for producing resonance-like signals for the prominent multiquark candidates
$P_{c}(4450)$ [216, 217], $Z_{c}(3900)^{\pm}$ [218, 219, 220] (see also Refs.
[221, 222, 223, 224]), $Z_{c}(4200,4430)^{\pm}$ [225] and $X(4050,4250)^{\pm}$
[226]. This means that it is essential to distinguish the signals from
resonances from those from kinematical singularities. Triangle singularity is
very sensitive to the involved kinematical variables such as masses and
energies. The examples mentioned in the above were all reported in the decays
of bottom hadrons ($B$ or $\Lambda$), which have fixed masses, or $e^{+}e^{-}$
collisions. The photoproduction or electroproduction processes at EicC have
completely different kinematical regions from these experiments; the photon is
space-like or nearly on shell, making the occurrence of triangle singularities
in the exclusive processes impossible. Furthermore, the dependence of the
signals on the energy and $Q^{2}$ can also be measured. In addition, the
double polarized beams facilitate the EicC to determine the quantum numbers,
such as spin and parity, of hadron resonances. In comparison with hadron
colliders, the EicC has a better signal to noise ratio. Furthermore, the EicC
covers all the mass regions for charmonium, bottomonium, $P_{c}$, $P_{b}$ and
excited heavy hadrons. As a result, the study of exotic mesons and baryons
will be one of the foci of EicC.

A search for the photoproduction of $Z_{c}(3900)$ [227] and $\chi_{c1}(3872)$
[172] has been performed by COMPASS with muon beam, giving valuable input for
the simulation of EicC. Within similar range of c.m. energy, EicC can search
for these and also other states with more than one order higher luminosity.
Furthermore, with solid angle coverage, especially hadron PID in the forward
angle and good vertex detector for decay topology, the acceptance and
reconstruction efficiency of EicC are expected to be significantly increased,
so the discovery potential of exotica states will be exploited.

####  2.4.3 Cross section estimates and simulations

In this subsection, more quantitative estimates of the production rates for a
selected list of exotic hadron candidates and simulations of the
$ep\rightarrow e\chi_{c1}(3872)p$, $ep\rightarrow eZ_{c}^{+}(3900)n$,
$ep\rightarrow eP_{c}\rightarrow eJ/\psi p$, and $ep\rightarrow
eP_{b}\rightarrow e\Upsilon p$ processes are reported. For more model
estimates of the exclusive productions of hidden-charm and bottom exotic
hadrons, we refer to Refs. [228, 229, 230, 231, 232, 233, 227, 234, 235, 236,
237, 204, 238, 172, 239, 240, 241, 242, 243, 244, 245, 205, 246, 247, 248,
249, 250].

Figure 2.21:  Left: The $ep\rightarrow epV\rightarrow epl^{+}l^{-}$ process. Right: The $ep\rightarrow e\chi_{c1}(3872)p$ and $ep\rightarrow eZ_{c}^{+}(3900)n$ process. Table 2.3:  Estimated event numbers that can be collected at EicC assuming an integrated luminosity of 50 fb-1. The lepton pairs $l^{+}l^{-}$ denote both $\mu^{+}\mu^{-}$ and $e^{+}e^{-}$. The event numbers are estimated using the assumed detection efficiencies listed in the third column, which are expected to be higher in the middle rapidity than that in the forward region.  Exotic states | Production/decay processes | Detection efficiency | Expected events
---|---|---|---
$P_{c}(4312)$ | $ep\rightarrow eP_{c}(4312)$ | $\sim$30% | 15$-$1450
$P_{c}(4312)\rightarrow pJ/\psi$
$J/\psi\rightarrow l^{+}l^{-}$
$P_{c}(4440)$ | $ep\rightarrow eP_{c}(4440)$ | $\sim$30% | 20$-$2200
$P_{c}(4440)\rightarrow pJ/\psi$
$J/\psi\rightarrow l^{+}l^{-}$
$P_{c}(4457)$ | $ep\rightarrow eP_{c}(4457)$ | $\sim$30% | 10$-$650
$P_{c}(4457)\rightarrow pJ/\psi$
$J/\psi\rightarrow l^{+}l^{-}$
$P_{b}(\text{narrow})$ | $ep\rightarrow eP_{b}(\text{narrow})$ | $\sim$30% | 0$-$20
$P_{b}(\text{narrow})\rightarrow p\Upsilon$
$\Upsilon\rightarrow l^{+}l^{-}$
$P_{b}(\text{wide})$ | $ep\rightarrow eP_{b}(\text{wide})$ | $\sim$30% | 0$-$200
$P_{b}(\text{wide})\rightarrow p\Upsilon$
$\Upsilon\rightarrow l^{+}l^{-}$
$\chi_{c1}(3872)$ | $ep\rightarrow e\chi_{c1}(3872)p$ | $\sim$50% | 0$-$90
$\chi_{c1}(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$
$J/\psi\rightarrow l^{+}l^{-}$
$Z_{c}(3900)^{+}$ | $ep\rightarrow eZ_{c}(3900)^{+}n$ | $\sim$60% | 90$-$9300
$Z_{c}^{+}(3900)\rightarrow\pi^{+}J/\psi$
$J/\psi\rightarrow l^{+}l^{-}$
Figure 2.22: The distributions of invariant masses, transverse momenta,
pseudo-rapidities and rapidities of the $P_{c}$ and $P_{b}$ states.  Figure
2.23:  The distributions of invariant masses, transverse momenta, pseudo-
rapidities and rapidities of the $Z_{c}(3900)^{+}$ in different $Q^{2}$
ranges.

Motivated by the heavy quark flavor symmetry for the potential between heavy
mesons and baryons, analogues of the pentaquark candidates $P_{c}$ in the
bottom sector, labeled as $P_{b}$ here, are expected to exist [251, 252, 237,
199, 253, 254, 255, 256, 257, 258]. A resonant state $P_{b}$ coupling to
$\Upsilon p$ with a mass around 11.12 GeV and a width ranging from tens of MeV
to 300 MeV is predicted by nearly all models. We simulate the exclusive
electroproduction of two $P_{b}$ at EicC with a width of 30 MeV ($P_{b}({\rm
narrow})$) and 300 MeV ($P_{b}({\rm wide})$), respectively, together with the
three narrow $P_{c}$ states with the resonance parameters reported by LHCb.

The process is depicted in the left panel of Fig. 2.21, where $V$ represents
the $J/\psi$ ($\Upsilon$) and the $P_{c}$ ($P_{b}$) couples to the $J/\psi N$
($\Upsilon N$) in the $s$-channel. The non-resonant background is modeled by
Pomeron-exchange to be discussed in Sec. 2.5.1. The $P_{c}/P_{b}$ states are
produced from the interaction between the virtual photon, emitted from the
electron beam, and the proton beam in the $s$-channel. The upper limit of
production rates at EicC can be determined by the upper limit of cross section
of $\gamma p\rightarrow P_{c}\rightarrow J/\psi p$ measured by the GlueX
Collaboration [173], by properly including the photon flux and $Q^{2}$
dependence of amplitudes. According to the analysis of the LHCb data, the
lower limit of the branching ratio $P_{c}\rightarrow J/\psi p$ is around 0.5%
[200], and we assume the same lower limit for the $P_{b}\rightarrow\Upsilon
p$. The lower limit of production rates at EicC is obtained by the vector
meson dominance (VMD) model, see, e.g. [245, 249].

After taking into account the detection efficiency and the dilepton decay
rates of the $J/\psi$ and $\Upsilon$, the expected production rates of these
exotic states at EicC are shown in Table 2.3. Here the detection efficiencies
in the third column are estimated from the simulated distribution of final-
state particles. The production rates on light nuclei are supposed to be
larger [241, 248], which however will have lower luminosity. The distributions
of the invariant mass spectra, the transverse momenta, the pseudo-rapidities,
and the rapidities of the $P_{b}/P_{c}$ states are presented in Fig. 2.22.
They are obviously characterized by the $s$-channel resonances, with small
transverse momenta and narrow ranges of pseudo-rapidity and rapidity,
apparently different from the non-resonant Pomeron-exchange contribution which
is smoothly spanned in the full range, as demonstrated in detailed
investigation [249, 247]. It is suggested to extract the pentaquark signal
from large non-resonant contribution with a proper kinematic cut [249].

As discussed above, the cross section of the open charm channel
$\bar{D}^{(*)}\Lambda_{c}$ is expected to be much bigger than the that of
$J/\psi p$, so is the case of the open bottom channel
$\bar{B}^{(*)}\Lambda_{b}$ in comparison with the $\Upsilon p$. In particular,
the branching fractions of the $P_{c}$ and $P_{b}$ states into open-flavor
channels are expected to be at least one-order-of-magnitude larger than those
of the $J/\psi p$ and $\Upsilon p$. In addition, as mentioned in Section
2.4.2, the open-charm ground state hadrons could be reconstructed at the level
of 10%. Thus, it is optimistic that the $P_{c}$ states can be studied in
detail through processes $ep\rightarrow J/\psi p$ and
$e\bar{D}^{(*)}\Lambda_{c}$ at EicC. If the open-bottom hadrons can also be
efficiently reconstructed, the hypothesized $P_{b}$ states may also be sought
at EicC.

In addition to the $P_{c}$ and $P_{b}$ states, the exclusive productions of
the $\chi_{c1}(3872)$ and $Z_{c}(3900)$ are also simulated. The
$\chi_{c1}(3872)$ is arguably the most interesting charmonium-like state. It
mass is $(3971.69\pm 0.17)$ MeV, coinciding with the $D^{0}\bar{D}^{*0}$
threshold exactly, and width is smaller than 1.2 MeV [148].444The mass quoted
in RPP is from averaging previous experiments. Recently, the LHCb
Collaboration reported precise determinations of the mass and width [259,
260]. In particular, a Flatté analysis, which is more proper than the Breit-
Wigner one for near-threshold states, is performed in Ref. [259].  The quantum
numbers are $1^{++}$ [261]. It is the first discovered and most studied exotic
meson candidate in the charmonium region [262]. The mass of
$Z_{c}(3900)^{\pm}$ is (3888.4$\pm$2.5) MeV with a width of (28.3$\pm$2.5) MeV
[148]. It was discovered in the $J/\psi\pi^{\pm}$ spectrum by BESIII [263] and
Belle [264] and confirmed by other experiments [265, 266]. The BESIII
collaboration identified its spin-parity as $1^{+}$ and isospin as 1 [267,
268]. As a charged state decaying into a charmonium, the quark content of
$Z_{c}(3900)$ contains at least $u\bar{d}c\bar{c}$, making it a prominent
candidate of tetraquarks or molecular states.

The right panel of Fig. 2.21 shows the $ep\rightarrow e\chi_{c1}(3872)p$ and
$ep\rightarrow eZ_{c}^{+}(3900)n$ processes with the yellow ellipse
representing the $t$-channel exchange. The $\gamma
p\rightarrow\chi_{c1}(3872)p$ can proceed through the exchange of vector
mesons (e.g. $\rho$, $\omega$ and $J/\psi$) in the $t$-channel. The $\gamma
p\rightarrow Z_{c}^{+}(3900)n$ can proceed through charged mesons (e.g.
$\pi^{+}$ and $a_{0}^{+}$) [228] or mesonic Regge trajectories [230] in the
$t$-channel. Around the averaged c.m. energy $W_{\gamma p}=13.7$ GeV, which is
in the EicC energy region, the COMPASS Collaboration has searched for the
$Z_{c}(3900)$, and the nonobservation sets an upper limit for the cross
section: $\mathcal{B}(Z_{c}(3900)^{\pm}\rightarrow
J/\psi\pi^{\pm})\sigma(\gamma N\rightarrow Z_{c}(3900)^{\pm}N)<52$ pb at the
90% confidence level [227]. In a similar energy region, the COMPASS
Collaboration has also obtained an upper limit for the $\chi_{c1}(3872)$ at
the same confidence level: $\mathcal{B}(\chi_{c1}(3872)\rightarrow
J/\psi\pi\pi)\sigma(\gamma N\rightarrow\chi_{c1}(3872)N)<2.9$ pb [172]. A
parametrization of these information with a reasonable formula is used as
input to the eSTARlight generator [269] to obtain the production rates at
EicC. The resulting estimated event numbers are shown in Table 2.3. In
particular, the upper bounds for the production of the $\chi_{c1}(3872)$ and
$Z_{c}(3900)^{+}$ are constrained by the COMPASS measurements. The lower
bounds are very roughly estimated by reducing these values by two orders of
magnitude, as inferred from the difference between the lower and upper bounds
of the $P_{c}$/$P_{b}$ production rates.

It is expected that a considerable amount of $Z_{c}^{+}(3900)$ could be
observed while the events of $\chi_{c1}(3872)$ are lower. The distributions of
the invariant mass distribution, the transverse momentum, the pseudo-rapidity
and the rapidity for the $Z_{c}^{+}(3900)$ are shown in Fig. 2.23 for
different $Q^{2}$ regions. The red cross histogram is the contribution from
$0<Q^{2}<1$ GeV2, the blue star one is that from $1<Q^{2}<5$ GeV2, and the
black circle one is the overall contribution. It is seen that most of the
events are within the low $Q^{2}$ range.

The above simulations show that the exotic hadrons produced at EicC are close
to the middle rapidity range. This is beneficial for detection. Thus, EicC
provides an excellent platform to study the nature of known, but barely
understood, charmonium-like states and search for new states.

Figure 2.24: The mechanism considered in Ref. [270] for the semi-inclusive
production of exotic hadrons (denoted as $X$) which couple strongly to a pair
of hadrons ($H$ and $H^{\prime}$) in lepton-proton collisions.

In addition to the exclusive processes discussed so far, exotic hadrons can be
searched for in semi-inclusive processes. In particular, for those hidden-
flavor exotic hadrons which couple strongly to a pair of heavy hadrons, such
as the $\chi_{c1}(3872)$ and the $Z_{c}(3900)$, one can achieve an order-of-
magnitude estimate of the production cross sections following the method of
Refs. [271, 272, 273, 274, 275]. The production can be factorized into a
short-distance part and a long-distance part. At short distances the hadron
pairs of interest are produced, which can be simulated using the Pythia event
generator [276]. The hadron pairs then merge to form the exotic hadrons which
couple to them strongly, and the long-distance piece can be computed at the
hadronic level. The mechanism is shown in Fig. 2.24. This mechanism, when
applied to hadron colliders, can produce cross sections for the prompt
production of the $\chi_{c1}(3872)$ if the momentum integration range for the
hadron-hadron Green’s function extends up to a few hundreds of MeV [272, 274,
275].

Figure 2.25:  Differential cross sections ${d\sigma}/{dk}$ (in units of
nb/GeV) for the semi-inclusive production of $D^{*0}\bar{D}^{*0}$ and
$\Sigma_{c}^{*+}\bar{D}^{0}$ through electron-proton scattering [270], where
$k$ is the c.m. momentum of the open-charm hadrons. The histograms are
obtained from the Pythia event generator while the curves are fitted according
to the momentum dependence $k^{2}$. The electron and proton energies are set
to 3.5 and 20 GeV, respectively.

