EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-108 LHCb-PAPER-2022-012 16 March 2022

Study of the coherent charmonium production in ultra-peripheral lead-lead
collisions

LHCb collaboration

The cross-sections of coherent ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ and ${\psi{(2S)}}$ production in ultra-peripheral PbPb collisions at
a nucleon-nucleon centre-of-mass energy of $5.02\text{\,Te\kern-1.00006ptV}$
are measured using a data sample corresponding to an integrated luminosity of
$228\pm 10\,\upmu\text{b}^{-1}$, collected by the LHCb experiment in 2018. The
differential cross-sections are measured separately as a function of
transverse momentum and rapidity in the nucleus-nucleus centre-of-mass frame
for ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and ${\psi{(2S)}}$
mesons. The integrated cross-sections are measured to be
$\sigma^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=5.965\pm 0.059\pm 0.232\pm 0.262\text{\,mb}$ and
$\sigma^{\mathrm{coh}}_{{\psi{(2S)}}}=0.923\pm 0.086\pm 0.028\pm
0.040\text{\,mb}$, where the first listed uncertainty is statistical, the
second systematic and the third due to the luminosity determination. The
cross-section ratio is measured to be
$\sigma^{\mathrm{coh}}_{{\psi{(2S)}}}/\sigma^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=0.155\pm 0.014\pm 0.003$, where the first uncertainty is statistical
and the second is systematic. These results are compatible with theoretical
predictions.

Submitted to JHEP

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

Ultra-peripheral collisions (UPCs) occur when two nuclei collide with an
impact parameter, the distance between their centres, larger than the sum of
their radii [1]. Because the nuclei do not overlap, strong interactions are
suppressed so that the two ions interact via their cloud of semi-real photons
and photon-nuclear interactions dominate. The number of photons in the
virtual-photon cloud is proportional to the square of electric charge, so
photon-nuclear interactions are significantly enhanced in lead-lead (PbPb)
collisions compared to proton-proton ($pp$) collisions. In UPCs,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons are
produced from the colourless exchange of a photon from one of the two nuclei
and a pomeron from the other. Coherent production occurs when the photon
interacts with a pomeron emitted by the entire nucleus, while for incoherent
production, the pomeron is emitted from a single nucleon within the nucleus.
Figure 1 illustrates the difference between coherent and incoherent
production.

Figure 1: Schematic diagrams for charmonium production in (a) coherent and (b)
incoherent UPC heavy-ion collisions [2]. For incoherent production, the
pomeron is emitted from a single nucleon, indicated by the blue dot, and the
nucleus typically dissociates, indicated by $A^{*}$.

This paper presents measurements of coherent
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ production
cross-sections, which are expected to probe the nuclear gluon distribution
functions at a momentum transfer of $Q^{2}\approx m^{2}/4$, where $m$ is the
mass of the meson. The photon-nuclear production of these mesons depends on
the longitudinal momentum fraction of gluons in the nucleus,
$x\approx(m/\sqrt{s_{\scriptscriptstyle\text{NN}}})e^{\pm y}$, where $y$ is
the rapidity of the meson and $\sqrt{s_{\scriptscriptstyle\text{NN}}}$ is the
nucleon-nucleon centre-of-mass energy. Thus, coherent photo-production of
charmonium mesons provides an excellent laboratory to study nuclear shadowing
effects and the initial states of collisions with small $x$, where
$10^{-5}\lesssim x\lesssim 10^{-2}$ at the LHC [3]. The charmonia produced in
this process have typical transverse momenta, $p_{\mathrm{T}}$, smaller than
$100\text{\,Me\kern-1.00006ptV\\!/}c$, with no other particles produced in the
collision. The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
${\psi{(2S)}}$ mesons are reconstructed through the dimuon final state using
the 2018 PbPb data sample collected by the LHCb experiment at
$\sqrt{s_{\scriptscriptstyle\text{NN}}}=5.02\text{\,Te\kern-1.00006ptV}$ and
corresponding to an integrated luminosity of $228\pm 10\,\upmu\text{b}^{-1}$.
The cross-section of coherent ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
production in UPCs at 5.02 TeV has been measured previously at ALICE [4] and
LHCb [5].

This study also measures the ratio between the coherent $\psi{(2S)}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-sections,
where the uncertainties due to systematic effects and the luminosity
determination largely cancel. This more precise measurement will help to
constrain theoretical predictions, where uncertainties arise from the choice
of the meson wave function in dipole scattering models [6, 7] and the
factorisation scale in perturbative QCD models [8].

The LHCb detector and simulation are described in Sec. 2. The selection of
signal candidates and the determination of cross-sections are described in
Sec. 3 and Sec. 4, respectively. The uncertainties due to systematic effects
are described in Sec. 5, while the results are presented in Sec. 6 and
conclusions are given in Sec. 7.

##  2 Detector, event reconstruction and simulation

The LHCb detector [9, 10] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the collision
region, a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three stations of
silicon-strip detectors and straw drift tubes placed downstream of the magnet.
The tracking system provides a measurement of the momentum, $p$, of charged
particles with a relative uncertainty that varies from 0.5% at low momentum to
1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum distance of a track
to a primary $pp$ collision vertex, the impact parameter (IP), is measured
with a resolution of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where
$p_{\mathrm{T}}$ is in $\text{\,Ge\kern-1.00006ptV\\!/}c$. Photons, electrons
and hadrons are identified by a calorimeter system consisting of
scintillating-pad (SPD) and preshower detectors, an electromagnetic and a
hadronic calorimeter. Muons are reconstructed as a long track passing through
the vertex detector and the three stations of silicon-strip tracking
detectors, and identified by a system composed of alternating layers of iron
and multiwire proportional chambers.

The pseudorapidity coverage is extended by forward shower counters (HeRSCheL)
consisting of five planes of scintillators with three planes at 114, 19.7 and
$7.5\text{\,m}$ upstream of the LHCb detector, and two planes downstream at 20
and $114\text{\,m}$. The HeRSCheL detector [11] significantly extends the
acceptance for detecting particles from dissociated nucleons by covering the
pseudorapidity range of $5\lesssim|\eta|\lesssim 10$, enhancing the
classification of central exclusive production and UPC events.

