# Vector meson photoproduction in UPCs with FoCal

A. Bylinkin1, J. Nystrand1 and D. Tapia Takaki2 1 Department of Physics and
Technology, University of Bergen, Bergen, Norway  2 Department of Physics and
Astronomy, University of Kansas, Lawrence, KS, USA<EMAIL_ADDRESS>
###### Abstract

We discuss the physics prospects of photon-induced measurements using the
high-granularity FoCal detector to be installed at the ALICE experiment,
covering the pseudorapidity interval $3.4\leq\eta\leq 5.8$. This new detector,
scheduled to be in operation from Run 4, will explore the small Bjorken-$x$
physics region in an unprecedented way. In this region the gluon saturation
phenomenon is expected to be dominant. Combined with the rest of the ALICE
subdetectors, including the zero degree calorimenters, FoCal will serve to
reconstruct in a model-independent way the measured photoproduction cross
sections for vectors mesons in a wide range of photon-target energies, down to
$x$ values of about $7\times 10^{-6}$ and $2\times 10^{-6}$ in ultra-
peripheral photon–proton and photon–lead collisions, respectively.

Keywords: ultra-peripheral collisions, UPC, gluon saturation, LHC Run 4
upgrades

##  1 Introduction

The FoCal detector [1] of ALICE (A Large Ion Collider Experiment) [2] at
CERN’s Large Hadron Collider will address fundamental physics questions about
the dynamics of partons inside nucleons and nuclei, exploring a unique
kinematic region at high energies where the growth of the gluon density is
expected to recede. This is a phenomenon known as gluon saturation that is
predicted by unitary principles in the theory of Quantum Chromodynamics (QCD),
the most compelling theory for describing the strong interactions of quarks
and gluons. Gluon saturated matter implies reaching a dynamical equilibrium
between the splitting and recombination of gluons. Such a regime can be
modelled with the introduction of non-linear QCD evolution equations [3, 4,
5]. It is of fundamental interest to experimentally determine the saturation
scale where this phenomenon appears, for different configurations in nucleons
and nuclei.

It is also of great interest to explore whether the existence of gluon
saturated protons implies the formation of a new state of matter with
universal features present in hadronic matter at high energies [6]. While
measurements at the HERA ep collider at the Deutsches Elektronen-Synchrotron
(DESY) have provided evidence for a significant rise of the gluon density in
protons as a function of energy or Bjorken-$x$ [7], the explored energies were
not sufficient to reach a consensus opinion on the formation of gluonic
saturated matter.

Early measurements from fixed target experiments, followed by studies at the
Relativistic Heavy-Ion Collider (RHIC) in Brookhaven National Laboratory (BNL)
[8, 9, 10, 11] indicate a reduction of the density of low momentum gluons in
heavy nuclei, as compared to the density in individual protons at high
energies. This experimental observation has been explained in terms of nuclear
gluon shadowing, a multi-re-scattering process present in nuclei that can be
linked to gluon saturation in nucleons [12]. The dynamics of the “initial
state” of nuclei before they collide impacts many measurements in nucleus-
nucleus collisions and is important for assessing the properties of the Quark
Gluon Plasma (QGP), a phase of QCD matter where quarks and gluons are
deconfined. For recent reviews see Refs. [13, 14].

The future Electron Ion Collider (EIC) [15] planned to be built at BNL will
focus on addressing several fundamental questions in nuclear physics [16],
including the study of gluon saturated matter. Being a dedicated QCD
accelerator, it promises detailed studies in a wide variety of reactions,
colliding species and energies. At the same time, several unique measurements
in small Bjorken-$x$ physics can only be carried out using the high energy
beams provided by the LHC. Since the end of the HERA collider, the highest
photon energies are being explored using ultra-peripheral heavy ion collisions
(UPC) at the LHC. The protons and heavy ions accelerated at the LHC carry
themselves electromagnetic fields which can be treated as a flux of quasi-real
photons. Using UPCs, the LHC can explore $\gamma\mathrm{p}$ center of mass
energies ($W_{\gamma\mathrm{p}}$) up to several TeV. The center of mass
energies for $\gamma\rm{A}$ and two-photon interactions go up to
$\sim\leavevmode\nobreak\ $700 GeV/nucleon and up to $\sim\leavevmode\nobreak\
$150 GeV, respectively. Altogether, such high energy collisions open the
possibility to study QCD physics and carry out searches for physics beyond the
standard model in an unexplored kinematic region [17, 18, 19, 20, 21, 22, 23].

The ALICE collaboration has recently completed a major detector and system
upgrade. The new data taking period has started this year with Run 3
(2022–2025) and will be followed by Run 4 (2029–2032). The current ALICE plans
are to collect data from Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.36$
TeV, pp collisions at various collision energies, p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=8$ TeV, and O–O interactions at
$\sqrt{s_{\mathrm{NN}}}=6.5$ TeV [24]. We anticipate an integrated luminosity
of about $\mathcal{L}$ = 13 nb-1 in Pb–Pb collisions in Runs 3 & 4, an order
of magnitude increase with respect to Runs 1 & 2\. The increase in the number
of reconstructed events for ALICE UPC measurements will be much more
significant because of the continuous detector read out implemented from Run
3. Despite the excellent performance and high-level productivity of the ALICE
UPC program, the use of dedicated triggers in Runs 1 & 2 has introduced
additional systematic uncertainties to the measurements, and some analyses
were not possible to be done due to trigger and bandwidth limitations. The
prospects for the UPC measurements at the LHC during Runs 3 & 4 are discussed
in Ref. [24]. Some future measurements have also been highlighted in Refs.
[25, 26].

In this context, and to fully exploit the potential of the ALICE experiment in
the study of the small-$x$ parton structure of nucleons and nuclei, a high-
granularity calorimeter (FoCal) has been proposed [1]. It is expected to be in
operation from Run 4. FoCal is a high-granularity, compact silicon-tungsten
(Si+W) sampling electromagnetic calorimeter with longitudinal segmentation
backed by a conventional high granularity metal and scintillating hadronic
calorimeter. It will be located at a distance of $7$ m from the interaction
point, outside the solenoid magnet. The Technical Design Report of FoCal is in
preparation and expected to be completed by summer 2023. FoCal has been
designed for measuring direct photons at forward rapidity with high precision,
as well as jets and $\gamma$-jet and jet-jet events, in pp and p–Pb collisions
[1].

The study of UPCs is also an integral part of the FoCal physics program. The
study of ultra-peripheral heavy-ion collisions with FoCal will contribute to a
more systematic and global study of nucleons and nuclei at high energies in
$\gamma\rm{p}$ and $\gamma\rm{Pb}$ interactions. Although the center of mass
energies in these interactions will be lower than in pp and PbPb, photon-
induced interactions have a clear advantage in probing the structure of the
target. In this document, we first present a brief review of recent UPC vector
meson measurements at the LHC. We then provide the expected yields of VM
photoproduction in the FoCal acceptance, and discuss some high-profile
measurements where FoCal will have a competitive advantage with respect to any
other LHC measurement.

