# Towards clarifying the possibility of observation of the LHCb hidden-charm
pentaquarks
$P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$, $P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$
in near-threshold charmonium photoproduction off protons and nuclei

E. Ya. Paryev
Institute for Nuclear Research, Russian Academy of Sciences,
Moscow 117312, Russia

###### Abstract

We study the near-threshold $J/\psi$ meson photoproduction from protons and
nuclei by considering incoherent direct non-resonant (${\gamma}p\to{J/\psi}p$,
${\gamma}n\to{J/\psi}n$) and two-step resonant (${\gamma}p\to
P_{ci}^{+}\to{J/\psi}p$, ${\gamma}n\to P_{ci}^{0}\to{J/\psi}n$, $i=1$, 2, 3,
4; $P_{c1}^{+,0}=P_{c}^{+,0}(4312)$, $P_{c2}^{+,0}=P_{c}^{+,0}(4337)$,
$P_{c3}^{+,0}=P_{c}^{+,0}(4440)$, $P_{c4}^{+,0}=P_{c}^{+,0}(4457)$) charmonium
production processes. We calculate the absolute excitation functions, energy
and momentum distributions for the non-resonant, resonant and for the combined
(non-resonant plus resonant) production of $J/\psi$ mesons on protons as well
as, using the nuclear spectral function approach, on carbon and tungsten
target nuclei at near-threshold incident photon energies by assuming the spin-
parity assignments of the hidden-charm resonances $P_{c}^{+,0}(4312)$,
$P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$ and $P_{c}^{+,0}(4457)$ as
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$
within three different realistic scenarios for the branching ratios of their
decays to the ${J/\psi}p$ and ${J/\psi}n$ modes (0.25, 0.5 and 1%). We show
that will be very hard to measure the $P_{ci}^{+}$ pentaquark states through
the scan of the $J/\psi$ total photoproduction cross section on a proton
target in the near-threshold energy region around the resonant photon energies
of 9.44, 9.554, 10.04 and 10.12 GeV if these branching ratios $\sim$ 1% and
less. We also demonstrate that at these photon beam energies the $J/\psi$
energy and momentum combined distributions considered reveal distinct
sensitivity to the above scenarios, respectively, at ”low” $J/\psi$ total
energies and momenta, which implies that they may be an important tool to
provide further evidence for the existence of the pentaquark $P_{ci}^{+}$ and
$P_{ci}^{0}$ resonances and to get valuable information on their decay rates
to the ${J/\psi}p$ and ${J/\psi}n$ final states. The measurements of these
distributions could be performed in the future at the 12 GeV CEBAF facility.

## 1\. Introduction

The study of the exotic hadronic states – the hidden-charm pentaquark
resonances – has received considerable interest in recent years and is one of
the most exciting topics of the nuclear and hadronic physics nowadays after
the discovery by the LHCb Collaboration pentaquark states $P_{c}^{+}(4380)$
and $P_{c}^{+}(4450)$ in the ${J/\psi}p$ invariant mass spectrum of the
$\Lambda^{0}_{b}\to K^{-}({J/\psi}p)$ decays [1] and, especially, after the
observation by the Collaboration of three new narrow resonances
$P_{c}^{+}(4312)$, $P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$ in these decays
[2], based on additional collected data and on an improved selection strategy,
instead of initially claimed $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$ states.
Very recently, the LHCb Collaboration discovered a new narrow hidden-charm
pentaquark denoted as $P_{c}^{+}(4337)$ in the invariant mass spectrum of the
${J/\psi}p$ in the $B_{s}^{0}\to{J/\psi}p{\bar{p}}$ decays [3]. On the other
hand, the search for the LHCb pentaquarks $P_{c}^{+}(4312)$, $P_{c}^{+}(4440)$
and $P_{c}^{+}(4457)$ by the GlueX Collaboration in Hall D at JLab through a
scan of the cross section of elastic reaction ${\gamma}p\to{J/\psi}p$ from
threshold of 8.21 GeV and up to incident photon energy $E_{\gamma}=11.8$ GeV
gave no evidence for them with present statistics and set the model-dependent
upper limits on branching ratios of $P_{c}^{+}(4312)\to{J/\psi}p$,
$P_{c}^{+}(4440)\to{J/\psi}p$ and $P_{c}^{+}(4457)\to{J/\psi}p$ decays of
several percent [4]. The preliminary results from a factor of 10 more data on
the $J/\psi$ photoproduction on a hydrogen target, collected in the Hall C
JLab E12-16-007 experiment (the so-called $J/\psi$-007 experiment), also show
no $P_{c}^{+}$ signal [5]. In this experiment the $e^{+}e^{-}$ pairs from the
$J/\psi$ decays were detected in coincidence using the two high momentum
spectrometers of Hall C: the Super High Momentum Spectrometer (SHMS) and High
Momentum Spectrometer (HMS) for the electron and the positron, respectively.
In recent publications [6] and [7] the role, respectively, of the LHCb
pentaquarks $P_{c}^{+}(4450)$ and $P_{c}^{+}(4312)$, $P_{c}^{+}(4440)$,
$P_{c}^{+}(4457)$ in charmonium photoproduction on protons and nuclei has been
investigated at near-threshold initial photon energies $E_{\gamma}\leq 11$
GeV. Here, the description was based on the consideration of the incoherent
direct (${\gamma}N\to{J/\psi}N$) and two-step (${\gamma}p\to
P_{c}^{+}(4450)\to{J/\psi}p$ and ${\gamma}p\to P_{c}^{+}(4312)\to{J/\psi}p$,
${\gamma}p\to P_{c}^{+}(4440)\to{J/\psi}p$, ${\gamma}p\to
P_{c}^{+}(4457)\to{J/\psi}p$) $J/\psi$ production processes. As a measure for
this role, the incident photon energy dependence of $J/\psi$ production cross
sections on protons and nuclei (excitation functions) has been adopted. It was
found that it is insignificant for the pentaquark resonances considered if
branching ratios of their decays to the ${J/\psi}p$ mode are less than a few
percent, which is in line with the results of the JLab experiments [4, 5].

In view of the aforementioned, to get a robust enough information for or
against the existence of the LHCb hidden-charm pentaquarks and to understand
their better, it is crucial to investigate the possibility of their
observation by measuring not only the excitation functions for $J/\psi$ meson
production from photon-induced reactions on protons and nuclei at near-
threshold photon energies, predicted in Refs. [6, 7], but also the $J/\psi$
energy and momentum distributions in these reactions, not predicted in the
previous papers [6, 7]. Their prediction is the main aim of the present study.
In it, we consider the contribution of the $P_{c}^{+,0}(4312)$,
$P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$ and $P_{c}^{+,0}(4457)$ resonances to
near-threshold $J/\psi$ photoproduction off protons and nuclei by adopting the
Breit-Wigner shape for this contribution and by employing the recent
experimental data [4] on the total and differential cross sections of the
${\gamma}p\to{J/\psi}p$ process to estimate the background contribution. The
consideration is mainly based on the model, developed in Refs. [6, 7]. We
present the predictions obtained within our present approach for the $J/\psi$
energy and momentum distributions in ${\gamma}p$ as well as in ${\gamma}$12C
and ${\gamma}$184W reactions at near-threshold incident photon energies. These
predictions may be useful in planning future $J/\psi$ photoproduction
experiments at the CEBAF facility.

## 2\. Theoretical framework

### 2.1. Direct non-resonant $J/\psi$ production processes

At near-threshold photon beam energies below 11 GeV of our interest 111)These
energies are well within the present capabilities of the upgraded CEBAF
facility at JLab, which is providing an opportunity to study the observed
[1–3] by the LHCb Collaboration exotic hidden-charm pentaquark states in
exclusive ${\gamma}p\to{J/\psi}p$ reactions in all experimental Halls A, B, C,
D [4, 5].), the following direct non-resonant elementary charmonium production
processes with the lowest free production threshold ($\approx$ 8.21 GeV)
contribute to the $J/\psi$ photoproduction on nuclei [6–8]:

$\gamma+p\to J/\psi+p,$ (1) $\gamma+n\to J/\psi+n.$ (2)

The modification of the masses of the final high-momentum $J/\psi$ meson and
proton (see below) in the nuclear medium will be ignored in the present study.

Disregarding the absorption of incident photons in the energy range of
interest as well as the $J/\psi$ meson quasielastic rescatterings on target
nucleons [9], and describing the charmonium final-state absorption in the
nuclear matter by the absorption cross section $\sigma_{{J/\psi}N}$ 222)For
which we will use the value $\sigma_{{J/\psi}N}=3.5$ mb [6–10].), we represent
the inclusive differential cross section for the production of $J/\psi$ mesons
with the momentum ${\bf p}_{J/\psi}$ on nuclei in the direct non-resonant
processes (1), (2) in the form [6–10]:

$\frac{d\sigma_{{\gamma}A\to{J/\psi}X}^{({\rm dir})}({\bf p}_{\gamma},{\bf
p}_{J/\psi})}{d{\bf
p}_{J/\psi}}=I_{V}[A,\sigma_{{J/\psi}N}]\left<\frac{d\sigma_{{\gamma}p\to{J/\psi}p}({\bf
p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right>_{A},$ (3)

where

$I_{V}[A,\sigma]=2{\pi}A\int\limits_{0}^{R}r_{\bot}dr_{\bot}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{\sqrt{R^{2}-r_{\bot}^{2}}}dz\rho(\sqrt{r_{\bot}^{2}+z^{2}})\exp{\left[-A{\sigma}\int\limits_{z}^{\sqrt{R^{2}-r_{\bot}^{2}}}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx\right]},$
(4) $\left<\frac{d\sigma_{{\gamma}p\to{J/\psi}p}({\bf p}_{\gamma},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right>_{A}=\int\int P_{A}({\bf p}_{t},E)d{\bf
p}_{t}dE\left[\frac{d\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right]$ (5)

and

$s^{*}=(E_{\gamma}+E_{t})^{2}-({\bf p}_{\gamma}+{\bf p}_{t})^{2},$ (6)
$E_{t}=M_{A}-\sqrt{(-{\bf p}_{t})^{2}+(M_{A}-m_{p}+E)^{2}}.$ (7)

Here, $d\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}},{\bf p}_{J/\psi})/d{\bf
p}_{J/\psi}$ is the off-shell differential cross section for the production of
$J/\psi$ meson in process (1) at the ”in-medium” ${\gamma}p$ c.m. energy
$\sqrt{s^{*}}$ 333)In Eq. (3), it is assumed that the cross sections for
$J/\psi$ meson production in ${\gamma}p$ and ${\gamma}n$ interactions are
equal to each other [6–8].); $\rho({\bf r})$ and $P_{A}({\bf p}_{t},E)$ are
normalized to unity the nucleon density and the nuclear spectral function
(they are given in Refs. [11, 12]) of target nucleus with mass number $A$,
having mass $M_{A}$ and radius $R$; ${\bf p}_{\gamma}$ and $E_{\gamma}$ are
the momentum and energy of the incident photon ($E_{\gamma}=|{\bf
p}_{\gamma}|=p_{\gamma}$) in the laboratory system; ${\bf p}_{t}$ and $E$ are
the momentum and binding energy of the intranuclear target proton,
participating in the reaction channel (1); $m_{p}$ is the free space proton
mass.