The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.

Simulated events are used to determine corrections for the detector
resolution, acceptance, and efficiency. The UPCs are modelled using STARlight
[12] with a specific LHCb configuration [13]. The STARlight generator models
coherent and incoherent vector-meson production in photon-nuclear
interactions. Decays of unstable particles are described by EvtGen [14] with
QED final-state radiation handled by Photos [15]. The interactions of the
generated particles with the detector are modelled using the Geant4 toolkit
[16, 17] as described in Ref. [18].

##  3 Selection of signal candidates

Signal candidates are reconstructed through the decays
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$
and ${\psi{(2S)}}\\!\rightarrow{\mu^{+}\mu^{-}}$, and are required to have a
rapidity within the range $2.0<y^{*}<4.5$, where the starred notation
indicates that the observable is defined in the nucleus-nucleus centre-of-mass
frame. All remaining selection criteria given here are defined in the
laboratory frame. One of the candidate muons must pass the hardware-level
trigger, which requires a muon $p_{\mathrm{T}}$ greater than
$900\text{\,Me\kern-1.00006ptV\\!/}c$. The dimuon candidates are selected with
a minimum-bias software trigger, requiring at least one track reconstructed by
the vertex detector; this software trigger is $100\%$ efficient with respect
to the following offline selection, since it has a looser multiplicity
requirement. The offline selection requires two muon candidates, both with
tracks that have $p_{\mathrm{T}}>700\text{\,Me\kern-1.00006ptV\\!/}c$ within
the pseudorapidity range $2.0<\eta<4.5$. The dimuon candidates are required to
have $p_{\mathrm{T}}<1\text{\,Ge\kern-1.00006ptV\\!/}c$ and an azimuthal
opening angle between the muons larger than $0.9\,\pi$. The mass of each
signal candidate, $m_{{\mu^{+}\mu^{-}}}$, is required to be within $\pm
65\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [19] or $\pm
77.35\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known $\psi{(2S)}$ mass
[19]. To suppress background from PbPb collisions with impact parameter
smaller than two times the nucleus radius, only events with less than 20 hits
in the SPD are retained, corresponding to very low occupancy events that make
up about 0.3% of all minimum-bias events. Additionally, a requirement based
upon a figure of merit that combines the signals from all HeRSCheL stations
[11], is used to discard events with significant activity in the HeRSCheL
acceptance region.

##  4 Cross-section determination

For comparison with theoretical predictions, the measured cross-sections are
transformed into the nucleus-nucleus centre-of-mass frame, from the laboratory
frame, to account for the non-zero crossing angle between the two Pb beams.
The differential cross-section for coherent charmonium production in a given
interval of rapidity or transverse momentum is determined as

$\frac{\mathrm{d}\sigma_{\psi}^{\mathrm{coh}}}{\mathrm{d}x}=\frac{N_{\psi}^{\mathrm{coh}}}{\mathcal{L}\times{\varepsilon_{\mathrm{tot}}}\times{\mathcal{B}}(\psi\\!\rightarrow{\mu^{+}\mu^{-}})\times\Delta
x}\,,$ (1)

where $\psi$ is either ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or
$\psi{(2S)}$, $x$ represents either the $y^{*}$ or $p_{\mathrm{T}}^{*}$,
$N_{\psi}^{\mathrm{coh}}$ is the coherent signal yield,
$\varepsilon_{\mathrm{tot}}$ is the total efficiency, $\mathcal{L}$ is the
integrated luminosity, $\Delta x$ is the width of either the $y^{*}$\- or
$p_{\mathrm{T}}^{*}$-interval, and
${\mathcal{B}}(\psi\\!\rightarrow{\mu^{+}\mu^{-}})$ is the branching fraction
of the charmonium decay. The branching fractions
${\mathcal{B}}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}})=(5.961\pm 0.033)\times 10^{-2}$ and
${\mathcal{B}}({\psi{(2S)}}\\!\rightarrow{e^{+}e^{-}})=(7.93\pm 0.17)\times
10^{-3}$ [19] are used. For the $\psi{(2S)}$ the more accurate dielectron
branching fraction is used, where lepton universality is assumed.

The ratio between the differential cross-sections of $\psi{(2S)}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in a given rapidity
interval is given by

$\frac{\mathrm{d}\sigma_{\psi{(2S)}}^{\mathrm{coh}}/\mathrm{d}y^{*}}{\mathrm{d}\sigma_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}^{\mathrm{coh}}/\mathrm{d}y^{*}}=\frac{N_{{\psi{(2S)}}}^{\mathrm{coh}}\times\varepsilon_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\times{\mathcal{B}}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}})}{N_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}^{\mathrm{coh}}\times\varepsilon_{{\psi{(2S)}}}\times{\mathcal{B}}({\psi{(2S)}}\\!\rightarrow{\mu^{+}\mu^{-}})}\,.$
(2)

The signal yields are extracted in two steps. First, an unbinned extended
maximum-likelihood fit to the dimuon mass distribution is performed to obtain
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ yields
within the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
mass windows, respectively. The yields of the non-resonant background are also
obtained from this fit. The yields of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ production include contributions from both coherent and incoherent
production, and feed-down from $\psi{(2S)}$ decays into
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons
(${\psi{(2S)}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X$).
Similarly, the $\psi{(2S)}$ yields include contributions from both coherent
and incoherent production, while the feed-down contribution from higher-order
charmonium excited states is negligible given the current statistical
precision. In the second step, fits to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$
$p_{\mathrm{T}}^{*}$ distributions are used to determine the yields due to
coherent production.

The yields of the signal candidates and the non-resonant background component
are determined by fitting the dimuon mass distribution using double-sided
Crystal-Ball functions to describe the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ and ${\psi{(2S)}}$ mass shapes and an exponential function for the
non-resonant background. The fit is performed in the range
$2.9<m_{\mu^{+}\mu^{-}}<4.0\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. The mass
distribution and the corresponding fit are shown in Fig. 2.

