# On the possibility of testing the two-peak structure of the LHCb hidden-
charm strange pentaquark $P_{cs}(4459)^{0}$ in near-threshold antikaon-induced
charmonium production on protons and nuclei

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

###### Abstract

Accounting for the LHCb observation that the reported hidden-charm strange
pentaquark $P_{cs}(4459)^{0}$ can split into two substructures,
$P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$, with a mass difference of 13 MeV as
well as the newly observed hidden-charm pentaquark resonance
$P_{cs}(4338)^{0}$ with strangeness, we study within the double-peak scenario
for the $P_{cs}(4459)^{0}$ state the near-threshold $J/\psi$ meson production
from protons and nuclei by considering incoherent direct non-resonant
(${K^{-}}p\to{J/\psi}\Lambda$) and two-step resonant (${K^{-}}p\to
P_{csi}^{0}\to{J/\psi}\Lambda$, $i=1$, 2, 3; $P_{cs1}^{0}=P_{cs}(4338)^{0}$,
$P_{cs2}^{0}=P_{cs}(4455)^{0}$, $P_{cs3}^{0}=P_{cs}(4468)^{0}$) charmonium
production processes with the main goal of clarifying the possibility to
observe within this scenario both above two substructures contributing to the
$P_{cs}(4459)^{0}$ state and the $P_{cs}(4338)^{0}$ resonance in this
production. 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 on carbon and tungsten target nuclei at near-threshold incident antikaon
beam momenta by assuming the spin-parity assignments of the hidden-charm
resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ as
$J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$ within four
different realistic choices for the branching ratios of their decays to the
${J/\psi}\Lambda$ mode (0.125, 0.25, 0.5 and 1%) as well as for two options
for the branching fraction of their decays to the $K^{-}p$ channel (0.01 and
0.001%). We show that these combined observables reveal clear sensitivity to
these scenarios. Hence, they may be an important tool to provide further
evidence for the existence of the above strange hidden-charm pentaquark
resonances.

## 1\. Introduction

The study of pentaquark states, comprising four quarks and an antiquark, has
received considerable interest in recent years. The first observation of the
hidden-charm pentaquark resonances $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$ in
the ${J/\psi}p$ invariant mass spectrum of the $\Lambda^{0}_{b}\to
K^{-}({J/\psi}p)$ decays has been reported by the LHCb experiment [1] 111)For
better readability of this paper, the old exotic hadron naming scheme is used
throughout it instead of the new one proposed in Ref. [2]. According to the
latter one, the $P_{\psi}^{N}$ would be the new name for the $P_{c}$ states.)
. With the inclusion of additional data, it was found that the
$P_{c}(4450)^{+}$ state is composed of two narrow overlapping peaks,
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$. In addition, a new narrow state, the
$P_{c}(4312)^{+}$ was discovered [3]. Recently, evidence for a new narrow
hidden-charm pentaquark denoted as $P_{c}(4337)^{+}$ was found in the
invariant mass spectrum of the ${J/\psi}p$ in the
$B_{s}^{0}\to{J/\psi}p{\bar{p}}$ decays [4]. The minimal quark structure of
the above pentaquarks is $|P^{+}_{c}>=|uudc{\bar{c}}>$. In a molecular
scenario, due to the closeness of the observed $P_{c}(4312)^{+}$ and
$P_{c}(4440)^{+}$, $P_{c}(4457)^{+}$ masses to the
${\Sigma^{+}_{c}}{\bar{D}}^{0}$ and ${\Sigma^{+}_{c}}{\bar{D}}^{*0}$
thresholds, the $P_{c}(4312)^{+}$ resonance can be, in particular, interpreted
as an S-wave ${\Sigma^{+}_{c}}{\bar{D}}^{0}$ bound state, while the
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ as S-wave
${\Sigma^{+}_{c}}{\bar{D}}^{*0}$ bound molecular states [5–16], though their
inner structure (charmed baryon–charmed antimeson two-body molecule or compact
pentaquark state) is still not determined and there are alternative
explanations too [17–20]. Due to lacking nearby meson-baryon thresholds, the
$P_{c}(4337)^{+}$ is difficult to accommodate within the molecular picture
[21, 22]. But, it is more natural to view it as the compact pentaquark state
[19, 23]. The LHCb Collaboration announced recently also the discovery of two
new narrow hidden-charm strange pentaquark states, $P_{cs}(4459)^{0}$ [24] and
$P_{cs}(4338)^{0}$ [25], with minimal quark content $|udsc{\bar{c}}>$ in the
invariant mass spectra of the ${J/\psi}\Lambda$ in the $\Xi^{-}_{b}\to
K^{-}({J/\psi}\Lambda)$ and $B^{-}\to{\bar{p}}({J/\psi}\Lambda)$ decays
222)Following the new naming convention in [2], in Ref. [25] these newly
observed states are named as $P_{{\psi}s}^{\Lambda}(4459)^{0}$ and
$P_{{\psi}s}^{\Lambda}(4338)^{0}$.) . The Breit-Wigner masses and total widths
of these exotic states were measured to be [24, 25]:

$P_{cs}(4459)^{0}:\,\,\,M=4458.8\pm 2.9^{+4.7}_{-1.1}~{}{\rm
MeV},\,\,\,\Gamma=17.3\pm 6.5^{+8.0}_{-5.7}~{}{\rm MeV};$ (1)
$P_{cs}(4338)^{0}:\,\,\,M=4338.2\pm 0.7\pm 0.4~{}{\rm MeV},\,\,\,\Gamma=7.0\pm
1.2\pm 1.3~{}{\rm MeV}.$ (2)

While the spin-parity quantum numbers $J^{P}$ of the $P_{cs}(4459)^{0}$
resonance were not experimentally determined due to a limited signal yield,
the spin of the $P_{cs}(4338)^{0}$ pentaquark candidate is determined to be
$J=1/2$ and negative parity is preferred [25]. Since the new strange
$P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$ pentaquarks are found close to
thresholds for the production of ordinary baryon-meson states
$\Xi_{c}{\bar{D}}^{*}$ and $\Xi_{c}{\bar{D}}$ 333)The pole mass of the
$P_{cs}(4459)^{0}$ state is just about 19 MeV below the
$\Xi_{c}^{0}{\bar{D}}^{*0}$ threshold [24]. The central value of the
$P_{cs}(4338)^{0}$ mass is only 0.8 and 2.9 MeV above, respectively, the
$\Xi_{c}^{+}D^{-}$ and $\Xi_{c}^{0}{\bar{D}}^{0}$ thresholds [26].) , it is
natural to interpret their as the hidden-charm singly-strange
$\Xi_{c}{\bar{D}}^{*}$ and $\Xi_{c}{\bar{D}}$ molecules. Such plausible
molecular interpretation of the $P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$
exotic states with strangeness was supported by many current theoretical
studies (see, e.g., Refs. [23–38]) and is commonly accepted. However, there
exists another explanations of the $P_{cs}$ states, based both on the compact
quark model [19, 39–42] and on other molecular picture [43], in which it was
found that due to the extra attraction obtained in the $\Xi_{c}{\bar{D}}^{*}$,
$\Xi_{c}{\bar{D}}$ pairs the $P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$ states
are associated, respectively, to the $\Xi_{c}^{\prime}{\bar{D}}$ and
$\Xi_{c}{\bar{D}}^{*}$. It should be pointed out that the existence of hidden-
charm strange pentaquark resonances $P_{cs}$ has been predicted before the
LHCb experiment [24] in some earlier papers (see, for example, Refs. [15,
44–53]). It is also worth noting that the large $Q$-values of the decays
$P_{cs}(4459)^{0}\to{J/\psi}\Lambda$ and $P_{cs}(4338)^{0}\to{J/\psi}\Lambda$
(in which the pentaquarks $P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$ were
observed) – respectively, 246 MeV and 126 MeV – and unnaturally small for such
$Q$-values widths of the $P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$ states
about of 17 MeV and 7 MeV tell us in favor of $\Xi_{c}{\bar{D}}^{*}$ and
$\Xi_{c}{\bar{D}}$ molecular interpretation of these states since in this
interpretation their decay into the ${J/\psi}\Lambda$ is naturally suppressed
due to the fact that two $c$ and ${\bar{c}}$ quarks, belonging, respectively,
to the $\Xi_{c}$ baryon and ${\bar{D}}^{*}$, ${\bar{D}}$ mesons, in hadronic
molecules are well-separated and it is unlikely for them to be close to each
other to form a single $J/\psi$ meson, which provides decay-suppressing
mechanism [26, 29]. Along the way of molecular scenario for the
$P_{cs}(4459)^{0}$ resonance, one can easily find its possible spin-parity
quantum numbers. Considering it as pure $\Xi_{c}{\bar{D}}^{*}$ molecule, in
which $\Xi_{c}$ baryon with $J^{P}=(1/2)^{+}$ and antimeson ${\bar{D}}^{*}$
with $J^{P}=(1)^{-}$ are in a relative S-wave, we get that the spin-parity of
the $P_{cs}(4459)^{0}$ is either $J^{P}=(1/2)^{-}$ or $J^{P}=(3/2)^{-}$ (cf.
Ref. [26]) 444)Similarly, identifying the $P_{cs}(4338)^{0}$ state as the
S-wave $\Xi_{c}(1/2)^{+}{\bar{D}}(0^{-})$ molecular state, we obtain that its
spin-parity combination would be $J^{P}=(1/2)^{-}$ – what exactly was observed
in the LHCb experiment [25].) . This conclusion was supported by more
elaborate theoretical approaches (see, e.g., Refs. [19, 30, 34, 36, 37, 39,
40, 41, 48, 54]), in which the $P_{cs}(4459)^{0}$ is considered as
$\Xi_{c}{\bar{D}}^{*}$ molecular pentaquark state [30, 34, 36, 37, 48, 54] or
as compact pentaquark state [19, 39, 40, 41]. In the absence of coupled-
channel effects the heavy-quark spin symmetry predicts the two
$\Xi_{c}{\bar{D}}^{*}$ states (with total spin $J=1/2$ and $J=3/2$,
respectively) to be degenerate [55]. Their inclusion breaks the spin
degeneracy of the $P_{cs}(4459)^{0}$ and generates a hyperfine splitting
between the two spin configurations [21, 27, 31, 32, 33, 35, 38, 50, 55, 56],
in which the $J^{P}=(1/2)^{-}$ $\Xi_{c}{\bar{D}}^{*}$ pentaquark will be
lighter than the $J^{P}=(3/2)^{-}$ one [27, 38, 50] and vice versa [21, 31,
32, 33, 35, 55, 56]. This means that in the mass region of the
$P_{cs}(4459)^{0}$ (slightly below the $\Xi_{c}{\bar{D}}^{*}$ threshold) may
exist two almost degenerate narrow overlapping $\Xi_{c}{\bar{D}}^{*}$
molecular states, similar to the case for the non-strange hidden-charm
pentaquark $P_{c}(4450)^{+}$, which can be equally replaced by two pentaquark
substructures $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ [3]. In particular, in
this region Ref. [50] predicts, within the chiral effective field theory, the
masses of the two $\Xi_{c}{\bar{D}}^{*}$ molecular states to be:

$\Xi_{c}{\bar{D}}^{*}~{}{\rm
with}~{}J^{P}=(1/2)^{-}:\,\,\,M=4456.9^{+3.2}_{-3.3}~{}{\rm MeV},$
$\Xi_{c}{\bar{D}}^{*}~{}{\rm
with}~{}J^{P}=(3/2)^{-}:\,\,\,M=4463.0^{+2.8}_{-3.0}~{}{\rm MeV}.$ (3)

Motivated by these findings, the LHCb Collaboration tested the hypothesis of
the two-peak structure of the $P_{cs}(4459)^{0}$ resonance [24] with the
predicted in [50] $J^{P}$ values. It was found [24] that this resonance can
also well be described by two pentaquark resonances with mass difference of 13
MeV, named in the present paper as $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$
(cf. [21, 26]). Their masses and widths were determined to be:

$P_{cs}(4455)^{0}:\,\,\,M=4454.9\pm 2.7~{}{\rm MeV},\,\,\,\Gamma=7.5\pm
9.7~{}{\rm MeV};$ $P_{cs}(4468)^{0}:\,\,\,M=4467.8\pm 3.7~{}{\rm
MeV},\,\,\,\Gamma=5.2\pm 5.3~{}{\rm MeV}.$ (4)

We see that the ’experimental’ masses (4) are in excellent agreement with the
calculated ones of Eq. (3). This fact and the theoretical predictions of the
double-peak structure of the $\Xi_{c}{\bar{D}}^{*}$ molecule in the energy
region of the $P_{cs}(4459)^{0}$ resonance, discussed above, make us suggest
that the $P_{cs}(4459)^{0}$ should indeed contain two substructures. And from
now on, we will assume the existence of the two $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$ peaks in spite of the fact that the LHCb analysis of the
current data sample cannot confirm or disprove the two-peak solution (4). One
may hope that the future more precise LHCb measurements and other ones,
stimulated, in particular, by the results of the present work, would indeed
shed light on whether the double-peak interpretation of the $P_{cs}(4459)^{0}$
state is correct.

