giving the same final state as TCS. Measuring the interference between TCS and
BH allows access to the real part of the Compton Form Factor, and can
therefore place constraints on the determination of GPDs.

####  4.7.1 EpIC Generator Settings and Fun4All Version

The EpIC generator [40], used for this study, is an MC event generator that
uses GPD models from the PARTONS framework [72], plus mFOAM (a general purpose
MC event simulator integrated with ROOT) to generate random events in phase
space. EpIC takes in an input .xml file, within which parameters such as beam
energy, kinematics, and the decay process are defined. Based on the input
information, EpIC then generates the four vectors of all particles as the
output. The EpIC generator is capable of generating pure TCS events, pure BH
events, and events that combine TCS, BH, and the interference term (INT). The
events simulated in this study were the combined set of TCS+BH+INT, extending
on the result using only pure TCS in the YR. The EpIC generator has the
capability to include radiative corrections, however, at the time of this
study these were still in development and thus have not been included.

EpIC was used to simulate TCS events at beam energy settings of 5 $\times$ 41
and 18 $\times$ 275, to study the anticipated two extremes in acceptances. The
electron beam helicity was set to negative in the event generator, and a total
of 1M events were generated for each energy setting.

The generated kinematics in EpIC was set to either match or slightly extend
upon the original kinematics studied in the YR, and are detailed below.

  * 1.

$0<-t<1$ GeV2 to capture the physics region of interest.

  * 2.

$2<Q^{\prime 2}<20$ GeV2 to ensure a hard scale for the scattering and to
minimize background from the low resonance region. $Q^{\prime 2}$ represents
the virtuality of the produced virtual photon (see Sec. 4.7.2 for the full
definition).

  * 3.

$0<\phi<2\pi$ to obtain a full lab frame azimuthal ($\phi$) angular coverage.

  * 4.

$0<\phi_{S}<2\pi$, (where $\phi_{S}$ represents the angle between the leptonic
plane, see Fig. 42, and the transverse component of the polarization of the
target nucleon), to obtain a full $\phi_{S}$ angular coverage.

  * 5.

$\frac{\pi}{6}<\theta<\frac{5\pi}{6}$, slightly widened from the range used in
the YR study (please note that at this stage in the YR studies, BH
singularities became apparent at extremes of theta. These have since been
rectified via recent updates to EpIC, and the restricted range from the YR is
thus widened here).

  * 6.

$0<Q^{2}<0.15$ GeV2 to select a quasi-real photon.

The Fun4All [19] simulation software prop 7.1 was used for the TCS studies.
This more recent version was used due to an improved scattered proton
acceptance compared to older Fun4All versions, as a result of the
implementation of the high acceptance setting (detailed in Sec. 4.5.2).

####  4.7.2 Event Selection, Reconstruction and Analysis

Event selection of the final state particles centers around the scattered
proton ($p^{\prime}$), the decay electron ($e^{-}$), and the decay positron
($e^{+}$).

For the $e^{-}e^{+}$ pair, the information from hits registered in the Fun4All
EEMC, FEMC, and BECAL was compared with momenta from the truth container and
separated by PID, taken from calorimeter cluster information. The virtual
photon $\gamma^{*}$ produced by the interaction was then calculated using this
decay $e^{-}e^{+}$ pair, via summation of four momenta
($\gamma^{*}=k(e^{+})+k^{\prime}(e^{-})$).

The energy determination for these particles could be improved by calibrating
the calorimeters using a similar method as described in Sec. 4.4, however, due
to time constraints the plots in this study have not been corrected for this
calibration. We plan to implement this in the near future and from preliminary
tests we anticipate the change to be relatively minor, around a $\approx$3%
correction. The scattered electron, $e^{\prime}$, in this study is calculated,
as opposed to being detected. This is due to the original findings of the YR
study, which indicated that the $e^{\prime}$ would be difficult to directly
detect, without implementation of a low $Q^{2}$ tagger and that it is instead
better to use the momenta of other final state particles compared to initial
beam momenta to calculate it [5].

In more recent versions of Fun4All, there has been an integration of a low
$Q^{2}$ tagger, which would mean that a missing mass study could be performed
with the scattered proton as the ’missing’ particle, as it is within detector
resolution to calculate this at high energies, however, this has not been
explored in this study.

To reconstruct the scattered proton, $p^{\prime}$, which is very forward-
going, the Roman Pots and B0 detectors were essential. The geometrical
acceptances for the Roman Pots and B0 detectors were handled in exactly the
same way as previously described in Sec. 4.5.2, i.e. the acceptance of the B0
layers was fully modeled in Fun4All directly and cuts to remove the
contribution of the beam pipe in each of the Roman Pots were added to the
analysis of the Fun4All output. The Roman Pots cuts for each simulation were
as given in Table 5. In the analysis of the Fun4All output to mimic detector
resolution effects (since these are not fully modeled in the forward region of
Fun4All yet) a $1\,\%$, smearing was applied to the truth proton. The track
direction of the proton, however, was not smeared in this study. A cut on
successfully detecting the scattered proton and the decay lepton pair in the
final state was also included in the event selection stage of the analysis.

In the analysis stage, several physics quantities of the reaction were
reconstructed, their definitions are below.

  * 1.

$Q^{2}=-q^{2}=-(e^{\prime}-e)^{2}$, where $e$ and $e^{\prime}$ represent the
four momenta of the beam and scattered electron respectively.

  * 2.

$Q^{\prime 2}=-q^{\prime 2}=-(k+k^{\prime})^{2}$, where $k$ and $k^{\prime}$
represent the four momenta of the decay positron and electron respectively.

  * 3.

$\tau=\frac{Q^{\prime 2}}{(s-M_{p}^{2})}$, where $s$ represents the center of
mass energy calculated via $(p+q)^{2}$ and $M_{p}^{2}$ represents the mass of
the proton [73].

