outside the Gaussian, from less well reconstructed events. The shoulder is
larger on the low-mass side, because of electron bremsstrahlung in the
detector material. With the higher field, the resolution (sigma) is more than
a factor of 2 better, allowing better resolution of the peak. The Gaussian
fits were done only to the central part of the peak, as shown by the solid
part of the red curves.

The $J/\psi$ is the other meson highlighted in the 2012 EIC White Paper [1].
It can be reconstructed from its decays into either the $\mu^{+}\mu^{-}$ or
$e^{+}e^{-}$ final state. Although the $J/\psi$ reconstruction is
straightforward, bremsstrahlung from electrons can produce a low-mass shoulder
in the $e^{+}e^{-}$ mass spectrum if the detector is too thick. Figure 53
shows the reconstructed $J/\psi$ spectrum expected from the all-silicon
detector for 1.5 and 3.0 T magnetic fields. The peaks were fitted to a
Gaussian function, with resolutions for the two fields of 22 MeV/$c^{2}$ and
13 MeV/$c^{2}$ respectively. Although the Gaussian describes the peak well,
there significant shoulders are visible. The shoulders have two components:
events that are less well reconstructed, and events where the electron
underwent bremsstrahlung in the beampipe or detector material. The latter only
contributes to the low-mass shoulder. The bremsstrahlung contribution should
be the same for the two fields.

Although a low-mass shoulder is visible in both spectra, the peaks stand out
clearly. The Gaussians are fitted to the data above the $J/\psi$ peak, and the
portion of the lower mass data where the peak is above the shoulder, as
indicated by the darker black line.

The origin of the shoulders is demonstrated in Fig. 53, which shows the
dilepton mass $M_{ee}$ vs. $p_{T}$. In addition to the pileup around
$M_{J/\psi}$, there is a clear diagonal band with lower $M_{ee}$ but higher
$p_{T}$. This band is expected due to bremsstrahlung. If one of the electrons
radiates a photon and loses energy while traversing the detector, the pair
will be reconstructed with lower pair mass, but higher $p_{T}$ [95]. These
events can be rejected by cuts on pair $p_{T}$ and $M_{ee}$, or with a photon
veto in the calorimeter. There is a further unsimulated source for these
events, the decay $J/\psi\rightarrow e^{+}e^{-}\gamma$. For photon energies
above 100 MeV, the branching ratio for this channel is 0.88% [96], or 15% of
the rate to $e^{+}e^{-}$.

Figure 54: A scatter plot showing the relationship between the reconstructed
$J/\psi\rightarrow e^{+}e^{-}$ mass and transverse momentum in $e$+$p$
collisions in a 3.0 T field. The vast majority of events are reconstructed
with $M_{ee}\approx M_{J/\psi}$, with a $p_{T}$ distribution expected for
coherent photoproduction. There is also a clear diagonal band extending to
lower $M_{ee}$ but higher $p_{T}$. In these events, one of the leptons
radiated a photon in the detector material, leading to the lower pair
invariant mass and higher $p_{T}$.

####  III.3.5 The $\Upsilon$ family

The three $\Upsilon$ states are relatively heavy, $\sim 10$ GeV/$c^{2}$, but
with rather small mass splittings - 563 MeV between the first and second, and
331 MeV between the second and third. Good momentum resolution is required to
effectively separate the three states. Figure 55 shows the $e^{+}e^{-}$ mass
spectrum expected from the three Upsilon states, in two different rapidity
ranges. Although the Bjorken-$x$ ranges are different for positive and
negative rapidity, the detector resolution should be similar at $+y$ and $-y$.
Table 10 shows the resolutions extracted from a Gaussian fit to the
$\Upsilon(1S)$ peaks. The resolution is about 40% better in the 3.0 T field
and in both cases worsens by about 20% at larger $|y|$. Nonetheless, either
magnetic field option provides adequate separation over the full range in $y$.

