# Photoproduction of $J/\psi$ in d+Au ultraperipheral collisions at the BNL
Relativistic Heavy Ion Collider

V. Guzey  National Research Center “Kurchatov Institute”, Petersburg Nuclear
Physics Institute (PNPI), Gatchina, 188300, Russia  E. Kryshen  National
Research Center “Kurchatov Institute”, Petersburg Nuclear Physics Institute
(PNPI), Gatchina, 188300, Russia  M. Strikman  Pennsylvania State University,
University Park, PA, 16802, USA  M. Zhalov  National Research Center
“Kurchatov Institute”, Petersburg Nuclear Physics Institute (PNPI), Gatchina,
188300, Russia

###### Abstract

We argue that the STAR data on $J/\psi$ photoproduction in $d+Au$
ultraperipheral collisions (UPCs) at the momentum transfer $t\neq 0$ primarily
probes elastic and possibly nucleon-dissociative $J/\psi$ photoproduction on
quasi-free nucleons, while being essentially insensitive to expectedly weak
nuclear modifications of the gluon distribution in the deuteron. Analyzing the
STAR detector capabilities for selection of UPC events, we explore the
possibility that the contribution of nucleon-dissociative $J/\psi$
photoproduction is negligibly small and obtain a very good description of the
$t$ dependence of the STAR total cross section.

Deuteron-nucleus relativistic scattering, ultraperipheral collisions, $J/\psi$
photoproduction

###### pacs:

24.85.+p, 25.20.-x,25.75.-q

##  I Introduction

Ultraperipheral collisions (UPCs) of relativistic ions have become a standard
tool for studies of quantum chromodynamics (QCD) Baltz:2007kq ; Klein:2019qfb
. In particular, photoproduction of $J/\psi$ vector mesons off heavy nuclear
targets in Pb-Pb UPCs at the Large Hadron Collider (LHC) provided new
constraints on the nuclear gluon distribution at small $x$, see Ref.
Guzey:2020ntc  for references to the experimental data. These results have
been interpreted within the leading twist approximation (LTA) Guzey:2013xba ;
Guzey:2013qza ; Guzey:2020ntc  and also in terms of proton shape fluctuations
in the context of the color glass condensate (CGC) framework and the color
dipole model Mantysaari:2017dwh ; Mantysaari:2020axf .

Recently the STAR collaboration at Brookhaven National Laboratory (BNL) for
the first time measured $J/\psi$ photoproduction in $d+Au$ UPCs at
$\sqrt{s_{NN}}=200$ GeV STAR:2021wwq . Since the flux of equivalent photons
from ions of Au is much larger than that from the deuteron (the photon flux
scales as $Z^{2}$ with $Z$ being the ion electric charge), these data are
treated as photoproduction off the deuteron target. The results for the
differential cross section as a function of the momentum transfer squared $t$
were compared in STAR:2021wwq  to theoretical predictions of the two
approaches mentioned above and it was found that the CGC saturation model
provides a better agreement, primarily at large $|t|$. Note that the
theoretical predictions were made for generic RHIC kinematics and without
taking into account the specifics of STAR detector capabilities for selection
of UPC events.

The purpose of this paper is to demonstrate that since the STAR $d+Au$ UPC
data corresponds to the photon-nucleon invariant energy $W\approx 25$ GeV and,
hence, to the parton momentum fraction $x\approx 0.015$, nuclear modifications
of the gluon distribution in the deuteron expectedly play an insignificant
role in interpretation of the data, see also the discussion in STAR:2021wwq ;
the enhancement in the $t\to 0$ limit is well-understood and explained by
nuclear coherence of the gluon density of the deuteron. Instead, the data at
$t\neq 0$ is sensitive primarily to elastic and possibly nucleon-dissociative
$J/\psi$ photoproduction on quasi-free nucleons, which were previously studied
at HERA at high $W$. In particular, we explore the scenario that the dominant
range of the produced nucleon-dissociative masses and the corresponding
rapidity interval were vetoed by the STAR detector system selecting the UPC
events, set the nucleon-dissociative contribution to zero, and obtain a very
good agreement with the STAR data on the total cross section over the entire
range of $t$. We also suggest that the difference between the LTA and CGC
predictions could come from different extrapolations of the elastic and
nucleon-dissociative $J/\psi$ photoproduction cross sections on the nucleon
target from HERA to RHIC energies.