As an example, in Fig. 2.25 we show the differential cross sections generated
using Pythia [276] for the semi-inclusive productions of the
$D^{*0}\bar{D}^{*0}$ and $\Sigma_{c}^{*+}\bar{D}^{0}$ pairs in electron-proton
collisions with the electron and proton beam energies set to 3.5 and 20 GeV,
respectively. The distribution can be well fitted by a $k^{2}$ dependence with
$k$ the c.m. momentum of the open-charm hadrons. The $XHH^{\prime}$ coupling
in Fig. 2.24 can be extracted from measurements or evaluated in
phenomenological models. In particular, for the hadronic molecular model, the
coupling is connected to the binding energy (see [159]). The loop in Fig. 2.24
is ultraviolet divergent, and the divergence in principal needs to be absorbed
into the multiplicative renormalization of the short-distance part. For an
order-of-magnitude estimate, the loop integral is evaluated using a Gaussian
regulator with a cutoff $\Lambda$ of 0.5 and 1 GeV. We list rough estimates
for the production cross sections of the $\chi_{c1}(3872)$,
$Z_{c}(3900)^{+,0}$, $X(4020)^{0}$ and the $P_{c}$ states observed by LHCb in
Table 2.4. The estimates for four more $P_{c}$ states predicted in the
hadronic molecular model using heavy quark spin symmetry [197, 277, 192] are
also presented with masses and couplings taken from Ref. [192]. Considering an
integrated luminosity of 50 fb-1, this leads to $\mathcal{O}(10^{5})$ events
for each of the $P_{c}$ states, and $\mathcal{O}(10^{6})$ for the
$\chi_{c1}(3872)$ and $\mathcal{O}(10^{7})$ events for $Z_{c}$ states. Notice
that neither branching fractions of further decays nor the detection
efficiency is taken into account here. For more details, we refer to Ref.
[270].

To conclude this section, it is promising that EicC can contribute
significantly to the study of heavy-flavor exotic hadrons, in particular to
charmonium-like states and hidden-charm pentaquarks, and thus contribute to
the understanding of how the emergent hadron spectrum is formed by the
nonperturbative strong interaction.

Table 2.4: Rough estimates of integrated cross sections (in units of pb) at EicC for the semi-inclusive production of a few selected states in the hadronic molecular (HM) model [270], where $\Lambda$ refers to the cutoff in the Gaussian regulator of the two-hadron Green’s function. The processes are $e^{-}+p\rightarrow\text{HM}$+all, where $\text{HM}=\chi_{c1}(3872)$, $Z_{c}(3900)^{0/+}$, $X(4020)^{0}$, and seven $P_{c}$ states. The energy configuration considered here is $E_{e}=3.5$ GeV and $E_{p}=20$ GeV. The branching fractions of further decays and the detection efficiency are not yet considered here.  | Channel |  $\Lambda=0.5$ GeV |  $\Lambda=1.0$ GeV
---|---|---|---
$\chi_{c1}(3872)$ | $D\bar{D}^{*}$ | 21 | 89
$Z_{c}(3900)^{0}$ | $(D\bar{D}^{*})^{0}$ | $0.3\times 10^{3}$ | $1.1\times 10^{3}$
$Z_{c}(3900)^{+}$ | $(D\bar{D}^{*})^{+}$ | $0.4\times 10^{3}$ | $1.3\times 10^{3}$
$X(4020)^{0}$ | $(D^{*}\bar{D}^{*})^{0}$ | $0.1\times 10^{3}$ | $0.5\times 10^{3}$
$P_{c}(4312)$ | $\Sigma_{c}\bar{D}$ | 0.8 | 4.1
$P_{c}(4440)$ | $\Sigma_{c}\bar{D}^{*}$ | 0.7 | 4.7
$P_{c}(4457)$ | $\Sigma_{c}\bar{D}^{*}$ | 0.5 | 1.9
$P_{c}(4380)$ | $\Sigma_{c}^{*}\bar{D}$ | 1.6 | 8.4
$P_{c}(4524)$ | $\Sigma_{c}^{*}\bar{D}^{*}$ | 0.8 | 3.9
$P_{c}(4518)$ | $\Sigma_{c}^{*}\bar{D}^{*}$ | 1.2 | 6.9
$P_{c}(4498)$ | $\Sigma_{c}^{*}\bar{D}^{*}$ | 1.2 | 9.8

###  2.5 Other important exploratory studies

####  2.5.1 Proton mass

The major part of the mass of observable matter is carried by the nucleons
(neutrons and protons) that constitute all the atomic nuclei in the Universe.
Nucleons themselves are constituted from the gluons and quarks of QCD. So a
key piece of the puzzle surrounding the origin of mass lies with understanding
how the proton’s mass emerges from the QCD Lagrangian, expressed in terms of
light valence-quarks, massless gluons and the interactions between them. The
current-quark masses are produced by the Higgs mechanism; but based on those
masses, a straightforward application of notions from relativistic quantum
mechanics delivers a proton mass that is two orders-of-magnitude too small.
Plainly, the source of the proton’s mass is far more subtle. Consider,
therefore, a sum rule connected with the trace of QCD’s energy-momentum tensor
(EMT) [278]:

$m_{p}=H_{m}+H_{a}\,,\quad H_{m}=\langle p|m\bar{\psi}\psi|p\rangle\,,\quad
H_{a}=\langle p|[\gamma_{m}m\bar{\psi}\psi+\beta(g)(E^{2}-B^{2})]|p\rangle\,,$
(2.6)

where: $\langle p|p\rangle=1$; and $m$ represents the light-quark current
masses, whose values are of the order $2$-$4\,$MeV at a renormalisation scale
$\zeta=2\,$GeV. The first of the two terms in Eq. (2.6) is that seeded by the
Higgs mechanism. The second term is more interesting in many ways: it is the
trace anomaly, whose origin is intimately connected with the need to
regularize and renormalize any quantum field theory defined in four
dimensional spacetime. (These and related issues are discussed further in Sec.
2.6.1.)

The $H_{m}$ term in Eq. (2.6) is unambiguous; but there are many ways to
decompose and rearrange $H_{a}$ [279, 280]. One popular approach is to focus
on the energy of a proton at rest and write [281, 282, 283]:

$m_{p}=H_{m}+H_{a}=H_{m}+H_{q}+H_{g}+\tfrac{1}{4}H_{a}\,,$ (2.7)

where

$H_{q}=\langle
p|\psi^{\dagger}(-i\textbf{D}\cdot\alpha)\psi|p\rangle\,,\quad\quad
H_{g}=\langle p|\tfrac{1}{2}(E^{2}+B^{2})|p\rangle$ (2.8)

are, respectively, the quark and gluon kinetic energies. Contemporary lattice-
QCD calculations reveal [284, 285, 286] that only about 9% of $m_{p}$ is
generated by the $u$-, $d$-, $s$-quark contributions in $H_{m}$. However, even
this contribution is nontrivial because it is built from products of the
Higgs-generated current-quark masses and enhancement factors that express
nonperturbative QCD dynamics, _viz_. $H_{m}\sim m_{\rm Higgs}\times\langle
p|\bar{\psi}\psi|p\rangle_{\rm np\,QCD}$. The remaining 91% is essentially
dynamical. It can be considered as the contribution generated by strong QCD
forces through the trace anomaly; or in the second decomposition, broken into
identified pieces that include those from quark and gluon kinetic energies.
Nonetheless, no matter how one chooses to cut the pie, a very large fraction
of the proton’s mass emerges as a dynamical consequence of strong interactions
within QCD.

As noted above and described in more detail below, the appearance of a nonzero
contribution to the trace of the QCD EMT, call it $\Theta$, is a
quintessentially quantum field theoretical effect. Empirically, the
expectation value of $\Theta$ is large in almost every hadron state. The
exception is the chiral-limit pseudoscalar meson, for which the expectation
value is expected to vanish. Providing a mathematically rigorous proof that
these outcomes are truly dynamical consequences of QCD would constitute a
solution to one of the seven Millennium Prize Problems [287]. Modern progress
in theory is seemingly bringing such a proof within reach; and numerical
simulations of lattice-regularized QCD are beginning to yield quantitative
results for the independent terms in Eq. (2.7).

Given the fundamental importance of the scale set by $m_{p}$, much attention
is now focused on the experimental confirmation of Eq. (2.6). In this context
it has been argued, using the operator product expansion and low-energy
theorems [288, 289, 290], that the heavy-vector-meson–proton scattering
amplitude near threshold is dominated by $H_{a}$ and sensitive to the $H_{m}$
correction. Recent theoretical study also suggests that this process at large
photon virtualities $Q^{2}$ can be used to extract the gluon part of the
proton gravitational form factor and sensitive to the trace anomaly effect at
subleading level[290]. A preliminary analysis of GlueX data on $J/\psi$
photoproduction in Hall-D at JLab is broadly consistent with the prediction.
However, a range of theoretical issues complicate the interpretation of such
$J/\psi$ measurements in this way and substantially more work is needed before
firm conclusions can be drawn.

On the other hand, the case for a connection between the $\Upsilon p$ near-
threshold scattering amplitude and the proton mass sum rule is theoretically
much cleaner. Experimentally, this system is inaccessible at JLab; and
existing measurements at other facilities are restricted to $W\gtrsim
90\,$GeV, as evident in Fig. 2.19, which is far above threshold. Consequently,
the EicC can here contribute uniquely, being able to explore collision
energies $W<20\,$GeV.

Figure 2.26:  Distribution of $ep\rightarrow ep\Upsilon\rightarrow
ep\mu^{+}\mu^{-}$ events, assuming $50\,{\rm fb}^{-1}$ integrated luminosity.
The $ep$ central energy $\sqrt{s}=16.755\,$GeV in the figure corresponds to
that of EicC. Cases with central energy$\sqrt{s}=15.0$, $17.755$ and
$20.0\,$GeV are also illustrated, but only $\sqrt{s}=16.755$ GeV is expected
for a running to accumulate 50 fb-1 luminosity. _Left panel_ – number of
events as a function of photon virtuality,$Q^{2}$; and _right panel_ – event
number as a function of $\sqrt{s}$.

Using the reaction $ep\rightarrow ep\Upsilon\rightarrow epl^{+}l^{-}$, as
pictured in Fig. 2.21 where $V=\Upsilon$ and the orange area represents a
$t$-channel Pomeron [291, 214], the distribution of events achievable with
EicC is shown in Fig. 2.26. It seems that the cross section estimation within
Pomeron exchange under a proper consideration of phase space [214] is indeed
consistent at the order-of-magnitude level with other models as shown in Fig.
2.20, in which dipole Pomeron model is used as the input of simulation.
Evidently, EicC could produce around 600 events under the proposed design,
$80$-$85$% of which lie in the $Q^{2}<1\,$GeV2 region, with more than $90$% at
$Q^{2}<10\,$GeV2. Moreover, if the two decay channels
$\Upsilon\rightarrow\mu^{+}\mu^{-}$, $e^{+}e^{-}$ are detected simultaneously,
the number of reconstructed events is even larger. Fig. 2.27 displays the
anticipated reconstruction profiles of the $\Upsilon$ in the distributions of
mass, transverse momentum, rapidity and quasi-rapidity. Though the detector
reconstruction efficiency would be lower than that of the resonant process in
Tab. 2.3, one can confidently assume a value of $20$%, given that the final
states are all charged particles. So it is feasible to investigate detailedly
the $t$-dependent cross sections at EicC.

Figure 2.27:  Reconstruction profiles in the distributions of mass, transverse
momentum, rapidity and quasi-rapidity, for the final state $\Upsilon$ in the
reaction $ep\rightarrow ep\Upsilon\rightarrow ep\mu^{+}\mu^{-}$.

This discussion demonstrates that EicC can deliver precision in the study of
$\Upsilon$ production and potentially thereby open a window onto the origins
of the proton’s mass. As a significant collateral benefit, $\Upsilon$
production is an important background that must be understood when analysing
data taken in searches for the hidden-bottom five-quark state, $P_{b}$.

####  2.5.2 Structure of light pseudoscalar mesons

Theoretical imperatives for investigating and revealing the structure of light
pseudoscalar mesons are detailed in Sec. 2.6. The case has many facets because
the pion is the lightest known hadron and it has a unique and crucial position
in nuclear and particle physics [292]. For example, the pion is the closest
approximation to a Nambu-Goldstone (NG) boson in Nature. It is massless in the
absence of Higgs-boson couplings into QCD and remains unusually light when
those couplings are switched on. In addition, this light pion is essential to
the formation of nuclei, carrying the strong force over length-scales large
enough to enforce stability against electromagnetic repulsion between the
protons within a nucleus. Thus, understanding pion structure is of the utmost
importance. Two clear paths are available: namely, the measurement of pion
elastic form factors and of pion structure functions.

Existing empirical knowledge of pion structure is poor. Elastic form factor
measurements do not extend beyond $Q^{2}=2.45\,$GeV2 [293, 294, 295, 296, 297,
298] and existing structure function measurements are more than thirty years
old [299, 300, 301, 302, 303, 304]. The kaon situation is worse; and that is
unsatisfactory for many more reasons. Largest amongst them being that the
standard model of particle physics has two sources of mass: explicit,
generated by Higgs boson couplings; and emergent, arising from strong
interaction dynamics, responsible for the $m_{N}\sim 1\,$GeV mass-scale that
characterises nuclei, and the source for more than 98% of visible mass.
Emergent hadronic mass is dominant for all nuclear physics systems; but the
Higgs mechanism applies modulations that are critical to the evolution of the
Universe, _e.g_. CP-violation, discovered in neutral kaon decays [305]. Thus,
knowledge of kaon structure is necessary because it provides a window onto the
interference between Higgs boson effects and emergent mass [306, 307].

The impediment to experimentally mapping the structure of light pseudoscalar
mesons is simply explained. These systems are unstable, they decay quickly:
so, how can one build a target? One answer is to use the Drell-Yan process at
high-energy accelerators. This is the mode exploited thirty years ago [299,
300, 301, 302, 303, 304]. Another approach is to measure the leading neutron
in high-energy $ep$ collisions [308, 309]. In these processes, the range of
light-front momentum fraction, $x$, has been somewhat limited and existing
errors are large.

|  |
---|---|---
Figure 2.28:  Sullivan processes: a nucleon’s pion cloud provides access to
(a) pion elastic form factor and (b) pion parton distribution functions. The
intermediate pion, $\pi^{\ast}(P=k-k^{\prime})$ with the Mandelstam variable
$t=P^{2}$, is off-shell.