Figure 55: The reconstructed $e^{+}e^{-}$ mass spectrum in the Upsilon region, with the $\Upsilon(1S)$, $\Upsilon(2S)$ $\Upsilon(3S)$ peaks, in the ratio predicted by eSTARlight, in 10 fb-1 of data, in (left) 1.5 T and (right) 3.0 T fields. The spectra are divided into two rapidity ranges, $|y|<1$ and $|y|>1$. Because the Upsilons are heavy, most of the production occurs in $|y|<2$.  Table 10: Mass resolution, for a Gaussian fit to the $\Upsilon(1S)$ peak shown in Fig. 55. Rapidity |  1.5 T field |  3.0 T field
---|---|---
$|y|<1$ | 66 MeV/$c^{2}$ | 40 MeV/$c^{2}$
$|y|>1$ | 79 MeV/$c^{2}$ | 51 MeV/$c^{2}$

####  III.3.6 Vector meson conclusions

A full EIC program will include studies of the vector mesons considered here
along with other vector mesons, likely including the $\rho^{\prime}$ and other
excited states. In most cases, the proposed silicon detector exceeds the
requirements for vector meson reconstruction. There are two problematic areas
- inadequate coverage in pseudorapidity to cover light vector meson over the
full range of Bjorken-$x$, and problematic reconstruction of very soft kaons
from $\phi$ decays. Rapidity is closely linked to Bjorken-$x$, so limitations
in rapidity will reduce acceptance at large and small $x$. The large $x$
limitation is problematic both for parton measurements, and for searches for
exotica, including the XYZ states [92] and pentaquarks, and for backward
production of mesons. However, the pseudorapidity coverage of tracking
detectors is largely limited by the positioning of the beampipe.

##  IV Summary and Conclusions

The EIC will be a high-luminosity, variable-energy collider with a broad and
compelling physics program. Completing this program will require detectors
capable of making precision measurements of many physics channels over a wide
kinematic range. We have presented a design for an all-silicon EIC tracking
detector and shown that it provides tracking with high resolution and good
vertex reconstruction over a large kinematic range, enabling the EIC’s broad
physics. We have also performed physics simulations for a variety of reactions
for which tracking is critical, including measurements of heavy quarks, jets,
and exclusive vector mesons. In most cases, the proposed all-silicon tracker
provides the necessary performance, and we have identified cases where
improved performance could enhance the kinematic coverage or enable new
measurements.

Our simulations have demonstrated the feasibility of a broad program of heavy-
quark studies that allow for measurements of gluonic PDFs, TMDs, and helicity
distributions in nucleons and nuclei, as well as cold nuclear matter effects
(see Sec. III.1). The proposed tracking system based on the ultra-thin (0.3%
$X_{0}$ per layer) and fine-pitch (10$\times$10 µm2) MAPS sensor technology
provides the momentum resolution as well as vertex reconstruction performance
necessary to enable these studies, with potential for improved sensitivity in
low momentum and/or charm baryon reconstruction with further reduced detector
thickness.

The charged-jet energy and angular resolution performance of the all-silicon
tracker was studied. Jets can be reconstructed with a resolution parameter
$R=1.0$, as the multiplicity in the collisions is small. While there is a
significant resolution loss when jet constituents are not all properly
included in charged jets, the charged jet resolutions with the silicon tracker
are nevertheless encouraging. Finally, we studied azimuthal differences
between jets and the scattered electron, which can provide access to parton
transport in nuclear matter, the quark TMD PDF and the Sivers effect in
transversely polarized e-p collisions, and the charged-jet fragmentation
function, which should be sensitive to the hadronization process.

The silicon detector can likewise reconstruct important vector meson decays.
Our simulations considered reconstructions of $\rho$, $\phi$, $J/\psi$ and
$\Upsilon$ decays, but these are representative of other vector mesons decays,
including the $\psi^{\prime}$ and $\rho^{*0}$ decays. The momentum resolution
will allow us to cleanly separate the different $\Upsilon$ states in the
dilepton spectrum. The major limitation of the proposed tracker is the limited
pseudorapidity coverage. Broad coverage is important to reconstruct the
photoproduction and electroproduction over the full range of Bjorken$-x$.
Unfortunately, this limitation appears to come primarily from the current
beampipe design; the non-zero beam crossing angle leads to a “X” shaped
interaction region, limiting the length of the free (for detector) region in
$|z|$, thereby limiting the acceptance of any tracking detector design. A
secondary consideration is that the detector should have low mass, to minimize
bremsstrahlung in dielectron events.

