HIP-2022-14/TH

IPPP/22/37

LTH 1305

August 27, 2024

Exclusive $J/\psi$ and $\Upsilon$ production in high energy $pp$ and $p$Pb
collisions

C. A. Fletta,b, S. P. Jonesc, A. D. Martinc, M. G. Ryskin and T. Teubnerd

aDepartment of Physics, University of Jyväskylä, P.O. Box 35, 40014 University
of Jyväskylä, Finland
bHelsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki,
Finland
cInstitute for Particle Physics Phenomenology, Durham University, Durham, DH1
3LE, U.K.

dDepartment of Mathematical Sciences, University of Liverpool, Liverpool, L69
3BX, U.K.

We present a formalism for determining the cross section for exclusive heavy
vector meson production ($J/\psi,\Upsilon$) as a function of rapidity, in both
high energy proton-proton and proton-heavy ion collisions, at next-to-leading
order in QCD. We compare and contrast the production in $pp$ and $p$Pb
collisions and show how data for these processes can give information on the
low $x$ gluon distribution of the proton and heavy ions at a range of
different scales.

##  1 Introduction

There is a long history of experimental and theoretical study of exclusive
vector meson production in high-energy proton-proton collisions. In
particular, data for the differential cross section $\text{d}\sigma(p+p\to
p+V+p)/\text{d}Y$ has come under theoretical scrutiny for vector mesons
$V=J/\psi,\Upsilon(1S)$, where the $+$ signs denote large rapidity gaps
between the rapidity $Y$ of the vector meson and the outgoing protons,
ensuring the exclusivity of the experimental measurements. The theoretical
description proceeds by first calculating the differential cross section for
the exclusive photoproduction subprocess $\gamma+p\to V+p$ for which the
dominant QCD diagram is sketched in simplified form in Fig. 1. At high energy
the process is driven by the behaviour of the gluon parton distribution of the
proton at small momentum fraction. In this work, we apply the theoretical
framework, based on collinear factorisation and developed to NLO in
perturbative QCD in [1, 2], to $p$Pb collisions. Other approaches include, for
example, models based on the Colour-Glass-Condensate (CGC) [3, 4] and
$k_{T}$-factorisation frameworks [5], as well as LO perturbative QCD [6]. Note
that the two exchanged gluons in Fig. 1 carry different fractions of the
incoming proton momentum, so we are dealing with a generalised, skewed gluon
distribution. The net momentum fraction transferred from the proton to the
vector meson is $2\xi$, thus for this exclusive process $2\xi$ plays the role
of the usual variable ‘$x$’ in DIS. For our high energy process we have
$x-\xi\ll x+\xi\ll 1$ and the cross section can be accurately estimated in
terms of the conventional gluon distribution $g(2\xi)$ and a skewing
correction [7, 8]. Recall that a detailed NLO analysis of the current
inclusive data determines the NLO gluon distribution down to $2\xi\simeq
10^{-4}$ [9]. For smaller momentum fractions the uncertainties become too
large, see also [10]. However, the exclusive $J/\psi$ data from LHC [11, 12]
are found to determine the gluon distribution down to $2\xi\sim 3\times
10^{-6}$.

In Section 2 we note that in going from the cross section for photoproduction,
$\gamma+p\to V+p$, to that for $p+p\to p+V+p$, we need to allow for photon
emissions from both incoming protons. Moreover, we have to evaluate the photon
flux and the survival factors of the rapidity gaps, i.e. the probabilities
that the rapidity gaps do not get populated by additional emissions.

Figure 1: A schematic diagram of high-energy exclusive vector meson
production, $\gamma^{*}+p\to V+p$. The factorized form follows since, in the
proton rest frame, the formation time $\tau_{f}\simeq
2E_{\gamma}/(Q^{2}+M^{2}_{V})$ is much greater than the $q\bar{q}$-proton
interaction time. $Q$ is the virtuality of the photon and the lightcone
momentum $P^{+}=(p+p^{\prime})/2$. For the photoproduction set-up considered
here, we have $Q^{2}=0$.

Section 3 concerns exclusive vector meson production in $p$Pb collisions. Data
for such a process appear to have the advantage of having a much larger cross
section; enhanced by the large charge of the heavy Pb ion, that is by a factor
$Z^{2}=6724$. However, when we come to evaluate the cross section we find the
theoretical formalism is much more complicated than that for $pp$ collisions.
In Section 4 we compare the differential cross section data from ALICE and CMS
at the proton-nucleon collision energy $\sqrt{s_{pN}}=5.02\leavevmode\nobreak\
$TeV for the process $p+{\rm Pb}\to p+V+{\rm Pb}$, where $V=J/\psi,\Upsilon$,
with our corresponding theoretical results. We then provide predictions at
$\sqrt{s_{pN}}=8.16\leavevmode\nobreak\ $TeV, before presenting our
conclusions and outlook in Section 5.

##  2 Exclusive $p+p\to p+V+p$ production

We begin with $pp$ collisions, and leave the discussion of $p$Pb collisions
until the next section. The procedure for calculating the cross section for
the high-energy exclusive process $p+p\to p+V+p$ (where the vector meson
$V=J/\psi$ or $\Upsilon$ and where the + signs represent rapidity gaps) is
described in [5, 13]. As mentioned above, we first calculate the cross section
for the exclusive photoproduction process

$\sigma_{W}(\gamma p)\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
\sigma(\gamma+p\to V+p)$ (1)

where $W$ is the $\gamma p$ centre-of-mass energy. This process is driven by
the two amplitudes sketched in Fig. 2. In the left diagram the photon is
radiated by the upper proton and then the vector meson, $V$, is created on the
lower proton via the $\gamma+p\to V+p$ reaction. The diagram on the right
shows the other possibility where the photon is emitted from the lower proton
while the upper proton acts as the target.

Figure 2: The two diagrams describing exclusive heavy vector meson production
in $p+p\to p+V+p$, at the LHC. The $W_{+}$ and $W_{-}$ contributions arise in
the ultraperipheral description of the $\gamma+p\to V+p$ subprocess, see the
text for details. In the $p$Pb mode, either the upper or lower proton is
replaced by a Pb-ion.

When the meson $V$ is detected in the forward region, the corresponding
$\gamma p$ energies are quite different

$W^{2}_{+}=M_{V}\sqrt{s}\leavevmode\nobreak\
e^{Y}\;\;\;\mbox{and}\;\;\;W^{2}_{-}=M_{V}\sqrt{s}\leavevmode\nobreak\
e^{-Y},$ (2)

where $Y$ is the rapidity of the vector meson in the laboratory (i.e. $pp$
centre-of-momentum) frame. Since the $\gamma p\to V+p$ cross section steeply
grows with energy, the dominant contribution comes from the first amplitude
which corresponds to a larger energy $W_{+}$.

The cross section for exclusive $V$ production in ultraperipheral $pp$
collisions is therefore given by the sum111The interference between the two
amplitudes of the two subprocesses is small since the transverse momentum of
the proton which radiates the photon is much smaller than that of the target
proton. Thus we may neglect the interference term at the accuracy we are
aiming at here. of the exclusive photoproduction cross sections,
$\sigma_{\pm}(\gamma p)$ of eqn. (1), at the two $\gamma p$ energies $W_{\pm}$

$\frac{d\sigma(p+p\to p+V+p)}{dY}\leavevmode\nobreak\ =\leavevmode\nobreak\
S^{2}(W_{+})\left(k_{+}\frac{dn_{p}}{dk_{+}}\right)\sigma_{+}(\gamma
p)\leavevmode\nobreak\ +\leavevmode\nobreak\
S^{2}(W_{-})\left(k_{-}\frac{dn_{p}}{dk_{-}}\right)\sigma_{-}(\gamma p).$ (3)

Note that the subprocess cross sections are weighted by the survival factors
$S^{2}(W_{\pm})$ which account for the probability that the rapidity gap
between the vector meson and the target proton is not populated by soft
interactions which would destroy the exclusivity of the event; and by the
photon fluxes $dn_{p}/dk_{\pm}$ for photons of energy
$k_{\pm}=x_{\pm}\sqrt{s}/2$, where $x_{\pm}$ are the fractions of the parent
proton energy carried by the photon.

The flux of photons emitted from a proton is well known [14]

$\frac{dn_{p}(x)}{dx}=\frac{\alpha^{\rm
QED}}{\pi^{2}x}\int\frac{d^{2}q_{\perp}}{q^{2}_{\perp}+x^{2}m^{2}_{p}}\left(\frac{q^{2}_{\perp}}{q^{2}_{\perp}+x^{2}m^{2}_{p}}(1-x)F_{E}(Q^{2})+\frac{x^{2}}{2}F_{M}(Q^{2})\right),$
(4)

where $q_{\perp}$ is the photon transverse momentum, $m_{p}$ is the proton
mass and $F_{E,M}$ are the proton form factors

$\displaystyle F_{E}(Q^{2})$
$\displaystyle=(4m^{2}_{p}G^{2}_{E}(Q^{2})+Q^{2}G_{M}(Q^{2}))/(4m^{2}_{p}+Q^{2})\,,$
(5) $\displaystyle F_{M}(Q^{2})$ $\displaystyle=G_{M}^{2}(Q^{2})\,.$ (6)

The scale $Q^{2}=(q^{2}_{\perp}+x^{2}m^{2}_{p})/(1-x)$, while $G_{E}$ and
$G_{M}$ are the ‘Sachs’ form factors, which may be expressed in dipole forms

$\displaystyle G_{E}(Q^{2})$
$\displaystyle=1/(1+Q^{2}/0.71\mbox{GeV}^{2})^{2},$ (7) $\displaystyle
G_{M}(Q^{2})$ $\displaystyle=2.79/(1+Q^{2}/0.71\mbox{GeV}^{2})^{2}.$ (8)

The rapidity gap survival factors, $S^{2}(W_{\pm})$, are calculated in impact
parameter, $b$, space using

$S^{2}(W_{\pm},b))\leavevmode\nobreak\ =\leavevmode\nobreak\
\exp(-\Omega(b,W_{\pm}))\,,$ (9)

where $\Omega(b,W)$ is the opacity (i.e. optical density) of the proton-proton
interaction at the energy $W$ and the vector valued impact parameter
$\boldsymbol{b}=(b_{x},b_{y})$, see e.g. [15].