The remainder of the paper is organized as follows. In Sec. II, within the
LTA, we present the derivation of the cross section of $J/\psi$
photoproduction off the deuteron, when the nuclear target can stay intact or
dissociate (so-called summed cross section). Our predictions for the $t$
dependence of the $d+Au$ UPC cross section and their comparison to the STAR
data are discussed in Sec. III. Finally, we summarize our findings in Sec. IV.

##  II Summed cross section off the deuteron

Applying the formalism of Ref. Guzey:2018tlk  to $J/\psi$ photoproduction on
the deuteron and using completeness of final nuclear states, one obtains the
following expression for the summed (coherent+incoherent) $\gamma+d\to
J/\psi+X$ differential cross section

$\displaystyle\frac{d\sigma_{d}^{\rm sum}(t)}{dt}=$
$\displaystyle\frac{\chi^{2}}{4\pi}\int d^{2}{\vec{b}}\int
d^{2}{\vec{b}^{\prime}}e^{i\vec{q}\cdot({\vec{b}}-\vec{b^{\prime}})}\langle
0|\Gamma_{d}^{\dagger}(\vec{b^{\prime}},\vec{r})\Gamma_{d}(\vec{b},\vec{r})|0\rangle\,,$
(1)

where $\chi$ is proportional to the $\gamma-J/\psi$ transition amplitude. In
Eq. (1), $\Gamma_{d}({\vec{b}},\vec{r})$ is the scattering amplitude on the
deuteron in impact parameter space; $\vec{b}$ and ${\vec{q}}$ are two-
dimensional vectors of the impact parameter and the momentum transfer,
respectively; the latter is taken to be transverse, $t=-|{\vec{q}}|^{2}$. The
matrix element $\langle 0|\dots|0\rangle$ stands for the integration with the
deuteron ground-state wave function $\Psi_{d}(\vec{r})$ squared

$\langle
0|\Gamma_{d}^{\dagger}(\vec{b^{\prime}},\vec{r})\Gamma_{d}(\vec{b},\vec{r})|0\rangle=\int
d^{3}\vec{r}\,|\Psi_{d}(\vec{r})|^{2}\Gamma_{d}^{\dagger}(\vec{b^{\prime}},\vec{r})\Gamma_{d}(\vec{b},\vec{r})\,,$
(2)

where $\vec{r}$ is the relative distance between the proton and the neutron in
the deuteron; averaging over the deuteron polarization is implied. In our
analysis, we used the Paris deuteron wave function Lacombe:1981eg .

In the Glauber multiple scattering approach Bauer:1977iq , the $\gamma+d\to
J/\psi+X$ nuclear scattering amplitude $\Gamma_{d}(\vec{b},\vec{r})$ can be
expressed in terms of the nucleon scattering amplitudes $\Gamma_{N}$
corresponding to the $\gamma+N\to J/\psi+N$ process (elastic production) and
$\Gamma_{Y}$ corresponding to the $\gamma+N\to J/\psi+Y$ process (nucleon
dissociation), respectively, as follows (we do not distinguish protons and
neutrons)

$\displaystyle\Gamma_{d}(\vec{b},\vec{r})$ $\displaystyle=$
$\displaystyle\Gamma_{N}(\vec{b}-\frac{\vec{r}}{2})+\Gamma_{N}(\vec{b}+\frac{\vec{r}}{2})-\Gamma_{N}(\vec{b}-\frac{\vec{r}}{2})\Gamma_{N}(\vec{b}+\frac{\vec{r}}{2})$
(3) $\displaystyle+$
$\displaystyle\Gamma_{Y}(\vec{b}-\frac{\vec{r}}{2})+\Gamma_{Y}(\vec{b}+\frac{\vec{r}}{2})-\Gamma_{N}(\vec{b}-\frac{\vec{r}}{2})\Gamma_{Y}(\vec{b}+\frac{\vec{r}}{2})$
$\displaystyle-$
$\displaystyle\Gamma_{Y}(\vec{b}-\frac{\vec{r}}{2})\Gamma_{N}(\vec{b}+\frac{\vec{r}}{2})-\Gamma_{Y}(\vec{b}-\frac{\vec{r}}{2})\Gamma_{Y}(\vec{b}+\frac{\vec{r}}{2})\,,$