Remarkably, another hidden-charm exotic state with open strangeness – the
strange hidden-charm tetraquark state $Z_{cs}(3985)^{-}$, decaying into
$D_{s}^{-}D^{*0}$ and $D_{s}^{*-}D^{0}$, has been observed in 2020 by the
BESIII Collaboration in the processes $e^{+}e^{-}\to
K^{+}(D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0})$ [57]. This state has the minimal quark
content $Z_{cs}(3985)^{-}=|{\bar{u}}sc{\bar{c}}>$. It is in the proximity of
the $D_{s}^{-}D^{*0}/D_{s}^{*-}D^{0}$ mass thresholds and can be interpreted
[58, 59] as strange molecular partner with hadronic molecular configuration
$D_{s}^{-}D^{*0}-D_{s}^{*-}D^{0}$ of the non-strange tetraquark resonance
$Z_{c}(3900)^{-}$, having the valence quark content
$Z_{c}(3900)^{-}=|{\bar{u}}dc{\bar{c}}>$ and possible hadronic molecular
structure $D^{-}D^{*0}-D^{*-}D^{0}$. Moreover, in 2022 year the BESIII
Collaboration reported [60] evidence for the neutral $Z_{cs}(3985)^{0}$ state
in the processes $e^{+}e^{-}\to K^{0}_{S}(D_{s}^{+}D^{*-}+D_{s}^{*+}D^{-})$,
whose mass and width are close to those of the charged $Z_{cs}(3985)^{-}$.
Hence, the $Z_{cs}(3985)^{0}$ can be regarded as the isospin partner of the
$Z_{cs}(3985)^{-}$. Earlier, the LHCb Collaboration reported the first
observation of exotic states $Z_{cs}(4000)^{+}$ and $Z_{cs}(4220)^{+}$ with a
new quark content $|u{\bar{s}}c{\bar{c}}>$ in the mass spectra of the
${J/\psi}K^{+}$ in $B^{+}\to{J/\psi}{\phi}K^{+}$ decays [61]. Furthermore, the
first results of searching for a heavier partner of the observed
$Z_{cs}(3985)^{-}$ state, denoted as $Z_{cs}^{{\prime}-}$, in the process
$e^{+}e^{-}\to K^{+}D_{s}^{*-}D^{*0}$ (c.c.) with the BESIII detector have
been reported in very recent publication [62]. In addition, one needs to note
that possible hidden-charm pentaquarks, having two and three strange quarks in
their valence composition (or pentaquarks with double and triple strangeness),
have been theoretically investigated in Refs. [19, 49, 63, 64, 65, 66]. They
are at present an experimentally virgin territory. But one may hope that they
will be discovered in the foreseeable future at the LHC as well. It should be
noticed that the investigation of magnetic moments of the hidden-charm
pentaquark states without strangeness, with strangeness and with double
strangeness, which play an important role in understanding their internal
structure, has been performed, in particular, in Ref. [67] within the QCD
light-cone sum rules. Extensive reviews of the recent experimental and
theoretical progresses in the field of heavy hadronic molecular states are
presented in Refs. [18, 68].

Confirming the $P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$ states, observed in
the $\Xi^{-}_{b}$ and $B^{-}$ decays at LHCb, in other processes would provide
independent and complementary verification of their existence as well as would
be helpful both to understand their nature and to pin down their resonance
parameters. In that way, the production of the single-peak $P_{cs}(4459)^{0}$
pentaquark in antikaon-induced and photon-induced reactions on nuclear targets
was discussed in Refs. [69, 70] and [71], correspondingly. In fact, in Ref.
[69] the contribution of this pentaquark to the elementary
$K^{-}p\to{J/\psi}\Lambda$ reaction has been determined, employing two
different theoretical approaches, i.e., the effective Lagrangian and the Regge
models. It was shown that the $P_{cs}(4459)^{0}$ can also be searched for
through a scan of the total and differential cross sections of this reaction.
In our work [70], we have calculated the absolute excitation functions, energy
and momentum distributions for the direct non-resonant
($K^{-}p\to{J/\psi}\Lambda$), two-step resonant ($K^{-}p\to
P_{cs}(4459)^{0}\to{J/\psi}\Lambda$) and for the combined (non-resonant plus
resonant) production of $J/\psi$ mesons off protons as well as off carbon and
tungsten target nuclei at near-threshold incident antikaon energies by
assuming the spin-parity assignment of the $P_{cs}(4459)^{0}$ resonance as
$J^{P}=(3/2)^{-}$ within six different scenarios for the branching ratio
$Br[P_{cs}(4459)^{0}\to{J/\psi}\Lambda]$ of the decay
$P_{cs}(4459)^{0}\to{J/\psi}\Lambda$, namely:
$Br[P_{cs}(4459)^{0}\to{J/\psi}\Lambda]=$ 1, 3, 5, 10, 15 and 50%. We also
have shown that the combined observables considered reveal definite
sensitivity to these scenarios, which means that they may be an important tool
to provide further evidence for the existence of the pentaquark
$P_{cs}(4459)^{0}$ resonance and to get valuable information on its decay rate
to the ${J/\psi}\Lambda$ final state. It should be noted that the energy of
the negative kaon beam, which will be available at the K10 beam line in the
extended J-PARC Hadron Experimental Facility [72], will be sufficient to
observe the $P_{cs}(4459)^{0}$ pentaquark via the $K^{-}p\to{J/\psi}\Lambda$
process. The possibility of searching for this pentaquark in ${\gamma}p\to
K^{+}P_{cs}(4459)^{0}$ reaction, assuming that it with the spin-parity
combinations $J^{P}=(1/2)^{-}$ or $J^{P}=(3/2)^{-}$ can be interpreted as a
$\Xi_{c}{\bar{D}}^{*}$ molecule or as a compact pentaquark, has been
investigated in Ref. [71]. It was shown that the total cross section of this
reaction, calculated in the latter picture of the $P_{cs}(4459)^{0}$, is quite
different from that, obtained in molecular scenario of the $P_{cs}(4459)^{0}$,
which can be used to test the nature of the $P_{cs}(4459)^{0}$ state [71]. And
finally, in the recent work [73], based on the hadronic molecular picture, the
authors have estimated the semi-inclusive electroproduction rates, in
particular, of the $P_{cs}(4459)^{0}$ and $P_{cs}(4338)^{0}$ multiquark states
at the proposed electron-ion colliders in China (EicC), US (EIC) and at the
proposed 24 GeV upgrade of CEBAF.

In the present article, assuming in view of the aforementioned that the
$P_{cs}(4459)^{0}$ resonance splits into two peaks, $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$, with $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$,
respectively, we consider the contribution both from these peaks and from the
$P_{cs}(4338)^{0}$ state to near-threshold $J/\psi$ meson production by
antikaons on protons and nuclei by adopting the standard Breit-Wigner shape
for this contribution and by employing an available at present information on
the total and differential cross sections of the direct
$K^{-}p\to{J/\psi}\Lambda$ process to estimate the background contribution.
The main goal of this consideration is also to clarify the possibility of
experimental observation of the above two peaks and the $P_{cs}(4338)^{0}$
state in this production. The consideration is strictly based on the model,
developed in Ref. [70]. We briefly recall its main assumptions and describe,
where necessary, the corresponding extensions. We present the predictions
obtained within this extended model for the $J/\psi$ excitation functions,
energy and momentum distributions in ${K^{-}}p$ as well as in ${K^{-}}$12C and
${K^{-}}$184W reactions at near-threshold incident antikaon momenta assuming
different scenarios for the branching fractions of the decays
$P_{cs}(4338/4455/4468)^{0}\to{K^{-}}p$ and
$P_{cs}(4338/4455/4468)^{0}\to{J/\psi}\Lambda$. These predictions may serve as
guidance for future dedicated experiment at the J-PARC facility devoted to the
study of possible two-peak structure of the $P_{cs}(4459)^{0}$ resonance.

## 2\. Theoretical framework

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

Direct non-resonant charmonium production on nuclear targets in the near-
threshold laboratory incident $K^{-}$ beam momentum region $8.844~{}{\rm
GeV/c}\leq p_{K^{-}}\leq 10.200~{}{\rm GeV/c}$ 555)In which the pentaquark
states $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ are
concentrated and where they can be observed in the $K^{-}p$ reactions (see
Figs. 1 and 2 below). We recall that the threshold (resonant) momenta
$p_{K^{-}}^{\rm R1}$, $p^{\rm R2}_{K^{-}}$ and $p^{\rm R3}_{K^{-}}$ for the
production of the $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$ resonances with central values of their masses
$M_{cs1}=4338.2$ MeV, $M_{cs2}=4454.9$ MeV and $M_{cs3}=4467.8$ MeV (cf. Eqs.
(2), (4)) on a free target proton being at rest are $p^{\rm R1}_{K^{-}}=9.417$
GeV/c, $p^{\rm R2}_{K^{-}}=9.965$ GeV/c and $p^{\rm R3}_{K^{-}}=10.026$ GeV/c,
respectively.) may proceed via the following elementary process, which has the
lowest free ${J/\psi}\Lambda$ production threshold momentum (8.844 GeV/c) and
in which the strangeness $S=-1$ is conserved [69, 70]:

$K^{-}+p\to J/\psi+\Lambda.$ (5)

Before going further, let us get a feeling about kinematic characteristics of
$J/\psi$ mesons – laboratory polar production angles and momenta (total
energies), allowed in this process in the simpler case of a free target proton
being at rest at an incident resonant antikaon momenta of 9.417, 9.965 and
10.026 GeV/c of our main interest. The kinematics of two-body reaction with a
threshold (as in our present case) indicate that the laboratory polar $J/\psi$
production angle $\theta_{J/\psi}$ varies from 0 to a maximal value
$\theta^{\rm max}_{J/\psi}$, i.e.:

$0\leq\theta_{J/\psi}\leq\theta^{\rm max}_{J/\psi},$ (6)

where

$\theta^{\rm max}_{J/\psi}={\rm
arcsin}[(\sqrt{s}p^{*}_{J/\psi})/(m_{J/\psi}p_{K^{-}})].$ (7)

Here, the $J/\psi$ c.m. momentum $p^{*}_{J/\psi}$ is determined by the
equation

$p_{J/\psi}^{*}=\frac{1}{2\sqrt{s}}\lambda(s,m_{J/\psi}^{2},m_{\Lambda}^{2}),$
(8)

in which the quantities $m_{J/\psi}$ and $m_{\Lambda}$ are the free space
charmonium and $\Lambda$ hyperon masses and the vacuum collision energy
squared $s$ and function $\lambda(x,y,z)$ are defined, respectively, by the
formulas

$s=(E_{K^{-}}+m_{p})^{2}-{p}_{K^{-}}^{2},\,\,\,\,E_{K^{-}}=\sqrt{m^{2}_{K^{-}}+p^{2}_{K^{-}}},$
(9)
$\lambda(x,y,z)=\sqrt{{\left[x-({\sqrt{y}}+{\sqrt{z}})^{2}\right]}{\left[x-({\sqrt{y}}-{\sqrt{z}})^{2}\right]}}.$
(10)

The quantities $m_{p}$ and $m_{K^{-}}$, entering into Eq. (9), denote the
vacuum proton and $K^{-}$ meson masses. From Eq. (7) one can get that

$\theta^{\rm max}_{J/\psi}=3.932^{\circ},\,\,\,\,\theta^{\rm
max}_{J/\psi}=5.377^{\circ}~{}{\rm and}\,\,\,\,\theta^{\rm
max}_{J/\psi}=5.507^{\circ}$ (11)

at initial resonant antikaon beam momenta of 9.417, 9.965 and 10.026 GeV/c,
respectively. Energy-momentum conservation in the reaction (5), proceeding on
a free target proton at rest, tells us that the kinematically allowed $J/\psi$
meson laboratory momentum $p_{J/\psi}$ and total energy $E_{J/\psi}$
($E_{J/\psi}=\sqrt{m^{2}_{J/\psi}+p^{2}_{J/\psi}})$ in this reaction vary
within the following momentum and energy ranges at given initial antikaon
momentum:

$p^{(2)}_{J/\psi}(0^{\circ})\leq p_{J/\psi}\leq p^{(1)}_{J/\psi}(0^{\circ}),$
(12) $E^{(2)}_{J/\psi}(0^{\circ})\leq E_{J/\psi}\leq
E^{(1)}_{J/\psi}(0^{\circ}),$ (13)

where the quantities $p^{(1,2)}_{J/\psi}(0^{\circ})$ and
$E^{(1,2)}_{J/\psi}(0^{\circ})$ are the $J/\psi$ momenta and energies at zero
polar charmonium production angle in the laboratory frame. They are defined as
follows [70]:

$p^{(1,2)}_{J/\psi}(0^{\circ})={\gamma_{\rm cm}}E^{*}_{J/\psi}(v_{\rm cm}\pm
v^{*}_{J/\psi}),$ (14) $E^{(1,2)}_{J/\psi}(0^{\circ})={\gamma_{\rm
cm}}(E^{*}_{J/\psi}\pm v_{\rm cm}p^{*}_{J/\psi}).$ (15)

Here,

${\gamma_{\rm cm}}=(E_{K^{-}}+m_{p})/\sqrt{s},\,\,\,\,v_{\rm
cm}=p_{K^{-}}/(E_{K^{-}}+m_{p}),\,\,\,\,v^{*}_{J/\psi}=p^{*}_{J/\psi}/E^{*}_{J/\psi},\,\,\,\,E^{*}_{J/\psi}=\sqrt{m^{2}_{J/\psi}+p^{*2}_{J/\psi}}.$
(16)

The sign ”+” in Eqs. (14), (15) corresponds to the first quantities
$p^{(1)}_{J/\psi}$, $E^{(1)}_{J/\psi}$ and sign ”-” - to the second ones
$p^{(2)}_{J/\psi}$, $E^{(2)}_{J/\psi}$. In line with Eqs. (14), (15), for
$p_{K^{-}}=9.417$ GeV/c the inequalities (12), (13) are:

$5.695~{}\leq p_{J/\psi}~{}\leq 7.898~{}{\rm GeV/c},$ (17) $6.483~{}\leq
E_{J/\psi}~{}\leq 8.484~{}{\rm GeV}.$ (18)

For $p_{K^{-}}=9.965$ GeV/c we obtain:

$5.488~{}\leq p_{J/\psi}~{}\leq 8.668~{}{\rm GeV/c},$ (19) $6.300~{}\leq
E_{J/\psi}~{}\leq 9.205~{}{\rm GeV}.$ (20)

And for $p_{K^{-}}=10.026$ GeV/c we also have:

$5.471~{}\leq p_{J/\psi}~{}\leq 8.748~{}{\rm GeV/c},$ (21) $6.286~{}\leq
E_{J/\psi}~{}\leq 9.280~{}{\rm GeV}.$ (22)

Obviously, the binding of target protons and their Fermi motion will distort
the distributions of the outgoing high-momentum (and high-energy) $J/\psi$
mesons 666)And $\Lambda$ hyperons, for which the kinematically allowed
momentum interval looks like $1.286~{}\leq p_{\Lambda}~{}\leq 4.522~{}{\rm
GeV/c}$, for instance, for $p_{K^{-}}=10$ GeV/c [70].) as well as lead to a
wider accessible momentum and energy intervals compared to those given above
by Eqs. (17)–(22). With this, we will ignore the modification of the final
high-momentum $J/\psi$ meson and $\Lambda$ hyperon in the nuclear medium in
the present work in the case when the reaction $K^{-}p\to{J/\psi}\Lambda$
proceeds on a proton embedded in a nuclear target.

For our purposes, we need the $J/\psi$ energy spectrum
$d\sigma_{{K^{-}}p\to{J/\psi}{\Lambda}}[\sqrt{s},p_{J/\psi}]/dE_{J/\psi}$,
arising from this reaction, taking place on a free target proton at rest, as a
function of the $J/\psi$ total energy $E_{J/\psi}$ belonging to the interval
(13). It was calculated [70] to be:

$\frac{d\sigma_{{K^{-}}p\to{J/\psi}{\Lambda}}[\sqrt{s},p_{J/\psi}]}{dE_{J/\psi}}=\left(\frac{2{\pi}\sqrt{s}}{p_{K^{-}}p^{*}_{J/\psi}}\right)\frac{d\sigma_{{K^{-}}p\to{J/\psi}{\Lambda}}[\sqrt{s},\theta_{J/\psi}^{*}(x_{0})]}{d{\bf\Omega}_{J/\psi}^{*}}~{}{\rm
for}~{}E^{(2)}_{J/\psi}(0^{\circ})\leq E_{J/\psi}\leq
E^{(1)}_{J/\psi}(0^{\circ}).$ (23)

Here,
$d\sigma_{{K^{-}}p\to{J/\psi}{\Lambda}}(\sqrt{s},\theta_{J/\psi}^{*})/d{\bf\Omega}_{J/\psi}^{*}$
is the on-shell differential cross section for the production of $J/\psi$
meson in reaction (5) under the polar angle $\theta_{J/\psi}^{*}$ in the
$K^{-}p$ c.m.s. This cross section is assumed to have the standard exponential
$t$-shape form, multiplied by the free total cross section
$\sigma_{{K^{-}}p\to{J/\psi}\Lambda}(\sqrt{s})$ of the reaction
${K^{-}}p\to{J/\psi}\Lambda$. The latter one assumes the form [70]:

$\sigma_{{K^{-}}p\to{J/\psi}\Lambda}(\sqrt{s})=7\cdot
10^{-6}\cdot\sigma_{{K^{-}}p\to{\phi}\Lambda}(\sqrt{{\tilde{s}}}),$ (24)

where the quantity $\sigma_{{K^{-}}p\to{\phi}\Lambda}(\sqrt{{\tilde{s}}})$
represents the free total cross section of the process
${K^{-}}p\to{\phi}\Lambda$ and the center-of-mass collision energies
$\sqrt{s}$ and $\sqrt{{\tilde{s}}}$ are linked by the relation:

$\sqrt{{\tilde{s}}}=\sqrt{s}-m_{J/\psi}+m_{\phi}.$ (25)

Here, the quantity $m_{\phi}$ is the vacuum $\phi$ meson mass. For the cross
section $\sigma_{{K^{-}}p\to{\phi}\Lambda}(\sqrt{{\tilde{s}}})$ we have
adopted the following parametrization:

$\sigma_{{K^{-}}p\to{\phi}\Lambda}(\sqrt{{\tilde{s}}})=Bp^{*}_{\phi}{\rm
e}^{-{\beta}p^{*}_{\phi}},$ (26)

suggested in Ref. [74]. Here, the $\phi$ c.m. momentum $p^{*}_{\phi}$ is
defined by the expression:

$p_{\phi}^{*}=\frac{1}{2\sqrt{{\tilde{s}}}}\lambda({\tilde{s}},m_{\phi}^{2},m_{\Lambda}^{2})$
(27)

and parameters $B$ and $\beta$ are: $B=315.31$ $\mu$b/(GeV/c), $\beta=1.45$
(GeV/c)-1. The expressions (23)–(27) will be used by us for calculating the
free space $J/\psi$ energy spectrum for incident resonant antikaon beam
momenta of 9.417, 9.965 and 10.026 GeV/c (see below) and the $J/\psi$
excitation function near threshold.

As was noticed above (see Eq. (11)), the $J/\psi$ mesons are produced at small
angles with respect to the $K^{-}$ beam direction at the considered initial
$K^{-}$ meson momenta. Therefore, in what follows, for calculation of the
momentum differential cross section for $J/\psi$ meson production in
${K^{-}}A$ interactions from the direct process (5) we will use Eq. (33) from
Ref. [70], which now reads:

$\frac{d\sigma_{{K^{-}}A\to{J/\psi}X}^{({\rm
dir})}(p_{K^{-}},p_{J/\psi})}{dp_{J/\psi}}=2{\pi}\left(\frac{Z}{A}\right)I_{V}[A,\sigma_{{J/\psi}N}]\int\limits_{\cos
20^{\circ}}^{1}d\cos{{\theta_{J/\psi}}}\left<\frac{d\sigma_{{K^{-}}p\to{J/\psi}{\Lambda}}(p_{K^{-}},p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf\Omega}_{J/\psi}}\right>_{A},$
(28)

where

$I_{V}[A,\sigma]=2{\pi}A\int\limits_{0}^{R}r_{\bot}dr_{\bot}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{\sqrt{R^{2}-r_{\bot}^{2}}}dz\rho(\sqrt{r_{\bot}^{2}+z^{2}})$
(29) $\times\exp{\left[-A\sigma_{K^{-}N}^{\rm
tot}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{z}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx-A{\sigma}\int\limits_{z}^{\sqrt{R^{2}-r_{\bot}^{2}}}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx\right]}.$

Here in Eq. (28),
$\left<\frac{d\sigma_{{K^{-}}p\to{J/\psi}\Lambda}(p_{K^{-}},p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf\Omega}_{J/\psi}}\right>_{A}$
is the off-shell inclusive differential cross section for the production of
$J/\psi$ mesons with momentum ${\bf p}_{J/\psi}$ in reaction (5), averaged
over the Fermi motion and binding energy of the protons in the nucleus (cf.
Eq. (5) from Ref. [70]). The quantities $\rho(r)$ as well as
$\sigma_{{K^{-}}N}^{\rm tot}$ and $\sigma=\sigma_{{J/\psi}N}$, entering into
Eq. (29), denote, respectively, the target nucleon density, normalized to
unity, as well as the total cross section of the free ${K^{-}}N$ interaction
and ${J/\psi}N$ absorption cross section 777)For which we use the values
$\sigma_{{K^{-}}N}^{\rm tot}=22$ mb and $\sigma_{{J/\psi}N}=3.5$ mb [70].) .
In our calculations we assume that the neutron and proton densities have the
same radial shape $\rho(r)$. For it we have assumed an harmonic oscillator and
a two-parameter Fermi distributions for 12C and 184W target nuclei,
respectively. For the rest of notation see Ref. [70].

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

At $K^{-}$ meson beam momenta below 10.2 GeV/c of interest, incident antikaons
can produce the neutral $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$ pentaquark resonances with single strangeness $S=-1$
directly in the first inelastic collisions with intranuclear protons:

$\displaystyle{K^{-}}+p\to P_{cs}(4338)^{0},$ $\displaystyle{K^{-}}+p\to
P_{cs}(4455)^{0},$ $\displaystyle{K^{-}}+p\to P_{cs}(4468)^{0}.$ (30)

Then the produced pentaquark resonances can decay into the final state
${J/\psi}\Lambda$, which will additionally contribute to the $J/\psi$ yield in
the ($K^{-}$,$J/\psi$) reactions on protons and nuclei:

$\displaystyle P_{cs}(4338)^{0}\to J/\psi+\Lambda,$ $\displaystyle
P_{cs}(4455)^{0}\to J/\psi+\Lambda,$ $\displaystyle P_{cs}(4468)^{0}\to
J/\psi+\Lambda.$ (31)

As above in the case of direct $J/\psi$ meson production, at first, we
consider here the charmonium production in the production/decay sequences
(30)/(31), taking place on a free target proton at rest. According to [70],
the free space total cross sections $\sigma_{{K^{-}}p\to
P_{csi}^{0}\to{J/\psi}\Lambda}(\sqrt{s},\Gamma_{csi})$ for resonant charmonium
production in these sequences can be represented as follows 888)Here, $i=$1,
2, 3 and $P^{0}_{cs1}$, $P^{0}_{cs2}$ and $P^{0}_{cs3}$ stand for
$P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ states,
respectively. Analogously, $\Gamma_{cs1}$, $\Gamma_{cs2}$ and $\Gamma_{cs3}$
will denote in this exploratory study, correspondingly, the central values of
their total decay widths, i.e.: $\Gamma_{cs1}=7.0$ MeV, $\Gamma_{cs2}=7.5$ MeV
and $\Gamma_{cs3}=5.2$ MeV.) :

$\sigma_{{K^{-}}p\to
P_{csi}^{0}\to{J/\psi}\Lambda}(\sqrt{s},\Gamma_{csi})=\sigma_{{K^{-}}p\to
P_{csi}^{0}}(\sqrt{s},\Gamma_{csi})\theta[\sqrt{s}-(m_{J/\psi}+m_{\Lambda})]Br[P_{csi}^{0}\to{J/\psi}\Lambda].$
(32)