The results as a function of the rapidity of the vector meson are tabulated
for different collider energies in [5] and [13] for $V=J/\psi$ and $\Upsilon$,
respectively.

##  3 Exclusive $p$ \+ Pb $\to p+V+$ Pb production

In a proton-lead collision the photon flux radiated coherently by the lead ion
is strongly enhanced by the factor $Z^{2}=6724$. So, as mentioned before, at
first sight the amplitude with the photon emitted by the lead ion (say, Fig. 2
left) should dominate. However the situation is not so simple.

###  3.1 The different contributions

Firstly, in the $\gamma$+Pb$\,\,\rightarrow\,V+$Pb process, the nucleons
situated at the same impact parameter (i.e. on the line directed along the
beam) may interact coherently as well. That is we have a competition between
the factor $Z^{2}=6724$ and a coherent factor of about $A^{4/3}=1232$, see
[16] for more details. Thus the proton induced rate is suppressed only by a
factor of 5 relative to that induced by Pb.

Moreover, as it is seen from the right hand side of (4), we have essentially a
logarithmic integral $\int dq^{2}_{\perp}/q^{2}_{\perp}$ in the interval from
$q_{\perp}\simeq xm_{p}$ up to the value $\sim 1/R$ limited by the form
factors. That is, within the leading-logarithmic approximation, the photon
flux is proportional to $\ln(1/(x\,m_{p}R))$, where
$x_{\pm}=(M_{V}/\sqrt{s})e^{\pm Y}$ is the proton momentum fraction carried by
the photon and $R$ is the radius of the photon emitter. Thus, for a large
rapidity $Y$, the flux radiated by the proton may be enhanced by the ratio

$\ln(1/(x_{-}m_{p}R_{p}))\leavevmode\nobreak\ /\leavevmode\nobreak\
\ln(1/(x_{+}m_{p}R_{\rm Pb}))\,,$ (10)

which for an $\Upsilon$ at $Y=2.5$ reaches 6.8 at $\sqrt{s}=14$ TeV. Here
$(xm_{p})^{2}$ is the minimal virtuality squared, $|t_{\rm min}|$, of the
photon which carries the fraction $x$ of the momentum of the parent proton.
The crucial point is that to produce a heavy $\Upsilon$ at large $Y$ the
photon must carry a large fraction of the nucleon beam momentum and the value
of $xm_{p}$ becomes comparable to the inverse ion radius, $1/R_{\rm Pb}$. That
is the logarithm in the denominator of the ratio becomes small.

Besides this, the amplitude where the photon was radiated by the proton may be
enhanced due to the energy dependence of the elementary photoproduction
amplitude.

Therefore, it is not evident that it is sufficient to consider only the
photons radiated by the heavy ion. In particular, for the case of forward
exclusive $\Upsilon$ production in the lead ion direction, the contribution
caused by the photon radiated off the proton beam may even dominate at large
$Y$. In our computations we will keep both amplitudes. The kinematic
configurations are illustrated (see Fig. 9) and discussed in Appendix B.

###  3.2 Including the probability of rapidity gap survival

Here we recall the structure of the calculations needed to account for the gap
survival probability in exclusive vector meson production in $p$Pb collisions.
We will follow Section 6 of [16], see also [17].

It is convenient to work in terms of the transverse coordinate; that is, to
work in the two-dimensional impact parameter $b$ space. Let us start with the
incoherent production of the vector meson in proton-ion collisions in which
the photon is emitted from the incoming proton. The situation is sketched in
Fig. 3(a). The figure shows the impact factors $\boldsymbol{b}_{\rm Pb}$,
$\boldsymbol{b}_{p}$ and $\boldsymbol{b}_{V}$ corresponding respectively to
the centre of the Pb ion, the proton and the produced vector meson. Note that
we show the extent in $b$ space of the Pb ion. It is convenient to take the
vertex of the meson production to be the origin of the transverse plane, that
is to take $\boldsymbol{b}_{V}=\boldsymbol{0}$.

Figure 3: The geometry of the $p+{\rm Pb}\to p+V+{\rm Pb}$ process in the
transverse ($x,y$) plane. In the left diagram the photon is radiated off the
incoming proton, whereas in the right diagram the photons are radiated by the
lead ion. The black dotted curves indicate that we add coherently the
contributions of each proton inside the lead ion. Note that the photons can be
emitted and absorbed at different points along the $z$ beam axis. Figure 4:
The geometry of the $p+{\rm Pb}\to p+V+{\rm Pb}$ process, where ($x,y$) is the
transverse plane and $z$ is in the direction of the beam. The black dotted
curves indicate that we add coherently the contributions of each nucleon
absorbing a photon inside the lead ion tube of the radius $r=\sqrt{4B_{V}}$
directed along the beam ($z$) axis. The transverse size of tube, $r$, is
driven by the $t$-slope (i.e. the size) of the $\gamma+N\to V+N$ amplitude.
The extra factor $4\pi B_{V}T(b_{\text{Pb}})$ in (15) in comparison with (11)
is the number of nucleons which may act coherently inside this tube of
transverse area $4\pi B_{V}$. Note that the photons can be emitted and
absorbed at different points along the $z$ beam axis.

Since we are looking for the cross section
($\sigma\propto\mathcal{A}^{*}\mathcal{A})$ integrated over the momentum
transferred $t$, the values of $b$ are the same in the $\mathcal{A}^{*}$ and
$\mathcal{A}$ amplitudes. Allowing for the photon flux and survival factors,
we can therefore write the expression for the cross section of the incoherent
$p$ \+ Pb$\,\,\to\,p\leavevmode\nobreak\ +\leavevmode\nobreak\
V\leavevmode\nobreak\ $\+ Pb interaction (for the case of the photon radiated
by the incoming proton) as

$\sigma_{\rm incoh}=\int d^{2}b_{p}d^{2}b_{\rm Pb}T(b_{\rm
Pb})\frac{xdn_{p}(x,b_{p})}{dx}|\mathcal{A}|^{2}S^{2}_{p}(b_{s})S^{2}_{V}(b_{sv})\,,$
(11)

where $n_{p}(x,b)$ is the flux of photons emitted off the proton,
$\mathcal{A}=\mathcal{A}(W,b_{a})$ is the $\gamma+p\to V+p$ amplitude in $b$
representation and $T(b_{\rm Pb})$ is the optical density of the Pb ion,
evaluated as given in the Appendix A. Equation (11) is written in the limit of
a small size amplitude $\mathcal{A}$, i.e. the size of the amplitude given by
the $t$-slope $B_{V}=B_{0}+4\alpha^{\prime}\ln\left(W/W_{0}\right)$, see
later, is much less than that for the heavy ion $R^{2}_{\rm Pb}\gg 4B_{V}$.222
Accounting, however, for the non-locality of $\mathcal{A}$ in the computation,
i.e. the size of the proton, we replace the optical density $T(b)$ (30) by the
convolution $T(b)=\frac{1}{4\pi B_{V}}\int d^{2}b^{\prime}\
T(b^{\prime})\exp[-(\boldsymbol{b}-\boldsymbol{b^{\prime}})^{2}/(4B_{V})]$.
Analogous replacements (with the corresponding $B$ slopes) were used in the
calculation of the survival factors $S^{2}$. For proton-nucleon interactions
the slope $B_{\rm el}=20$ GeV-2 is used for $\sqrt{s_{p\text{N}}}=8.16$ TeV
and $B_{\rm el}=19.1$ GeV-2 for $\sqrt{s_{p\text{N}}}=5.02$ TeV. The survival
factors are

$S^{2}_{p}(b_{s})=\exp\left(-T(b_{s})\sigma(pN)\right)\,,$ (12)

which is the probability not to fill the gap by secondary emissions produced
in additional proton-lead interactions, where $b_{s}=|\boldsymbol{b}_{\rm
Pb}-\boldsymbol{b}_{p}|$, and

$S^{2}_{V}(b_{sv})=\exp\left(-T(b_{sv})\sigma(VN)\right)\,,$ (13)

which is the probability not to fill the gap by secondaries produced in
additional vector meson-Pb interactions, where $b_{sv}=|\boldsymbol{b}_{\rm
Pb}-\boldsymbol{b}_{V}|=b_{\rm Pb}$ with our choice
$\boldsymbol{b}_{V}=\boldsymbol{0}$. Finally, $\sigma(pN)$ (and $\sigma(VN)$)
are the cross sections of the proton-nucleon (vector meson-nucleon)
interaction inside the ion. Recall that the expressions (12, 13) are similar
to the probability not to have an additional interaction in the target and the
product $T(b)\sigma$ plays the role of the optical density of the target, that
is the opacity $\Omega$ in (9).