Also, as previously in [6–8], we will suppose that the off-shell differential
cross section
$d\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}},{\bf p}_{J/\psi})/d{\bf
p}_{J/\psi}$ for $J/\psi$ production in process (1) is the same as the
corresponding on-shell cross section
$d\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s},{\bf p}_{J/\psi})/d{\bf p}_{J/\psi}$
determined for the off-shell kinematics of this process and in which the
vacuum c.m. energy squared $s$, given by the formula

$s=W^{2}=(E_{\gamma}+m_{p})^{2}-{\bf
p}_{\gamma}^{2}=m_{p}^{2}+2m_{p}E_{\gamma},$ (8)

is replaced by the in-medium expression (6). The above off-shell differential
cross section is then (cf. [8, 10]):

$\frac{d\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}},{\bf p}_{J/\psi})}{d{\bf
p}_{J/\psi}}=\frac{\pi}{I_{2}(s^{*},m_{J/\psi},m_{p})E_{J/\psi}}$ (9)
$\times\frac{d\sigma_{{\gamma}p\to{J/\psi}{p}}(\sqrt{s^{*}},\theta_{J/\psi}^{*})}{d{\bf\Omega}_{J/\psi}^{*}}\frac{1}{(\omega+E_{t})}\delta\left[\omega+E_{t}-\sqrt{m_{p}^{2}+({\bf
Q}+{\bf p}_{t})^{2}}\right],$

where

$I_{2}(s^{*},m_{J/\psi},m_{p})=\frac{\pi}{2}\frac{\lambda(s^{*},m_{J/\psi}^{2},m_{p}^{2})}{s^{*}},$
(10)
$\lambda(x,y,z)=\sqrt{{\left[x-({\sqrt{y}}+{\sqrt{z}})^{2}\right]}{\left[x-({\sqrt{y}}-{\sqrt{z}})^{2}\right]}},$
(11) $\omega=E_{\gamma}-E_{J/\psi},\,\,\,\,{\bf Q}={\bf p}_{\gamma}-{\bf
p}_{J/\psi},\,\,\,\,E_{J/\psi}=\sqrt{m^{2}_{J/\psi}+{\bf p}_{J/\psi}^{2}}.$
(12)

Here, the off-shell c.m. charmonium angular distribution in reaction (1)
$d\sigma_{{\gamma}p\to{J/\psi}{p}}(\sqrt{s^{*}},\theta_{J/\psi}^{*})/d{\bf\Omega}_{J/\psi}^{*}$
as a function of the $J/\psi$ production c.m. polar angle
$\theta_{J/\psi}^{*}$ is given by [10]:

$\frac{d\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}},\theta^{*}_{J/\psi})}{d{\bf\Omega}_{J/\psi}^{*}}=a{\rm
e}^{b_{J/\psi}(t-t^{+})}\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}}),$ (13)

where $\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}})$ is the off-shell total
cross section for $J/\psi$ meson production in this reaction and

$t=m_{J/\psi}^{2}-2E^{*}_{\gamma}E^{*}_{J/\psi}+2p^{*}_{\gamma}p^{*}_{J/\psi}\cos{\theta^{*}_{J/\psi}},$
(14)
$E_{\gamma}^{*}=p^{*}_{\gamma},\,\,\,\,E_{J/\psi}^{*}=\sqrt{m^{2}_{J/\psi}+p^{*2}_{J/\psi}},$
(15)
$p_{\gamma}^{*}=\frac{1}{2\sqrt{s^{*}}}\lambda(s^{*},0,E_{t}^{2}-p_{t}^{2}),$
(16)
$p_{J/\psi}^{*}=\frac{1}{2\sqrt{s^{*}}}\lambda(s^{*},m_{J/\psi}^{2},m_{p}^{2}),$
(17)
$t^{+}=t(\cos{\theta^{*}_{J/\psi}}=1)=m_{J/\psi}^{2}-2E^{*}_{\gamma}E^{*}_{J/\psi}+2p^{*}_{\gamma}p^{*}_{J/\psi},$
(18) $t-t^{+}=2p^{*}_{\gamma}p^{*}_{J/\psi}(\cos{\theta^{*}_{J/\psi}}-1).$
(19)

The angle of charmonium production in the ${\gamma}p$ c.m. system,
$\theta^{*}_{J/\psi}$, is expressed through its production angle,
$\theta_{J/\psi}$, in the laboratory system ($\cos{\theta_{J/\psi}}={\bf
p}_{\gamma}{\bf p}_{J/\psi}/p_{\gamma}p_{J/\psi}$) by means of equation [8,
10]:

$\cos{\theta_{J/\psi}^{*}}=\frac{p_{\gamma}p_{J/\psi}\cos{\theta_{J/\psi}}+(E_{\gamma}^{*}E_{J/\psi}^{*}-E_{\gamma}E_{J/\psi})}{p_{\gamma}^{*}p_{J/\psi}^{*}}.$
(20)

The condition of normalization

$\int\limits_{4\pi}a{\rm e}^{b_{J/\psi}(t-t^{+})}d{\bf\Omega}_{J/\psi}^{*}=1$
(21)

gives for the parameter $a$ in Eq. (13) the following expression:

$a=\frac{p^{*}_{\gamma}p^{*}_{J/\psi}b_{J/\psi}}{\pi}\left[1-{\rm
e}^{-4p^{*}_{\gamma}p^{*}_{J/\psi}b_{J/\psi}}\right]^{-1}.$ (22)

Parameter $b_{J/\psi}$ in Eqs. (13), (21), (22) is an exponential $t$-slope of
the differential cross section of the reaction ${\gamma}p\to{J/\psi}p$ in the
near-threshold energy region. It should be pointed out that the differential
cross section of this reaction was also very recently measured in the
$J/\psi$-007 experiment [13] as a function of the photon energy in the range
of $9.1~{}{\rm GeV}\leq E_{\gamma}\leq 10.6~{}{\rm GeV}$ with the aim to
explore the impact of the collected data on the determination of the proton’s
gravitational form factors, the proton-mass radius, and the contribution of
the trace anomaly to the proton mass. Since the $t$-slope $b_{J/\psi}$ was not
determined in [13], we will adopt for it the GlueX result [4], namely:
$b_{J/\psi}$ $\approx$ 1.67 GeV-2. We will use this value in our calculations.

Now consider the off-shell total cross section
$\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s^{*}})$ for $J/\psi$ production in
process (1). In line with the above-mentioned, it is the same as the vacuum
cross section $\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s})$, in which the vacuum
c.m. energy squared s, defined by the formula (8), is replaced by the in-
medium expression (6). For the vacuum total cross section
$\sigma_{{\gamma}p\to{J/\psi}p}(\sqrt{s})$ at near-threshold photon energies
we have used the following parametrization [7] of the available here
experimental information [4] on it from the GlueX experiment, based on the
near-threshold predictions of the two gluon and three gluon exchange model
[14]:

$\sigma_{{\gamma}p\to{J/\psi}p}({\sqrt{s}})=\sigma_{2g}({\sqrt{s}})+\sigma_{3g}({\sqrt{s}}),$
(23)

where 2$g$ and 3$g$ exchanges cross sections $\sigma_{2g}({\sqrt{s}})$ and
$\sigma_{3g}({\sqrt{s}})$ are given in Ref. [7] by formulas (7) and (8),
respectively.

Figure 1: (Color online) The total cross section for the background reaction
${\gamma}p\to{J/\psi}p$ as a function of the photon energy $E_{\gamma}$.
Dashed and dotted-dashed curves are, respectively, calculations on the basis
of the two gluon and three gluon exchange model [14]. Solid curve is the
incoherent sum of the above two calculations. The GlueX experimental data are
from Ref. [4]. The arrow indicates the threshold energy of 8.21 GeV for direct
non-resonant charmonium photoproduction off a free target proton at rest.
Figure 2: (Color online) Plot of the allowed final $J/\psi$ meson and proton
momenta in the direct non-resonant ${\gamma}p\to{J/\psi}p$ reaction, occurring
in the laboratory system in the free space at initial photon energy of 9.44
GeV, as functions of their production angles with respect to the photon beam
direction in this system. Figure 3: (Color online) The same as in Fig. 2, but
for the initial photon energy of 10.12 GeV.

Fig. 1 shows that the GlueX near-threshold data are well fitted by only the
combination (23) of the two gluon and three gluon exchange cross sections and
in the resonance incident photon energy range $\sim$ 9.5–10.0 GeV the main
contribution to the elastic $J/\psi$ production comes from the three gluon
exchanges.

At the initial photon energies of interest the $J/\psi$ mesons are produced at
small laboratory polar angles (see below). Therefore, we will calculate the
$J/\psi$ momentum distributions from considered target nuclei for the
laboratory solid angle
${\Delta}{\bf\Omega}_{J/\psi}$=$0^{\circ}\leq\theta_{J/\psi}\leq 20^{\circ}$,
and $0\leq\varphi_{J/\psi}\leq 2{\pi}$ (cf. [10]). Then, integrating the
differential cross section (3) over this solid angle, we can represent the
differential cross section for charmonium production from the direct non-
resonant processes (1) and (2) into this solid angle as follows:

$\frac{d\sigma_{{\gamma}A\to{J/\psi}X}^{({\rm
dir})}(p_{\gamma},p_{J/\psi})}{dp_{J/\psi}}=\int\limits_{{\Delta}{\bf\Omega}_{J/\psi}}d{\bf\Omega}_{J/\psi}\frac{d\sigma_{{\gamma}A\to{J/\psi}X}^{({\rm
dir})}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}p_{J/\psi}^{2}$
(24) $=2{\pi}I_{V}[A,\sigma_{{J/\psi}N}]\int\limits_{\cos
20^{\circ}}^{1}d\cos{{\theta_{J/\psi}}}\left<\frac{d\sigma_{{\gamma}p\to{J/\psi}{p}}(p_{\gamma},p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf\Omega}_{J/\psi}}\right>_{A}.$

Before going further, we now consider, adopting the relativistic kinematics,
more simpler case of the free space production of $J/\psi$ mesons and protons
in the process ${\gamma}p\to{J/\psi}p$, proceeding on a free target proton
being at rest, to get an idea about their kinematic characteristics allowed in
this process at incident photon energies considered. The kinematics of two-
body reaction with a threshold (as in our present case) tell us that the
laboratory polar $J/\psi$ and final proton production angles $\theta_{J/\psi}$
and $\theta_{p}$ vary from 0 to a maximal values $\theta^{\rm max}_{J/\psi}$
and $\theta^{\rm max}_{p}$, correspondingly, i.e.:

$0\leq\theta_{J/\psi}\leq\theta^{\rm max}_{J/\psi},$ (25)
$0\leq\theta_{p}\leq\theta^{\rm max}_{p};$ (26)

where

$\theta^{\rm max}_{J/\psi}={\rm
arcsin}[(\sqrt{s}p^{*}_{J/\psi})/(m_{J/\psi}p_{\gamma})],$ (27) $\theta^{\rm
max}_{p}={\rm arcsin}[(\sqrt{s}p^{*}_{p})/(m_{p}p_{\gamma})].$ (28)

Here, the $J/\psi$ c.m. momentum $p^{*}_{J/\psi}$ is determined by Eq. (17),
in which the in-medium c.m. energy squared $s^{*}$ should be replaced by the
vacuum collision energy squared $s$, defined by the formula (8), and
$p^{*}_{p}$ is the final proton c.m. momentum. It is equal to the $J/\psi$
c.m. momentum $p^{*}_{J/\psi}$. From Eqs. (27), (28) one can get, for example,
that