At high-luminosity facilities, the Sullivan process, illustrated in Fig. 2.28,
becomes a very good method for for gaining access to meson targets. The
approach capitalizes on the feature that a proton is always surrounded by a
meson cloud and can sometimes be viewed as a correlated $\pi^{+}n$
$(K^{+}\Lambda)$ system. Theory predicts [310] that such processes provide
reliable access to a pion target on $-t\lesssim 0.6\,$GeV2; and for the kaon,
on $-t\lesssim 0.9\,$GeV2. The 12 GeV-upgraded JLab facility will exploit
these reactions to extend the $Q^{2}$ reach of existing $\pi$ and $K$ form
factor measurements [311, 312] and the $x$-range of available data on the pion
structure function [313]. Efforts will also be made to measure the kaon
structure function [314]. Given the wider kinematic range of EicC, its
capacity to achieve these goals will be far greater. At the EicC, the
scattering angle and energy of the final state baryons can be measured
precisely, to tag the Sullivan process and to measure the invariant mass of
the virtual light mesons. Technologies of the far-forward hadron calorimeter
and trackers could be learned from projects at HERA (H1 and ZEUS), and from
similar experiments at the US-EIC. The final state neutron at the EicC has
energy around 15 GeV, a theta angle around 25 mrad, and $p_{T}$ $<$ 0.5 GeV.
Assuming the energy resolution to be 35%/$\sqrt{(E/GeV)}$ and the spatial
resolution to be 1 cm for the far-forward neutron detector, one can estimate
the resolution for the invariant mass of the virtual meson,
$\frac{\sigma_{|m^{2}(\pi^{*})|}}{|m^{2}(\pi^{*})|}$, to be around 27% at the
EicC.

Figure 2.29:  Projected statistical errors on EicC pion form factor
measurements compared with those at the US EIC [306] and JLab 12 GeV program
[315]. Also depicted: extant JLab data obtained by the Fπ-Collaboration [294,
295, 296, 297, 298]; and statistical error projections for kaon form factor
measurements at EicC.

Application and experience have delivered a reliable approach to obtaining the
pion elastic form factor, $F_{\pi}(Q^{2})$, from the Sullivan process [316,
317, 318, 319]. The longitudinal electroproduction cross-section is expressed
such that $F_{\pi}(Q^{2})$ is the only unknown, which can be determined by
comparison between data and the model. The $\pi^{+}/\pi^{-}$ production ratio
extracted from electron-deuteron beam collisions in the same kinematics as
charged pion data from electron-proton collisions is used to ensure that the
longitudinal cross-section has been isolated [292]. Fig. 2.29 shows that a
high-luminosity, high-energy EicC can deliver precise results on pion and kaon
electromagnetic form factors, providing detailed maps on the domain
$Q^{2}/{\rm GeV}^{2}\in[10,30]$, which has the highest physics discovery
potential.

The estimates in Fig. 2.29 were prepared using the following cuts:
$|t|<0.6\,$GeV and $W/{\rm GeV}\in(2,10)$. Separating the data into ten bins
on $Q^{2}/{\rm GeV}^{2}\in[10,30]$, the statistical errors on $F_{\pi}(Q^{2})$
data are competitive with all other existing proposals. Regarding the size of
the estimated statistical errors, the same is true for the EicC kaon form
factor measurement.

Fig. 2.28(b) shows that a Sullivan process can also be used to measure $\pi$
and $K$ structure functions. Compared with elastic form factor measurements,
the cross-section is much larger because the meson target is shattered. Deep
inelastic $e\pi$ interactions are ensured by selecting events in which the
transverse momentum of the tagged final-state neutron is small and its
longitudinal momentum exceeds 50% of that of the incoming proton.

For kaon structure functions, the tagged outgoing system is the
$\Lambda$-baryon.

The projected statistical precision of an EicC pion structure function
measurement is sketched in Fig. 2.30. It assumes roughly one year of running
and is based on a Monte-Carlo simulation of leading-neutron tagged DIS with
the pion valence-quark distribution function taken from Ref. [320]. Fig. 2.30
indicates that EicC can map the domain $x_{\pi}\in(0.01,0.95)$ with precision,
yielding data that could be crucial to resolving the pion structure function
controversy described in Sec. 2.6.3.

Figure 2.30:  Projected statistical uncertainty on pion structure function in
a certain $Q^{2}$ bin as a function of the four-momentum-transfer $-t$ and the
Bjorken-x of pion. The cuts for producing the projection are also shown in the
plot. In order to select the scattering between electron and virtual pion, a
cut on $x_{L}>0.75$ is made [309]. Moreover, the detection efficiency for the
final state neutron is assumed to be $50\%$. The measurement at EicC could be
performed on a large kinematic domain with uncertainty that is uniformly
$\lesssim 5$%.

####  2.5.3 Intrinsic charm

Great progress has been made in the literature in understanding the
fundamental structure of the nucleon in recent years. However, we still know
little about the properties and distributions of the heavy quarks in nucleon.
As a potential constituent of the nucleon, information for those heavy-flavor
content plays an important role in testing Standard Model (SM), and in
searching for new physics beyond the SM.

Heavy content in nucleon is predicted theoretically in QCD theory. In the
light-cone framework, the wave function of a proton can be expanded in terms
of superposition of Fock states as following,

$\displaystyle|p\rangle=c_{1}|uud\rangle+c_{2}|uudg\rangle+c_{3}|uud{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c\bar{c}}\rangle+\cdots.$
(2.9)

Here, $|F\rangle$ are the Fock states, and the coefficients $c_{j}$
($j=1,2,\cdots$) are proportional to the wave function amplitudes of Fock
components [321, 322, 323]. It indicates that there are heavy charm quarks
arise in the proton, even the states such as $|uudc\bar{c}\rangle$ are
extremely rare. On the view point of QCD theory, the extra
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c\bar{c}}$
pair in the proton can be generated in two distinct processes involving in
perturbative and nonperturbative effects.

One probable charm content arisen in a proton is the “extrinsic charm (EC)”.
In this case, a gluon radiates from the valence quark in the proton, and then
splits to a $c\bar{c}$ pair associated with large transverse momentum. The
gluon must be hard enough in order to produce a heavy $c\bar{c}$ pair. The
process is therefore governed by the QCD evolution corresponding to the short
distance effects. In this scenario, the charm and anti-charm quarks have the
same significant features in the proton. The EC behaves like a sea quark and
is generally softer than the gluon by a factor of $(1-x)$, where $x$ is the
momentum fraction of EC in the proton. The parton distribution function (PDF)
describes the extrinsic charm density in the proton, which is related to the
momentum fraction $x$ and the factorization scale $\mu_{F}$. Using an initial
form at a specific scale, such as the charm quark mass $m_{c}$, the EC PDF can
be evolved to any factorization scale $\mu_{F}$ with the help of the
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [324,
325, 326, 327]. In practice, the initial form of the EC PDF could be modeled,
whose parameters can be fixed from a comprehensive global analysis of hard
scattering data generated from a variety of fixed-target or collision
experiments. The extrinsic charm distribution is assumed to be concentrated in
the small region of momentum faction, e.g. $x\in[0,0.1]$, since it drops down
quickly at $x>0.1$. The EC component is usually neglected in early studies on
the processes with small and moderate center-of-mass energy (CME) in
collisions, especially for the hadronic colliders where the small $x$ events
are generally not recorded. With much higher CME of colliders today, the EC
becomes more and more important in high energy processes.

Another probable charm content arisen in a proton is the “intrinsic charm
(IC)”, which is quite different from the extrinsic one and has strong hints
from experimental observations. In year 1979, the production of charmed
particles was reported by the CERN and ACCDHW Collaborations via
$pp\rightarrow D^{+}(\Lambda_{c}^{+})X$ with the center-of-mass energy
$\sqrt{s}=53$ and 63 GeV, respectively [328, 329, 330, 331]. The cross section
was measured to be $0.1\sim 0.5\,\rm mb$, which is larger than theoretical
predictions ($10\sim 50\,\rm\mu b$ [332]) by about one order of magnitude.
Moreover, the $D^{+}$ was found to be generated abundantly in the forward
region, which is hard to be explained by simply using the ordinary extrinsic
charm. In order to narrow the gap between experimental data and theoretical
predictions, the idea of IC was proposed by S. J. Brodsky et al. in Ref.[333,
334, 335]. Theoretically, this fluctuation of the proton can be achieved in
two ways, one is the interaction between valance quarks and multiple soft
gluons, another one is vacuum polarization which is usually extremely rare and
can be neglected. The intrinsic charm generated in this manner exhibits
remarkable differences from the extrinsic charm, and the probability of
finding IC in a proton is proportional to ${\cal O}(1/m_{c}^{2})$. In contrast
to the EC, the IC can be generated at or even below the energy scale of the
heavy quark mass threshold. The intrinsic charm and anti-charm are not
necessary to have the same distributions in the proton. The IC has a “valence-
like” character, which has small contributions in low momentum fraction $x$
and dominates in the relatively large $x$ region. The IC distribution is non-
perturbative, and many models have been suggested in the literature, whose
inputs can be fixed by global fitting of various known experimental data.

Some previously done measurements could indicate the existence of IC in the
nucleon. In 1980s, the charm structure function $F_{2}^{c}(x,\mu)$ was
measured by the European Muon Collaboration (EMC) [336, 189]. An enhancement
at large $x$ beyond theoretical expectation was reported by EMC, and such a
gap could be compensated by taking the IC into consideration [337]. The
inclusive photon production with heavy-flavor $Q$ jets in hadronic collisions
would also provide valuable information for the IC distributions in the
nucleon. The partonic process $g+Q\rightarrow\gamma+Q+X$ gives the main
contributions to the photon events, which depends strongly on the heavy
contents in an incident hadron. Data from the DØ experiment at the Tevatron
[338] shows disagreement between the prediction without IC and the measurement
for $\gamma+c+X$ in large transverse momentum of the photon. By including IC
contributions, the discrepancy is reduced but still prominent in the larger
$p_{T}^{\gamma}$ region, in particular when $p_{T}^{\gamma}$ is larger than 80
GeV.

Up to now, a definite conclusion on the existence of IC is still missing.
Additional experiments are necessary for investigating the IC. The measurement
of inclusive charmed hadron production at hadron colliders is an additional
promising way to investigate the IC [339, 340]. The doubly heavy baryons
(especially $\Xi_{cc}$) production associated with initial charm quark at the
parton level will be suitable to investigate the intrinsic content, since the
IC impacts significantly on these production channels. The $\Xi_{cc}$
production in hadronic collisions involve three typical mechanisms, i.e. the
gluon-gluon fusion $(g+g)$, the gluon-charm collision $(g+c)$, and the charm-
charm collision $(c+c)$, at the proton-proton (or proton-antiproton)
colliders. Conventionally, contributions from the gluon-gluon fusion are
expected to be dominant in the hadronic production of $\Xi_{cc}$. However,
other production mechanisms may also have sizable contributions. For the
$(g+c)$ and $(c+c)$ production mechanisms, the initial $c$ quark can be either
extrinsic or intrinsic in the incident protons. Because the proportion of the
IC components in the nucleon is small, which is only up to $\sim 1\%$, the
intrinsic charm usually gives no significant contribution in most of the high-
energy processes. However, in specific kinematic regions, the IC may lead to
unexpected conspicuous consequences in the $\Xi_{cc}$ production.

The hadronic production of $\Xi_{cc}$ baryon was investigated at the LHC, the
Tevatron, and the Fixed-target Experiments at hadron collider (FixExp@HC)
[341, 342, 343]. Using the generator GENXICC [344, 345, 346, 347], one may
simulate the hadronic production of $\Xi_{cc}$ with both extrinsic and
intrinsic charm being considered. Because the intrinsic component at large $x$
will decrease at the high factorization scale, the $(g+g)$ channel becomes
dominant in the $\Xi_{cc}$ hadroproduction with high CME. The IC mechanisms
shall have a significant impact on the hard processes with moderate
factorization scales at those colliders with a relatively lower center-of-mass
energy. Therefore, colliders with high luminosity at lower center-of-mass
energy, such as the fixed-target experiment (like After@LHC [348, 349, 350,
351, 352] operating at a center-of-mass energy $\sqrt{s}=115\;\rm GeV$) and
future electron-ion colliders (e.g., EIC US [30] and EicC, etc.), would be
ideally suited to discover or constrain the intrinsic content in nucleon.

The EicC will provide a brand new mode to study the production of doubly
charmed baryon. Two important subprocesses occur in the electron-nucleus
($e$-N) collision with exchange of a virtual photon between the electron and
nucleus, i.e., $\gamma+g\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ and
$\gamma+c\rightarrow\Xi_{cc}+\bar{c}$, which are classified by the virtuality
$Q^{2}$ of the photon. Numerically, we observe that the intrinsic charm
enhanced the total cross sections by nearly 3 times to that without intrinsic
components with $A_{\rm in}=1\%$ in the $e$-Au collision mode at the EicC.
Moreover, at the instantaneous luminosity of $2.0\times 10^{33}\;\rm
cm^{-2}s^{-1}$, the estimated number of $\Xi_{cc}$ events in one year is about
$4.0\times 10^{5}$ by adopting the Non-relativistic Quantum Chromodynamics
[353]. It makes the precise investigation of the properties of the doubly
charmed baryon accessible at the EicC.

The observation of the doubly charmed baryon $\Xi_{cc}^{++}$ had been reported
by the LHCb collaboration [354] in 2017. However, the $\Xi_{cc}^{+}$ with a
similar production rate but a bit shorter lifetime than $\Xi_{cc}^{++}$, has
not been discovered yet. The shorter lifetime means more loss events of
$\Xi_{cc}^{+}$ before tested by the detectors, which requires more efforts in
experimental measurements. Recently, the LHCb collaborations reported zero
results of searching for $\Xi_{cc}^{+}$ [355] again with higher integrated
luminosity than that in 2013 by an order of magnitude. This is in
contradiction with the observation of $\Xi_{cc}^{+}$ baryon by the SELEX in
2002 [356] and in 2005 [357]. This contrary may be interpreted by the
difference of kinematic cuts between those two experiments. The LHCb
experiment may lose more events in the small $p_{t}$ region than that in the
SELEX, which are related closely to the intrinsic content in a nucleon.
Therefore, more different kinds of experiments, such as the After@LHC, the
EicC, and the LHeC, etc., are needed for clarifying the puzzle in searching
for the $\Xi_{cc}^{+}$. Conversely, experimental studies on $\Xi_{cc}$
production at those colliders could help to further progress in verifying the
existence of IC.

In conclusion, different high-energy colliders (such as the After@LHC, the
EicC, etc.), are expected to increase the probability for the discovery of IC.
Moreover, those different experimental platforms could provide important
cross-checks to ensure the correctness of experimental measurements and
theoretical analysis. And it shall shed light on the intrinsic heavy mechanism
and fundamental structure of the nucleon.