If we were to neglect the survival factors (i.e. setting $S^{2}=1$) then the
cross section would be

$\sigma_{\rm incoh}=\frac{xdn_{p}(x)}{dx}\sigma(\gamma+p\to V+p)\cdot A\,,$
(14)

where $A=208$ is the lead atomic number.

As mentioned above, besides the incoherent interaction, we need to consider
coherent production as well. Here the situation is sketched in Fig. 4. The
coherent cross section is of the form

$\sigma_{\rm coh}=4\pi B_{V}F^{2}_{\rm Pb}(t_{\rm min})\int
d^{2}b_{p}d^{2}b_{\rm Pb}T^{2}(b_{\rm
Pb})\frac{xdn_{p}(x,b_{p})}{dx}|\mathcal{A}|^{2}S^{2}_{p}(b_{s})S^{2}_{V}(b_{\rm
Pb})\,,$ (15)

where the dimension of the extra $T$ factor is compensated by the $t$-slope of
the $\gamma+p\to V+p$ amplitude333Recall that the $t$ behaviour of the
photoproduction cross section $d\sigma(\gamma+p\to V+p)/dt\propto\exp(B_{V}t)$
corresponds to the amplitude
$\mathcal{A}(b)=\mathcal{A}(b=0)\exp[-b^{2}/(2B_{V})]$.. The amplitude
$\mathcal{A}$ is normalized to $\int
d^{2}b|\mathcal{A}(b)|^{2}=\sigma(\gamma+p\to V+p)$.

The form factor $F_{\rm Pb}$ in (15) accounts for the nucleon distribution in
the lead ion. The point is that the coherence of the interaction with
different nucleons should not be destroyed by the longitudinal component of
the momentum transferred. This component is represented by $t_{\rm
min}=-(xm_{p})^{2}/(1-x)$. Since the value of $|t_{\rm min}|$ is small in our
kinematics, here we use just the exponential parametrization $F_{\rm
Pb}(t)={\rm exp}(t\langle r^{2}_{\rm Pb}\rangle/6)$, with $\langle r^{2}_{\rm
Pb}\rangle$ being the mean radius squared of the lead ion.

We are particularly interested in the case when the photons are radiated by
the lead ion. Then we have only coherent radiation to consider. Here the
situation is sketched in the right panel of Fig. 3. Now the cross section has
the form

$\sigma_{\rm Pb}=\sigma(\gamma+p\to V+p)\int d^{2}b_{\rm Pb}\frac{xdn_{\rm
Pb}(x,b_{\rm Pb})}{dx})S^{2}_{p}(b_{\rm Pb})S^{2}_{V}(b_{\rm Pb})\,.$ (16)

Calculating the photon flux, $n(x,b)$, given by (4) we keep just the electric
($F_{E}$) term since the magnetic ($F_{M}$) contribution contains an
additional $x^{2}$ factor, while we work at small $x$. Besides this, the
magnetic contribution is concentrated at low impact parameters where the gap
survival factor $S^{2}_{p}$ is extremely small. (We have checked that the
magnetic ($F_{M}$) contribution does not exceed 1%.) For the lead ion the form
factor $F_{E}$ corresponds to the proton distribution in lead, see (31).

Note, however, that expression (4) cannot be transformed to the coordinate,
$b$, representation directly. First, we have to transform the amplitude of
photon emission and to account for the polarization structure of the
amplitude. The point is that the photon polarization vector
$\boldsymbol{e}_{\gamma}$ is directed parallel to the photon transverse
momentum $\boldsymbol{q}_{\perp}$. That is, the amplitude should be a vector.
In $b$ space it will be the vector
$\boldsymbol{a}_{\gamma}=\boldsymbol{b}\leavevmode\nobreak\ a(b)$. Calculating
the Fourier transform gives

$\boldsymbol{b}\leavevmode\nobreak\
a(b)=\frac{1}{4\pi^{2}}\int\frac{d^{2}q_{\perp}e^{i\boldsymbol{b}\cdot\boldsymbol{q}_{\perp}}}{(q^{2}_{\perp}+x^{2}m^{2}_{p})}\boldsymbol{q}_{\perp}\sqrt{\frac{\alpha^{\rm
QED}}{\pi x}(1-x)F_{E}(Q^{2})}\,,$ (17)

and after the angular integration we do not obtain the usual zero-order Bessel
$J_{0}(\boldsymbol{b}\cdot\boldsymbol{q_{\perp}})$, but rather
$J_{1}(\boldsymbol{b}\cdot\boldsymbol{q_{\perp}})$ [18]. Since
$J_{1}(\boldsymbol{b}\cdot\boldsymbol{q_{\perp}})$ vanishes as $b\to 0$, the
typical values of the impact parameter become larger.

Finally the photon flux in the $b$ representation outside the heavy ion takes
the form

$\frac{d^{3}n_{\rm Pb}}{dxd^{2}b_{\gamma}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \frac{Z^{2}\alpha^{\rm
QED}}{x\pi^{2}b_{\gamma}^{2}}\leavevmode\nobreak\
(xm_{p}b_{\gamma})^{2}\leavevmode\nobreak\ K_{1}^{2}(xm_{p}b_{\gamma})\,,$
(18)

where $K_{1}(z)$ is the modified Bessel function.

###  3.3 The uncertainty in the evaluation of $S^{2}$

Note that the expression for $S_{V}$ in (13) is written in the spirit of the
vector meson dominance (VMD) model [19, 20]. That is, we assume that the
photon to $\Upsilon$ transition takes place before the collision and then the
completely dressed $\Upsilon$ meson interacts with the heavy ion. This is
reasonable when the meson goes in the proton beam direction (see [21] for a
detailed discussion). However, the assumption is not justified for a very
forward (large rapidity) $\Upsilon$ going in the direction of the lead ion.
Indeed, the $\gamma\to b\bar{b}$ vertex is point-like and for $Y>2-3$ the
quarks do not have sufficient time to form the normal $\Upsilon$ wave
function. From the beginning the size of the $b\bar{b}$ pair is too small, and
the corresponding cross section [22, 23]

$\sigma(VN)\propto\alpha^{2}_{s}\langle r^{2}_{b\bar{b}}\rangle$ (19)

is smaller than the cross section of the normally dressed $\Upsilon$ meson.

In addition, the absorption cross section for the $\Upsilon$ meson,
$\sigma(VN)$, is not known experimentally. Moreover it depends on the $VN$
collision energy $s_{VN}=W^{2}$. For our numerical estimate we take the Regge
behaviour

$\sigma_{VN}(W)=\sigma_{0}\left(\frac{W^{2}}{M^{2}_{V}}\right)^{\alpha_{P}(0)-1}$
(20)

and assume that $\sigma_{0}\propto 1/M^{2}_{V}$. The normalization is fixed to
the $J/\psi$ absorptive cross section, $\sigma(J/\psi N)\simeq 4$ mb, at
$W\simeq 10$ GeV for a completely dressed $J/\psi$ meson measured by the N50
collaboration [24]. For the Pomeron intercept we take $\alpha_{P}(0)-1=0.25$.
This is consistent with the DIS data and the intercept of the BFKL Pomeron
after the resummation of the next-to-leading logarithmic corrections [25, 26,
27, 28]. Since the expected value of $\sigma(\Upsilon N)$ is small, the
survival probability due to $\Upsilon$ absorption, $S^{2}_{V}$, is rather
close to unity.

To demonstrate the effect, we compare the obtained results with those taking
$\sigma(\Upsilon N)=0$. The difference never exceeds 6-10%. Accounting for the
‘undressed meson’ problem this means that for the mesons going with large
rapidity in the lead ion direction the true value of $S^{2}$ may be about 3-5%
larger.

Finally, there is some uncertainty due to the value of the proton-nucleon
cross section $\sigma(pN)$. Recall that at 8 TeV TOTEM had measured
$\sigma_{\rm tot}(pp)=(103\pm 2.3)$ mb [29], while ATLAS-ALFA gives $(96\pm
1)$ mb [30]. Here we take $\sigma(pN)=100$ mb for our numerics. The
corresponding uncertainty is about 3-4%.