##  2 Brief review of UPC vector meson photoproduction at the LHC

The ALICE, ATLAS, CMS and LHCb collaborations have carried out UPC
measurements in $\gamma\rm{p}$, $\gamma\rm{A}$, and two-photon interactions.
The cross sections for these processes at the LHC are very high. The studies
of photoproduced vector mesons have been of particular interest. One important
goal of these studies is to determine the nuclear gluon distributions. To
leading order perturbative QCD, there is a direct proportionality between the
photoproduction cross section of heavy vector mesons and the square of the
gluon distribution function [27]. Various attempts have been made in order to
translate the photonuclear production cross section to a gluon distribution,
including work towards next-to-leading order calculations of parton
distribution functions and generalized parton distributions [28, 29, 30].

Exclusive photoproduction of $\rm{J/\psi}$ off proton targets has been studied
at the LHC by ALICE in p–Pb collisions [31, 32] and by LHCb in pp collisions
[33, 34, 35]. For a $\rm{J/\psi}$ produced at rapidity $y$ it follows from the
kinematics that the photon-proton center of mass energy squared is

$\rm{W_{\gamma p}^{2}=2E_{p}M_{J/\psi}e^{\pm y}}.$ (1)

Here, $E_{p}$ is the proton energy in the laboratory frame. The $\pm$ in the
exponent corresponds to the cases when the photon emitting beam moves in the
positive and negative direction, respectively. In pp collisions, it is equally
probable that either beam particle emits the photon, whereas in p–Pb
collisions the Pb nucleus is the dominant photon source. This two-fold
ambiguity in determining $W_{\gamma p}$ is the main reason why p–Pb is the
preferred configuration to study photoproduction off a proton target. The
formula also shows that by going to higher rapidities higher values of
$W_{\gamma p}$ are obtained. From the $\gamma\rm{p}$ center of mass energy the
corresponding Bjorken-$x$ is determined, $x=\rm{(M_{J/\psi}/W_{\gamma
p})^{2}}$.

Exclusive $\rm{J/\psi}$ production on proton target has been studied at HERA
up to center of mass energies of 300 GeV, corresponding to $x\sim 10^{-4}$.
The ALICE measurement in p–Pb collisions has extended the energy range to
$W_{\gamma p}=706$ GeV.

Results from HERA showed that the photoproduction cross section,
$\sigma(\gamma p\rightarrow\rm{J/\psi}p)$, increases with center of mass
energy following a power law. The result from ALICE, extending down to $x\sim
10^{-5}$, showed no deviation from this behaviour, although the statistical
error was large.

While the LHCb collaboration has done studies for exclusive $\rm{J/\psi}$ in
pp collisions at $\sqrt{s}=7$ TeV [33, 34] and $\sqrt{s}=13$ TeV [35], their
analyses of $\sigma(\gamma\mathrm{p})$ are strongly model dependent because of
the ambiguity in the photon direction mentioned above. The photoproduction
cross section in a pp collision can be expressed as

$\frac{d\sigma}{dy}=n(+y)\sigma(\rm\gamma\rm{p,+y})+\rm{n(-y})\sigma(\gamma\rm{p,-y})\,,$
(2)

where $\rm{n(+y)}$ and $\rm{n(-y)}$ are the photon fluxes for positive and
negative rapidity, respectively. For each of the measured $d\sigma/dy$ points,
they report two “solutions” for the photon-proton cross section
$\sigma(\rm\gamma p)$: one for the photon emitter and one for the photon
target, corresponding to two different $W_{\gamma\mathrm{p}}$. It is well
known that the corresponding photon fluxes decrease rapidly as the rapidity
increases, thus most of their reported solutions are effectively from low
$W_{\gamma\mathrm{p}}$. In their 7 TeV measurements they have assumed that the
proton dissociation background is not $W_{\gamma\mathrm{p}}$ dependent, in
contradiction to the measurements performed by H1 and ALICE [36, 31]. The 13
TeV pp results from LHCb incorporated the use of a forward detector called
HeRSCheL to mitigate the background from proton dissociation, while the photon
direction ambiguity remains [35].

For these reasons, only measurements in UPC p–Pb collisions can provide a
model independent way for studying the energy dependence of
$\sigma({\gamma\mathrm{p}})$. While LHCb and ATLAS have not reported
measurements in UPC p–Pb collisions, CMS has reported results on exclusive
$\rho^{0}$ [37] and $\Upsilon$ [38] photoproduction. FoCal serves this program
by extending the exclusive $\rm{J/\psi}$ measurements further forward in
rapidity than what it is possible using the ALICE muon spectrometer and LHCb,
reaching down to $x\sim 7\times 10^{-6}$.

The measurements of coherent $\rm{J/\psi}$ photoproduction cross section in
UPC Pb–Pb collisions by the ALICE, CMS and LHCb collaborations have been found
to be compatible with moderate nuclear gluon shadowing, and in good agreement
with the central values of the EPS09 nuclear parton parameterization [39, 40,
41, 42]. Theoretical effort is underway for incorporating the data in parton
distributions [30], albeit some theoretical uncertainties remain [43].
Contrary to pp collisions, it might be possible to determine the photon
direction of the photon emitter and the photon target in Pb–Pb collisions via
the study of the neutron dependence using zero degree calorimeters (ZDCs).
Only ALICE, ATLAS and CMS have ZDCs, although no separation of the photon
sources has been reported at present. FoCal is special since can reach down to
$x\sim 2\times 10^{-6}$ for coherent $\rm{J/\psi}$ photoproduction by doing a
neutron-dependent analysis.

##  3 Physics measurements in p–Pb UPCs with FoCal

The cross sections and expected yields for exclusive production of heavy VMs,
$\rm{p+Pb\rightarrow p+Pb+VM}$, in the dielectron decay channel have been
calculated using STARlight[44], assuming the geometrical acceptance of FoCal
(both electrons within $3.4\leq\eta\leq 5.8$). The yields have been calculated
for both the p-Pb and Pb-p configurations, assuming an integrated luminosity
of $150\,\rm{nb^{-1}}$ in both cases. The results of these calculations are
shown in Table 1.

Hereafter, for the FoCal projections shown in all figures, we have considered
a 60% detector efficiency for the quarkonia measurements as discussed in the
Letter of Intent [1]. A 7% systematic uncertainty is added for the projected
points for exclusive $\rm{J/\psi}$ photoproduction, and 15% for $\rm{J/\psi}$
mesons with proton dissociation following the Run 2 measurements performed by
ALICE. These numbers include the uncertainty on the luminosity determination.
To obtain the uncertainty of the projected numbers, the statistical
uncertainties calculated from the expected yields have been added in
quadrature with the systematical uncertainties reported in ALICE Run 2
analyses. In addition, we only show data point projections when the yield,
before applying efficiency corrections, is at least 80 events per bin in
rapidity. The projected points are also smeared randomly according to the
Gaussian distribution with the width corresponding to the expected statistics
of each measurement. Finally, the FoCal energy reach corresponding to its
acceptance is shown as a box on the figures.