Here, $\theta(x)$ is the step function and the quantities
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]$ ($i=$1, 2, 3) represent the branching
ratios of the decays (31). In Eq. (32), the quantities $\sigma_{{K^{-}}p\to
P_{csi}^{0}}(\sqrt{s},\Gamma_{csi})$ are the total cross sections for
production of the $P_{csi}^{0}$ resonances with the possible spin-parity
quantum numbers $J^{P}=(1/2)^{-}$ for $P_{cs1}^{0}$ and $P_{cs2}^{0}$, and
$J^{P}=(3/2)^{-}$ for $P_{cs3}^{0}$ in reactions (30) 999)Which might be
assigned to them within the hadronic molecular scenario for their internal
structure (see Introduction section above).) . These cross sections can be
described, using the spectral functions $S_{csi}^{0}(\sqrt{s},\Gamma_{csi})$
of resonances and knowing the branching fractions $Br[P^{0}_{csi}\to{K^{-}}p]$
of their decays to the $K^{-}p$ mode, as follows [70]:

$\sigma_{{K^{-}}p\to
P_{csi}^{0}}(\sqrt{s},\Gamma_{csi})=f_{csi}\left(\frac{\pi}{p^{*}_{K^{-}}}\right)^{2}Br[P_{csi}^{0}\to{K^{-}}p]S_{csi}^{0}(\sqrt{s},\Gamma_{csi})\Gamma_{csi},\,\,i=1,2,3,$
(33)

where the c.m. 3-momentum in the incoming ${K^{-}}p$ channel, $p^{*}_{K^{-}}$,
is defined by the formula

$p_{K^{-}}^{*}=\frac{1}{2\sqrt{{s}}}\lambda({s},m_{K^{-}}^{2},m_{p}^{2})$ (34)

and the ratios of the spin factors $f_{cs1}=2$, $f_{cs2}=2$, $f_{cs3}=4$. In
line with Ref. [70], we suppose that the free spectral functions
$S_{csi}^{0}(\sqrt{s},\Gamma_{csi})$ of the intermediate $P_{csi}^{0}$
resonances, entering into Eq. (33), are described by the non-relativistic
Breit-Wigner distributions 101010)In view of the fact that the
$P_{cs}(4338)^{0}$ state is very close to the $\Xi_{c}^{0}{\bar{D}}^{0}$ and
$\Xi_{c}^{+}D^{-}$ thresholds, its lineshape may be distorted from the
conventional BW distribution due to the double threshold effects [29, 75]. But
since the possible pole positions for the $P_{cs}(4338)^{0}$ (what concerns
the masses and widths), found in [29, 75], are close (within the
uncertainties) to those BW parameters determined in the LHCb analysis, the
generation of the $P_{cs}(4338)^{0}$ shape with BW one carried out in the
present work is reasonable enough for our exploratory purposes.) :

$S_{csi}^{0}(\sqrt{s},\Gamma_{csi})=\frac{1}{2\pi}\frac{\Gamma_{csi}}{(\sqrt{s}-M_{csi})^{2}+{\Gamma}_{csi}^{2}/4}.$
(35)

Inspection of Eqs. (32), (33) tells us that to evaluate the $J/\psi$
production cross sections from the production/decay sequences (30)/(31),
taking place on a vacuum proton (and on a bound in a nuclear medium proton),
one needs to know the branching ratios $Br[P^{0}_{csi}\to{K^{-}}p]$ and
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]$ ($i=$1, 2, 3) of the $P^{0}_{csi}$ decays
to the $K^{-}p$ and ${J/\psi}\Lambda$ channels. We focus now on them.

The branching ratios $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$ of the decays (31)
were not determined experimentally. Therefore, we must rely on the theoretical
predictions for them as well as on the similarity of the behaviors and decay
properties of the $P_{cs}$ and $P_{c}$ systems (cf. Introduction and Refs.
[15, 35]). Thus, the decay rates of the modes $P_{c}^{+}\to{J/\psi}p$ and
$P_{cs}^{0}\to{J/\psi}\Lambda^{0}$ are expressed in Ref. [15] in terms of the
single model parameter $\Lambda$, which should be constrained from the future
experiments, and these rates are comparable to each other. 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 experiment [76] at JLab. Preliminary
results from a factor of 10 more data, collected in the $J/\psi$–007
experiment [77] also at JLab, focused on the large $t$ region 111111)In which
the rather flat resonant production of $J/\psi$ through the $P_{c}^{+}$ in
photon-induced reactions on the protons is expected to be enhanced relative to
the suppressed here mostly forward diffractive production.) in searching for
the LHCb hidden-charm pentaquarks [24], also observe no signals for them and
set more stringent upper limits on the cross sections for production of the
$P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ states in
${\gamma}p$ collisions almost an order of magnitude below the respective GlueX
limits [76]. With this, with upper limits on the branching fractions
$Br[P_{c}(4312/4440/4457)^{+}\to{J/\psi}p]$ already available from the GlueX
experiment [76] and within the representation of these cross sections, in
which they are proportional, respectively, to the
$Br^{2}[P_{c}(4312/4440/4457)^{+}\to{J/\psi}p]$, the upper limits on the above
branching fractions, which are expected from the $J/\psi$-007 experiment [77],
were estimated to be at the level of 1% in Ref. [78]. Based on the branching
ratios and fractions measured by the LHCb and GlueX Collaborations, the
authors of Ref. [79] 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 as
well as the fact that the ratios of the branching fractions
$Br[P_{c}^{+}\to{J/\psi}p]$ between three states $P_{c}(4312)^{+}$,
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ were predicted to be $\sim$ 1 in Ref.
[16], we adopted in our study [78] for the branching fractions of
$P^{+}_{c}\to{J/\psi}p$ decays for each individual $P_{c}^{+}$ state three
following conservative options: $Br[P^{+}_{c}\to{J/\psi}p]=0.25$, 0.5 and 1%.
With this and in view of the aforesaid, it is natural to assume for all
branching ratios $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$ of the decays (31) the
same three main options, namely: $Br[P_{csi}^{0}\to{J/\psi}\Lambda]=0.25$, 0.5
and 1% ($i=$1, 2, 3) as those given above and used for the
$P^{+}_{c}\to{J/\psi}p$ decays, and additional one with reduced value of
0.125% of these ratios in order to see better the size of their impact on the
resonant $J/\psi$ yield in ${K^{-}}p\to{J/\psi}\Lambda$, ${K^{-}}$12C
$\to{J/\psi}X$ and ${K^{-}}$184W $\to{J/\psi}X$ reactions 121212)It should be
pointed out that in view of the aforementioned these options are more
realistic than those of 3, 5, 10, 15 and 50% employed in Ref. [70] for the
$P_{cs}(4459)^{0}\to{J/\psi}\Lambda$ decays.) . It is worth mentioning that
the investigations in Refs. [34, 54] support in some sense such choice for the
branching fractions of the decays (31). Thus, investigation of the strong
decays of the $P_{cs}(4459)^{0}$ pentaquark, assuming that it is a pure
$S$-wave $\Xi_{c}{\bar{D}}^{*}$ molecular state with two possible spin-parity
assignments $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$, has been performed in
Ref. [34] through hadronic loops with the help of the effective Lagrangians.
In comparison with the LHCb data [24], the $S$-wave $\Xi_{c}{\bar{D}}^{*}$
molecule with $J^{P}=(1/2)^{-}$ assignment for the $P_{cs}(4459)^{0}$ is
supported by the study [34]. Using the results for the total decay width and
for the partial width of the decay of the $P_{cs}(4459)^{0}$ with
$J^{P}=(1/2)^{-}$ into the ${J/\psi}\Lambda$, the branching ratio of this
decay could be evaluated at the level of 1%. Further, the author in Ref. [54],
interpreting the pentaquark $P_{cs}(4459)^{0}$ observed by the LHCb
Collaboration [24] as the hadronic molecule with the dominant $S$-wave
channels $\Xi_{c}{\bar{D}}^{*}$ and $\Xi_{c}^{*}{\bar{D}}$ with the spin-
parity combination $J^{P}=(3/2)^{-}$, predicted adopting the one-boson-
exchange model that the partial decay width for the process
$P_{cs}(4459)^{0}\to{J/\psi}\Lambda$ is around 0.1 MeV in the resonance
region. With this value and with the $P_{cs}(4459)^{0}$ resonance total decay
width of about 20 MeV also predicted in [54] in this region, we obtain that
the branching fraction $Br[P_{cs}(4459)^{0}\to{J/\psi}\Lambda]$ amounts to
0.5%. Moreover, 1% branching ratio of the $P_{cs}(4459)^{0}\to{J/\psi}\Lambda$
decay was chosen as the main one in Ref. [69]. In previous works [69, 70],
devoted to the study of the $P_{cs}(4459)^{0}$ resonance production in
antikaon-induced reactions on protons and nuclei, the value of 0.01% was used
for the branching fraction $Br[P_{cs}(4459)^{0}\to{K^{-}}p]$. It was chosen to
be similar to that characterizing [80] the $P_{c}^{0}\to{\pi^{-}}p$ decay.
Based on the molecular scenario, the partial decay widths of the
$P_{cs}(4459)^{0}$ with $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$ into $K^{-}p$
final state via hadronic loops were evaluated in Ref. [71] to be of the order
of 2.05 and 0.24 KeV, respectively. If we adopt an experimental magnitude of
17.3 MeV for the $P_{cs}(4459)^{0}$ total decay width, we obtain with the
above partial decay widths that the branching fraction
$Br[P_{cs}(4459)^{0}\to{K^{-}}p]$ would be $\sim$ 0.01% and 0.001% for the
$P_{cs}(4459)^{0}$ with $J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$,
correspondingly. In line with the aforementioned, it is also natural to employ
in our cross-section calculations for the branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]$ ($i=$1, 2, 3) the main value of 0.01%. To see the
effect of the branching ratios on the resonant $J/\psi$ yield in the reactions
of interest, we will also use for them in these calculations the value of
0.001%.

Before going further, we determine the resonant $J/\psi$ energy distributions
$d\sigma_{{K^{-}}p\to
P_{csi}^{0}\to{J/\psi}\Lambda}[\sqrt{s},p_{J/\psi}]/dE_{J/\psi}$ ($i=$1, 2, 3)
from the two-step processes (30)/(31), proceeding on the free target proton at
rest, in addition to that (23) from the background
${K^{-}}p\to{J/\psi}\Lambda$ reaction. 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 $K^{-}$ meson momentum. The results of the calculations, performed in
Ref. [70] assuming that the decays of the $P_{csi}^{0}$ to the
${J/\psi}\Lambda$ are dominated by the lowest partial waves with relative
orbital angular momentum $L=0$, show that these distributions can be
represented in the following way:

$\frac{d\sigma_{{K^{-}}p\to
P_{csi}^{0}\to{J/\psi}\Lambda}[\sqrt{s},p_{J/\psi}]}{dE_{J/\psi}}=\sigma_{{K^{-}}p\to
P_{csi}^{0}}(\sqrt{s},\Gamma_{csi})\theta[\sqrt{s}-(m_{J/\psi}+m_{\Lambda})]\times$
(36)
$\times\left(\frac{\sqrt{s}}{2p_{K^{-}}p^{*}_{J/\psi}}\right)Br[P_{csi}^{0}\to{J/\psi}\Lambda]~{}{\rm
for}~{}E^{(2)}_{J/\psi}(0^{\circ})\leq E_{J/\psi}\leq
E^{(1)}_{J/\psi}(0^{\circ}).$

Eq. (36) shows that the free space $J/\psi$ energy distributions, which arise
from the production/decay chains (30)/(31), exhibit a totally flat behavior
within the allowed energy range (13).