###  2.6 QCD theory and phenomenology

####  2.6.1 Synergies

High level theory and phenomenology are required in order to inspire, guide,
and capitalize on the wide ranging program of EicC experiments. Rigorous QCD
theory must be complemented by insightful phenomenology so that every
discovery opportunity can be seized. The lattice formulation maintains the
closest connections with the QCD Lagrangian. Here, steady progress continues
and testable predictions are being delivered. Tight links with QCD are also
provided by continuum Schwinger function methods, e.g. the Dyson-Schwinger
equations (DSEs). With the added flexibility of continuum methods, DSEs
provide access to a wider range of observables. Thus, with their complementary
content, lattice QCD and DSEs have a natural synergy that can be exploited for
the benefit of EicC physics.

####  2.6.2 Lattice QCD

Lattice QCD is a fundamental method to study strong interactions non-
perturbatively from first principles. It directly computes the QCD path
integral in a discretized finite-volume Euclidean space-time, and then the
finite lattice spacing and volume provide natural ultraviolet and infrared
cut-offs. The input parameters in lattice QCD are the quark masses and the
coupling constant, which is related to the lattice spacing by the
renormalization group. Today, with the help of powerful computing facilities,
lattice QCD can provide important information on many aspects of EicC physics
and comparisons with experimental results.

##### Nucleon spin structure

As discussed in Sec. 2.1, the proton’s spin can be decomposed into the
helicity and orbital angular momentum of quarks and gluons. The contribution
from quark helicity is the sum of quark polarization along the proton spin
direction, which can be obtained from lattice QCD by computing the first
Mellin moment of the polarized parton distribution function of quarks. There
have been many lattice calculations, see Refs. [358, 359] for reviews. Recent
lattice results [360, 361, 362] are mutually consistent and agree well with
global fits; and strange flavor results are more precise than the global fits.
Both experiment [363] and lattice calculation [364] indicate that the gluon
helicity contributes considerably to proton spin, but improved lattice QCD
calculations are needed in order to deliver precise predictions [365].

The QCD energy-momentum tensor admits several expressions for the quark and
gluon orbital angular momentum contribution to the proton spin. With the
Jaffe-Manohar decomposition [366], both the orbital and total angular momentum
of quarks and gluons can be defined in a gauge invariant manner with
appropriate light-cone gauge links. On the other hand, gauge invariant and
frame independent total angular momentum of quarks and gluons can be defined
through the symmetrized energy-momentum tensor based on Ji’s decomposition
[60], expressed in Eq. (2.3). The difference between the two is that the
quark-gluon interaction term is allocated to the gluon (quark) orbital angular
momentum in the Jaffe-Manohar (Ji) decomposition. The gauge invariant total
angular momenta of quarks and gluons have been calculated in lattice QCD [367,
360, 368]: the gluon angular momentum result needs improved precision.
Exploratory lattice studies have been made for the quark orbital angular
momentum and preliminary results have been obtained [369, 370].

Lattice QCD is playing a valuable role in the study of nucleon spin structure,
with important recent progress. Quark helicity results for different flavors
have reached 10% precision, and preliminary estimates have been obtained for
the gluon helicity, quark orbital angular momentum and quark (gluon) total
angular momentum. It is expected that lattice QCD will be able to provide more
accurate and extensive information on nucleon spin structure.

##### Proton mass decomposition

From the sum rule of the QCD energy-momentum tensor, Eq. (2.6), the invariant
mass of proton can be decomposed into quark mass term and the trace anomaly
[278]. The quark mass term is a product, involving the chiral condensate and
quark current masses; hence, it is directly related to the mass generated by
the Higgs mechanism and scale invariant. This part only contributes about 9%
of the total proton mass [283] for three light flavors; so most of the proton
mass can be considered as arising from the trace anomaly [278].

In the chiral limit, the pion matrix element of the trace anomaly is zero,
whereas it is nonzero for the nucleon, yielding the entire nucleon mass. As
yet, there is no direct calculation of the trace anomaly using lattice QCD,
but the proton mass sum rule has been used to predict the trace anomaly
contribution to $m_{p}$ [283]. A calculation of the trace anomaly contribution
to $m_{p}$ is being pursued using lattice QCD; and success will enable a
comparison between QCD theory and the EicC measurements described in Sec.
2.5.1.

Using one definition of the quark kinetic energy [281], its contribution to
$m_{p}$ can be obtained from the quark light-front momentum fraction, i.e. the
first moment of the unpolarized quark distribution functions; similarly, for
the gluon kinetic energy contribution. Lattice results for the momentum
fraction [283] agree with phenomenological analyses [358]; but uncertainties
still need to be further suppressed, especially for the sea quarks and gluons.

##### 1-D and 3-D structure of nucleons

Most lattice QCD calculations of PDF-related quantities have been limited to
the first few PDF moments, mainly due to the fact that light-cone PDFs are not
directly accessible in Euclidean space-time. However, novel theoretical
developments in recent years [371, 372, 373] are enabling lattice QCD to
directly calculate PDFs, including 1-D Bjorken-x dependent PDFs, generalized
parton distribution functions(GPDs), and also transverse moment dependent
parton distribution functions(TMDs)

Large momentum effective theory (LaMET) is one approach [374]. It uses an
equal time correlation function to reach a light cone correlation function
using standard methods of effective field theory matching and running. Namely,
from the equal time correlations that can be computed in lattice QCD, one
obtains the so-called quasi-PDFs; and when the nucleon momentum is large
enough, quasi-PDFs can be matched perturbatively to the physical PDFs, with
computable large-momentum power corrections.

LaMET has been employed in some lattice QCD calculations, with notable
progress on the non-singlet quark PDFs ($u(x)-d(x)$), including unpolarized,
polarized and transverse distributions. The latest results agree with global
fits [375, 376, 377]. The small $x$ region is challenging for laMET because it
requires simulations with high nucleon momenta, which is difficult in lattice
QCD. At present, lattice QCD can compute valence-quark PDFs on $x\gtrsim 0.1$
and provide some constraints at small-$x$ for the sea quarks via PDF moments
[378]. LaMET can also be used to compute GPDs, e.g. the pion’s unpolarized
quark GPD has been explored [379] and this may be extended to the nucleon
case.

TMD calculations using lattice QCD are more complicated than those of PDFs and
GPDs. An additional soft function is required to factorize processes involving
small transverse momentum, like Drell-Yan production and semi-inclusive DIS.
This function involves two light-like gauge links along reversed light-cone
directions; hence, cannot be simulated directly in Euclidean space. Recently,
it was argued that this difficulty can be overcome [380]; and as shown in Fig.
2.31, several lattice QCD attempts at compuation of the soft function have
been carried out [381, 382]. This opens a path to prediction of TMD-related
quantities using lattice QCD.

Figure 2.31: Lattice results for the TMD soft function: intrinsic part (left
panel) [382] and rapidity dependent part (right panel) [381, 382]. For the
rapidity dependent part, the results from Ref. [382] (red/blue points) are
consistent with those from Ref. [381] (green/brown points). The lattice
calculation reaches much larger transverse separation $b_{\perp}$ than
perturbative calculations.

##### Partonic structure of the nucleus

Nucleons within the nucleus appear to interact weakly with each other via
long-range forces and the binding energy per nucleon is small in comparison
with the nucleon mass; yet, as illustrated in Fig. 2.14, various experiments
have indicated that the PDFs of bound nucleons are different from those in
free space [383, 127, 20]. This highlights the importance understanding
nuclear structure from the gluon-quark level. Naturally, for quarks and gluons
contained within nucleons and nuclei, the non-perturbative nature of the
bound-state problem makes the theoretical study very difficult. Lattice QCD
simulations can shed light on this problem.

Lattice calculation at physical kinematics are very challenging owing to the
signal-to-noise problem. In a first step, Ref. [384] reported a study of the
PDFs of ${}^{3}\text{He}$, extracting the first Mellin moment of the
unpolarized isovector quark PDFs at an unphysical quark mass corresponding to
$m_{\pi}\sim 800$ MeV. The ratio of the quark momentum fraction in
${}^{3}\text{He}$ to that in a free nucleon was found to be consistent with
unity. Although no EMC effect was observed, this study together with an
earlier lattice analysis of the ${}^{4}\text{He}$ binding energy [385] show
that lattice methods are reaching a level of practicable maturity for very
light nuclei. Calculation precision can be controlled to a level of few
percent by using a relatively heavy pion mass; and even in this unphysical
realm, results from the HALQCD collaboration suggest that some nuclear physics
remains [386].

Further reduction of both statistical and systematic uncertainties requires
more effort from the lattice QCD community. One can anticipate that, with the
continuing development of computer hardware and software, lattice calculations
may come to play a unique role in hunting the EMC effect.

##### Exotic hadrons

The methods to study hadron spectroscopy in lattice QCD are relatively mature.
One computes correlation functions of operators with the desired quantum
numbers and obtains the spectrum from their time dependence. The interpolating
operators encode the hadron structure information. Usually the operators are
constructed by a quark–anti-quark pair (meson) or three quarks (baryon). To
explore the structure of exotic hadrons, such as those discussed in Sec. 2.4,
one needs to build operators that express the exotic’s likely composition,
e.g. multi-quark states, hybrids and glueballs, etc. Most hadrons are unstable
resonances, which appear as poles of hadron scattering amplitudes. Again,
owing to the Euclidean space-time used in lattice QCD, real-time dependent
matrix elements related to the scattering processes cannot be computed
directly. One way to circumvent this problem is to use the finite volume
method developed by M. Lüscher [387], by which the scattering information can
be extracted from the energies of analogous systems in a finite box. This
approach offers a path forward for lattice QCD in calculations of the
structure and properties of exotic states.

EicC is ideal for the study of heavy flavor hadrons. Many charmed exotic
hadrons have been observed in experiments, e.g. the $XYZ$ particles and the
pentaquark candidates $P_{c}$s, but the structure of these states are not
known yet. Some studies on $XYZ$ particles have been performed in Lattice QCD
[388, 389, 390, 391]. However, the results are generally contaminated by
systematic uncertainties, which come from finite lattice spacing, finite
volume, unphysical light quark mass, ignoring coupled channel effects, and so
on. At the same time, it is expected that the bottom counterpart of the charm
exotic states should also exist; yet the number of observed exotic bottom
hadrons is fewer than in the charm sector. With continuing steady improvement,
lattice QCD will be able to contribute to resolving the puzzles associated
with heavy flavor hadrons.

####  2.6.3 Continuum theory and phenomenology

##### Mass and matter

Sec. 2.5.1 highlights that the masses of the neutron and proton, the kernels
of all visible matter, are roughly 100-times larger than the Higgs-generated
masses of the light $u$\- and $d$-quarks. In contrast, Nature’s composite
Nambu-Goldstone bosons are (nearly) massless. In these states, the strong
interaction’s $m_{N}\approx 1\,$GeV mass-scale is effectively hidden.
Furthermore, in the chiral limit, when the Higgs-generated current masses in
QCD’s Lagrangian are omitted, the $\pi$ and $K$ mesons are exactly massless
and perturbation theory suggests that strong interactions cannot distinguish
between quarks with negative or positive helicity. Such a chiral symmetry
would have many consequences, but none of them is realised in Nature. Instead,
the symmetry is broken by interactions. Dynamical chiral symmetry breaking
(DCSB) entails that the massless quarks in QCD’s Lagrangian acquire a large
effective mass [392, 393, 394] and ensures that the interaction energy between
those quarks cancels their masses exactly so that the chiral-limit pion is
massless [395, 396, 397].

DCSB underpins the notion of constituent-quark masses and, hence, sets the
characterising mass-scale for the spectrum of mesons and baryons constituted
from $u$, $d$ quarks and/or antiquarks. Moreover, restoring the Higgs
mechanism, then DCSB is responsible for, _inter alia_ : the measured pion mass
($m_{\pi}\approx 0.15\,m_{N}$); and the large mass-splitting between the pion
and its valence-quark spin-flip partner, the $\rho$-meson
($m_{\rho}>5\,m_{\pi}$). There are many other corollaries, extending also to
the physics of hadrons with strangeness, wherein the competition between
dynamical and Higgs-driven mass-generation has numerous observable
consequences. The competition extends to the charm and bottom quark sectors;
and much can be learnt by tracing its evolution.

Such phenomena, their origins and corollaries, entail that the question of how
did the Universe evolve is inseparable from the questions of how does the
$m_{N}\approx 1\,$GeV mass-scale that characterizes atomic nuclei appear; why
does it have the observed value; and, enigmatically, why does the dynamical
generation of $m_{N}$ have no apparent effect on the composite Nambu-Goldstone
bosons in QCD, _i.e_. whence the near-absence of the pion mass? A decisive
challenge is to determine whether the answers to these questions are contained
in QCD, or whether, even here, the Standard Model is incomplete.

These questions are being addressed using DSEs, which provide a symmetry-
preserving approach to solving the continuum hadron bound-state problem [398,
292, 399, 400, 401]. In connection with the emergence of hadronic mass, the
framework has delivered a significant advance with the prediction of a
process-independent QCD effective charge [402]. Depicted in Fig. 2.32,
$\hat{\alpha}(k)$ saturates at infrared momenta,
$\hat{\alpha}(0)/\pi=0.97(4)$, owing to the emergence of a gluon mass-scale:
$m_{0}/{\rm GeV}=0.43(1)$. These and other features of $\hat{\alpha}(k)$
suggest that QCD is rigorously defined; if so, then it is unique amongst known
four-dimensional quantum field theories. Numerous consequences can be tested
with EicC experiments, a few of which are subsequently described.

Figure 2.32:  QCD’s process-independent running-coupling,
$\hat{\alpha}(k^{2})/\pi$, obtained by combining the best available results
from continuum and lattice analyses [402]. World data on the process-dependent
charge, $\alpha_{g_{1}}$, defined via the Bjorken sum rule, are also depicted.
(Sources are detailed elsewhere [402]. Image courtesy of D. Binosi.)

##### 1-D hadron structure

_Meson Form Factors_.
The best known and most rigorous QCD predictions are those made for the
electromagnetic form factors of pseudoscalar mesons, _e.g_. the pion and kaon.
These hadrons are abnormally light; yet, their properties provide the cleanest
window onto the emergence of mass within the Standard Model [403]. This
connection is expressed most forcefully, in the behaviour of meson form
factors at large momentum transfers. On this domain, QCD relates measurements
simultaneously to low- and high-energy features of QCD, _viz_. to subtle
features of meson wave functions and to the character of quark-quark
scattering at high-energy [321, 404, 322]. These relationships are expressed
concretely in advanced DSE analyses, with predictions that expose the crucial
role of emergent mass [405].

Throughout the modern history of nuclear and particle physics, much attention
has focused on finding evidence for power-law scaling in experimental data.
This is an important step; but it should be remembered that QCD is not found
in scaling laws. Instead, since quantum field theory requires deviations from
strict scaling, then QCD is to be found in the existence and character of
scaling violations. Considering meson elastic electromagnetic form factors,
theory predicts that (_i_) scaling violations will become apparent at momentum
transfers $Q^{2}\gtrsim 10$ GeV2 [405] and (_ii_) the magnitude of any given
form factor on a sizeable domain above $Q^{2}=10$ GeV2 is determined by the
physics of emergent mass. Hence, experiments focused in this area are of the
greatest importance; and as discussed in connection with Fig. 2.29, EicC can
here make crucial contributions.