###  3.4 Numerical results for (photon flux)$\times$(gap survival)

For proton-lead collisions, eqn. (3) is replaced by the form

$\displaystyle\frac{d\sigma(p+{\rm Pb}\to p+V+{\rm
Pb})}{dY}\leavevmode\nobreak\ =\leavevmode\nobreak\ $ $\displaystyle
S^{2}(W_{\rm Pb})\left(k_{+}\frac{dn_{\rm Pb}}{dk_{+}}\right)\sigma_{\gamma
p}(W_{\rm Pb})\leavevmode\nobreak\ $ $\displaystyle+\leavevmode\nobreak\
S^{2}(W_{p})\left(k_{-}\frac{dn_{p}}{dk_{-}}\right)\biggl{(}\sigma_{\rm
incoh}(W_{p})+\sigma_{\rm coh}(W_{p})\biggr{)}R^{2}_{A}\,,$ (21)

where the nuclear modification factor $R_{A}(2\xi)=g_{\rm
Pb}(2\xi)/(Ag_{p}(2\xi))$ accounts for the fact that the gluon
distribution444Recall that the value of $\sigma(\gamma+p\to
V+p)\propto(2\xi\,g(2\xi))^{2}$ is almost completely driven by the gluon
distribution. in the lead ion may differ from the sum of the gluon
distributions in free nucleons. The value of the nucleon modification factor
$R_{A}(2\xi)$ at the corresponding scale $\mu=M_{V}/2$ is taken from the
EPPS16 NLO analysis [31]. It is convenient to introduce the so-called
‘effective fluxes’, $f_{\rm Pb}$ and $f_{p}$, which include the original
photon flux $xdn/dx$ times the survival and nuclear modification effects555It
is not completely correct to use the word ‘flux’ for a quantity which includes
these additional effects; however we use it since it enables us to shorten the
description of the computations.. The effective flux radiated by the lead ion
is

$f_{\rm Pb}(2\xi)=\frac{\sigma_{\rm Pb}}{\sigma(\gamma+p\to V+p)}\,,$ (22)

where $\sigma_{\rm Pb}$ is given by (16). The effect of the flux radiated by
the proton is given by the sum of coherent and incoherent fluxes,

$f_{\rm incoh}(2\xi)=\frac{\sigma_{\rm incoh}}{\sigma(\gamma+p\to
V+p)}R^{2}_{A}(2\xi)\ $ (23)

and

$f_{\rm coh}(2\xi)=\frac{\sigma_{\rm coh}}{\sigma(\gamma+p\to
V+p)}R^{2}_{A}(2\xi)\,,$ (24)

where $\sigma_{\rm incoh}$ and $\sigma_{\rm coh}$ are given by (11) and (15),
respectively. Here, $2\xi=2M_{V}^{2}/(2W^{2}-M_{V}^{2})\approx
M_{V}^{2}/W^{2}$ for $W^{2}\gg M^{2}_{V}$, i.e. $2\xi\ll 1$, is the fractional
momentum transfer provided by the two-gluon exchange.

Finally, the cross section of heavy vector meson production can be written as

$\frac{d\sigma(p+{\rm Pb}\to p+V+{\rm Pb})}{dY}\leavevmode\nobreak\ =f_{\rm
Pb}(W_{\rm Pb})\sigma_{\gamma p}(W_{\rm Pb})\leavevmode\nobreak\
+\leavevmode\nobreak\ \biggl{(}f_{\rm incoh}(W_{p})+f_{\rm
coh}(W_{p})\biggr{)}\sigma_{\gamma p}(W_{p})\,.$ (25)

Here the values of $W_{\rm Pb}$ and $W_{p}$ correspond to the energies of the
$\gamma+p\to V+p$ process initiated by the photon emitted off the lead ion and
proton beam, respectively.

The values of the effective fluxes are presented in Tables 1 and 2 in Appendix
B for $\Upsilon$ and $J/\psi$ production at proton-nucleon collision energy
$\sqrt{s_{pN}}=5.02$ TeV. The analogous values at $\sqrt{s_{pN}}=8.16$ TeV are
given in Tables 3 and 4.

##  4 Comparison with data

In this section, we compare our theoretical predictions with the rapidity
differential cross section data from ALICE [32, 33] and CMS [34] for the
process $p+{\rm Pb}\to p+V+{\rm Pb}$, where $V=J/\psi,\Upsilon$. Using eqn.
(25) we compute $\text{d}\sigma(p+{\rm Pb}\to p+J/\psi+{\rm Pb})/\text{d}Y$,
with $\sigma_{\gamma p}$ evaluated using, as input, the gluon PDF fit obtained
from our previous analyses [35].

We work at NLO within the collinear factorisation scheme and express the
amplitude for exclusive heavy vector meson photoproduction as

$\mathcal{A}=\frac{4\pi\sqrt{4\pi\alpha}e_{q}(\epsilon_{V}^{*}\cdot\epsilon_{\gamma})}{N_{c}}\left(\frac{8\,\langle
O_{1}\rangle_{V}}{M_{V}^{3}}\right)^{1/2}\int_{-1}^{1}\text{d}x\biggl{(}C_{g}(x,\xi)F_{g}(x,\xi)+C_{q}(x,\xi)F_{q}(x,\xi)\biggr{)}\,,$
(26)

where $F_{g}$ and $F_{q}$ are the gluon and quark singlet Generalised Parton
Distributions (GPDs), $C_{g}$ and $C_{q}$ are the gluon and quark coefficient
functions, see [1, 2], and $x-\xi,\leavevmode\nobreak\ x+\xi$ are the parton
momentum fractions in the lightcone direction $P^{+}$. The corresponding gluon
and quark coefficient functions for exclusive heavy vector meson
electroproduction were calculated in [36, 37]. The dependence on the
factorisation and renormalisation scales, $\mu_{F},\mu_{R}$, and on the four-
momentum transfer squared, $t$, is not shown. The set-up is illustrated in
Fig. 5. The non-relativistic QCD (NRQCD) matrix element $\langle
O_{1}\rangle_{V}$ is fixed by the experimental value of the heavy vector meson
decay width to a dilepton pair, see [38].

We take $\mu_{R}=\mu_{F}$ and use the ‘optimal’ factorization scale
$\mu_{F}=M_{V}/2$, see [39]. That is, the exclusive $J/\psi$ and $\Upsilon$
photoproduction data probe the gluon densities at two different scales.
Besides this, we implement the $Q_{0}$ subtraction [2] needed to avoid the
double counting between the low $k_{T}<Q_{0}$ contributions in the NLO
coefficient functions and that hidden in the PDF inputs. This choice provides
a sufficiently good scale stability of the prediction666Recall that exactly
the same approach (that is use of the optimal scale $\mu_{F}=M_{V}/2$ and the
$Q_{0}$ subtraction) was applied in [35] to determine the low-$x$ gluon PDF
from exclusive $J/\psi$ data, used in the present analysis..

Figure 5:  The LO and the NLO quark contributions to $\gamma+p\to V+p$
amplitude. The parton momentum fractions are $x+\xi$ and $x-\xi$ and $k$ is
the loop momentum.

The cross section, $\sigma_{\gamma p}$, integrated over the Mandelstam
variable $t$, is given by

$\sigma_{\gamma
p}(W)=\frac{1}{B_{V}(W)}\left(\frac{\text{d}\sigma}{\text{d}t}\left(\gamma
p\rightarrow
Vp\right)\biggl{|}_{t=0}\right)=\frac{1}{B_{V}(W)}\frac{(\text{Im}\mathcal{A})^{2}(1+\rho^{2})}{16\pi
W^{4}}\,,$ (27)

where

$\rho\leavevmode\nobreak\ \leavevmode\nobreak\ =\leavevmode\nobreak\
\leavevmode\nobreak\ \frac{{\rm Re}\mathcal{A}}{{\rm
Im}\mathcal{A}}\leavevmode\nobreak\ \leavevmode\nobreak\ =\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm tan}\left(\frac{\pi}{2}\leavevmode\nobreak\
\frac{\partial\ln({\rm Im}\mathcal{A}/W^{2})}{\partial\ln W^{2}}\right)$ (28)

is the real part correction, see e.g. [40], and $B_{V}(W)$ is the Regge-
motivated energy dependent slope parameter given by

$B_{V}(W)=\left(B_{0}+4\alpha^{\prime}_{P}\ln\left(\frac{W}{W_{0}}\right)\right)\,\,\,\text{GeV}^{-2}\,.$
(29)

Here, $B_{0}=4.9$ for $V=J/\psi$ [41] and $B_{0}=4.63$ for $V=\Upsilon$, with
$\alpha^{\prime}_{P}=0.06$ and $W_{0}=90$ GeV [42]. The smaller value of
$B_{0}$ for the case $V=\Upsilon$ is due to the need for reducing the $J/\psi$
slope parameter by the factor
$4\alpha^{\prime}\ln\left(M_{\Upsilon}/M_{J/\psi}\right)$.

(a)

(b)

(c)

(d)

Figure 6: Theoretical predictions for coherent exclusive $J/\psi$ and
$\Upsilon$ photoproduction rapidity differential cross sections in $p$Pb
collisions with $\sqrt{s_{pN}}=5.02$ TeV. Panels (a) and (c) show the
decomposition of the total cross section into the $\sigma_{\rm Pb}$ and
$\sigma_{p}$ contributions, see text for details. Panels (b) and (d) compare
our predictions with existing data from ALICE and/or CMS [32, 33, 34]. We
emphasise that no fit is made to the data shown, the width of the uncertainty
bands is obtained by the $\pm 1\sigma$ statistical uncertainties and the
normalisation errors of the datasets used in our previous fit [35] to
determine the low $x$ gluon PDF, which is used in the present analysis.

Fig. 6 shows our rapidity differential cross section predictions at
$\sqrt{s_{pN}}=5.02$ TeV for the process $p+{\rm Pb}\to p+V+{\rm Pb}$. The
upper row shows the predictions for $V=J/\psi$ and the lower one for
$V=\Upsilon$. In the left panels, we show the decomposition of the total cross
section result into the $\gamma p$ contribution (the first term on the right
hand side of eqn. (25), labelled by $\sigma_{\text{Pb}}$ in the figure) and
the $\gamma$Pb contribution (the second term on the right hand side of eqn.
(25), labelled by $\sigma_{p}$ in the figure). In the right panels, we show
our results compared with existing data from ALICE and/or CMS [32, 33, 34].
The error bands are indicative of the uncertainty due to our previous low $x$
gluon PDF fit [35] only and does not account for theoretical uncertainties in
the present formalism. We emphasise that no fit is performed to the $p$Pb data
here, and the width of the bands is given by propagating the errors on the fit
parameters using the full covariance matrix obtained from our previous gluon
PDF fit made to the low $x$ exclusive $J/\psi$ data in $pp$ collisions in
[35].

