aainstitutetext: School of Physics,
Beihang University, Beijing 102206, Chinabbinstitutetext: School of Space and
Environment,
Beihang University, Beijing 102206, Chinaccinstitutetext:  Beijing Key
Laboratory of Advanced Nuclear Materials and Physics,
Beihang University, Beijing 102206, Chinaddinstitutetext: Peng Huanwu
Collaborative Center for Research and Education, Beihang University, Beijing
100191, Chinaeeinstitutetext: Southern Center for Nuclear-Science Theory
(SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou
516000, Guangdong Province, China

# Production rates of hidden-charm pentaquark molecules in $\Lambda_{b}$
decays

Ya-Wen Pan, b Ming-Zhu Liu a,c,d,e and Li-Sheng Geng<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The partial decay widths and production mechanism of the three pentaquark
states, $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$, discovered by the LHCb
Collaboration in 2019, are still under debate. In this work, we employ the
contact-range effective field theory approach to construct the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$, $\bar{D}^{*}\Lambda_{c}$,
$\bar{D}\Lambda_{c}$, $J/\psi p$, and $\eta_{c}p$ coupled-channel interactions
to dynamically generate the multiplet of hidde-charm pentaquark molecules by
reproducing the masses and widths of $P_{c}(4312)$, $P_{c}(4440)$, and
$P_{c}(4457)$. Assuming that the pentaquark molecules are produced in the
$\Lambda_{b}$ decay via the triangle diagrams, where $\Lambda_{b}$ firstly
decays into $D_{s}^{(\ast)}\Lambda_{c}$, then $D_{s}^{(\ast)}$ decays into
$\bar{D}^{(\ast)}K$, and finally the molecules are dynamically generated by
the $\bar{D}^{(\ast)}\Lambda_{c}$ interactions, we calculate the branching
fractions of the decays $\Lambda_{b}\to P_{c}K$ using the effective Lagrangian
approach. With the partial decay widths of these pentaquark molecules, we
further estimate the branching fraction of the decays $\Lambda_{b}\to(P_{c}\to
J/\psi p)K$ and $\Lambda_{b}\to(P_{c}\to\bar{D}^{*}\Lambda_{c})K$. Our results
show that the pentaquark states $P_{c}(4312)$, $P_{c}(4440)$, and
$P_{c}(4457)$ as hadronic molecules can be produced in the $\Lambda_{b}$
decay, and on the other hand their heavy quark spin symmetry partners are
invisible in the $J/\psi p$ invariant mass distribution because of the small
production rates. Our studies show that is possible to observe some of the
pentaquark states in the $\Lambda_{b}\to\bar{D}^{*}\Lambda_{c}K$ decays.

##  1 Introduction

In 2015, two pentaquark states $P_{c}(4380)$ and $P_{c}(4450)$ were observed
by the LHCb Collaboration in the $J/\psi p$ invariant mass distributions of
the $\Lambda_{b}\to J/\psi pK$ decay Aaij:2015tga . Four years later, they
updated the data sample and found that the original $P_{c}(4450)$ state splits
into two states, $P_{c}(4440)$ and $P_{c}(4457)$, and a new state
$P_{c}(4312)$ emerges below the $\bar{D}\Sigma_{c}$ threshold Aaij:2019vzc .
Recently the LHCb Collaboration found the evidence of the hidden-charm
pentaquark state $P_{c}(4337)$ in the $B_{s}$ meson decay LHCb:2021chn , as
well as the hidden-charm pentaquark states with strangeness $P_{cs}(4459)$ in
the $\Xi_{b}$ decay LHCb:2020jpq , the existence of which need to be confirmed
because at present the significance of the observation is only about
3$\sigma$. Very recently the LHCb Collaboration reported another pentaquark
state $P_{cs}(4338)$ in the $B$ decay with a high significance LHCb:2022ogu .
In this work, we only focus on the three pentaquark states $P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$, which have been extensively studied in a
series of theoretical works. We note that although the
$\bar{D}^{(*)}\Sigma_{c}$ molecular interpretations for these pentaquark
states are the most popular Chen:2019asm ; He:2019ify ; Chen:2019bip ;
Xiao:2019aya ; Sakai:2019qph ; Yamaguchi:2019seo ; He:2019rva ; Liu:2019zvb ;
Valderrama:2019chc ; Meng:2019ilv ; Du:2019pij ; Ling:2021lmq ; Dong:2021juy ;
Ozdem:2021ugy ; Pan:2022xxz ; Zhang:2023czx ; Pan:2022whr ; Liu:2023wfo ,
there exist other explanations, e.g., hadro-charmonia Eides:2019tgv , compact
pentaquark states Ali:2019npk ; Wang:2019got ; Cheng:2019obk ; Weng:2019ynv ;
Zhu:2019iwm ; Pimikov:2019dyr ; Ruangyoo:2021aoi , virtual states Fernandez-
Ramirez:2019koa , triangle singularities Nakamura:2021qvy , and cusp effects
Burns:2022uiv .

From the perspective of masses, the three pentaquark states can be nicely
arranged into the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ multiplet. However, their
widths obtained in the hadronic molecule picture always deviate a bit from the
experimental data. In Ref. Xie:2022hhv , we found that their partial decay
widths into three-body final states $\bar{D}^{(\ast)}\Lambda_{c}\pi$ are only
at the order of a few hundreds of keV, which indicates that the two-body decay
modes are dominant. The chiral unitary study found that the partial decay
widths of $P_{c}\to J/\psi p(\eta_{c}p)$ account for the largest portion of
their total decay widths Xiao:2019aya , while the study based on the triangle
diagrams shows that the three $P_{c}$ states mainly decay into
$\bar{D}^{(*)}\Lambda_{c}$ Lin:2017mtz . In Refs. Yamaguchi:2019seo ;
He:2019rva ; Burns:2021jlu , the authors argued that the one-pion exchange is
responsible for the $\bar{D}^{(*)}\Sigma_{c}\to\bar{D}^{(*)}\Lambda_{c}$
interactions, and therefore dominantly contributes to the widths of the
pentaquark states. From these studies, we conclude that these three molecules
mainly decay via two modes: hidden-charm $J/\psi p(\eta_{c}p)$ and open-charm
$\bar{D}^{(*)}\Lambda_{c}$. Considering the upper limit of the branching
fraction $\mathcal{B}(P_{c}\to J/\psi p)<2\%$ measured in the photoproduction
processes Meziani:2016lhg ; GlueX:2019mkq , the partial decays
$P_{c}\to\bar{D}^{(*)}\Lambda_{c}$ are expected to play a dominant role.
However, such small upper limits cannot be easily reconciled with the current
LHCb data Cao:2019kst . In this work, we employ the contact-range effective
field theory(EFT) approach to revisit the partial decay widths of the hidden-
charm pentaquark molecules by studying their two- and three-body decays.

Up to now, the hidden-charm pentaquark states have only been observed in the
exclusive $b$ decays in proton-proton collisions. The productions of
pentaquark states in other processes have been proposed. In Refs. Wang:2015jsa
; HillerBlin:2016odx ; Karliner:2015voa ; Wang:2019krd ; Wu:2019adv , the
authors claimed that the hidden-charm pentaquark states can be produced in the
$J/\psi$ photoproduction off proton. This process could distinguish whether
these pentaquark states are genuine states or anomalous triangle
singularities. Moreover, it is suggested that the hidden-charm pentaquark
states can be produced in the $e^{+}e^{-}$ collisions Li:2017ghe  and
antiproton-deuteron collisions Voloshin:2019wxx . Based on Monte Carlo
simulations, the inclusive production rates of these pentaquark states are
estimated in proton-proton collisions Chen:2021ifb ; Ling:2021sld  and
electron-proton collisions Shi:2022ipx , which are helpful for future
experimental searches for the pentaquark states. In this work, based on the
LHCb data, we primarily focus on the production mechanism of the pentaquark
states in the $\Lambda_{b}$ decays.

The production mechanism of the pentaquark states in the $\Lambda_{b}$ decays
can be classified into two categories. In mechanism I, the mother particle $M$
weakly decays into three particles $A$, $B$ and $C$, and the hadronic molecule
under study can be dynamically generated via the rescattering of any two
particles of $A$, $B$ and $C$. This mechanism has already been applied to
study the production rates of $X(3872)$ as a $\bar{D}D^{*}$ molecule via the
weak decays $B\to\bar{D}D^{*}K$ Braaten:2004fk ; Braaten:2004ai . For the
pentaquark states, it was proposed that the weak decays of
$\Lambda_{b}\to\bar{D}^{(*)}\Sigma_{c}K$ and $\Lambda_{b}\to J/\psi pK$ can
dynamically generate the hidden-charm pentaquark molecules via the
$\bar{D}^{(*)}\Sigma_{c}$ rescattering Du:2019pij ; Du:2021fmf  and $J/\psi p$
rescatterring Roca:2015dva , respectively, which can well describe the
experimental invariant mass distribution of $J/\psi p$, while their absolute
production rates are not quantitatively estimated. In particular, as pointed
out in Ref. Burns:2022uiv , the branching fractions
$Br(\Lambda_{b}\to\bar{D}^{(*)}\Sigma_{c}K)$ are so tiny that the pentaquark
molecules are rather difficult to be produced via the weak decays
$\Lambda_{b}\to\bar{D}^{(*)}\Sigma_{c}K$. Therefore, whether these pentaquark
molecules can be produced via Mechanism I remains unsettled.

In Mechanism II, the mother particle $M$ weakly decays into two states $A$ and
$B$, then $A$ decays into $C$ and $D$, and finally the final-state interaction
of $B$ and $C$ dynamically generates the molecules of interest Wang:2013cya ;
Guo:2013zbw ; Hsiao:2019ait ; Liu:2022dmm ; Wu:2023rrp . A typical example is
that the $X(3872)$ as a $\bar{D}D^{*}$ molecule can be generated through the
weak decays $B\to\bar{D}^{(*)}D_{s}^{(*)}$ following $D_{s}^{(*)}$ decaying
into $D^{(*)}K$ Wu:2023rrp . In Ref. Wu:2019rog , Wu et al. proposed that
$\Lambda_{b}$ weakly decays into $\Sigma_{c}$ and ${D}_{s}^{(*)}$, then
${D}_{s}^{(*)}$ decays into $\bar{D}^{(*)}$ and $K$, and the pentaquark
molecules are finally generated via the $\bar{D}^{(*)}\Sigma_{c}$
interactions. We note that the $\Lambda_{b}$ decaying into $\Sigma_{c}^{(*)}$
is highly suppressed due to the fact the light quark pair transition between a
symmetric and antisymmetric spin-flavor configuration is forbidden Falk:1992ws
; Gutsche:2018utw , which indicates that the production of pentaquark
molecules is difficult (if not impossible) via the weak decays of
$\Lambda_{b}\to{D}_{s}^{(*)}\Sigma_{c}$ 111In Ref. Wu:2019rog , the
$\Lambda_{b}\to\Sigma_{c}$ transition is assumed to be proportional to the
$\Lambda_{b}\to\Lambda_{c}$ transition, characterized by an unknown parameter
$R$. By reproducing the experimental production rates of the pentaquark
molecules, $R$ is found to be about 0.1. . In Ref. Burns:2022uiv , the authors
select the Color favorable weak decays
$\Lambda_{b}\to{D}_{s}^{(*)}\Lambda_{c}$ to produce the pentaquark molecules
as well as to analyse their mass distributions, but did not explicitly
calculate their productions rates. Following Refs. Liu:2022dmm ; Wu:2023rrp ,
we take the effective Lagrangian approach to calculate the production rates of
the pentaquark molecules in $\Lambda_{b}$ decays with no free parameters, and
try to answer the questions whether the three pentaquark states $P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$ as hadronic molecules can be produced in the
$\Lambda_{b}$ decays, as well as why their HQSS partners have not been
observed in the same decays.

This work is organized as follows. We first calculate the two-body partial
decay widths of the pentaquark molecules obtained by the contact range EFT,
and the amplitudes of their production mechanism in $\Lambda_{b}$ decays via
the triangle diagrams using the effective Lagrangian approach in section 2.
The results and discussions on the widths of the pentaquark molecules and the
branching fractions of the decays $P_{c}\to J/\psi p$ and
$P_{c}\to\bar{D}^{(*)}\Lambda_{c}$, as well as the branching fractions of the
weak decays $\Lambda_{b}\to P_{c}K$, $\Lambda_{b}\to(P_{c}\to J/\psi p)K$, and
$\Lambda_{b}\to(P_{c}\to\bar{D}^{*}\Lambda_{c})K$ are provided in section 3,
followed by a short summary in the last section.

##  2 Theoretical framework

\begin{overpic}[scale={.59}]{triangle.eps} \put(75.0,7.0){$P_{c}^{1/2}$}
\put(38.0,8.0){$\Lambda_{c}$} \put(35.0,36.0){$D_{s}$}
\put(14.0,27.0){$\Lambda_{b}$ } \put(75.0,36.0){$K$}
\put(60.0,22.0){$\bar{D}^{*}$} \end{overpic}

\begin{overpic}[scale={.59}]{triangle.eps} \put(75.0,7.0){$P_{c}^{1/2}$}
\put(38.0,8.0){$\Lambda_{c}$} \put(34.0,36.0){$D_{s}^{*}$}
\put(14.0,27.0){$\Lambda_{b}$ } \put(75.0,36.0){$K$}
\put(60.0,22.0){$\bar{D}^{*}$} \end{overpic}

\begin{overpic}[scale={.59}]{triangle.eps} \put(75.0,7.0){$P_{c}^{1/2}$}
\put(38.0,8.0){$\Lambda_{c}$} \put(34.0,36.0){$D_{s}^{*}$}
\put(14.0,27.0){$\Lambda_{b}$ } \put(75.0,36.0){$K$}
\put(60.0,22.0){$\bar{D}$} \end{overpic}

Figure 1: Triangle diagrams accounting for $\Lambda_{b}\to
D_{s}^{(\ast)}\Lambda_{c}\to P_{c}^{1/2}K$ with $P_{c}^{1/2}$ representing one
of the pentaquark molecules of spin $1/2$

In this work, we employ the triangle diagrams to describe the productions of
pentaquark molecules. We suppose that the colour favored weak decays
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(*)-}$ are responsible for the short-range
interactions because the branching fractions
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}D_{s}^{(*)-})$ are large among the
nonleptonic decays of $\Lambda_{b}$. Then the $D_{s}^{(\ast)-}$ mesons decay
into $\bar{D}^{(\ast)}$ and $K$ mesons, and the pentaquark molecules with spin
$1/2$ and $3/2$ are dynamically generated via the
$\bar{D}^{(\ast)}\Lambda_{c}$ interactions as shown in Fig. 1 and Fig. 2,
respectively, where $P_{c}^{1/2}$ and $P_{c}^{3/2}$ denote the pentaquark
molecules of spin $1/2$ and $3/2$, respectively. As shown in a number of
previous studies Liu:2019tjn ; Xiao:2019aya ; Du:2019pij ; Yamaguchi:2019seo ;
PavonValderrama:2019nbk ; Lin:2019qiv ; He:2019rva ; Yalikun:2021bfm ;
Dong:2021juy ; Zhang:2023czx  and also explicitly shown later, there exists a
complete multiplet of hidden-charm pentaquark molecules dominantly generated
by the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ interactions. We denote the seven
pentaquark molecules as $P_{c1}$, $P_{c2}$, … ,$P_{c7}$, following the order
of Scenario A of Table I in Ref. Liu:2019tjn . Considering only $S$-wave
$\bar{D}^{(*)}\Lambda_{c}$ interactions, the pentaquark molecule $P_{c7}$ can
not be produced via Fig. 1 or Fig. 2, which means that the production rate of
$P_{c7}$ in the $\Lambda_{b}$ decay will be very low. Therefore, we only focus
on the productions of the remaining six pentaquark molecules in the
$\Lambda_{b}$ decays in this work.

\begin{overpic}[scale={.59}]{triangle.eps} \put(75.0,7.0){$P_{c}^{3/2}$}
\put(38.0,8.0){$\Lambda_{c}$} \put(35.0,36.0){$D_{s}$}
\put(14.0,27.0){$\Lambda_{b}$ } \put(75.0,36.0){$K$}
\put(60.0,22.0){$\bar{D}^{*}$} \end{overpic}

\begin{overpic}[scale={.59}]{triangle.eps} \put(75.0,7.0){$P_{c}^{3/2}$}
\put(38.0,8.0){$\Lambda_{c}$} \put(34.0,36.0){$D_{s}^{*}$}
\put(14.0,27.0){$\Lambda_{b}$ } \put(75.0,36.0){$K$}
\put(60.0,22.0){$\bar{D}^{*}$} \end{overpic}

Figure 2: Triangle diagrams accounting for $\Lambda_{b}\to
D_{s}^{(\ast)}\Lambda_{c}\to P_{c}^{3/2}K$ with $P_{c}^{3/2}$ denoting one of
the pentaquark molecule of spin $3/2$.