where $\vec{b}-\vec{r}/2$ and $\vec{b}+\vec{r}/2$ denote the coordinates of
the two nucleons in the deuteron. Equation (3) corresponds to the interaction
with one and two nucleons of the nuclear target and generalizes the classic
result of the Gribov-Glauber theory of nuclear shadowing for hadron-deuteron
scattering Glauber:1955qq ; Gribov:1968jf  by including the contribution of
nucleon dissociation.

The nucleon scattering amplitudes are parametrized in the following form
Guzey:2018tlk

$\displaystyle\Gamma_{N}(\vec{b})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{16\pi d\sigma_{\rm el}(t=0)/dt}}{4\pi B_{\rm
el}\chi}e^{-|\vec{b}|^{2}/(2B_{\rm el})}\,,$
$\displaystyle\Gamma_{Y}(\vec{b})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{16\pi d\sigma_{\rm
diss}(t=0)/dt}}{4\pi\chi}\int\frac{d^{2}\vec{q^{\prime}}}{2\pi}e^{-i\vec{q^{\prime}}\cdot\vec{b}}f_{pd}(t^{\prime})\,,$
(4)

where $d\sigma_{\rm el}(t)/dt$ is the cross section of the $\gamma+N\to
J/\psi+N$ elastic production on the nucleon and $B_{\rm el}=4.3\pm 0.2$ GeV-2
Alexa:2013xxa  is the slope of its $t$ dependence; $d\sigma_{\rm diss}(t)/dt$
is the cross section of the nucleon-dissociative $J/\psi$ photoproduction
$\gamma+N\to J/\psi+Y$ with $f_{pd}(t)$ giving its $t$ dependence, see details
below.

Substituting Eq. (3) in Eq. (1), one notices that the resulting terms can be
grouped together as follows (we need to consider only the terms even in powers
of $\Gamma_{Y}$ since the terms odd in $\Gamma_{Y}$ do not contribute to the
matrix element in Eq. (2)). The terms
$\Gamma_{N}^{\dagger}(\vec{b^{\prime}}\pm\vec{r}/2)\Gamma_{N}(\vec{b}\pm\vec{r}/2)$
decouple from the integration with $|\Psi_{d}(\vec{r})|^{2}$ and correspond to
the $d\sigma_{\rm el}(t)/dt$ cross section, while the terms
$\Gamma_{Y}^{\dagger}(\vec{b^{\prime}}\pm\vec{r}/2)\Gamma_{Y}(\vec{b}\pm\vec{r}/2)$
correspond to $d\sigma_{\rm diss}/dt$. The interference terms
$\Gamma_{N}^{\dagger}(\vec{b^{\prime}}\pm\vec{r}/2)\Gamma_{N}(\vec{b}\mp\vec{r}/2)$,
when integrated with the deuteron wave function squared, result in the product
of the deuteron charge form factor $F_{d}(4t)$ and the $d\sigma_{\rm
el}(t)/dt$ cross section on the nucleon. Note that $4t$-argument of the
deuteron form factor is a characteristic feature of the summed cross section
and the nuclear shadowing correction, which originates from taking into
account the effect of the deuteron center of mass. A similar contribution
results from the
$\Gamma_{Y}^{\dagger}(\vec{b^{\prime}}\pm\vec{r}/2)\Gamma_{Y}(\vec{b}\mp\vec{r}/2)$
interference terms.