Table 1: Cross sections and yields for VMs calculated from STARlight. The top three lines correspond to the case when the proton is moving towards FoCAL, corresponding to the low energy regime ($11\leq W_{\gamma\rm{p}}\leq 36$ GeV for the $\rm{J/\psi}$). The lower three lines corresponds to the opposite configuration when the Pb-nucleus moves towards FoCAL, corresponding to the high energy regime ($1100\leq W_{\gamma\rm{p}}\leq 3600$ GeV for the $\rm{J/\psi}$). VM | $\sigma(\rm p+Pb\rightarrow p+Pb+VM)$ | $\sigma(3.4\leq\eta_{1,2}\leq 5.8)$ | Yield
---|---|---|---
|  |  p $\rightarrow$ FoCal |  p $\rightarrow$ FoCal
$\rho^{0}$ | 35 mb | 140 nb | 21,000
$\phi$ | 1.7 mb | 51 nb | 7,700
$\rm{J/\psi}$ |  98 $\mu$b | 400 nb | 60,000
$\psi(2S)$ |  16 $\mu$b | 8.9 nb | 1,300
$\Upsilon(1S)$ | 220 nb | 0.38 nb | 60
|  |  Pb $\rightarrow$ FoCal |  Pb $\rightarrow$ FoCal
$\rho^{0}$ | 35 mb | 17 nb | 2,600
$\phi$ | 1.7 mb | 5.3 nb | 800
$\rm{J/\psi}$ |  98 $\mu$b | 36 nb | 5,400
$\psi(2S)$ |  16 $\mu$b | 0.53 nb | 80
$\Upsilon(1S)$ | 220 nb | 0.67 pb |  $\sim$ 0
Figure 1:  ALICE data [45] (full red squares) on exclusive photoproduction of
$\rm{J/\psi}$ off protons as a function of $W_{\gamma\mathrm{p}}$, obtained in
p–Pb UPCs at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, compared to a power-law fit
(green band), to the data from HERA [46, 36] (full violet triangles), to the
STARlight projection (full black circles) in the FoCal acceptance, and to
theoretical models (see text). The LHCb solutions (open circles and squares)
for pp collisions at $\sqrt{s}=7$ TeV and $13$ TeV are also shown [33, 34,
35]. The uncertainties of the ALICE data are the quadratic sum of the
statistical and systematic uncertainties. The uncertainties of the STARlight
prediction are the quadratic sum of the statistical uncertainty of the
expected number of events (Table 1) multiplied by the reconstruction
efficiency and the systematic uncertainty taken at the level of ALICE Run 2
measurements [45]. The FoCal energy reach corresponding to its acceptance is
illustrated with a box.

Figure 1 shows the energy dependence of exclusive $\rm{J/\psi}$
photoproduction measured by ALICE, LHCb, H1 and ZEUS, and the projected data
expected by FoCal. The ALICE measurements have been obtained using ultra-
peripheral p–Pb collisions where there is no ambiguity in the photon
direction, which is not the case for symmetric systems such as pp collisions.
As mentioned above, LHCb has made several assumptions for extracting the
energy dependence of exclusive $\rm{J/\psi}$ in pp. Observing a deviation of a
power-law trend would require the most model-independent calculation possible.
For this figure, the projected FoCal data are shown for the case when the Pb
travels in the direction of the FoCal detector. The STARlight cross section is
a parameterization based on the HERA data, which follows a power-law trend. At
high energies, there are three sets of non-linear QCD models, namely, the Hot
Spot model (CCT) [47], the NLO BFKL [48] and the CGC-based calculations [49].
The data are fitted to the power-law function $N(W_{\gamma p}/W_{0})^{\delta}$
with $W_{0}=90$ GeV following the HERA measurements and $N$ and $\delta$ being
free parameters. The green band shows the experimental uncertainty obtained
from the fit to the existing ALICE data alone. The blue band corresponds to
the uncertainty obtained from the power-law fit to the ALICE data combined
with the STARlight points projected in the FoCal acceptance. Figure 2 shows
the ratio of the existing ALICE data, the STARlight projection and theoretical
models to the power-law fit to the existing ALICE data on exclusive
$\rm{J/\psi}$ photoproduction cross sections. The incorporation of the FoCal
data leads to a significant reduction in the measured uncertainty at high
energies, even in the case of no saturation scenario (STARlight projection).

Figure 2:  Ratio of the ALICE data, STARlight projection and theoretical
models shown in Fig. 1 to the power-law fit through the ALICE data points. The
green band shows the uncertainty of the power-law fit to the existing ALICE
data while the blue band shows the reduction of the fit uncertainty at high
energy when the ALICE data is combined with the STARlight projected points.
The FoCal energy reach corresponding to its acceptance is illustrated with a
box.

The FoCal detector will provide access to an unexplored kinematic regime at
small-$x$ where a different trend in the growth of the cross section might
occur. For this reason the projection of the $\rm{J/\psi}$ photoproduction
cross section in the FoCal acceptance was also obtained using the NLO BFKL
model [48], as it is shown in Figure 3. Since saturation models are expected
to deviate from the power-law dependence, the following function is used to
fit the existing ALICE data together with the projected data points:

$\sigma(\gamma p)\approx\frac{N}{\frac{1}{W_{\gamma p}^{\delta}}+A},$ (3)

which has three free parameters: $N$ that gives the overall normalization,
$\delta$ that describes the power-law rise of the cross section at lower
energies, and $A$ that determines the saturation behavior (broken power-law)
at higher $W_{\gamma p}$ values. As one can see, such functional shape
provides a good description of both the existing ALICE data and the NLO BFKL
projection.

Figure 3:  ALICE data [45] (full red squares) on exclusive photoproduction of
$\rm{J/\psi}$ off protons as a function of $W_{\gamma\mathrm{p}}$, obtained in
p–Pb UPCs at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, compared to a power-law fit
(green band), to the data from HERA [46, 36] (full purple triangles), to the
NLO BFKL projection (full black stars) in the FoCal acceptance, and to
theoretical models (see text). LHCb solutions (open circles and squares) for
pp collisions at $\sqrt{s}=7$ TeV and $13$ TeV are also shown [33, 34, 35].
The uncertainties of the ALICE data are the quadratic sum of the statistical
and systematic uncertainties. The uncertainties of the NLO BFKL prediction are
the quadratic sum of the statistical uncertainty of the expected number of
events (Table 1) multiplied by the reconstruction efficiency and the
systematic uncertainty taken at the level of ALICE Run 2 measurements [45].
The FoCal energy reach corresponding to its acceptance is illustrated with a
box.

To illustrate it in a more quantitative manner, Figure 4 shows the ratio of
the NLO BFKL projection to the same power-law fit to the ALICE data (shown in
Figure 3). Thus, one can conclude that if saturation occurs as predicted by
these models the expected precision of the FoCal measurement on exclusive
$\rm{J/\psi}$ photoproduction in UPC p–Pb collisions would be sufficient to
observe the deviation from the power-law trend at high energies.

Figure 4:  Ratio of the ALICE data, NLO BFKL projection and theoretical models
shown in Fig. 3 to the power-law fit through the ALICE data points. The green
band shows the uncertainty of the power-law fit to the existing ALICE data
while the blue band shows the reduction of the fit uncertainty at high energy
when the ALICE data is combined with the NLO BFKL projected points and fitted
to the formula shown in Eq. 3. The FoCal energy reach corresponding to its
acceptance is illustrated with a box.