  * 4.

$-t=-(p-p^{\prime})^{2}$, where $p$ and $p^{\prime}$ represent the four
momenta of the beam and scattered proton respectively.

See Fig. 42 for visualization of the four momenta. Due to the background
events caused by the J$/\psi$ channel, there should also be a windowed cut
applied to $Q^{\prime}$ around the J$/\psi$ mass, ($\approx 3.1$ GeV), which
would appear as a gap in the $Q^{\prime 2}$ phase space between $\approx 9-12$
GeV2, however, a fuller analysis must be conducted to discern this range
correctly, wherein a $3\sigma$ cut would be taken around the J$/\psi$ mass
peak in a set of J$/\psi$ generated data. This cut has not been included in
this analysis due to time constraints, however, its effect would only be a
slight reduction in statistics, and would not greatly affect the overall shape
of the resulting distributions.

####  4.7.3 Results

Several of the results obtained from the analysis of the Fun4All output are
given in Fig. 45 -45. This includes TCS physics variables, the acceptance of
the ECCE detector with regards to reconstructing these variables, and the
kinematic phase space available for this reaction with the generated settings
and the ECCE detector.

Please note that for any detector acceptance plots, the acceptance is
calculated per bin and defined as the number of reconstructed events from the
Fun4All output divided by the number of events outputted by the EpIC generator
directly.

Figure 43:  5 $\times$ 41 - TCS Differential cross-section versus the momentum
transfer to the struck parton $-t$ reconstructed using the beam and scattered
protons $t=(p-p^{\prime})^{2}$ (left) and detector acceptance for $-t$
reconstructed using the beam and scattered protons (right). Note acceptance is
given as a value where 1 corresponds to 100%

Figure 44: Left: 5 $\times$ 41 acceptance vs pseudorapidity ($\eta$) of the
scattered proton from TCS events. Right: 18 $\times$ 275 acceptance vs
pseudorapidity ($\eta$) of the scattered proton. Note acceptance is given as a
value where 1 corresponds to 100%

Figure 45:  18 $\times$ 275 - TCS Differential cross-section versus the
momentum transfer to the struck parton $-t$ (left) and detector acceptance for
$-t$ (right). Note acceptance is given as a value where 1 corresponds to 100%

####  4.7.4 Discussion and Summary

As outlined in Sec. 8.4.4 of the YR, it is important to reconstruct the
momentum transfer to the struck parton $t$ via $-t=-(p-p^{\prime})^{2}$, i.e.
utilizing the four-momentum information from the target ($p$) and scattered
proton ($p^{\prime}$) in TCS. This method provides a better resolution than
using the reconstructed photon information. The YR also showed that the
$p^{\prime}$ is detected at very low transverse momenta ($p_{T}$), and very
high pseudorapidities ($\eta$), i.e. in the far forward direction. The far
forward nature of the $p^{\prime}$ is supported in the ECCE study by Fig. 45,
which shows the detector acceptance for an $\eta$ range: $4.3<\eta<8.4$ at low
energies, indicative of events captured with an acceptance of around 15-25% in
both the B0 and the RP. For high energies, we see an acceptance across an
$\eta$ range: $6.3<\eta<8.4$, indicative of events captured only in the RP,
not the B0, with an acceptance again of around 15-25%. This result supports
the need for both the B0 detector and Roman Pots in the detection of the
scattered proton, as, similar to the $ep$ DVCS studies (Sec. 4.4), we see a
high count of lower $\eta$ protons at the 5 $\times$ 41 beam energies in the
B0, and for higher energy settings a larger count is shown in the Roman Pots.

Utilizing the information from the Roman Pots, and for lower energies
information from the B0, $t$ was successfully reconstructed across the full
range with an acceptance of around 10-24% for beam energy 5 $\times$ 41 as
shown in Fig. 45. For the 18 $\times$ 275 beam energy setting in Fig. 45, we
show again a full reconstruction of $t$, with an acceptance of around 10-22%.
The cross-section measurement for $t$ was calculated as in Eq. 10, however
with the variables $Q^{2}\rightarrow Q^{\prime 2}$ and $x_{B}\rightarrow\tau$.
The cross-section for beam energy 5$\times$41 was averaged over $2<Q^{\prime
2}<20$ GeV2, $0.003<\tau<0.05$ and $0<\phi<\pi$, determined by a phase space
analysis of generated and reconstructed data, see D. Beam energy 18$\times$275
simulations differed only in that $0<\tau<0.02$.

In summary, many of the main requirements observed for the TCS measurement
outlined by the YR have been confirmed by the ECCE analysis. The forward
acceptance is the main driving factor for the projected cross-section and
statistics of this reaction.

It appeared initially that the detector performed better overall for the 5
$\times$ 41 energy setting than for 18 $\times$ 275, however with the new
upgrade to the simulation software it has been shown that the two are much
more comparable. An important next step would be to study an intermediate
setting of 10 $\times$ 100.

Another step to be taken in future studies is to calculate asymmetries rather
than cross-sections, as in the leading order, the background contribution from
pure BH can be removed. There is also a further background contribution
stemming from measuring the final state $e^{-}e^{+}$ pair, where there is the
potential that these are in fact misidentified pions, which could be dealt
with by studying the $\mu^{-}\mu^{+}$ channel. This may also make for a
simpler analysis procedure, due to issues with separating the decay electron
$e^{-}$ from the scattered electron $e^{\prime}$. A final step would be to
integrate analysis of the performance of the low $Q^{2}$ tagger in detecting
the scattered electron, and performing a missing mass study on the scattered
proton, or the total reaction, to further discern how well each component of
the detector system performs.