###  2.1 Effective Lagrangians

\begin{overpic}[scale={.4}]{quark2.eps} \put(5.0,25.0){$b$ }
\put(5.0,3.0){$u$} \put(93.0,3.0){$u$} \put(5.0,14.0){$d$}
\put(93.0,14.0){$d$} \put(93.0,25.0){$c$} \put(81.0,45.0){$c$}
\put(71.0,55.0){$\bar{s}$} \put(37.0,32.0){$W^{-}$} \end{overpic} Figure 3:
External $W$-emission accounting for $\Lambda_{b}\to D_{s}^{(*)}\Lambda_{c}$
at quark level.

In this work, we adopt the effective Lagrangian approach to calculate the
triangle diagrams of Figs. 1 and 2. In the following, we spell out the
relevant Lagrangians.

First, we focus on the weak decays of
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(\ast)-}$. At quark level, the decays of
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(*)-}$ can occur via the external
$W$-emission mechanism shown in Fig. 3, which is usually the largest in terms
of the topological classification of weak decays Chau:1982da ; Chau:1987tk ;
Molina:2019udw . As shown in Ref. Wu:2023rrp , the color favored weak decays
$B\to D_{s}^{(\ast)}{D}^{(\ast)}$ are significant to produce the
$\bar{D}^{*}D^{(*)}$ molecules in $B$ decays, which share similar topologies
to the weak decays $\Lambda_{b}\to\Lambda_{c}D_{s}^{(*)-}$ at quark level.

The effective Hamiltonian describing the weak decays of
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(*)-}$ has the following form

$\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}[c_{1}(\mu)\mathcal{O}_{1}(\mu)+c_{2}(\mu)\mathcal{O}_{2}(\mu)]+h.c.,$
(1)

where $G_{F}$ is the Fermi constant, $V_{bc}$ and $V_{cs}$ are the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements, $c_{1,2}(\mu)$ are the Wilson
coefficients, and $\mathcal{O}_{1}(\mu)$ and $\mathcal{O}_{2}(\mu)$ are the
four-fermion operators of $(s\bar{c})_{V-A}(c\bar{b})_{V-A}$ and
$(\bar{c}c)_{V-A}(s\bar{b})_{V-A}$ with $(\bar{q}q)_{V-A}$ standing for
$\bar{q}\gamma_{\mu}(1-\gamma_{5})q$ Cheng:1996cs ; Ali:1998eb ; Li:2012cfa .
The Wilson coefficients $c_{1,2}(\mu)$ include the short-distance quantum
chromodynamics (QCD) dynamic scaling from $\mu=M_{W}$ to $\mu=m_{c}$.

In the naive factorisation approach Bauer:1986bm , the amplitudes of
$\Lambda_{b}\to\Lambda_{c}D_{s}^{(\ast)-}$ can be expressed as the products of
two current hadronic matrix elements

$\displaystyle\mathcal{A}\left(\Lambda_{b}\to\Lambda_{c}D_{s}^{-}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}\left\langle
D_{s}^{-}|(s\bar{c})|0\right\rangle\left\langle\Lambda_{c}|(c\bar{b})|\Lambda_{b}\right\rangle$
(2)
$\displaystyle\mathcal{A}\left(\Lambda_{b}\to\Lambda_{c}D_{s}^{\ast-}\right)$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}\left\langle
D_{s}^{\ast-}|(s\bar{c})|0\right\rangle\left\langle\Lambda_{c}|(c\bar{b})|\Lambda_{b}\right\rangle$
(3)

where the effective Wilson coefficient $a_{1}$ is expressed as
$a_{1}=c_{1}(\mu)+c_{2}(\mu)/N_{c}$ with $N_{c}=3$ the number of colors
Bauer:1986bm ; Li:2012cfa .

The matrix elements between a pseudoscalar meson or vector meson and the
vacuum have the following form:

$\displaystyle\left\langle D_{s}^{-}|(s\bar{c})|0\right\rangle$
$\displaystyle=$ $\displaystyle
i\,f_{D_{s}^{-}}p^{\mu}_{D_{s}^{-}}\,,~{}~{}\left\langle
D_{s}^{\ast-}|(s\bar{c})|0\right\rangle=m_{D_{s}^{\ast-}}f_{D_{s}^{\ast-}}\epsilon_{\mu}^{*}\,,$
(4)

where $f_{D_{s}^{-}}$ and $f_{D_{s}^{\ast-}}$ are the decay constants for
$D_{s}^{-}$ and $D_{s}^{\ast-}$, respectively, and $\epsilon_{\mu}^{*}$
denotes the polarization vector of $D_{s}^{\ast-}$.

The $\Lambda_{b}\to\Lambda_{c}$ transition form factors are parameterized as
follows Gutsche:2018utw

$\displaystyle\langle B(p^{\prime})|V_{\mu}-A_{\mu}|B(p)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}(p^{\prime})[f_{1}^{V}(q^{2})\gamma_{\mu}-f_{2}^{V}(q^{2})\frac{i\sigma_{\mu\nu}q^{\nu}}{m}+f_{3}^{V}(q^{2})\frac{q^{\mu}}{m}$
$\displaystyle-(f_{1}^{A}(q^{2})\gamma_{\mu}-f_{2}^{A}(q^{2})\frac{i\sigma_{\mu\nu}q^{\nu}}{m}+f_{3}^{A}(q^{2})\frac{q^{\mu}}{m})\gamma^{5}]u(p)$

where
$\sigma^{\mu\nu}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$
and $q=p-p^{\prime}$. As a result, the weak decays
$\Lambda_{b}\rightarrow\Lambda_{c}{D}_{s}^{(*)}$ can be characterised by the
following Lagrangian Cheng:1996cs :

$\begin{split}\mathcal{L}_{\Lambda_{b}\Lambda_{c}D_{s}}&=i\bar{\Lambda}_{c}(A+B\gamma_{5})\Lambda_{b}D_{s}\,,\\\
\mathcal{L}_{\Lambda_{b}\Lambda_{c}D_{s}^{*}}&=\bar{\Lambda}_{c}(A_{1}\gamma_{\mu}\gamma_{5}+A_{2}\frac{p_{2\mu}}{m}\gamma_{5}+B_{1}\gamma_{\mu}+B_{2}\frac{p_{2\mu}}{m})\Lambda_{b}D_{s}^{*\mu}\,.\end{split}$
(6)

where $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $A$, and $B$ are:

$\begin{split}A&=\lambda
f_{D_{s}}[(m-m_{2})f_{1}^{V}+\frac{m_{1}^{2}}{m}f_{3}^{V}]\,,\quad B=\lambda
f_{D_{s}}[(m+m_{2})f_{1}^{A}-\frac{m_{1}^{2}}{m}f_{3}^{A}]\,,\\\
A_{1}&=-\lambda
f_{D_{s}^{*}}m_{1}[f_{1}^{A}-f_{2}^{A}\frac{m-m_{2}}{m}]\,,\quad B_{1}=\lambda
f_{D_{s}^{*}}m_{1}[f_{1}^{V}+f_{2}^{V}\frac{m+m_{2}}{m}]\,,\\\ A_{2}&=2\lambda
f_{D_{s}^{*}}m_{1}f_{2}^{A}\,,\quad\quad\quad\quad\quad\quad\quad\quad
B_{2}=-2\lambda f_{D_{s}^{*}}m_{1}f_{2}^{V}\,.\end{split}$ (7)

with $\lambda=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cs}a_{1}$ and $m,m_{1},m_{2}$
referring to the masses of $\Lambda_{b}$, ${D}_{s}^{(*)}$, and $\Lambda_{c}$,
respectively. The form factors can be expressed in a double-pole form:

$f_{i}^{V/A}(q^{2})=F_{i}^{V/A}(0)\frac{F_{0}}{1-a\,\varphi+b\,\varphi^{2}}$
(8)

with $\varphi=q^{2}/m^{2}$. The values of $F_{0}$, $a$ and $b$ in the
$\Lambda_{b}\rightarrow\Lambda_{c}$ transition form factors are taken from
Ref. Gutsche:2015mxa  and shown in Table 1.

| $F_{1}^{V}$ | $F_{2}^{V}$ | $F_{3}^{V}$ | $F_{1}^{A}$ | $F_{2}^{A}$ | $F_{3}^{A}$
---|---|---|---|---|---|---
$F(0)$ | 0.549 | 0.110 | $-0.023$ | 0.542 | 0.018 | $-0.123$
$a$ | 1.459 | 1.680 | 1.181 | 1.443 | 0.921 | 1.714
$b$ | 0.571 | 0.794 | 0.276 | 0.559 | 0.255 | 0.828
Table 1: Values of $F(0)$, $a$, $b$ in the $\Lambda_{b}\rightarrow\Lambda_{c}$
transition form factors Gutsche:2015mxa .

In this work, we take $G_{F}=1.166\times 10^{-5}~{}{\rm GeV}^{-2}$,
$V_{cb}=0.041$, $V_{cs}=0.987$, $f_{D_{s}^{-}}=250$ MeV, and
$f_{D_{s}^{\ast-}}=272$ MeV as in Refs. ParticleDataGroup:2020ssz ;
Verma:2011yw ; Gutsche:2018utw ; FlavourLatticeAveragingGroup:2019iem . We
note that the value of $a_{1}$ as a function of the energy scale $\mu$ differs
from process to process Chen:1999nxa ; Cheng:2010ry . Therefore, we take the
branching fraction $\mathcal{B}(\Lambda_{b}\to\Lambda_{c}D_{s}^{-})=(1.10\pm
0.10)\%$ to determine the effective Wilson coefficient to be $a_{1}=0.883$. We
note that in Ref. Wu:2023rrp , the effective Wilson Coefficient $a_{1}$ are
determined to be 0.79 and 0.81 by reproducing the branching fractions of the
decays $B^{+}\to\bar{D}^{0}D_{s}^{+}$ and $B^{+}\to\bar{D}^{0}D_{s}^{\ast+}$,
respectively. These values are consistent with the value obtained from the
weak decay $\Lambda_{b}\to\Lambda_{c}D_{s}^{-}$, showing that the naive
factorisation approach works well for the external $W$-emission mechanism. Due
to the non availability of experimental data for the branching fraction
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}D_{s}^{*-})$, we assume that the
effective Wilson coefficient in Eq.(3) is the same as that in Eq.(2). With the
so obtained effective Wilson coefficient $a_{1}$ we predict the branching
fraction $\mathcal{B}(\Lambda_{b}\to\Lambda_{c}D_{s}^{*-})=(2.47\pm 0.26)\%$,
consistent with the results of Refs. Gutsche:2018utw ; Chua:2019yqh . As a
matter of fact, the experimental branching fraction
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}D_{s}^{-})$ helps reduce the uncertainty
in the weak vertices.

The Lagrangians describing the $D_{s}^{(*)}$ mesons decaying into
$\bar{D}^{(*)}$ and $K$ mesons are:

$\begin{split}\mathcal{L}_{KD_{s}D^{*}}&=ig_{KD_{s}D^{*}}D^{*\mu}[\bar{D}_{s}\partial_{\mu}K-(\partial_{\mu}\bar{D}_{s})K]+H.c.\,,\\\
\mathcal{L}_{KD_{s}^{*}D^{*}}&=-g_{KD_{s}^{*}D^{*}}\epsilon^{\mu\nu\alpha\beta}[\partial_{\mu}\bar{D}_{\nu}^{*}\partial_{\alpha}D^{*}_{s\beta}\bar{K}+\partial_{\mu}D_{\nu}^{*}\partial_{\alpha}\bar{D}^{*}_{s\beta}K]+H.c.\,,\end{split}$
(9)

where $g_{KD_{s}D^{*}}$ and $g_{KD_{s}^{*}D^{*}}$ are the kaon couplings to
$D_{s}D^{\ast}$ and $D_{s}^{\ast}D^{*}$, respectively. Unfortunately, there
exists no experimental data to determine the values of these couplings. The
coupling $g_{D_{s}D^{\ast}K}$ is estimated to be $16.6$ and $10$ assuming
SU(3)-flavor symmetry Xie:2022lyw  and SU(4)-flavor symmetry Azevedo:2003qh ,
respectively, while the QCD sum rule yields $5$ Bracco:2006xf ; Wang:2006ida .
In view of this large variance, we adopt the couplings estimated by SU(4)
symmetry, which are in between those estimated utilizing SU(3) symmetry and by
the QCD sum rule, i.e., $g_{D_{s}D^{\ast}K}=g_{D_{s}^{\ast}DK}=10$ and
$g_{D_{s}^{\ast}D^{\ast}K}=7.0$ GeV-1 Azevedo:2003qh .

The effective Lagrangians describing the interactions between pentaquark
molecules and their constituents $\bar{D}^{(*)}\Lambda_{c}$ are written as

$\begin{split}\mathcal{L}_{P_{c}^{1/2}\Lambda_{c}\bar{D}}&=g_{P_{c}^{1/2}\Lambda_{c}\bar{D}}P_{c}^{1/2}\Lambda_{c}\bar{D}\,,\\\
\mathcal{L}_{P_{c}^{1/2}\Lambda_{c}\bar{D}^{*}}&=g_{P_{c}^{1/2}\Lambda_{c}\bar{D}^{*}}\bar{\Lambda}_{c}\gamma_{5}(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m_{P_{c}^{1/2}}^{2}})\gamma^{\nu}P_{c}^{1/2}D^{*\mu}\,,\\\
\mathcal{L}_{P_{c}^{3/2}\Lambda_{c}\bar{D}^{*}}&=g_{P_{c}^{3/2}\Lambda_{c}\bar{D}^{*}}\bar{\Lambda}_{c}P_{c\mu}^{3/2}D^{*\mu}\,.\end{split}$
(10)

where $g_{P_{c}^{1/2}\Lambda_{c}\bar{D}}$,
$g_{P_{c}^{1/2}\Lambda_{c}\bar{D}^{*}}$, and
$g_{P_{c}^{3/2}\Lambda_{c}\bar{D}^{*}}$ are the couplings of the $P_{c}^{1/2}$
and $P_{c}^{3/2}$ pentaquark molecules to their constituents
$\bar{D}^{(*)}\Lambda_{c}$. One should note that although these pentaquark
molecules are dominantly generated by the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
interactions Liu:2019tjn , the $\bar{D}^{(*)}\Lambda_{c}$ and
$J/\psi(\eta_{c})p$ coupled channels also play a relevant role Guo:2019kdc ;
Xiao:2019aya ; Lin:2019qiv ; Yamaguchi:2019seo ; He:2019rva ; Burns:2021jlu .
Below, we estimate the couplings of the pentaquark molecules to their
constituents $\bar{D}^{(*)}\Lambda_{c}$ and $J/\psi(\eta_{c})p$ by the contact
range EFT approach, which is widely applied to study the dynamical generation
of hadronic molecules Ji:2022vdj ; Pan:2022whr .