$\theta^{\rm max}_{J/\psi}=5.570^{\circ},\,\,\,\,\theta^{\rm
max}_{p}=18.686^{\circ}$ (29)

at initial photon beam energy of $E_{\gamma}=9.44$ GeV and

$\theta^{\rm max}_{J/\psi}=6.768^{\circ},\,\,\,\,\theta^{\rm
max}_{p}=22.892^{\circ}$ (30)

at photon energy of $E_{\gamma}=10.12$ GeV. Energy-momentum conservation in
the reaction (1), taking place in a vacuum, leads to two different solutions
for the laboratory $J/\psi$ meson and final proton momenta $p_{J/\psi}$ and
$p_{p}$ at given laboratory polar production angles $\theta_{J/\psi}$ and
$\theta_{p}$, belonging, correspondingly, to the angular intervals (25) and
(26):

$p^{(1,2)}_{J/\psi}(\theta_{J/\psi})=\frac{p_{\gamma}\sqrt{s}E^{*}_{J/\psi}\cos{\theta_{J/\psi}}\pm(E_{\gamma}+m_{p})\sqrt{s}\sqrt{p^{*2}_{J/\psi}-{\gamma^{2}_{\rm
cm}}{v^{2}_{\rm
cm}}m^{2}_{J/\psi}\sin^{2}{\theta_{J/\psi}}}}{(E_{\gamma}+m_{p})^{2}-p^{2}_{\gamma}\cos^{2}{\theta_{J/\psi}}},$
(31)
$p^{(1,2)}_{p}(\theta_{p})=\frac{p_{\gamma}\sqrt{s}E^{*}_{p}\cos{\theta_{p}}\pm(E_{\gamma}+m_{p})\sqrt{s}\sqrt{p^{*2}_{p}-{\gamma^{2}_{\rm
cm}}{v^{2}_{\rm
cm}}m^{2}_{p}\sin^{2}{\theta_{p}}}}{(E_{\gamma}+m_{p})^{2}-p^{2}_{\gamma}\cos^{2}{\theta_{p}}}.$
(32)

Here, ${\gamma_{\rm cm}}=(E_{\gamma}+m_{p})/\sqrt{s}$, $v_{\rm
cm}=p_{\gamma}/(E_{\gamma}+m_{p})$, the $J/\psi$ total c.m. energy
$E^{*}_{J/\psi}$ is defined above by Eq. (15),
$E^{*}_{p}=\sqrt{m^{2}_{p}+p^{*2}_{p}}$ and sign ”+” in the numerators of Eqs.
(31), (32) corresponds to the first solutions $p^{(1)}_{J/\psi}$,
$p^{(1)}_{p}$ and sign ”-” - to the second ones $p^{(2)}_{J/\psi}$,
$p^{(2)}_{p}$. Looking at the expressions (31) and (32), one can come to the
conclusion that the first solutions $p^{(1)}_{J/\psi}$ and $p^{(1)}_{p}$ as
well as the second ones $p^{(2)}_{J/\psi}$ and $p^{(2)}_{p}$ have different
dependencies, respectively, on the production angles $\theta_{J/\psi}$ and
$\theta_{p}$ within the angular intervals [0, $\theta_{J/\psi}^{\rm max}]$ and
[0, $\theta_{p}^{\rm max}]$. Namely, the former drop and the latter ones
increase as the production angles $\theta_{J/\psi}$ and $\theta_{p}$ increase
in these intervals (cf. Figs. 2 and 3) and

$p^{(1)}_{J/\psi}(\theta_{J/\psi}^{\rm
max})=p^{(2)}_{J/\psi}(\theta_{J/\psi}^{\rm
max})=p_{J/\psi}(\theta_{J/\psi}^{\rm max}),$ (33)
$p^{(1)}_{p}(\theta_{p}^{\rm max})=p^{(2)}_{p}(\theta_{p}^{\rm
max})=p_{p}(\theta_{p}^{\rm max}),$ (34)

where

$p_{J/\psi}(\theta_{J/\psi}^{\rm
max})=(p_{\gamma}m_{J/\psi}^{2}\cos{\theta_{J/\psi}^{\rm
max}})/(\sqrt{s}E^{*}_{J/\psi}),$ (35) $p_{p}(\theta_{p}^{\rm
max})=(p_{\gamma}m_{p}^{2}\cos{\theta_{p}^{\rm max}})/(\sqrt{s}E^{*}_{p}).$
(36)

According to Eqs. (35), (36), for $E_{\gamma}=9.44$ GeV we get then that
$p_{J/\psi}(\theta_{J/\psi}^{\rm max})=6.600$ GeV/c and $p_{p}(\theta_{p}^{\rm
max})=1.593$ GeV/c. For $E_{\gamma}=10.12$ GeV we obtain
$p_{J/\psi}(\theta_{J/\psi}^{\rm max})=6.745$ GeV/c and $p_{p}(\theta_{p}^{\rm
max})=1.471$ GeV/c (cf. Figs. 2 and 3). These figures show that the
kinematically allowed charmonium laboratory momenta and total energies in the
direct non-resonant ${\gamma}p\to{J/\psi}p$ process, taking place on the free
target proton at rest, at given initial photon energy vary within the
following momentum and energy ranges:

$p^{(2)}_{J/\psi}(0^{\circ})\leq p_{J/\psi}\leq p^{(1)}_{J/\psi}(0^{\circ}),$
(37) $E^{(2)}_{J/\psi}(0^{\circ})\leq E_{J/\psi}\leq
E^{(1)}_{J/\psi}(0^{\circ}),$ (38)

where the quantities $p^{(1,2)}_{J/\psi}(0^{\circ})$ are defined above by Eq.
(31) and
$E^{(1,2)}_{J/\psi}(0^{\circ})=\sqrt{m_{J/\psi}^{2}+[p^{(1,2)}_{J/\psi}(0^{\circ})]^{2}}$.

Finally, we calculate the $J/\psi$ energy distribution
$d\sigma_{{\gamma}p\to{J/\psi}{p}}[\sqrt{s},p_{J/\psi}]/dE_{J/\psi}$ from the
reaction ${\gamma}p\to{J/\psi}p$ within the kinematically allowed interval
(38). Integration of the more general differential cross section (9) over the
angle $\theta_{J/\psi}$, when this angle varies in the allowed angular region
(25), in the limits: ${\bf p}_{t}\to 0$, $E_{t}\to m_{p}$ and $s^{*}\to s$
yields:

$\frac{d\sigma_{{\gamma}p\to{J/\psi}{p}}[\sqrt{s},p_{J/\psi}]}{dE_{J/\psi}}=2\pi\int\limits_{\cos{\theta_{J/\psi}^{\rm
max}}}^{1}d\cos{\theta_{J/\psi}}p_{J/\psi}E_{J/\psi}\frac{d\sigma_{{\gamma}p\to{J/\psi}p}[\sqrt{s},{\bf
p}_{J/\psi}]}{d{\bf p}_{J/\psi}}=$ (39)
$=\left(\frac{2{\pi}\sqrt{s}}{p_{\gamma}p^{*}_{J/\psi}}\right)\frac{d\sigma_{{\gamma}p\to{J/\psi}{p}}[\sqrt{s},\theta_{J/\psi}^{*}(x_{0})]}{d{\bf\Omega}_{J/\psi}^{*}}~{}{\rm
for}~{}E^{(2)}_{J/\psi}(0^{\circ})\leq E_{J/\psi}\leq
E^{(1)}_{J/\psi}(0^{\circ}),$

where

$x_{0}=\frac{[p^{2}_{\gamma}+p^{2}_{J/\psi}+m^{2}_{p}-(\omega+m_{p})^{2}]}{2p_{\gamma}p_{J/\psi}},\,\,\,\,\,p_{J/\psi}=\sqrt{E^{2}_{J/\psi}-m^{2}_{J/\psi}}$
(40)

and the quantity $\cos{\theta_{J/\psi}^{*}(x_{0})}$ is defined by Eq. (20), in
which one has to perform the replacement: $\cos{\theta_{J/\psi}}\to x_{0}$,
and the photon and $J/\psi$ c.m. momenta $p^{*}_{\gamma}$ and $p^{*}_{J/\psi}$
are defined by formulas (16) and (17), correspondingly, in which one needs
also to make the substitutions: $E_{t}\to m_{p}$, $p_{t}\to 0$ and $s^{*}\to
s$. We will adopt the expression (39) for evaluating the free space $J/\psi$
energy distribution from the direct process (1), proceeding on a proton at
rest, for incident photon beam resonant energies of 9.44, 9.554, 10.04 and
10.12 GeV (see below).

Figure 4: (Color online) The non-resonant total cross section $\sigma_{1}$ for
the reaction ${\gamma}p\to{J/\psi}p$ (solid curve) and the incoherent sum
(dotted curve) of it and the total cross section $\sigma_{2}$ (short-dashed
curve) for the resonant $J/\psi$ production in the processes ${\gamma}p\to
P^{+}_{c}(4312)\to{J/\psi}p$, ${\gamma}p\to P^{+}_{c}(4337)\to{J/\psi}p$,
${\gamma}p\to P^{+}_{c}(4440)\to{J/\psi}p$ and ${\gamma}p\to
P^{+}_{c}(4457)\to{J/\psi}p$, calculated assuming that the resonances
$P^{+}_{c}(4312)$, $P^{+}_{c}(4337)$, $P^{+}_{c}(4440)$ and $P^{+}_{c}(4457)$
with the spin-parity quantum numbers $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$,
$J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$ decay to ${J/\psi}p$ with the lower
allowed relative orbital angular momentum $L=0$ with all four branching
fractions $Br[P^{+}_{ci}\to{J/\psi}p]=$1%, as functions of photon energy. The
left and four right arrows indicate, correspondingly, the threshold energy
$E^{\rm th}_{\gamma}=8.21$ GeV for the reaction ${\gamma}p\to{J/\psi}p$
proceeding on a free target proton being at rest and the resonant energies
$E^{\rm R1}_{\gamma}=9.44$ GeV, $E^{\rm R2}_{\gamma}=9.554$ GeV, $E^{\rm
R3}_{\gamma}=10.04$ GeV and $E^{\rm R4}_{\gamma}=10.12$ GeV.