_Meson Structure Functions_.
At a similar level of rigor is the QCD prediction for the behavior of meson
structure functions. The momentum distributions of light valence quarks within
the pion have the following behaviour at large-$x$ [406, 407, 408]: ${\mathpzc
u}^{\pi}(x;\zeta\approx m_{0})\sim(1-x)^{2}$. The most recent measurements of
${\mathpzc u}^{\pi}(x;\zeta)$ are thirty years old [299, 300, 301, 302, 303,
304]; and conclusions drawn from those experiments have proved controversial
[409]. For example, using a leading-order (LO) perturbative-QCD analysis, the
E615 experiment [304] reported: ${\mathpzc u}_{\rm
E615}^{\pi}(x;\zeta_{5}=5.2\,{\rm GeV})\sim(1-x)^{1}$, in striking conflict
with the expected behavior. Subsequent calculations [410] confirmed the QCD
prediction and eventually prompted reconsideration of the E615 analysis,
demonstrating that E615 data may be viewed as consistent with QCD [411, 412].
However, uncertainty over ${\mathpzc u}^{\pi}(x)$ remains because more recent
analyses of available data have failed to consistently treat higher-order
effects and, crucially, modern data are lacking.

|  |
---|---|---
Figure 2.33:  _Left panel_. Blue solid curve – theory prediction for
${\mathpzc u}^{\pi}(x;\zeta_{2}=2\,{\rm GeV})$ [413, 414]; and cyan short-
dashed curve – phenomenological result from Ref. [415]. _Right panel_. Theory
predictions for the pion’s glue and sea-quark distributions [413, 414]: green
solid curve – ${\mathpzc g}^{\pi}(x;\zeta_{2})$; and red dot-dashed,
${\mathpzc S}^{\pi}(x;\zeta_{2})$. The associated momentum fractions are:
$\langle 2x{\mathpzc q}^{\pi}(x;\zeta_{2})\rangle=0.48(4)$, $\langle
x{\mathpzc g}^{\pi}(x;\zeta_{2})\rangle=0.41(2)$, $\langle x{\mathpzc
S}^{\pi}(x;\zeta_{2})\rangle=0.11(2)$. For comparison, phenomenological
results from Ref. [415]: $p=\,$glue – dark-green long-dashed; and $p=\,$sea –
brown dashed. (The bands around the theory curves express the impact of the
uncertainty in $\hat{\alpha}(k^{2}=0)$, Fig. 2.32. Normalisation: $\langle
x[2{\mathpzc u}^{\pi}(x)+g^{\pi}(x)+S^{\pi}(x]\rangle=1$.)

Pressure is also being applied by modern advances in theory. Lattice QCD is
beginning to yield results for the pointwise behaviour of the pion’s valence-
quark distribution [416, 417, 418, 419]. Furthermore, continuum analyses [413,
414] have yielded the first parameter-free predictions for the valence, glue
and sea distributions within the pion; and revealed that, like the pion’s
leading-twist parton distribution amplitude (PDA) [420], the valence-quark
distribution function is hardened as a direct consequence of emergent mass.

The valence, glue and sea distributions from Refs. [413, 414] are drawn in
Fig. 2.33. Also shown are the phenomenological extractions from Ref. [415].
Even though the valence distribution fitted in Ref. [415] yields a momentum
fraction compatible with the theory prediction, its $x$-profile is very
different. (Ref. [415] did not include the higher-order corrections discussed
in Ref. [412].) It is significant, therefore, that the continuum result for
${\mathpzc u}^{\pi}(x;\zeta_{5})$ matches that obtained using lattice QCD
[419]. Consequently, the Standard Model prediction: ${\mathpzc
u}^{\pi}(x;\zeta=m_{0})\sim(1-x)^{2}$, is stronger than ever before; and as
demonstrated by Fig. 2.30, EicC design specifications would enable it to
deliver clear experimental validation.

If the pion’s valence-quark distributions are controversial, then its glue and
sea distributions can be described as uncertain or worse. Fig. 2.33 – right
panel compares the predictions from Refs. [413, 414] with fits from a combined
analysis of $\pi$-nucleon Drell-Yan and HERA leading-neutron electroproduction
data [415]. The gluon distribution predicted in Refs. [413, 414] and that
fitted in Ref. [415] are markedly different on $x\lesssim 0.05$; and both glue
DFs in Fig. 2.33 disagree with those inferred previously [320, 421]. These
remarks highlight that the pion’s gluon content is empirically uncertain.
Thus, new measurements which are directly sensitive to the pion’s gluon
content are necessary. Prompt photon and $J/\psi$ production could address
this need [422, 423]. The sea DFs in Fig. 2.33 have different profiles on the
entire $x$-domain. Hence, it is fair to describe the sea-quark distribution as
empirically unknown. This problem could be addressed with DY data obtained
from $\pi^{\pm}$ beams on isoscalar targets [424, 422]. Equivalent tagged DIS
measurements at the EicC could also provide this information. Evidently,
precision measurements that are sensitive to meson glue and sea-quark
distributions are highly desirable. Here, too, EicC can have a significant
impact.

_Balancing Emergent Mass and the Higgs Mechanism_.
The ability of form factor and structure function measurements to expose
emergent hadronic mass is enhanced if one includes similar kaon data because
theory has revealed that $s$-quark physics lies at the boundary between
dominance of strong (emergent) mass generation and weak (Higgs-connected) mass
[403], as highlighted in Fig. 2.34-left panel. Hence, comparisons between
distributions within systems constituted solely from light valence quarks and
those associated with systems containing $s$-quarks are ideally suited to
exposing measurable signals of emergent mass in counterpoint to Higgs-driven
effects.

|  |
---|---|---
Figure 2.34:  _Left panel_ – Twist-two parton distribution amplitudes at
$\zeta=2\,$GeV$=:\zeta_{2}$. A solid (green) curve – pion $\Leftarrow$
emergent mass generation is dominant; B dot-dashed (blue) curve – $\eta_{c}$
meson $\Leftarrow$ Higgs mechanism is the primary source of mass generation; C
solid (thin, purple) curve – asymptotic profile, $\varphi_{\rm as}(x)$; and D
dashed (black) curve – “heavy-pion”, _i.e_. a pion-like pseudo-scalar meson in
which the valence-quark current masses take values corresponding to a strange
quark $\Leftarrow$ the boundary, where emergent and Higgs-driven mass
generation are equally important. _Right Panel_ – Solid curve: ratio of
strange-to-normal matter in the $K^{+}$; green dashed curve and band: results
obtained using the QCD hard-scattering formula [321, 404, 322].

One example can be found in the contrast between the parton distribution
functions of the pion and kaon, first theory predictions for which are now
available [425, 426]. Another is illustrated in Fig. 2.34 – right panel, which
displays a flavor-separation of the charged-kaon, $K^{+}$, elastic
electromagnetic form factor: $s_{K}:=F_{K}^{\bar{s}}$ is the $s$-quark
contribution to the form factor and $u_{K}:=F_{K}^{u}$ is the analogous
$u$-quark term. The rate of growth from $Q^{2}=0$, the peak height and
location, and the logarithmic decay away from the peak are all expressions of
emergent mass and its modulation by Higgs-boson effects [427].

Analogous predictions exist or are being completed for nucleon form factors,
following the approach in Ref. [428], and parton distribution functions [429].
Simulations indicate that the planned EicC could be used to validate the
connection between emergent mass and these observables; and also many others
that are members of the same class.

##### Meson fragmentation functions

Along with the promised rewards described in Sec. 2.2, many new challenges are
faced in extracting 3-D images from new-generation experiments.
Phenomenological models of a wide variety of parton distribution functions
will be crucial. They can provide guidance on the size of the cross-sections
to be measured and the best means by which to analyse them [112]. On the other
hand, as experiences with meson structure functions have shown, in order to
profit fully from such experiments, one must use computational frameworks that
can reliably connect measurements with qualities of QCD. Here, continuum
calculations can provide valuable insights [430, 431, 432, 433, 434].

Regarding TMDs, there is an additional complication. Namely, every cross-
section that can yield a given hadronic TMD involves a related parton
fragmentation function (PFF) [435], the structure of which must be known in
detail. Therefore, the future of momentum imaging depends critically on making
significant progress with the measurement and calculation of PFFs.
Notwithstanding these demands, there are currently no realistic computations
of PFFs. Even a formulation of the problem remains uncertain. Here are both a
challenge and an opportunity for EicC.

If EicC can provide precise data on quark fragmentation into a pion or kaon,
it will deliver results that directly test those aspects of QCD calculations
which incorporate and express emergent phenomena, _e.g_. confinement, DCSB,
and bound-state formation. The fact that only bound-states emerge from such
processes is one of the cleanest available manifestations of confinement. As a
collider, EicC measurements will potentially have enormous advantages over
earlier and existing fixed-target experiments. By capitalising on energy
range, versatility, and detection capabilities in the collider environment,
EicC should be able to first cleanly single out the pion or kaon target and
subsequently the fragmentation process tag, thereby delivering an array of
information that will best test theory aimed at the calculation and
interpretation of PFFs.

##  Chapter 3 Accelerator conceptual design

To achieve the proposed scientific goals in chapter 2, the EicC project will
construct a high performance polarized electron-proton collider which can
reach the luminosity of $2.0\times 10^{33}$ $\mathrm{cm^{-2}s^{-1}}$ at the
center-of-mass energy of 16.76 GeV while ensuring an average polarization for
electrons and protons of about $80\%$ and $70\%$, respectively. For
convenience, unless stated, the following polarization all represents the
average polarization. Moreover, the center-of-mass energy of the EicC has a
flexible range from 15 GeV to 20 GeV in order to serve different experimental
purposes. To keep a balance between the physics goals and the overall cost, we
will take full advantage of the existing HIAF accelerator complex and its
ancillary facilities. Therefore, only a new figure-8 ion collider ring, a
polarized electron injector, and a racetrack electron collider ring will be
constructed. This chapter is organized as follows: In Section 3.1, we present
a comprehensive overview of the EicC accelerator facility, including its
design goals, main layout, key accelerator parameters, and potential technical
challenges. Section 3.2 will mainly discuss the basic operation modes of the
ion accelerator complex and the electron accelerator complex of the EicC. The
following section 3.3 to section 3.5 will provide a detailed description of
three key conceptual design ingredients for the EicC accelerator facility,
i.e. beam cooling, beam polarization, and interaction regions (IRs). Last but
not the least, several topics of the technical pre-research in the
accelerators will be presented in Section 3.6.

###  3.1 Overall design and key parameters

The various scientific goals pose the various requirements that must be met in
the EicC accelerator facility. Taking the proton beam as the reference beam
(also for the following sections), these requirements are explained as
follows:

  1. 1.

The center-of-mass energy should range from 15 GeV to 20 GeV, which
corresponds to the energy of 2.8 GeV to 5.0 GeV for electron beams in the
electron collider ring and the energy of 19.08 GeV for proton beams in the ion
collider ring. The center-of-mass energy of other ion beams can be obtained
according to the maximum momentum rigidity which is 86 $\mathrm{T\cdot m}$.

  2. 2.

The luminosity should reach up to $2.0\times 10^{33}$
$\mathrm{cm}^{-2}\mathrm{s}^{-1}$ and should be optimized mainly at the
center-of-mass energy of 16.76 GeV.

  3. 3.

At the interaction point (IP), the electron beams should be polarized
longitudinally with the polarization of $80\%$, while the proton beams should
be polarized both longitudinally and transversely with the polarization of
$70\%$. Besides the polarized proton beams, the ion accelerator complex should
also provide polarized deuterium (${}^{2}\mathrm{D}^{+}$) beams, polarized
Helium-3 (${}^{3}\mathrm{He}^{2+}$) beams as well as unpolarized heavy ion
beams.

  4. 4.

The design of the IRs should fully accommodate the design of the EicC
detectors of which a large number of components, such as the polarimetry,
luminosity monitors, forward detectors, and so on, will be integrated into the
beamlines.

Figure 3.1:  The layout of EicC accelerator facility.

As shown in the Fig.3.1, the EicC was carefully designed to meet all
requirements listed above. The ion accelerator complex will fully utilize the
existing HIAF complex which helps us to largely reduce the construction cost.
On top of that, a new ion collider ring (pRing) will be constructed. The high-
intensity heavy-ion booster ring (BRing) in the HIAF can produce proton beams
of energy up to 9.3 GeV, which covers the injection energy of 2 GeV in the
pRing. Furthermore, in the BRing, there will be sufficient free space reserved
for the installation of a DC electron cooler for beam cooling and the Siberian
snake for spin preservation. Therefore, the BRing in the HIAF is planned to be
upgraded and fully employed as the booster in the EicC ion accelerator
complex. To avoid the depolarization resonances during the acceleration
process, a figure-8 ring will be adopted for the pRing. This design ensures
the spin tune of the pRing is always zero and makes it possible to achieve the
high polarization of proton beams during the acceleration process with a large
energy range. Meanwhile, it also allows us to control the polarization
direction of proton beams more efficiently. The figure-8 ring design is one of
the essential technical solutions to achieve the high polarization in the EicC
ion accelerator complex. Besides the polarization design, it is also important
to achieve high luminosity in the EicC. To this end, a multistage beam cooling
scheme is proposed. Firstly, the pre-cooling of proton beams will be performed
in the BRing. Secondly, a high energy bunched beam electron cooler based on an
energy recovery linac (ERL) will be installed in the pRing and employed to
cool the proton beams after they are accelerated to the energy of 19.08 GeV.
The beam cooling will be employed to suppress the intra-beam scattering effect
over the whole collision process, which is the key to achieve the required
luminosity.

The EicC electron accelerator complex mainly consists of an electron injector
and an electron collider ring (eRing). The electron injector is a
superconducting radio frequency (SRF) linac which is considered to be the best
choice to deliver the electron beams with the energy range from 2.8 GeV to 5.0
GeV. Since the eRing adopts the full energy injection scheme which means there
is no acceleration process, and the electron beams will experience the self-
polarization process, the eRing is designed as a racetrack ring which is
widely used in the electron facilities. Meanwhile, the polarization direction
of the electron beams produced by the electron injector should be matched with
the eRing.