###  2.2 Contact-range EFT approach

In this subsection, we introduce the contact-range EFT approach. The
scattering amplitude $T$ is responsible for the dynamical generation of the
pentaquark molecules via the Lippmann-Schwinger equation

$\displaystyle T(\sqrt{s})=(1-VG(\sqrt{s}))^{-1}V,$ (11)

where $V$ is the coupled-channel potential determined by the contact-range EFT
approach (see Appendix A.1), and $G(\sqrt{s})$ is the two-body propagator. In
this work, we consider the following coupled channels
$\bar{D}^{*}\Sigma_{c}^{*}-\bar{D}^{*}\Sigma_{c}-\bar{D}\Sigma_{c}-\bar{D}^{*}\Lambda_{c}-\bar{D}\Lambda_{c}-J/\psi
p-\eta_{c}p$ with $J^{P}=1/2^{-}$ and
$\bar{D}^{*}\Sigma_{c}^{*}-\bar{D}^{*}\Sigma_{c}-\bar{D}^{*}\Lambda_{c}-J/\psi
p$ with $J^{P}=3/2^{-}$. Since the mass splitting between
$\bar{D}^{*}\Sigma_{c}^{*}$ and $\eta_{c}p$ is about 600 MeV, we take a
relativistic propagator:

$\displaystyle
G(\sqrt{s})=2m_{1}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{\omega_{1}+\omega_{2}}{2~{}\omega_{1}\omega_{2}}\frac{F(q^{2},k)}{(\sqrt{s})^{2}-(\omega_{1}+\omega_{2})^{2}+i\varepsilon},$
(12)

where $\sqrt{s}$ is the total energy in the center-of-mass (c.m.) frame of
$m_{1}$ and $m_{2}$, $\omega_{i}=\sqrt{m_{i}^{2}+q^{2}}$ is the energy of the
particle and the c.m. momentum $k$ is

$\displaystyle
k(\sqrt{s})=\frac{\sqrt{{\sqrt{s}^{2}}-(m_{1}+m_{2})^{2}}\sqrt{\sqrt{s}^{2}-(m_{1}-m_{2})^{2}}}{2{\sqrt{s}}}.$
(13)

A regulator of Gaussian form
$F(q^{2},k)=e^{-2q^{2}/\Lambda^{2}}/e^{-2k^{2}/\Lambda^{2}}$ is used to
regulate the loop function. We note that the loop function can also be
regularized by other methods such as the momentum cut off scheme and
dimensional regularization scheme Oset:1997it ; Jido:2003cb ; Wu:2010jy ;
Hyodo:2011ur ; Debastiani:2017ewu .

The dynamically generated pentaquark molecules correspond to poles on the
unphysical sheet, which is defined as Oller:1997ti ; Roca:2005nm ,

$\displaystyle
G_{II}(\sqrt{s})=G_{I}(\sqrt{s}){+~{}i\frac{2m_{1}}{4\pi}\frac{k(\sqrt{s})}{\sqrt{s}}},$
(14)

where $m_{1}$ stands for the mass of the baryon.

With the potentials obtained in Eq. (29) and Eq. (30) of the Appendix, we
search for poles in the vicinity of the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
channels, and then determine the couplings between the pentaquark molecules
and their constituents from the residues of the corresponding poles,

$\displaystyle
g_{i}g_{j}=\lim_{{\sqrt{s}}\to{\sqrt{s_{0}}}}\left({\sqrt{s}}-{\sqrt{s_{0}}}\right)T_{ij}(\sqrt{s}),$
(15)

where $g_{i}$ denotes the coupling of channel $i$ to the dynamically generated
molecules and ${\sqrt{s_{0}}}$ is the pole position.

Using the couplings $g_{i}$ obtained above, one can estimate the partial decay
widths of the pentaquark molecules Yu:2018yxl

$\displaystyle\Gamma_{i}=g_{i}^{2}\frac{1}{2\pi}\frac{m_{i}}{m_{P_{c}}}p_{i}$
(16)

where $m_{i}$ stands for the mass of the baryon of channel $i$, $m_{P_{c}}$ is
the mass of the pentaquark molecule (the real part of the pole position), and
$p_{i}$ is the momentum of the baryon (meson) of channel $i$ in the $P_{c}$
rest frame.

###  2.3 Decay Amplitudes

With the above effective Lagragians, we obtain the following decay amplitudes
for $\Lambda_{b}\to P_{c}^{1/2}K$ of Fig. 1

$\displaystyle\begin{split}\mathcal{M}_{1}^{a}=&i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}[g_{P_{c}^{1/2}\Lambda_{c}\bar{D}^{\ast}}\bar{u}(p_{2})\gamma^{\nu}\gamma_{5}(g_{\mu\nu}-\frac{p_{2\mu}p_{2\nu}}{m^{2}_{p_{c}}})(q_{2}\\!\\!\\!\\!\\!/+m_{2})\,i(A+B\gamma_{5})u(k_{0})]\\\
&[-g_{KD^{\ast}D_{s}}(q_{1}+p_{1})_{\alpha}](-g^{\mu\alpha}+\frac{q^{\mu}q^{\alpha}}{m^{2}_{E}})\frac{1}{q^{2}_{1}-m^{2}_{1}}\frac{1}{q^{2}_{2}-m^{2}_{2}}\frac{1}{q^{2}-m^{2}_{E}}\,,\\\
\mathcal{M}_{1}^{b}=&i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}[g_{P_{c}^{1/2}\Lambda_{c}\bar{D}^{\ast}}\bar{u}(p_{2})\gamma^{\nu}\gamma_{5}(g_{\mu\nu}-\frac{p_{2\mu}p_{2\nu}}{m^{2}_{P_{c}}})(q_{2}\\!\\!\\!\\!\\!/+m_{2})\\\
&[(A_{1}\gamma_{\alpha}\gamma_{5}+A_{2}\frac{q_{2\alpha}}{m}\gamma_{5}+B_{1}\gamma_{\alpha}+B_{2}\frac{q_{2\alpha}}{m})u(k_{0})][-g_{KD^{\ast}D^{\ast}_{s}}\varepsilon_{\rho\lambda\eta\tau}q^{\rho}q^{\eta}_{1}]\\\
&(-g^{\mu\lambda}+\frac{q^{\mu}q^{\lambda}}{m^{2}_{E}})(-g^{\alpha\tau}+\frac{q_{1}^{\alpha}q_{1}^{\tau}}{m^{2}_{1}})\frac{1}{q^{2}_{1}-m^{2}_{1}}\frac{1}{q^{2}_{2}-m^{2}_{2}}\frac{1}{q^{2}-m^{2}_{E}}\,,\\\
\mathcal{M}_{1}^{c}=&i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}[g_{P_{c}^{1/2}\Lambda_{c}\bar{D}}\bar{u}(p_{2})(q_{2}\\!\\!\\!\\!\\!/+m_{2})(A_{1}\gamma_{\alpha}\gamma_{5}+A_{2}\frac{q_{2\alpha}}{m}\gamma_{5}+B_{1}\gamma_{\alpha}+B_{2}\frac{q_{2\alpha}}{m})u(k_{0})]\\\
&[-g_{KDD^{\ast}_{s}}(-q+p_{1})_{\tau}](-g^{\alpha\tau}+\frac{q_{1}^{\alpha}q_{1}^{\tau}}{m^{2}_{1}})\frac{1}{q^{2}_{1}-m^{2}_{1}}\frac{1}{q^{2}_{2}-m^{2}_{2}}\frac{1}{q^{2}-m^{2}_{E}}\,.\end{split}$
(17)

where $k_{0}$, $q_{1}$, $q_{2}$, $q$, $p_{1}$, and $p_{2}$ refer to the
momenta of $\Lambda_{b}$, $D_{s}^{(\ast)}$, $\Lambda_{c}$, $\bar{D}^{(\ast)}$,
$K$, and $P_{c}^{1/2}$, respectively, and $\bar{u}(p_{2})$ and $u(k_{0})$
represent the spinors of $P_{c}^{1/2}$ and $\Lambda_{b}$. Similarly, we write
the decay amplitudes for $\Lambda_{b}\to P_{c}^{3/2}K$ of Fig. 2 as follows

$\displaystyle\begin{split}\mathcal{M}_{3}^{a}=&i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}[g_{P_{c}^{3/2}\Lambda_{c}\bar{D}^{\ast}}\bar{u}_{\mu}(p_{2})](q_{2}\\!\\!\\!\\!\\!/+m_{2})[i(A+B\gamma_{5})u(k_{0})]\\\
&[-g_{KD^{\ast}D_{s}}(q_{1}+p_{1})_{\nu}](-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{E}})\,\frac{1}{q^{2}_{1}-m^{2}_{1}}\frac{1}{q^{2}_{2}-m^{2}_{2}}\frac{1}{q^{2}-m^{2}_{E}}\,,\\\
\mathcal{M}_{3}^{b}=&i^{3}\int\frac{d^{4}q}{(2\pi)^{4}}[g_{P_{c}^{3/2}\Lambda_{c}\bar{D}^{\ast}}\bar{u}_{\mu}(p_{2})(q_{2}\\!\\!\\!\\!\\!/+m_{2})(-i)(A_{1}\gamma_{\alpha}\gamma_{5}+A_{2}\frac{q_{2\alpha}}{m}\gamma_{5}+B_{1}\gamma_{\alpha}+B_{2}\frac{q_{2\alpha}}{m})u(k_{0})]\\\
&[-g_{KD^{\ast}D^{\ast}_{s}}\varepsilon_{\rho\lambda\eta\tau}q^{\rho}q^{\eta}_{1}](-g^{\mu\lambda}+\frac{q^{\mu}q^{\lambda}}{m^{2}_{E}})(-g^{\alpha\tau}+\frac{q_{1}^{\alpha}q_{1}^{\tau}}{m^{2}_{1}})\frac{1}{q^{2}_{1}-m^{2}_{1}}\frac{1}{q^{2}_{2}-m^{2}_{2}}\frac{1}{q^{2}-m^{2}_{E}}\,.\end{split}$
(18)

With the amplitudes for the decays of $\Lambda_{b}\to P_{c}^{1/2}K$ and
$\Lambda_{b}\to P_{c}^{3/2}K$ given above, one can compute the corresponding
partial decay widths

$\displaystyle\Gamma=\frac{1}{2J+1}\frac{1}{8\pi}\frac{|\vec{p}|}{m_{\Lambda_{b}}^{2}}{|\overline{M}|}^{2}$
(19)

where $J$ is the total angular momentum of the initial $\Lambda_{b}$ baryon
and $|\vec{p}|$ is the momentum of either final state in the rest frame of the
$\Lambda_{b}$ baryon.

Regarding the three-body decays of the pentaquark molecules, we have
systematically investigated two decay modes: tree diagrams and triangle
diagrams, and found that the former can almost saturate their total three-body
decay widths Xie:2022hhv . In this work, with the new couplings between the
pentaquark molecules and $\bar{D}^{(*)}\Sigma_{c}^{(*)}$, we update the widths
of the three-body decays $P_{c}\to\bar{D}^{(*)}\Lambda_{c}\pi$, where these
hidden-charm pentaquark molecules decay into $\bar{D}^{(*)}\Lambda_{c}\pi$ via
the off-shell $\Sigma_{c}^{(*)}$ baryons decaying into $\Lambda_{c}\pi$. The
details for the calculations can be found in our previous work Xie:2022hhv .

##  3 Results and discussions

Because the three-body partial decay widths of the pentaquark states
$P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ as hadronic molecules are less
than $1$ MeV, we can neglect their three-body decays and assume that their
two-body decays saturate their total widths. Therefore, we suppose that these
three pentaquark molecules are dynamically generated via the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$, $\bar{D}^{(\ast)}\Lambda_{c}$,
$\eta_{c}N$ and $J/\psi N$ coupled-channel potentials. In the heavy quark
limit, the contact potentials of this coupled-channel system are parameterized
by seven parameters as shown in Eq.(29) and Eq.(30). In this work, we set the
potential $V_{J/\psi(\eta_{c})N\to J/\psi(\eta_{c})N}=0$, resulting in six
unknown parameters. The unknown couplings of the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}\to\bar{D}^{(*)}\Sigma_{c}^{(*)}$ potentials are
well described by the light meson saturation approach Liu:2019zvb , which is
widely applied to study heavy hadronic molecules Peng:2020xrf ; Dong:2021juy ;
Peng:2021hkr ; Dong:2021bvy . Therefore, we expect that the light meson
saturation approach(see Appendix) is also valid for the
$\bar{D}^{(*)}\Lambda_{c}\to\bar{D}^{(*)}\Lambda_{c}$ interaction. With the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}\to\bar{D}^{(*)}\Sigma_{c}^{(*)}$ potentials
determined in Ref. Liu:2019tjn , we obtain the
$\bar{D}^{(*)}\Lambda_{c}\to\bar{D}^{(*)}\Lambda_{c}$ potentials using the
light meson saturation approach, and then search for poles near the
$\bar{D}^{(*)}\Lambda_{c}$ threshold, but find none, indicating that there
exists no genuine states generated by the $\bar{D}^{(*)}\Lambda_{c}$
interactions, consistent with Refs. Yang:2011wz ; Xiao:2019aya ; He:2019rva .
Very recently, Duan et al. argued that there exist the enhancements at the
$\bar{D}^{(*)}\Lambda_{c}$ thresholds induced by the triangle and box
singularities Duan:2023dky . Therefore, even taking into account the
$\bar{D}^{(*)}\Lambda_{c}$, $\eta_{c}N$, and $J/\psi N$ channels, the number
of hidden-charm pentaquark molecules does not change. These channels mainly
affect the imaginary part of the pole positions, i.e., the widths of the
pentaquark states.

Hadron | $I(J^{P})$ | M (MeV) | Hadron | $I(J^{P})$ | M (MeV)
---|---|---|---|---|---
$p$ | $\frac{1}{2}(1/2^{+})$ | $938.27$ | $n$ | $\frac{1}{2}(1/2^{+})$ | $939.57$
$\Sigma_{c}^{++}$ | $1(1/2^{+})$ | $2453.97$ | $\Sigma_{c}^{+}$ | $1(1/2^{+})$ | $2452.65$
$\Sigma_{c}^{\ast++}$ | $1(3/2^{+})$ | $2518.41$ | $\Sigma_{c}^{\ast+}$ | $1(3/2^{+})$ | $2517.4$
$\Sigma_{c}^{0}$ | $1(1/2^{+})$ | $2453.75$ | $\Sigma_{c}^{\ast 0}$ | $1(3/2^{+})$ | $2518.48$
$\Lambda_{c}^{+}$ | $0(1/2^{+})$ | $2286.46$ | $\Lambda_{b}$ | $0(1/2^{+})$ | $5619.60$
$\pi^{\pm}$ | $1(0^{-})$ | $139.57$ | $\pi^{0}$ | $1(0^{-})$ | $134.98$
$K^{\pm}$ | $1/2(0^{-})$ | $493.677$ | $K^{0}$ | $1/2(0^{-})$ | $497.611$
$\bar{D}^{0}$ | $\frac{1}{2}(0^{-})$ | $1864.84$ | $D^{-}$ | $\frac{1}{2}(0^{-})$ | $1869.66$
$\bar{D}^{\ast 0}$ | $\frac{1}{2}(1^{-})$ | $2006.85$ | $D^{\ast-}$ | $\frac{1}{2}(1^{-})$ | $2010.26$
$D_{s}^{\pm}$ | $0(0^{-})$ | $1968.35$ | $D_{s}^{\ast\pm}$ | $0(1^{-})$ | $2112.2$
$J/\psi$ | $0(1^{-})$ | $3096.90$ | $\eta_{c}$ | $0(0^{-})$ | $2983.90$
Table 2: Masses and quantum numbers of hadrons relevant to this work
ParticleDataGroup:2020ssz .