For $V=J/\psi$, the $\gamma$ Pb contribution is less than a percent at mid-
rapidity, while for $V=\Upsilon$, it is as much as 7%. The mass of the
$\Upsilon$ is $\sim 3$ times that of the $J/\psi$ and so (with $k_{+}\propto
M_{V}$ and $W_{+}\propto\sqrt{M_{V}}$), the typical photon energy in exclusive
$\Upsilon$ production is now much larger than in exclusive $J/\psi$ production
for a given $Y_{\text{lab}}$. Therefore, the photon flux radiated off Pb is
relatively suppressed for $\Upsilon$ production. We remark that for
$Y_{\text{lab}}<-3$ in Fig. 6(c), the photon radiated by Pb has a somewhat
greater $x$ (with respect to the nucleon in Pb) and so the value of
$b_{\text{Pb}}=b_{\gamma}$ (in eq. (18)) is rather small. This means that the
integrated flux from Pb is not large and $S^{2}(b_{\text{Pb}})$ is small. On
the other hand, the photon radiated by the proton has a much smaller $x$ (with
respect to the proton) leading to a larger $S^{2}$ and a larger integrated
flux. This is evident from Tab. 3 where, for $Y_{\text{lab}}<-3$, the value of
$f_{\text{Pb}}$ is an order of magnitude smaller than $f_{p}$. The energy
dependence of $\sigma(\gamma+N\to V+N)$ does not compensate this difference
for $\sqrt{s_{pN}}=5.02$ TeV. This accounts for the $W_{p}$ contribution being
greater than the $W_{\text{Pb}}$ one in this region.

Our predictions agree favourably with the ALICE data, particularly at
backward, central and semi-forward rapidities. For $V=J/\psi$, our prediction
undershoots the data for $Y_{\text{lab}}\geq 2.5$. In this region, however,
our prediction is well constrained. As shown above, the dominant contribution
comes from the photon radiated by Pb, occuring at large $b$, with $S^{2}$ very
close to unity where we have practically no uncertainty in the photon flux. On
the other hand, the ‘gamma-proton’ centre of mass energy is relatively small,
and this is precisely the region where our gluon PDF fit agreed well with the
HERA data [35]. The earlier ALICE data from [32] for $Y_{\text{lab}}\geq 2.5$
show an unexpected plateauing behaviour and are arguably inconsistent with
those in [33]777 The behaviour of the earlier ALICE data in Fig. 6b are
incompatible with other theoretical predictions, too, see e.g. Fig. 4 of [6]..

Figure 7: Theoretical predictions for coherent exclusive $J/\psi$ and
$\Upsilon$ photoproduction rapidity differential cross sections in $p$Pb
collisions with $\sqrt{s_{pN}}=8.16$ TeV. The bands and curves are as
described in the caption of Fig. 6. Note that currently there are no data for
the process $p+{\rm Pb}\to p+V+{\rm Pb}$, with $V=J/\psi,\Upsilon$, at this
centre-of-mass energy, but measurements for $\Upsilon$ photoproduction are
shortly anticipated [43, 44].

In Fig. 7, we show the analogous set of plots at $\sqrt{s_{pN}}=8.16$ TeV.
There are currently no data for $p+{\rm Pb}\to p+V+{\rm Pb}$ at this centre-
of-mass energy, but forthcoming measurements for the case $V=\Upsilon$ from
CMS are anticipated [43, 44]. Our results for the rapidity distribution are
qualitatively consistent with the shapes predicted by other approaches,
however, the normalisation of our prediction is larger than that obtained from
CGC models [3, 4].

In both Figs. 6 and 7, note that the larger
$\text{d}\sigma/\text{d}Y_{\text{lab}}$ at positive $Y_{\text{lab}}$ is
indicative of the proton direction corresponding to the positive
$Y_{\text{lab}}$. At negative $Y_{\text{lab}}$ the distribution is skewed due
to a smaller energy per nucleon in the Pb-ion. Indeed, at large rapidities in
the Pb direction (i.e. large negative $Y_{\text{lab}}$), the contribution due
to the photon radiated from Pb is strongly suppressed, especially in the
$\Upsilon$ case, since the momentum fraction,
$x\propto(M_{V}/\sqrt{s})\exp(Y)$ carried by the photon becomes large and the
transverse momentum cutoff $xm_{p}$ becomes comparable with the inverse lead
ion radius $1/R_{\text{Pb}}$, i.e. here we deal with the impact parameter
$b_{\text{Pb}}\sim R_{\text{Pb}}$. Therefore, in this region, the survival
factor $S^{2}(b_{\text{Pb}})$ suppresses the $\sigma_{\text{Pb}}$
contribution, and the second term of (25), i.e. the photons radiated by the
proton, starts to dominate.

##  5 Conclusions

We have predicted the cross sections for exclusive $J/\psi$ and $\Upsilon$
meson production in proton-lead ion collisions at the LHC for centre-of-mass
energies $\sqrt{s_{pN}}=5.02$ TeV and $\sqrt{s_{pN}}=8.16$ TeV, using the low
$x$ gluon distribution extracted in [35] from the data on $p+p\to p+J/\psi+p$.
We account for the gap survival probability caused by both the additional
proton-lead interactions and the interaction of the secondary vector meson
inside the heavy ion.

As expected, the dominant contribution to the cross section in exclusive $p$Pb
collisions comes from the amplitude where the photon is radiated by the lead
ion (for all rapidities $Y_{\text{lab}}\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}{-3}$).
However, our detailed study finds that the naively expected dominance of the
$Z^{2}$ enhancement of the cross section, when we go from exclusive production
from photons radiated by a proton to that for photons radiated by the heavy
ion, is considerably reduced. Nevertheless, we find that the enhancement is
still sufficient to enable forthcoming data for exclusive heavy vector meson
production in $p$Pb collisions to provide additional constraints on the free
proton gluon PDF at low to moderate values of $x$ and scale. Such data could
be used in a combined fit with those in $pp$ collisions anticipated from the
High-Luminosity phase of the LHC, as well as in the upcoming $ep$ programme of
the Electron-Ion collider, to provide refined constraints on the gluon PDF.

Appendix A: Optical density of the Pb ion

The optical density of the Pb ion may be written in the form

$T(b_{\rm Pb})=\int_{-\infty}^{+\infty}dr_{z}\,(\rho_{p}(r)+\rho_{n}(r)),$
(30)

with $r=\sqrt{r_{z}^{2}+r^{2}_{t}}$. For the nucleon density in lead,
$\rho(r)$, we use the Woods-Saxon form [45]

$\rho_{N}(r)=\frac{\rho_{0}}{1+\exp{((r-R)/d)}}\;,$ (31)

where the parameters $d$ and $R$, respectively, characterise the skin
thickness and the radius of the nucleon density in the heavy ion;
$r=(r_{z},r_{t})$. For 208Pb we take the recent results of [46, 47]

$\displaystyle R_{p}$ $\displaystyle=6.680\,{\rm fm}\;,$ $\displaystyle d_{p}$
$\displaystyle=0.447\,{\rm fm}\;,$ $\displaystyle R_{n}$
$\displaystyle=(6.67\pm 0.03)\,{\rm fm}\;,$ $\displaystyle d_{n}$
$\displaystyle=(0.55\pm 0.01)\,{\rm fm}\;.$ (32)

The nucleon densities, $\rho$, are normalized to

$\int\rho_{p}(r)d^{3}r=Z\;,\qquad\int\rho_{n}(r)d^{3}r=N_{n}\;,$ (33)

for which the corresponding proton (neutron) densities are $\rho_{0}=0.063$
(0.093) ${\rm fm}^{-3}$.

Appendix B: Tables of effective fluxes

The values of the effective fluxes used in eqn. (25) are presented in Tables 1
and 2 below for $\Upsilon$ and $J/\psi$ production at proton-nucleon collision
energy $\sqrt{s_{pN}}=5.02$ TeV. The analogous values at $\sqrt{s_{pN}}=8.16$
TeV are also given, in Tables 3 and 4.

The first column is the rapidity, $Y_{\rm lab}$, of the vector meson measured
in the laboratory frame. We account for the asymmetry of proton-ion
collisions. In the case of lead we have (in the laboratory frame) a proton
beam momentum equal to 4 TeV (6.5 TeV), while the momentum of a nucleon in
lead is $4\leavevmode\nobreak\ (Z/A)=1.58\leavevmode\nobreak\ $TeV
($6.5\leavevmode\nobreak\ (Z/A)=2.56\leavevmode\nobreak\ $TeV) for
$\sqrt{s_{pN}}=5.02$ (8.16) TeV. Positive $Y_{\rm lab}>0$ corresponds to the
vector meson going in the proton direction.

The second and fourth columns are the $\gamma$-nucleon collision energies
$W_{\rm Pb}$ and $W_{p}$. The effective flux $f_{\rm Pb}$ is given in the
third column while the fluxes $f_{\rm incoh}$, $f_{\rm coh}$ and their sum are
shown in the fifth, sixth and seventh column, respectively.