For calculation of the $J/\psi$ inclusive differential cross section (momentum
spectrum) arising from the production and decay of the intermediate resonances
$P_{csi}^{0}$ in $K^{-}A$ collisions we will use the following expression
based on Eq. (78) from Ref. [70]:

$\frac{d\sigma_{{K^{-}}A\to{J/\psi}X}^{({\rm
sec,i})}(p_{K^{-}},p_{J/\psi})}{dp_{J/\psi}}=2{\pi}\left(\frac{Z}{A}\right)I_{V}[A,\sigma^{\rm
in}_{{P_{csi}}N}]$ (37) $\times\int\limits_{\cos
20^{\circ}}^{1}d\cos{{\theta_{J/\psi}}}\left<\frac{d\sigma_{{K^{-}}p\to
P_{csi}^{0}\to{J/\psi}{\Lambda}}(p_{K^{-}},p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf\Omega}_{J/\psi}}\right>_{A},\,\,\,\,i=1,2,3;$

where $\left<\frac{d\sigma_{{K^{-}}p\to
P_{csi}^{0}\to{J/\psi}{\Lambda}}(p_{K^{-}},p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf\Omega}_{J/\psi}}\right>_{A}$
is the off-shell inclusive differential cross section for the production of
$J/\psi$ mesons with momentum ${\bf p}_{J/\psi}$ in production/decay chain
${K^{-}}p\to P_{csi}^{0}\to{J/\psi}{\Lambda}$, averaged over the Fermi motion
and binding energy of the protons in the nucleus (cf. Eq. (71) from Ref.
[70]). The quantity $I_{V}[A,\sigma^{\rm in}_{P_{csi}N}]$ in Eq. (37) is
defined above by Eq. (29), in which one needs to make the substitution
$\sigma\to\sigma^{\rm in}_{P_{csi}N}$. Here, $\sigma^{\rm in}_{P_{csi}N}$ is
the $P_{csi}^{0}$–nucleon inelastic total cross section. For this cross
section we will use in our calculations the same value of 23.7 mb as that
adopted in Ref. [70] for the inelastic cross section $\sigma^{\rm
in}_{P_{cs}(4459)^{0}N}$ and obtained here in the molecular picture of the
$P_{cs}(4459)^{0}$.

Figure 1: (Color online.) The non-resonant total cross section $\sigma$ for
the reaction ${K^{-}}p\to{J/\psi}\Lambda$ (black solid curve), calculated on
the basis of Eqs. (24)–(27). Incoherent sum (blue dotted curve) of it and the
sum $\sigma_{1}$+$\sigma_{2}$+$\sigma_{3}$ (red short-dashed curve) of the
total cross sections $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$ for the
resonant $J/\psi$ production in the processes ${K^{-}}p\to
P_{cs}(4338)^{0}\to{J/\psi}\Lambda$, ${K^{-}}p\to
P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$, calculated in line with Eq. (32) assuming
that the resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$ with the spin-parity quantum numbers $J^{P}=(1/2)^{-}$,
$J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$ decay to the $K^{-}p$ and
${J/\psi}\Lambda$ channels with all three individual branching fractions 0.01%
and 1, 0.5, 0.25 and 0.125% (respectively, upper left, upper right, lower left
and lower right panels), as functions of the incident $K^{-}$ meson momentum.
The left and three right arrows indicate, correspondingly, the threshold
momentum $p^{\rm th}_{K^{-}}=8.844$ GeV/c for the reaction
${K^{-}}p\to{J/\psi}\Lambda$, taking place on a free target proton being at
rest, and the resonant antikaon momenta $p^{\rm R1}_{K^{-}}=9.417$ GeV/c,
$p^{\rm R2}_{K^{-}}=9.965$ GeV/c and $p^{\rm R3}_{K^{-}}=10.026$ GeV/c. Figure
2: (Color online.) The same as in Fig. 1, but calculated for all three
branching ratios of $P^{0}_{csi}$ ($i=1$, 2, 3) decays to the $K^{-}p$ mode of
0.001%. Figure 3: (Color online.) The direct non-resonant $J/\psi$ energy
distribution in the free space elementary process
${K^{-}}p\to{J/\psi}\Lambda$, calculated in line with Eq. (23) at initial
$K^{-}$ meson resonant momentum of 9.417 GeV/c in the laboratory system (black
solid curve). The resonant $J/\psi$ energy distributions in the two-step
processes ${K^{-}}p\to P_{cs}(4338)^{0}\to{J/\psi}\Lambda$, ${K^{-}}p\to
P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$, calculated in line with Eq. (36) at the
same incident antikaon momentum of 9.417 GeV/c assuming that the resonances
$P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ with the spin-
parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(3/2)^{-}$,
correspondingly, all decay to the $K^{-}p$ and ${J/\psi}\Lambda$ modes with
branching fractions of 0.01% and 0.125% (respectively, red dashed, blue
dotted, dark cyan dashed-doted curves). Incoherent sum of the direct non-
resonant $J/\psi$ energy distribution and resonant ones, calculated supposing
that the resonances $P_{cs}(4338)^{0}$ as well as $P_{cs}(4338)^{0}$,
$P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ with the same spin-parity combinations
all decay to the $K^{-}p$ and ${J/\psi}\Lambda$ with branching fractions 0.01%
and 0.125, 0.25, 0.5, 1% (respectively, dark yellow short-dashed, wine short-
dashed-dotted, olive dashed-dotted, green dotted as well as navy short-dotted,
pink dotted, royal dashed-dotted-dotted, magenta short-dashed-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 ($6.483\leq E_{J/\psi}\leq 8.484~{}{\rm GeV}$) for the
considered direct non-resonant and resonant $J/\psi$ production off a free
target proton at rest at given initial $K^{-}$ meson resonant momentum of
9.417 GeV/c. Figure 4: (Color online.) The direct non-resonant $J/\psi$ energy
distribution in the free space elementary process
${K^{-}}p\to{J/\psi}\Lambda$, calculated in line with Eq. (23) at initial
$K^{-}$ meson resonant momentum of 9.965 GeV/c in the laboratory system (black
solid curve). The resonant $J/\psi$ energy distributions in the two-step
processes ${K^{-}}p\to P_{cs}(4338)^{0}\to{J/\psi}\Lambda$, ${K^{-}}p\to
P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$, calculated in line with Eq. (36) at the
same incident antikaon momentum of 9.965 GeV/c assuming that the resonances
$P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ with the spin-
parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(3/2)^{-}$,
correspondingly, all decay to the $K^{-}p$ and ${J/\psi}\Lambda$ modes with
branching fractions of 0.01% and 0.125% (respectively, red dashed, blue
dotted, dark cyan dashed-doted curves). Incoherent sum of the direct non-
resonant $J/\psi$ energy distribution and resonant ones, calculated supposing
that the resonances $P_{cs}(4455)^{0}$ as well as $P_{cs}(4338)^{0}$,
$P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ with the same spin-parity combinations
all decay to the $K^{-}p$ and ${J/\psi}\Lambda$ with branching fractions 0.01%
and 0.125, 0.25, 0.5, 1% (respectively, dark yellow short-dashed, wine short-
dashed-dotted, olive dashed-dotted, green dotted as well as navy short-dotted,
pink dotted, royal dashed-dotted-dotted, magenta short-dashed-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 ($6.300\leq E_{J/\psi}\leq 9.205~{}{\rm GeV}$) for the
considered direct non-resonant and resonant $J/\psi$ production off a free
target proton at rest at given initial $K^{-}$ meson resonant momentum of
9.965 GeV/c. Figure 5: (Color online.) The direct non-resonant $J/\psi$ energy
distribution in the free space elementary process
${K^{-}}p\to{J/\psi}\Lambda$, calculated in line with Eq. (23) at initial
$K^{-}$ meson resonant momentum of 10.026 GeV/c in the laboratory system
(black solid curve). The resonant $J/\psi$ energy distributions in the two-
step processes ${K^{-}}p\to P_{cs}(4338)^{0}\to{J/\psi}\Lambda$, ${K^{-}}p\to
P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$, calculated in line with Eq. (36) at the
same incident antikaon momentum of 10.026 GeV/c assuming that the resonances
$P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ with the spin-
parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$, $J^{P}=(3/2)^{-}$,
correspondingly, all decay to the $K^{-}p$ and ${J/\psi}\Lambda$ modes with
branching fractions of 0.01% and 0.125% (respectively, red dashed, blue
dotted, dark cyan dashed-doted curves). Incoherent sum of the direct non-
resonant $J/\psi$ energy distribution and resonant ones, calculated supposing
that the resonances $P_{cs}(4468)^{0}$ as well as $P_{cs}(4338)^{0}$,
$P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ with the same spin-parity combinations
all decay to the $K^{-}p$ and ${J/\psi}\Lambda$ with branching fractions 0.01%
and 0.125, 0.25, 0.5, 1% (respectively, dark yellow short-dashed, wine short-
dashed-dotted, olive dashed-dotted, green dotted as well as navy short-dotted,
pink dotted, royal dashed-dotted-dotted, magenta short-dashed-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 ($6.286\leq E_{J/\psi}\leq 9.280~{}{\rm GeV}$) for the
considered direct non-resonant and resonant $J/\psi$ production off a free
target proton at rest at given initial $K^{-}$ meson resonant momentum of
10.026 GeV/c. Figure 6: (Color online.) The same as in Fig. 3, but calculated
for all three branching ratios of $P^{0}_{csi}$ ($i=1$, 2, 3) decays to the
$K^{-}p$ mode of 0.001%. Figure 7: (Color online.) The same as in Fig. 4, but
calculated for all three branching ratios of $P^{0}_{csi}$ ($i=1$, 2, 3)
decays to the $K^{-}p$ mode of 0.001%. Figure 8: (Color online.) The same as
in Fig. 5, but calculated for all three branching ratios of $P^{0}_{csi}$
($i=1$, 2, 3) decays to the $K^{-}p$ mode of 0.001%. Figure 9: (Color online.)
The direct non-resonant momentum distribution of $J/\psi$ mesons, produced in
the reaction ${K^{-}}{\rm{}^{12}C}\to{J/\psi}X$ in the laboratory polar
angular range of 0∘–20∘ and calculated in line with Eq. (28) at initial
antikaon resonant momentum of 9.417 GeV/c in the laboratory system (black
solid curve). The resonant momentum distributions of $J/\psi$ mesons, produced
in the two-step processes ${K^{-}}p\to P_{cs}(4338)^{0}\to{J/\psi}\Lambda$,
${K^{-}}p\to P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$ and calculated in line with Eq. (37) at
the same incident antikaon momentum of 9.417 GeV/c assuming that the
resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ with
the spin-parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and
$J^{P}=(3/2)^{-}$, correspondingly, all decay to the $K^{-}p$ and
${J/\psi}\Lambda$ modes with branching fractions 0.01% and 0.125%
(respectively, red dashed, blue dotted, dark cyan dashed-doted curves) and
their incoherent sum (orange solid curve). Incoherent sum of the direct non-
resonant $J/\psi$ momentum distribution and resonant ones, calculated
supposing that the resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$,
$P_{cs}(4468)^{0}$ with the same spin-parity combinations all decay to the
$K^{-}p$ and ${J/\psi}\Lambda$ with branching fractions 0.01% and 0.125, 0.25,
0.5 and 1% (respectively, navy short-dotted, pink dotted, royal dashed-dotted-
dotted and magenta 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 ${K^{-}}{\rm{}^{184}W}\to{J/\psi}X$ in the laboratory
polar angular range of 0∘–20∘ and calculated in line with Eq. (28) at initial
antikaon resonant momentum of 9.417 GeV/c in the laboratory system (black
solid curve). The resonant momentum distributions of $J/\psi$ mesons, produced
in the two-step processes ${K^{-}}p\to P_{cs}(4338)^{0}\to{J/\psi}\Lambda$,
${K^{-}}p\to P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$ and calculated in line with Eq. (37) at
the same incident antikaon momentum of 9.417 GeV/c assuming that the
resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ with
the spin-parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and
$J^{P}=(3/2)^{-}$, correspondingly, all decay to the $K^{-}p$ and
${J/\psi}\Lambda$ modes with branching fractions 0.01% and 0.125%
(respectively, red dashed, blue dotted, dark cyan dashed-doted curves) and
their incoherent sum (orange solid curve). Incoherent sum of the direct non-
resonant $J/\psi$ momentum distribution and resonant ones, calculated
supposing that the resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$,
$P_{cs}(4468)^{0}$ with the same spin-parity combinations all decay to the
$K^{-}p$ and ${J/\psi}\Lambda$ with branching fractions 0.01% and 0.125, 0.25,
0.5 and 1% (respectively, navy short-dotted, pink dotted, royal dashed-dotted-
dotted and magenta 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 calculated for the initial antikaon
momentum of 10.026 GeV/c. Figure 12: (Color online.) The same as in Fig. 10,
but calculated for the initial antikaon momentum of 10.026 GeV/c. Figure 13:
(Color online.) The direct non-resonant momentum distribution of $J/\psi$
mesons, produced in the reaction ${K^{-}}{\rm{}^{184}W}\to{J/\psi}X$ in the
laboratory polar angular range of 0∘–20∘ and calculated in line with Eq. (28)
at initial antikaon resonant momentum of 10.026 GeV/c in the laboratory system
(black solid curve). The resonant momentum distributions of $J/\psi$ mesons,
produced in the two-step processes ${K^{-}}p\to
P_{cs}(4338)^{0}\to{J/\psi}\Lambda$, ${K^{-}}p\to
P_{cs}(4455)^{0}\to{J/\psi}\Lambda$ and ${K^{-}}p\to
P_{cs}(4468)^{0}\to{J/\psi}\Lambda$ and calculated in line with Eq. (37) at
the same incident antikaon momentum of 10.026 GeV/c assuming that the
resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ with
the spin-parity assignments $J^{P}=(1/2)^{-}$, $J^{P}=(1/2)^{-}$ and
$J^{P}=(3/2)^{-}$, correspondingly, all decay to the $K^{-}p$ and
${J/\psi}\Lambda$ modes with branching fractions 0.001% and 0.125%
(respectively, red dashed, blue dotted, dark cyan dashed-doted curves) and
their incoherent sum (orange solid curve). Incoherent sum of the direct non-
resonant $J/\psi$ momentum distribution and resonant ones, calculated
supposing that the resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$,
$P_{cs}(4468)^{0}$ with the same spin-parity combinations all decay to the
$K^{-}p$ and ${J/\psi}\Lambda$ with branching fractions 0.001% and 0.125,
0.25, 0.5 and 1% (respectively, navy short-dotted, pink dotted, royal dashed-
dotted-dotted and magenta short-dashed-dotted curves), all as functions of the
$J/\psi$ momentum $p_{J/\psi}$ in the laboratory frame.

