maximum energy), it is essential that the far-forward detection
works optimally for high-energy protons. Here, the greatest challenge is to
detect low-$p_{T}$ protons which stay within the beam envelope. This
capability can be improved by using a secondary focus, which essentially
provides full acceptance for $x>10^{-2}$, and significantly improves the
low-$p_{T}$ acceptance for lower $x$. For lower proton beam energies, a
secondary focus is still useful, although less crucial. However, at lower
energies, high-$p_{T}$ protons will start experiencing losses in the apertures
of the accelerator quadrupole magnets, leading to a reduced acceptance for
detectors downstream of these magnets. Embedding a tracking detector, such as
is envisioned with the B0 tracker in IP6, provides increased coverage of
high-$p_{T}$ protons at the lower beam energies. This issue can be alleviated
by using magnet technologies that allow for higher peak fields, which makes it
possible to increase the apertures, but there are other technical constraints
that could make this challenging, especially at IP6, and more study is needed
to determine what level of improvement is possible. In conjunction with a
secondary focus, this would further enhance the capabilities of the EIC to do
transverse proton imaging.

##### Determination of transverse momentum in exclusive reactions:

Another important consideration for exclusive reactions is reconstruction of
$t$. In principle it can be done either by using the recoiling system detected
in the far-forward detectors, or from the scattered electron and produced
particle (charged meson or DVCS photon) detected in the central detector.
There are advantages to both methods. For example, the former method is very
straightforward, but requires a good understanding of the beam effects (e.g.
angular divergence). Ideally one would want to be able to apply both, but this
requires that the central detector can provide a sufficiently good
$p_{T}$-resolution. This is a challenge for a tracker, but even more so for
the EM calorimetry if one wants to be able to determine $t$ ($\Delta_{\perp}$)
from the DVCS photon (or the photons from $\pi^{0}$ production). However,
while such a dual capability is useful for protons, it becomes essential for
ions, where the ability to determine $t$ from the ion is more limited and
vanishes entirely when the ion is not detected (high A and low $p_{T}$). Being
able to determine $t$ from the DVCS photon would thus greatly enhance the
ability to do transverse imaging of ions.

##### Light ion detection:

Coherent exclusive scattering on light ions differs from protons in two ways.
First, the available ion beam energies are restricted a range between 100
GeV/A and $275\times Z/A$ GeV/A, as well as a discrete energy at 41 GeV/A.
Second, scattered ions travel much closer to the beam, making low-$p_{T}$
acceptance very challenging (and conversely, the high-$p_{T}$ acceptance much
less so, even for the high-$t$ tails). This is the combined result of two
effects: cross sections for ions peak at lower $t$, and a given $t$
corresponds to a lower $p_{T}$ per nucleon. The former means that in contrast
to the proton, clean imaging of light ions requires an acceptance down to
$p_{T}\sim 0$, and the latter that implementing such an acceptance is
particularly difficult. A secondary focus is thus essential for high-quality
measurements of coherent scattering on light ions. However, if the central
detector has the ability to reconstruct the $p_{T}$ from the produced photon
or meson as discussed above, a secondary focus would allow for a hybrid method
where ions with higher $p_{T}$ are detected (the incoherent backgrounds become
more challenging as one moves towards the first diffraction minimum), while
the low-$p_{T}$ part is reconstructed by vetoing the breakup (which is
generally easier to do with light than heavy ions). A hybrid measurement would
not be as clean as one where all recoiling ions are detected, but it would
make it possible to reach lower $x$ and higher A, extending the discovery
potential of the EIC.

###  VII.3 Spectator detection

Detection of nuclear breakup is essential for a broad range of EIC physics
topics. From a detection perspective, these broadly fall into two categories:
spectator nucleons and nuclear fragments. In the first case, the spectator
nucleon typically experiences a very small change in momentum, but its
magnetic rigidity (A/Z) is very different from that of the original beam. A
proton spectator will thus initially continue moving with the beam, but will
separate quickly from it after passing the first dipole magnet. The detection
challenge here thus lies primarily in providing adequate magnet apertures. An
key example of spectator proton tagging are measurements of neutron structure
in deuterium and ${}^{3}He$.

In the case of nuclear fragments, they may be detected as a way of vetoing
breakup or part of the direct measurement. The former case was discussed above
(Sec. VII.2 in the context of light and medium nuclei, but coherent processes
on heavy ions are different in that even with a secondary focus, the
high-$p_{T}$ tails cannot be measured directly as the ion always stays inside
the beam envelope. A secondary focus can, however, make it possible to detect
residual ions that have lost a single nucleon (A-1 tagging). Adding such a
capability will significantly improve the efficiency for vetoing the large
incoherent backgrounds, making a reasonably clean measurement possible.