### 2.2. Two-step resonant $J/\psi$ production processes

At photon energies below 11 GeV, incident photons can produce the observed
[1–3] experimentally non-strange charged $P^{+}_{c}(4312)$, $P^{+}_{c}(4337)$,
$P^{+}_{c}(4440)$, $P^{+}_{c}(4457)$ pentaquark resonances with quark
structure $|P^{+}_{c}>=|uudc{\bar{c}}>$ and predicted [15], but non-observed
yet their neutral isospin partners $P^{0}_{c}(4312)$, $P^{0}_{c}(4337)$,
$P^{0}_{c}(4440)$, $P^{0}_{c}(4457)$ 444)The minimal quark content of the
$P_{c}^{0}$ states is $|P^{0}_{c}>=|uddc{\bar{c}}>$. Following the observation
of the narrow pentaquarks $P^{+}_{c}(4312)$, $P^{+}_{c}(4440)$ and
$P^{+}_{c}(4457)$ by the LHCb Collaboration [1, 2], it was proposed to search
for the $P_{c}^{0}$ states in ${\pi^{-}}p\to{J/\psi}n$ reaction [16].)
directly in the first inelastic collisions with intranuclear protons and
neutrons 555)We remind that, for example, the threshold (resonant) energies
$E^{\rm R1}_{\gamma}$, $E^{\rm R2}_{\gamma}$, $E^{\rm R3}_{\gamma}$ and
$E^{\rm R4}_{\gamma}$ for the photoproduction of the $P_{c}^{+}$ resonances
with pole masses $M_{c1}^{+}=4311.9$ MeV, $M_{c2}^{+}=4337.0$ MeV,
$M_{c3}^{+}=4440.3$ MeV and $M_{c4}^{+}=4457.3$ MeV [2, 3] on a free target
proton being at rest are $E^{\rm R1}_{\gamma}=9.44$ GeV, $E^{\rm
R2}_{\gamma}=9.554$ GeV, $E^{\rm R3}_{\gamma}=10.04$ GeV and $E^{\rm
R4}_{\gamma}=10.12$ GeV, respectively.) :

$\displaystyle{\gamma}+p\to P^{+}_{c}(4312),$ $\displaystyle{\gamma}+p\to
P^{+}_{c}(4337),$ $\displaystyle{\gamma}+p\to P^{+}_{c}(4440),$
$\displaystyle{\gamma}+p\to P^{+}_{c}(4457);$ (41) $\displaystyle{\gamma}+n\to
P^{0}_{c}(4312),$ $\displaystyle{\gamma}+n\to P^{0}_{c}(4337),$
$\displaystyle{\gamma}+n\to P^{0}_{c}(4440),$ $\displaystyle{\gamma}+n\to
P^{0}_{c}(4457).$ (42)

Furthermore, the produced pentaquark resonances can decay into the final
states ${J/\psi}p$ and ${J/\psi}n$, which will additionally contribute to the
$J/\psi$ yield in the ($\gamma$,$J/\psi$) reactions on protons and nuclei:

$\displaystyle P^{+}_{c}(4312)\to J/\psi+p,$ $\displaystyle P^{+}_{c}(4337)\to
J/\psi+p,$ $\displaystyle P^{+}_{c}(4440)\to J/\psi+p,$ $\displaystyle
P^{+}_{c}(4457)\to J/\psi+p;$ (43) $\displaystyle P^{0}_{c}(4312)\to
J/\psi+n,$ $\displaystyle P^{0}_{c}(4337)\to J/\psi+n,$ $\displaystyle
P^{0}_{c}(4440)\to J/\psi+n,$ $\displaystyle P^{0}_{c}(4457)\to J/\psi+n.$
(44)

The branching ratios $Br[P^{+}_{ci}\to{J/\psi}p]$ 666)Here, $i=$1, 2, 3, 4 and
$P^{+}_{c1}$, $P^{+}_{c2}$, $P^{+}_{c3}$ and $P^{+}_{c4}$ stand for
$P^{+}_{c}(4312)$, $P^{+}_{c}(4337)$, $P^{+}_{c}(4440)$ and $P^{+}_{c}(4457)$,
respectively. Analogously, $P^{0}_{c1}$, $P^{0}_{c2}$, $P^{0}_{c3}$ and
$P^{0}_{c4}$ will denote below the $P^{0}_{c}(4312)$, $P^{0}_{c}(4337)$,
$P^{0}_{c}(4440)$ and $P^{0}_{c}(4457)$ states.) of the decays (43) have not
been determined yet. Model-dependent upper limits on branching fractions
$Br[P^{+}_{c}(4312)\to{J/\psi}p]$, $Br[P^{+}_{c}(4440)\to{J/\psi}p]$ and
$Br[P^{+}_{c}(4457)\to{J/\psi}p]$ of several percent were set by the GlueX
Hall-D experiment [4] at JLab, having a moderate statistics (about 470
$J/\psi$ events). Preliminary results from a factor of 10 more data (about
4000 $J/\psi$ events), collected in the $J/\psi$–007 Hall-C experiment [17] at
JLab as well, focused on the large $t$ region 777)In which the rather flat
resonant production of $J/\psi$ through the $P_{c}^{+}$ is expected to be
enhanced relative to the suppressed here mostly forward diffractive
production.) in searching for the LHCb hidden-charm pentaquarks [1, 2], also
observe no signals for them and will set more stringent upper limits on the
above branching fractions and on pentaquark-$J/\psi$ couplings. Based on the
branching ratios and fractions measured by the LHCb and GlueX Collaborations,
the authors of Ref. [18] obtain that a lower limit of
$Br[P^{+}_{c}\to{J/\psi}p]$ is of the order of 0.05% $\sim$ 0.5%. Taking into
account these findings, we will adopt in our study for the four branching
ratios $Br[P^{+}_{ci}\to{J/\psi}p]$ of the decays (43) three following
conservative options: $Br[P^{+}_{ci}\to{J/\psi}p]=0.25$, 0.5 and 1%
($i=1,2,3,4$), and in line with Ref. [15], will assume that
$Br[P^{0}_{ci}\to{J/\psi}n]=Br[P^{+}_{ci}\to{J/\psi}p]$. This will allow us to
get a better impression of the size of the effect of branching fractions
$Br[P^{+}_{ci}\to{J/\psi}p]$ and $Br[P^{0}_{ci}\to{J/\psi}n]$ on the resonant
$J/\psi$ yield in ${\gamma}p\to{J/\psi}p$ as well as in ${\gamma}$12C
$\to{J/\psi}X$ and ${\gamma}$184W $\to{J/\psi}X$ reactions. Moreover, we will
also suppose, analogously to [15], for the $P_{ci}^{0}$ states the same pole
masses $M_{ci}^{0}$ and total decay width $\Gamma_{ci}^{0}$ as those
$M_{ci}^{+}$ and $\Gamma_{ci}^{+}$ for their hidden-charm charged counterparts
$P_{ci}^{+}$, i.e.: $M_{ci}^{0}=M_{ci}^{+}$ and
$\Gamma_{c1}^{0}=\Gamma_{c1}^{+}=9.8$ MeV,
$\Gamma_{c2}^{0}=\Gamma_{c2}^{+}=29.0$ MeV,
$\Gamma_{c3}^{0}=\Gamma_{c3}^{+}=20.6$ MeV,
$\Gamma_{c4}^{0}=\Gamma_{c4}^{+}=6.4$ MeV [2, 3].

In line with Refs. [6, 7, 19], we suppose that the in-medium spectral
functions $S_{ci}^{+}(\sqrt{s^{*}},\Gamma_{ci}^{+})$ and
$S_{ci}^{0}(\sqrt{s^{*}},\Gamma_{ci}^{0})$ of the intermediate $P_{ci}^{+}$
and $P_{ci}^{0}$ resonances are described by the non-relativistic Breit-Wigner
distributions 888)We ignore, for reasons of the simplification of
calculations, the modification of the $P_{ci}^{+}$ and $P_{ci}^{0}$ masses and
total decay widths in the nuclear matter in our present study.) :

$S_{ci}^{+}(\sqrt{s^{*}},\Gamma_{ci}^{+})=\frac{1}{2\pi}\frac{\Gamma_{ci}^{+}}{(\sqrt{s^{*}}-M_{ci}^{+})^{2}+({\Gamma}_{ci}^{+})^{2}/4}$
(45)

and

$S_{ci}^{0}(\sqrt{s^{*}},\Gamma_{ci}^{0})=\frac{1}{2\pi}\frac{\Gamma_{ci}^{0}}{(\sqrt{s^{*}}-M_{ci}^{0})^{2}+({\Gamma}_{ci}^{0})^{2}/4}.$
(46)

The in-medium total cross sections for production of these resonances with the
possible spin-parity quantum numbers $J^{P}=(1/2)^{-}$ for $P_{c1}^{+}$ and
$P_{c1}^{0}$, $J^{P}=(1/2)^{-}$ for $P_{c2}^{+}$ and $P_{c2}^{0}$,
$J^{P}=(1/2)^{-}$ for $P_{c3}^{+}$ and $P_{c3}^{0}$, and $J^{P}=(3/2)^{-}$ for
$P_{c4}^{+}$ and $P_{c4}^{0}$ 999)Which might be assigned to them within the
hadronic molecular scenario for their internal structure (cf. [7, 15,
20–22]).) in reactions (41), (42) can be determined, using the spectral
functions (45), (46) and known branching fractions
$Br[P_{ci}^{+}\to{\gamma}p]$ and $Br[P_{ci}^{0}\to{\gamma}n]$ ($i=1$, 2, 3,
4), as follows [6, 7, 19]:

$\sigma_{{\gamma}p\to
P_{ci}^{+}}(\sqrt{s^{*}},\Gamma_{ci}^{+})=f_{ci}\left(\frac{\pi}{p^{*}_{\gamma}}\right)^{2}Br[P_{ci}^{+}\to{\gamma}p]S_{ci}^{+}(\sqrt{s^{*}},\Gamma_{ci}^{+})\Gamma_{ci}^{+},\,\,i=1,2,3,4$
(47)

and

$\sigma_{{\gamma}n\to
P_{ci}^{0}}(\sqrt{s^{*}},\Gamma_{ci}^{0})=f_{ci}\left(\frac{\pi}{p^{*}_{\gamma}}\right)^{2}Br[P_{ci}^{0}\to{\gamma}n]S_{ci}^{0}(\sqrt{s^{*}},\Gamma_{ci}^{0})\Gamma_{ci}^{0},\,\,i=1,2,3,4.$
(48)

Here, the c.m. 3-momentum in the incoming ${\gamma}N$ channel,
$p^{*}_{\gamma}$, is defined above by Eq. (16) 101010)For simplicity, we
assume that the neutron mass $m_{n}$ is equal to the proton mass $m_{p}$.) and
the ratios of the spin factors $f_{c1}=1$, $f_{c2}=1$, $f_{c3}=1$, $f_{c4}=2$.
Since we are mainly interested in the resonance $P_{c}$ region, which is not
far from the ${J/\psi}N$ production threshold, we suppose in line with [7, 19,
23] that the hidden-charm pentaquarks $P_{ci}^{+}$ and $P_{ci}^{0}$ decays to
${J/\psi}p$ and ${J/\psi}n$ modes are dominated by the lowest partial waves
with zero relative orbital angular momentum $L$. In this case, adopting the
vector-meson dominance model, one can obtain that the branching ratios
$Br[P_{ci}^{0}\to{\gamma}n]$ and $Br[P_{ci}^{+}\to{\gamma}p]$ are equal to
each other (cf. [24])

$Br[P_{ci}^{0}\to{\gamma}n]=Br[P_{ci}^{+}\to{\gamma}p]$ (49)

and the latter for $P^{+}_{c}(4312)$, $P^{+}_{c}(4440)$ and $P^{+}_{c}(4457)$
are expressed in the framework of this model via the branching fractions
$Br[P^{+}_{c}(4312)\to{J/\psi}p]$, $Br[P^{+}_{c}(4440)\to{J/\psi}p]$ and
$Br[P^{+}_{c}(4457)\to{J/\psi}p]$ by formula (24) from Ref. [7] and within
this model we get that

$Br[P^{+}_{c}(4337)\to{\gamma}p]=1.48\cdot
10^{-3}Br[P^{+}_{c}(4337)\to{J/\psi}p].$ (50)