Based on the layout of the EicC accelerator facility, two IRs are available at
two intersections of the long straight sections of the pRing and the eRing, as
shown in the Fig. 3.1. These two identical IRs will allow us to install two
independent detector systems for different scientific goals simultaneously or
the same physics goal but with different experimental techniques for cross-
validation. However, the current design only considers one detector to be
installed in one of these IRs, with the other one reserved for the future
upgrade

The luminosity is one of the most important characteristics of a collider.
Different from other EIC designs in the world which commonly choose high
collision frequency and low bunch intensity, the EicC accelerator facility
will adopt the unique scheme of low collision frequency and high bunch
intensity to achieve a fourfold increase of luminosity to reach the required
luminosity of $2\times 10^{33}$ $\mathrm{cm}^{-2}\mathrm{s}^{-1}$, satisfying
the physics goals of the EicC project. This is an essential part of the
luminosity design of the EicC. For the collision between a proton with
constant current $I_{p}$ and an electron beam with constant current $I_{e}$,
the luminosity can be expressed as

$\displaystyle N_{p}f_{c}\propto I_{p}\equiv\mathrm{const},$ (3.1)
$\displaystyle N_{e}f_{c}\propto I_{e}\equiv\mathrm{const},$ (3.2)
$\displaystyle
L=\frac{N_{p}N_{e}f_{c}}{4\pi\sigma_{x}\sigma_{y}}H\propto\frac{I_{p}I_{e}}{\sigma_{x}\sigma_{y}f_{c}}.$
(3.3)

Here $N_{p}$ and $N_{e}$ are the numbers of particles in one proton bunch and
electron bunch, respectively. $f_{c}$ is the collision frequency. $\sigma_{x}$
and $\sigma_{y}$ are the transverse beam size in horizontal and vertical
directions, respectively. $H$ is the hourglass factor, which remains constant
when the ratio of the $\beta$ function to the bunch length is fixed. Although
there is a positive correlation between $\sigma_{x}$ or $\sigma_{y}$ and the
number of particles in the bunch, it can be proven that the influence of
$\sigma_{x}$ and $\sigma_{y}$ on the luminosity is less than that of $f_{c}$.
Therefore, a decrease of $f_{c}$ can increase the luminosity. In fact,
constant average beam current and constant intra-beam scattering in the bunch
is a major factor of collision lifetime, i.e.

$\frac{1}{\tau}\propto\frac{N_{p}}{\varepsilon_{x}\varepsilon_{y}\sigma_{\delta}\sigma_{s}},$
(3.4)

from which it can be derived that if the bunch intensity is increased by $m$
times, the increase of the horizontal emittance $\varepsilon_{x}$, the
vertical emittance $\varepsilon_{y}$, the longitudinal bunch length
$\sigma_{\delta}$ and the momentum spread $\sigma_{s}$ reads

$\varepsilon_{x}\propto m^{\frac{1}{3}},\ \varepsilon_{y}\propto
m^{\frac{1}{3}},\ \sigma_{\delta}\propto m^{\frac{1}{6}},\
\sigma_{s}\propto\beta_{x,y}^{*}\propto m^{\frac{1}{6}}.$ (3.5)

Substituting the Eq.3.5 to the Eq.3.3, the relationship between the luminosity
and the beam intensity is

$L\propto\sqrt{m},$ (3.6)

indicating that the luminosity increases by $\sqrt{m}$ times while the bunch
intensity increases by $m$ times.

Based on the optimized luminosity scheme introduced above, the main parameters
of the EicC accelerator facility are generated, as listed in the Tab.3.1 in
which proton beams are used as the reference beam and the optimization is
performed at the center-of-mass energy of 16.76 GeV. Furthermore, several
technical limitations are considered in the Tab.3.1, as is shown below.

  1. 1.

The power density of the synchrotron radiation in the eRing should be less
than $20$ $\mathrm{kW/m}$.

  2. 2.

The beam-beam interaction parameter of proton beams is less than 0.03.

  3. 3.

The beam-beam interaction parameter of electron beams is less than 0.1.

To meet the scientific goals, the parameters of collisions between electron
beams and several heavy ion beams are also shown in the Tab.3.2. In the
following sections of this chapter, the design and the implementation of each
parameter in the Tab.3.1 will be discussed in detail.

Table 3.1:  Main parameters for the EicC. Particle | $\rm{e}$ | $\rm{p}$
---|---|---
Circumference(m) | 809.44 | 1341.58
Kinetic energy(GeV) | 3.5 | 19.08
Momentum(GeV/c) | 3.5 | 20
Total energy(GeV) | 3.5 | 20.02
CM energy(GeV) | 16.76
$\rm{f_{collision}(MHz)}$ | 100
Polarization | 80% | 70%
$\rm{B\rho(T\cdot m)}$ | 11.7 | 67.2
Particles per bunch$(\times\rm{10^{9}})$ | 170 | 125
$\rm{\varepsilon_{x},\varepsilon_{y}(nm\cdot rad,rms)}$ | 60/60 | 300/180
$\rm{\beta^{*}_{x}/\beta^{*}_{y}(m)}$ | 0.2/0.06 | 0.04/0.02
Bunch length(m, rms) | 0.02 | 0.04
Beam-Beam Parameter $\rm{\xi_{y}}$ | 0.09/0.05 | 0.004/0.004
Laslett tune shift | - | 0.09
Energy loss per turn(MeV) | 0.32 | -
Total SR power(MW) | 0.86 | -
SR linear power density(kW/m) | 3.3 | -
Current(A) | 2.7 | 2
Crossing angle(mrad) | 50
Hourglass | 0.78
Luminosity at nucleon level ($\rm{cm^{-2}s^{-1}}$) | $\rm{2.0\times 10^{33}}$

Table 3.2:  Main parameters for the EicC.

Particle | $\rm{e}$ | $\rm{d}$ | $\rm{{}^{3}He^{++}}$ | $\rm{{}^{7}Li^{3+}}$ | $\rm{{}^{12}C^{6+}}$ | $\rm{{}^{40}Ca^{20+}}$ | $\rm{{}^{197}Au^{79+}}$ | $\rm{{}^{208}Pb^{82+}}$ | $\rm{{}^{238}U^{92+}}$
---|---|---|---|---|---|---|---|---|---
Kinetic energy(GeV/u) | 3.5 | 12.00 | 16.30 | 10.16 | 12.00 | 12.00 | 9.46 | 9.28 | 9.09
Momentum(GeV/c/u) | 3.5 | 12.90 | 17.21 | 11.05 | 12.90 | 12.90 | 10.35 | 10.17 | 9.98
Total energy(GeV/u) | 3.5 | 12.93 | 17.23 | 11.09 | 12.93 | 12.93 | 10.39 | 10.21 | 10.02
CM energy(GeV/u) | - | 13.48 | 15.55 | 12.48 | 13.48 | 13.48 | 12.09 | 11.98 | 11.87
$\rm{f_{collision}(MHz)}$ | - | 499.25 | 499.82 | 498.79 | 499.25 | 499.25 | 498.54 | 498.47 | 498.39
Polarization | 80% | Yes | Yes | No | No | No | No | No | No
$\rm{B\rho(T\cdot m)}$ | 11.67 | 86.00 | 86.00 | 86.00 | 86.00 | 86.00 | 86.00 | 86.00 | 86.00
Particles per bunch$(\times\rm{10^{9}})$ | 40 | 6.1 | 3.0 | 2.04 | 1.00 | 0.30 | 0.07 | 0.065 | 0.055
$\rm{\varepsilon_{x},\varepsilon_{y}(nm\cdot rad,rms)}$ | 20 | 100/60 | 100/60 | 100/60 | 100/60 | 100/60 | 100/60 | 100/60 | 100/60
$\rm{\beta^{*}_{x}/\beta^{*}_{y}(m)}$ | 0.2/0.06 | 0.04/0.02 | 0.04/0.02 | 0.04/0.02 | 0.04/0.02 | 0.04/0.02 | 0.04/0.02 | 0.04/0.02 | 0.04/0.02
Bunch length(m, rms) | 0.01 | 0.015 | 0.015 | 0.02 | 0.015 | 0.015 | 0.02 | 0.02 | 0.02
Beam-Beam Parameter $\rm{\xi_{y}}$ | 0.007 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002
Laslett tune shift | - | 0.07 | 0.06 | 0.04 | 0.06 | 0.06 | 0.06 | 0.06 | 0.06
Current(A) | 3.3 | 0.49 | 0.48 | 0.49 | 0.48 | 0.48 | 0.44 | 0.43 | 0.40
Crossing angle(mrad) | 50
Hourglass | - | 0.94 | 0.94 | 0.92 | 0.94 | 0.94 | 0.92 | 0.92 | 0.92
Luminosity at nucleon level ($\rm{cm^{-2}s^{-1}}$) | - | $\rm{8.48\times 10^{32}}$ | $\rm{6.29\times 10^{32}}$ | $\rm{9.75\times 10^{32}}$ | $\rm{8.35\times 10^{32}}$ | $\rm{8.35\times 10^{32}}$ | $\rm{9.37\times 10^{32}}$ | $\rm{9.22\times 10^{32}}$ | $\rm{8.92\times 10^{32}}$

###  3.2 Accelerator facilities

The EicC mainly consists of an ion accelerator complex and an electron
accelerator complex, as shown in the Fig.3.1. The Fig.3.2 illustrates the
operation mode of the two accelerator complexes. There are a lot of
differences between the ion accelerator complex and the electron accelerator
complex, not only on the complicated beam manipulations but also on the key
design and technical challenges, all of which will be described in detail in
the following subsections. In Section 3.2.1, the ion accelerator complex of
the EicC will be described, mainly including the acceleration scheme of the
pRing, the collective effects, and so on. The details of the electron
accelerator complex, including some key designs and beam manipulations of the
eRing, will be presented in Section 3.2.2. The topic of beam cooling and beam
polarization will be discussed in Section 3.3 and Section 3.4, respectively.
The design of the IRs, which is one of the most important accelerator designs
in the EicC accelerator facility, will be presented in Section 3.5.

Figure 3.2:  The beam path of the EicC accelerator facility.

####  3.2.1 Ion accelerator complex

In the proposed ion accelerator complex of the EicC accelerator facility, the
existing Ion Linac (iLinac) and the BRing of the HIAF will be employed
respectively as the ion injector and the booster to provide injection beams
for the pRing. The iLinac will be operated in pulsed mode, in which the
polarized proton beams can be accelerated to the energy of 48 MeV, and then
injected into the BRing with a matched polarization direction. In the BRing, a
two-plane painting injection scheme will be adopted for the beam accumulation
with coasting beams to increase the beam intensity. In this scheme, a tilted
electrostatic septum will be employed for painting simultaneously in the
horizontal and vertical directions during the injection. As a result, the beam
intensity will be increased by 100 times and as high as $4\times 10^{12}$
$\mathrm{ppp}$. The proton beams of the energy of 48 MeV in the BRing will be
captured into two bunches and accelerated to the energy of 2 GeV. During the
acceleration, the Siberian snakes will be used to maintain the polarization of
the proton beams[436], since the beams will cross several depolarization
resonances.

The proton beams of the energy of 2 GeV which reach the extraction energy of
the BRing in the ion accelerator complex will experience the first stage of
beam cooling provided by a DC electron cooler. There are several advantages of
performing beam cooling at the energy of 2 GeV. Firstly, the beam cooling
effect is still strong around such energy while the space charge effect is
relatively weak. Secondly, as a well-developed technology widely used for the
beam cooling, the DC electron cooler can effectively reduce the technical
difficulties and the construction costs. At this stage, to obtain a more
efficient beam cooling, a longitudinal bunch rotation at the energy of 2 GeV
will be performed before the beam cooling to minimize the bunch momentum
spread as much as possible. After the bunch rotation, the bunched proton beams
will be debunched into coasting beams to improve the beam cooling efficiency.
After the beam cooling, the coasting proton beams in the BRing will be bunched
once again into one bunch and compressed to match the injection settings of
the pRing. The polarization direction of the proton beams extracted will get
matched to the polarization direction of the pRing by using a spin rotator.

The pRing will accelerate the injected proton beams of the energy of 2 GeV to
reach the energy of 19.08 GeV. For such a wide energy range, the Siberian
snakes are unable to keep the polarization of the proton beams. Therefore, the
pRing will be designed as a figure-8 shape, in which the spin tune purely
contributed by the pRing itself remains zero. The figure-8 design, along with
a solenoid in the long straight sections, ensures that the proton beams will
not cross any depolarization resonances during acceleration. Moreover, such a
design will facilitate the control of the beam polarization direction. The
bucket-to-bucket injection scheme will be adopted for the injection of 14
bunches from the booster BRing to the pRing. The beam intensity will increase
to $\rm{5.6\times 10^{13}}$ $\mathrm{ppp}$, equivalent to an average beam
current of 2 A. After the injection, the beams will be split into 448 bunches
step by step and finally accelerated to the energy of 19.08 GeV. The collision
frequency introduced by these 448 bunches is 100 MHz, which meets the
requirements of the luminosity design.

To increase the luminosity, the pRing will install a 500 MHz RF system to
shorten the bunch length. After the proton bunches are accelerated to the
energy of 19.08 GeV, a bunch rotation will be performed. And during the bunch
rotation, the bunch length will be sharply shortened while the momentum spread
will be increased, which is quite different from the bunch rotation in the
BRing. Along with the bunch rotation, the 500 MHz RF system will be turned on
to make the bunch length as short as possible while keeping particle numbers
in the bunch unchanged, so that the requirements of the luminosity design will
be satisfied.

The longitudinal manipulation of the proton beams is followed by the second
stage of beam cooling which is supported by high energy bunched beam electron
cooler based on an energy-recovery linac (ERL). The intra-beam scattering
effect will be also suppressed during the whole collision process by this
cooler to improve the luminosity life.

The luminosity optimization scheme of low collision frequency, high beam
intensity could bring strong single-bunch collective effects. Therefore,
besides the threshold values of the average current, and beam-beam parameters,
the impedance limitations introduced by the single-bunch collective effects
are also significant in the collider rings since it determines the feasibility
of luminosity optimization scheme in the pRing. In principle, the longitudinal
microwave instability, which is caused by the longitudinal broadband
impedance, is one of the most concerned longitudinal collective effects. When
the particle numbers in the bunch exceed the corresponding limits, the
longitudinal microwave instability will lead to an increase in the momentum
spread, which can further induce bunch lengthening and weaken the luminosity.
The typical growth time of the instability is shorter than one synchrotron
period. In the pRing, it has been calculated that the longitudinal broadband
impedance ($Z^{\parallel}/n$) should be lower than 87.4 $\Omega$, which is
achievable in practice. In the transverse planes, the transverse mode coupling
instability caused by the transverse broadband impedance is the most likely to
occur among the single-bunch collective effects. When the bunch intensity
exceeds the threshold, particles in the bunch will be lost quickly.
Calculation results have shown that the transverse broadband impedance
threshold in the pRing is 30.3 $\mathrm{M\Omega/m}$, which is also achievable
in practice. From the view of the pRing, the luminosity optimization scheme
with low collision frequency, the high beam intensity is feasible for the EicC
accelerator facility.