###  4.8 XYZ spectroscopy

Spectroscopy of mesons with charmed quarks has provided some of the most
surprising recent results and raised many interesting questions. These new
states are commonly referred to as “XYZ” mesons and have unexpectedly small
widths and masses inconsistent with quark model expectations. Instead many of
these states are characterized by masses very close to two-meson decay
thresholds. As a result, there are many possible means of describing the
dynamics of these structures, for example, tetraquark states, di-meson
molecular states, glueballs, hybrids, or kinematical effects due to thresholds
and re-scattering interactions. For an overview of the subject, see Ref. [74].

In general, most of the new states have only been seen via single production
mechanisms, such as B decays or $e^{+}e^{-}$ annihilation. This makes it
difficult to resolve the dynamics contributing to the states. Photoproduction
experiments offer an alternative production method with the advantage of, in
principle, being able to produce all states within the center-of-mass range
without the same potential for kinematic rescattering effects. This will be
limited by the small production cross-sections for states with heavy quarks.
However, it has been shown that production rates for many of these states are
sufficiently high to be measurable with the EIC [75]. In this section, we show
that the proposed ECCE detector can deliver the event reconstruction required
for investigating this exciting physics program.

There are currently dozens of these potential new charmonium resonances. To
make this study manageable, we limit the states under consideration to three:
$\chi_{c1}(3872)$ or $X(3872)$, $Y(4260)$, and the well-established quark
model state $\psi(2s)$. All of these states have decay branches to
J$/\psi\pi^{+}\pi^{-}$, and so we focus on reconstruction of this final state
with the J$/\psi$ decaying to $e^{+}e^{-}$. This allows us to compare the
expected production of the exotic states to a regular quark-antiquark meson
and check if we can distinguish the invariant mass peaks of these relatively
close states.

As previously mentioned, with photoproduction we should produce many different
mesons of exotic and non-exotic character. For example, the production of
$Z_{c}$ isovector states will be of similar magnitude and decay to
J$/\psi\pi$. We might also imagine production on deuteron producing both
charge states of this manifestly exotic particle.

We may also search for poorly established or yet unknown states by looking
through the many different final states accessible with the EIC, such as
J$/\psi$ \+ vector mesons or kaons.

####  4.8.1 Simulations

To test the performance of the ECCE detector for XYZ production, an event
generator was developed coupling realistic photoproduction amplitudes to low
$Q^{2}$ virtual photons produced by electron scattering. The photoproduction
helicity amplitudes were calculated following the formalism and parameters
given in [75]. The models therein are expected to give an order of magnitude
estimates for meson production cross sections. Details of the generator are
given in C.

For the J/$\psi\pi^{+}\pi^{-}$ final state events, we produced events via
$\psi(2s)$, $\chi_{c1}(3872)$ and $Y$(4260) production. The number of events
generated based on an integrated luminosity of $10$ fb-1 are summarised in
Table 7. In each case the branching ratio of J$/\psi$ to $e^{+}e^{-}$ and of
the meson to J$/\psi\pi^{+}\pi^{-}$ were included. For the latter, the
branching ratios assumed in [75] were used.

Table 7: Generated event yields corresponding to 10 fb-1 for the beam energy settings $E_{e}\times E_{p}$. $ep$ Setting | $\chi_{c1}(3872)$ | Y(4260) | $\psi(2s)$ | Total
---|---|---|---|---
5x41 | 96933 | 9104 | 71070 | 177107
5x100 | 114906 | 22384 | 164942 | 302232
10x100 | 125706 | 37511 | 270920 | 434137
18x275 | 104291 | 86199 | 648881 | 839371

As all six final state particles were charged, we used the reconstructed
tracks given in the Fun4All DSTs SvtxTrackMap. Particle ID was taken from
matched truth values. As we are primarily interested in high production rates,
we included events with very low $Q^{2}$, and hence the electron was usually
scattered below $2\degree$. For these events, we investigated the benefits of
a possible low $Q^{2}$ tagger. Similarly, the recoil proton was usually
incident upon the far forward detector region and so we investigated the
acceptances given by the nominal Roman Pot and B0 detector systems. For the
tracks in the central detector, the main uncertainty is from the $P_{t}$
threshold of the tracks, which can have quite a large effect on the decay
pions we hope to detect.

Here, we focus on the results for the 5$\times$100 beam setting unless
otherwise stated.

####  4.8.2 Far Forward Models

The far forward and far backward detectors were partially implemented in the
simulations. The Roman Pot configuration used in the simulation is based on
the high divergence $e+p$ scattering beam configuration, where its actual
acceptance represents the $10\sigma$ beam boundary. Here, for the far forward
detectors: Roman Pots and B0, a realistic physical coverage was used leading
to reasonable estimates for proton acceptances. However, no realistic
reconstruction was in place, so there are no genuine momentum components for
deducing resolutions of variables requiring proton detection.

The hit distribution on the first layer of Roman Pots, at 26 m, is shown in
Fig. 46. A further cut was applied on the position: $-1<y<1$,
$-88.22<x<-78.22$ to remove the region where large backgrounds from beam
divergence may occur, this cut is effectively around $10\sigma$ of beam
divergence.

To summarize, for events for 5$\times$41, 13% of events hit the first Roman
Pot, and 9% survive the cut; for 5$\times$100 it is 56 and 37%; for
10$\times$100 58 and 39%; and for 18$\times$275 99 and 28%.

Figure 46:  Hit distributions on the first Roman Pot layer for beam settings,
left-to-right 5$\times$100 and 18$\times$275\. The top row plots show full
event distribution; the bottom row plots show the RP acceptance cut applied to
remove possible beam backgrounds/contributions.

Fig. 47 shows the hit distributions for the 4 layers of the B0 detector for
each beam momentum configuration. It is clear the B0 detectors play a far more
important role at the lower CM energies.

Figure 47:  Hit distributions on the four B0 layers for beam settings, top-to-
bottom 5$\times$100 and 18$\times$275; with left-to-right front-to-back.