Recent theoretical calculations suggest that the ratio of the cross sections
between $\psi(2S)$ and $\rm{J/\psi}$ has a larger sensitivity to gluon
saturation than the individual cross section themselves [26]. This is partly
due to the different wave functions, and the radius dependence of the color
dipole for the two mesons. Figure 5 shows this measured ratio from H1 and ZEUS
data [50, 51], and the projected STARlight events within the FoCal acceptance.
The LHCb collaboration has not reported the measurement for this ratio with
systematic uncertainties [33, 34, 35], so no LHCb points are shown in Fig. 5.
The HERA points are compared to two different types of model calculations: the
calculations based on the color dipole model (GBW) and on the BGK model,
respectively [26]. For these two types of calculations, the linear and the
non-linear (saturation) predictions are shown. There is a very visible
separation between the linear and the non-linear calculation, making this
ratio a very promising measurement to observe gluon saturation at the LHC. The
HERA data at lower energies does not provide any conclusive determination. The
projected FoCal points obtained from STARlight agree with the linear-based
calculations, which is as expected since it does not include any saturation
effects. While this figure does not show the theoretical uncertainty bands
displayed on Figure 2, the comparison between these two different sets of
calculations, the GBW and BGK, provide some confidence that this effect can be
seen regardless of which of the two commonly used QCD models is considered.
The figure also shows the kinematic region that can be explored utilizing the
ALICE muon spectrometer during Run 3. Although there is a sensitivity to
observe saturation effects with the Run 3 data alone, a comprehensive science
program would require exploring the highest energy points which is only
possible with the use of FoCal. Also, there is an added value of having
independent measurements using two different detectors at different rapidity
intervals in Runs 3&4.

Figure 5:  Energy dependence of the ratio between $\psi(2S)$ and $\rm{J/\psi}$
photoproduction cross sections. The points correspond to the HERA data (full
triangles) measured by the H1 and ZEUS collaborations, respectively [50, 51],
and to the STARlight projections (full circles). For the STARlight projected
points only the statistical uncertainty is shown based on the number of
expected events multiplied by the reconstruction efficiency. Theoretical
calculations based on BKG and GBW saturation models and their corresponding
linearized versions [26], are also shown. The two boxes illustrate the
kinematic regions that can be explored utilizing the ALICE muon detectors
during Run 3 and the ALICE FoCal detector during Run 4, respectively.

Besides the study of exclusive photoproduction of vector mesons, the
dissociative process has received recent interest. According to the Good-
Walker formalism [52, 53], dissociative diffraction probes the variance of the
different target configurations, contrary to exclusive processes that only
probes the average of such substructures. It has been shown that the momentum
transfer distribution of dissociative $\rm{J/\psi}$ in proton targets measured
by H1 [36] can be described by shape fluctuations (the BM model) [54, 55]. In
addition, the CCT model [47] considers that the proton target configurations
measured using the dissociative $\rm{J/\psi}$ process can be described by
gluonic hot spots, which as the energy increases, their number increases and
they start overlapping. When the hot spot overlap is significant, the
different configurations look similar and the variance should significantly
decrease in the gluon saturation regime. Figure 6 shows the ratio of
dissociative-to-exclusive $\rm{J/\psi}$ photoproduction as a function of
$W_{\gamma\rm{p}}$, including the H1 data [36] and ALICE preliminary data, and
comparisons to the BM and CCT predictions. The FoCal projected points shown
uses the STARlight yield for the exclusive process, and the BM model
prediction for the dissociative process. Note that STARlight does not make
predictions for dissociative photoproduction. Other more recent predictions
for the energy dependence of the proton geometry fluctuations have also been
reported [56]. The FoCal data are uniquely positioned to probe the
fluctuations of the proton target configurations for gluonic saturated matter.
The observation of a significant reduction of the measured cross section as
energy increases will provide a clear signature of gluon saturation at high
energies.

Figure 6:  Energy dependence of the ratio of dissociative to exclusive
$\rm{J/\psi}$ photoproduction cross sections. The points correspond to the H1
measurement (full triangles), the preliminary ALICE data (full squares) and to
the projected data (open and full circles) for the measurements with the ALICE
muon detectors during Run 3 and the ALICE FoCal detector during Run 4,
respectively. The two boxes correspond to the kinematic regions that can be
explored in the FoCal measurements. Theoretical calculations based on the CCT
and BM saturation models are also shown.

Using the FoCal detector for the case of ultra-peripheral Pb–p collisions in
which the proton travels in its direction might also be of interest. At low
energy $W_{\gamma\rm{p}}$, there are several theoretical motivations related
to a better description of nucleon mass, quarkonia production and others [57].
Figure 7 (top) shows the projected $\rm{J/\psi}$ photoproduction cross
sections predicted by STARlight, together with the previous LHC, HERA and
fixed-target measurements. With FoCal the energies that can be achieved are
much lower than those from the current collider experiments, probing as low as
$W_{\gamma\rm{p}}$ = 12 GeV, where an onset of near-threshold effects in the
$\rm{J/\psi}$ photoproduction can already be seen. This measurement will also
provide a cross check, with unprecedented experimental precision, of the data
reported by the E401 experiment which is found to be significantly below the
current theoretical predictions in this kinematic region. Recently, the GlueX
experiment at the Thomas Jefferson National Accelerator Facility (JLab) has
performed precise measurements of near threshold $\rm{J/\psi}$ photoproduction
[58]. Their data are also shown in Fig. 7 as a function of $W_{\gamma\rm{p}}$.
There is a significant gap between these low-energy points and the existing
LHC data. Before the start of the future EIC, this kinematic region can be
explored by having a special Pb–p run at the LHC. We consider a potential
special run such that the proton and lead beams would be colliding at
$\sqrt{s_{\mathrm{NN}}}=1.26$ TeV. It is interesting to point out that with
only two days of machine operation, corresponding to a collected luminosity
$\mathcal{L}$ = 20 nb-1, the number of recorded events should be sufficient to
precisely map out this kinematic region, see Figure 7 (bottom). In this center
of mass energy range, the photon energy is low leading to very large fluxes
from the lead ion. This explains the large yields even for modest integrated
luminosities in this configuration.

Figure 7:  ALICE data [45] (full red squares) on exclusive photoproduction of
$\rm{J/\psi}$ off protons as a function of $W_{\gamma\mathrm{p}}$, obtained in
Pb–p UPCs at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, compared to a power-law fit
(green band), to the data from HERA [46] (full blue triangles), to the
STARlight projection (full black circles) in the FoCal acceptance, and to
theoretical models (see text). LHCb solutions (open squares) for pp collisions
at $\sqrt{s}=7$ TeV and $13$ TeV are also shown together with various
measurements performed by fixed target experiments. The lower panel shows in
addition the STARlight projection for the Pb–p UPCs at
$\sqrt{s_{\mathrm{NN}}}=1.26$ TeV (full stars).