Using Eqs. (47)–(49), we have

$\sigma_{{\gamma}p\to
P_{ci}^{+}}(\sqrt{s^{*}},\Gamma_{ci}^{+})=\sigma_{{\gamma}n\to
P_{ci}^{0}}(\sqrt{s^{*}},\Gamma_{ci}^{0}).$ (51)

According to Eq. (47), for example, the free total cross sections
$\sigma_{{\gamma}p\to P_{ci}^{+}\to{J/\psi}p}(\sqrt{s},\Gamma_{ci}^{+})$ for
resonant charmonium production in the two-step processes (41)/(43), taking
place on the target proton at rest, can be represented as follows [6, 7]:

$\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}p}(\sqrt{s},\Gamma_{ci}^{+})=\sigma_{{\gamma}p\to
P_{ci}^{+}}(\sqrt{s},\Gamma_{ci}^{+})\theta[\sqrt{s}-(m_{J/\psi}+m_{p})]Br[P_{ci}^{+}\to{J/\psi}p].$
(52)

Here, $\theta(x)$ is the step function and the c.m. 3-momentum in the incoming
${\gamma}p$ channel, $p^{*}_{\gamma}$, entering into Eq. (47), is determined
above by the formula (16), in which one has to make the replacements
$E_{t}^{2}-p_{t}^{2}\to m^{2}_{p}$ and $s^{*}\to s$. In line with Eqs. (47)
and (50), we see that these cross sections are proportional to
$Br^{2}[P_{ci}^{+}\to{J/\psi}p]$. This fact enables us to evaluate upper
limits on the branching fractions $Br[P_{c}^{+}(4312)\to{J/\psi}p]$,
$Br[P_{c}^{+}(4440)\to{J/\psi}p]$ and $Br[P_{c}^{+}(4457)\to{J/\psi}p]$, which
are expected from preliminary results of the JLab E12-16-007 experiment [17].
According to them, upper limits on the cross sections (52) for
$P_{c}^{+}(4312)$, $P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$ states almost an
order of magnitude below the respective GlueX limits [4]. With this and within
the representation of Eq. (52), we readily obtain the following relation
between upper limits on the above branching fractions, which are expected from
the $J/\psi$-007 experiment, and those already available from the GlueX
experiment:
$Br_{J/\psi-007}[P_{ci}^{+}\to{J/\psi}p]\approx(1/\sqrt{10})Br_{\rm
GlueX}[P_{ci}^{+}\to{J/\psi}p]$ ($i=1$, 3, 4). Model-dependent upper limits on
the latter ratios of 4.6%, 2.3% and 3.8% for $P_{c}^{+}(4312)$,
$P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$, assuming for each $P_{ci}^{+}$ spin-
parity combination $J^{P}=(3/2)^{-}$, were set by the GlueX Collaboration [4].
So that, following the above relation, we get that
$Br_{J/\psi-007}[P_{c}^{+}(4312)\to{J/\psi}p]\approx$1.46%,
$Br_{J/\psi-007}[P_{c}^{+}(4440)\to{J/\psi}p]\approx$0.73% and
$Br_{J/\psi-007}[P_{c}^{+}(4457)\to{J/\psi}p]\approx$1.20%. This means that
our choice of 1% for upper value of the branching ratios
$Br[P_{ci}^{+}\to{J/\psi}p]$ for all 4 states is quite reasonable and
justified.

Accounting for the fact that the most of the narrow $P_{ci}^{+}$ amd
$P_{ci}^{0}$ resonances ($i=1$, 2, 3, 4), having vacuum total decay widths in
their rest frames of 9.8, 29.0, 20.6 and 6.4 MeV [2, 3], respectively, decay
to ${J/\psi}p$ and ${J/\psi}n$ outside of the considered target nuclei [6] as
well as the results presented both in Refs. [6, 7, 10] and above by Eqs. (3),
(4), (51), (52), we can obtain the following expression for the $J/\psi$
inclusive differential cross section arising from the production and decay of
intermediate resonances $P_{ci}^{+}$ and $P_{ci}^{0}$ in ${\gamma}A$
reactions:

$\frac{d\sigma_{{\gamma}A\to{J/\psi}X}^{({\rm sec})}({\bf p}_{\gamma},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}=\frac{d\sigma_{{\gamma}A\to
P_{ci}^{+}\to{J/\psi}p}^{({\rm sec})}({\bf p}_{\gamma},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}+\frac{d\sigma_{{\gamma}A\to
P_{ci}^{0}\to{J/\psi}n}^{({\rm sec})}({\bf p}_{\gamma},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}=$ (53) $=I_{V}[A,\sigma^{\rm
in}_{{P_{c}}N}]\left<\frac{d\sigma_{{\gamma}p\to P_{ci}^{+}\to{J/\psi}p}({\bf
p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right>_{A},i=1,2,3,4,$

where

$\left<\frac{d\sigma_{{\gamma}p\to P_{ci}^{+}\to{J/\psi}p}({\bf
p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right>_{A}=\int\int
P_{A}({\bf p}_{t},E)d{\bf p}_{t}dE\left[\frac{d\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}p}(\sqrt{s^{*}},{\bf p}_{J/\psi})}{d{\bf
p}_{J/\psi}}\right],$ (54)

and

$\frac{d\sigma_{{\gamma}p\to P_{ci}^{+}\to{J/\psi}p}(\sqrt{s^{*}},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}=\sigma_{{\gamma}p\to
P_{ci}^{+}}(\sqrt{s^{*}},\Gamma_{ci}^{+})\theta[\sqrt{s^{*}}-(m_{J/\psi}+m_{p})]\times$
(55) $\times\frac{1}{\Gamma_{ci}^{+}(\sqrt{s^{*}},{\bf p}_{\gamma})}\int d{\bf
p}_{p}\frac{d\Gamma_{P_{ci}^{+}\to{J/\psi}p}(\sqrt{s^{*}},{\bf
p}_{J/\psi},{\bf p}_{p})}{d{\bf p}_{J/\psi}d{\bf p}_{p}},$
$\frac{d\Gamma_{P_{ci}^{+}\to{J/\psi}p}(\sqrt{s^{*}},{\bf p}_{J/\psi},{\bf
p}_{p})}{d{\bf p}_{J/\psi}d{\bf
p}_{p}}=\frac{1}{2E_{ci}^{+}}\frac{1}{2J+1}|M_{P_{ci}^{+}\to{J/\psi}p}|^{2}(2\pi)^{4}\delta(E_{ci}^{+}-E_{J/\psi}-E_{p})\times$
(56) $\times\delta({\bf p}_{ci}^{+}-{\bf p}_{J/\psi}-{\bf
p}_{p})\frac{1}{(2\pi)^{3}{2E_{J/\psi}}}\frac{1}{(2\pi)^{3}{2E_{p}}},$
$\Gamma_{ci}^{+}(\sqrt{s^{*}},{\bf
p}_{\gamma})=\Gamma_{ci}^{+}/\gamma_{ci}^{+},$ (57)
$E_{ci}^{+}=E_{\gamma}+E_{t},\,\,\,\,\,{\bf p}_{ci}^{+}={\bf p}_{\gamma}+{\bf
p}_{t},\,\,\,\,\,\gamma_{ci}^{+}=E_{ci}^{+}/\sqrt{s^{*}}.$ (58)

Here, $E_{p}$ is the final proton total energy ($E_{p}=\sqrt{m^{2}_{p}+{\bf
p}^{2}_{p}}$) and $|M_{P_{ci}^{+}\to{J/\psi}p}|^{2}$ is summarized over spin
states of initial and final particles matrix element squared describing the
decays (43) for given $i$. The quantity $I_{V}[A,\sigma^{\rm in}_{P_{c}N}]$ in
Eq. (53) is defined above by Eq. (4), in which one needs to make the
substitution $\sigma\to\sigma^{\rm in}_{P_{c}N}$. Here the quantity
$\sigma^{\rm in}_{P_{c}N}$ denotes the inelastic total cross sections of the
free $P_{c}N$ interaction. Our estimates [6] 111111)These estimates also show
that we can neglect quasielastic $P_{ci}^{+}{N}$ and $P_{ci}^{0}{N}$
rescatterings in their way out of the target nucleus.), based on the
${J/\psi}p$ molecular scenario for the $P_{c}^{+}$ pentaquarks, show that this
quantity can be evaluated as $\sigma^{\rm in}_{P_{c}N}\approx 33.5$ mb. We
will use this value throughout our calculations. In view of the aforesaid, the
hidden-charm pentaquarks $P_{ci}^{+}$ (and $P_{ci}^{0}$) decays to ${J/\psi}p$
(and ${J/\psi}n$) are dominated by the lowest partial $s$-waves with zero
relative orbital angular momentum. This implies that the matrix elements
squared $|M_{P_{ci}^{+}\to{J/\psi}p}|^{2}$ (and
$|M_{P_{ci}^{0}\to{J/\psi}n}|^{2}$) lead to an isotropic angular distributions
of the $P_{ci}^{+}\to{J/\psi}p$ (and $P_{ci}^{0}\to{J/\psi}n$) decays for the
considered spin-parity assignments of the $P_{ci}^{+}$ (and $P_{ci}^{0}$)
states. With this, we can readily obtain the following relation between
$|M_{P_{ci}^{+}\to{J/\psi}p}|^{2}$ and the partial width
$\Gamma_{P_{ci}^{+}\to{J/\psi}p}$ of the $P_{ci}^{+}\to{J/\psi}p$ decay (cf.
[10]):

$\frac{1}{2J+1}\frac{|M_{P_{ci}^{+}\to{J/\psi}p}|^{2}}{(2\pi)^{2}}=\frac{2s^{*}}{\pi{p^{*}_{J/\psi}}}\Gamma_{P_{ci}^{+}\to{J/\psi}p}.$
(59)

With it, we find for the expression (55) a more simpler form (cf. Eq. (9)):

$\frac{d\sigma_{{\gamma}p\to P_{ci}^{+}\to{J/\psi}p}(\sqrt{s^{*}},{\bf
p}_{J/\psi})}{d{\bf p}_{J/\psi}}=\sigma_{{\gamma}p\to
P_{ci}^{+}}(\sqrt{s^{*}},\Gamma_{ci}^{+})\theta[\sqrt{s^{*}}-(m_{J/\psi}+m_{p})]\times$
(60)
$\times\frac{1}{I_{2}(s^{*},m_{J/\psi},m_{p})}Br[P_{ci}^{+}\to{J/\psi}p]\frac{1}{4E_{J/\psi}}\frac{1}{(\omega+E_{t})}\delta\left[\omega+E_{t}-\sqrt{m_{p}^{2}+({\bf
Q}+{\bf p}_{t})^{2}}\right],$

where the quantities $\omega$ and ${\bf Q}$ are defined above by Eq. (12). We
will employ this expression in our calculations of the $J/\psi$ momentum
distribution from the processes (41)–(44) in ${\gamma}A$ reactions.
Integrating the differential cross section (53) over the angular range of
${\Delta}{\bf\Omega}_{J/\psi}$=$0^{\circ}\leq\theta_{J/\psi}\leq 20^{\circ}$,
$0\leq\varphi_{J/\psi}\leq 2{\pi}$ of our interest, we represent this
distribution for given $i$ in this angular range in the following form (cf.
Eq. (24)):