The higher terms ${\cal O}(\Gamma_{N}^{3})$ give a small negative contribution
to the summed cross section and, hence, describe the effect of nuclear
shadowing. While their effect is negligibly small in the considered
kinematics, it is still instructive to present the LTA shadowing correction to
the summed nuclear cross section since it has not been discussed in the
literature. For a review and extensive references on early results on
photoproduction of light and heavy vector mesons off light and heavy nuclei,
see Ref. Bauer:1977iq . To understand the structure of the answer, one notices
that, e.g., in the
$\Gamma_{N}^{\dagger}(\vec{b^{\prime}}-\vec{r}/2)\Gamma_{N}(\vec{b}-\vec{r}/2)\Gamma_{N}(\vec{b}+\vec{r}/2)$
term, the first factor is proportional to the elastic $\gamma+N\to J/\psi+N$
amplitude, while the product of the last two factors is proportional to the
$(d\sigma_{\rm diff}/dt)/\sigma_{\rm tot}$ ratio of the diffractive and total
cross sections of inclusive deep inelastic scattering (DIS) on the nucleon
$\gamma^{\ast}+N\to X+N$ Frankfurt:2011cs .

Neglecting a vanishingly small contribution ${\cal O}(\Gamma_{Y}^{2})$ to the
shadowing term, one obtains the following expression for the summed cross
section

$\displaystyle\frac{d\sigma_{d}^{\rm
sum}(t)}{dt}=2\left(1+F_{d}(4t)\right)\left(\frac{d\sigma_{\rm
el}(t)}{dt}+\frac{d\sigma_{\rm diss}(t)}{dt}\right)$ (5) $\displaystyle-$
$\displaystyle 8\frac{d\sigma_{\rm el}(t=0)}{dt}e^{-B_{el}t/2}$
$\displaystyle\times$ $\displaystyle\frac{d\sigma_{\rm
diff}/dt(t=0)}{\sigma_{\rm
tot}}\int\frac{d^{2}\vec{q^{\prime}}}{2\pi}e^{-B(|\vec{q^{\prime}}|^{2}+\vec{q^{\prime}}\cdot\vec{q}+t/2)}F_{d}(4t^{\prime})\,,$

where $B\approx 6$ GeV-2 H1:2006uea  is the slope of the $t$ dependence of the
diffractive $\gamma^{\ast}+N\to X+N$ differential cross section. Note that the
deuteron quadrupole form factor does not contribute to the summed cross
section, see Ref. Franco:1969qj .

Equation (5) has a transparent physical interpretation. The first line is the
impulse approximation (IA), where the overall factor of two reflects the
contributions of two nucleons (proton and neutron) in the deuteron and the
factor of $F_{d}(4t)$ originates from the interference of the proton and
neutron contributions at small $t$. Note that the slope of the $t$ dependence
of this term is twice as large as that for the coherent cross section on the
deuteron in IA. In the $t\to 0$ limit, $d\sigma_{d}^{\rm
sum}(t)/dt=4(d\sigma_{\rm el}(t)/dt+d\sigma_{\rm diss}(t)/dt)$. The second and
third lines in Eq. (5) give the negative shadowing correction, whose strength
is driven by the $(d\sigma_{\rm diff}/dt)/\sigma_{\rm tot}$ ratio and whose
$t$ dependence is determined by the geometry of the photon-deuteron scattering
in transverse plane, i.e., by the deuteron form factor and the slopes $B_{el}$
and $B$.

In LTA, the shadowing correction in $J/\psi$ photoproduction can be expressed
in terms of the nuclear shadowing factor for the gluon distribution in the
deuteron $g_{d}(x,Q^{2})$ at small $x$. The latter is given by the following
relation Frankfurt:2011cs ,

$\displaystyle S_{g}(x)$ $\displaystyle=$
$\displaystyle\frac{g_{d}(x,Q^{2})}{2g_{N}(x,Q^{2})}$ (6) $\displaystyle=$
$\displaystyle 1-\frac{d\sigma_{\rm diff}/dt(t=0)}{\sigma_{\rm tot}}\int
dt^{\prime}e^{-Bt^{\prime}}F_{d}(4t^{\prime})\,,$

where $g_{N}(x,Q^{2})$ is the free nucleon gluon distribution. Note that all
involved cross sections implicitly depend on energy (Bjorken $x$).