Figure 2: Dimuon mass distribution for signal candidates in the rapidity range
$2.0<y^{*}<4.5$. The data are overlaid with the result of the fit.

The coherent yields are determined with separate unbinned maximum-likelihood
fits to the $\ln(p_{\mathrm{T}}^{*2})$ distributions for the signal candidates
inside both the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ mass windows. The shapes of coherent, incoherent and $\psi{(2S)}$
feed-down components are taken from simulation. The shape of the non-resonant
background does not depend upon the dimuon mass and is obtained from the mass
window, $3.2<m_{{\mu^{+}\mu^{-}}}<3.6\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
where there is no signal component. The yields of the non-resonant background
are determined as the integral of the non-resonant component from the dimuon
mass fit in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ mass windows.

Figure 3 shows the $\ln(p_{\mathrm{T}}^{*2})$ distributions of selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ candidates in
the rapidity interval $2<y^{*}<4.5$. Fits to the $\ln(p_{\mathrm{T}}^{*2})$
distributions are performed in each $y^{*}$ interval to extract the
corresponding ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
yields, as reported in Table 1. The coherent yield of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ production for
each $p_{\mathrm{T}}^{*}$ interval is calculated by subtracting the background
components from the measured yield for that interval as reported in Tables 2
and 3. The contributions from background components are determined by an
overall fit to the $\ln(p_{\mathrm{T}}^{*2})$ distributions.

${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$

$\psi{(2S)}$

Figure 3: The $\ln(p_{\mathrm{T}}^{*2})$ distribution of dimuon candidates in the $2.0<y^{*}<4.5$ range for (left) ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and (right) $\psi{(2S)}$ candidates. The data are overlaid with the result of the fit. Table 1: Total and coherent ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ yields from the invariant mass and transverse momentum fits in different rapidity intervals. Interval | $N^{\mathrm{tot}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ | $N^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ | $N^{\mathrm{tot}}_{{\psi{(2S)}}}$ | $N^{\mathrm{coh}}_{{\psi{(2S)}}}$
---|---|---|---|---
$2.0<y^{*}<4.5$ | $23\,355\pm 183$ | $20\,193\pm 199$ | $513\pm 43$ | $471\pm 44$
$2.0<y^{*}<2.5$ | $2\,457\pm\phantom{0}60$ | $2\,070\pm\phantom{0}66$ | $75\pm 15$ | $65\pm 15$
$2.5<y^{*}<3.0$ | $6\,845\pm 100$ | $5\,926\pm 108$ | $147\pm 26$ | $137\pm 26$
$3.0<y^{*}<3.5$ | $7\,875\pm 106$ | $6\,883\pm 115$ | $168\pm 26$ | $161\pm 26$
$3.5<y^{*}<4.0$ | $5\,019\pm\phantom{0}82$ | $4\,362\pm\phantom{0}90$ | $102\pm 18$ | $85\pm 18$
$4.0<y^{*}<4.5$ | $1\,166\pm\phantom{0}38$ | $956\pm\phantom{0}44$ | $24\pm\phantom{0}8$ | $21\pm\phantom{0}8$

The total efficiency $\varepsilon_{\mathrm{tot}}$ is determined as the product
of the acceptance efficiency ($\varepsilon_{\text{acc}}$), the muon acceptance
efficiency ($\varepsilon_{\mu\text{-acc}}$), the tracking efficiency
($\varepsilon_{\text{trk}}$), the selection efficiency
($\varepsilon_{\text{sel}}$), the particle identification (PID) efficiency
($\varepsilon_{\text{PID}}$), the trigger efficiency
($\varepsilon_{\text{trg}}$) and the HeRSCheL-veto efficiency
($\varepsilon_{\text{her}}$). Each efficiency is evaluated separately for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons in each
$y^{*}$ and $p_{\mathrm{T}}^{*}$ interval for the differential cross-section
measurements. Efficiencies are evaluated from simulation calibrated to data.
The value of $\varepsilon_{\mu\text{-acc}}$ is determined at generator level
as the fraction of events with both muon candidates passing
$p_{\mathrm{T}}>700\text{\,Me\kern-1.00006ptV\\!/}c$ and $2.0<\eta<4.5$. The
signal candidates are required to pass the
$p_{\mathrm{T}}<1\text{\,Ge\kern-1.00006ptV\\!/}c$ selection and fall in the
mass windows defined in Sec. 3, for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\psi{(2S)}$ mesons separately. For $\varepsilon_{\text{trk}}$,
$\varepsilon_{\text{PID}}$ and $\varepsilon_{\text{trg}}$, the simulation does
not always describe the data well. Efficiency corrections from data using the
tag-and-probe method [20] are determined from
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events in PbPb collision data.
The HeRSCheL-veto criteria is chosen to retain a signal efficiency of 90%
according to a set of separately selected pure signal and background data
samples. Dependencies of the efficiency correction on $y^{*}$ and
$p_{\mathrm{T}}^{*}$ of the dimuon system are studied and found to be
negligible in the invariant mass range from $2.9$ to
$4.0\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.