$\frac{d\sigma_{{\gamma}A\to{J/\psi}X}^{({\rm
sec})}(p_{\gamma},p_{J/\psi})}{dp_{J/\psi}}=\int\limits_{{\Delta}{\bf\Omega}_{J/\psi}}d{\bf\Omega}_{J/\psi}\frac{d\sigma_{{\gamma}A\to{J/\psi}X}^{({\rm
sec})}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}p_{J/\psi}^{2}$
(61) $=2{\pi}I_{V}[A,\sigma^{\rm in}_{{P_{c}}N}]\int\limits_{\cos
20^{\circ}}^{1}d\cos{{\theta_{J/\psi}}}\left<\frac{d\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}{p}}(p_{\gamma},p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf\Omega}_{J/\psi}}\right>_{A},\,\,\,i=1,2,3,4.$

Before going to the next step, we calculate the free space resonant $J/\psi$
energy distribution $d\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}{p}}[\sqrt{s},p_{J/\psi}]/dE_{J/\psi}$ from the two-step
processes (41)/(43), proceeding on the free target proton at rest, in addition
to that from the background ${\gamma}p\to{J/\psi}p$ reaction (cf. Eq. (39)).
The energy-momentum conservation in these precesses leads to the conclusion
that the kinematical characteristics of $J/\psi$ mesons produced in them and
in this reaction are the same at given incident photon energy. The full on-
shell differential cross section $d\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}{p}}[\sqrt{s},{\bf p}_{J/\psi}]/d{\bf p}_{J/\psi}$ can be
obtained from more general one (60) in the limits: ${\bf p}_{t}\to 0$,
$E_{t}\to m_{p}$ and $s^{*}\to s$. Its integration over the laboratory polar
angle $\theta_{J/\psi}$, when this angle belongs to the allowed angular
interval (25), gives:

$\frac{d\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}{p}}[\sqrt{s},p_{J/\psi}]}{dE_{J/\psi}}=2\pi\int\limits_{\cos{\theta_{J/\psi}^{\rm
max}}}^{1}d\cos{\theta_{J/\psi}}p_{J/\psi}E_{J/\psi}\frac{d\sigma_{{\gamma}p\to
P_{ci}^{+}\to{J/\psi}p}[\sqrt{s},{\bf p}_{J/\psi}]}{d{\bf p}_{J/\psi}}=$ (62)
$=\sigma_{{\gamma}p\to
P_{ci}^{+}}(\sqrt{s},\Gamma_{ci}^{+})\theta[\sqrt{s}-(m_{J/\psi}+m_{p})]\times$
$\times\left(\frac{\sqrt{s}}{2p_{\gamma}p^{*}_{J/\psi}}\right)Br[P_{ci}^{+}\to{J/\psi}p]~{}{\rm
for}~{}E^{(2)}_{J/\psi}(0^{\circ})\leq E_{J/\psi}\leq
E^{(1)}_{J/\psi}(0^{\circ}).$

Eq. (62) shows that the free space $J/\psi$ energy distribution, which arises
from the production/decay chains (41)/(43), exhibits a completely flat
behavior within the allowed energy range (38).

Figure 5: (Color online) The direct non-resonant $J/\psi$ energy distribution
in the free space elementary process ${\gamma}p\to{J/\psi}p$, calculated in
line with Eq. (39) at initial photon resonant energy of 9.44 GeV in the
laboratory system (solid curve). The resonant $J/\psi$ energy distributions in
the two-step processes ${\gamma}p\to P_{c}^{+}(4312)\to{J/\psi}p$,
${\gamma}p\to P_{c}^{+}(4337)\to{J/\psi}p$, ${\gamma}p\to
P_{c}^{+}(4440)\to{J/\psi}p$ and ${\gamma}p\to P_{c}^{+}(4457)\to{J/\psi}p$,
calculated in line with Eq. (62) at the same incident photon energy of 9.44
GeV assuming that the resonances $P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$,
$P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$ with the spin-parity assignments
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$,
correspondingly, all decay to the ${J/\psi}p$ with branching fractions 0.25%
(respectively, red dashed, blue dotted, dark cyan dashed-doted and magenta
dashed-dotted-dotted curves). Incoherent sum of the direct non-resonant
$J/\psi$ energy distribution and resonant ones, calculated supposing that the
resonances $P_{c}^{+}(4312)$ and $P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$,
$P_{c}^{+}(4440)$, $P_{c}^{+}(4457)$ with the same spin-parity combinations
all decay to the ${J/\psi}p$ with branching fractions 0.25, 0.5 and 1%
(respectively, dark yellow short-dashed, wine short-dashed-dotted, olive
dashed-dotted and navy short-dotted, pink dotted, royal dashed-dotted-dotted
curves), all as functions of the total $J/\psi$ energy $E_{J/\psi}$ in the
laboratory system. The vertical dotted lines indicate the range of $J/\psi$
allowed energies in this system for the considered direct non-resonant and
resonant $J/\psi$ production off a free target proton at rest at given initial
photon resonant energy of 9.44 GeV. Figure 6: (Color online) The direct non-
resonant $J/\psi$ energy distribution in the free space elementary process
${\gamma}p\to{J/\psi}p$, calculated in line with Eq. (39) at initial photon
resonant energy of 9.554 GeV in the laboratory system (solid curve). The
resonant $J/\psi$ energy distributions in the two-step processes ${\gamma}p\to
P_{c}^{+}(4312)\to{J/\psi}p$, ${\gamma}p\to P_{c}^{+}(4337)\to{J/\psi}p$,
${\gamma}p\to P_{c}^{+}(4440)\to{J/\psi}p$ and ${\gamma}p\to
P_{c}^{+}(4457)\to{J/\psi}p$, calculated in line with Eq. (62) at the same
incident photon energy of 9.554 GeV assuming that the resonances
$P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$, $P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$
with the spin-parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$,
$J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$, correspondingly, all decay to the
${J/\psi}p$ with branching fractions 0.25% (respectively, red dashed, blue
dotted, dark cyan dashed-doted and magenta dashed-dotted-dotted curves).
Incoherent sum of the direct non-resonant $J/\psi$ energy distribution and
resonant ones, calculated supposing that the resonances $P_{c}^{+}(4337)$ and
$P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$, $P_{c}^{+}(4440)$, $P_{c}^{+}(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p$ with
branching fractions 0.25, 0.5 and 1% (respectively, dark yellow short-dashed,
wine short-dashed-dotted, olive dashed-dotted and navy short-dotted, pink
dotted, royal dashed-dotted-dotted curves), all as functions of the total
$J/\psi$ energy $E_{J/\psi}$ in the laboratory system. The vertical dotted
lines indicate the range of $J/\psi$ allowed energies in this system for the
considered direct non-resonant and resonant $J/\psi$ production off a free
target proton at rest at given initial photon resonant energy of 9.554 GeV.
Figure 7: (Color online) The direct non-resonant $J/\psi$ energy distribution
in the free space elementary process ${\gamma}p\to{J/\psi}p$, calculated in
line with Eq. (39) at initial photon resonant energy of 10.04 GeV in the
laboratory system (solid curve). The resonant $J/\psi$ energy distributions in
the two-step processes ${\gamma}p\to P_{c}^{+}(4312)\to{J/\psi}p$,
${\gamma}p\to P_{c}^{+}(4337)\to{J/\psi}p$, ${\gamma}p\to
P_{c}^{+}(4440)\to{J/\psi}p$ and ${\gamma}p\to P_{c}^{+}(4457)\to{J/\psi}p$,
calculated in line with Eq. (62) at the same incident photon energy of 10.04
GeV assuming that the resonances $P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$,
$P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$ with the spin-parity assignments
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$,
correspondingly, all decay to the ${J/\psi}p$ with branching fractions 0.25%
(respectively, red dashed, blue dotted, dark cyan dashed-doted and magenta
dashed-dotted-dotted curves). Incoherent sum of the direct non-resonant
$J/\psi$ energy distribution and resonant ones, calculated supposing that the
resonances $P_{c}^{+}(4440)$ and $P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$,
$P_{c}^{+}(4440)$, $P_{c}^{+}(4457)$ with the same spin-parity combinations
all decay to the ${J/\psi}p$ with branching fractions 0.25, 0.5 and 1%
(respectively, dark yellow short-dashed, wine short-dashed-dotted, olive
dashed-dotted and navy short-dotted, pink dotted, royal dashed-dotted-dotted
curves), all as functions of the total $J/\psi$ energy $E_{J/\psi}$ in the
laboratory system. The vertical dotted lines indicate the range of $J/\psi$
allowed energies in this system for the considered direct non-resonant and
resonant $J/\psi$ production off a free target proton at rest at given initial
photon resonant energy of 10.04 GeV. Figure 8: (Color online) The direct non-
resonant $J/\psi$ energy distribution in the free space elementary process
${\gamma}p\to{J/\psi}p$, calculated in line with Eq. (39) at initial photon
resonant energy of 10.12 GeV in the laboratory system (solid curve). The
resonant $J/\psi$ energy distributions in the two-step processes ${\gamma}p\to
P_{c}^{+}(4312)\to{J/\psi}p$, ${\gamma}p\to P_{c}^{+}(4337)\to{J/\psi}p$,
${\gamma}p\to P_{c}^{+}(4440)\to{J/\psi}p$ and ${\gamma}p\to
P_{c}^{+}(4457)\to{J/\psi}p$, calculated in line with Eq. (62) at the same
incident photon energy of 10.12 GeV assuming that the resonances
$P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$, $P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$
with the spin-parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$,
$J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$, correspondingly, all decay to the
${J/\psi}p$ with branching fractions 0.25% (respectively, red dashed, blue
dotted, dark cyan dashed-doted and magenta dashed-dotted-dotted curves).
Incoherent sum of the direct non-resonant $J/\psi$ energy distribution and
resonant ones, calculated supposing that the resonances $P_{c}^{+}(4457)$ and
$P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$, $P_{c}^{+}(4440)$, $P_{c}^{+}(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p$ with
branching fractions 0.25, 0.5 and 1% (respectively, dark yellow short-dashed,
wine short-dashed-dotted, olive dashed-dotted and navy short-dotted, pink
dotted, royal dashed-dotted-dotted curves), all as functions of the total
$J/\psi$ energy $E_{J/\psi}$ in the laboratory system. The vertical dotted
lines indicate the range of $J/\psi$ allowed energies in this system for the
considered direct non-resonant and resonant $J/\psi$ production off a free
target proton at rest at given initial photon resonant energy of 10.12 GeV.
Figure 9: (Color online) The direct non-resonant momentum distribution of
$J/\psi$ mesons, produced in the reaction ${\gamma}{\rm{}^{12}C}\to{J/\psi}X$
in the laboratory polar angular range of 0∘–20∘ and calculated in line with
Eq. (24) at initial photon resonant energy of 9.44 GeV in the laboratory
system (solid curve). The resonant momentum distributions of $J/\psi$ mesons,
produced in the two-step processes ${\gamma}p(n)\to
P_{c}^{+}(4312)(P_{c}^{0}(4312))\to{J/\psi}p(n)$, ${\gamma}p(n)\to
P_{c}^{+}(4337)(P_{c}^{0}(4337))\to{J/\psi}p(n)$, ${\gamma}p(n)\to
P_{c}^{+}(4440)(P_{c}^{0}(4440))\to{J/\psi}p(n)$ and ${\gamma}p(n)\to
P_{c}^{+}(4457)(P_{c}^{0}(4457))\to{J/\psi}p(n)$ and calculated in line with
Eq. (61) at the same incident photon energy of 9.44 GeV assuming that the
resonances $P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$ and
$P_{c}^{+,0}(4457)$ with the spin-parity assignments $J^{P}=(1/2)^{-}$,
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$, correspondingly,
all decay to the ${J/\psi}p(n)$ with branching fractions 0.25% (respectively,
red dashed, blue dotted, dark cyan dashed-doted and magenta dashed-dotted-
dotted curves) and their incoherent sum (orange dotted curve). Incoherent sum
of the direct non-resonant $J/\psi$ momentum distribution and resonant ones,
calculated supposing that the resonances $P_{c}^{+,0}(4312)$,
$P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$, $P_{c}^{+,0}(4457)$ with the same
spin-parity combinations all decay to the ${J/\psi}p(n)$ with branching
fractions 0.25, 0.5 and 1% (respectively, green short-dashed, navy short-
dotted and pink short-dashed-dotted curves), all as functions of the $J/\psi$
momentum $p_{J/\psi}$ in the laboratory frame. Figure 10: (Color online) The
direct non-resonant momentum distribution of $J/\psi$ mesons, produced in the
reaction ${\gamma}{\rm{}^{184}W}\to{J/\psi}X$ in the laboratory polar angular
range of 0∘–20∘ and calculated in line with Eq. (24) at initial photon
resonant energy of 9.44 GeV in the laboratory system (solid curve). The
resonant momentum distributions of $J/\psi$ mesons, produced in the two-step
processes ${\gamma}p(n)\to P_{c}^{+}(4312)(P_{c}^{0}(4312))\to{J/\psi}p(n)$,
${\gamma}p(n)\to P_{c}^{+}(4337)(P_{c}^{0}(4337))\to{J/\psi}p(n)$,
${\gamma}p(n)\to P_{c}^{+}(4440)(P_{c}^{0}(4440))\to{J/\psi}p(n)$ and
${\gamma}p(n)\to P_{c}^{+}(4457)(P_{c}^{0}(4457))\to{J/\psi}p(n)$ and
calculated in line with Eq. (61) at the same incident photon energy of 9.44
GeV assuming that the resonances $P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$,
$P_{c}^{+,0}(4440)$ and $P_{c}^{+,0}(4457)$ with the spin-parity assignments
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$,
correspondingly, all decay to the ${J/\psi}p(n)$ with branching fractions
0.25% (respectively, red dashed, blue dotted, dark cyan dashed-doted and
magenta dashed-dotted-dotted curves) and their incoherent sum (orange dotted
curve). Incoherent sum of the direct non-resonant $J/\psi$ momentum
distribution and resonant ones, calculated supposing that the resonances
$P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$,
$P_{c}^{+,0}(4457)$ with the same spin-parity combinations all decay to the
${J/\psi}p(n)$ with branching fractions 0.25, 0.5 and 1% (respectively, green
short-dashed, navy short-dotted and pink short-dashed-dotted curves), all as
functions of the $J/\psi$ momentum $p_{J/\psi}$ in the laboratory frame.
Figure 11: (Color online) The same as in Fig. 9, but for the initial photon
energy of 10.12 GeV. Figure 12: (Color online) The same as in Fig. 10, but for
the initial photon energy of 10.12 GeV.