The estimated far forward detection is also shown in Table 8 as a percentage
of the total number of events for Roman Pots and B0 detectors. Note, we take
the number of B0 hits as the number in the highest occupancy layer.

Table 8: Percentage of protons detected in the far forward detector systems. Setting | 5$\times$41 | 5$\times$100 | 10$\times$100 | 18$\times$275
---|---|---|---|---
Roman Pot | 9 | 37 | 39 | 28
B0 | 66 | 31 | 30 | 1

####  4.8.3 Particle Acceptances

First, we show the event distributions and acceptances for the forward-going
recoiling proton which low $Q^{2}$ $t$-channel exchange production process of
concern here are all in the far-forward detectors. The overall average
acceptance comes to around 63%.

The majority of scattered electrons will also miss the main detectors and
require detection in a low $Q^{2}$ tagger in the far backward region. Around
5% of the higher $Q^{2}$ events do make it into the backward electron arm of
the central detector. Overall, the two systems could detect around 52% of the
electrons, with 45% potentially in the tagger. We observe unphysical
acceptances greater than 1 when we plot in terms of pseudorapidity, this is
presumably due to bin migration effects in this non-linear variable, perhaps
due to beam divergence effects applied by the simulation afterburner.

To investigate the potential for a low $Q^{2}$ tagger to improve these
spectroscopy measurements, we investigated scattered electrons reconstructed
in the nominal tagger (cut $\eta<-6.5$) and the main detector. For the
5$\times$100 setting, the tagger supplies an order of magnitude more events
with complete reconstruction and will be an important addition to the
spectroscopy program particularly for measuring quantum numbers and spin
density matrix elements.

One nice feature of these high-mass meson production processes at the lower CM
energies is that the meson decay products populate the detector relatively
uniformly allowing excellent acceptance for the states of interest. The
$e^{+}e^{-}$ decay products from the J$/\psi$ are particularly well
reconstructed and shown in Fig. 49. With very symmetric responses for both
lepton charge states, the average acceptance is around 95%.

Figure 48:  5$\times$100 generated (red) and reconstructed (blue) J$/\psi$
decay $e^{-}$ distributions of momentum (P), pseudorapidity ($\eta$), and
angles ($\phi$).

Figure 49:  5$\times$100 generated (red) and reconstructed (blue) $\pi^{+}$
distributions of momentum (P), pseudorapidity ($\eta$), and angles ($\phi$).

Kinematically, the detection of the pions is more challenging due to their
lower momentum, with significant numbers below 200 MeV, which is close to the
tracking threshold. The distributions for the $\pi^{+}$ are shown in Fig. 49.
These show that the ECCE detector is capable of detecting pions with high
efficiency above this threshold. Overall acceptance is higher at central
angles, with some fall-off towards the forward detector systems.

####  4.8.4 Particle Resolutions

Good particle resolution is important to separate out background processes.
The proton and electron reconstructed tracks were not available for the far
forward/backward detectors, so we do not consider their effects here.

For the case of the pions, the difference in reconstructed to generated
momenta is shown in Fig. 50 (Top). The widths of these distributions give an
estimate for the resolutions and we find around 0.4% for momentum;
$1.1\degree$ for $\theta$ and $2\degree$ for $\phi$ averaged over all events.

Figure 50:  5x100 $\pi^{+}$ (Top) and $e^{-}$ (Bottom), resolutions, $\Delta
P$, $\Delta\theta$ and $\Delta\phi$ ($\degree$), calculated as the difference
between reconstructed and truth values.

For the J$/\psi$ decay leptons, the difference in reconstructed to generated
momenta is shown in Fig. 50 (Bottom). Here, the estimated resolution is closer
to 1% for momentum with a slight radiative tail, and $0.25\degree$ for
$\theta$ and $0.5\degree$ for $\phi$ averaged over all events. The
significantly better angular resolution is probably related to the higher
average momentum of the tracks.

####  4.8.5 Event Acceptances and Resolutions

The effect of the detector systems on the overall physics observables related
to the meson photoproduction was considered. The study only involved the
exclusive process where all final state particles were detected.

The $Q^{2}$, and $W$ distributions are shown in Fig. 51 top row plots. The
acceptances are shown in the bottom row. We do not consider the resolutions
for these variables, as they depend on particles detected in the far
forward/backward systems.

Overall, the full particle acceptance for this reaction at 5$\times$100 is
found to be around 13%.

Also shown in Fig. 51 (bottom row plots), are the reconstructed decay angle
distributions of the mesonic states. These are very uniform, suggesting ECCE
is suitable for performing high-level analysis of the meson decay and
therefore accessing quantum numbers and Spin Density Matrix Elements.

Figure 51: Top: 5x100 generated (red) and reconstructed (blue) distributions
of $Q^{2}$ and $W$, for events where all particles were detected. Bottom:
5$\times$100 generated (red) and reconstructed (blue) distributions of the
produced meson decay angles in the Gottfried-Jackson reference frame,
$\cos{\theta}$ and $\phi$ (°), for events where all particles were detected.

Finally, we show the reconstructed invariant mass distributions and
resolutions in Fig. 52. The resolutions for $M(e^{+}e^{-})$ and
$M(e^{+}e^{-}\pi^{+}\pi^{-})$ are both around 30 MeV. This should be
sufficiently narrow for distinguishing many of the final state mesons in the
mass region. The good resolution for the J$/\psi$ mass also helps reduce the
background from the signal without this meson.

Figure 52:  Top: reconstructed invariant masses for meson decay products, the
three states of interest are clearly observed on the right plot. Bottom: shows
the difference in reconstructed to truth masses.