## 3\. Results and discussion

The direct non-resonant $J/\psi$ production total cross section (24) on the
target proton at rest (black solid curves), the total cross section for the
resonant $J/\psi$ production in the processes (30)/(31) calculated on the
basis of Eq. (32) for the considered spin-parity assignments of the
$P_{csi}^{0}$ resonances ($i=$1, 2, 3) as well as for the branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01% and $Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$
0.125, 0.25, 0.5 and 1% for all three $P_{csi}^{0}$ states (red short-dashed
curves) and the combined (non-resonant plus resonant) $J/\psi$ production
total cross section (blue dotted curves) are presented in Fig. 1 as functions
of the $K^{-}$ meson laboratory momentum. The same as that shown in Fig. 1,
but determined for the three branching ratios $Br[P_{csi}^{0}\to{K^{-}}p]=$
0.001%, is presented in Fig. 2 to see the sensitivity of the combined $J/\psi$
production total cross section to these ratios. One can see from these figures
that the $P_{cs}(4338)^{0}$ state as well as $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$ resonances manifest itself, respectively, as distinct
narrow single peak as well as two narrow overlapping peaks at antikaon
resonant momenta $p^{\rm R1}_{K^{-}}=9.417$ GeV/c as well as at $p^{\rm
R2}_{K^{-}}=9.965$ GeV/c and $p^{\rm R3}_{K^{-}}=10.026$ GeV/c in the combined
cross section, only if $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01% and
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$ 1% for all three $P_{csi}^{0}$ states. In
this case, in the resonance regions the resonant contributions are much larger
than the non-resonant ones of about 0.5 nb and the peak values of the
strengths of these three peaks reach a well measurable values $\sim$ 2–3 nb.
Hence, the background reaction $K^{-}p\to{J/\psi}\Lambda$ will not influence
the direct observation of the hidden-charm strange pentaquarks
$P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ production at
these momenta and in this case. If $Br[P_{csi}^{0}\to{J/\psi}\Lambda]\leq$
0.5%, then the resonant $J/\psi$ yields are comparable to (or are less than)
the non-resonant ones in the resonance regions. As a result, the
$P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ states do not
appear as pronounced peaks in the combined cross section. This implies that
will be hard to measure in this case the $P_{csi}^{0}$ pentaquark states in
$J/\psi$ total production cross section on a proton target in the antikaon-
induced reactions. From Fig. 2 we see that the resonant $J/\psi$ production
cross section is small compared to the non-resonant contribution at all
antikaon momenta considered for four adopted values for the three branching
ratios $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$, when $Br[P_{csi}^{0}\to{K^{-}}p]=$
0.001%. In this case, the combined total cross section of the reaction
${K^{-}}p\to{J/\psi}\Lambda$ has no distinct peak structures, corresponding to
the $P_{csi}^{0}$ states, and it is practically not distinguished from that
for the background reaction. In view of the above, it is natural to expect
that the $P_{csi}^{0}$ signals could be well distinguished from the background
reaction via the detailed scan of the $J/\psi$ total production cross section
on a proton target in the near-threshold $K^{-}$ momentum regions around
resonant momenta of 9.417, 9.965 and 10.026 GeV/c in the future dedicated
experiment at J-PARC facility, if branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]$ $\sim$ 0.01% and
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]$ $\sim$ 1%. To see experimentally such
signals in the combined total cross section of the reaction
${K^{-}}p\to{J/\psi}\Lambda$, it is enough to have the $K^{-}$ momentum
resolution (and the momentum binning) of the order of 5 MeV/c. Thus, the c.m.
energy ranges $M_{csi}-{\Gamma_{csi}}/2<\sqrt{s}<M_{csi}+{\Gamma_{csi}}/2$
($i=$1, 2, 3) correspond to the laboratory antikaon momentum regions of 9.401
GeV/c $<p_{K^{-}}<$ 9.433 GeV/c, 9.947 GeV/c $<p_{K^{-}}<$ 9.982 GeV/c and
10.013 GeV/c $<p_{K^{-}}<$ 10.038 GeV/c, i.e. ${\Delta}p_{K^{-}}=$ 32, 35 and
25 MeV/c for the $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$, respectively. This means that to resolve the peaks in the
upper left panel of Fig. 1 the $K^{-}$ beam momentum resolution (and the
momentum bin size) of the order of 5 MeV/c are required. One may hope that
this requirement will be satisfied at the K10 beam line at the J-PARC facility
[72]. To further motivate the conducting of the relevant experiment at this
facility, it is important to estimate the expected yields of the $P_{csi}^{0}$
signals from the reactions $K^{-}p\to P_{csi}^{0}\to{J/\psi}\Lambda$,
$J/\psi\to e^{+}e^{-}$, $\Lambda\to p{\pi^{-}}$ at least in the case,
corresponding to the upper left panel of Fig. 1. Assuming incident antikaon
beam intensity $\sim$ 106 $K^{-}$/s in the peak antikaon momenta and 4g/cm2
(57 cm) for the thickness of the liquid hydrogen target [72], one could reach
an integrated luminosity of 76 pb-1 for a year of data taking. For the
pentaquark yield (for the total number of the $P_{csi}^{0}$ events) estimates
in a one-year beam time, one needs to multiply the above luminosity by the
$P_{csi}^{0}$ production cross sections of 1.2, 1.2 and 2.3 nb at resonant
$K^{-}$ momenta of 9.417, 9.965 and 10.026 GeV/c as well as by the detection
efficiency and by the appropriate branching ratios $Br[{J/\psi}\to
e^{+}e^{-}]$ $\approx$ 6% and $Br[\Lambda\to p{\pi^{-}}]$ $\approx$ 64%. Even
with a relatively low (and realistic) 10% detection efficiency, we estimate
about of 350, 350 and 670 events per year for the $P_{cs}(4338)^{0}$,
$P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ signals, respectively. For elastic
(background) $J/\psi{\Lambda}$ production in the reaction
$K^{-}p\to{J/\psi}\Lambda$ we have $\sigma=0.54$ nb at considered $K^{-}$
momenta. This leads to $\approx$ 160 events per year. We see that the elastic
background is sufficiently small compared to the resonant yields. Therefore,
highly intense, energetic and resolution $K^{-}$ beam, which will be available
at this beam line, should allow to get a definite result for or against the
existence of the genuine $P_{cs}(4338)^{0}$ pentaquark state in the nature and
the presence of two resonances in the peak of the ${J/\psi}\Lambda$ event
distribution, associated to the $P_{cs}(4459)^{0}$, through the scan of the
$J/\psi$ total production cross section on a proton target in the resonance
regions, only if branching ratios $Br[P_{csi}^{0}\to{K^{-}}p]$ $\sim$ 0.01%
and $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$ $\sim$ 1%. It is obvious that to see
the $P_{csi}^{0}$ pentaquarks experimentally in the cases, when more ”softer”
constraints than those given above are imposed on the branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]$ and $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$, one
needs to consider such observable, which is appreciably sensitive to the
$P_{cs}^{0}$ signal in some region of the available phase space. For instance,
the large $t$ region of the differential cross section $d\sigma/dt$ in the
$J/\psi$-007 experiment [77], 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 Refs. [81–83].