##  4 Physics measurements in Pb–Pb UPCs with FoCal

Exclusive photonuclear production of VMs has been studied by several
experiments at the LHC [39, 59, 60, 61]. The cross sections and expected
yields for exclusive production of heavy VMs in heavy-ion collisions,
$\rm{Pb+Pb\rightarrow Pb+Pb+V}$, in the dielectron decay channel have been
calculated from STARlight[44], assuming the geometrical acceptance of FoCal
(both electrons within $3.4\leq\eta\leq 5.8$). The yields have been calculated
assuming an integrated luminosity $\mathcal{L}$ = $7.0\,{\rm nb^{-1}}$ for LHC
Run 4. The results are shown in Table 2. The expected statistics for the
$\rm{J/\psi}$ is very large, and it should thus be possible to map out the
detailed shape of the rapidity distribution with high precision over the range
$3.4<y<5.8$.

A recent study, based on a next-to-leading order calculation, has found that
there is a strong contribution to the cross section not just from gluons but
also from quarks [30]. Figure 8 shows the prediction from this model (NLO)
compared with the existing ALICE and LHCb data. The figure also includes the
STARlight projection into the FoCal acceptance, scaled by a factor 0.42 as
documented in the recent CERN Yellow Report [24]. The latter is included since
STARlight, which does not include gluon shadowing, is known to overpredict the
measured cross section at midrapidity. As can be seen in the figure, the
interference between the quark and gluon contributions is largest in the
region covered by the FoCal acceptance.

Both STARlight and the NLO calculations show a bump in the cross section at
very forward rapidities ($y>5.3$), see Figure 8 (bottom). This bump comes from
the second diffractive peak in the nuclear form factor, which cuts off the
cross section for very low photon energies. This means that in this region the
$\rm{J/\psi}$ transverse momentum spectrum will be modified, leading to a
larger $<p_{T}>$ than for the bulk of the events at lower rapidities.

Table 2: Cross sections and yields for VMs calculated from STARlight assuming an integrated luminosity $\mathcal{L}$ = 7.0 nb-1. VM | $\sigma(\rm Pb+Pb\rightarrow Pb+Pb+VM)$ | $\sigma(3.4\leq\eta_{1,2}\leq 5.8)$ | Yield
---|---|---|---
$\rho^{0}$ | 5.0 b |  20 $\mu$b | 140,000
$\phi$ | 440 mb |  10 $\mu$b | 70,000
$\rm{J/\psi}$ | 39 mb |  53 $\mu$b | 370,000
$\psi(2S)$ | 7.5 mb |  1.1 $\mu$b | 7,500
$\Upsilon(1S)$ |  94 $\mu$b | 5.0 nb | 35

Figure 8:  ALICE data [45] (full red squares) for the rapidity dependence of
the measured cross section for coherent $\rm{J/\psi}$ photoproduction in
ultra-peripheral Pb–Pb collisions compared to the recent theoretical
calculation (see text) [30], to the LHCb data (open circles) and to the
STARlight projection (full black circles) in the FoCal acceptance. The box
corresponds to the acceptance of the FoCal detector. The bottom figure shows
the same data in the logarithmic scale, where the contribution from the form
factor is more visible in the most forward region.

Contrary to analyses based on pp data at forward rapidity, it has been
suggested that in ultra-peripheral Pb–Pb collisions it is possible to
distinguish the photon direction by studying the emission of forward neutrons
that accompany the vector meson [24, 62]. The method to identify which of the
two Pb nucleus is the photon emitter or the photon target relies on the
measurement of $\rm{J/\psi}$ events in terms of the following nuclear break up
configurations: 0n0n (no neutron on both sides of the interaction point), Xn0n
or 0nXn (at least one neutron on one side of the interaction point and no
neutrons on the opposite side), and XnXn (at least one neutron on both sides
of interaction point). The measured cross section can be expressed as

$\frac{d\sigma}{dy}=\frac{d\sigma\rm(0n0n)}{dy}+2\frac{d\sigma\rm(0nXn)}{dy}+\frac{d\sigma\rm(XnXn)}{dy}\,,$
(4)

where for coherent $\rm{J/\psi}$ is reasonable to consider that the cross
sections are the same for the Xn0n and 0nXn configurations. The relative
fractions of the neutron break up modes can be computed using STARlight [44]
or nOOn [63]. Table 3 presents the predicted STARlight cross sections and the
expected yields.

Table 3: Cross sections and yields for coherent $\rm{J/\psi}$ photoproduction as a function of the different break up scenarios, calculated from STARlight assuming an integrated luminosity $\mathcal{L}$ = 7.0 nb-1. The small difference in the $\rm{J/\psi}$ yield compared to Table 2 is due to rounding errors in the calculation. Neutron configuration | $\sigma(\rm Pb+Pb\rightarrow\rm{J/\psi}+Pb+Pb)$ | $\sigma(3.4\leq\eta_{1,2}\leq 5.8)$ | Yield
---|---|---|---
0n0n | 28.8 mb |  47 $\mu$b | 329,000
0nXn + Xn0n | 7.3 mb |  5.0 $\mu$b | 35,000
XnXn | 3.0 mb |  2.0 $\mu$b | 14,000

For each of three differential equations shown on the left hand side of Eq. 4,
one can express them in terms of the total cross section
$\rm\sigma(\gamma+Pb)$ and the corresponding photon flux for both rapidity
directions using Eq. 2, thus resulting in three coupled equations. The
solution of this linear system results in the determination of
$\sigma(\rm\gamma\rm{Pb,+y})$ and $\sigma(\rm\gamma\rm{Pb,-y})$. In practice,
there are several ways how to solve the system of coupled equations. As
described in [24, 62], at least the 0n0n and 0nXn are needed since the XnXn
component only contributes to less than 4% of the yield. Also, the XnXn sample
might suffer from a large background from non-UPC events. It is also possible
to extract the cross sections $\sigma(\rm\gamma\rm{Pb,+y})$ and
$\sigma(\rm\gamma\rm{Pb,-y})$ by considering all the measurements together
using a $\chi^{2}$ fit or employing the singular value decomposition (SVD)
method [64].

For this study, we have considered the SVD method and that the experimental
uncertainties of the measurement are about 5% for 0n0n, about 6% for 0nXn (or
Xn0n) and about 14% for XnXn. As done above, a 60% efficiency correction is
applied beyond the acceptance effect. We have also required that there should
be about 80 $\rm{J/\psi}$ events in each of the neutron configurations. This
imposes a careful selection of the number of rapidity intervals, thus the
$\rm\sigma(\gamma+Pb)$ ranges, that can be studied. This results in different
rapidity bin sizes, specially for the lowest and highest energy points. We
found that in order to optimize the extraction of the high energy points (a
small fraction of the total sample since the photon flux decreases rapidly
with rapidity), only five rapidity intervals can be considered, resulting in
10 $\rm\sigma(\gamma+Pb)$ points. Figure 9 shows the $\sigma(\gamma\rm{Pb})$
cross section for coherent $\rm{J/\psi}$ photoproduction in ultra-peripheral
Pb–Pb collisions as function of $W_{\gamma\rm{Pb}}$, using the SVD method to
reconstruct the photon direction. The uncertainty of the measurement results
from propagating the experimental uncertainties of the $\rm d{\sigma}/{dy}$s
in the various neutron configurations. Clearly a different analysis of the
data could also be done so to optimize the low energy points, while here we
preferred to illustrate the projected statistics and uncertainties by
optimizing the extraction of the high energy points where the effects of non-
linear QCD should be dominant. Figure 9 also shows the predictions of the
impulse approximation, and the CCT and b-BK-A models [47, 65, 66]. The latter
models include gluon saturation effects, while STARlight consider Glauber-like
corrections not present in the impulse approximation. At present, no
experimental data exist for this measurement. Note that LHCb does not have
ZDCs installed at present or for Run 4, making ALICE, thanks to the FoCal
detector, the only experiment capable of carrying out this analysis, reaching
down to $x$ values of about $2\times 10^{-6}$.