####  4.8.6 Summary

The study presented shows the ECCE detector to be very promising for studies
of exotic meson spectroscopy with the EIC. In particular, at mid-center-of-
mass energies, the meson decay products are nicely distributed throughout the
central detector. Four particle invariant mass resolutions provide sufficient
separation to distinguish narrow states with mass differences greater than
order 0.1 GeV, compared to typical masses of 4 GeV.

Excellent far-forward and backward detector systems will also be essential for
reconstructing the overall reaction kinematics to allow for partial wave
analysis and investigations of production mechanisms.

Given that, the fully reconstructed events yields were estimated to be around
10k for $X$, and 3k for $Y$ production for 10 fb-1. This is very competitive
with previously published experiments, as shown in Table 9.

Table 9: Expected event yields at the 5$\times$100 beam energy configuration for a luminosity of 10 fb-1, compared to previous publications for the J$/\psi\pi^{+}\pi^{-}$ final state. Note yields from published Y results are estimated from the publications rather than given explicitly. Lab. | ECCE 5$\times$100 | CDF[76] | LHCb[77] | D$\Phi$[78] | ATLAS[79]
---|---|---|---|---|---
$\chi_{c1}(3872)$ | 10000 | 2292 | 4230 | 522 | 30000
Lab. | ECCE 5$\times$100 | BABAR[80] | BABAR[81] | BELLE[82] | BESIII[83]
Y(4260) | 3000 | 125 | 200 | 600 | 7000

##  5 Insights for EIC detector 2 at IP8

The EIC accelerator site at BNL is capable of instrumenting a second
interaction region at IP8 (Interaction Point 8, where the sPHENIX experiment
is currently located) in addition to the primary interaction region planned
for IP6 (where the STAR experiment is currently located). Although the current
scope of the EIC project consists of only one detector at the IP6 location,
the community is enthusiastic about the possibility of instrumenting IP8 with
the second detector. The Exclusive, Diffractive, and Tagging working group
performed physics impact studies to look for complementarities to the physics
measurement of the first detector. Here, the general concept of these impact
studies is to keep the design of the central detector the same as the IP6 and
modify the beamline components according to the official preliminary design of
IP8 (documented in Ref. [27]). It is worth noting that the design of the
second detector at IP8 is still in the early conceptual stage.

Preliminary designs of the secondary IP feature a larger electron-ion beam
crossing angle (35 mrad) and a region of high dispersion followed by a
secondary focus. Figure 53 shows a schematic diagram of IP8 with a 41 GeV
proton beam being steered through the Far-forward detector stack.

The full ECCE simulation package, Fun4All, is capable of simulating physics
processes using the IP6 and IP8 configurations. Currently, there are two main
differences between the two configurations:

  1. 1.

The addition of the secondary focus at IP8;

  2. 2.

The crossing angle is 35 mrad at IP8, compared to 25 mrad at IP6.
Consequently, the ZDC acceptance is larger than $\pm 8$ mrad at IP8, compared
to $\pm 5$ mrad at IP6.

It is important to note that the baseline magnet configuration only allows
$\pm 5$ mrad ZDC acceptance due to magnet aperture constraints at IP8.
However, an alternative (improved) magnet design (with Nb3Sn) brings the
possibility of an enlarged ZDC acceptance up to $\pm 8$ mrad. The studies in
this section are based on this optimized scenario [84].

Near the second focus, particles that are close in rigidity (momentum/charge)
to the beam are separated, while the beam itself is focused on a small beam
spot. This allows a set of Roman pot (RP) detectors (3&4 as shown in Fig. 53)
to be placed close to the beam where they can detect particles that are
slightly off from the beam rigidity.

Several physics processes can benefit significantly from the secondary focus
capability. Examples include veto-tagging of incoherent diffractive vector
meson production by detecting nuclear remnants, better kinematic acceptance
for measuring the pion structure function (Sec. 4.2), and better acceptance
for $e$A-DVCS (Deeply Virtual Compton Scattering) Sec. 4.5.

In this section, we will discuss some studies using a very preliminary design
and simulation of IP8. These include the basic Roman Pot acceptance in
rigidity and angle, the impact on diffractive studies, and the impact on
acceptance in exclusive physics processes.

Figure 53: 41 GeV beam proton steering in IP8 configuration.

###  5.1 Roman Pot Acceptance

For this study, the BeAGLE (Benchmark eA Generator for LEptoproduction) event
generator [22] is used to simulate $e$Zr exclusive J$/\psi\rightarrow
e^{-}+e^{+}$ events colliding with beam energies given by 18 GeV for the
electrons and 122.22 GeV/nucleon for the Zr.

The occupancy of hits registered in the RPs is studied in the first two RP
layers (consisting of silicon trackers) in the IP6 configuration as well as
all four RP layers in the IP8 configuration. Of particular interest are the
RPs near the secondary focus in the IP8 configuration (third and fourth
layers). Occupancies from layers two and four are similar to layers one and
three, respectively.

Figure 54: Roman Pot occupancy Y versus X for layer 1 (top plot) and layer 3
(bottom plot) near the secondary focus. The $10\sigma$ beam cut is visible at
the center. Note the different scales on the two plots.

The Geant4 truth hits are plotted versus the X and Y local coordinates of the
particular layer in Fig. 54. The rectangular $10\sigma$ beam cut is visible in
the center and is much smaller for the RP near the secondary focus. A sharper
focus of particles is evident for the third RP layer. For layer 1,
$10\sigma_{x}=5$ cm and $10\sigma_{y}=0.7$ cm. For layer 3, $10\sigma_{x}=0.4$
cm and $10\sigma_{y}=0.16$ cm.

To better distinguish the identity of the ions detected in the RPs, the truth
hits are matched to the generator-level particle, which is used to construct
the rigidity ratio $x_{L}=(p/Z)/(p/Z)_{\textrm{beam}}$ and polar angle
$\theta$. The RP occupancies in $x_{L}$ vs $\theta$ (after applying the
$10\sigma$ cut) for all four layers of RP are shown in Fig. 55. Note that the
distribution in $x_{L}$ for layer 3 is much narrower than for layer 1.