Accounting for the above-mentioned, we focus now on the $J/\psi$ energy
distribution from the considered ${K^{-}}p\to{J/\psi}\Lambda$ elementary
reaction. Our model allows one to calculate the direct non-resonant $J/\psi$
energy distribution from this reaction, the resonant ones from the
production/decay sequences (30)/(31), proceeding on the free target proton
being at rest. They were calculated in line with Eqs. (23), (36),
respectively, for incident antikaon resonant momenta of 9.417, 9.965 and
10.026 GeV/c. The resonant $J/\psi$ energy distributions were determined for
the employed spin-parity assignments of the $P_{cs}(4338)^{0}$,
$P_{cs}(4455)^{0}$, $P_{cs}(4468)^{0}$ resonances for branching fractions
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01% and $Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$
0.125% for all three states. These dependencies, together with the incoherent
sum of the non-resonant $J/\psi$ energy distribution and resonant ones,
calculated supposing that all the resonances $P_{cs}(4338)^{0}$ and
$P_{csi}^{0}$ ($i=1$, 2, 3), $P_{cs}(4455)^{0}$ and $P_{csi}^{0}$ ($i=1$, 2,
3), $P_{cs}(4468)^{0}$ and $P_{csi}^{0}$ ($i=1$, 2, 3) decay to the
${J/\psi}\Lambda$ final state with four adopted options for the branching
ratios $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$, as functions of the $J/\psi$ total
energy $E_{J/\psi}$ are shown, respectively, in Figs. 3, 4, 5. The same as
that shown in these figures, but calculated for the three branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001%, is presented, respectively, in Figs. 6,
7, 8 to see the sensitivity of the combined $J/\psi$ energy distribution to
these ratios. It can be seen from all these figures that while the resonant
$J/\psi$ production differential cross sections show a quite flat behavior at
all allowed energies $E_{J/\psi}$, the non-resonant cross section drops
quickly as the energy $E_{J/\psi}$ decreases. At incident $K^{-}$ meson
resonant momenta of 9.417, 9.965 and 10.026 GeV/c of our interest and in the
case when $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01% ($i=$ 1, 2, 3), its strength is
essentially larger than those of the resonant $J/\psi$ production cross
sections, calculated for the value of the branching ratios
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$ 0.125%, for ”high” allowed $J/\psi$ total
energies greater, correspondingly, than $\approx$ 7.75, 8.25 and 8.5 GeV.
Whereas at ”low” $J/\psi$ total energies (below, respectively, 7.75, 8.25 and
8.5 GeV) the contribution from the resonance, decaying to the
${J/\psi}\Lambda$ mode with the branching ratio of 0.125%, with the centroid,
correspondingly, at momenta of 9.417, 9.965 and 10.026 GeV/c, is much larger
than the non-resonant one. When $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001%,
analogous $J/\psi$ total energies are somewhat smaller, and they,
respectively, are $\approx$ 7.25, 7.75 and 8.0 GeV. Thus, for example, in the
first and second cases (what concerns the branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]$ considered), for the $J/\psi$ mesons with total
energies about of 7.0 and 6.5 GeV their resonant production cross section,
calculated for the branching fraction of 0.125% of its decay to the
${J/\psi}\Lambda$, is enhanced compared to the non-resonant one by sizeable
factors about of one, two and three order of magnitude at incident antikaon
momenta of 9.417, 9.965 and 10.026 GeV/c, respectively. Furthermore, this
contribution is also essentially larger than those, arising from the decays of
another two pentaquarks to the ${J/\psi}\Lambda$ with the branching ratios
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$ 0.125%, at the aforementioned ”low”
$J/\psi$ total energies. As a consequence, for each $K^{-}$ beam momentum
considered the $J/\psi$ meson combined energy distribution, appearing from the
direct $J/\psi$ meson production and from the decay of the pentaquark
resonance located at this momentum to the ${J/\psi}\Lambda$ channel, reveals
here a clear sensitivity to the employed variations in the branching ratio of
this decay. Thus, for instance, for the $J/\psi$ mesons with total energy of
6.5 GeV and for the lowest incident antikaon momentum of 9.417 GeV/c this
$J/\psi$ combined distribution is enhanced for the values of this ratio of
0.125, 0.25, 0.5, 1% by substantial factors of about 10.0, 20.0, 37.5, 80.0,
respectively, as compared to that from the directly produced $J/\psi$ mesons
in the case when $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001%. And for the highest
initial $K^{-}$ meson momentum of 10.026 GeV/c of our interest, at which the
resonance $P_{cs}(4468)^{0}$ appears as a weak peak structure in the total
cross section of the exclusive reaction ${K^{-}}p\to{J/\psi}\Lambda$ only if
$Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$ 1%, the analogous factors become much
larger and they are of about 0.25$\cdot$103, 0.5$\cdot$103, 1$\cdot$103,
2$\cdot$103, respectively. Similarly, for the $J/\psi$ mesons with total
energy of 7.0 GeV, for incident $K^{-}$ meson momenta of 9.417 and 10.026
GeV/c these factors are of about 13, 25, 50, 100 and 0.35$\cdot$103,
0.7$\cdot$103, 1.3$\cdot$103, 2.6$\cdot$103, correspondingly, if
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01%. Moreover, what is important, one can see
that the above ”partial” combined energy distribution of the $J/\psi$ mesons,
calculated at given values of the branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]$ and $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$, is
practically indistinguishable from their ”total” combined energy distribution,
arising from the direct and resonant $J/\psi$ meson production through all
production/decay sequences (30)/(31), and determined at the same values of
these branching ratios. This means that the differences between the combined
results, obtained by using a value of the branching fractions of the decays
$P_{csi}^{0}\to{J/\psi}\Lambda$ of 0.125% 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 0.125 and 0.25%, 0.25 and 0.5%, 0.5
and 1%, are quite large and experimentally measurable at ”low” $J/\psi$ total
energies and for both adopted options for the branching fractions
$Br[P_{csi}^{0}\to{K^{-}}p]$ ($i=$ 1, 2, 3). Furthermore, for each initial
antikaon resonant momentum considered the observation here of the specific
hidden-charm LHCb pentaquark with strangeness will be practically not
influenced by the presence of the another two strange 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 and $\sim$ 0.1–1.0 nb/GeV at ”low” $J/\psi$ total energies $E_{J/\psi}$
for branching fractions $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001 and 0.01%,
respectively, their measurement requires both high luminosities and large-
acceptance detectors. Such measurement might be performed in the near future
at the J-PARC facility within the planned here experiments at the K10 beam
line [72]. Accounting for the estimates of the $P_{csi}^{0}$ ($i=$1, 2, 3)
yields given before and the above-mentioned $J/\psi$ production differential
cross sections at ”low” $J/\psi$ total energies, we can evaluate the numbers
of events expected in this measurement in the $J/\psi$ energy range of 6.5–7.5
GeV for the $K^{-}p\to P_{csi}^{0}\to{J/\psi}\Lambda$, $J/\psi\to e^{+}e^{-}$,
$\Lambda\to p{\pi^{-}}$ reactions. They are about of 2.9–29 and 29–290 per
year for $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001 and 0.01%, respectively. The
number of background events is lower by several orders of magnitude. This
means that the observation of the hidden-charm strange pentaquarks
$P_{csi}^{0}$ via the low-energy $J/\psi$ meson production on proton target is
quite optimistic at the J-PARC at least in the case when
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01% and $Br[P_{csi}^{0}\to{J/\psi}\Lambda]=$
0.125, 0.25, 0.5 and 1%. It should be pointed out that most likely it would be
difficult to determine the spin-parity quantum numbers of the $P_{csi}^{0}$
states by using the combined total and integrated over production angles
differential cross sections considered incoherently in the present work since
that observables, based on the squared absolute values of the resonant and
non-resonant production amplitudes, depend weakly on them. Such consideration
is justified because the role of the interference effects between resonant and
non-resonant contributions in the $K^{-}p\to{J/\psi}\Lambda$ reaction as well
as between different resonance states in the observables considered is
expected to be insignificant due to the fact that the $t$-channel $J/\psi$
production contributes only to the forward angles in the c.m. frame while the
$s$-channel resonances contribute in a full solid angle [84]. A way of
determining the $P_{csi}^{0}$ quantum numbers should be based on the model,
which takes into account the interference effects via adding the resonant and
non-resonant production amplitudes coherently. The results of such procedure
depend, in particular, on these numbers. To probe them, another observables
like angular distributions [69, 85] or polarization observables (single and
double) [86, 87], accounting for the spin correlations between the polarized
target proton and the polarized recoil $J/\psi$ meson and $\Lambda$ hyperon
(spin of the incident $K^{-}$ meson is equal to zero), are needed. The
consideration of this aspect is beyond the scope of the present paper. One may
hope that it will be studied in the future work when the appropriate $K^{-}p$
data will appear to control that interference.

The momentum dependencies of the absolute non-resonant, resonant and combined
$J/\psi$ meson differential cross sections, respectively, from the direct (5),
two-step (30)/(31) and direct plus two-step $J/\psi$ production processes in
$K^{-}$12C and $K^{-}$184W interactions, calculated on the basis of Eqs. (28),
(37) for laboratory polar angles of 0∘–20∘, for incident $K^{-}$ meson lowest
resonant momentum of 9.417 GeV/c and for branching fractions
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.01%, are shown, respectively, in Figs. 9 and
10. The same as in these figures, but determined for the initial highest
antikaon resonant momentum of 10.026 GeV/c, is given in Figs. 11 and 12. And
finally, the same as that shown in Fig. 12, but calculated for the three
branching ratios $Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001%, is presented in Fig. 13
to see the sensitivity of the combined $J/\psi$ momentum distribution to these
ratios. The resonant momentum distributions of the $J/\psi$ mesons in the two-
step processes (30)/(31), taking place on the bound protons of carbon and
tungsten nuclei, were obtained for four employed values of the branching
ratios $Br[P_{csi}^{0}\to{J/\psi}\Lambda]$. It is seen from Figs. 9–12 that
the total contribution to the $J/\psi$ production on both these nuclei,
stemming from all the intermediate $P_{csi}^{0}$ states, decaying to the
${J/\psi}\Lambda$ mode with branching fractions of 0.125% (orange solid
curves), shows practically flat behavior, and it is significantly larger than
that from the background process (5) (black solid curves) in the
”low”-momentum regions of 4.5–6.0 GeV/c and 4.5–6.5 GeV/c for considered
antikaon beam momenta of 9.417 and 10.026 GeV/c, respectively. The results,
presented in Fig. 13, show that such ”low”-momentum region also exists and
amounts to 4.5–6.0 GeV/c for the initial $K^{-}$ momentum of 10.026 GeV/c and
$Br[P_{csi}^{0}\to{K^{-}}p]=$ 0.001%. And in these ”low”-momentum regions,
what is of primary importance, the decays $P_{cs}(4455)^{0}\to{J/\psi}\Lambda$
and $P_{cs}(4468)^{0}\to{J/\psi}\Lambda$ are dominant. As a consequence, the
combined $J/\psi$ yield in them is practically completely governed by the
presence of the $P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ states in its
production. For given value of the branching ratios
$Br[P_{csi}^{0}\to{K^{-}}p]$, the strength of the yield is almost completely
determined by the branching ratios
$Br[P_{cs}(4455/4468)^{0}\to{J/\psi}\Lambda]$, used in the calculations, with
a value, which, on the one hand, depends weakly on the antikaon momentum and
which, on the other hand, increases by a factor of about four for both
antikaon beam momenta considered, when going from carbon target nucleus to
tungsten one. The value is still large enough to be measured, as one may hope,
in the future experiment at the J-PARC. This results in the well separated and
experimentally distinguishable differences between all combined calculations,
corresponding to the employed options for these ratios, for both target
nuclei, for both antikaon momenta considered and for both adopted options for
the branching fractions $Br[P_{csi}^{0}\to{K^{-}}p]$ ($i=$ 1, 2, 3).
Therefore, the $J/\psi$ meson production differential cross section
measurements on light and especially on heavy nuclear targets in the above
$J/\psi$ ”low”-momentum regions at antikaon momenta in the resonance regions
will open an opportunity to confirm or refute the $P_{cs}(4455)^{0}$ and
$P_{cs}(4468)^{0}$ states and, if they will be confirmed, to determine their
branching ratios to the ${J/\psi}\Lambda$ – at least to distinguish between
realistic options of 0.125, 0.25, 0.5 and 1%.

Taking into account the above considerations, we can conclude that the near-
threshold $J/\psi$ energy and momentum distribution measurements in antikaon-
induced reactions both on protons and on nuclear targets will provide further
evidence for the existence of the hidden-charm strange pentaquark
$P_{cs}(4338)^{0}$, for the presence of two resonances in the peak of the
${J/\psi}\Lambda$ mass spectrum around 4459 MeV, and will shed light on their
decay rates to the channel ${J/\psi}\Lambda$.

## 4\. Conclusions

Accounting for the LHCb observation that the reported hidden-charm strange
pentaquark $P_{cs}(4459)^{0}$ can split into two substructures,
$P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$, with a mass difference of 13 MeV as
well as the newly observed hidden-charm pentaquark resonance
$P_{cs}(4338)^{0}$ with strangeness, in this paper we have studied within the
double-peak scenario for the $P_{cs}(4459)^{0}$ state the near-threshold
$J/\psi$ meson production from protons and nuclei by considering incoherent
direct non-resonant (${K^{-}}p\to{J/\psi}\Lambda$) and two-step resonant
(${K^{-}}p\to P_{csi}^{0}\to{J/\psi}\Lambda$, $i=1$, 2, 3;
$P_{cs1}^{0}=P_{cs}(4338)^{0}$, $P_{cs2}^{0}=P_{cs}(4455)^{0}$,
$P_{cs3}^{0}=P_{cs}(4468)^{0}$) charmonium production processes with the main
goal of clarifying the possibility to observe within this scenario both above
two substructures contributing to the $P_{cs}(4459)^{0}$ state and the
$P_{cs}(4338)^{0}$ resonance in this production. 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 antikaon beam momenta by assuming the spin-parity
assignments of the hidden-charm resonances $P_{cs}(4338)^{0}$,
$P_{cs}(4455)^{0}$ and $P_{cs}(4468)^{0}$ as $J^{P}=(1/2)^{-}$,
$J^{P}=(1/2)^{-}$ and $J^{P}=(3/2)^{-}$ within four different realistic
choices for the branching ratios of their decays to the ${J/\psi}\Lambda$ mode
(0.125, 0.25, 0.5 and 1%) as well as for two options for the branching
fraction of their decays to the $K^{-}p$ channel (0.01 and 0.001%). It was
shown that when the latter fraction is assumed to be 0.01% then will be very
hard to measure the $P_{csi}^{0}$ pentaquark states through the scan of the
$J/\psi$ total production cross section on a proton target in the near-
threshold momentum region around the resonant antikaon momenta of 9.417, 9.965
and 10.026 GeV/c if the considered branching ratios of the ${J/\psi}\Lambda$
decay mode $\sim$ 0.5% and less. It was also found that if the branching
fraction of the $P_{csi}^{0}\to{K^{-}}p$ decays is suggested to be 0.001% then
these hidden-charm pentaquark states will be not accessible in such
measurements. It was further demonstrated that at these $K^{-}$ meson beam
momenta the $J/\psi$ energy and momentum combined distributions of interest
reveal noticeable sensitivity to all the above scenarios, respectively, at
”low” $J/\psi$ total energies and momenta, which means that they may be an
important tool to provide further evidence for the existence of the strange
hidden-charm pentaquark resonances $P_{cs}(4338)^{0}$, $P_{cs}(4455)^{0}$,
$P_{cs}(4468)^{0}$ and to get valuable information on their decay rates to the
$K^{-}p$ initial and ${J/\psi}\Lambda$ final states. The measurements of these
distributions could be performed in the future at the J-PARC facility. The
present model’s predictions can also serve as a good starting point in
supporting of these measurements.

## References

  * [1] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 115, 072001 (2015);
arXiv:1507.03414 [hep-ex].

  * [2] T. Gershon (LHCb Collaboration), arXiv:2206.15233 [hep-ex].
  * [3] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 122, 222001 (2019);
arXiv:1904.03947 [hep-ex].

  * [4] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 128, 062001 (2022);
arXiv:2108.04720 [hep-ex].

  * [5] G.-J. Wang et al., Phys. Rev. D 102, 036012 (2020);
arXiv:1911.09613 [hep-ph].

  * [6] F.-Z. Peng et al., arXiv:2211.09154 [hep-ph].
  * [7] X.-Y. Wang, X.-R. Chen, and J. He, Phys. Rev. D 99, 114007 (2019).
  * [8] H. X. Chen, W. Chen and S.-L. Zhu, Phys. Rev. D 100, 051501 (2019);
arXiv:1903.11001 [hep-ph].

  * [9] R. Chen, Z. F. Sun, X. Liu and S.-L. Zhu, Phys. Rev. D 100, 011502 (2019);
arXiv:1903.11013 [hep-ph].