Equation (6) allows us to rewrite the expression for the summed cross section
in the following final compact form

$\displaystyle\frac{d\sigma_{d}^{\rm sum}(t)}{dt}$ $\displaystyle=$
$\displaystyle 2\left(1+F_{d}(4t)\right)\left(\frac{d\sigma_{\rm
el}(t)}{dt}+\frac{d\sigma_{\rm diss}(t)}{dt}\right)$ (7) $\displaystyle-$
$\displaystyle 8\frac{d\sigma_{\rm el}(t=0)}{dt}(1-S_{g}(x))e^{-(B_{\rm
el}+B)t/2}\,,$

where we neglected the numerically unimportant
$e^{-B\vec{q^{\prime}}\cdot\vec{q}}$ term in Eq. (5).

Since the deuteron is the lightest nucleus and the STAR kinematics corresponds
to $W\approx 25$ GeV STAR:2021wwq , $x=M_{J/\psi}^{2}/W^{2}\approx 0.015$, and
$Q^{2}\approx M_{J/\psi}^{2}/4\approx 2.4-3$ GeV2 ($M_{J/\psi}$ is the mass of
$J/\psi$ and $Q$ is a generic value of the resolution scale probed in $J/\psi$
photoproduction), the gluon shadowing correction in this case is very small,
$S_{g}(x)=0.99$. i.e., on the order of 1% Frankfurt:2011cs . This explicit
numerical estimate supports a similar conclusion in Ref. STAR:2021wwq .

##  III LTA predictions for $t$ dependence of $d+Au$ UPC cross section and
comparison to STAR data

It is customary to present UPC cross sections as a function of the rapidity of
the produced final state, in the considered case, the $J/\psi$ rapidity $y$.
In the STAR notation, the UPC cross section and the cross section at the
photon level are related via the photon flux $\Phi_{T,\gamma}$

$\frac{d^{2}\sigma^{d+Au\to
J/\psi+X}}{dtdy}=\Phi_{T,\gamma}\frac{d^{2}\sigma^{\gamma+d\to
J/\psi+X}}{dtdy}\,,$ (8)

where $t=-|\vec{p}_{T}|^{2}$ and $\vec{p}_{T}$ is the $J/\psi$ transverse
momentum. Note that $J/\psi$ photoproduction on the gold nuclei can be safely
neglected because the photon flux due to the deuteron ions is suppressed
approximately by the factor of $1/(79^{2})=1/6240$ compared to that due to the
gold ions. Since the STAR results are given for $d^{2}\sigma^{\gamma+d\to
J/\psi+X}/(dtdy)$, our predictions can be readily compared to the data. This
is presented in Fig. 1, which shows predictions of Eq. (7) for the summed
$\gamma+d\to J/\psi+X$ cross section as a function of $|t|$ (blue dot-dashed
curve) and the STAR data STAR:2021wwq  for the total cross section (filled
circles). The error bars represent statistical and systematic errors added in
quadrature. A comparison of the LTA prediction to the filled circles shows
that while the theory describes well the data within experimental errors at
small $|t|$, it significantly overestimates the data for $|t|>0.2$ GeV2.

This discrepancy cannot be attributed to the gluon nuclear shadowing since it
affects the summed cross section only at the level of a few percent. In fact,
with $1-2$% precision, Eq. (5) for $|t|>0.2$ GeV2 can be written in the
following approximate simple form

$\frac{d\sigma_{d}^{\rm sum}(t)}{dt}\approx 2\frac{d\sigma_{\rm
el}(t)}{dt}+2\frac{d\sigma_{\rm diss}(t)}{dt}\,,$ (9)

thus emphasizing that the gluon nuclear shadowing or details of the nuclear
structure (deuteron wave function) do not play a role here.

Figure 1: The summed $\gamma+d\to J/\psi+X$ differential cross section as a
function of $|t|$: the LTA predictions (blue dot-dashed curve) and the STAR
data STAR:2021wwq  for the total cross section. The experimental statistical
and systematic errors have been added in quadrature.