Table 2: Total and coherent ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ yields in different $p_{\mathrm{T}}^{*}$ intervals within the rapidity range $2.0<y^{*}<4.5$. Interval $\mathrm{[Me\kern-1.00006ptV\\!/}c]$ | $N_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}^{\mathrm{tot}}$ | $N_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}^{\mathrm{coh}}$
---|---|---
$0<p_{\mathrm{T}}^{*}<200$ | $21\,153\pm 175$ | $20\,180\pm 175$
$0<p_{\mathrm{T}}^{*}<\phantom{0}20$ | $2\,216\pm\phantom{0}58$ | $2\,204\pm\phantom{0}58$
$20<p_{\mathrm{T}}^{*}<\phantom{0}40$ | $5\,647\pm\phantom{0}92$ | $5\,619\pm\phantom{0}92$
$40<p_{\mathrm{T}}^{*}<\phantom{0}60$ | $5\,931\pm\phantom{0}83$ | $5\,885\pm\phantom{0}83$
$60<p_{\mathrm{T}}^{*}<\phantom{0}80$ | $3\,928\pm\phantom{0}65$ | $3\,863\pm\phantom{0}65$
$80<p_{\mathrm{T}}^{*}<100$ | $1\,848\pm\phantom{0}44$ | $1\,759\pm\phantom{0}44$
$100<p_{\mathrm{T}}^{*}<120$ | $497\pm\phantom{0}23$ | $381\pm\phantom{0}24$
$120<p_{\mathrm{T}}^{*}<140$ | $225\pm\phantom{0}16$ | $88\pm\phantom{0}17$
$140<p_{\mathrm{T}}^{*}<160$ | $289\pm\phantom{0}17$ | $137\pm\phantom{0}18$
$160<p_{\mathrm{T}}^{*}<180$ | $328\pm\phantom{0}18$ | $167\pm\phantom{0}20$
$180<p_{\mathrm{T}}^{*}<200$ | $244\pm\phantom{0}16$ | $77\pm\phantom{0}17$
Table 3: Total and coherent $\psi{(2S)}$ yields in different $p_{\mathrm{T}}^{*}$ intervals within the rapidity range $2.0<y^{*}<4.5$. Interval $\mathrm{[Me\kern-1.00006ptV\\!/}c]$ | $N_{{\psi{(2S)}}}^{\mathrm{tot}}$ | $N_{{\psi{(2S)}}}^{\mathrm{coh}}$
---|---|---
$0<p_{\mathrm{T}}^{*}<200$ | $475\pm 41$ | $468\pm 41$
$0<p_{\mathrm{T}}^{*}<\phantom{0}30$ | $77\pm 35$ | $77\pm 35$
$30<p_{\mathrm{T}}^{*}<\phantom{0}70$ | $275\pm 39$ | $274\pm 39$
$70<p_{\mathrm{T}}^{*}<\phantom{0}90$ | $91\pm 14$ | $91\pm 14$
$90<p_{\mathrm{T}}^{*}<110$ | $27\pm\phantom{0}8$ | $27\pm\phantom{0}8$
$110<p_{\mathrm{T}}^{*}<150$ | $0\pm\phantom{0}5$ | $0\pm\phantom{0}5$
$150<p_{\mathrm{T}}^{*}<200$ | $5\pm\phantom{0}4$ | $2\pm\phantom{0}4$

##  5 Systematic uncertainties

Systematic uncertainties on the cross-section measurements arise from the
efficiency and background determination, signal and background shapes,
momentum resolution, integrated luminosity and knowledge of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$
and ${\psi{(2S)}}\\!\rightarrow{\mu^{+}\mu^{-}}$ branching fractions. For the
$\psi{(2S)}$ to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section
ratio measurement, only systematic uncertainties from the charmonia decay
branching fractions are considered. Those from efficiency and background
determination, signal and background shapes integrated luminosity are highly
correlated and cancel. A summary of the systematic uncertainties is presented
in Table 4.

The systematic uncertainties related to the efficiencies are driven by the
sizes of the simulation and data samples. They vary from (0.5 – 2.0)% for the
tracking efficiency, (0.9 – 1.6)% for the PID efficiency and (2.1 – 3.7)% for
the trigger efficiency, across different $y^{*}$ and $p_{\mathrm{T}}^{*}$
intervals. The uncertainty associated with the HeRSCheL efficiency is a
constant 1.4%.

The uncertainty on the background shape is estimated by varying the shape
parameters within their fitted uncertainties. The maximum difference on the
extracted signal yields is 1.2%, and is assigned as the background
uncertainty.

The momentum resolution is expected to shift events from one
$p_{\mathrm{T}}^{*}$ interval to another. The uncertainties due to the
momentum resolution are evaluated by comparing the $p_{\mathrm{T}}$ spectra
between generated and reconstructed events. The evaluated relative
uncertainties vary from 0.9 to 34% for different $p_{\mathrm{T}}^{*}$
intervals. The largest uncertainty corresponds to the $p_{\mathrm{T}}^{*}$
interval between 140 to 160 MeV as shown in Table 8 (Appendix A), where very
small signal yields are observed.

The slight discrepancy betwee and the fit visible at
$\ln(p_{\mathrm{T}}^{*2})\sim-4$
[$\ln{(\text{\,Ge\kern-1.00006ptV\\!/}c^{2})}$] for both
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$, as shown in
Fig. 3, is expected to originate from a mis-modelling of the predicted signal
shape from simulation. A systematic uncertainty on the signal shape model is
estimated by evaluating the difference between the fitted signal yields with
respect to an alternative empirical signal shape. The obtained difference,
about 0.04%, is included as the signal shape uncertainty.

The uncertainties on the branching fractions result in relative uncertainties
on the measured cross-sections of 0.6% and 2.1% [19], respectively. The
relative uncertainty on the luminosity is 4.4% [21].

Table 4: Summary of the systematic uncertainties. Source | Relative uncertainty [%]
---|---
| $\sigma_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}^{\mathrm{coh}}$ | $\sigma_{\psi{(2S)}}^{\mathrm{coh}}$
Tracking efficiency | 0.5–2.0 | 0.5–2.0
PID efficiency | 0.9–1.6 | 0.9–1.6
Trigger efficiency | 2.7–3.7 | 2.1–2.5
HeRSCheL efficiency | 1.4 | 1.4
Background estimation | 1.2 | 1.2
Signal shape | 0.04 | 0.04
Momentum resolution | 0.9–34 | 1.3–27
Branching fraction | 0.6 | 2.1
Luminosity | 4.4 | 4.4

##  6 Results and discussion

The integrated cross-sections of coherent
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and ${\psi{(2S)}}$ production
in PbPb collisions are measured in the rapidity region $2.0<y^{*}<4.5$ as

$\displaystyle\sigma^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ $\displaystyle=5.965\pm 0.059\pm 0.232\pm 0.262\text{\,mb}\,,$
$\displaystyle\sigma^{\mathrm{coh}}_{{\psi{(2S)}}}$ $\displaystyle=0.923\pm
0.086\pm 0.028\pm 0.040\text{\,mb}\,,$

where the first listed uncertainty is statistical, the second is systematic
and the third is due to the luminosity determination. The ratio of the
coherent $\psi{(2S)}$ to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production cross-sections is measured to be

$\displaystyle\sigma^{\mathrm{coh}}_{{\psi{(2S)}}}/\sigma^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ $\displaystyle=0.155\pm 0.014\pm 0.003\,,$

where the first uncertainty is statistical and the second is systematic. The
luminosity uncertainty cancels in the ratio measurement.