Figure 9:  STARlight projection (full black circles) for the cross section of
coherent $\rm{J/\psi}$ photoproduction in ultra-peripheral Pb–Pb collisions
shown as a function of the photon-lead center-of-mass energy
$W_{\gamma\rm{Pb}}$. To separate the two photon directions, this analysis will
use both the FoCal and ZDC detectors. Predictions from the impulse
approximation, CCT and b-BK models are also shown. The two boxes indicate the
FoCal detector acceptance for the two explored kinematic regions.

##  5 Other possible UPC measurements with FoCal

There are several additional UPC measurements where FoCal can contribute. Such
measurements include the study of low-mass vector mesons such as $\rho^{0}$
and $\phi(1020)$ [67], dielectron production from two-photon interactions,
excited $\rho$ states [68], open charm [69], inclusive photonuclear dijets
[70] and diffractive dijets [71]. The light-by-light scattering process could
also be explored [72] in future performance studies. UPC processes have also
been suggested as a way to probe the Einstein-Podolsky-Rosen relationship
[73]. It would be interested to utilize new techniques such as quantum
tomography [74] to analyze the UPC data in new ways. Finally, photon-induced
interactions might be of future interest in the search for physics beyond the
standard model.

##  6 Summary

The high-granularity FoCal calorimeter at ALICE will provide access to an
unexplored kinematic region where gluon saturation phenomena should be very
visible. We have provided the expected yield of vector meson photoproduction
in the dielectron decay channel. The expected statistics will be very large in
both ultra-peripheral p–Pb and Pb–Pb collisions. The future measurements of
the energy dependence of $\sigma(\gamma\rm{p})$ and $\sigma(\gamma\rm{Pb})$
for the photoproduction of $\rm{J/\psi}$ mesons should provide a clear
signature of gluon saturation if it is there. Moreover, FoCal will enable
measuring the energy dependence of the ratio of exclusive
$\psi(2S)$–to-$\rm{J/\psi}$, which is a measurement with high sensitivity. The
study of the dissociative production of exclusive $\rm{J/\psi}$ is sensitive
to quantum fluctuations of the proton constituents, thus also sensitive of
gluon saturation. Finally, we have discussed the prospects of studying the low
energy points of $\rm{J/\psi}$ photoproduction in UPC p–Pb collisions, and the
impact of having a short special run in p–Pb to explore the photoproduction of
$\rm{J/\psi}$ near threshold. These prospects have been discussed in the
context of existing and future measurements at LHC, concluding that FoCal
provides a unique physics program for these processes, probing the high-energy
limit of QCD in an unprecedented way.

## Acknowledgments

We thank colleagues from the ALICE UPC and FoCal groups for interesting
discussions, and Martin Hentschinski, Heikki Mäntysaari and the authors of
[43] for providing their theory predictions. We acknowledge support by the
Norwegian Research Council and the U.S. Department of Energy, Office of
Science, Nuclear Physics.