At this point it is important to discuss the input for the elastic and proton-
dissociative cross sections of $J/\psi$ photoproduction on the nucleon. In our
analysis, we used the low-energy (LE) H1 parametrization based on the data
covering the $25\ {\rm GeV}<W<80$ GeV interval Alexa:2013xxa . In particular,
we combined results of the fit for the $W$ dependence in Table 3 of Ref.
Alexa:2013xxa  with that for the $t$ dependence in Table 2. The resulting
parameterization reads

$\displaystyle\frac{d\sigma_{\rm el}(t)}{dt}$ $\displaystyle=$ $\displaystyle
N_{el}\left(\frac{W}{90}\right)^{\delta_{el}}b_{el}e^{-b_{el}|t|}\,,$
$\displaystyle\frac{d\sigma_{\rm diss}(t)}{dt}$ $\displaystyle=$
$\displaystyle
N_{pd}\left(\frac{W}{90}\right)^{\delta_{pd}}\frac{f_{pd}(t)}{\int
dt^{\prime}f_{pd}(t^{\prime})}\,,$ (10)

where $N_{el}=81\pm 3$ nb, $\delta_{el}=0.67\pm 0.03$, and $b_{el}=4.3\pm
0.02$ GeV-2 for the elastic case; $N_{pd}=62\pm 12$ nb, $\delta_{pd}=0.42\pm
0.05$, and $f_{pd}(t)=1/(1+b_{pd}|t|/n)^{n}$ with $n=3.58$ and $b_{pd}=1.6\pm
0.2$ GeV-2 for the nucleon-dissociative case.

Note that the H1 analysis included the proton-dissociative masses in the range
$m_{p}<M_{Y}<10$ GeV Alexa:2013xxa . A simple estimate in the case of the RHIC
kinematics shows that the nucleon-dissociative final state covers the rapidity
interval $\Delta y=(1/2)\ln[(M_{Y}^{2}+|t|)/(m_{p}^{2}+|t|)]\approx 2$, which
is adjacent to the $|y_{N}|=\ln[2E_{N}/(m_{p}^{2}+|t|)^{1/2}]\approx 5$
rapidity of the final-state nucleon scattered quasi-elastically. When
comparing our calculations to the RHIC UPC data, one has to keep in mind that
in selection of UPC events, the STAR detector covers a wide range of
rapidities vetoing events with dissociation. This includes $|y|<1$ covered by
the Time Projection Chamber (TPC), $3.4<|y|<5$ covered by two Beam-Beam
Counters (BBCs), and Zero Degree Calorimeters (ZDCs) tagging forward neutrons.
Therefore, the nucleon-dissociative state $Y$ is predominantly produced in the
rapidity region covered by the STAR detector and, hence, can be effectively
rejected.

To explore the possibility that the range of $M_{Y}$ included in the STAR
analysis is significantly smaller than that in the H1 case, we set
$d\sigma_{\rm diss}/dt=0$ in Eq. (7). Note that since $d\sigma_{\rm
diss}/dt\gg d\sigma_{\rm el}/dt$ for sufficiently large $|t|$, the discrepancy
mentioned above should originate from modeling of $d\sigma_{\rm diss}/dt$. The
resulting prediction is given by the red solid curve in Fig. 2 and leads to a
very good agreement with the STAR data over the entire range of $t$. For
comparison, we also show our result from Fig. 1 by the blue dot-dashed curve,
which includes the nucleon-dissociative events.

Figure 2:  The summed $\gamma+d\to J/\psi+X$ differential cross section as a
function of $|t|$: the LTA predictions with the $d\sigma_{\rm diss}/dt=0$
scenario (red solid curve) vs. the STAR data STAR:2021wwq  for the total cross
section (filled circles). The data for the neutron-tagged cross section are
shown by open circles. Our result from Fig. 1 is given by the blue dot-dashed
curve.

Note that the H1 measurement of proton-dissociative $J/\psi$ photoproduction
Alexa:2013xxa  is the only collider data in the relevant range of $W=25$ GeV;
the earlier ZEUS data ZEUS:2002vvv  correspond to higher $W$, $80\ {\rm
GeV}<W<120$ GeV.