The measured differential cross-sections as a function of $y^{*}$ and
$p_{\mathrm{T}}^{*}$ for coherent ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\psi{(2S)}$ are shown in Figs. 4 and 5, respectively. The cross-
section ratio of coherent $\psi{(2S)}$ to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production as a function of
rapidity is shown in Fig. 6. The numerical values of the results are reported
in Tables 5 – 9 in Appendix A. These results are compared to several
theoretical predictions in Figs. 4, 5 and 6 which can be grouped into models
based on perturbative-QCD (pQCD) [8, 22] and colour-glass-condensate (CGC) [7,
23, 24, 25, 26, 6, 27, 28] calculations.

Figure 4: Differential cross-section as a function $y^{*}$ for coherent (left)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and (right) $\psi{(2S)}$
production, compared to theoretical predictions. These models are grouped as
(red lines) perturbative-QCD calculations and (blue lines) colour-glass-
condensate models.

Figure 5: Differential cross-section as a function of $p_{\mathrm{T}}^{*}$
within the rapidity range $2<y^{*}<4.5$ for coherent (left)
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and (right) $\psi{(2S)}$
production compared to theoretical predictions. Figure 6: Differential cross-
section ratio of $\psi{(2S)}$ to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ as a function of $y^{*}$, compared to theoretical predictions. These
models are separated into (red lines) perturbative QCD calculations and (blue
lines) colour-glass-condensate models.

The models provided by Guzey et al. [8, 22] are based on pQCD calculations
under the leading-logarithm approximation. The exclusive
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ photo-production cross-
section on a proton target is calculated at leading order. The final cross-
section is calculated with the weak and strong leading twist nuclear shadowing
(LTA) models (LTA_W and LTA_S, respectively) [29], together with the EPS09
[30] and EPPS16 [31, 32] nuclear parton distribution functions. These models
are compatible with the data, with excellent agreement at high rapidity and a
slight trend of underestimation at low rapidity for both
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$, as shown in
Fig. 4. The underestimation at low rapidity results in an overall lower
prediction in the differential cross-section as a function of
$p_{\mathrm{T}}^{*}$, shown in Fig. 5. However, excellent agreement between
the prediction and data for the $\psi{(2S)}$ to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ratio measurement can be
observed in Fig. 6.

The models by Krelina et al. [24, 23] can be considered as variations of the
colour-dipole model [33, 34, 35] based on CGC theory. In the models proposed
by Mäntysaari et al. [6, 28, 27] the cross-section is also calculated using
the colour-dipole model, including a subnucleon scale fluctuation based on CGC
theory. The models by Gonçalves et al. [25, 26] depend on the dipole-hadron
scattering amplitude and vector-meson wave function. The
Golec–Biernat–Wusthoff (GBW) [36, 37] and Kopeliovich–Schafer–Tarasov (KST)
[38] models include light-front colour dipoles. In Figs. 4 and 6, a second
core-section model is shown, where Buchmüller–Tye (BT) [39] and power-like
(POW) [40, 41] potentials provide the best description of the existing data
for proton-electron generation of charmonium. The Glauber–Gribov (GG)
methodology [42, 43, 44] uses a parameterisation of the dipole-proton cross-
section and its propagation to the dipole-nucleus scattering amplitudes,
together with a nuclear-profile model consisting of sub-nucleon degrees of
freedom in the form of hot–spots (hs) [24, 23]. The boosted–Gaussian (BG) and
the Gauss-LC (GLC) models [25] are two different vector wave functions. Both
assume the meson mainly consists of a quark–anti-quark pair, and the spin and
polarization are the same as that of the photon. In the BG model, the vector-
meson wave function is used to predict the absence of subnuclear fluctuations.
The impact-parameter-CGC (bCGC) [45, 46] and the impact-parameter-saturation
(IP-SAT) [47] models are based on dipole-hadron scattering. To calculate the
dipole-hadron scattering amplitude, the bCGC model not only considers the
solution of the Balitsky–Fadin–Kuraev–Lipatov equation [48, 49, 50, 51] and
the Balitski–Kovchegov equation [52, 53], but also considers the impact
parameter dependence on the saturation scale. The IP-SAT model assumes that
the dipole-hadron scattering amplitude depends on a gluon distribution evolved
through the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equation [54, 55, 56].
These models based on CGC are compatible with the data, with large variations
between different models as shown in Figs. 4, 5 and 6. Better agreement can be
seen at low rapidity than at high rapidity. A relatively larger
underestimation at high rapidity and as a function of $p_{\mathrm{T}}^{*}$ can
be seen for the $\psi{(2S)}$ meson in Figs. 4 and 5. This can be understood to
be due to higher theoretical uncertainty, since the wave function of the
$\psi{(2S)}$ meson has a more complicated structure than that of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson [6, 28, 45].

##  7 Conclusion

The coherent ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
production cross-sections in PbPb ultra-peripheral collisions at a centre-of-
mass energy of
$\sqrt{s_{\scriptscriptstyle\text{NN}}}=5.02\text{\,Te\kern-1.00006ptV}$ are
studied using a data sample corresponding to an integrated luminosity of
$228\pm 10\,\upmu\text{b}^{-1}$ collected by the LHCb detector. The
differential cross-sections, as a function of $y^{*}$ and
$p_{\mathrm{T}}^{*}$, are measured separately for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons in the
ranges $2.0<y^{*}<4.5$ and
$0<p_{\mathrm{T}}^{*}<0.2\text{\,Ge\kern-1.00006ptV\\!/}c$. The ratio of the
cross-sections between the coherent $\psi{(2S)}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production, as a function of
rapidity, is also determined for the first time in PbPb collisions and is
found to be compatible with theoretical models.