Figure 55: Top left and right plots show RP occupancy $x_{L}$ versus $\theta$
in BeAGLE for layer 1 and layer 3 (near the secondary focus), while the
$10\sigma$ beam cut is not applied; bottom left and right show the same for
the case when the $10\sigma$ beam cut is applied. $x_{L}$ is defined as the
rigidity fraction: $(p/Z)/(p/Z)_{\textrm{beam}}$. Note the different scales on
the plots.

None of the remnant heavy ions are visible in layer 1, while most of them
remain in layer 3, which is near the secondary focus. However, the lighter
ions (${}^{1}\textrm{H}_{1}$, ${}^{3}\textrm{H}_{1}$,
${}^{3}\textrm{He}_{2}$), with rigidities very different from the beam, are
clearly visible in layer 1, while they are not detectable for the layers near
the secondary focus.

The derived $x_{L}$ acceptances corresponding to $\theta<1$ mrad are shown in
Fig. 56. It is clear that the Roman pots near the secondary focus greatly
increase the $x_{L}$ acceptance to about 0.015 from the beam rigidity.

Figure 56: Roman Pot $x_{L}$ acceptance (left) for layer 1 and (right) for
layer 3 near the secondary focus for the range $\theta<1$ mrad in BeAGLE.
$x_{L}$ is defined as the rigidity fraction: $(p/Z)/(p/Z)_{\textrm{beam}}$.
Note the different scales on the x-axis.

In order to get a more comprehensive view of this acceptance, a toy Monte
Carlo was made with scattered protons simulated uniformly over $0.5<x_{L}<1.5$
and $\theta<10$ mrad. The occupancy for all four Roman Pot layers is shown in
Fig. 57. Recall that layers 3 and 4 are near the secondary focus in IP8, while
1 and 2 have a similar acceptance to that seen in IP6.

Figure 57: Roman Pot $x_{L}$ vs. $\theta$ occupancy for all four layers (3 and
4 are near the secondary focus) in the proton Monte Carlo. $x_{L}$ is defined
as the rigidity fraction: $(p/Z)/(p/Z)_{\textrm{beam}}$.

The left panel of Fig. 58 shows the projected acceptance as a function of
$x_{L}$ for the band $\theta<1$ mrad. Note that this is not the full range of
coverage, but rather focuses on smaller polar angles. It can be seen that the
coverage is not complete. The right panel of Fig. 58 shows the improved
acceptance for the “High Acceptance” machine parameters, which allows the
Roman pots to be moved closer to the beam. The coverage is significantly
improved. It should be noted that these studies are preliminary and that the
design of the IR and forward detectors for IP8 are expected to evolve. It
should also be noted that these studies used the beam parameters and Roman
plot placement appropriate to the proton beam, even for the Zr beam studies,
as Zr beam parameters are not known at this time. Plots such as these will be
useful for optimizing the detector placement and machine parameter decisions
moving forward.

Figure 58: Roman Pot $x_{L}$ acceptance overlaid for layers 1 (conventional)
and 3 (secondary focus) for the range $\theta<1$ mrad for the high divergence
(left) and high acceptance (right) settings of the beam configuration. Proton
Monte Carlo data was used. Figure 59: Background veto efficiency effectiveness
of the secondary focus Roman pots in IP8 as a function of $-t$ for the $e$Zr
diffractive J/$\psi$ electroproduction study.

###  5.2 Effect of Secondary Focus on Veto Tagging in Diffraction

Sec. 5.1 illustrates the ability of the secondary focus at IP8 to allow the
detection of nuclear remnants close to beam rigidity (momentum/charge). We can
use this ability to improve our efficiency in tagging incoherent diffractive
events vs. coherent diffractive events.

Coherent $e$A diffractive J$/\psi$ production is an important measurement at
the EIC, as it allows access to the spatial distribution of gluons in the
nucleus [5]. In order to make this measurement, incoherent diffractive events
must be vetoed as they swamp the signal. The ability to veto incoherent $e$A
diffractive J/$\psi$ production was studied using both the IP6 and IP8
configurations. Note: B0 photon detection and the beampipe were not yet
implemented in this study. Fig. 59 shows the impact of the secondary focus at
IP8 in the case of $e$Zr diffractive events. The line at 1 corresponds to the
amount of background remaining when all of the cuts are made except the
secondary focus. The points show the relative effect of the cut using RP
layers 3 and 4. In particular, they correspond to the ratio of the background
after all cuts are made (including the secondary focus RP layer 3 and 4 cuts)
to the background before that cut. The additional background rejection is
significant, particularly at larger values of $|t|$.

###  5.3 Pion SF IP8

Figure 60:  Top plots: B0 occupancy of the simulated leading neutron for a
range of energies 5$\times$41 GeV (left) and 10$\times$100 GeV (right) at IP8.
Bottom plots: ZDC acceptance of the simulated leading neutron for a range of
energies 5$\times$41 GeV (left) and 10$\times$100 GeV (right) at IP8.

The B0 occupancy in Fig. 60 top row plots show a marginal decrease from IP6 to
IP8. Similar to IP6, the ZDC acceptance for IP8 in Fig. 60 bottom row plots
show a significant drop in neutron detection for the lowest energy setting
(i.e. 5$\times$41) due to the increased occupancy in the B0. This drop is more
prominent at IP6 than at IP8.

| IP6 | IP8
---|---|---
Energy [GeV] | Detector |  $\Delta$t | Detector |  $\Delta$t
| Fraction |  | Fraction |
5$\times$41 | 59$\%$ | 0.019 | 78$\%$ | 0.018
5$\times$100 | 100$\%$ | 0.007 | 100$\%$ | 0007
10$\times$100 | 100$\%$ | 0.007 | 100$\%$ | 0.007
18$\times$275 | 100$\%$ | 0.005 | 100$\%$ | 0.008
Table 10: The neutron detection fractions in the ZDC from the above plots are
laid out for a range of energies (5$\times$41 GeV, 10$\times$100 GeV) at IP6
and IP8 as well as the deviation of t from the detected value of t (i.e.
$\Delta$t).