The comparison of the LTA results to the RHIC data presented in Fig. 2 should
be taken with some degree of caution. First, the RHIC data corresponds to the
rapidity range $|y|<1$, where the UPC cross section of $J/\psi$
photoproduction at RHIC energies has a rapid $y$ dependence, see, e.g.
Strikman:2005ze ; Frankfurt:2006tp . Second, to validate our hypothesis of the
absence of the nucleon-dissociative contribution, it is important to model the
distribution over the momentum of the recoiled nucleons and to utilize the
“n-tagged data” (open circles in Fig. 2), which is beyond the scope of the
present article.

To explain the latter point, we notice that the incoherent cross section is
given by the difference between the summed and coherent cross sections, which
in IA reads

$\displaystyle\frac{d\sigma_{d}^{\rm inc}(t)}{dt}$ $\displaystyle=$
$\displaystyle\frac{d\sigma_{d}^{\rm sum}(t)}{dt}-\frac{d\sigma_{d}^{\rm
coh}(t)}{dt}$ (11) $\displaystyle=$ $\displaystyle 2\frac{d\sigma_{\rm
el}(t)}{dt}\left(1+F_{d}(4t)-2|F_{d}(t)|^{2}\right)$ $\displaystyle+$
$\displaystyle 2\frac{d\sigma_{\rm diss}(t)}{dt}\left(1+F_{d}(4t)\right)\,.$

Thus, our hypothesis of the absence of a noticeable nucleon-dissociative
contribution seems to be supported by “n-tagged data” for $|t|<0.1$ GeV2 since
the difference between the solid and open circles in Fig. 2 can be explained
by the contribution of coherent $J/\psi$ photoproduction off the deuteron. In
the $0.1<|t|<0.4$ GeV2 range, the measured cross sections with and without
neutron-tagging agree within experimental errors and are also described by our
calculations neglecting the nucleon-dissociation contribution. At the same
time, the data at $|t|>0.4$ GeV2 require further analysis.

We discussed in the Introduction that the STAR data on $J/\psi$
photoproduction in $d+Au$ UPCs is described well within the CGC framework with
nucleon shape fluctuations Mantysaari:2019jhh . We also explained above that
the effect of leading twist gluon nuclear shadowing as well as details of the
deuteron structure are largely unimportant for $|t|\geq 0.2$ GeV and that the
summed cross section on the deuteron should be reproduced sufficiently
accurately by approximate Eq. (9). Therefore, the difference in predictions of
the LTA and CGC approaches could come from different extrapolations of the
$d\sigma_{\rm el}(t)/dt$ and $d\sigma_{\rm diss}(t)/dt$ cross sections on the
nucleon from HERA to RHIC energies. Note also that the CGC approach considers
cross sections integrated over $M_{Y}$, which makes it challenging to do a
comparison with the data containing various cuts on $M_{Y}$.

##  IV Summary

In this paper, we argue that the STAR data on $J/\psi$ photoproduction in
$d+Au$ UPCs for $t\neq 0$ primarily probes elastic and probably nucleon-
dissociative $J/\psi$ photoproduction on quasi-free nucleons and is
essentially insensitive to nuclear modifications of the gluon distribution in
the deuteron. Exploring the possibility that the nucleon-dissociative
contribution is negligibly small, we obtain a very good description of the $t$
dependence of the STAR total cross section. We point out that the difference
between the LTA and CGC predictions could come from different extrapolations
of the elastic and nucleon-dissociative $J/\psi$ photoproduction cross
sections on the nucleon from HERA to RHIC energies.

###### Acknowledgements.

The authors would like to thank Zhoudunming Tu for stimulating our interest in
this work, discussion of the STAR results, and useful comments on an earlier
draft of this letter. The research of M.S. was supported by the US Department
of Energy Office of Science, Office of Nuclear Physics under Award No. DE-
FG02-93ER40771. M.S. thanks Theory Division of CERN for hospitality while this
work was done.

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