Column eight gives the sum $f_{\rm incoh}+f_{\rm coh}$ for the case $R_{A}=1$.
The factor of $R_{A}$ enters only the term where the photon is radiated by the
proton and, as shown in Figs. 6 and 7, this contribution is relatively small
(especially for $J/\psi$) and can be seen at large backward rapidities only.
The contribution due to the term where the photon is radiated from the proton
with and without inclusion of the nuclear modification factor, $R_{A}$, is
shown in Fig. 8. This difference at the total cross section level amounts to
at most a few percent only for the $J/\psi$ case because the contribution when
the photon is radiated from the lead ion dominates. Therefore, the nuclear
modification factor is hardly seen in $p$Pb collisions, where the first term
of (25) dominates. This factor can be much better observed (and measured) in
the Pb-Pb set-up, see e.g. [48] for a baseline description of the Pb + Pb
$\rightarrow$ Pb + $J/\psi$ \+ Pb process in the collinear factorisation
framework to NLO.

Figure 8: The contribution due to the term where the photon is emitted from
the proton, $\sigma_{p}$, with and without inclusion of the nuclear
modification factor, $R_{A}$. The left panel is for the $J/\psi$ production at
$\sqrt{s_{pN}}=5.02$ TeV and the right panel is for $\Upsilon$ production at
the same centre-of-mass energy. Figure 9: The four kinematic configurations
for exclusive heavy vector meson, $V$, ultraperipheral production in $p$Pb
collisions at a given rapidity $Y$ occurring via a photoproduction subprocess
at two energies, $W_{+}$ and $W_{-}$. The incoming photon is a Weizsäcker-
Williams photon that has been emitted from a proton (lead nucleus) and
interacts with an incoming lead nucleus (proton). The vector meson can travel
in the direction of the photon ($W_{+}$), or against the direction of the
photon ($W_{-})$.

Note that in asymmetric $p$Pb collisions, the laboratory frame does not
coincide with the centre of momentum frame of the $p$Pb system and so eqn. (2)
should be adjusted. For the case where the vector meson is detected at
positive rapidities (corresponding to the configurations in the upper row of
Fig. 9),

$W^{2}_{+}=M_{V}\sqrt{s}\leavevmode\nobreak\
e^{+(Y_{\text{lab}}-Y_{0})}=W_{p}^{2}\;\;\;\mbox{and}\;\;\;W^{2}_{-}=M_{V}\sqrt{s}\leavevmode\nobreak\
e^{-(Y_{\text{lab}}-Y_{0})}=W_{\text{Pb}}^{2}\,,$ (34)

where $Y_{0}=0.465$ is the shift of the nucleon-nucleon centre of mass with
respect to the laboratory frame.

$Y_{\rm lab}$ |  $W_{\rm Pb}$ (GeV) | $f_{\rm Pb}$ |  $W_{p}$ (GeV) | $f_{\rm incoh}$ | $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}(R_{A}=1)$
---|---|---|---|---|---|---|---
-5.125 | 3570 | 0.464 $\times 10^{-12}$ | 13.31 | 6.23 | 0.0 | 6.23 | 7.16
-4.875 | 3150 | 0.577 $\times 10^{-10}$ | 15.08 | 5.77 | 0.52 $\times 10^{-21}$ | 5.77 | 6.91
-4.625 | 2780 | 0.930 $\times 10^{-8}$ | 17.09 | 6.37 | 0.22 $\times 10^{-10}$ | 6.37 | 6.66
-4.375 | 2450 | 0.725 $\times 10^{-6}$ | 19.37 | 6.83 | 0.51 $\times 10^{-5}$ | 6.83 | 6.41
-4.125 | 2160 | 0.262 $\times 10^{-4}$ | 21.95 | 7.11 | 0.39 $\times 10^{-2}$ | 7.11 | 6.16
-3.875 | 1910 | 0.481 $\times 10^{-3}$ | 24.87 | 7.21 | 0.15 | 7.36 | 6.03
-3.625 | 1690 | 0.500 $\times 10^{-2}$ | 28.18 | 7.17 | 1.2 | 8.35 | 6.60
-3.375 | 1490 | 0.326 $\times 10^{-1}$ | 31.93 | 7.03 | 3.9 | 10.9 | 8.38
-3.125 | 1310 | 0.146 | 36.18 | 6.75 | 7.5 | 14.2 | 10.9
-2.875 | 1160 | 0.482 | 41.00 | 6.29 | 11 | 16.8 | 13.2
-2.625 | 1020 | 1.26 | 46.46 | 5.64 | 12 | 17.8 | 14.7
-2.375 | 902 | 2.71 | 52.64 | 4.97 | 12 | 17.4 | 15.5
-2.125 | 796 | 5.03 | 59.65 | 4.36 | 12 | 16.3 | 15.6
-1.875 | 703 | 8.33 | 67.60 | 3.81 | 11 | 14.9 | 15.4
-1.625 | 620 | 12.6 | 76.60 | 3.35 | 10 | 13.4 | 14.8
-1.375 | 547 | 17.7 | 86.80 | 2.96 | 9.1 | 12.1 | 14.1
-1.125 | 483 | 23.5 | 98.35 | 2.64 | 8.2 | 10.9 | 13.3
-0.875 | 426 | 29.9 | 111.45 | 2.36 | 7.4 | 9.77 | 12.4
-0.625 | 376 | 36.7 | 126.29 | 2.11 | 6.7 | 8.79 | 11.5
-0.375 | 332 | 43.8 | 143.10 | 1.89 | 6.0 | 7.91 | 10.6
-0.125 | 293 | 51.2 | 162.16 | 1.69 | 5.4 | 7.09 | 9.74
0.125 | 258 | 58.6 | 183.75 | 1.51 | 4.8 | 6.33 | 8.83
0.375 | 228 | 66.2 | 208.21 | 1.33 | 4.3 | 5.61 | 7.94
0.625 | 201 | 73.9 | 235.94 | 1.17 | 3.8 | 4.93 | 7.06
0.875 | 178 | 81.6 | 267.35 | 1.02 | 3.3 | 4.29 | 6.21
1.125 | 157 | 89.4 | 302.95 | 0.88 | 2.8 | 3.68 | 5.37
1.375 | 138 | 97.2 | 343.29 | 0.74 | 2.4 | 3.11 | 4.58
1.625 | 122 | 105 | 389.00 | 0.62 | 1.9 | 2.57 | 3.82
1.875 | 108 | 113 | 440.79 | 0.51 | 1.6 | 2.08 | 3.11
2.125 | 95.1 | 121 | 499.48 | 0.40 | 1.2 | 1.63 | 2.46
2.375 | 83.9 | 128 | 565.99 | 0.31 | 0.93 | 1.24 | 1.88
2.625 | 74.0 | 136 | 641.35 | 0.23 | 0.67 | 0.903 | 1.38
2.875 | 65.3 | 144 | 726.74 | 0.17 | 0.46 | 0.630 | 0.971
3.125 | 57.7 | 152 | 823.50 | 0.12 | 0.30 | 0.419 | 0.650
3.375 | 50.9 | 160 | 933.15 | 0.08 | 0.19 | 0.266 | 0.415
3.625 | 44.9 | 167 | 1057.40 | 0.05 | 0.11 | 0.161 | 0.253
3.875 | 39.6 | 175 | 1198.19 | 0.03 | 0.60 $\times 10^{-1}$ | 0.939 $\times 10^{-1}$ | 0.149
4.125 | 35.0 | 183 | 1357.73 | 0.02 | 0.32 $\times 10^{-1}$ | 0.535 $\times 10^{-1}$ | 0.852 $\times 10^{-1}$
4.375 | 30.9 | 191 | 1538.51 | 0.01 | 0.16 $\times 10^{-1}$ | 0.300 $\times 10^{-1}$ | 0.480 $\times 10^{-1}$
4.625 | 27.2 | 199 | 1743.36 | 0.01 | 0.80 $\times 10^{-2}$ | 0.165 $\times 10^{-1}$ | 0.267 $\times 10^{-1}$
4.875 | 24.0 | 206 | 1975.48 | 0.01 | 0.38 $\times 10^{-2}$ | 0.892 $\times 10^{-2}$ | 0.145 $\times 10^{-1}$
5.125 | 21.2 | 214 | 2238.52 | 0.00 | 0.18 $\times 10^{-2}$ | 0.459 $\times 10^{-2}$ | 0.749 $\times 10^{-2}$

Table 1: The effective photon flux $f_{\rm Pb}$ and $f_{\rm incoh},f_{\rm
coh}$ radiated in the case of $\Upsilon$ production at proton-nucleon
collision energy $\sqrt{s_{pN}}=5.02$ TeV by the lead ion and by the proton
beam.