## References

  * [1] ALICE Collaboration, “Letter of Intent: A Forward Calorimeter (FoCal) in the ALICE experiment,”.
  * [2] ALICE Collaboration, K. Aamodt et al., “The ALICE experiment at the CERN LHC,” JINST 3 (2008) S08002.
  * [3] L. V. Gribov, E. M. Levin, and M. G. Ryskin, “Semihard Processes in QCD,” Phys. Rept. 100 (1983) 1–150.
  * [4] A. H. Mueller, “Small x Behavior and Parton Saturation: A QCD Model,” Nucl. Phys. B 335 (1990) 115–137.
  * [5] K. J. Golec-Biernat and M. Wusthoff, “Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction,” Phys. Rev. D 59 (1998) 014017, arXiv:hep-ph/9807513.
  * [6] F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, “The Color Glass Condensate,” Ann. Rev. Nucl. Part. Sci. 60 (2010) 463–489, arXiv:1002.0333 [hep-ph].
  * [7] H1, ZEUS Collaboration, H. Abramowicz et al., “Combination of measurements of inclusive deep inelastic ${e^{\pm}p}$ scattering cross sections and QCD analysis of HERA data,” Eur. Phys. J. C 75 no. 12, (2015) 580, arXiv:1506.06042 [hep-ex].
  * [8] BRAHMS Collaboration, I. Arsene et al., “Evolution of the nuclear modification factors with rapidity and centrality in d+Au collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV,” Phys. Rev. Lett. 93 (2004) 242303, arXiv:nucl-ex/0403005.
  * [9] STAR Collaboration, J. Adams et al., “Forward neutral pion production in p+p and d+Au collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV,” Phys. Rev. Lett. 97 (2006) 152302, arXiv:nucl-ex/0602011.
  * [10] PHENIX Collaboration, A. Adare et al., “Suppression of back-to-back hadron pairs at forward rapidity in $d+$Au Collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV,” Phys. Rev. Lett. 107 (2011) 172301, arXiv:1105.5112 [nucl-ex].
  * [11] PHOBOS Collaboration, B. Alver et al., “High transverse momentum triggered correlations over a large pseudorapidity acceptance in AuAu collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV,” Phys. Rev. Lett. 104 (2010) 062301, arXiv:0903.2811 [nucl-ex].
  * [12] N. Armesto, “Nuclear shadowing,” J. Phys. G 32 (2006) R367–R394, arXiv:hep-ph/0604108.
  * [13] W. Busza, K. Rajagopal, and W. van der Schee, “Heavy Ion Collisions: The Big Picture, and the Big Questions,” Ann. Rev. Nucl. Part. Sci. 68 (2018) 339–376, arXiv:1802.04801 [hep-ph].
  * [14] A. Morreale and F. Salazar, “Mining for Gluon Saturation at Colliders,” Universe 7 no. 8, (2021) 312, arXiv:2108.08254 [hep-ph].
  * [15] A. Accardi et al., “Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all,” Eur. Phys. J. A 52 no. 9, (2016) 268, arXiv:1212.1701 [nucl-ex].
  * [16] R. Abdul Khalek et al., “Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report,” Nucl. Phys. A 1026 (2022) 122447, arXiv:2103.05419 [physics.ins-det].
  * [17] A. Baltz, “The Physics of Ultraperipheral Collisions at the LHC,” Phys. Rept. 458 (2008) 1–171, arXiv:0706.3356 [nucl-ex].
  * [18] J. G. Contreras and J. D. Tapia Takaki, “Ultra-peripheral heavy-ion collisions at the LHC,” Int. J. Mod. Phys. A30 (2015) 1542012.
  * [19] S. Klein and J. Nystrand, “Ultraperipheral nuclear collisions,” Phys. Today 70 no. 10, (2017) 40–47.
  * [20] S. R. Klein and H. Mäntysaari, “Imaging the nucleus with high-energy photons,” Nature Rev. Phys. 1 no. 11, (2019) 662–674, arXiv:1910.10858 [hep-ex].
  * [21] S. Klein and P. Steinberg, “Photonuclear and Two-photon Interactions at High-Energy Nuclear Colliders,” Ann. Rev. Nucl. Part. Sci. 70 (2020) 323–354, arXiv:2005.01872 [nucl-ex].
  * [22] W. Schäfer, “Photon induced processes: from ultraperipheral to semicentral heavy ion collisions,” Eur. Phys. J. A 56 no. 9, (2020) 231.
  * [23] L. Frankfurt, V. Guzey, A. Stasto, and M. Strikman, “Selected topics in diffraction with protons and nuclei: past, present, and future,” Rept. Prog. Phys. 85 no. 12, (2022) 126301, arXiv:2203.12289 [hep-ph].
  * [24] Z. Citron et al., “Report from Working Group 5: Future physics opportunities for high-density QCD at the LHC with heavy-ion and proton beams,” CERN Yellow Rep. Monogr. 7 (2019) 1159–1410, arXiv:1812.06772 [hep-ph].
  * [25] S. Klein et al., “New opportunities at the photon energy frontier,” arXiv:2009.03838 [hep-ph].
  * [26] M. Hentschinski et al., “White Paper on Forward Physics, BFKL, Saturation Physics and Diffraction,” arXiv:2203.08129 [hep-ph].
  * [27] M. G. Ryskin, “Diffractive J/$\psi$ electroproduction in LLA QCD,” Z. Phys. C 57 (1993) 89–92.
  * [28] V. Guzey, E. Kryshen, M. Strikman, and M. Zhalov, “Evidence for nuclear gluon shadowing from the ALICE measurements of PbPb ultraperipheral exclusive $J/{\psi}$ production,” Phys. Lett. B726 (2013) 290–295, arXiv:1305.1724 [hep-ph].
  * [29] C. A. Flett, S. P. Jones, A. D. Martin, M. G. Ryskin, and T. Teubner, “Exclusive production of heavy quarkonia as a probe of the low x and low scale gluon PDF,” arXiv:1912.09128 [hep-ph].
  * [30] C. A. Flett, S. P. Jones, A. D. Martin, M. G. Ryskin, and T. Teubner, “How to include exclusive $J/\psi$ production data in global PDF analyses,” arXiv:1908.08398 [hep-ph].
  * [31] ALICE Collaboration, B. B. Abelev et al., “Exclusive $\mathrm{J/}\psi$ photoproduction off protons in ultra-peripheral p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV,” Phys. Rev. Lett. 113 no. 23, (2014) 232504, arXiv:1406.7819 [nucl-ex].
  * [32] ALICE Collaboration, S. Acharya et al., “Energy dependence of exclusive $\mathrm{J}/\psi$ photoproduction off protons in ultra-peripheral p–Pb collisions at $\sqrt{s_{\mathrm{\scriptscriptstyle NN}}}=5.02$ TeV,” Eur. Phys. J. C 79 no. 5, (2019) 402, arXiv:1809.03235 [nucl-ex].
  * [33] LHCb Collaboration, R. Aaij et al., “Exclusive $J/\psi$ and $\psi$(2S) production in pp collisions at $\sqrt{s}=7$ TeV,” J. Phys. G 40 (2013) 045001, arXiv:1301.7084 [hep-ex].
  * [34] LHCb Collaboration, R. Aaij et al., “Updated measurements of exclusive $J/\psi$ and $\psi$(2S) production cross-sections in pp collisions at $\sqrt{s}=7$ TeV,” J. Phys. G 41 (2014) 055002, arXiv:1401.3288 [hep-ex].
  * [35] LHCb Collaboration, R. Aaij et al., “Central exclusive production of $J/\psi$ and $\psi(2S)$ mesons in $pp$ collisions at $\sqrt{s}=13\leavevmode\nobreak\ $TeV,” JHEP 10 (2018) 167, arXiv:1806.04079 [hep-ex].
  * [36] H1 Collaboration, C. Alexa et al., “Elastic and Proton-Dissociative Photoproduction of J/$\psi$ Mesons at HERA,” Eur. Phys. J. C 73 no. 6, (2013) 2466, arXiv:1304.5162 [hep-ex].
  * [37] CMS Collaboration, A. M. Sirunyan et al., “Measurement of exclusive $\rho(770)^{0}$ photoproduction in ultraperipheral pPb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV,” Eur. Phys. J. C 79 no. 8, (2019) 702, arXiv:1902.01339 [hep-ex].
  * [38] CMS Collaboration, A. M. Sirunyan et al., “Measurement of exclusive $\Upsilon$ photoproduction from protons in pPb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV,” Eur. Phys. J. C 79 no. 3, (2019) 277, arXiv:1809.11080 [hep-ex]. [Erratum: Eur.Phys.J.C 82, 343 (2022)].
  * [39] ALICE Collaboration, B. Abelev et al., “Coherent $J/\psi$ photoproduction in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV,” Phys. Lett. B 718 (2013) 1273–1283, arXiv:1209.3715 [nucl-ex].