## Appendices

##  Appendix A Numerical results

Table 5: The differential cross-section for coherent ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production as a function of $y^{*}$. Interval |  $\mathrm{d}{\sigma}^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}/\mathrm{d}y^{*}$ $[\mathrm{mb}]$ | Uncertainties $[\mathrm{mb}]$
---|---|---
|  | Stat. | Syst. | Lumi. | Total
$2.0<y^{*}<2.5$ | 3.392 | 0.108 | 0.165 | 0.147 | 0.247
$2.5<y^{*}<3.0$ | 2.896 | 0.053 | 0.117 | 0.127 | 0.181
$3.0<y^{*}<3.5$ | 2.395 | 0.040 | 0.089 | 0.105 | 0.144
$3.5<y^{*}<4.0$ | 1.922 | 0.039 | 0.072 | 0.084 | 0.117
$4.0<y^{*}<4.5$ | 1.181 | 0.054 | 0.049 | 0.052 | 0.089
$2.0<y^{*}<4.5$ | 5.965 | 0.059 | 0.232 | 0.262 | 0.355
Table 6: The differential cross-section of coherent $\psi{(2S)}$ production as a function of $y^{*}$. Interval |  $\mathrm{d}{\sigma}^{\mathrm{coh}}_{{\psi{(2S)}}}/\mathrm{d}y^{*}$ $[\mathrm{mb}]$ | Uncertainties $[\mathrm{mb}]$
---|---|---
|  | Stat. | Syst. | Lumi. | Total
$2.0<y^{*}<2.5$ | 0.656 | 0.155 | 0.024 | 0.029 | 0.160
$2.5<y^{*}<3.0$ | 0.430 | 0.082 | 0.014 | 0.019 | 0.085
$3.0<y^{*}<3.5$ | 0.373 | 0.060 | 0.012 | 0.016 | 0.064
$3.5<y^{*}<4.0$ | 0.268 | 0.057 | 0.009 | 0.012 | 0.059
$4.0<y^{*}<4.5$ | 0.184 | 0.071 | 0.007 | 0.008 | 0.072
$2.0<y^{*}<4.5$ | 0.923 | 0.086 | 0.028 | 0.040 | 0.099
Table 7: The differential cross-section ratio between $\psi{(2S)}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ coherent production as a function of $y^{*}$. The uncertainty due to luminosity determination cancels in the ratio. Interval | $\mathrm{d}{\sigma}^{\mathrm{coh}}_{{\psi{(2S)}}}/\mathrm{d}{\sigma}^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ | Uncertainties
---|---|---
|  | Stat. | Syst. | Total
$2.0<y^{*}<2.5$ | 0.193 | 0.046 | 0.004 | 0.046
$2.5<y^{*}<3.0$ | 0.149 | 0.028 | 0.003 | 0.029
$3.0<y^{*}<3.5$ | 0.156 | 0.025 | 0.003 | 0.026
$3.5<y^{*}<4.0$ | 0.139 | 0.030 | 0.003 | 0.030
$4.0<y^{*}<4.5$ | 0.156 | 0.061 | 0.003 | 0.061
$2.0<y^{*}<4.5$ | 0.155 | 0.014 | 0.003 | 0.015
Table 8: The double differential cross-section of coherent ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production as a function of $p_{\mathrm{T}}^{*}$ in rapidity range $2.0<y^{*}<4.5$. Interval $[\text{\,Me\kern-1.00006ptV\\!/}c]$ |  $\mathrm{d}^{2}{\sigma}^{\mathrm{coh}}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}/\mathrm{d}p_{\mathrm{T}}^{*}\mathrm{d}y^{*}$ $[\mathrm{mb/(Ge\kern-1.00006ptV\\!/}c)]$ | Uncertainties $[\mathrm{mb/(Ge\kern-1.00006ptV\\!/}c)]$
---|---|---
|  | Stat. | Syst. | Lumi. | Total
$0<p_{\mathrm{T}}^{*}<\phantom{0}20$ | 13.391 | 0.352 | 0.908 | 0.587 | 1.138
$20<p_{\mathrm{T}}^{*}<\phantom{0}40$ | 33.940 | 0.556 | 2.007 | 1.489 | 2.560
$40<p_{\mathrm{T}}^{*}<\phantom{0}60$ | 35.077 | 0.495 | 1.462 | 1.538 | 2.179
$60<p_{\mathrm{T}}^{*}<\phantom{0}80$ | 22.645 | 0.381 | 0.492 | 0.993 | 1.172
$80<p_{\mathrm{T}}^{*}<100$ | 9.945 | 0.249 | 0.472 | 0.436 | 0.689
$100<p_{\mathrm{T}}^{*}<120$ |  02.028 | 0.128 | 0.311 | 0.089 | 0.347
$120<p_{\mathrm{T}}^{*}<140$ |  00.432 | 0.083 | 0.138 | 0.019 | 0.163
$140<p_{\mathrm{T}}^{*}<160$ |  00.781 | 0.103 | 0.273 | 0.034 | 0.293
$160<p_{\mathrm{T}}^{*}<180$ |  00.986 | 0.118 | 0.213 | 0.043 | 0.247
$180<p_{\mathrm{T}}^{*}<200$ |  00.464 | 0.102 | 0.080 | 0.020 | 0.131
$0<p_{\mathrm{T}}^{*}<200$ | 11.904 | 0.103 | 0.233 | 0.522 | 0.581
Table 9: The double differential cross-section of coherent $\psi{(2S)}$ production as a function of $p_{\mathrm{T}}^{*}$ in rapidity range $2.0<y^{*}<4.5$. Interval $[\text{\,Me\kern-1.00006ptV\\!/}c]$ |  $\mathrm{d}^{2}{\sigma}^{\mathrm{coh}}_{{\psi{(2S)}}}/\mathrm{d}p_{\mathrm{T}}^{*}\mathrm{d}y^{*}$ $[\mathrm{mb/(Ge\kern-1.00006ptV\\!/}c)]$ | Uncertainties $[\mathrm{mb/(Ge\kern-1.00006ptV\\!/}c)]$
---|---|---
|  | Stat. | Syst. | Lumi. | Total
$0<p_{\mathrm{T}}^{*}<\phantom{0}30$ | 2.073 | 0.942 | 0.141 | 0.091 | 0.957
$30<p_{\mathrm{T}}^{*}<\phantom{0}70$ | 5.447 | 0.775 | 0.254 | 0.239 | 0.850
$70<p_{\mathrm{T}}^{*}<\phantom{0}90$ | 3.476 | 0.535 | 0.110 | 0.152 | 0.567
$90<p_{\mathrm{T}}^{*}<110$ | 1.136 | 0.337 | 0.108 | 0.050 | 0.357
$110<p_{\mathrm{T}}^{*}<150$ | 0.000 | 0.093 | 0.000 | 0.000 | 0.093
$150<p_{\mathrm{T}}^{*}<200$ | 0.025 | 0.051 | 0.006 | 0.001 | 0.051
$0<p_{\mathrm{T}}^{*}<200$ | 1.833 | 0.160 | 0.052 | 0.080 | 0.187