One of the biggest challenges in the optical design of colliders is the
correction of the large chromaticity introduced by the very small $\beta$
function at the IPs, with the requirement of large dynamic aperture. As shown
in the Fig.3.1, there are four arc sections in the pRing. These arc sections
are connected via two short straight sections and two long straight sections.
The long straight sections are employed for the IPs, the beam cooling
sections, the spin rotators as well as the RF devices. Accelerator control
devices, such as the chromaticity correction sextupole magnets and so on, are
placed on the short straight sections and the arc sections. The chromaticity
correction scheme based on sextupole magnets on the arc sections and the
straight sections with dispersion will be employed in the pRing. Each arc
section consists of eight FODO cells with $90^{\circ}$-phase advance per FODO
cell, where 12 sextupole magnets will be installed. Dispersion exist in the
short straight sections, and the optical parameters are designed symmetrically
to the center of the short straight section. By doing this, the larger
$\beta_{y}$-value appears at the position $\pm\pi/2$ from the center, which
can enhance the capability of chromaticity correction contributed by the
sextupole magnets here. The chromaticity correction scheme, which is based on
52 sextupole magnets in total placed on the arc sections and the short
straight sections, can correct the chromaticity of the pRing to zero, as well
as keep the dynamic aperture larger than $8\sigma$. This scheme satisfies the
requirement of the EicC accelerator facility.

####  3.2.2 Electron accelerator complex

A recirculating superconducting radio frequency (SRF) linear accelerator will
be employed as the injector of the electron accelerator complex of the EicC
accelerator facility. The recirculating SRF linac is considered to be the best
choice for the electron injector since it has the advantages not only of
linear accelerators, i.e. high accelerating gradient, greater compactness but
also of circular accelerators, i.e. high efficiency and low cost. The
polarized electron beams generated from the polarized photocathode electron
gun will get matched to the electron injector, and then accelerated to the
extraction energy range from 2.8 GeV to 5.0 GeV via passing through the RF
cavities several turns to provide electron beams for the full energy injection
scheme in the eRing. In this process, the polarized electron beams will be
bunched into micro bunches with the bunch length of picoseconds. Taking into
account the limitations of beam power and beam dump, the recirculating SRF
linac can provide the electron beams of micro-Ampere beam intensity.
Furthermore, the polarization direction of the electron beams extracted from
the electron injector should get matched to the eRing, which can be achieved
via a spin rotator.

Because of the full energy injection scheme and the self-polarization effect
in the electron beams, the eRing is designed to be racetrack-shaped as many
other electron colliders. The RF system of the eRing will adopt the frequency
of 500 MHz to generate 1350 stable buckets. The polarized electron beams will
be injected into 270 stable buckets, forming into 270 equally spaced electron
bunches, to meet the collision frequency of 100 MHz in the luminosity design.
Moreover, as the transverse and longitudinal beam sizes will decrease rapidly
because of the damping effects introduced by the synchrotron radiation, the
full energy injection scheme will be adopted to reach the bunch intensity of
$1.70\times 10^{11}$ ppb required by the luminosity design. After beam
accumulation is finished, the electron bunches in the eRing will be reinforced
or replaced by the recirculating SRF linac on line to maintain the luminosity.

In addition to the limitations of average beam current, synchrotron radiation
power density, and beam-beam interaction parameters as listed in the Tab.3.1,
the luminosity optimization scheme with low collision frequency, high beam
intensity could also bring longitudinal microwave instability and transverse
mode coupling instability. This is similar to the case in the pRing. It can be
calculated theoretically that the thresholds of the longitudinal and the
transverse broadband impedance ($Z^{\parallel}/n$ and $Z^{\bot}$) in the eRing
are $0.040$ $\Omega$ and $0.259$ $\mathrm{M}\Omega/\mathrm{m}$, respectively.
The thresholds are considered to be reachable, if we take into account the
case in the KEKB electron-positron collider in Japan with the corresponding
thresholds of $0.012$ $\Omega$ and $0.235$ $\mathrm{M}\Omega/\mathrm{m}$.
Therefore, the luminosity optimization scheme of low collision frequency, the
high beam intensity is feasible in the eRing.

Since the eRing is designed to be racetrack-shaped, the chromaticity
correction scheme base on the sextupole magnets on the arc sections will be
adopted. Specifically, there are two arc sections in the eRing, as shown in
the Fig.3.1. Each of them consists of 20 FODO cells with $120^{\circ}$ phase
advance. Two pairs of FODO cells at both ends of one arc section are used for
optical matching, while the other 16 FODO cells are employed for chromaticity
correction. Each three of the 16 FODO cells form a super periodical structure
to cancel out the non-linear effects caused by the sextupole magnets within
the arc sections. By using such a chromaticity correction scheme, the
chromaticity can be corrected to zero while keeping the dynamic aperture
larger than $20\sigma$, which satisfies the accelerator design requirements.

###  3.3 Beam cooling

High luminosity, which is the primary goal of an electron-ion collider, is
thus the most essential parameter for the design of the EicC accelerator
facility. It can be proven from the luminosity formula that an efficient way
to increase the luminosity is to decrease the six-dimensional emittance of ion
beams. Since there is no synchrotron radiation damping effect in the heavy ion
synchrotrons, an external cooling mechanism is required to reduce the ion beam
emittance. For decades, now electron cooling has become one of the most
effective and well-developed methods to reduce the ion beam emittance. Since
electron cooling has a greater effect on the ion beams of low energy and low
emittance, the EicC accelerator facility will adopt the staged electron
cooling scheme to shorten the cooling time and improve the cooling efficiency.
In the first cooling stage, an electron cooler[437], based on conventional
electrostatic high voltage acceleration, will be installed in the BRing to
reduce the transverse emittance and the momentum spread of the medium-energy
ion beams to reach the design value. In the second cooling stage, a high
energy bunched beam electron cooler, based on an energy recovery linac (ERL),
will be installed in the pRing to further reduce the transverse emittance and
the momentum spread of the high-energy ion beams. The intra-beam scattering
effect in the collision will be suppressed by this cooler to keep the
emittance and the momentum spread to be the design value, which can ensure
high luminosity and long luminosity life required by the scientific goals. Due
to a reduced emittance after the low-energy cooling in the first stage, the
cooling time for the high-energy beam can also be largely reduced, leading to
a shortened total cooling time and enhanced cooling efficiency. The staged
electron cooling scheme for the ion beams of the EicC accelerator facility is
shown in the Fig.3.3.

Figure 3.3: The layout of the staged electron cooling scheme for the EicC
accelerator facility.

The Tab.3.3 lists the details of the staged beam cooling scheme in the EicC
accelerator facility. Taking the polarized proton beam as an example, it will
be first accelerated to the energy of 48 MeV by the iLinac, then injected into
the BRing for further accumulation, and accelerated rapidly to 2 GeV via the
RF cavities. The proton beam will be debunched into a coasting beam after the
momentum spread is reduced by a bunch rotation manipulation. The coasting beam
will be cooled via DC electron beams of the energy of 1.09 MeV which are
produced from a conventional electrostatic electron cooler. The cooling
process can reduce the emittance and the momentum spread of the proton beam to
the design value. After that, the high intensity polarized proton beam in the
BRing will be injected to the pRing for further accumulation and acceleration.
When the proton beam is accelerated to reach the energy of 19.08 GeV, the
second cooling stage, which is based on high energy, high-intensity and high-
quality electron bunches produced by an energy recovery linac (ERL), will be
performed to cool the beam again and suppress the emittance growth and the
bunch lengthening caused by intra-beam scattering effects over the whole
collision process. For other ion beams with lower energy and higher cooling
efficiency, the staged electron cooling process requires a shorter cooling
time compared to proton beams.

Table 3.3: The staged electron cooling scheme of the EicC accelerator facility. |  Position |  Function |  Proton energy(GeV) |  Electron energy(MeV) |  Cooler
---|---|---|---|---|---
Phase 1 |  BRing |  Reduction of beam emittance |  2 |  1.09 |  DC
Phase 2 |  pRing |  Beam emittance reduction and intra-beam scattering suppression |  19.08 |  10.4 |  Bunched beam (ERL)

The DC electron cooler in the BRing consists of an electron gun, an
accelerating section, a cooling section, a decelerating section, a collector,
several solenoids, and several correctors, as shown in the Fig.3.4.

Figure 3.4: Layout of the low energy DC electron cooler.

The electron beams emitted from the cathode of the electron gun can be
extracted to the accelerating section via a potential difference introduced at
the anode. After acceleration via electrostatic high voltage to obtain the
same average speed as the ion beams, the electron beams will be transported to
the cooling section, in which they will interact with the ion beams and absorb
part of the heat from the ion beams via the Coulomb interaction. Then the
electron beams will pass through the decelerating section and be collected in
the collector. The repeated electron beams with low temperatures can finally
reduce the emittance and the momentum spread of the ion beams to the design
value. Nowadays the technology of the DC electron cooling has already been
well-developed and there are many DC electron coolers around the world, with
the energy ranging from a few tens of KeV to a few MeV. For instance, in the
Recycler Ring, Fermi-Lab, USA, the energy of the electron beam is as high as
4.3 MeV. In the COSY electron cooler in the Jülich, Germany, the electron beam
reaches the energy of 2 MeV. And also the first RF linac-based electron cooler
(bunched beam cooling) with an electron energy of 1.6 and 2.0 MeV was
successfully commissioned at RHIC. For the EicC accelerator facility, the
energy of the electron beam is designed to be 1.09 MeV for the DC electron
cooler, which is technically achievable.

In the pRing, beam cooling requires electron beams with a maximum energy of
10.4 MeV. Since the conventional DC electron cooler are unable to accelerate
the electron beams to this energy, it is necessary to employ the electron
beams accelerated by RF cavities, which is high energy bunched beam electron
cooling. However, the specific electron beams cannot be used in the beam
cooling for a very long time and should be always replaced by new electron
beams, because the electron cooling requires the electron beams of very high
quality (low emittance, low energy spread, high beam intensity). Moreover, the
discarded electron beams should be collected to avoid radiation protection
issues, since there is almost no energy loss during beam cooling. The quality
of the electron beams in the synchrotrons hardly satisfies the requirement of
the beam cooling. It is limited by the equilibrium conditions of the
synchrotron radiation effects. The electron linear accelerator can accelerate,
transport beams effectively, and keep the high beam quality. However, the high
power level of the RF cavity increases the cost of construction and operation.
Meanwhile, the collection of such high power electron beams can cause severe
issues of radiation protection, such as neutron activation, and environmental
pollution. In comparison, the energy recovery linac (ERL) cannot only
accumulate the electron beam intensity as effectively as a synchrotron but
also keep high beam quality as the linear accelerators, which satisfies the
requirement of the high energy bunched beam cooling. The electron beams in the
ERL can be sent back into the RF cavities with a decelerating phase, where the
power of the high energy electron beams can be transferred to the power of the
microwave acceleration field which can be used to accelerate newly injected
electron beams. The energy recovery from the discarded electron beams in the
ERL cannot only largely reduce the technical difficulty and the costs of the
RF power, but also decrease the power deposition in the beam collector,
solving the possible problems of the radiation protection and environmental
pollution caused by the high power electron beam. The ERL-based high energy
bunched beam electron cooler employed in the EicC accelerator facility is
shown in the Fig.3.5. It consists of a photocathode electron gun, merger line,
superconducting RF cavities, arc sections, two 25-meter-long cooling sections,
beam matching sections, circulating sections, and collectors.

Figure 3.5: The high energy bunched beam cooler based on the ERL.

The high intensity, high-quality electron bunches generated from the
photocathode will be firstly accelerated via the pre-accelerating cavity to 2
MeV, then transported into the main accelerating section of the ERL through
the merger line while keeping the initial emittance and energy spread. The
electron bunches are further accelerated by the superconducting RF cavities to
reach the energy of 10.4 MeV. The electron bunches are transported through the
arc sections to the cooling sections to travel along with the ion beams. Then,
the electron bunches interact with the ion bunches within the cooling sections
in the strong solenoid magnetic field. After that, the electron bunches are
stored in the circulating ring until making 53 bunches in the circulating
ring, with the frequency between the bunches being 100 MHz. Then, the first
stored electron bunch is kicked out of the circulating ring by an ultra-fast
kicker cavity, with its energy recovered in the superconducting RF cavities.
Finally, the residual power can be deposited in the collectors. The newly-
injected electron beams will be accelerated by the recycled energy in the RF
cavities and replace the bunch which had been kicked out of the circulating
ring every 6.25 MHz. This process will be repeated until the emittance and the
momentum spread of the ion beams meet the design requirements. The ERL and its
circulating ring are shown in the Tab.3.4.

Table 3.4: The main parameters of ERL and circulating ring. Gun type | SRF |  RMS energy spread | $5\times 10^{-4}$
---|---|---|---
PRF | 6.25 MHz |  Cooling energy | 10.4 MeV
Main RF frequency | 700 MHz |  Cooling section length | 50 m
Charge per bunch | 4 nC |  Cryostat number | 1
Injection energy | 2 MeV |  Cryostat length | 1.5 m
Beam current in ERL | 25 mA |  Beam current in circulating ring | 400 mA
RMS bunch length in the cooling sections | 150 ps |  Circumference of circulating ring | 159 m
Transverse RMS normalized emittance | 2.5 $\pi\cdotp\mathrm{mm}\cdotp\mathrm{mrad}$ |  Bunch frequency in circulating ring | 100 MHz

In recent years, compared to many other types of high energy particle
accelerators, the ERL has been paid more attention and experienced rapid
development and wide application, since it has the advantages of less power
consumption, high beam quality, and so on. The ERLs are being considered to
replace the conventional linear or ring accelerator for the application of the
free-electron laser (FEL), the synchrotron radiation sources, the colliders,
as well as the electron coolers. Up to now, there exist several established
ERL facilities around the world, including the Novosibirsk ERL at BINP in
Russia, the CEBAF-ER and the IR ERL in the Jefferson laboratories (JLab) in
the USA, the S-DALINAC ERL at Technical University Darmstadt in Germany and
the ERL of the High Energy Accelerator Research Organization (KEK) in Japan,
and so on. BNL has also tried to build test ERL and CeC ERL for high-energy
beam cooling before. With the technology development of the photocathode gun
and the superconducting radio frequency (SRF), the difficulties for the
construction of the high energy, high-intensity ERL has been reduced a lot.
There are several ERLs under construction or proposed, including the bERLinPro
of the Helmholtz-Zentrum Berlin(HZB) and the MESA at the Johannes Gutenberg-
Universität Mainz in Germany, Cornell-BNL-ERL-Test-Accelerator (CBETA), the
light source in the Cornell University, the US-EIC in the BNL and the LHeC in
CERN. Some of them will be employed as high energy bunched beam electron
cooler, which can lay a solid foundation for the construction of the high
energy bunched beam electron cooler based on the ERL in the EicC accelerator
facility while largely reducing the technical risk of the beam cooling in the
project.

###  3.4 Beam polarization

As a dual-polarized electron-ion collider, the beam polarization is another
important part of the design of the EicC accelerator facility, besides the
luminosity and the beam cooling which are typical designs of a collider. The
physics goals of the EicC project pose the following requirements on the beam
polarization design of the EicC accelerator facility.