###  3.1  Widths of hidden-charm pentaquark molecules

In Table 2, we tabulate the masses and quantum numbers of relevant particles.
Following our previous work Liu:2019tjn , we study two scenarios A and B to
determine the parameters. In Scenario A, the spins of $P_{c}(4440)$ and
$P_{c}(4457)$ are $1/2$ and $3/2$, respectively, while they are $3/2$ and
$1/2$ in Scenario B. For the cutoff in the Gaussian regulator, we choose the
value of $\Lambda=1.5$ GeV Xie:2022hhv . To quantify the agreement with the
experimental data, we use the $\chi^{2}$ defined as

$\displaystyle\chi^{2}=\sum_{i=1}^{3}\frac{(M_{exp}^{i}-M_{fit}^{i})^{2}}{{d_{M}^{i}}^{2}}+\sum_{i=1}^{3}\frac{(\Gamma_{exp}^{i}-\Gamma_{fit}^{i})^{2}}{{d_{\Gamma}^{i}}^{2}}$
(20)

where $M_{exp}^{i}(\Gamma_{exp}^{i})$ and $M_{fit}^{i}(\Gamma_{fit}^{i})$ are
the masses(widths) measured by the LHCb Collaboration and those obtained in
the contact-range EFT approach, $d_{M}^{i}$ and $d_{\Gamma}^{i}$ are the
uncertainties of experimental masses and widths, and the superscripts with
$i=1$, $i=2$ and $i=3$ represent $P_{c}(4312)$, $P_{c}(4440)$ and
$P_{c}(4457)$, respectively. The masses and decay widths of the three states
are Aaij:2019vzc

$\displaystyle M_{P_{c}(4312)}$ $\displaystyle=$ $\displaystyle 4311.9\pm
0.7^{+6.8}_{-0.6}~{}\mbox{MeV}\,,\quad\Gamma_{P_{c}(4312)}=9.8\pm
2.7^{+3.7}_{-4.5}~{}\mbox{MeV}\,,$ $\displaystyle M_{P_{c}(4440)}$
$\displaystyle=$ $\displaystyle 4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mbox{MeV}\,,\quad\Gamma_{P_{c}(4440)}=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mbox{MeV}\,,$ (21) $\displaystyle M_{P_{c}(4457)}$
$\displaystyle=$ $\displaystyle 4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mbox{MeV}\,,\quad\Gamma_{P_{c}(4457)}=6.4\pm
2.0^{+5.7}_{-1.9}~{}\mbox{MeV}\,.$

Scenario | $C_{a}$ | $C_{b}$ | $C_{a}^{\prime}$ | $C_{b}^{\prime}$ | $g_{1}$ | $g_{2}$ | $\chi^{2}$
---|---|---|---|---|---|---|---
A | -52.533 | 6.265 | -11.366 | 2.669 | 20.360 | -19.590 | 1.155
B | -56.030 | -5.418 | -12.120 | -5.136 | 58.963 | 28.587 | 2.980
Table 3:  Parameters $C_{a}$, $C_{b}$, $C_{a}^{\prime}$, $C_{b}^{\prime}$,
$g_{1}$ and $g_{2}$ (in units of GeV-1) determined by fitting to the masses
and widths of $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ and the
corresponding $\chi^{2}$ in Scenario A and Scenario B.

Given the fact that the light meson saturation is valid for the
$\bar{D}^{(*)}\Lambda_{c}\to\bar{D}^{(*)}\Lambda_{c}$ and
$\bar{D}^{(*)}\Sigma_{c}^{(*)}\to\bar{D}^{(*)}\Sigma_{c}^{(*)}$ potentials 222
As indicated in Ref Liu:2019zvb , the ratio of $C_{a}$ to $C_{b}$ estimated by
the light meson saturation approach is consistent with that obtained by the
contact-range EFT approach. , we adopt the ratio $C_{a}^{\prime}/C_{a}=0.216$
obtained in the light meson saturation approach. As a result, there only
remain five parameters. With the above preparations we determine the values of
the parameters $C_{a}$, $C_{b}$, $C_{b}^{\prime}$, $g_{1}$ and $g_{2}$ as well
as the $\chi^{2}$ for Scenario A and Scenario B and show them in Table 3. In
the following, we compare the fitted parameters ($C_{a}$, $C_{b}$, and
$C_{b}^{\prime}$) with those obtained in the light meson saturation approach
(see the Appendix for details). With the light meson saturation, we obtain the
ratio $C_{b}/C_{a}=0.12$, while the ratio determined in the EFT approach (by
fitting to the data) is $C_{b}/C_{a}=-0.12$ and $C_{b}/C_{a}=0.10$ for
Scenario A and Scenario B, respectively. It seems that the light meson
saturation approach prefers Scenario B, consistent with the single-channel
analysis Liu:2019zvb . Moreover, the light meson saturation approach yields
$C_{b}^{\prime}/C_{b}=0.61$, while the value determined in the EFT approach is
$C_{b}^{\prime}/C_{b}=0.43$ for Scenario A but $C_{b}^{\prime}/C_{b}=0.95$ for
Scenario B. One can see that they are quite different for either Scenario A or
Scenario B. From the perspective of light meson saturation, only the $\rho$
meson exchange is considered to saturate the
$\bar{D}^{(\ast)}\Sigma_{c}\to\bar{D}^{(*)}\Lambda_{c}$ interactions. However,
the one-pion exchange plays a significant role in the
$\bar{D}^{(\ast)}\Sigma_{c}\to\bar{D}^{(*)}\Lambda_{c}$ potentials
Yamaguchi:2019seo ; He:2019rva ; Burns:2021jlu . Since the one-pion exchange
is not considered, the $C_{b}^{\prime}$ obtained in the light meson saturation
approach is not consistent with the $C_{b}^{\prime}$ determined in the EFT
approach for either Scenario A or Scenario B.

Scenario | A
---|---
Name | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Molecule | $\bar{D}\Sigma_{c}$ | $\bar{D}\Sigma^{*}_{c}$ | $\bar{D}^{\ast}\Sigma_{c}$ | $\bar{D}^{\ast}\Sigma_{c}$ | $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ | $\bar{D}^{\ast}\Sigma_{c}^{\ast}$
$J^{P}$ | $\frac{1}{2}^{-}$ | $\frac{3}{2}^{-}$ | $\frac{1}{2}^{-}$ | $\frac{3}{2}^{-}$ | $\frac{1}{2}^{-}$ | $\frac{3}{2}^{-}$
Pole (MeV) | 4310.6+3.5$i$ | 4372.8 +2.7$i$ | 4440.6+8.6$i$ | 4458.4+0.7$i$ | 4500.0+9.9$i$ | 4513.2+7.7$i$
$g_{P_{c}\Sigma_{c}^{*}\bar{D}^{*}}$ | - | - | - | - | 2.686 | 2.194
$g_{P_{c}\Sigma_{c}\bar{D}^{*}}$ | - | - | 2.554 | 1.082 | 0.141 | 0.218
$g_{P_{c}\Sigma_{c}^{*}\bar{D}}$ | - | 2.133 | - | 0.179 | - | 0.237
$g_{P_{c}\Sigma_{c}\bar{D}}$ | 2.089 | - | 0.254 | - | 0.139 | -
$g_{P_{c}\Lambda_{c}\bar{D}^{*}}$ | 0.234 | 0.074 | 0.177 | 0.050 | 0.110 | 0.241
$g_{P_{c}\Lambda_{c}\bar{D}}$ | 0.014 | - | 0.158 | - | 0.207 | -
$g_{P_{c}J/\psi N}$ | 0.251 | 0.454 | 0.584 | 0.103 | 0.434 | 0.532
$g_{P_{c}\eta_{c}N}$ | 0.420 | - | 0.261 | - | 0.527 | -
Scenario | B
Name | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Molecule | $\bar{D}\Sigma_{c}$ | $\bar{D}\Sigma^{*}_{c}$ | $\bar{D}^{\ast}\Sigma_{c}$ | $\bar{D}^{\ast}\Sigma_{c}$ | $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ | $\bar{D}^{\ast}\Sigma_{c}^{\ast}$
$J^{P}$ | $\frac{1}{2}^{-}$ | $\frac{3}{2}^{-}$ | $\frac{1}{2}^{-}$ | $\frac{3}{2}^{-}$ | $\frac{1}{2}^{-}$ | $\frac{3}{2}^{-}$
Pole (MeV) | 4309.9+4$i$ | 4365.8+6.2$i$ | 4458.4+4.5$i$ | 4441.4+1.1$i$ | 4521.6+7.5$i$ | 4522.5+3.7$i$
$g_{P_{c}\Sigma_{c}^{*}\bar{D}^{*}}$ | - | - | - | - | 1.841 | 1.621
$g_{P_{c}\Sigma_{c}\bar{D}^{*}}$ | - | - | 1.679 | 2.462 | 0.107 | 0.143
$g_{P_{c}\Sigma_{c}^{*}\bar{D}}$ | - | 2.451 | - | 0.099 | - | 0.171
$g_{P_{c}\Sigma_{c}\bar{D}}$ | 2.072 | - | 0.161 | - | 0.131 | -
$g_{P_{c}\Lambda_{c}\bar{D}^{*}}$ | 0.392 | 0.090 | 0.247 | 0.159 | 0.232 | 0.223
$g_{P_{c}\Lambda_{c}\bar{D}}$ | 0.020 | - | 0.191 | - | 0.281 | -
$g_{P_{c}J/\psi N}$ | 0.263 | 0.704 | 0.277 | 0.168 | 0.314 | 0.312
$g_{P_{c}\eta_{c}N}$ | 0.413 | - | 0.164 | - | 0.328 | -

Table 4: Pole positions(in units of MeV) of six hidden-charm pentaquark
molecules and the couplings to their constituents in Scenario A and Scenario
B.

In Table 4, we present the pole positions of the hidden-charm pentaquark
molecules and the couplings to their constituents. From the obtained pole
positions of $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$, it is obvious
that scenario A, yielding results consistent with the experimental data, is
better than Scenario B, which is quite different from the single-channel study
Liu:2019tjn . Our study shows that the coupled-channel effects can help
distinguish the two possible scenarios. In a similar approach but without the
$\bar{D}^{(*)}\Lambda_{c}$ channels, Scenario A is still slightly better than
Scenario B Xie:2022hhv . We note in passing that the chiral unitary model
Xiao:2019aya  also prefers scenario A. We further note that the coefficients
in the contact-range potentials of Eq.(29) and Eq.(30) are derived assuming
the HQSS, while the HQSS breaking is not taken into account. In Ref.
Yamaguchi:2019seo , it was shown that the tensor term of the one-pion exchange
potentials plays a crucial role in describing the widths of the pentaquark
molecules, while the $D$-wave potentials are neglected in this work.
Therefore, we can not conclude which scenario is more favorable at this stage.

Up to now, the spins of $P_{c}(4440)$ and $P_{c}(4457)$ are still undetermined
experimentally, which motivated many theoretical discussions on how to
determine their spins Pan:2019skd ; Zhang:2023czx ; Liu:2023wfo . One crucial
issue is that the strength of $\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}$
potentials of $J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$ are undetermined. We can see
that the $J^{P}=1/2^{-}$ $\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}$
potential is stronger than the $J^{P}=3/2^{-}$
$\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}$ potential in Scenario A, while
their order reverses in Scenario B. In Refs. Burns:2021jlu ; Burns:2022uiv ,
Burns et al. proposed another case, named as Scenario C, which actually
corresponds to a special case of Scenario B, where the $J^{P}=1/2^{-}$
$\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}$ potential is not strong enough
to form a bound state, and therefore $P_{c}(4457)$ is interpreted as a kinetic
effect rather than a genuine state. From their values of $C_{a}$ and $C_{b}$
Burns:2022uiv , the ratio $C_{b}/C_{a}$ is determined to be around 0.5, which
implies the emergence of a large spin-spin interaction, inconsistent with the
principle of EFTs. It is no surprise that such a large spin-spin interaction
breaks the completeness of the multiplet picture of hidden-charme pentaquark
molecules Liu:2019tjn ; Xiao:2019aya ; Du:2019pij ; Yamaguchi:2019seo ;
PavonValderrama:2019nbk ; Lin:2019qiv ; He:2019rva ; Yalikun:2021bfm ;
Dong:2021juy ; Zhang:2023czx . Therefore, we strongly recommend that lattice
QCD simulations could study the potentials of $J^{P}=1/2^{-}$
$\bar{D}^{*}\Sigma_{c}$ and $J^{P}=3/2^{-}$ $\bar{D}^{*}\Sigma_{c}$ to address
this issue333 A recent lattice QCD study shows that there exists a
$J^{P}=1/2^{-}$$\bar{D}^{*}\Sigma_{c}$ bound state with a binding energy of
$6$ MeV Xing:2022ijm . .

Scenario | A
---|---
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Two-body decay | 7.00 | 5.40 | 17.20 | 1.40 | 19.80 | 15.40
Three-body decay | 0.20 | 1.47 | 0.03 | 0.33 | 3.83 | 6.85
Total decay | 7.20 | 6.87 | 17.23 | 1.73 | 23.63 | 22.25
Scenario | B
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Two-body decay | 8.00 | 12.40 | 2.20 | 9.00 | 15.00 | 7.40
Three-body decay | 0.16 | 1.00 | 0.01 | 2.44 | 13.58 | 9.58
Total decay | 8.16 | 13.40 | 2.21 | 11.44 | 28.58 | 16.98
Table 5: Two-body decay widths, three-body decay widths, and total decay
widths (in units of MeV) of hidden-charm pentaquark molecules in Scenario A
and Scenario B.

The imaginary parts of the pole positions in Table 4 specify the two-body
partial decay widths of the pentaquark molecules, which can also be calculated
via the triangle diagrams using the effective Lagrangian approach Xiao:2019mvs
; Lin:2019qiv . From the results of Table 4, we can calculate the two-body
decay widths of these pentaquark molecules and tabulate them in Table 5.
Moreover, with the newly obtained couplings
$g_{P_{c}\bar{D}^{(*)}\Sigma_{c}^{(*)}}$, we update their three-body decay
widths as shown in Table 5. Comparing with the results in Ref. Xie:2022hhv ,
we find that the new results vary a bit because the pole positions affect the
phase space of the three-body decays and the values of the couplings
$g_{P_{c}\bar{D}^{(*)}\Sigma_{c}^{(*)}}$. Assuming that the two-body and
three-body decays are dominant decay channels for the pentaquark molecules, we
can obtain their total decay widths by summing the two decay modes. Our
results indicate that the widths of $P_{c5}$ and $P_{c6}$ as the
$\bar{D}^{*}\Sigma_{c}$ molecules are larger than those of $P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$ reported by the LHCb Collaboration, and their
three-body decay widths account for a large proportion of their total widths.
In addition, we can see that the three-body decay widths of $P_{c}(4312)$,
$P_{c}(4440)$, and $P_{c}(4457)$ account for only a small proportion of their
total widths, which confirms our assumption that their total widths are almost
saturated by the two-body decays.

###  3.2  Production rates of hidden-charm pentaquark molecules

From the values of the couplings given in Table 4, one can see that the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ channel plays a dominant role in generating
these pentaquark molecules. Yet their productions in the $\Lambda_{b}$ decay
can not proceed via the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ interactions as
discussed above. It is important to investigate the productions of hidden-
charm pentaquark molecules in the $\Lambda_{b}$ decays via the
$\bar{D}^{(*)}\Lambda_{c}$ interactions although the couplings of the
pentaquark states to $\bar{D}^{(*)}\Lambda_{c}$ are small. With the couplings
$g_{P_{c}\bar{D}^{(*)}\Lambda_{c}}$ given in Table 4, we employ the effective
Lagrangian approach to calculate the decays of $\Lambda_{b}\to P_{c}K$
illustrated in Fig. 1 and Fig. 2.