$Y_{\rm lab}$ |  $W_{\rm Pb}$ (GeV) | $f_{\rm Pb}$ |  $W_{p}$ (GeV) | $f_{\rm incoh}$ | $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}(R_{A}=1)$
---|---|---|---|---|---|---|---
-5.125 | 2040 | 0.102 $\times 10^{-3}$ | 7.62 | 5.68 | 0.18 $\times 10^{-1}$ | 5.70 | 4.16
-4.875 | 1800 | 0.145 $\times 10^{-2}$ | 8.63 | 5.50 | 0.29 | 5.79 | 4.06
-4.625 | 1590 | 0.121 $\times 10^{-1}$ | 9.78 | 5.21 | 1.3 | 6.54 | 4.49
-4.375 | 1400 | 0.662 $\times 10^{-1}$ | 11.08 | 4.80 | 3.1 | 7.90 | 5.44
-4.125 | 1240 | 0.257 | 12.56 | 4.10 | 4.5 | 8.63 | 6.41
-3.875 | 1090 | 0.759 | 14.23 | 3.31 | 5.0 | 8.30 | 7.03
-3.625 | 964 | 1.81 | 16.12 | 2.55 | 4.6 | 7.17 | 7.21
-3.375 | 851 | 3.63 | 18.27 | 1.93 | 3.9 | 5.80 | 7.04
-3.125 | 751 | 6.38 | 20.70 | 1.46 | 3.1 | 4.58 | 6.65
-2.875 | 663 | 10.1 | 23.46 | 1.12 | 2.5 | 3.58 | 6.13
-2.625 | 585 | 14.7 | 26.58 | 0.87 | 1.9 | 2.80 | 5.56
-2.375 | 516 | 20.2 | 30.12 | 0.69 | 1.5 | 2.22 | 4.98
-2.125 | 455 | 26.2 | 34.13 | 0.56 | 1.2 | 1.78 | 4.42
-1.875 | 402 | 32.8 | 38.68 | 0.47 | 1.0 | 1.46 | 3.89
-1.625 | 355 | 39.8 | 43.83 | 0.39 | 0.82 | 1.21 | 3.40
-1.375 | 313 | 47.0 | 49.66 | 0.34 | 0.68 | 1.02 | 2.95
-1.125 | 276 | 54.4 | 56.27 | 0.29 | 0.57 | 0.860 | 2.54
-0.875 | 244 | 61.9 | 63.77 | 0.25 | 0.48 | 0.733 | 2.18
-0.625 | 215 | 69.5 | 72.26 | 0.22 | 0.40 | 0.626 | 1.85
-0.375 | 190 | 77.2 | 81.88 | 0.20 | 0.34 | 0.535 | 1.57
-0.125 | 168 | 84.9 | 92.78 | 0.17 | 0.28 | 0.456 | 1.32
0.125 | 148 | 92.7 | 105.13 | 0.15 | 0.24 | 0.389 | 1.11
0.375 | 130 | 100 | 119.13 | 0.13 | 0.20 | 0.330 | 0.922
0.625 | 115 | 108 | 134.99 | 0.12 | 0.16 | 0.279 | 0.764
0.875 | 102 | 116 | 152.97 | 0.10 | 0.13 | 0.235 | 0.630
1.125 | 89.7 | 124 | 173.34 | 0.09 | 0.11 | 0.197 | 0.517
1.375 | 79.1 | 132 | 196.42 | 0.08 | 0.86 $\times 10^{-1}$ | 0.164 | 0.423
1.625 | 69.8 | 139 | 222.57 | 0.07 | 0.69 $\times 10^{-1}$ | 0.136 | 0.344
1.875 | 61.6 | 147 | 252.20 | 0.06 | 0.54 $\times 10^{-1}$ | 0.112 | 0.279
2.125 | 54.4 | 155 | 285.78 | 0.05 | 0.42 $\times 10^{-1}$ | 0.918 $\times 10^{-1}$ | 0.225
2.375 | 48.0 | 163 | 323.83 | 0.04 | 0.33 $\times 10^{-1}$ | 0.748 $\times 10^{-1}$ | 0.181
2.625 | 42.4 | 170 | 366.95 | 0.04 | 0.25 $\times 10^{-1}$ | 0.605 $\times 10^{-1}$ | 0.145
2.875 | 37.4 | 178 | 415.81 | 0.03 | 0.19 $\times 10^{-1}$ | 0.486 $\times 10^{-1}$ | 0.115
3.125 | 33.0 | 186 | 471.18 | 0.02 | 0.14 $\times 10^{-1}$ | 0.388 $\times 10^{-1}$ | 0.911 $\times 10^{-1}$
3.375 | 29.1 | 194 | 533.91 | 0.02 | 0.11 $\times 10^{-1}$ | 0.307 $\times 10^{-1}$ | 0.717 $\times 10^{-1}$
3.625 | 25.7 | 202 | 605.00 | 0.02 | 0.78 $\times 10^{-2}$ | 0.242 $\times 10^{-1}$ | 0.562 $\times 10^{-1}$
3.875 | 22.7 | 209 | 685.56 | 0.01 | 0.56 $\times 10^{-2}$ | 0.190 $\times 10^{-1}$ | 0.437 $\times 10^{-1}$
4.125 | 20.0 | 217 | 776.84 | 0.01 | 0.40 $\times 10^{-2}$ | 0.148 $\times 10^{-1}$ | 0.339 $\times 10^{-1}$
4.375 | 17.7 | 225 | 880.27 | 0.01 | 0.29 $\times 10^{-2}$ | 0.114 $\times 10^{-1}$ | 0.262 $\times 10^{-1}$
4.625 | 15.6 | 233 | 997.48 | 0.01 | 0.20 $\times 10^{-2}$ | 0.884 $\times 10^{-2}$ | 0.202 $\times 10^{-1}$
4.875 | 13.8 | 240 | 1130.29 | 0.01 | 0.14 $\times 10^{-2}$ | 0.673 $\times 10^{-2}$ | 0.154 $\times 10^{-1}$
5.125 | 12.1 | 248 | 1280.79 | 0.00 | 0.96 $\times 10^{-3}$ | 0.505 $\times 10^{-2}$ | 0.116 $\times 10^{-1}$

Table 2: The effective photon flux $f_{\rm Pb}$ and $f_{\rm incoh},f_{\rm
coh}$ radiated in the case of $J/\psi$ production at proton-nucleon collision
energy $\sqrt{s_{pN}}=5.02$ TeV by the lead ion and by the proton beam.

$Y_{\rm lab}$ |  $W_{\rm Pb}$ (GeV) | $f_{\rm Pb}$ |  $W_{p}$ (GeV) | $f_{\rm incoh}$ | $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}(R_{A}=1)$
---|---|---|---|---|---|---|---
-5.125 | 4550 | 0.548 $\times 10^{-8}$ | 16.97 | 7.14 | 0.90 $\times 10^{-11}$ | 7.14 | 7.52
-4.875 | 4010 | 0.482 $\times 10^{-6}$ | 19.23 | 7.70 | 0.35 $\times 10^{-5}$ | 7.70 | 7.27
-4.625 | 3540 | 0.190 $\times 10^{-4}$ | 21.79 | 8.07 | 0.33 $\times 10^{-2}$ | 8.07 | 7.02
-4.375 | 3130 | 0.374 $\times 10^{-3}$ | 24.69 | 8.22 | 0.15 | 8.37 | 6.88
-4.125 | 2760 | 0.410 $\times 10^{-2}$ | 27.98 | 8.22 | 1.2 | 9.47 | 7.49
-3.875 | 2430 | 0.279 $\times 10^{-1}$ | 31.70 | 8.11 | 4.2 | 12.3 | 9.51
-3.625 | 2150 | 0.129 | 35.92 | 7.85 | 8.4 | 16.3 | 12.4
-3.375 | 1900 | 0.437 | 40.70 | 7.37 | 12 | 19.5 | 15.2
-3.125 | 1670 | 1.16 | 46.12 | 6.65 | 14 | 20.8 | 17.2
-2.875 | 1480 | 2.54 | 52.26 | 5.92 | 15 | 20.7 | 18.3
-2.625 | 1300 | 4.78 | 59.22 | 5.23 | 14 | 19.5 | 18.7
-2.375 | 1150 | 7.99 | 67.11 | 4.61 | 13 | 18.0 | 18.5
-2.125 | 1010 | 12.2 | 76.04 | 4.08 | 12 | 16.4 | 18.0
-1.875 | 896 | 17.2 | 86.17 | 3.65 | 11 | 14.9 | 17.4
-1.625 | 790 | 23.0 | 97.64 | 3.28 | 10 | 13.6 | 16.6
-1.375 | 697 | 29.3 | 110.64 | 2.97 | 9.4 | 12.4 | 15.7
-1.125 | 615 | 36.1 | 125.38 | 2.69 | 8.6 | 11.3 | 14.8
-0.875 | 543 | 43.2 | 142.07 | 2.45 | 7.9 | 10.3 | 13.9
-0.625 | 479 | 50.5 | 160.99 | 2.23 | 7.2 | 9.43 | 12.9
-0.375 | 423 | 58.0 | 182.42 | 2.03 | 6.6 | 8.60 | 12.0
-0.125 | 373 | 65.6 | 206.71 | 1.84 | 6.0 | 7.82 | 11.1
0.125 | 329 | 73.2 | 234.23 | 1.67 | 5.4 | 7.07 | 10.1
0.375 | 291 | 80.9 | 265.42 | 1.50 | 4.9 | 6.37 | 9.21
0.625 | 257 | 88.7 | 300.76 | 1.34 | 4.4 | 5.69 | 8.31
0.875 | 226 | 96.5 | 340.81 | 1.19 | 3.9 | 5.04 | 7.43
1.125 | 200 | 104 | 386.19 | 1.04 | 3.4 | 4.42 | 6.57
1.375 | 176 | 112 | 437.61 | 0.91 | 2.9 | 3.83 | 5.73
1.625 | 156 | 120 | 495.87 | 0.78 | 2.5 | 3.27 | 4.93
1.875 | 137 | 128 | 561.90 | 0.65 | 2.1 | 2.75 | 4.17
2.125 | 121 | 135 | 636.72 | 0.54 | 1.7 | 2.26 | 3.45
2.375 | 107 | 143 | 721.49 | 0.44 | 1.4 | 1.81 | 2.79
2.625 | 94.4 | 151 | 817.56 | 0.35 | 1.1 | 1.41 | 2.19
2.875 | 83.3 | 159 | 926.42 | 0.27 | 0.79 | 1.06 | 1.66
3.125 | 73.5 | 167 | 1049.77 | 0.20 | 0.57 | 0.766 | 1.20
3.375 | 64.9 | 174 | 1189.54 | 0.14 | 0.39 | 0.530 | 0.837
3.625 | 57.2 | 182 | 1347.93 | 0.10 | 0.25 | 0.348 | 0.555
3.875 | 50.5 | 190 | 1527.40 | 0.07 | 0.15 | 0.219 | 0.351
4.125 | 44.6 | 198 | 1730.77 | 0.04 | 0.87 $\times 10^{-1}$ | 0.132 | 0.212
4.375 | 39.3 | 206 | 1961.22 | 0.03 | 0.48 $\times 10^{-1}$ | 0.766 $\times 10^{-1}$ | 0.124
4.625 | 34.7 | 213 | 2222.35 | 0.02 | 0.25 $\times 10^{-1}$ | 0.435 $\times 10^{-1}$ | 0.710 $\times 10^{-1}$
4.875 | 30.6 | 221 | 2518.26 | 0.01 | 0.13 $\times 10^{-1}$ | 0.244 $\times 10^{-1}$ | 0.400 $\times 10^{-1}$
5.125 | 27.0 | 229 | 2853.56 | 0.01 | 0.62 $\times 10^{-2}$ | 0.135 $\times 10^{-1}$ | 0.222 $\times 10^{-1}$