## References

  * [1] C. A. Bertulani, S. R. Klein, and J. Nystrand, _Physics of ultra-peripheral nuclear collisions_ , Ann. Rev. Nucl. Part. Sci. 55 (2005) 271, arXiv:nucl-ex/0502005
  * [2] S. Sarkar, H. Satz, and B. Sinha, eds., The physics of the quark-gluon plasma,  vol. 785, Springer, Berlin Heidelberg, 2010
  * [3] S. P. Jones, A. D. Martin, M. G. Ryskin, and T. Teubner, _Exclusive $J/\psi$ and $\Upsilon$ photoproduction and the low $x$ gluon_, J. Phys. G43 (2016) 035002, arXiv:1507.06942
  * [4] ALICE collaboration, S. Acharya et al., _Coherent J/ $\psi$ photoproduction at forward rapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV_, Phys. Lett. B798 (2019) 134926, arXiv:1904.06272
  * [5] LHCb collaboration, R. Aaij et al., _Study of coherent $J/\psi$ production in lead-lead collisions at $\sqrt{s_{NN}}=5TeV$_, arXiv:2107.03223
  * [6] H. Kowalski, L. Motyka, and G. Watt, _Exclusive diffractive processes at HERA within the dipole picture_ , Phys. Rev. D74 (2006) 074016, arXiv:hep-ph/0606272
  * [7] V. P. Gonçalves and M. V. T. Machado, _Vector meson production in coherent hadronic interactions: an update on predictions for RHIC and LHC_ , Phys. Rev. C84 (2011) 011902, arXiv:1106.3036
  * [8] V. Guzey, E. Kryshen, and M. Zhalov, _Coherent photoproduction of vector mesons in ultraperipheral heavy ion collisions: Update for run2 at the CERN Large Hadron Collider_ , Phys. Rev. C93 (2016) 055206, arXiv:1602.01456
  * [9] LHCb Collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [10] LHCb Collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [11] K. C. Akiba et al., _The _HeRSCheL_ detector: high-rapidity shower counters for LHCb_, JINST 13 (2018) P04017, arXiv:1801.04281
  * [12] S. R. Klein et al., _STARlight: A Monte Carlo simulation program for ultra-peripheral collisions of relativistic ions_ , Comput. Phys. Commun. 212 (2017) 258, arXiv:1607.03838
  * [13] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [14] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [15] N. Davidson, T. Przedzinski, and Z. Was, _PHOTOS interface in C++: Technical and physics documentation_ , Comp. Phys. Comm. 199 (2016) 86, arXiv:1011.0937
  * [16] Geant4 Collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [17] Geant4 Collaboration, S. Agostinelli et al., _Geant4 : A simulation toolkit_, Nucl. Instrum. Meth. A506 (2003) 250
  * [18] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [19] Particle Data Group, P. A. Zyla et al., _Review of particle physics_ , Prog. Theor. Exp. Phys. 2020 (2020) 083C01
  * [20] LHCb collaboration, R. Aaij et al., _Measurement of the track reconstruction efficiency at LHCb_ , JINST 10 (2015) P02007, arXiv:1408.1251
  * [21] LHCb collaboration, R. Aaij et al., _Precision luminosity measurements at LHCb_ , JINST 9 (2014) P12005, arXiv:1410.0149
  * [22] V. Guzey, M. Strikman, and M. Zhalov, _Accessing transverse nucleon and gluon distributions in heavy nuclei using coherent vector meson photoproduction at high energies in ion ultraperipheral collisions_ , Phys. Rev. C95 (2017) 025204, arXiv:1611.05471
  * [23] J. Cepila, J. G. Contreras, and M. Krelina, _Coherent and incoherent $\mathrm{J/}\psi$ photonuclear production in an energy-dependent hot-spot model_, Phys. Rev. C97 (2018) 024901, arXiv:1711.01855
  * [24] B. Z. Kopeliovich, M. Krelina, J. Nemchik, and I. K. Potashnikova, _Heavy quarkonium production in ultraperipheral nuclear collisions_ , arXiv:2008.05116
  * [25] V. P. Gonçalves et al., _Color dipole predictions for the exclusive vector meson photoproduction in $pp$, $p\mathrm{Pb}$, and $\mathrm{Pb}\mathrm{Pb}$ collisions at run2 LHC energies_, Phys. Rev. D96 (2017) 094027, arXiv:1710.10070
  * [26] V. P. Gonçalves and M. V. T. Machado, _The QCD pomeron in ultraperipheral heavy ion collisions: IV. Photonuclear production of vector mesons_ , Eur. Phys. J. C40 (2005) 519, arXiv:hep-ph/0501099
  * [27] H. Mäntysaari and B. Schenke, _Probing subnucleon scale fluctuations in ultraperipheral heavy ion collisions_ , Phys. Lett. B772 (2017) 832, arXiv:1703.09256
  * [28] T. Lappi and H. Mäntysaari, _Diffractive vector meson production in ultraperipheral heavy ion collisions from the color glass condensate_ , PoS DIS2014 (2014) 069, arXiv:1406.2877