In Table 6, we present the branching fractions of $\Lambda_{b}\to P_{c}K$ in
Scenario A and Scenario B. Our results show that the branching factions of the
three pentaquark states discovered by the LHCb Collaboration:
$\mathcal{B}(\Lambda_{b}\to P_{c}(4312)K)=35.18\times 10^{-6}$,
$\mathcal{B}(\Lambda_{b}\to P_{c}(4440)K)=15.30\times 10^{-6}$, and
$\mathcal{B}(\Lambda_{b}\to P_{c}(4457)K)=0.48\times 10^{-6}$ in Scenario A
and $\mathcal{B}(\Lambda_{b}\to P_{c}(4312)K)=98.88\times 10^{-6}$,
$\mathcal{B}(\Lambda_{b}\to P_{c}(4440)K)=5.21\times 10^{-6}$, and
$\mathcal{B}(\Lambda_{b}\to P_{c}(4457)K)=27.23\times 10^{-6}$ in Scenario B.
From the order of magnitude of the obtained branching fractions, we can
conclude that the pentaquark molecules can be produced via the triangle
diagrams shown in Fig. 1 and Fig. 2. The branching fractions of the decay
$\Lambda_{b}\to P_{c}(4312)K$ are larger than those of $\Lambda_{b}\to
P_{c}(4440)K$ and $\Lambda_{b}\to P_{c}(4457)K$ in both cases, and the
branching fractions involving $P_{c}(4440)$ and $P_{c}(4457)$ for
$J^{P}=1/2^{-}$ are always larger than those for $J^{P}=3/2^{-}$. Such results
reflect that the branching fractions of the decays $\Lambda_{b}\to P_{c}K$ are
related to the couplings $g_{P_{c1}\bar{D}^{(*)}\Lambda_{c}}$, especially the
coupling $g_{P_{c1}\bar{D}^{*}\Lambda_{c}}$, which shows that the
$\bar{D}^{*}\Lambda_{c}$ interactions play an important role in producing the
pentaquark molecules in the $\Lambda_{b}$ decays. Similarly, we predict the
branching fractions of $\Lambda_{b}$ decaying into $P_{c2}$, $P_{c5}$ and
$P_{c6}$ plus a kaon as shown in Table 6, the order of magnitude of which are
similar to those involving $P_{c}(4440)$ and $P_{c}(4457)$.

Scenario | A
---|---
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
$\mathcal{B}(\Lambda_{b}\to P_{c}K)$ | 35.18 | 1.49 | 15.30 | 0.48 | 6.37 | 9.01
Scenario | B
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
$\mathcal{B}(\Lambda_{b}\to P_{c}K)$ | 98.88 | 2.27 | 27.23 | 5.21 | 21.69 | 7.43
Table 6: Branching fractions ($10^{-6}$) of $\Lambda_{b}$ decaying into a $K$
meson and a hidden-charm pentaquark molecule in Scenario A and Scenario B.

Up to now, there exist no available experimental data for the branching
fractions of the decays $\Lambda_{b}\to P_{c}K$. The LHCb Collaboration
measured the relevant ratios of branching fractions for the three pentaquark
states: $R_{P_{c}(4312)}=(0.30\pm 0.07^{+0.34}_{-0.09})\%$,
$R_{P_{c}(4440)}=(1.11\pm 0.33^{+0.22}_{-0.11})\%$, and
$R_{P_{c}(4457)}=(0.53\pm 0.16^{+0.15}_{-0.13})\%$, where $R$ is defined as

$\displaystyle R_{P_{c}}=\frac{\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{+}K^{-})\cdot\mathcal{B}(P_{c}^{+}\to J/\psi
p)}{\mathcal{B}(\Lambda_{b}^{0}\to J/\psi pK^{-})}$ (22)

According to RPP ParticleDataGroup:2020ssz , the branching fraction of
$\Lambda_{b}^{0}\to J/\psi pK^{-}$ is $\mathcal{B}(\Lambda_{b}^{0}\to J/\psi
pK^{-})=3.2^{+0.6}_{-0.5}\times 10^{-4}$, and then we obtain the product of
the branching fractions of the decays $\Lambda_{b}\to P_{c}K$ and $P_{c}\to
J/\psi p$

$\displaystyle\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{+}(4312)K^{-})\cdot\mathcal{B}(P_{c}^{+}(4312)\to J/\psi p)$
$\displaystyle=0.96^{+1.13}_{-0.39}\times 10^{-6}\,,$ (23)
$\displaystyle\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{+}(4440)K^{-})\cdot\mathcal{B}(P_{c}^{+}(4440)\to J/\psi p)$
$\displaystyle=3.55^{+1.43}_{-1.24}\times 10^{-6}\,,$
$\displaystyle\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{+}(4457)K^{-})\cdot\mathcal{B}(P_{c}^{+}(4457)\to J/\psi p)$
$\displaystyle=1.70^{+0.77}_{-0.71}\times 10^{-6}\,.$

To compare with the experimental data, we have to obtain the branching
fractions of $P_{c}$ decaying into $J/\psi p$, which are not yet determined
experimentally. We note that the GlueX and JLab Collaborations investigated
the production rates of pentaquark states in the photoproduction process and
only gave the upper limits of $\mathcal{B}(P_{c}\to J/\psi p)<2.0\%$
Meziani:2016lhg ; GlueX:2019mkq , which indicates that the branching fractions
of $\mathcal{B}(\Lambda_{b}^{0}\to P_{c}^{+}K^{-})$ are at the order of
$10^{-4}$, approaching to the values of $\mathcal{B}(\Lambda_{b}^{0}\to J/\psi
pK^{-})$. Such large values highlight the inconsistency between the LHCb
results and the GlueX/JLab results. Therefore, more precise experimental data
are needed to settle this issue.

Scenario | A
---|---
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
$\Gamma_{2}(\Sigma_{c}\bar{D}^{\ast})$ | - | - | - | - |  0.50 (2.52 $\%$) |  1.38 (8.87 $\%$)
$\Gamma_{3}(\Sigma^{*}_{c}\bar{D})$ | - | - | - |  1.14 (73.23 $\%$) | - |  2.62 (16.87 $\%$)
$\Gamma_{4}(\Sigma_{c}\bar{D})$ | - | - |  2.87 (16.59 $\%$) | - |  1.04 (5.29 $\%$) | -
$\Gamma_{5}(\Lambda_{c}\bar{D}^{\ast})$ |  0.83 (11.71 $\%$) |  0.19 (3.48 $\%$) |  1.44 (8.33 $\%$) |  0.12 (7.82 $\%$) |  0.65 (3.32 $\%$) |  3.24 (20.81 $\%$)
$\Gamma_{6}(\Lambda_{c}\bar{D})$ |  0.01 (0.14 $\%$) | - |  1.60 (9.22 $\%$) | - |  2.98 (15.14 $\%$) | -
$\Gamma_{7}(J/\psi N)$ |  1.43 (20.22 $\%$) |  5.16 (96.52 $\%$) |  9.29 (53.64 $\%$) |  0.29 (18.95 $\%$) |  5.46 (27.73 $\%$) |  8.31 (53.45 $\%$)
$\Gamma_{8}(\eta_{c}N)$ |  4.81 (67.93 $\%$) | - |  2.12 (12.23 $\%$) | - |  9.06 (45.98 $\%$) | -
Scenario | B
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
$\Gamma_{2}(\Sigma_{c}\bar{D}^{\ast})$ | - | - | - | - |  0.36 (2.17 $\%$) |  0.64 (8.30 $\%$)
$\Gamma_{3}(\Sigma^{*}_{c}\bar{D})$ | - | - | - |  0.31 (13.63 $\%$) | - |  1.41 (18.19 $\%$)
$\Gamma_{4}(\Sigma_{c}\bar{D})$ | - | - |  1.23 (12.88 $\%$) | - |  0.98 (5.92 $\%$) | -
$\Gamma_{5}(\Lambda_{c}\bar{D}^{\ast})$ |  2.27 (26.71 $\%$) |  0.26 (2.10 $\%$) |  2.97 (30.92 $\%$) |  1.17 (52.05 $\%$) |  3.05 (18.48 $\%$) |  2.82 (36.37 $\%$)
$\Gamma_{6}(\Lambda_{c}\bar{D})$ |  0.02 (0.23 $\%$) | - |  2.40 (25.03 $\%$) | - |  5.65 (34.18 $\%$) | -
$\Gamma_{7}(J/\psi N)$ |  1.57 (18.45 $\%$) |  12.28 (97.90 $\%$) |  2.13 (22.26 $\%$) |  0.77 (34.33 $\%$) |  2.92 (17.67 $\%$) |  2.88 (37.14 $\%$)
$\Gamma_{8}(\eta_{c}N)$ |  4.65 (54.61 $\%$) | - |  0.85 (8.86 $\%$) | - |  3.57 (21.58 $\%$) | -
Table 7: Two-body partial decay widths (in units of MeV) of hidden-charm
pentaquark molecules as well as their branching fractions in Scenario A and
Scenario B .

Using Eq.(16), we calculate the two-body partial decay widths of hidden-charm
pentaquark molecules, and then estimate the branching fractions of the decays
$P_{c}\to J/\psi p$. The results are shown in Table 7, where the three-body
decay widths of the pentaquark molecules are not included. One can see that
the partial decay widths of $P_{c}\to\bar{D}^{(*)}\Lambda_{c}$ are less than
those of $P_{c}\to J/\psi p$ in Scenario A, but their order reverses in
Scenario B. In Ref. Xiao:2020frg , the estimated branching fractions of the
decays $P_{c}\to\bar{D}^{(*)}\Lambda_{c}$ are much smaller than those of the
decays $P_{c}\to J/\psi p$, consistent with Scenario A. In terms of the meson
exchange theory, the branching fractions of the decays
$P_{c}\to\bar{D}^{(*)}\Lambda_{c}$ are larger than those of the decays
$P_{c}\to J/\psi p$, where the heavy meson ($\bar{D}^{(*)}$) exchange and the
light meson ($\pi(\rho)$) exchange are responsible for the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}\to J/\psi p$ and
$\bar{D}^{(*)}\Sigma_{c}^{(*)}\to\bar{D}^{(*)}\Lambda_{c}$ interactions,
respectively Lin:2019qiv . It is obvious that the transitions
$\bar{D}^{(*)}\Sigma_{c}^{(*)}\to J/\psi p$ are heavily suppressed, resulting
in the small partial decay widths of $P_{c}\to J/\psi p$ in the same
theoretical framework Lin:2019qiv . We note that the meson exchange theory has
been tested for light mesons exchanges, but remains to be verified for heavy
meson exchanges, especially when both heavy and light mesons can be exchanged.
The meson exchange theory dictates that charmed mesons are responsible for the
very short range interaction, but they can not adequately describe such short-
range interactions because one gluon exchange may play a role. In Ref.
Yamaguchi:2019djj , the authors found that the strength of the short range
potential provided by the one gluon exchange is much stronger than that
provided by the heavy meson exchange. In the present work, the hidden-charm
meson-baryon potentials are provided by the contact-range EFT constrained by
HQSS with the low-energy constants determined by fitting to data, which are
plausible but the underlying mechanism needs to be clarified.

Scenario | A
---|---
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Ours | 7.11 | 1.44 | 8.21 | 0.09 | 1.77 | 4.82
ChUA Xiao:2020frg  | 1.82 | 8.62 | 0.13 | 0.83 | 0.04 | 2.36
Exp | 0.96 | - | 3.55 | 1.70 | - | -
Scenario | B
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Ours | 18.24 | 2.22 | 6.06 | 1.79 | 3.83 | 2.76
ChUA Xiao:2020frg  | - | - | - | - | - | -
Exp | 0.96 | - | 1.70 | 3.55 | - | -
Table 8: Branching fractions ($10^{-6}$) of the decays
$\Lambda_{b}\to(P_{c}\to J/\psi p)K$ in Scenario A and Scenario B.

With the obtained branching fractions $\mathcal{B}(\Lambda_{b}\to P_{c}K)$ in
Table 6 and $\mathcal{B}(P_{c}\to J/\psi p)$ in Table 7, we further calculate
the branching fractions $\mathcal{B}[\Lambda_{b}\to(P_{c}\to J/\psi p)K]$ for
Scenario A and Scenario B as shown Table 8. Our results show that the
branching fractions for $P_{c}(4312)$ and $P_{c}(4440)$ are of the same order
as their experimental counterparts, but the branching fraction for
$P_{c}(4457)$ is smaller by one order of magnitude. For Scenario B, the
branching fractions for $P_{c}(4440)$ and $P_{c}(4457)$ are of the same order
as their experimental counterparts, but the branching fraction for
$P_{c}(4312)$ is larger by one order of magnitude. We can see that our model
can not simultaneously describe the branching fractions of these three
pentaquark states. In Ref. Xiao:2020frg , the ChUA estimated the couplings
$g_{P_{c}\bar{D}^{(*)}\Lambda_{c}}$ and the branching fractions
$\mathcal{B}(P_{c}\to J/\psi p)$, which actually corresponds to Scenario A of
our results. Using the values estimated by ChUA we recalculate the branching
fractions $\mathcal{B}[\Lambda_{b}\to(P_{c}\to J/\psi p)K]$ as shown in Table
8. The branching fractions for $P_{c}(4312)$ and $P_{c}(4457)$ are of the same
order as their experimental counterparts, but the branching fraction for
$P_{c}(4440)$ is smaller by one order of magnitude. Obviously, the branching
fractions for $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ in our model are
related to the couplings of the pentaquark molecules to
$\bar{D}^{(*)}\Lambda_{c}$ and $J/\psi p$. Nevertheless, the production
mechanism of these three pentaquark states via the triangle diagrams shown in
Fig. 1 and Fig. 2 is capable of qualitatively reproducing the experimental
data, which further corroborates the hadronic molecular picture of these
pentaquark states.

In Table 8, we show the branching fractions for the HQSS partners of
$P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$, where only the two-body decay
modes contribute to the branching fractions of the decays $P_{c}\to J/\psi p$.
As shown in Table 5, the three-body decay widths of $P_{c2}$, $P_{c5}$, and
$P_{c6}$ are up to several MeV. If taking into account the three-body decay
widths, the branching fractions of $P_{c2}$, $P_{c5}$, and $P_{c6}$ decaying
into $J/\psi p$ become $\mathcal{B}(P_{c2}\to J/\psi p)=75\%$,
$\mathcal{B}(P_{c5}\to J/\psi p)=23\%$, and $\mathcal{B}(P_{c6}\to J/\psi
p)=37\%$ in Scenario A and $\mathcal{B}(P_{c2}\to J/\psi p)=91\%$,
$\mathcal{B}(P_{c5}\to J/\psi p)=10\%$, and $\mathcal{B}(P_{c6}\to J/\psi
p)=17\%$ in Scenario B. As result, the corresponding branching fraction of the
decays $\Lambda_{b}\to(P_{c}\to J/\psi p)K$ reduce to
$\mathcal{B}[\Lambda_{b}\to(P_{c2}\to J/\psi p)K]=1.11\times 10^{-6}$,
$\mathcal{B}[\Lambda_{b}\to(P_{c5}\to J/\psi p)K]=1.47\times 10^{-6}$ and
$\mathcal{B}[\Lambda_{b}\to(P_{c6}\to J/\psi p)K]=3.37\times 10^{-6}$ in
Scenario A and $\mathcal{B}[\Lambda_{b}\to(P_{c2}\to J/\psi p)K]=2.08\times
10^{-6}$, $\mathcal{B}[\Lambda_{b}\to(P_{c5}\to J/\psi p)K]=2.22\times
10^{-6}$ and $\mathcal{B}[\Lambda_{b}\to(P_{c6}\to J/\psi p)K]=1.26\times
10^{-6}$ in Scenario B. We can see that the branching fractions of the
pentaquark states $P_{c2}$, $P_{c5}$, and $P_{c6}$ as hadronic molecules are
smaller than those of $P_{c}(4312)$ and sum of $P_{c}(4440)$ and $P_{c}(4457)$
in Scenario A and Scenario B, which is consistent with the fact that these
three HQSS partners have not been seen in the LHCb data sample of 2019.