## 3\. Results

The free space direct non-resonant $J/\psi$ production total cross section
(23) (solid curve), the total cross section for the resonant $J/\psi$
production in the processes (41)/(43) determined on the basis of Eq. (52) for
the considered spin-parity assignments of the hidden-charm resonances
$P_{ci}^{+}$ ($i=1$, 2, 3, 4) and for branching ratios
$Br[P_{ci}^{+}\to{J/\psi}p]=1$% for all four $P_{ci}^{+}$ states (short-dashed
curve) and the combined (non-resonant plus resonant) $J/\psi$ production total
cross section (dotted curve) are presented in Fig. 4 as functions of photon
energy. It can be seen from this figure that the $P_{c}^{+}(4312)$ and
$P_{c}^{+}(4337)$ as well as $P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$
resonances exhibit itself as two narrow overlapping peaks, respectively, at
$E_{\gamma}=9.44$ and $E_{\gamma}=9.554$ GeV as well as at $E_{\gamma}=10.04$
and $E_{\gamma}=10.12$ GeV. The strengths of these four peaks reach a value
$\sim$ 0.1–0.2 nb. Whereas, the non-resonant contribution in the resonance
region is of about 1 nb. As a result, the combined total cross section of the
reaction ${\gamma}p\to{J/\psi}p$ has no distinct peak structures,
corresponding to the $P_{ci}^{+}$ states, and it is practically not
distinguished from that for the background reaction. If
$Br[P_{ci}^{+}\to{J/\psi}p]=0.25$ and 0.5%, then the resonant $J/\psi$ yield
will be even more less than the non-resonant one. This means that will be very
hard to measure the $P_{ci}^{+}$ pentaquark states in $J/\psi$ total
photoproduction cross section on a proton target in the near-threshold energy
region. Evidently, to see their experimentally one needs to consider such
observable, which is appreciably sensitive to the $P_{c}^{+}$ signal in some
region of the available phase space. For example, the large $t$ region of the
differential cross section $d\sigma/dt$ in the $J/\psi$-007 experiment [17],
where the $t$-dependence of the background $J/\psi$ meson production is
suppressed while its resonant production is rather flat. This is also
supported by the findings of Ref. [25], where the photoproduction of initially
claimed by the LHCb Collaboration hidden charm pentaquark states
$P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$ with the spin-parity assignments of
$(3/2^{-},5/2^{+})$ or $(3/2^{+},5/2^{-})$, respectively, on the proton target
was considered by including the $t$-channel diffractive Pomeron exchanges and
the $s$-channel pentaquark productions. Here, by assuming that the pentaquark
states decay into the ${J/\psi}p$ mode with fraction of 5% was, in particular,
shown that the contributions from the $P_{c}^{+}$ states calculated at
resonant c.m. energies $W=4.38$ GeV and 4.45 GeV for the two spin-parity
combinations considered make the differential cross section of the
${\gamma}p\to{J/\psi}p$ reaction strongly deviated from the diffractive one at
off-forward angles in the c.m. frame. This cross section indeed is rather flat
at these angles and overestimates at them significantly the contributions from
the diffractive Pomeron exchanges. Furthermore, the predictions for the
differential cross section of the ${\gamma}p\to{J/\psi}p$ reaction, obtained
in Ref. [26] within the approach in which the Pomeron-exchange model with the
parameters determined from fitting the available total cross section data up
to $W=300$ GeV is used to calculate the non-resonant amplitudes as well as the
partial decay widths of nucleon resonances with hidden charm,
$N^{*}_{c{\bar{c}}}$, predicted by the considered meson-baryons ($MB$)
coupled-channel models to estimate the $N^{*}_{c{\bar{c}}}\to MB$ transition
matrix elements and the vector-meson dominance model to evaluate ${\gamma}p\to
N^{*}_{c{\bar{c}}}$ as ${\gamma}p\to Vp\to N^{*}_{c{\bar{c}}}$ with
$V=\rho,\omega,J/\psi$ are adopted, demonstrate that the $N^{*}_{c{\bar{c}}}$
can be readily identified in the near-threshold differential cross section of
the ${\gamma}p\to{J/\psi}p$ process at large angles where the contribution
from Pomeron exchanges becomes insignificant. It should also be noted that an
earlier prediction of the differential cross section of this process, made in
Ref. [27] at the resonant energy point $W=4.412$ GeV by considering the non-
resonant (${\gamma}p\to{J/\psi}p$) and resonant (${\gamma}p\to
N^{*}_{c{\bar{c}}}(4412)\to{J/\psi}p$) ${J/\psi}p$ photoproduction using,
respectively, the two gluon and three gluon exchange model [14] and vector-
meson dominance model to generate vector mesons $\rho,\omega,J/\psi$ from
photon which rescatter with the target proton to form intermediate hidden
charmed nucleon resonance $N^{*}_{c{\bar{c}}}(4412)$, shows as well that this
cross section is quite weakly dependent on the c.m.s. $J/\psi$ production
angle. It should be additionally pointed out that the feasibility of detecting
the $P_{c}^{+}(4450)$ resonance with the spin-parity quantum numbers
$J^{P}=3/2^{-}$ and $J^{P}=5/2^{+}$ in near-threshold $J/\psi$ photoproduction
off protons in the CLAS12 experiment at JLab was also discussed in Ref. [23]
in the framework of a two-component model containing the directly produced
resonance, diffractive background and accounting for the experimental
resolution effects. The contribution of the $P_{c}^{+}(4450)$ state, produced
through the vector-meson dominance mechanism, was parametrized using the
Breit-Wigner ansatz and the non-resonant contribution was described by the
Pomeron exchanges. The fit of the available at that time data points for
differential cross section of the ${\gamma}p\to{J/\psi}p$ reaction, with
$|t|\leq~{}1.5$ GeV2, covering energy range from threshold to
$E_{\gamma}\sim~{}120$ TeV in the lab frame, with this model showed that the
upper limits for branching ratio $Br[P_{c}^{+}(4450)\to{J/\psi}p]$ of the
$P_{c}^{+}(4450)$ pentaquark range from 23% to 30% for $J=3/2$, depending on
the experimental resolution, and from 8% to 17% for $J=5/2$. These are
essentially larger than those of several percent set later on by the GlueX
Collaboration [4]. Finally, it is worth noting that the photoproduction of the
$J/\psi$ off the proton near threshold was studied in Ref. [28] using a novel
final ${J/\psi}p$ production mechanism via the open charm
$\Lambda_{c}^{+}{\bar{D}}^{0}$ and $\Lambda_{c}^{+}{\bar{D}}^{*0}$
intermediate states. The authors found that the existing experimental data [4]
on ${\gamma}p\to{J/\psi}p$ can be well described within the suggested
mechanism. Moreover, they identified a clear experimental signature for this
mechanism: within it must be pronounced cusps at the
$\Lambda_{c}^{+}{\bar{D}}^{0}$ and $\Lambda_{c}^{+}{\bar{D}}^{*0}$ thresholds
in the energy dependence of the total cross section of the
${\gamma}p\to{J/\psi}p$ reaction, and found that the data [4] consistent with
this feature within their accuracy. One may hope that further measurements of
the $J/\psi$ photoproduction off the proton at JLab with higher statistics
than GlueX will provide a deeper understanding of the $J/\psi$ photoproduction
mechanism.