For $D^{0}$, $\Lambda_{c}$ and most quarkonia reconstruction, a stronger
magnetic field choice would be preferred as it enables better momentum and
therefore mass resolution for these resonances. However, the impact on the low
$p_{T}$ threshold due to the increased magnetic field strength is much less
than the gain in signal significance due to the better mass resolution. For
$D^{0}$ mesons, the $p_{T}$-integrated signal significance is improved by
$\sim$50% comparing 3.0 T vs. 1.5 T magnetic field setting (See Sec.
III.1.2.2). For $D^{*+}\rightarrow D^{0}\pi^{+}$ reconstruction, the $p_{T}$
threshold for the soft pion may cause acceptance loss for low $p_{T}$ $D^{*+}$
mesons. Our initial study shows reconstruction using fewer tracking layers for
the soft pions enables $D^{*+}$ to be still a viable channel in the 3.0 T
magnetic field configuration (See Sec. III.1.2.2). The efficiency for
$\phi\rightarrow K^{+}K^{-}$ is significantly higher for the 1.5 T field than
at 3.0 T, because the higher field causes the tracks to curl up more tightly.

One limitation, common to all of these analyses is that the Bjorken-$x$ of the
struck parton is strongly correlated with the final state rapidity, with
interactions at very large or very small $x$ corresponding to large $|y|$,
where detector acceptance may be limited by the interaction region geometry.
This, along with other issues identified in the simulations presented here,
will require further examination of the tracking system or overall
spectrometer design to see if modifications or optimizations can improve these
measurements.

The all-silicon tracker geometry presented in this document will be revised as
details of the EIC overall detector are established and silicon-pixel R&D
efforts progress, $e.g.$ optimizing an asymmetric tracker geometry in case the
nominal interaction point is shifted away from ($0,0,0$). The electron and
hadron beams at the EIC are expected to cross each other at an angle of 25
mrad in the interaction region. Since the solenoid axis will be aligned with
the electron beam direction, this crossing angle translates into a 25 mrad
angular difference between the magnetic field and the hadron-beam direction,
which causes the momentum resolution in the hadron direction to develop a
dependence on the azimuthal angle. For tracks with $\phi\approx 0$ (in a
coordinate system with the $z$ axis aligned along the hadron beampipe), the
field integral is smaller, and the momentum resolution degrades. Conversely,
for tracks with $\phi\approx\pi$ the field integral is larger, and the
momentum resolution improves. This effect is more significant at higher
pseudorapidities and hadron momenta. For example, for hadrons with
$\eta\approx 3.6$ and $p\approx 50\,{\rm GeV}/c$, this effect can lead to an
improvement (deterioration) of the momentum resolution with respect to the
nominal momentum resolution by $\approx 40\%$ ($\approx 90\%$) on either side
of the beam pipe in the horizontal plane. While we quantified the effect of
the beam-crossing angle on the momentum resolution, we have not propagated the
resulting azimuthal dependence in the physics studies presented here. This is
in line with other studies for the Yellow Report, but does present an
important area for continued study at forward pseudorapidities.

In summary, we have presented a concept all-silicon tracker that makes use of
state-of-the-art technology, and have demonstrated the applicability of such
tracker to fulfill several physics studies at the EIC.

##  V Acknowledgements

This work is supported in part by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under contract numbers DE-AC02-05CH11231
and DE-FG02-89ER40531, the DOE Office of Science Distinguished Scientists
Fellow Award MPO/IEWO606342, the DOE National Nuclear Security Administration
under the Nuclear Science Security Consortium award DE-NA0003180, by the EIC
Generic Detector R&D program under project eRD16, and by the Strategic
Priority Research Program of Chinese Academy of Sciences, under grant number
XDB34000000.

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