These hidden-charm pentaquark molecules can be seen in the $J\psi p$ invariant
mass distribution, and one can also expect to see them in the
$\bar{D}^{*}\Lambda_{c}$ invariant mass distribution. Therefore, with the same
approach we calculate the branching fractions of the decays
$\Lambda_{b}\to(P_{c}\to\bar{D}^{*}\Lambda_{c})K$ and the results are shown in
Table 9. We can see that the branching fractions of the pentaquark molecules
in the decays $\Lambda_{b}\to(P_{c}\to J/\psi p)K$ and
$\Lambda_{b}\to(P_{c}\to\bar{D}^{*}\Lambda_{c})K$ are similar except for
$P_{c2}$. The branching fraction
$\mathcal{B}[\Lambda_{b}\to(P_{c2}\to\bar{D}^{*}\Lambda_{c})K]$ is smaller
than the branching fraction $\mathcal{B}[\Lambda_{b}\to(P_{c2}\to J/\psi p)K]$
by two order of magnitude. We encourage experimental searches for these
pentaquark states in the $\bar{D}^{*}\Lambda_{c}$ invariant mass distributions
of the $\Lambda_{b}$ decays.

Scenario | A
---|---
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Ours | 4.12 | 0.05 | 1.27 | 0.00 | 0.21 | 1.87
Scenario | B
Molecule | $P_{c1}$ | $P_{c2}$ | $P_{c3}$ | $P_{c4}$ | $P_{c5}$ | $P_{c6}$
Ours | 26.41 | 0.05 | 8.42 | 2.71 | 4.01 | 2.70
Table 9: Branching fractions($10^{-6}$) of the decays
$\Lambda_{b}\to(P_{c}\to\bar{D}^{*}\Lambda_{c})K$ in Scenario A and Scenario
B.

##  4 Summary and outlook

The three pentaquark states $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$
can be nicely arranged into a complete multiplet of
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecules, while their partial decay
widths and production rates in the $\Lambda_{b}$ decay remain undetermined. In
this work, we employed the contact-range effective field theory approach to
dynamically generate the pentaquark molecules via the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$, $\bar{D}^{(*)}\Lambda_{c}$, $J/\psi p$, and
$\eta_{c}p$ coupled-channel interactions, where the six relevant unknown
parameters were determined by fitting to the experimental data. With the
obtained pole positions, we estimated the couplings of the pentaquark
molecules to their constituents $J/\psi p$ and $\bar{D}^{*}\Lambda_{c}$, and
then calculated the productions rates of these molecules in the $\Lambda_{b}$
decays via the triangle diagrams, where the $\Lambda_{b}$ baryon weakly decays
into $\Lambda_{c}{D}_{s}^{(*)}$, then the ${D}_{s}^{(*)}$ mesons decay into
$\bar{D}^{(*)}K$, and finally the pentaquark molecules are dynamically
generated by the $\bar{D}^{*}\Lambda_{c}$ interactions. In this work, with no
extra parameters (except those contained in the contact range EFT approach and
determined by their masses and widths) we took the effective Lagrangian
approach to calculate the triangle diagrams and their production rates in the
$\Lambda_{b}$ decays.

Our results showed that the masses of the three pentaquark states are well
described either in Scenario A or Scenario B, which confirmed our previous
conclusion that we can not determine the favorable scenario in terms of their
masses alone. However, we found that Scenario A is more favored than Scenario
B once their widths are taken into account. Moreover, our results showed that
their couplings to $\bar{D}^{(*)}\Lambda_{c}$ are smaller than those to
$J/\psi p$ in Scenario A, but larger in Scenario B. For the branching
fractions of the decays $\Lambda_{b}\to P_{c}K$, that of the $P_{c}(4312)$ is
the largest and those of $P_{c}(4440)$ and $P_{c}(4457)$ with $J=1/2$ are
always larger than those with $J=3/2$ in both Scenario A and Scenario B.
Moreover, we predicted the following branching fractions:
$\mathcal{B}[\Lambda_{b}\to P_{c2}K]=(1\sim 2)\times 10^{-6}$,
$\mathcal{B}[\Lambda_{b}\to P_{c5}K]=(6\sim 22)\times 10^{-6}$ and
$\mathcal{B}[\Lambda_{b}\to P_{c6}K]=(7\sim 9)\times 10^{-6}$, respectively.

With the couplings between the molecules and their constituents determined, we
estimated the branching fractions $\mathcal{B}(P_{c}\to J/\psi p)$, and then
obtained the branching fraction $\mathcal{B}[\Lambda_{b}\to(P_{c}\to J/\psi
p)K]$. Our results showed that such branching fractions for $P_{c}(4312)$ and
$P_{c}(4440)$ are consistent with the experimental data, while that for
$P_{c}(4457)$ is larger than the experimental data in Scenario A. For Scenario
B, the branching fractions for $P_{c}(4440)$ and $P_{c}(4457)$ are consistent
with the experimental data, while that for $P_{c}(4312)$ is larger than the
experimental data. Given the complicated nature of these decays and the
various physical processes involved, we deem the agreements with the existing
data acceptable. Therefore, we conclude that the three pentaquark states as
hidden-charm meson-baryon molecules can be dynamically generated via the
$\bar{D}^{(*)}\Lambda_{c}$ interactions in the $\Lambda_{b}$ decay, which
further corroborated the molecular interpretations of the pentaquark states.
Moreover, the branching fractions of the HQSS partners of $P_{c}(4312)$,
$P_{c}(4440)$, and $P_{c}(4457)$ are estimated to be the order of $10^{-6}$,
smaller than that of $P_{c}(4312)$ and the sum of those of $P_{c}(4440)$ and
$P_{c}(4457)$. Therefore, we can attribute the non-observation of the other
HQSS partners in the decay $\Lambda_{b}\to(P_{c}\to J/\psi p)K$ to their small
production rates. As a byproduct, we further predicted the production rates of
the pentaquark molecules in the decays
$\Lambda_{b}\to(P_{c}\to\bar{D}^{*}\Lambda_{c})K$.

##  Appendix A Appendix

###  A.1 Contact-range potentials

To systematically generate the complete multiplet of hidden-charm pentaquark
molecules, we take into account the $\bar{D}^{(*)}\Lambda_{c}$ channels in the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ coupled-channel systems, where the HQSS plays
an important role. First, we express the spin wave function of the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ pairs in terms of the spins of the heavy
quarks $s_{1h}$ and $s_{2h}$ and those of the light quark(s) (often referred
to as brown mucks Isgur:1991xa ; Flynn:1992fm ) $s_{1l}$ and $s_{2l}$, where 1
and 2 denote $\bar{D}^{(*)}$ and $\Sigma_{c}^{(*)}$, respectively, via the
following spin coupling formula,

$\displaystyle|s_{1l},s_{1h},j_{1};s_{2l},s_{2h},j_{2};J\rangle=$ (24)
$\displaystyle\sqrt{(2j_{1}+1)(2j_{2}+1)(2s_{L}+1)(2s_{H}+1)}\left(\begin{matrix}s_{1l}&s_{2l}&s_{L}\\\
s_{1h}&s_{2h}&s_{H}\\\
j_{1}&j_{2}&J\end{matrix}\right)|s_{1l},s_{2l},s_{L};s_{1h},s_{2h},s_{H};J\rangle.$

The total light quark spin $s_{L}$ and heavy quark spin $s_{H}$ are given by
$s_{L}=s_{1l}\otimes s_{2l}$ and $s_{H}=s_{1H}\otimes s_{2H}$, respectively.

More explicitly, for the $\bar{D}^{(\ast)}\Sigma_{c}$ states, the
decompositions read

$\displaystyle|\Sigma_{c}\bar{D}(1/2^{-})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}0_{H}\otimes{1/2}_{L}+\frac{1}{2\sqrt{3}}1_{H}\otimes{1/2}_{L}+\sqrt{\frac{2}{3}}1_{H}\otimes{3/2}_{L}\,,$
(25) $\displaystyle|\Sigma_{c}^{\ast}\bar{D}(3/2^{-})\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}0_{H}\otimes{3/2}_{L}+\frac{1}{\sqrt{3}}1_{H}\otimes{1/2}_{L}+\frac{\sqrt{\frac{5}{3}}}{2}1_{H}\otimes{3/2}_{L}\,,$
$\displaystyle|\Sigma_{c}\bar{D}^{\ast}(1/2^{-})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}0_{H}\otimes{1/2}_{L}+\frac{5}{6}1_{H}\otimes{1/2}_{L}-\frac{\sqrt{2}}{3}1_{H}\otimes{3/2}_{L}\,,$
$\displaystyle|\Sigma_{c}\bar{D}^{\ast}(3/2^{-})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}0_{H}\otimes{3/2}_{L}-\frac{1}{3}1_{H}\otimes{1/2}_{L}+\frac{\sqrt{5}}{3}1_{H}\otimes{3/2}_{L}\,,$
$\displaystyle|\Sigma_{c}^{\ast}\bar{D}^{\ast}(1/2^{-})\rangle$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}0_{H}\otimes{1/2}_{L}-\frac{\sqrt{2}}{3}1_{H}\otimes{1/2}_{L}-\frac{1}{3}1_{H}\otimes{3/2}_{L}\,,$
$\displaystyle|\Sigma_{c}^{\ast}\bar{D}^{\ast}(3/2^{-})\rangle$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\frac{5}{3}}}{2}0_{H}\otimes{3/2}_{L}+\frac{\sqrt{5}}{3}1_{H}\otimes{1/2}_{L}-\frac{1}{6}1_{H}\otimes{3/2}_{L}\,.$

The total light quark spin ${1/2}_{L}$ of the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
system is given by the coupling of the light quark spins,
${1/2}_{1l}\otimes{1}_{2l}$. Since the light quark spin of $\Lambda_{c}$ is
$0$, the total light quark spin ${1/2}_{L}^{\prime}$ of the
$\bar{D}^{(*)}\Lambda_{c}$ system is given by ${1/2}_{1l}\otimes{0}_{2l}$. The
decompositions of the $\bar{D}^{(\ast)}\Lambda_{c}$ states are written as

$\displaystyle|\bar{D}\Lambda_{c}(J^{P}=1/2^{-})\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{0}_{H}\otimes{1/2}_{L}^{\prime}+\frac{\sqrt{3}}{2}{1}_{H}\otimes{1/2}_{L}^{\prime}\,,$
(26) $\displaystyle|\bar{D}^{*}\Lambda_{c}(J^{P}=1/2^{-})\rangle$
$\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}{0}_{H}\otimes{1/2}_{L}^{\prime}+\frac{1}{2}{1}_{H}\otimes{1/2}_{L}^{\prime}\,,$
$\displaystyle|\bar{D}^{*}\Lambda_{c}(J^{P}=3/2^{-})\rangle$ $\displaystyle=$
$\displaystyle{1}_{H}\otimes{1/2}_{L}^{\prime}\,.$

In the heavy quark limit, the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
interactions are independent of the spin of the heavy quark, and therefor the
potentials can be parameterized by two coupling constants describing the
interactions between light quarks of spin 1/2 and 3/2, respectively, i.e.,
$F_{1/2}=\langle 1/2_{L}|V|1/2_{L}\rangle$ and $F_{3/2}=\langle
3/2_{L}|V|3/2_{L}\rangle$:

$\displaystyle V_{\Sigma_{c}\bar{D}}(1/2^{-})$ $\displaystyle=$
$\displaystyle\frac{1}{3}F_{1/2L}+\frac{2}{3}F_{3/2L}\,,$ (27) $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}}(3/2^{-})$ $\displaystyle=$
$\displaystyle\frac{1}{3}F_{1/2L}+\frac{2}{3}F_{3/2L}\,,$ $\displaystyle
V_{\Sigma_{c}\bar{D}^{\ast}}(1/2^{-})$ $\displaystyle=$
$\displaystyle\frac{7}{9}F_{1/2L}+\frac{2}{9}F_{3/2L}\,,$ $\displaystyle
V_{\Sigma_{c}\bar{D}^{\ast}}(3/2^{-})$ $\displaystyle=$
$\displaystyle\frac{1}{9}F_{1/2L}+\frac{8}{9}F_{3/2L}\,,$ $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}^{\ast}}(1/2^{-})$ $\displaystyle=$
$\displaystyle\frac{8}{9}F_{1/2L}+\frac{1}{9}F_{3/2L}\,,$ $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}^{\ast}}(3/2^{-})$ $\displaystyle=$
$\displaystyle\frac{5}{9}F_{1/2L}+\frac{4}{9}F_{3/2L}\,.$

which can be rewritten as a combination of $C_{a}$ and $C_{b}$, i.e.,
$F_{1/2}=C_{a}-2C_{b}$ and $F_{3/2}=C_{a}+C_{b}$ Liu:2019tjn .

In the heavy quark limit, the
$\bar{D}^{(\ast)}\Lambda_{c}\to\bar{D}^{(\ast)}\Lambda_{c}$ interactions are
parameterised by one coupling constant, i.e., $F_{1/2L}^{\prime}=\langle
1/2_{L}^{\prime}|V|1/2_{L}^{\prime}\rangle$:

$\displaystyle
V_{\bar{D}\Lambda_{c}}(1/2^{-})=V_{\bar{D}^{*}\Lambda_{c}}(1/2^{-})=V_{\bar{D}^{*}\Lambda_{c}}(3/2^{-})=F_{1/2L}^{\prime},$
(28)

where the parameter $F_{1/2L}^{\prime}$ can be rewritten as $C_{a}^{\prime}$.

The potentials of $J/\psi N\to J/\psi N$, $J/\psi N\to\eta_{c}N$ and
$\eta_{c}N\to\eta_{c}N$ are suppressed due to the Okubo-Zweig-Iizuka (OZI)
rule, which is also supported by lattice QCD simulations Skerbis:2018lew . In
this work, we set $V_{J/\psi(\eta_{c})N\to J/\psi(\eta_{c})N}=0$.

In the heavy quark limit, the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to
J/\psi(\eta_{c})N$ and $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to J/\psi\Delta$
potentials are allowed, while the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to
J/\psi\Delta$ potentials are suppressed due to isospin symmetry breaking. From
HQSS, the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to J/\psi(\eta_{c})N$
interactions are only related to the spin of the light quark $1/2$, denoted by
one coupling: $g_{2}=\langle\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}|1_{H}\otimes
1/2_{L}\rangle=\langle\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}|0_{H}\otimes
1/2_{L}\rangle$. Similarly, we can express the $\bar{D}^{(*)}\Lambda_{c}\to
J/\psi(\eta_{c})N$ interactions by another parameter:
$g_{1}=\langle\bar{D}^{(\ast)}\Lambda_{c}|1_{H}\otimes
1/2_{L}\rangle=\langle\bar{D}^{(\ast)}\Lambda_{c}|0_{H}\otimes
1/2_{L}\rangle$. As for the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(*)}\Lambda_{c}$ interactions,
they are dependent on only one coupling constant in the heavy quark limit.
Therefore, we parameter the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(*)}\Lambda_{c}$ potential by
one coupling:
$C_{b}^{\prime}=\langle\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}|\bar{D}^{(\ast)}\Lambda_{c}\rangle$.