  * [10] F. K. Guo, H. J. Jing, U. G. Meissner and S. Sakai, Phys. Rev. D 99, 091501 (2019);
arXiv:1903.11503 [hep-ph].

  * [11] M. Z. Liu et al., Phys. Rev. Lett. 122, 242001 (2019);
arXiv:1903.11560 [hep-ph].

  * [12] J. R. Zhang, Eur. Phys. J. C 79, 1001 (2019);
arXiv:1904.10711 [hep-ph].

  * [13] C. W. Xiao, J. Nieves and E. Oset, Phys. Rev. D 100, 014021 (2019);
arXiv:1904.01296 [hep-ph].

  * [14] L. Meng, B. Wang, G.-J. Wang and S.-L. Zhu, Phys. Rev. D 100, 014031 (2019);
arXiv:1905.04113 [hep-ph].

  * [15] T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 100, 094031 (2019);
arXiv:1910.03984 [hep-ph].

  * [16] C. J. Xiao et al., Phys. Rev. D 100, 014022 (2019);
arXiv:1904.00872 [hep-ph].

  * [17] J. B. Cheng and Y. R. Liu, Phys. Rev. D 100, 054002 (2019);
arXiv:1905.08605 [hep-ph].

  * [18] L. Meng, B. Wang, G.-J. Wang and S.-L. Zhu, arXiv:2204.08716 [hep-ph].
  * [19] C.-R. Deng, Phys. Rev. D 105, 116021 (2022);
arXiv:2202.13570 [hep-ph].

  * [20] Z.-G. Wang, Int. J. Mod. Phys. A 35, 2050003 (2020);
arXiv:1905.02892 [hep-ph].

  * [21] M.-J. Yan et al., Eur. Phys. J. C 82, 574 (2022);
arXiv:2108.05306 [hep-ph].

  * [22] S. X. Nakamura, A. Hosaka and Y. Yamaguchi, Phys. Rev. D 104, L091503 (2021);
arXiv:2109.15235 [hep-ph].

  * [23] X.-W. Wang and Z.-G. Wang, Chin. Phys. C 47, 013109 (2023);
arXiv:2207.06060 [hep-ph].

  * [24] R. Aaij et al. (LHCb Collaboration), Sci. Bull. 66,1278 (2021);
arXiv:2012.10380 [hep-ex].

  * [25] LHCb Collaboration, arXiv:2210.10346 [hep-ex].
  * [26] M. Karliner and J. L. Rosner, Phys. Rev. D 106, 036024 (2022);
arXiv:2207.07581 [hep-ph].

  * [27] F.-L. Wang and X. Liu, Phys. Lett. B 835, 137583 (2022);
arXiv:2207.10493 [hep-ph].

  * [28] M. Z. Liu, Y. W. Pan and L. S. Geng, Phys. Rev. D 103, 034003 (2021);
arXiv:2011.07935 [hep-ph].

  * [29] L. Meng, B. Wang and S.-L. Zhu, Phys. Rev. D 107, 014005 (2023);
arXiv:2208.03883 [hep-ph].

  * [30] P. G. Ortega, D. R. Entem and F. Fernandez, Phys. Lett. B 838, 137747 (2023);
arXiv:2210.04465 [hep-ph].

  * [31] J.-T. Zhu, S.-Y. Kong and J. He, Phys. Rev. D 107, 034029 (2023);
arXiv:2211.06232 [hep-ph].

  * [32] K. Chen, Z.-Y. Lin and S.-L. Zhu, arXiv:2211.05558 [hep-ph].
  * [33] Z.-Y. Yang et al., arXiv:2211.08211 [hep-ph].
  * [34] F. Yang, Y. Huang and H. Q. Zhu, Sci. China-Phys. Mech. Astron. 64, 121011 (2021);
arXiv:2107.13267 [hep-ph].

  * [35] X. Hu and J. Ping, Eur. Phys. J. C 82, 118 (2022);
arXiv:2109.09972 [hep-ph].

  * [36] H. X. Chen, W. Chen, X. Liu, and X. H. Liu, Eur. Phys. J. C 81, 409 (2021);
arXiv:2011.01079 [hep-ph].

  * [37] C. W. Xiao, J. J. Wu and B. S. Zou, Phys. Rev. D 103, 054016 (2021);
arXiv:2102.02607 [hep-ph].

  * [38] M.-L. Du, Z.-H. Guo and J. A. Oller, Phys. Rev. D 104, 114034 (2021);
arXiv:2109.14237 [hep-ph].

  * [39] Z.-G. Wang, Int. J. Mod. Phys. A 36, 2150071 (2021);
arXiv:2011.05102 [hep-ph].

  * [40] K. Azizi, Y. Sarac, and H. Sundu, Phys. Rev. D 103, 094033 (2021);
arXiv:2101.07850 [hep-ph].

  * [41] P.-P. Shi, F. Huang and W.-L. Wang, Eur. Phys. J. A 57, 237 (2021);
arXiv:2107.08680 [hep-ph].

  * [42] E. Santopinto and A. Giachino, Phys. Rev. D 96, 014014 (2017);
arXiv:1604.03769 [hep-ph].

  * [43] A. Feijoo, W.-F. Wang, C.-W. Xiao, J. J. Wu, E. Oset, J. Nieves and B. -S. Zou,
Phys. Lett. B 839, 137760 (2023); arXiv:2212.12223 [hep-ph].

  * [44] J. J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. Lett. 105, 232001 (2010);
arXiv:1007.0573 [nucl-th].

  * [45] J. J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. C 84, 015202 (2011);
arXiv:1011.2399 [nucl-th].

  * [46] Z. G. Wang, Eur. Phys. J. C 76, 142 (2016);
arXiv:1509.06436 [hep-ph].

  * [47] R. Chen, J. He and X. Liu, Chin. Phys. C 41, 103105 (2017);
arXiv:1609.03235 [hep-ph].

  * [48] C. W. Xiao, J. Nieves, and E. Oset, Phys. Lett. B 799, 135051 (2019);
arXiv:1906.09010 [hep-ph].

  * [49] A. Ali et al., JHEP 2019, 256 (2019);
arXiv:1907.06507 [hep-ph].

  * [50] B. Wang, L. Meng and S.-L. Zhu, Phys. Rev. D 101, 034018 (2020);
arXiv:1912.12592 [hep-ph].

  * [51] H. X. Chen, L. S. Geng, W. H. Liang et al., Phys. Rev. C 93, 065203 (2016);
arXiv:1510.01803 [hep-ph].

  * [52] A. Feijoo, V. K. Magas, A. Ramos and E. Oset, Eur. Phys. J. C 76, 446 (2016);
arXiv:1512.08152 [hep-ph].

  * [53] J. X. Lu, E. Wang, J. J. Xie, L. S. Geng and E. Oset, Phys. Rev. D 93, 094009 (2016);
arXiv:1601.00075 [hep-ph].

  * [54] R. Chen, Phys. Rev. D 103, 054007 (2021); [arXiv:2011.07214 [hep-ph]];
R. Chen, Eur. Phys. J. C 81, 122 (2021) [arXiv:2101.10614 [hep-ph]].

  * [55] F. Z. Peng, M. J. Yan, M. Sanchez Sanchez, and M. P. Valderrama,
Eur. Phys. J. C 81, 666 (2021); arXiv:2011.01915 [hep-ph].

  * [56] J.-T. Zhu, L.-Q. Song and J. He, Phys. Rev. D 103, 074007 (2021);
arXiv:2101.12441 [hep-ph].

  * [57] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 126, 102001 (2021);
arXiv:2011.07855 [hep-ex].

  * [58] L. Meng, B. Wang and S.-L. Zhu, Phys. Rev. D 102, 111502 (2020);
arXiv:2011.08656 [hep-ph].

  * [59] Z. Yang et al., Phys. Rev. D 103, 074029 (2021);
arXiv:2011.08725 [hep-ph].

  * [60] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 129, 112003 (2022);
arXiv:2204.13703 [hep-ex].

  * [61] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 127,082001 (2021);
arXiv:2103.01803 [hep-ex].

  * [62] M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 47, 033001 (2023);
arXiv:2211.12060 [hep-ex].

  * [63] F. L. Wang, R. Chen and X. Liu, Phys. Rev. D 103, 034014 (2021);

  * [64] F. L. Wang, X. D. Yang, R. Chen and X. Liu, Phys. Rev. D 103, 054025 (2021);
arXiv:2101.11200 [hep-ph].

  * [65] K. Azizi, Y. Sarac and H. Sundu, Eur. Phys. J. C 82, 543 (2022);
arXiv:2112.15543 [hep-ph].

  * [66] K. Azizi, Y. Sarac and H. Sundu, Phys. Rev. D 107, 014023 (2023);
arXiv:2210.03471 [hep-ph].
J. A. Marse-Valera, V. K. Magas and A. Ramos, Phys. Rev. Lett. 130, 091903
(2023);
arXiv:2210.02792 [hep-ph].

  * [67] U. Ozdem, arXiv:2303.10649 [hep-ph].
U. Ozdem, Phys. Lett. B 836, 137635 (2023); arXiv:2208.07684 [hep-ph].
U. Ozdem, Eur. Phys. J. C 81, 277 (2021); arXiv:2102.01996 [hep-ph].

  * [68] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu and S.-L. Zhu, Rept. Prog. Phys. 86, 026201 (2023);
arXiv:2204.02649 [hep-ph].

  * [69] S. Clymton, H.-J. Kim and H.-C. Kim, Phys. Rev. D 104, 014023 (2021);
arXiv:2102.08737 [hep-ph].

  * [70] E. Ya. Paryev, Nucl. Phys. A 1023, 122452 (2022);
arXiv:2205.00728 [hep-ph].

  * [71] C. Cheng, F. Yang and Y. Huang, Phys. Rev. D 104, 116007 (2021);
arXiv:2110.04746 [hep-ph].

  * [72] H. Ohnishi, F. Sakuma, T. Takahashi, Prog. Part. Nucl. Phys. 113, 103773 (2020);
arXiv:1912.02380 [nucl-ex].
K. Aoki et al., arXiv:2110.04462 [nucl-ex].

  * [73] P.-P. Shi, F.-K. Guo and Z. Yang, Phys. Rev. D 106, 114026 (2022);
arXiv:2208.02639 [hep-ph].

  * [74] C. Gobbi, C. B. Dover and A. Gal, Phys. Rev. C 50, 1594 (1994).
  * [75] S. X. Nakamura and J. J. Wu, arXiv:2208.11995 [hep-ph].
  * [76] A. Ali et al. (The GlueX Collaboration), Phys. Rev. Lett. 123, 072001 (2019);
arXiv:1905.10811 [nucl-ex].

  * [77] S. Joosten. Quarkonium production near threshold at JLab and EIC,
9th Workshop of the APS Topical Group on hadron physics (2021).
URL
https://indico.jlab.org/event/412/contributions/8266/attachments/6888/9385/20210416-GHP-
Jpsi-Threshold.pdf

  * [78] E. Ya. Paryev, Nucl. Phys. A 1029, 122562 (2023);
arXiv:2211.16037 [hep-ph].

  * [79] X. Cao, J.-P. Dai, Phys. Rev. D 100, 054033 (2019);
arXiv:1904.06015 [hep-ph].

  * [80] X.-Y. Wang, J. He, X.-R. Chen, Q. Wang, and X. Zhu, Phys. Lett. B 797, 134862 (2019);
arXiv:1906.04044 [hep-ph].

  * [81] Q. Wang, X.-H. Liu and Q. Zhao, Phys. Rev. D 92, 034022 (2015);
arXiv:1508.00339 [hep-ph].

  * [82] J. J. Wu, T.-S.H. Lee and B. S. Zou, Phys. Rev. C 100, 035206 (2019);
arXiv:1906.05375 [nucl-th].

  * [83] Y. Huang et al., J. Phys. G, Nucl. Part. Phys. 41, 115004 (2014);
arXiv:1305.4434 [nucl-th].

  * [84] X. Cao et al., Phys. Rev. D 101, 074010 (2020);
arXiv:1912.12054 [hep-ph].

  * [85] B. G. Yu, T. K. Choi and C.-R. Ji, Phys. Rev. C 70, 045205 (2004);
arXiv:nucl-th/0312075.

  * [86] B. G. Yu, T. K. Choi and C.-R. Ji, J. Phys. G 32, 387 (2006);
arXiv:nucl-th/0408006.

  * [87] D. Winney et al., Phys. Rev. D 100, 034019 (2019);
arXiv:1907.09393 [hep-ph].