Taking into account the aforementioned, now we consider the $J/\psi$ energy
distribution from the considered ${\gamma}p\to{J/\psi}p$ elementary reaction.
The model developed by us allows to calculate the direct non-resonant $J/\psi$
energy distribution from this reaction, the resonant ones from the
production/decay chains (41)/(43), proceeding on the free target proton being
at rest. They were calculated according to Eqs. (39), (62), respectively, for
incident photon resonant energies of 9.44, 9.554, 10.04 and 10.12 GeV. The
resonant $J/\psi$ energy distributions were determined for the considered
spin-parity assignments of the $P_{c}^{+}(4312)$, $P_{c}^{+}(4337)$,
$P_{c}^{+}(4440)$, $P_{c}^{+}(4457)$ resonances for branching fractions
$Br[P_{ci}^{+}\to{J/\psi}p]=$ 0.25% for all four states. These dependencies,
together with the incoherent sum of the non-resonant $J/\psi$ energy
distribution and resonant ones, calculated assuming that all the resonances
$P_{c}^{+}(4312)$ and $P_{ci}^{+}$ ($i=1$, 2, 3, 4), $P_{c}^{+}(4337)$ and
$P_{ci}^{+}$ ($i=1$, 2, 3, 4), $P_{c}^{+}(4440)$ and $P_{ci}^{+}$ ($i=1$, 2,
3, 4), $P_{c}^{+}(4457)$ and $P_{ci}^{+}$ ($i=1$, 2, 3, 4) decay to the
${J/\psi}p$ mode with three adopted options for the branching ratios
$Br[P_{ci}^{+}\to{J/\psi}p]$, as functions of the $J/\psi$ total energy
$E_{J/\psi}$ are shown, respectively, in Figs. 5, 6, 7, 8. It is seen from
these figures that the resonant $J/\psi$ production cross sections show a flat
behavior at all allowed energies $E_{J/\psi}$. Whereas the non-resonant cross
section drops fastly as $E_{J/\psi}$ decreases. At incident photon resonant
energies of 9.44, 9.554, 10.04 and 10.12 GeV of interest its strength is
essentially larger than those of the resonant $J/\psi$ production cross
sections, calculated for the value of the branching ratios
$Br[P_{ci}^{+}\to{J/\psi}p]=$ 0.25% for ”high” allowed $J/\psi$ total energies
greater than $\approx$ 7.25 GeV. Whereas at ”low” $J/\psi$ total energies
(below 7.25 GeV) and for each considered photon energy the contribution from
the resonance with the centroid at this energy, decaying to the ${J/\psi}p$
with the branching ratio of 0.25%, is much larger than the non-resonant one.
Thus, for instance, in this case for the $J/\psi$ mesons with total energy of
6.5 GeV their resonant production cross section is enhanced compared to the
non-resonant one by sizeable factors of about 2.9, 3.6, 9.5 and 22.5 at
initial photon energies of 9.44, 9.554, 10.04 and 10.12 GeV, respectively.
Moreover, this contribution is also substantially larger than those, arising
from the decays of another three pentaquarks to the ${J/\psi}p$ channel with
the branching ratios $Br[P_{ci}^{+}\to{J/\psi}p]=$ 0.25%, at the above-
mentioned ”low” $J/\psi$ total energies. As a result, at each considered
photon energy the $J/\psi$ meson combined energy distribution, deriving from
the direct $J/\psi$ meson production and from the decay of the pentaquark
resonance located at this energy to the ${J/\psi}p$ mode, reveals here a clear
sensitivity to the adopted variations in the branching ratio of this decay.
Thus, for example, for the $J/\psi$ mesons with total energy of 6.5 GeV and
for the lowest considered incident photon energy of 9.44 GeV this $J/\psi$
combined distribution is enhanced for the values of this ratio of 0.25, 0.5
and 1% by notable factors of about 4.0, 12.5 and 46.8, respectively, as
compared to that from the directly produced $J/\psi$ mesons. And for the
highest initial photon energy of 10.12 GeV of our interest, at which the
resonance $P_{c}^{+}(4457)$ appears as peak structure in the total cross
section of the exclusive reaction ${\gamma}p\to{J/\psi}p$, the analogous
factors become much larger and they are of about 23.5, 90.8 and 360.3,
respectively. Furthermore, one can see that the above ”partial” combined
energy distribution of the $J/\psi$ mesons is practically indistinguishable
from their ”total” combined differential energy distribution, arising from the
direct and resonant $J/\psi$ meson production via the production/decay chains
(41)/(43). This implies, on the one hand, that the differences between the
combined results, obtained by using a conservative value of the branching
fractions of the decays $P_{ci}^{+}\to{J/\psi}p$ of 0.25% and the non-resonant
background, as well as between the combined results, determined by employing
the values of the branching ratios of these decays of 0.25 and 0.5%, 0.5 and
1%, are quite sizeable and experimentally measurable at ”low” charmonium total
energies. On the other hand, at each incident photon resonant energy
considered the observation here of the specific hidden-charm LHCb pentaquark
will be practically not influenced by the presence of the another three
hidden-charm pentaquark states and by the background reaction. Since the
$J/\psi$ production differential cross sections have a small absolute values
$\sim$ 0.01–0.1 nb/GeV at ”low” $J/\psi$ total energies $E_{J/\psi}$, their
measurement requires both high luminosities and large-acceptance detectors.
Such measurement might be performed in the near future at the JLab in Hall A
within the planned here high-statistics ($\sim$ 800k $J/\psi$ events in
photoproduction) and high-precision E12-12-006 experiment using the SoLID
detector [5, 17].

The momentum dependencies of the absolute non-resonant, resonant and combined
$J/\psi$ meson differential cross sections, correspondingly, from the direct
(1), (2), two-step (41)/(43), (42)/(44) and direct plus two-step $J/\psi$
production processes in $\gamma$12C and $\gamma$184W interactions, calculated
on the basis of Eqs. (24), (61) for laboratory polar angles of 0∘–20∘ and for
incident photon lowest resonant energy of 9.44 GeV, are shown, respectively,
in Figs. 9 and 10. The same as in these figures, but for initial highest
photon resonant energy of 10.12, is presented in Figs. 11 and 12. The resonant
momentum differential cross sections for the production of $J/\psi$ mesons in
the two-step processes ${\gamma}p\to P_{ci}^{+}\to{J/\psi}p$ and ${\gamma}n\to
P_{ci}^{0}\to{J/\psi}n$ ($i=1$, 2, 3, 4), proceeding on the intranuclear
nucleons of carbon and tungsten target nuclei, were obtained for three
employed values of the branching ratios $Br[P_{ci}^{+}\to{J/\psi}p]$ and
$Br[P_{ci}^{0}\to{J/\psi}n]$. It can be seen from these figures that the total
contribution to the $J/\psi$ production on both these nuclei, coming from the
intermediate $P_{ci}^{+}$ and $P_{ci}^{0}$ states decaying to the ${J/\psi}p$
and ${J/\psi}n$ modes with branching fractions of 0.25%, shows practically
flat behavior, and it is significantly larger than that from the background
processes (1), (2) in the ”low”-momentum regions of 4.5–5.5 GeV/c and 4.5–6
GeV/c for considered photon beam energies of 9.44 and 10.12 GeV, respectively.
As a result, in them the combined charmonium yield is completely governed by
the presence of the $P_{ci}^{+}$ and $P_{ci}^{0}$ states in its production.
Its strength is almost totally determined by the branching ratios
$Br[P_{ci}^{+}\to{J/\psi}p]$ and $Br[P_{ci}^{0}\to{J/\psi}n]$ used in the
calculations with a value, which is still large enough to be measured, as one
may hope, at the CEBAF (cf. [13]), and which increases by a factor of about
ten for both photon beam energies considered when going from carbon target
nucleus to tungsten one 121212)It is interesting to note that the
photoproduction of $J/\psi$-3He bound state ([3He]J/ψ) on a 4He target has
been investigated in Ref. [29] using the impulse approximation, several
$\gamma+N\to J/\psi+N$ models based on the Pomeron-exchange and accounting for
the pion-exchange mechanism at low energies, and various $J/\psi$-nucleus
potentials. The upper boundary of the predicted total cross sections was found
to be very small – it is about 0.1–0.3 pb. The possibility of photoproduction
of a six quark-$J/\psi$ bound state ([$q^{6}$]J/ψ) on the 3He target has been
studied in Ref. [29] as well. The upper boundary of the predicted total cross
sections of $\gamma+^{3}{\rm He}\to[q^{6}]_{J/\psi}+N$ was obtained to be
slightly larger than in the preceding case – it is about 2–4 pb, depending on
the model of $\gamma+N\to J/\psi+N$ used in the calculations. These
predictions may facilitate the planning of possible measurements of [3He]J/ψ
and [$q^{6}$]J/ψ bound states at JLab.) . This leads to the well separated and
experimentally distinguishable differences between all combined calculations,
corresponding to the adopted options for these ratios, for both target nuclei
and for both photon energies considered. Since the $J/\psi$ meson production
differential cross sections at photon beam energy of 9.44 GeV are larger than
those at the energy of 10.12 GeV by a factor of about 20 in the above
”low”-momentum regions, their measurements on light and especially on heavy
nuclear targets at photon energies in the ”low”-energy resonance region will
open an opportunity to determine accurately the above branching ratios – at
least to distinguish between their realistic options of 0.25, 0.5 and 1%. Such
measurements could also be performed in the future at the JLab in the
framework of the proposed here E12-12-006 experiment [5, 17].

Accounting for the above considerations, we can conclude that the near-
threshold $J/\psi$ energy and momentum distribution measurements in photon-
induced reactions both on protons and on nuclear targets will provide further
evidence for the existence of the pentaquark $P_{ci}^{+}$, $P_{ci}^{0}$
resonances, and will shed light on their decay rates to the channels
${J/\psi}p$ and ${J/\psi}n$.

## 4\. Epilogue

In this paper we studied the near-threshold $J/\psi$ meson photoproduction
from protons and nuclei by considering incoherent direct non-resonant
(${\gamma}p\to{J/\psi}p$, ${\gamma}n\to{J/\psi}n$) and two-step resonant
(${\gamma}p\to P_{ci}^{+}\to{J/\psi}p$, ${\gamma}n\to P_{ci}^{0}\to{J/\psi}n$,
$i=1$, 2, 3, 4; $P_{c1}^{+,0}=P_{c}^{+,0}(4312)$,
$P_{c2}^{+,0}=P_{c}^{+,0}(4337)$, $P_{c3}^{+,0}=P_{c}^{+,0}(4440)$,
$P_{c4}^{+,0}=P_{c}^{+,0}(4457)$) charmonium production processes. We have
calculated the absolute excitation functions, energy and momentum
distributions for the non-resonant, resonant and for the combined (non-
resonant plus resonant) production of $J/\psi$ mesons on protons as well as,
using the nuclear spectral function approach, on carbon and tungsten target
nuclei at near-threshold incident photon energies by assuming the spin-parity
assignments of the hidden-charm resonances $P_{c}^{+,0}(4312)$,
$P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$ and $P_{c}^{+,0}(4457)$ as
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$
within three different realistic scenarios for the branching ratios of their
decays to the ${J/\psi}p$ and ${J/\psi}n$ modes (0.25, 0.5 and 1%). It was
shown that will be very hard to measure the $P_{ci}^{+}$ pentaquark states
through the scan of the $J/\psi$ total photoproduction cross section on a
proton target in the near-threshold energy region around the resonant photon
energies of 9.44, 9.554, 10.04 and 10.12 GeV if these branching ratios $\sim$
1% and less. It was also demonstrated that at these photon beam energies the
$J/\psi$ energy and momentum combined distributions considered reveal distinct
sensitivity to the above scenarios, respectively, at ”low” $J/\psi$ total
energies and momenta, which implies that they may be an important tool to
provide further evidence for the existence of the pentaquark $P_{ci}^{+}$ and
$P_{ci}^{0}$ resonances and to get valuable information on their decay rates
to the ${J/\psi}p$ and ${J/\psi}n$ final states. The measurements of these
distributions could be performed in the near future at the JLab in Hall A
within the planned here high-statistics ($\sim$ 800k $J/\psi$ events in
photoproduction) and high-precision E12-12-006 experiment using the SoLID
detector.

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