In the heavy quark limit, the contact-range potentials of
$\bar{D}^{*}\Sigma_{c}^{*}-\bar{D}^{*}\Sigma_{c}-\bar{D}\Sigma_{c}-\bar{D}^{*}\Lambda_{c}-\bar{D}\Lambda_{c}-J/\psi
N-\eta_{c}N$ system with $J^{P}=1/2^{-}$ can be expressed as

$V^{J=1/2}=\\\ \\\
\begin{pmatrix}C_{a}-\frac{5}{3}C_{b}&-\frac{\sqrt{2}}{3}C_{b}&-\sqrt{\frac{2}{3}}C_{b}&\sqrt{\frac{2}{3}}C_{b}^{\prime}&\sqrt{2}C_{b}^{\prime}&-\frac{\sqrt{2}}{3}g_{2}&\sqrt{\frac{2}{3}}g_{2}\\\
-\frac{\sqrt{2}}{3}C_{b}&C_{a}-\frac{4}{3}C_{b}&\frac{2}{\sqrt{3}}C_{b}&-\frac{2}{\sqrt{3}}C_{b}^{\prime}&C_{b}^{\prime}&\frac{5}{6}g_{2}&\frac{1}{2\sqrt{3}}g_{2}\\\
-\sqrt{\frac{2}{3}}C_{b}&\frac{2}{\sqrt{3}}C_{b}&C_{a}&C_{b}^{\prime}&0&\frac{1}{2\sqrt{3}}g_{2}&\frac{1}{2}g_{2}\\\
\sqrt{\frac{2}{3}}C_{b}^{\prime}&-\frac{2}{\sqrt{3}}C_{b}^{\prime}&C_{b}^{\prime}&C_{a}^{\prime}&0&\frac{1}{2}g_{1}&\frac{\sqrt{3}}{2}g_{1}\\\
\sqrt{2}C_{b}^{\prime}&C_{b}^{\prime}&0&0&C_{a}^{\prime}&\frac{\sqrt{3}}{2}g_{1}&-\frac{1}{2}g_{1}\\\
-\frac{\sqrt{2}}{3}g_{2}&\frac{5}{6}g_{2}&\frac{1}{2\sqrt{3}}g_{2}&\frac{1}{2}g_{1}&\frac{\sqrt{3}}{2}g_{1}&0&0\\\
\sqrt{\frac{2}{3}}g_{2}&\frac{1}{2\sqrt{3}}g_{2}&\frac{1}{2}g_{2}&\frac{\sqrt{3}}{2}g_{1}&-\frac{1}{2}g_{1}&0&0\end{pmatrix}$
(29)

and the contact potentials of
$\bar{D}^{*}\Sigma_{c}^{*}-\bar{D}^{*}\Sigma_{c}-\bar{D}\Sigma_{c}^{*}-\bar{D}^{*}\Lambda_{c}-J/\psi
N$ system with $J^{P}=3/2^{-}$ are written as

$V^{J=3/2}=\begin{pmatrix}C_{a}-\frac{2}{3}C_{b}&-\frac{\sqrt{5}}{3}C_{b}&\sqrt{\frac{5}{3}}C_{b}&\sqrt{\frac{5}{3}}C_{b}^{\prime}&\frac{\sqrt{5}}{3}g_{2}\\\
-\frac{\sqrt{5}}{3}C_{b}&C_{a}+\frac{2}{3}C_{b}&\frac{1}{\sqrt{3}}C_{b}&\frac{1}{\sqrt{3}}C_{b}^{\prime}&-\frac{1}{3}g_{2}\\\
\sqrt{\frac{5}{3}}C_{b}&\frac{1}{\sqrt{3}}C_{b}&C_{a}&-C_{b}^{\prime}&\frac{1}{\sqrt{3}}g_{2}\\\
\sqrt{\frac{5}{3}}C_{b}^{\prime}&\frac{1}{\sqrt{3}}C_{b}^{\prime}&-C_{b}^{\prime}&C_{a}^{\prime}&g_{1}\\\
\frac{\sqrt{5}}{3}g_{2}&-\frac{1}{3}g_{2}&\frac{1}{\sqrt{3}}g_{2}&g_{1}&0\end{pmatrix}$
(30)

###  A.2 light meson saturation

Following Ref. Peng:2020xrf , we expect the EFT couplings
$C_{a}(C_{a}^{\prime})$ and $C_{b}(C_{b}^{\prime})$ to be saturated by scalar
and vector meson exchanges

$\displaystyle C_{a}^{\rm sat(\prime)}(\Lambda\sim m_{\sigma},m_{V})$
$\displaystyle\propto$ $\displaystyle C_{a}^{S}+C_{a}^{V}\,,$ (31)
$\displaystyle C_{b}^{\rm sat(\prime)}(\Lambda\sim m_{\sigma},m_{V})$
$\displaystyle\propto$ $\displaystyle C_{b}^{V}\,.$ (32)

The value of the saturated couplings is expected to be proportional to the
light meson $M$ potential $V_{M}(\vec{q})$ at $|\vec{q}|=0$ once we have
removed the Dirac-delta term Peng:2020xrf . According to the one-boson
exchange(OBE) model, the $\sigma$ and $\rho(\omega)$ exchanges are responsible
for the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
potentials, while $\sigma$,$\omega$ exchanges and $\rho$ exchange are allowed
for the $\bar{D}^{(*)}\Lambda_{c}\to\bar{D}^{(*)}\Lambda_{c}$ potentials and
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(*)}\Lambda_{c}$ potentials
due to isospin symmetry. This gives us Liu:2019zvb

$\displaystyle C_{a}^{{\rm sat}(\sigma)}(\Lambda\sim m_{\sigma})$
$\displaystyle\propto$ $\displaystyle-\frac{g_{\sigma 1}g_{\sigma
2}}{m_{\sigma}^{2}}\,,$ (33) $\displaystyle C_{a}^{{\rm sat}(V)}(\Lambda\sim
m_{\rho})$ $\displaystyle\propto$
$\displaystyle\frac{g_{V1}g_{V2}}{m_{V}^{2}}\,(1+\vec{\tau}_{1}\cdot\vec{T}_{2})\,,$
(34) $\displaystyle C_{b}^{{\rm sat}(V)}(\Lambda\sim m_{\rho})$
$\displaystyle\propto$
$\displaystyle\frac{f_{V1}f_{V2}}{6M^{2}}(1+\vec{\tau}_{1}\cdot\vec{T}_{2})\,,$
(35) $\displaystyle C_{a}^{{\rm sat}(\sigma)\,\prime}(\Lambda\sim m_{\sigma})$
$\displaystyle\propto$ $\displaystyle-\frac{g_{\sigma 1}g_{\sigma
3}}{m_{\sigma}^{2}}\,,$ (36) $\displaystyle C_{a}^{{\rm
sat}(V)\,\prime}(\Lambda\sim m_{\omega})$ $\displaystyle\propto$
$\displaystyle\frac{g_{V1}g_{V3}}{m_{V}^{2}}\,,$ (37) $\displaystyle
C_{b}^{{\rm sat}(V)\prime}(\Lambda\sim m_{\rho})$ $\displaystyle\propto$
$\displaystyle\frac{f_{V1}f_{V3}}{6M^{2}}(\vec{\tau}_{1}\cdot\vec{t}_{2})\,.$
(38)

where $V=\rho,\omega$ and we have made the simplification that
$m_{\rho}=m_{\omega}=m_{V}$. The proportionality constant is unknown and
depends on the details of the renormalization process. In this work, we assume
that these proportionality constants are the same. The $g_{\sigma_{1}}$,
$g_{\sigma_{2}}$, and $g_{\sigma_{3}}$ denote the couplings of the
$\bar{D}^{(*)}$ mesons, $\Sigma_{c}^{(*)}$ baryons, and $\Lambda_{c}$ baryon
to the sigma meson, and $g_{v1}$, $g_{v2}$, and $g_{v3}$ ($f_{v1}$, $f_{v2}$,
and $f_{v3}$ ) denote the electric-type (magnetic-type) couplings between the
$\bar{D}^{(*)}$ mesons, $\Sigma_{c}^{(*)}$ baryons, and $\Lambda_{c}$ baryon
and a light vector meson. $M$ is a mass scale to render $f_{v}$ dimensionless.
Following Refs. Liu:2011xc ; Liu:2019zvb , we tabulate the values of these
couplings in Table 10. The $\vec{\tau}_{1}\cdot\vec{T}_{2}$ and
$\vec{\tau}_{1}\cdot\vec{t}_{2}$ are the isospin factors of
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
potentials and
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(*)}\Lambda_{c}$ potentials,
which are $\vec{\tau}_{1}\cdot\vec{T}_{2}=-2$ and
$\vec{\tau}_{1}\cdot\vec{t}_{2}=-\sqrt{3}$ for the total isospin $I=1/2$.

Coupling | Value for $P$/$P^{*}$ | Coupling | Value for $\Sigma_{Q}$/$\Sigma_{Q}^{*}$ | Coupling | Value for $\Lambda_{Q}$
---|---|---|---|---|---
$g_{\sigma 1}$ | 3.4 | $g_{\sigma 2}$ | 6.8 | $g_{\sigma 3}$ | 3.4
$g_{v1}$ | 2.6 | $g_{v2}$ | 5.8 | $g_{v3}$ | 2.9
$\kappa_{v1}$ | 2.3 | $\kappa_{v2}$ | 1.7 | $\kappa_{v3}$ | 1.2
Table 10: Couplings of the light mesons of the OBE model ( $\sigma$, $\rho$,
$\omega$) to the heavy-meson and heavy-baryon. For the magnetic-type coupling
of the $\rho$ and $\omega$ vector mesons we have used the decomposition
$f_{V}=\kappa_{V}\,g_{V}$, with $V=\rho,\omega$. $M=940$ MeV refers to the
mass scale involved in the magnetic-type couplings Liu:2011xc ; Liu:2019zvb .

###### Acknowledgements.

We are grateful to Eulogio Oset, Fu-Sheng Yu, Chun-Wen Xiao, Jun-Xu Lu, and Qi
Wu for useful discussions. This work is supported in part by the National
Natural Science Foundation of China under Grants No.11975041 and
No.11961141004. Ming-Zhu Liu acknowledges support from the National Natural
Science Foundation of China under Grant No.12105007 and China Postdoctoral
Science Foundation under Grants No. 2022M710317, and No. 2022T150036.