Finally, there are several measurements that rely on detection of the
spectator nucleons, the residual nucleus, or nuclear fragments in the final
state. One example is the case when the struck nucleon and its partner are in
a short-range correlation with a high relative momentum. In this case, the
spectator nucleon will not only have a different A/Z compared with the
original ion, but also a large $p_{T}$. The breakup kinematics can then be
best constrained if the residual A-2 nucleus can be detected, which is
facilitated by a forward spectrometer with a secondary focus. A related topic
is detection of rare isotopes produced in the interaction, which is discussed
in Sec. VII.7. Additional detail, including discussion of the theoretical
framework for several of the tagged measurements can be found in Ref. Strikman
and Weiss (2018); Cosyn and Weiss (2020)

##### Neutron structure through spectator tagging

Light ion beams can be used as an effective free neutron target via spectator
tagging. Deuterium is the simplest system, while ${}^{3}He$ can be polarized
(70%) and thus give access to the neutron spin structure. Spectator tagging
can be applied to any primary measurement ($F_{2}$, DVCS, etc), but a key
common challenge is to account for final-state interactions (FSI). However,
recent studies Jentsch  _et al._ (2021) have shown that free neutron structure
can be extracted by on-shell extrapolation to the non-physical pole, where the
neutron is by definition free and unaffected by FSI. In contrast to the pion,
this approach is much more robust for the heavier nucleon where the
extrapolation takes place over a shorter interval. The extrapolation is done
by fitting the measured $t$ distribution, but focuses on the low-to-modest
values of $t$ part, where the extrapolation has minimal model dependence.

Experimentally, this measurement relies on a high-resolution determination of
the $p_{T}$ distribution, and of having sufficiently large magnet apertures to
tag a spectator proton with low $p_{T}$ Jentsch  _et al._ (2021). As a cross
check, it is also possible to apply the same method to the bound proton by
tagging the neutron from deuterium in the ZDC.

##### Proton and neutron spectators from deuteron beams

Deuteron beams can be used as an effective free neutron target via spectator
tagging, where the undisturbed proton is measured to isolate scattering from
the proton. To isolate nearly on-shell neutrons, the goal is to tag protons
which had low initial momenta (corresponding to low $-t$) in the deuteron rest
frame. Measurements will be made over a range of $t$, so that the
extrapolation to the on-shell neutron can be performed over different ranges
of $t$ to ensure stability of the extrapolation. As noted above, detection of
these protons in the Off-Momentum Detector and Roman Pots is relatively
straightforward and the key issue is minimizing the loss of acceptance in the
apertures of the accelerator magnets.

Similar studies of the proton structure of the proton structure can be
performed with neutron tagging used to isolate scattering from a low-momentum
proton. In this case, the results can be compared to the known proton
structure, and these studies can be used to study the $t$-dependence and test
the extrapolation to the on-shell proton. For the low $t$ values required for
these measurements, the neutrons have $x_{L}$ near unity and small $P_{T}$ and
are detected in the ZDC.

##### Double tagging from 3He breakup

While the deuteron is the most common target used to study unpolarized neutron
structure, polarized 3He serves as the most effective target for measuring
neutron spin structure, as the neutron carries most of the spin in 3He.
Inclusive measurements can provide some information, with the protons acting
mainly as a dilution to the asymmetries associated with scattering from the
polarized neutron. But double tagging of the two spectator protons in 3He can
be used to isolate scattering from the neutron without dilution or corrections
for the proton contributions Friscic  _et al._ (2021). In this case, the goal
is to measure spectators with low momenta in the 3He rest frame which have
momenta close to the beam momentum per nucleon and small $P_{T}$, but lower
mass and therefore roughly 2/3 of the rigidity of the 3He beam. One can also
examine events with one large-momentum proton to identify high-momentum
neutrons in the initial state to look at the spin structure as a function of
initial neutron momentum, which is relevant for understanding the spin EMC
effect.

##### Tagged Pion structure - nucleon spectators from proton beams

Measurement of the $\pi^{+}$ electromagnetic form factor can be accomplished
at the EIC by the detection of the neutron spectator in coincidence with the
scattered electron and $\pi^{+}$, i.e. an exclusive reaction with
$e^{\prime}-\pi^{+}-n$ triple coincidence. The neutron is emitted with 80-98%
of the proton beam momentum, and is detected in the ZDC. The pion form factor
measurement only requires $-t$ measurements up to about 0.4 GeV2, so a
moderate acceptance ZDC is sufficient to catch the events of interest. Very
good ZDC angular resolution is required for two reasons. First, to separate
the small exclusive $\pi^{+}$ cross section from dominant inclusive
backgrounds, a cut may be placed on the detected neutron angle in comparison
to the reconstructed neutron angle (from $e^{\prime}$ and $\pi^{+}$ using
momentum conservation). Second, a $t$ reconstruction resolution better than
$\sim$0.02 GeV2 is necessary for a quality form factor measurement and 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 at the EIC.

##### Tagged Kaon structure - hyperon spectators from proton beam

Measurements of the $K^{+}$ electromagnetic form factor at high $Q^{2}$ via
the Sullivan process would yield valuable information on nonperturbative
DCSB–Higgs-boson interference effects in hard exclusive processes. The
reaction of interest is $e+p\rightarrow e^{\prime}+K^{+}+\Lambda$, where the
$\Lambda$ is emitted with $>70$% of the proton beam momentum. We expect that
lower beam energies are optimal, to ensure a high $\Lambda$ decay fraction, as
non-decayed $\Lambda$ will be impossible to distinguish from neutron hits.

The $\Lambda$ needs to be identified from its decay products to ensure the
clean identification of the exclusive events from inclusive backgrounds, and
to reconstruct $t=(p_{p}-p_{\Lambda})^{2}$ with sufficiently high resolution.