Table 3: The effective photon flux $f_{\rm Pb}$ and $f_{\rm incoh},f_{\rm
coh}$ radiated in the case of $\Upsilon$ production at proton-nucleon
collision energy $\sqrt{s_{pN}}=8.16$ TeV by the lead ion and by the proton
beam.

$Y_{\rm lab}$ |  $W_{\rm Pb}$ (GeV) | $f_{\rm Pb}$ |  $W_{p}$ (GeV) | $f_{\rm incoh}$ | $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}$ |  $f_{\rm incoh}$ \+ $f_{\rm coh}(R_{A}=1)$
---|---|---|---|---|---|---|---
-5.125 | 2600 | 0.102 $\times 10^{-1}$ | 9.71 | 5.80 | 1.4 | 7.19 | 4.94
-4.875 | 2300 | 0.577 $\times 10^{-1}$ | 11.00 | 5.37 | 3.3 | 8.70 | 5.99
-4.625 | 2030 | 0.230 | 12.47 | 4.63 | 5.0 | 9.62 | 7.11
-4.375 | 1790 | 0.695 | 14.13 | 3.76 | 5.6 | 9.36 | 7.86
-4.125 | 1580 | 1.68 | 16.01 | 2.91 | 5.2 | 8.15 | 8.11
-3.875 | 1390 | 3.43 | 18.14 | 2.21 | 4.4 | 6.64 | 7.97
-3.625 | 1230 | 6.10 | 20.55 | 1.68 | 3.6 | 5.27 | 7.56
-3.375 | 1080 | 9.73 | 23.29 | 1.30 | 2.8 | 4.13 | 7.01
-3.125 | 957 | 14.3 | 26.39 | 1.01 | 2.2 | 3.25 | 6.39
-2.875 | 845 | 19.7 | 29.90 | 0.80 | 1.8 | 2.57 | 5.76
-2.625 | 746 | 25.7 | 33.89 | 0.66 | 1.4 | 2.09 | 5.14
-2.375 | 658 | 32.2 | 38.40 | 0.55 | 1.2 | 1.72 | 4.55
-2.125 | 581 | 39.2 | 43.51 | 0.46 | 0.97 | 1.43 | 4.00
-1.875 | 512 | 46.4 | 49.30 | 0.40 | 0.81 | 1.21 | 3.49
-1.625 | 452 | 53.7 | 55.87 | 0.34 | 0.68 | 1.03 | 3.03
-1.375 | 399 | 61.3 | 63.31 | 0.30 | 0.58 | 0.880 | 2.62
-1.125 | 352 | 68.9 | 71.74 | 0.27 | 0.49 | 0.758 | 2.25
-0.875 | 311 | 76.6 | 81.29 | 0.24 | 0.42 | 0.653 | 1.92
-0.625 | 274 | 84.3 | 92.11 | 0.21 | 0.35 | 0.562 | 1.63
-0.375 | 242 | 92.0 | 104.37 | 0.19 | 0.30 | 0.484 | 1.38
-0.125 | 214 | 99.8 | 118.27 | 0.17 | 0.25 | 0.415 | 1.16
0.125 | 188 | 108 | 134.02 | 0.15 | 0.21 | 0.356 | 0.974
0.375 | 166 | 115 | 151.86 | 0.13 | 0.17 | 0.303 | 0.814
0.625 | 147 | 123 | 172.08 | 0.12 | 0.14 | 0.258 | 0.678
0.875 | 130 | 131 | 195.00 | 0.10 | 0.12 | 0.218 | 0.564
1.125 | 114 | 139 | 220.96 | 0.09 | 0.95 $\times 10^{-1}$ | 0.184 | 0.467
1.375 | 101 | 146 | 250.38 | 0.08 | 0.77 $\times 10^{-1}$ | 0.155 | 0.387
1.625 | 89.0 | 154 | 283.72 | 0.07 | 0.62 $\times 10^{-1}$ | 0.130 | 0.319
1.875 | 78.6 | 162 | 321.50 | 0.06 | 0.49 $\times 10^{-1}$ | 0.109 | 0.263
2.125 | 69.3 | 170 | 364.30 | 0.05 | 0.39 $\times 10^{-1}$ | 0.904 $\times 10^{-1}$ | 0.217
2.375 | 61.2 | 178 | 412.81 | 0.04 | 0.31 $\times 10^{-1}$ | 0.750 $\times 10^{-1}$ | 0.178
2.625 | 54.0 | 185 | 467.77 | 0.04 | 0.24 $\times 10^{-1}$ | 0.619 $\times 10^{-1}$ | 0.145
2.875 | 47.7 | 193 | 530.06 | 0.03 | 0.19 $\times 10^{-1}$ | 0.508 $\times 10^{-1}$ | 0.119
3.125 | 42.1 | 201 | 600.63 | 0.03 | 0.14 $\times 10^{-1}$ | 0.416 $\times 10^{-1}$ | 0.963 $\times 10^{-1}$
3.375 | 37.1 | 209 | 680.61 | 0.02 | 0.11 $\times 10^{-1}$ | 0.338 $\times 10^{-1}$ | 0.778 $\times 10^{-1}$
3.625 | 32.8 | 216 | 771.23 | 0.02 | 0.83 $\times 10^{-2}$ | 0.273 $\times 10^{-1}$ | 0.627 $\times 10^{-1}$
3.875 | 28.9 | 224 | 873.92 | 0.02 | 0.62 $\times 10^{-2}$ | 0.219 $\times 10^{-1}$ | 0.501 $\times 10^{-1}$
4.125 | 25.5 | 232 | 990.28 | 0.01 | 0.46 $\times 10^{-2}$ | 0.174 $\times 10^{-1}$ | 0.398 $\times 10^{-1}$
4.375 | 22.5 | 240 | 1122.13 | 0.01 | 0.33 $\times 10^{-2}$ | 0.138 $\times 10^{-1}$ | 0.315 $\times 10^{-1}$
4.625 | 19.9 | 248 | 1271.54 | 0.01 | 0.24 $\times 10^{-2}$ | 0.108 $\times 10^{-1}$ | 0.248 $\times 10^{-1}$
4.875 | 17.5 | 255 | 1440.85 | 0.01 | 0.17 $\times 10^{-2}$ | 0.852 $\times 10^{-2}$ | 0.195 $\times 10^{-1}$
5.125 | 15.5 | 263 | 1632.69 | 0.01 | 0.12 $\times 10^{-2}$ | 0.663 $\times 10^{-2}$ | 0.152 $\times 10^{-1}$

Table 4: The effective photon flux $f_{\rm Pb}$ and $f_{\rm incoh},f_{\rm
coh}$ radiated in the case of $J/\psi$ production at proton-nucleon collision
energy $\sqrt{s_{pN}}=8.16$ TeV by the lead ion and by the proton beam.

## Acknowledgements

We thank Petja Paakkinen for providing the numerical values for the EPPS16
gluon modification factor at the scales $\mu^{2}=m_{c}^{2}$ and
$\mu^{2}=m_{b}^{2}$ used in this work. CAF is supported by the Helsinki
Institute of Physics core funding project QCD-THEORY. SPJ is supported by a
Royal Society University Research Fellowship (Grant URF/R1/201268). The work
of TT is supported by the STFC Consolidated Grant ST/T000988/1. This work is
also supported in part by the STFC Grant ST/T001011/1.

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