  * [29] L. Frankfurt, V. Guzey, and M. Strikman, _Leading twist nuclear shadowing phenomena in hard processes with nuclei_ , Phys. Rept. 512 (2012) 255, arXiv:1106.2091
  * [30] K. J. Eskola, H. Paukkunen, and C. A. Salgado, _EPS09: A new generation of NLO and LO nuclear parton distribution functions_ , JHEP 04 (2009) 065, arXiv:0902.4154
  * [31] K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, _EPPS16: Nuclear parton distributions with LHC data_ , Eur. Phys. J. C77 (2017) 163, arXiv:1612.05741
  * [32] S. Dulat et al., _New parton distribution functions from a global analysis of quantum chromodynamics_ , Phys. Rev. D93 (2016) 033006, arXiv:1506.07443
  * [33] B. Z. Kopeliovich and B. G. Zakharov, _Quantum effects and color transparency in charmonium photoproduction on nuclei_ , Phys. Rev. D44 (1991) 3466
  * [34] B. Z. Kopeliovich, J. Nemchick, N. N. Nikolaev, and B. G. Zakharov, _Decisive test of color transparency in exclusive electroproduction of vector mesons_ , Phys. Lett. B324 (1994) 469, arXiv:hep-ph/9311237
  * [35] J. Nemchik, N. N. Nikolaev, E. Predazzi, and B. G. Zakharov, _Color dipole phenomenology of diffractive electroproduction of light vector mesons at HERA_ , Z. Phys. C75 (1997) 71, arXiv:hep-ph/9605231
  * [36] K. J. Golec-Biernat and M. Wusthoff, _Saturation effects in deep inelastic scattering at low $Q^{2}$ and its implications on diffraction_, Phys. Rev. D59 (1998) 014017, arXiv:hep-ph/9807513
  * [37] K. J. Golec-Biernat and M. Wusthoff, _Saturation in diffractive deep inelastic scattering_ , Phys. Rev. D60 (1999) 114023, arXiv:hep-ph/9903358
  * [38] B. Z. Kopeliovich, A. Schafer, and A. V. Tarasov, _Nonperturbative effects in gluon radiation and photoproduction of quark pairs_ , Phys. Rev. D62 (2000) 054022, arXiv:hep-ph/9908245
  * [39] W. Buchmuller and S. H. H. Tye, _Quarkonia and quantum chromodynamics_ , Phys. Rev. D24 (1981) 132
  * [40] A. Martin, _A fit of upsilon and charmonium spectra_ , Phys. Lett. B93 (1980) 338
  * [41] N. Barik and S. N. Jena, _Fine-hyperfine splittings of quarkonium levels in an effective power-law potential_ , Phys. Lett. B97 (1980) 265
  * [42] V. N. Gribov, _Glauber corrections and the interaction between high-energy hadrons and nuclei_ , Sov. Phys. JETP 29 (1969) 483
  * [43] J. Hufner, B. Kopeliovich, and J. Nemchik, _Glauber multiple scattering theory for the photoproduction of vector mesons off nuclei and the role of the coherence length_ , Phys. Lett. B383 (1996) 362, arXiv:nucl-th/9605007
  * [44] B. Z. Kopeliovich, _Gribov inelastic shadowing in the dipole representation_ , Int. J. Mod. Phys. A31 (2016) 1645021, arXiv:1602.00298
  * [45] H. Kowalski, L. Motyka, and G. Watt, _Exclusive diffractive processes at HERA within the dipole picture_ , Phys. Rev. D74 (2006) 074016, arXiv:hep-ph/0606272
  * [46] G. Watt and H. Kowalski, _Impact parameter dependent colour glass condensate dipole model_ , Phys. Rev. D 78 (2008) 014016, arXiv:0712.2670
  * [47] J. Bartels, K. J. Golec-Biernat, and H. Kowalski, _A modification of the saturation model: DGLAP evolution_ , Phys. Rev. D 66 (2002) 014001, arXiv:hep-ph/0203258
  * [48] C. B. Dover, P. J. Moffa, and J. P. Vary, _Inelastic heavy ion scattering in a folding model_ , Phys. Lett. B56 (1975) 4
  * [49] L. N. Lipatov, _Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories_ , Sov. J. Nucl. Phys. 23 (1976) 338
  * [50] E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, _Multi-reggeon processes in the Yang-Mills theory_ , Sov. Phys. JETP 44 (1976) 443
  * [51] I. I. Balitsky and L. N. Lipatov, _The pomeranchuk singularity in quantum chromodynamics_ , Sov. J. Nucl. Phys. 28 (1978) 822
  * [52] I. Balitsky, _Operator expansion for high-energy scattering_ , Nucl. Phys. B463 (1996) 99, arXiv:hep-ph/9509348
  * [53] Y. V. Kovchegov, _Small-x $F_{2}$ structure function of a nucleus including multiple pomeron exchanges_, Phys. Rev. D60 (1999) 034008, arXiv:hep-ph/9901281
  * [54] Y. L. Dokshitzer, _Calculation of the structure functions for deep inelastic scattering and ${e^{+}}$ ${e^{-}}$ annihilation by perturbation theory in quantum chromodynamics._, Sov. Phys. JETP 46 (1977) 641
  * [55] G. Altarelli and G. Parisi, _Asymptotic freedom in parton language_ , Nucl. Phys. B126 (1977) 298
  * [56] V. N. Gribov and L. N. Lipatov, _Deep inelastic ${e}$ ${p}$ scattering in perturbation theory_, Sov. J. Nucl. Phys. 15 (1972) 438