Fig. 61 shows the $t$-distribution for the two energies at IP8 for a range of
$Q^{2}$ bins. The results are similar to those of IP6, with the drop in events
at the higher $Q^{2}$ bins for the lower energy. This is best shown in Table
10, where the detection fraction of the ZDC and the deviation of $t$ from the
detected $t$ (i.e. $\Delta{t}$) are broken down for four energies at IP6 and
IP8.

Figure 61:  The $-t$ distribution for a range of energies at IP8 (5$\times$41
GeV, 10$\times$100 GeV) at IP8. There are four $Q^{2}$ bins presented (7, 15,
30, 60 GeV2) of bin width $\pm$5 GeV2.

##  6 Summary

This article presents a collection of simulation studies using the ECCE
detector concept in the context of the EIC’s exclusive, diffractive, and
tagging physics program. This program is a wide umbrella that covers a diverse
set of reactions (as listed in Table 4), and ultimately provides answers to
the physics questions asked by the NAS white paper (in Sec. 1).

The unifying theme of this program is the key role played by the far-forward
and far-backward detector systems. These detector systems are used, either to
ensure exclusivity, isolate diffractive reactions, or detect a particle that
serves as a tag for a particular reaction of interest.

Full simulation studies are the best way to study and establish the expected
performance of the individual detector responses in far-forward systems. The
preliminary results (Sec. 4) demonstrate the design proposed by ECCE exceeds
the requirements (in acceptance and resolution) underlined by the YR. Here, it
is important to point out that the proposed technologies represent a snapshot
in time in the development of the EIC, where further modification and
improvements will be made in the near future.

For the physics impact studies, slightly different choices and assumptions
about reconstruction are made and based on the general projected detector
performances (based on full simulation results). The results represent the
expected physics impact/significance with an integrated luminosity of 10-1 fb
(which corresponds to the first few years of EIC commissioning and operation).
The results of these studies can be used for future comparison. As the EIC
detector’s design becomes more refined, and as inevitable trade-offs and
compromises are made when turning an idea into reality, future simulations can
be compared with these results to understand the impact on eventual physics
the EIC can deliver.

Similarly, the IP8 studies of Sec. 5 can serve as a guidepost for the future
development of a second EIC detector. The secondary focus at IP8 holds
potential that may at some point be exploited.

The EIC will usher in a new era of exploration of the rich quark-gluon
structure of nucleons and nuclei. As shown in these studies, the ECCE detector
concept can deliver impactful results on a host of interesting questions
through its exclusive, diffractive, and tagging physics program.

##  7 Acknowledgements

We thank the EIC Silicon Consortium for cost estimate methodologies concerning
silicon tracking systems, technical discussions, and comments. We acknowledge
the important prior work of projects eRD16, eRD18, and eRD25 concerning
research and development of MAPS silicon tracking technologies.

We thank the EIC LGAD Consortium for technical discussions and acknowledge the
prior work of project eRD112.

We acknowledge support from the Office of Nuclear Physics in the Office of
Science in the Department of Energy, the National Science Foundation, the Los
Alamos National Laboratory Laboratory Directed Research and Development (LDRD)
20200022DR, the Natural Sciences and Engineering Research Council of Canada
(NSERC), and the UK Research and Innovation Science and Technology Facilities
Council.

This research used resources from the Compute and Data Environment for Science
(CADES) at the Oak Ridge National Laboratory, which is supported by the Office
of Science of the U.S. Department of Energy under Contract No. DE-
AC05-00OR22725.

And we gratefully acknowledge the support of Brookhaven National Lab and the
Thomas Jefferson National Accelerator Facility which are operated under
contracts DE-SC0012704 and DE-AC05-06OR23177 respectively.

##  Appendix A Detector layout inside B0 magnet

To better visualize the layout of the trackers and calorimeter inside of the
B0 magnet (see Fig. 2), a CAD drawing (Fig. 62) was created based on the
official IP6 design specification from Ref. [27]. The shape of the trackers
and calorimeter resembles the ‘PAC-man’, which is optical to accommodate the
incoming (upstream) electron and outgoing (downstream) ion beam pipes. Note
that the trackers and calorimeters are mounted on the guiding rods, which can
be slid out for servicing. There is no access from the backside of the B0
magnet due to special constraints.

Figure 62: B0 design. Dimensions are given in mm.

##  Appendix B DVCS off Helium-4 and the TOPEG generator

The TOPEG event generator, which was originally used for the 4He DVCS studies
in the YR, was used for our DVCS off-helium study. Full details on the TOPEG
generator can be found in the YR and in [38] (as well as in the subsequent
references provided within). For the ECCE studies, the model used neglects the
real part of the $H$ generalized parton distribution in the full coherent DVCS
4He implementation (generator model 3, according to the TOPEG nomenclature).
This model allows for a reasonable computation time, without sacrificing
necessary physics precision.

Complete generator card data corresponding to the settings used for the
results shown in this document can be found in Table 11.

Parameter | Values
---|---
Epz (GeV2) | 5
Hepz (GeV2) | 163.958
N TFoam Cells | 104
N Cell Samples | 300
y | 0.05 $\leq$ y $\leq$ 0.85
Q2 (GeV2) | 2.0 $\leq$ Q2 $\leq$ 30
W2 (GeV2) | W2 $\geq$ 16
$\theta^{e}_{max}$ (rad) |  $\theta^{e}_{max}\geq$ 2.35
t (GeV2) | 0.01 $\leq$ t $\leq$ 0.5
eBeam helicity |  $\pm$ 1
Table 11: TOPEG generator configuration used for these studies. Further value
ranges are also under current study.