1\. The polarized electron beams will collide with the polarized proton beams,
the polarized deuterium beams, and the polarized Helium-3 beams, respectively.
The required polarization here is about $80\%$ for the electron beams, $70\%$
for the proton beams. Other ion beams are non-polarized.

2\. At the IPs, the polarization direction of the electron beams should be
longitudinal, while for the proton beams, the deuterium beams, and the
Helium-3 beams, the polarization direction could be arbitrary.

3\. The measuring errors of the polarization for the proton beams, the
deuterium beams, and the Helium-3 beams should be less than $5\%$, while for
the electron beams it is less than $2\%$.

For these requirements and goals listed above, the scheme of the beam
polarization is designed and optimized based on the layout in the Fig.3.1 and
the operation mode in the Fig.3.2, including polarization control and
polarization measurement, which is shown in the Fig.3.6.

Figure 3.6: The polarization design of the EicC accelerator facility.

####  3.4.1 Ion polarization

For the ion accelerator complex, the atomic beam polarized ion source (ABPIS)
can produce both polarized and non-polarized proton beams, while other
polarized ion beams are similar to the proton beams. There are four successive
processes to transform hydrogen atoms into polarized proton beams, i.e. the
dissociation, the separation in a sextupole magnet, the transition in a weak
RF field, and ionization. Since the dissociated hydrogen atom has an isotropic
spin distribution, the hydrogen atoms of high energy level and with an
electron spin of $1/2$ can be selected and guided into the next devices.
During the transition, an atom with a proton spin of $-1/2$ will jump
selectively into the atomic state of $1/2$ proton spin, forming into high
intensity and high polarized atomic beams. Furthermore, with the selection of
the polarization direction of the weak RF field, the polarization direction of
the atomic beams in the transition module can be controlled to match the
polarization directions of the following beamlines and accelerators. Proton
beams of high intensity and high polarization can thus be produced via the
ionization of such atomic beams. The ABPIS is one of the most widely-employed
ion sources and can generate the proton beams of polarization as high as
$90\%$[438].

The polarized proton beams extracted from the polarized ion source will be
measured via a Lamb-shift polarimeter. The Lamb-shift polarimeter scans the
magnetic field in the rf spin filter to quench the different meta-stable
hydrogen atoms and measures the Lyman - $\alpha$ photons from the quenching
downstream. The meta-stable hydrogen atoms neutralized from the polarized
protons will be quenched at special spin filter magnetic field values and emit
Lyman - $\alpha$ photons. The polarized proton beams will be accelerated in
the iLinac to reach the energy of 48 MeV needed by the injection of the BRing,
and finally injected to the BRing via a beamline for injection. During this
process, the polarization of the proton beams remains unaffected since there
is no spin resonance. However, to measure the polarization direction of the
beams injected into the BRing, a polarimeter will be still installed in the
beamline for the polarization direction matching to the BRing. The polarimeter
mainly records the counting rate of the Coulomb elastic scattering between the
polarized proton beams and the target at different angles, from which the
polarization at a certain angle can be calculated.

High intensity polarized proton beams of the energy of 48 MeV will be
accelerated to 2 GeV in the BRing. During the acceleration, the polarization
of the beam will significantly decline because the spin tune will experience
several depolarization resonances. To maintain high polarization, two Siberian
snakes will be installed at the two ends of the electron cooling section of
the BRing to change the spin tune of the ring. The Siberian snakes, i.e. the
solenoids whose magnetic fields are synchronized with the beam energy, can
keep the spin tune always to be $1/2$ which is far from the depolarization
resonances. In this way, the problem of the decline of the polarization of the
proton beams caused by the depolarization resonances can be solved. Besides, a
polarimeter will be installed, which can be used for the on-line measurement
of the polarization direction and the polarization of the beams and provide
useful information for adjustment and optimization of the polarization-related
parameters of the BRing. The polarimeter should operate at the energy range
from 48 MeV to 2 GeV in which a high counting rate should remain and the
requirement of high-precision control for the BRing should be met.

Figure 3.7: The spin rotation in a spin rotator.

The polarized proton beams of 2 GeV energy will be extracted via a fast
extraction scheme from the BRing and injected into the pRing after measuring
and matching the polarization direction. The polarization matching can be
implemented via a spin rotator. Based on a set of magnets consisting of
solenoid-horizontal bending dipole magnet-solenoid-horizontal bending dipole
magnet, the polarization direction can be guided arbitrarily. This is one of
the most essential polarization control devices in the EicC accelerator
facility. The polarization direction control scheme is illustrated in the
Fig.3.7, taking as the example of a vertical polarization rotating to the
longitudinal polarization. In the Fig.3.7, the blue line denotes the rotation
of polarization direction contributed by the solenoids, and the purple line
represents the rotation from the horizontal bending dipole magnets. The
initial and final polarization directions are shown in red. When passing
through a solenoid, the polarization direction of the polarized beam will
rotate with a certain angle with the beam direction as the axis. In the
horizontal bending dipole magnets, the polarization direction will rotate
around the vertical direction. With once more such rotations in another
solenoid and horizontal bending dipole magnet, the polarization direction of
the beam can be transferred from the vertical direction to the horizontal
direction. Similarly, the polarization direction of the beam will rotate to an
arbitrary direction by this spin rotator with carefully-chosen rotation angles
in the solenoids and the bending dipole magnets. Moreover, the polarization
direction can be well-controlled in a wide energy range while leaving the
closed orbit unchanged, because the four spin angles provide sufficient
polarization control variables, which makes it possible to provide multiple
solutions for the adjustment of a certain polarization direction. With the
help of the spin rotator, the polarization direction of the extracted high
intensity polarized proton beams can be matched with one of the arc sections
in the pRing, thus avoiding the depolarization caused by the mismatch of the
polarization direction. Another polarimeter will be installed near the
injection section of the pRing to ensure an efficient and complete matching of
the polarization direction during the injection.

After injected into the pRing, the polarized proton beams will be further
accelerated to the energy of 19.08 GeV. During the acceleration, the proton
beams will experience several depolarization resonances, which means it is
very difficult to maintain the polarization of the proton beams in the
conventional racetrack-shaped accelerators. Furthermore, for the large energy
range in the pRing, there are quite a lot of technical challenges in Siberian
snakes consisting of solenoids. However, the alternative Siberian snakes
consisting of dipole magnets will cause the closed orbit distortion. These
make it difficult to avoid the depolarization resonances by only using the
Siberian snakes. To solve the issues, the pRing of the EicC accelerator
facility is designed to be a novel figure-8 structure, in which the spin
precession in the arc sections of one side can be always exactly canceled by
the ones of the other side, keeping the total spin tune in the ring to be
zero. The spin precession angle of the proton along the reference orbit is
G$\gamma$ times the bending angle. The total bending angle of figure-8 ring is
zero, so the spin tune will be zero independent of the beam energy for the
convenience of spin control and the spin tune is far away from all intrinsic
depolarization resonances. Meanwhile, a small solenoid added in the long
straight section to introduce additional spin precession angle can ensure the
spin tune to be far away from the imperfection depolarization resonances. It
can be shown that in the optimized condition the solenoid at the IPs can also
meet requirements for adjusting the spin tune, which is a novel scheme for the
ion accelerator complex design in the EicC project.

Due to the arbitrary polarization direction of the proton beams needed at the
IPs, two spin rotators will be placed on either side of the long straight
section containing the IPs and the arc sections at both ends of the long
straight section. As discussed above, each spin rotator can perform
polarization direction rotation to generate the arbitrary direction needed by
the experiments. Moreover, by using the spin rotator, the polarization
direction will get matched to the direction of the arc sections before the
proton beams enter the arc sections upstream of the IPs. With this scheme, the
requirements of the arbitrary polarization directions and the polarization
matching can be satisfied easily.

A polarimeter will also be installed in the pRing for the online measurement
of the polarization direction and the polarization of the proton beams at the
IPs, as well as providing measuring data of the polarization for various
physics experiments. The measuring accuracy of the polarimeter should be as
high as $5\%$, which meets the physics experiment requirements.

####  3.4.2 Electron polarization

In the electron accelerator complex, the polarized electron beams can be
produced via the photocathode polarized electron gun. Since the energy band of
the electron excitation in the cathode plated by Cs and GaAs can overlap with
the energy band of the vacuum, the electrons will become free electrons when
transiting to the excitation energy band. With a laser of a specified energy
and polarization direction, the electrons of the specified spin state will
transit to the excitation energy band, forming highly-polarized electron
beams. The polarization direction of the electron beams can be controlled by
changing the polarization direction of the laser. Up to now, the polarization
of the beam generated from the polarized electron gun can be as high as
$90\%$[439], which can meet the requirement of the EicC project. Before
electron beams are injected into the electron injector, their polarization
characteristics will be measured via a polarimeter. This polarimeter can be
also employed to adjust the parameters of the photocathode polarized electron
gun according to the requirements of physics experiments and accelerator
operations.

After the polarized electron beams are injected into the electron injector, it
will be accelerated to reach the energy range from 2.8 GeV to 5.0 GeV. During
the acceleration, the depolarization resonances will not occur since the
electron beam travels through different arc sections in the different turns,
causing a lack of resonances between the periodic transverse and longitudinal
motion, as well as the spin periodic precession[440]. Therefore, it is not
necessary to place any polarization preservation devices in the electron
injector.

The polarized electron beam will be accelerated to its maximum energy in the
electron injector and then injected into the eRing. During this process, the
polarization direction of the electron beam should get matched to one of the
arc sections of the eRing. The polarization matching can be realized via a
spin rotator in the ion accelerator complex.

The polarization direction and the polarization of the beam will be measured
before the beam is injected into the eRing. It is different from the case in
the ion accelerator complex or in the low energy electron accelerators, where
the internal-target measuring scheme is adopted. The differential Compton
cross section is a function of the initial electron and photon
polarizations[441]. The polarization direction and the polarization of the
electron beams can be obtained indirectly via detecting the backscattered
photons after the scattering of circularly polarized photons on polarized
electrons. This is a non-interception measurement scheme that will not affect
the beam quality.

The polarized electron beams should preserve the polarization of $80\%$ during
the whole collision. The eRing is designed to be racetrack-shaped as other
facilities according to the experiment results of the existing facilities in
the world[442]. Such a design has the advantage that the depolarization
effects can be canceled by the self-polarization effect introduced by the
synchrotron radiation of the electron beams, resulting in high equilibrium
polarization at the level of $80\%$ during the whole collision.When the beam
energy lies nearly on the depolarization resonance energy and it is not enough
for the self-polarization effect to cancel the depolarization effects, the
spin tune can be moved away from the depolarization resonance via a solenoid
at the IR. At the IR, the polarization direction of the electron beam is
longitudinal, so the spin tune is moved via the solenoid while the
polarization direction remains longitudinal. When colliding with the polarized
proton beams, the polarization direction of the polarized electron beams
should be longitudinal, but the polarization direction matching to the arc
sections of the eRing is transverse. Therefore, it is necessary to place two
spin rotators between the long straight sections containing the IPs and the
arc sections to satisfy the matching of the polarization direction and the
requirement of the physics goals. The spin rotator here is similar to the one
in the ion accelerator complex. The online measurement of the polarization
direction and the polarization in the eRing, as well as its beamline for
injection, is also designed based on the Compton backscatter, only with an
accuracy of $2\%$ required by the physics experiments.

###  3.5 Design of the interaction regions (IR)

To achieve the physics goals, one full-acceptance detector will be built at
one IR for detection and identification of reaction products, such as charged
particles, i.e. electrons $e$, muon $\mu$, $\pi$-meson $\pi$, k-meson $K$,
proton $p$ and so on, as well as neutral particles, i.e. photon $\gamma$,
neutron $n$, etc. Another IR will be reserved for the future upgrade. This
detector consists of a central detector with end caps and forward detectors.
The central detector with end caps will be placed around a superconducting
solenoid and can be divided into two parts: the barrel part and the end cap
part on both sides, which is used to detect the reaction products in a large
angular range. It consists of a series of detectors, including vertex
detectors, tracking detectors, time-of-flight (TOF) detectors, Cherenkov
detectors, electromagnetic calorimeters, and hadron calorimeters. Free space
with a length of 8 m is required for the installation of the central detector
with end caps. The forward detectors will be mainly used for the detection of
the final-state particles emitted from the reaction in a small or ultra-small
angle. Those detectors are far from the IP and dipole magnets are required to
separate the particles. The particle detection of the electron forward
detectors is realized with the help of dipole magnets and several long drift
sections after the first quadrupole magnet. For the ion forward detectors, a
dipole magnet should be installed in front of the first quadrupole magnet to
make it possible to identify the reaction products at a small angle.
Furthermore, other dipole magnets and long drift sections will be placed
behind the first quadrupole magnet to allow the identification of the reaction
products in ultra-small angles. Compared to conventional colliders, such
specifications from the detectors raise many more requirements of the
interaction region design of the EicC accelerator facility.

Typically, a conventional collider poses the following requirements about the
layout of the interaction region and its optics design.

  1. 1.

The beam transverse size or the $\beta^{*}$ at the IP should be as small as
possible.

  2. 2.

The devices in the interaction region should meet the required minimum
installation length of the central detector.

  3. 3.

The $\beta$ function in the first quadruple magnet in the interaction region
should be not too large.

  4. 4.

The optics design in the interaction region should satisfy the chromaticity
correction of the collider ring.

  5. 5.

The cross angle at the IP should be larger than the minimum value required for
the bunch separation, and smaller than the maximum acceptable value decided by
the crab cavity.

  6. 6.

In the optics design of the interaction region, the size of each magnet should
not exceed the space limit introduced by the detectors, and there should be no
interference among the magnets.

Furthermore, in order to identify nearly $100\%$ of the reaction products with
higher resolution, the full-acceptance detector poses more requirements on the
design of the interaction region in the EicC accelerator facility.

  1. 1.

In the interaction region of the eRing, a large deflection of electron beams
should be avoided to reduce the impact of the synchrotron radiation background
generated by the electron beams themselves on the detectors in the interaction
region.

  2. 2.

The interaction region of the pRing should be designed as close as possible to
the arc sections to reduce the hadron background produced by the collision
among the proton beams and the residual gas molecules which could affect the
detectors in the interaction region.

  3. 3.

In the pRing, a dipole magnet should be placed downstream of the IP to improve
the detecting resolution of the reaction products at small angles.

  4. 4.

In the pRing, a set of dipole magnets should be placed after the first
focusing quadrupole magnet to improve the detecting resolution of the reaction
products at ultra-small angles.

  5. 5.

In the eRing, at least one set of dipole magnets should be placed after the
first focusing quadrupole magnet to improve the detecting resolution of the
reaction products at small angles [443].

  6. 6.

In the interaction region, the transverse aperture of the magnets should be
larger than the clear zone, which is at least ten times of the transverse RMS
beam size, to let the debris at larger scattering angles in the small-angle
reaction products pass through the vacuum pipe of the magnets and reach the
forward detectors downstream of the IP.