  * [40] ALICE Collaboration, E. Abbas et al., “Charmonium and $e^{+}e^{-}$ pair photoproduction at mid-rapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\rm NN}}$=2.76 TeV,” Eur. Phys. J. C 73 no. 11, (2013) 2617, arXiv:1305.1467 [nucl-ex].
  * [41] ALICE Collaboration, S. Acharya et al., “Coherent $J/\psi$ and $\psi^{\prime}$ photoproduction at midrapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.02$ TeV,” Eur. Phys. J. C 81 no. 8, (2021) 712, arXiv:2101.04577 [nucl-ex].
  * [42] CMS Collaboration, V. Khachatryan et al., “Coherent $J/\psi$ photoproduction in ultra-peripheral PbPb collisions at $\sqrt{s_{NN}}=$ 2.76 TeV with the CMS experiment,” Phys. Lett. B 772 (2017) 489–511, arXiv:1605.06966 [nucl-ex].
  * [43] K. J. Eskola, C. A. Flett, V. Guzey, T. Löytäinen, and H. Paukkunen, “Exclusive $J/\psi$ photoproduction in ultraperipheral Pb+Pb collisions at the LHC to next-to-leading order perturbative QCD,” arXiv:2203.11613 [hep-ph].
  * [44] S. R. Klein, J. Nystrand, J. Seger, Y. Gorbunov, and J. Butterworth, “STARlight: A Monte Carlo simulation program for ultra-peripheral collisions of relativistic ions,” Comput. Phys. Commun. 212 (2017) 258–268, arXiv:1607.03838 [hep-ph].
  * [45] ALICE Collaboration, S. Acharya et al., “Energy dependence of exclusive $\mathrm{J}/\psi$ photoproduction off protons in ultra-peripheral p–Pb collisions at $\sqrt{s_{\mathrm{\scriptscriptstyle NN}}}=5.02$ TeV,” Eur. Phys. J. C79 no. 5, (2019) 402, arXiv:1809.03235 [nucl-ex].
  * [46] ZEUS Collaboration, S. Chekanov et al., “Exclusive photoproduction of J/$\psi$ mesons at HERA,” Eur. Phys. J. C 24 (2002) 345–360, arXiv:hep-ex/0201043.
  * [47] J. Cepila, J. Contreras, and J. D. Tapia Takaki, “Energy dependence of dissociative $\mathrm{J/}\psi$ photoproduction as a signature of gluon saturation at the LHC,” Phys. Lett. B 766 (2017) 186–191, arXiv:1608.07559 [hep-ph].
  * [48] I. Bautista, A. Fernandez Tellez, and M. Hentschinski, “BFKL evolution and the growth with energy of exclusive $J/\psi$ and $\Upsilon$ photoproduction cross sections,” Phys. Rev. D 94 no. 5, (2016) 054002, arXiv:1607.05203 [hep-ph].
  * [49] N. Armesto and A. H. Rezaeian, “Exclusive vector meson production at high energies and gluon saturation,” Phys. Rev. D 90 no. 5, (2014) 054003, arXiv:1402.4831 [hep-ph].
  * [50] H1 Collaboration, C. Adloff et al., “Diffractive photoproduction of $\psi$(2S) mesons at HERA,” Phys. Lett. B 541 (2002) 251–264, arXiv:hep-ex/0205107.
  * [51] ZEUS Collaboration, “Measurement of the cross-section ratio $\sigma_{\psi(2S)}/\sigma_{J/\psi(1S)}$ in exclusive photoproduction at HERA,” arXiv:2206.13343 [hep-ex].
  * [52] M. L. Good and W. D. Walker, “Diffraction disssociation of beam particles,” Phys. Rev. 120 (1960) 1857–1860.
  * [53] H. I. Miettinen and J. Pumplin, “Diffraction Scattering and the Parton Structure of Hadrons,” Phys. Rev. D 18 (1978) 1696.
  * [54] H. Mäntysaari and B. Schenke, “Evidence of strong proton shape fluctuations from incoherent diffraction,” Phys. Rev. Lett. 117 no. 5, (2016) 052301, arXiv:1603.04349 [hep-ph].
  * [55] H. Mäntysaari and B. Schenke, “Revealing proton shape fluctuations with incoherent diffraction at high energy,” Phys. Rev. D 94 no. 3, (2016) 034042, arXiv:1607.01711 [hep-ph].
  * [56] A. Kumar and T. Toll, “Energy dependence of the proton geometry in exclusive vector meson production,” Phys. Rev. D 105 no. 11, (2022) 114011, arXiv:2202.06631 [hep-ph].
  * [57] Y.-P. Xie and V. P. Goncalves, “Near threshold heavy vector meson photoproduction at LHC and EicC,” Eur. Phys. J. C 81 no. 7, (2021) 645, arXiv:2103.12568 [hep-ph].
  * [58] GlueX Collaboration, A. Ali et al., “First Measurement of Near-Threshold J/$\psi$ Exclusive Photoproduction off the Proton,” Phys. Rev. Lett. 123 no. 7, (2019) 072001, arXiv:1905.10811 [nucl-ex].
  * [59] ALICE Collaboration, E. Abbas et al., “Charmonium and $e^{+}e^{-}$ pair photoproduction at mid-rapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\rm NN}}$=2.76 TeV,” Eur. Phys. J. C73 no. 11, (2013) 2617, arXiv:1305.1467 [nucl-ex].
  * [60] ALICE Collaboration, J. Adam et al., “Coherent $\rho^{0}$ photoproduction in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV,” JHEP 09 (2015) 095, arXiv:1503.09177 [nucl-ex].
  * [61] ALICE Collaboration, “Coherent $\psi$(2S) photo-production in ultra-peripheral Pb Pb collisions at $\sqrt{s_{\mathrm{N}N}}$ = 2.76 TeV,” Phys. Lett. B751 (2015) 358–370, arXiv:1508.05076 [nucl-ex].
  * [62] V. Guzey, M. Strikman, and M. Zhalov, “Disentangling coherent and incoherent quasielastic $J/\psi$ photoproduction on nuclei by neutron tagging in ultraperipheral ion collisions at the LHC,” Eur. Phys. J. C 74 no. 7, (2014) 2942, arXiv:1312.6486 [hep-ph].
  * [63] M. Broz, J. G. Contreras, and J. D. Tapia Takaki, “A generator of forward neutrons for ultra-peripheral collisions: nOOn,” Comput. Phys. Commun. 253 (2020) 107181, arXiv:1908.08263 [nucl-th].
  * [64] A. Hocker and V. Kartvelishvili, “SVD approach to data unfolding,” Nucl. Instrum. Meth. A 372 (1996) 469–481, arXiv:hep-ph/9509307.
  * [65] J. Cepila, J. G. Contreras, M. Krelina, and J. D. Tapia Takaki, “Mass dependence of vector meson photoproduction off protons and nuclei within the energy-dependent hot-spot model,” Nucl. Phys. B 934 (2018) 330–340, arXiv:1804.05508 [hep-ph].
  * [66] D. Bendova, J. Cepila, and J. G. Contreras, “Dissociative production of vector mesons at electron-ion colliders,” Phys. Rev. D 99 no. 3, (2019) 034025, arXiv:1811.06479 [hep-ph].
  * [67] V. P. Gonçalves, F. S. Navarra, and D. Spiering, “Exclusive $\rho$ and $J/\Psi$ photoproduction in ultraperipheral $pA$ collisions: Predictions of the gluon saturation models for the momentum transfer distributions,” Phys. Lett. B 791 (2019) 299–304, arXiv:1811.09124 [hep-ph].
  * [68] M. Klusek-Gawenda and J. D. T. Takaki, “Exclusive four-pion photoproduction in ultra-peripheral heavy-ion collisions at RHIC and LHC energies,” Acta Phys. Polon. B 51 (2020) 1393, arXiv:2005.13624 [hep-ph].
  * [69] S. R. Klein, J. Nystrand, and R. Vogt, “Heavy quark photoproduction in ultraperipheral heavy ion collisions,” Phys. Rev. C 66 (2002) 044906, arXiv:hep-ph/0206220.
  * [70] ATLAS Collaboration, ATLAS Collaboration, “Photo-nuclear dijet production in ultra-peripheral Pb+Pb collisions.” ATLAS-CONF-2017-011, 2017\.
  * [71] CMS Collaboration, “Azimuthal correlations within exclusive dijets with large momentum transfer in photon-lead collisions,” arXiv:2205.00045 [nucl-ex].
  * [72] M. Klusek-Gawenda, “$\gamma\gamma\to\gamma\gamma$ in heavy ion collisions – new results and prospects,” PoS ICHEP2020 (2021) 329.
  * [73] S. R. Klein and J. Nystrand, “Does particle decay cause wave function collapse: An Experimental test,” Phys. Lett. A 308 (2003) 323–328, arXiv:quant-ph/0206060.
  * [74] J. C. Martens, J. P. Ralston, and J. D. Tapia Takaki, “Quantum tomography for collider physics: Illustrations with lepton pair production,” Eur. Phys. J. C 78 no. 1, (2018) 5, arXiv:1707.01638 [hep-ph].