In TOPEG, simulated data is constrained by kinematic limits set by the user in
the generator input settings. Notably attempts to generate events at $t<$0.01
GeV2 and $Q$2 $\leq$ 1 GeV2 often encountered issues.

The matching kinematic phase space plots are also given in Figure 63. Cross
sections are calculated by integrating over the largest possible phase space
which is filled with events, as seen in these figures.

Figure 63: 2D kinematic coverage plots for DVCS-e4He. The left-hand side plots
show the generated phase space from TOPEG directly. The right-hand side plots
show the kinematic coverage as reconstructed from the Fun4All output for the
ECCE detector. The color scales indicate raw counts and are not normalized to
each other.

##  Appendix C XYZ production event generator

The event generator was custom developed for spectroscopy reactions at the
EIC. It consists of two main parts: photoproduction helicity amplitudes; and
virtual photon production.

###  C.1 JPAC Photoproduction Amplitudes

The helicity amplitudes were calculated following the formalism and parameters
given in Ref.[75]. The models therein are expected to give an order of
magnitude estimates for meson production cross sections. For the
J$/\psi\pi^{+}\pi^{-}$ events, we assumed only $\psi(2s)$, $\chi_{c1}(3872)$
and Y(4260) states were produced. For the $\chi_{c1}(3872)$ production we used
the pion exchange amplitudes and for $\psi(2s)$ and Y(4260) we assumed only
pomeron exchange. As these models consist of high and low energy limits, for
the current study we chose to combine the two via a simple linear
interpolation of the helicity amplitudes between the high and low regions
given.

###  C.2 Virtual photoproduction

To make estimates of exclusive electroproduction with low-$Q^{2}$ quasi-real
virtual photons we first generate a beam of virtual photons that interacted
with the proton beam producing the meson which we subsequently decayed to
specific final states which were then run through the ECCE detector
simulation,

$\frac{d^{4}\sigma}{dsdQ^{2}d\phi
dt}=\frac{d^{2}\sigma_{e,\gamma*e^{\prime}}}{dsdQ^{2}}\frac{d^{2}\sigma_{\gamma*+p\rightarrow
V+p}(s,Q^{2})}{d\phi dt}$ (11)

the virtual photon flux factor was sampled from

$\frac{d^{2}\sigma_{e,\gamma*e^{\prime}}}{dsdQ^{2}}=\frac{\alpha}{2\pi}\frac{K.L}{E}\frac{1}{Q^{2}}\frac{1}{(s-M^{2}+Q^{2})}$
(12) $K=\frac{W^{2}-M^{2}}{2M}\quad
L=\frac{1+(1-y)^{2}}{y}-\frac{2m_{e}^{2}y}{Q^{2}}$ (13)

and the two-body photoproduction cross-section was calculated as

$\frac{d^{2}\sigma_{\gamma^{*}+p\rightarrow V+p}(s,Q^{2})}{d\phi
dt}=\frac{1}{128\pi^{2}s}\frac{1}{|\mathbf{p}_{\gamma^{*}_{CM}}|^{2}}{|{M}(s,t)|^{2}}$
(14)

with $M(s,t)$ the photoproduction amplitude. This cross-section was modified
by an additional $Q^{2}$ dependence taken from [85]. Eqn. (11) was integrated
numerically to give the total cross section for determining event rates.

The generation algorithm proceeded as

  * 1.

Generate the scattered electron by sampling from 2D distribution in Eq. 12 in
the rest frame of the proton.

  * 2.

Sample the intermediate particle masses from Breit-Wigner distributions with
parameters taken from PDG values.

  * 3.

Given s and the particle masses accept/reject the event based on the n-body
mass phase space.

  * 4.

Given s, $Q^{2}$ and final state masses accept/reject on the production t from
Eq. 14

  * 5.

Sample random $\phi$ angle and complete the kinematics of the produced meson
and recoiling proton.

  * 6.

Decay produced meson to J$/\psi$ and 2 pions using flat decay angle
distributions.

  * 7.

Decay J$/\psi$ to $e^{+}e^{-}$ using flat decay angle distributions.

  * 8.

Boost all stable particles to the lab system.

Prior to tracking in Geant4 the ECCE afterburner is applied to the 4-vectors
to apply crossing angles and divergences.

##  Appendix D TCS $ep$ and the EpIC generator 2D phase space

Figs. 64 and 65 are a representation of the phase space coverage of the ECCE
detector, as compared with the generated data from EpIC.

Figure 64: 5 $\times$ 41 - $Q^{\prime 2}$ versus TCS $\tau$ ($x_{B}$) for
generated EpIC data (left) and reconstructed Fun4All data (right). Figure 65:
18 $\times$ 275 - $Q^{\prime 2}$ versus TCS $\tau$ ($x_{B}$) generated EpIC
data (left) and reconstructed Fun4All data (right).

##  Appendix E LAGER generator for exclusive J$/\psi$ production

The LAGER generator [39] was used to produce event samples for the ECCE
studies presented. LAGER is described as a modular accept-reject generator,
capable of simulating both fixed target and collider kinematics, and has
previously been used for vector meson studies at EIC kinematics, with
significant recent developmental effort in support of DVMP studies.

The event samples are processed through eic-smear and the resulting ROOT trees
are provided to Fun4All, which simulates the full ECCE detector response in
Geant4. The final output of this process is the Fun4All DST files. All studies
presented were performed at the IP6 detector location using Prop.4 (aka July
detector designd). The kinematic presented in this study corresponds to
electron and proton beam energies of 18 GeV and 275 GeV, respectively.

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