## References

  * (1) LHCb collaboration, _Observation of $J/\psi p$ Resonances Consistent with Pentaquark States in $\Lambda_{b}^{0}\to J/\psi K^{-}p$ Decays_, _Phys. Rev. Lett._ 115 (2015) 072001 [1507.03414].
  * (2) LHCb collaboration, _Observation of a narrow pentaquark state, $P_{c}(4312)^{+}$, and of two-peak structure of the $P_{c}(4450)^{+}$_, _Phys. Rev. Lett._ 122 (2019) 222001 [1904.03947].
  * (3) LHCb collaboration, _Evidence for a new structure in the $J/\psi p$ and $J/\psi\bar{p}$ systems in $B_{s}^{0}\to J/\psi p\bar{p}$ decays_, _Phys. Rev. Lett._ 128 (2022) 062001 [2108.04720].
  * (4) LHCb collaboration, _Evidence of a $J/\psi\Lambda$ structure and observation of excited $\Xi^{-}$ states in the $\Xi^{-}_{b}\to J/\psi\Lambda K^{-}$ decay_, _Sci. Bull._ 66 (2021) 1278 [2012.10380].
  * (5) LHCb collaboration, _Observation of a J/ $\psi$$\Lambda$ Resonance Consistent with a Strange Pentaquark Candidate in B-→J/$\psi$$\Lambda$p¯ Decays_, _Phys. Rev. Lett._ 131 (2023) 031901 [2210.10346].
  * (6) R. Chen, Z.-F. Sun, X. Liu and S.-L. Zhu, _Strong LHCb evidence supporting the existence of the hidden-charm molecular pentaquarks_ , _Phys. Rev._ D100 (2019) 011502 [1903.11013].
  * (7) J. He, _Study of $P_{c}(4457)$, $P_{c}(4440)$, and $P_{c}(4312)$ in a quasipotential Bethe-Salpeter equation approach_, _Eur. Phys. J._ C79 (2019) 393 [1903.11872].
  * (8) H.-X. Chen, W. Chen and S.-L. Zhu, _Possible interpretations of the $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$_, _Phys. Rev._ D100 (2019) 051501 [1903.11001].
  * (9) C.W. Xiao, J. Nieves and E. Oset, _Heavy quark spin symmetric molecular states from ${\bar{D}}^{(*)}\Sigma_{c}^{(*)}$ and other coupled channels in the light of the recent LHCb pentaquarks_, _Phys. Rev._ D100 (2019) 014021 [1904.01296].
  * (10) S. Sakai, H.-J. Jing and F.-K. Guo, _Decays of $P_{c}$ into $J/\psi N$ and $\eta_{c}N$ with heavy quark spin symmetry_, _Phys. Rev._ D100 (2019) 074007 [1907.03414].
  * (11) Y. Yamaguchi, H. García-Tecocoatzi, A. Giachino, A. Hosaka, E. Santopinto, S. Takeuchi et al., _$P_{c}$ pentaquarks with chiral tensor and quark dynamics_, _Phys. Rev. D_ 101 (2020) 091502 [1907.04684].
  * (12) J. He and D.-Y. Chen, _Molecular states from $\Sigma^{(*)}_{c}\bar{D}^{(*)}-\Lambda_{c}\bar{D}^{(*)}$ interaction_, _Eur. Phys. J._ C79 (2019) 887 [1909.05681].
  * (13) M.-Z. Liu, T.-W. Wu, M. Sánchez Sánchez, M.P. Valderrama, L.-S. Geng and J.-J. Xie, _Spin-parities of the $P_{c}(4440)$ and $P_{c}(4457)$ in the one-boson-exchange model_, _Phys. Rev. D_ 103 (2021) 054004 [1907.06093].
  * (14) M. Pavon Valderrama, _One pion exchange and the quantum numbers of the P c(4440) and Pc(4457) pentaquarks_, _Phys. Rev._ D100 (2019) 094028 [1907.05294].
  * (15) L. Meng, B. Wang, G.-J. Wang and S.-L. Zhu, _The hidden charm pentaquark states and $\Sigma_{c}\bar{D}^{(*)}$ interaction in chiral perturbation theory_, _Phys. Rev._ D100 (2019) 014031 [1905.04113].
  * (16) M.-L. Du, V. Baru, F.-K. Guo, C. Hanhart, U.-G. Meißner, J.A. Oller et al., _Interpretation of the LHCb $P_{c}$ States as Hadronic Molecules and Hints of a Narrow $P_{c}(4380)$_, _Phys. Rev. Lett._ 124 (2020) 072001 [1910.11846].
  * (17) X.-Z. Ling, J.-X. Lu, M.-Z. Liu and L.-S. Geng, _$P_{c}$(4457) $\to P_{c}$ (4312) $\pi/\gamma$ in the molecular picture_, _Phys. Rev. D_ 104 (2021) 074022 [2106.12250].
  * (18) X.-K. Dong, F.-K. Guo and B.-S. Zou, _A survey of heavy-antiheavy hadronic molecules_ , _Progr. Phys._ 41 (2021) 65 [2101.01021].
  * (19) U. Özdem, _Magnetic dipole moments of the hidden-charm pentaquark states: $P_{c}(4440)$, $P_{c}(4457)$ and $P_{cs}(4459)$_, _Eur. Phys. J. C_ 81 (2021) 277 [2102.01996].
  * (20) Y.-W. Pan, T.-W. Wu, M.-Z. Liu and L.-S. Geng, _Three-body molecules D¯D¯* $\Sigma$c: Understanding the nature of Tcc, Pc(4312), Pc(4440), and Pc(4457)_, _Phys. Rev. D_ 105 (2022) 114048 [2204.02295].
  * (21) Z. Zhang, J. Liu, J. Hu, Q. Wang and U.-G. Meißner, _Revealing the nature of hidden charm pentaquarks with machine learning_ , _Sci. Bull._ 68 (2023) 981 [2301.05364].
  * (22) Y.-W. Pan, T.-W. Wu, M.-Z. Liu and L.-S. Geng, _Hadronic molecules composed of a doubly charmed tetraquark state and a charmed meson_ , _Eur. Phys. J. C_ 82 (2022) 908 [2208.05385].
  * (23) Z.-W. Liu, J.-X. Lu, M.-Z. Liu and L.-S. Geng, _Distinguishing the spins of Pc(4440) and Pc(4457) with femtoscopic correlation functions_ , _Phys. Rev. D_ 108 (2023) L031503 [2305.19048].
  * (24) M.I. Eides, V.Y. Petrov and M.V. Polyakov, _New LHCb pentaquarks as hadrocharmonium states_ , _Mod. Phys. Lett. A_ 35 (2020) 2050151 [1904.11616].
  * (25) A. Ali and A.Y. Parkhomenko, _Interpretation of the narrow $J/\psi p$ Peaks in $\Lambda_{b}\to J/\psi pK^{-}$ decay in the compact diquark model_, _Phys. Lett. B_ 793 (2019) 365 [1904.00446].
  * (26) Z.-G. Wang, _Analysis of the $P_{c}(4312)$, $P_{c}(4440)$, $P_{c}(4457)$ and related hidden-charm pentaquark states with QCD sum rules_, _Int. J. Mod. Phys. A_ 35 (2020) 2050003 [1905.02892].
  * (27) J.-B. Cheng and Y.-R. Liu, _$P_{c}(4457)^{+}$ , $P_{c}(4440)^{+}$, and $P_{c}(4312)^{+}$: molecules or compact pentaquarks?_, _Phys. Rev._ D100 (2019) 054002 [1905.08605].
  * (28) X.-Z. Weng, X.-L. Chen, W.-Z. Deng and S.-L. Zhu, _Hidden-charm pentaquarks and $P_{c}$ states_, _Phys. Rev. D_ 100 (2019) 016014 [1904.09891].
  * (29) R. Zhu, X. Liu, H. Huang and C.-F. Qiao, _Analyzing doubly heavy tetra- and penta-quark states by variational method_ , _Phys. Lett._ B797 (2019) 134869 [1904.10285].
  * (30) A. Pimikov, H.-J. Lee and P. Zhang, _Hidden charm pentaquarks with color-octet substructure in QCD Sum Rules_ , _Phys. Rev._ D101 (2020) 014002 [1908.04459].
  * (31) W. Ruangyoo, K. Phumphan, C.-C. Chen, A. Limphirat and Y. Yan, _P c resonances in the compact pentaquark picture_, _J. Phys. G_ 49 (2022) 075001 [2105.14249].
  * (32) JPAC collaboration, _Interpretation of the LHCb $P_{c}$(4312)+ Signal_, _Phys. Rev. Lett._ 123 (2019) 092001 [1904.10021].
  * (33) S.X. Nakamura, _$P_{c}(4312)^{+}$ , $P_{c}(4380)^{+}$, and $P_{c}(4457)^{+}$ as double triangle cusps_, _Phys. Rev. D_ 103 (2021) 111503 [2103.06817].
  * (34) T.J. Burns and E.S. Swanson, _Production of Pc states in $\Lambda$b decays_, _Phys. Rev. D_ 106 (2022) 054029 [2207.00511].
  * (35) J.-M. Xie, X.-Z. Ling, M.-Z. Liu and L.-S. Geng, _Search for hidden-charm pentaquark states in three-body final states_ , _Eur. Phys. J. C_ 82 (2022) 1061 [2204.12356].
  * (36) Y.-H. Lin, C.-W. Shen, F.-K. Guo and B.-S. Zou, _Decay behaviors of the $P_{c}$ hadronic molecules_, _Phys. Rev. D_ 95 (2017) 114017 [1703.01045].
  * (37) T.J. Burns and E.S. Swanson, _Experimental constraints on the properties of $P_{c}$ states_, _Eur. Phys. J. A_ 58 (2022) 68 [2112.11527].
  * (38) Z.E. Meziani et al., _A Search for the LHCb Charmed ’Pentaquark’ using Photo-Production of $J/{\psi}$ at Threshold in Hall C at Jefferson Lab_, 1609.00676.
  * (39) GlueX collaboration, _First Measurement of Near-Threshold J/ $\psi$ Exclusive Photoproduction off the Proton_, _Phys. Rev. Lett._ 123 (2019) 072001 [1905.10811].
  * (40) X. Cao and J.-p. Dai, _Confronting pentaquark photoproduction with new LHCb observations_ , _Phys. Rev. D_ 100 (2019) 054033 [1904.06015].
  * (41) Q. Wang, X.-H. Liu and Q. Zhao, _Photoproduction of hidden charm pentaquark states $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$_, _Phys. Rev. D_ 92 (2015) 034022 [1508.00339].
  * (42) A.N. Hiller Blin, C. Fernández-Ramírez, A. Jackura, V. Mathieu, V.I. Mokeev, A. Pilloni et al., _Studying the P c(4450) resonance in J/$\psi$ photoproduction off protons_, _Phys. Rev. D_ 94 (2016) 034002 [1606.08912].
  * (43) M. Karliner and J.L. Rosner, _Photoproduction of Exotic Baryon Resonances_ , _Phys. Lett. B_ 752 (2016) 329 [1508.01496].
  * (44) X.-Y. Wang, X.-R. Chen and J. He, _Possibility to study pentaquark states $P_{c}(4312),P_{c}(4440)$, and $P_{c}(4457)$ in $\gamma p\rightarrow J/\psi p$ reaction_, _Phys. Rev._ D99 (2019) 114007 [1904.11706].
  * (45) J.-J. Wu, T.S.H. Lee and B.-S. Zou, _Nucleon resonances with hidden charm in $\gamma$p reactions_, _Phys. Rev. C_ 100 (2019) 035206 [1906.05375].
  * (46) S.-Y. Li, Y.-R. Liu, Y.-N. Liu, Z.-G. Si and X.-F. Zhang, _Hidden-charm Pentaquark Production at $e^{+}e^{-}$ Colliders_, _Commun. Theor. Phys._ 69 (2018) 291 [1706.04765].
  * (47) M.B. Voloshin, _Hidden-charm pentaquark formation in antiproton-deuterium collisions_ , _Phys. Rev. D_ 99 (2019) 093003 [1903.04422].
  * (48) C.-h. Chen, Y.-L. Xie, H.-g. Xu, Z. Zhang, D.-M. Zhou, Z.-L. She et al., _Exotic states Pc(4312), Pc(4440), and Pc(4457) in pp collisions at s=7, 13 TeV_ , _Phys. Rev. D_ 105 (2022) 054013 [2111.03241].
  * (49) P. Ling, X.-H. Dai, M.-L. Du and Q. Wang, _Prompt production of the hidden charm pentaquarks in the LHC_ , _Eur. Phys. J. C_ 81 (2021) 819 [2104.11133].
  * (50) P.-P. Shi, F.-K. Guo and Z. Yang, _Semi-inclusive electroproduction of hidden-charm and double-charm hadronic molecules_ , _Phys. Rev. D_ 106 (2022) 114026 [2208.02639].
  * (51) E. Braaten, M. Kusunoki and S. Nussinov, _Production of the X(3870) in B meson decay by the coalescence of charm mesons_ , _Phys. Rev. Lett._ 93 (2004) 162001 [hep-ph/0404161].
  * (52) E. Braaten and M. Kusunoki, _Exclusive production of the X(3872) in B meson decay_ , _Phys. Rev. D_ 71 (2005) 074005 [hep-ph/0412268].
  * (53) M.-L. Du, V. Baru, F.-K. Guo, C. Hanhart, U.-G. Meißner, J.A. Oller et al., _Revisiting the nature of the P c pentaquarks_, _JHEP_ 08 (2021) 157 [2102.07159].
  * (54) L. Roca, J. Nieves and E. Oset, _LHCb pentaquark as a $\bar{D}^{*}\Sigma_{c}-\bar{D}^{*}\Sigma_{c}^{*}$ molecular state_, _Phys. Rev. D_ 92 (2015) 094003 [1507.04249].
  * (55) Q. Wang, C. Hanhart and Q. Zhao, _Decoding the riddle of $Y(4260)$ and $Z_{c}(3900)$_, _Phys. Rev. Lett._ 111 (2013) 132003 [1303.6355].
  * (56) F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang and Q. Zhao, _Production of the X(3872) in charmonia radiative decays_ , _Phys. Lett. B_ 725 (2013) 127 [1306.3096].
  * (57) Y.-K. Hsiao, Y. Yu and B.-C. Ke, _Resonant $a_{0}(980)$ state in triangle rescattering $D_{s}^{+}\rightarrow\pi^{+}\pi^{0}\eta$ decays_, _Eur. Phys. J. C_ 80 (2020) 895 [1909.07327].
  * (58) M.-Z. Liu, X.-Z. Ling, L.-S. Geng, En-Wang and J.-J. Xie, _Production of Ds0*(2317) and Ds1(2460) in B decays as D(*)K and Ds(*) $\eta$ molecules_, _Phys. Rev. D_ 106 (2022) 114011 [2209.01103].
  * (59) Q. Wu, M.-Z. Liu and L.-S. Geng, _Unified description of the productions of $\bar{D}^{*}D$ and $\bar{D}^{*}D^{*}$ molecules in $B$ decays_, 2304.05269.
  * (60) Q. Wu and D.-Y. Chen, _Production of $P_{c}$ states from $\Lambda_{b}$ decay_, _Phys. Rev._ D100 (2019) 114002 [1906.02480].
  * (61) A.F. Falk and M. Neubert, _Second order power corrections in the heavy quark effective theory. 2. Baryon form-factors_ , _Phys. Rev. D_ 47 (1993) 2982 [hep-ph/9209269].
  * (62) T. Gutsche, M.A. Ivanov, J.G. Körner and V.E. Lyubovitskij, _Nonleptonic two-body decays of single heavy baryons $\Lambda_{Q}$, $\Xi_{Q}$, and $\Omega_{Q}$ $(Q=b,c)$ induced by $W$ emission in the covariant confined quark model_, _Phys. Rev. D_ 98 (2018) 074011 [1806.11549].
  * (63) M.-Z. Liu, Y.-W. Pan, F.-Z. Peng, M. Sánchez Sánchez, L.-S. Geng, A. Hosaka et al., _Emergence of a complete heavy-quark spin symmetry multiplet: seven molecular pentaquarks in light of the latest LHCb analysis_ , _Phys. Rev. Lett._ 122 (2019) 242001 [1903.11560].
  * (64) M. Pavon Valderrama, _One pion exchange and the quantum numbers of the P c(4440) and Pc(4457) pentaquarks_, _Phys. Rev. D_ 100 (2019) 094028 [1907.05294].
  * (65) Y.-H. Lin and B.-S. Zou, _Strong decays of the latest LHCb pentaquark candidates in hadronic molecule pictures_ , _Phys. Rev._ D100 (2019) 056005 [1908.05309].
  * (66) N. Yalikun, Y.-H. Lin, F.-K. Guo, Y. Kamiya and B.-S. Zou, _Coupled-channel effects of the $\Sigma$c(*)D¯(*)-$\Lambda$c(2595)D¯ system and molecular nature of the Pc pentaquark states from one-boson exchange model_, _Phys. Rev. D_ 104 (2021) 094039 [2109.03504].
  * (67) L.-L. Chau, _Quark Mixing in Weak Interactions_ , _Phys. Rept._ 95 (1983) 1.
  * (68) L.-L. Chau and H.-Y. Cheng, _Analysis of Exclusive Two-Body Decays of Charm Mesons Using the Quark Diagram Scheme_ , _Phys. Rev. D_ 36 (1987) 137.
  * (69) R. Molina, J.-J. Xie, W.-H. Liang, L.-S. Geng and E. Oset, _Theoretical interpretation of the $D^{+}_{s}\to\pi^{+}\pi^{0}\eta$ decay and the nature of $a_{0}(980)$_, _Phys. Lett. B_ 803 (2020) 135279 [1908.11557].
  * (70) H.-Y. Cheng, _Nonleptonic weak decays of bottom baryons_ , _Phys. Rev. D_ 56 (1997) 2799 [hep-ph/9612223].
  * (71) A. Ali, G. Kramer and C.-D. Lu, _Experimental tests of factorization in charmless nonleptonic two-body B decays_ , _Phys. Rev. D_ 58 (1998) 094009 [hep-ph/9804363].
  * (72) H.-n. Li, C.-D. Lu and F.-S. Yu, _Branching ratios and direct CP asymmetries in $D\to PP$ decays_, _Phys. Rev. D_ 86 (2012) 036012 [1203.3120].
  * (73) M. Bauer, B. Stech and M. Wirbel, _Exclusive Nonleptonic Decays of D, D(s), and B Mesons_ , _Z. Phys. C_ 34 (1987) 103.
  * (74) T. Gutsche, M.A. Ivanov, J.G. Körner, V.E. Lyubovitskij, P. Santorelli and N. Habyl, _Semileptonic decay $\Lambda_{b}\to\Lambda_{c}+\tau^{-}+\bar{\nu_{\tau}}$ in the covariant confined quark model_, _Phys. Rev. D_ 91 (2015) 074001 [1502.04864].
  * (75) Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * (76) R.C. Verma, _Decay constants and form factors of s-wave and p-wave mesons in the covariant light-front quark model_ , _J. Phys. G_ 39 (2012) 025005 [1103.2973].
  * (77) Flavour Lattice Averaging Group collaboration, _FLAG Review 2019: Flavour Lattice Averaging Group (FLAG)_ , _Eur. Phys. J. C_ 80 (2020) 113 [1902.08191].
  * (78) Y.-H. Chen, H.-Y. Cheng, B. Tseng and K.-C. Yang, _Charmless hadronic two-body decays of B(u) and B(d) mesons_ , _Phys. Rev. D_ 60 (1999) 094014 [hep-ph/9903453].
  * (79) H.-Y. Cheng and C.-W. Chiang, _Two-body hadronic charmed meson decays_ , _Phys. Rev. D_ 81 (2010) 074021 [1001.0987].
  * (80) C.-K. Chua, _Color-allowed bottom baryon to $s$-wave and $p$-wave charmed baryon nonleptonic decays_, _Phys. Rev. D_ 100 (2019) 034025 [1905.00153].
  * (81) J.-M. Xie, M.-Z. Liu and L.-S. Geng, _Production rates of Ds+Ds- and DD¯ molecules in B decays_ , _Phys. Rev. D_ 107 (2023) 016003 [2207.12178].
  * (82) R.S. Azevedo and M. Nielsen, _J / psi kaon cross-section in meson exchange model_ , _Phys. Rev. C_ 69 (2004) 035201 [nucl-th/0310061].
  * (83) M.E. Bracco, A. Cerqueira, Jr., M. Chiapparini, A. Lozea and M. Nielsen, _D* D(s) K and D*(s) D K vertices in a QCD Sum Rule approach_ , _Phys. Lett. B_ 641 (2006) 286 [hep-ph/0604167].
  * (84) Z.G. Wang and S.L. Wan, _Analysis of the vertices D* D(s)K, D(s)* DK, D0 D(s)K, and D(s)0 DK with the light-cone QCD sum rules_ , _Phys. Rev. D_ 74 (2006) 014017 [hep-ph/0606002].
  * (85) Z.-H. Guo and J. Oller, _Anatomy of the newly observed hidden-charm pentaquark states: $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$_, _Phys. Lett. B_ 793 (2019) 144 [1904.00851].
  * (86) T. Ji, X.-K. Dong, M. Albaladejo, M.-L. Du, F.-K. Guo, J. Nieves et al., _Understanding the 0++ and 2++ charmonium(-like) states near 3.9 GeV_ , _Sci. Bull._ 68 (2023) 688 [2212.00631].
  * (87) E. Oset and A. Ramos, _Nonperturbative chiral approach to s wave anti-K N interactions_ , _Nucl. Phys._ A635 (1998) 99 [nucl-th/9711022].
  * (88) D. Jido, J.A. Oller, E. Oset, A. Ramos and U.G. Meissner, _Chiral dynamics of the two Lambda(1405) states_ , _Nucl. Phys._ A725 (2003) 181 [nucl-th/0303062].
  * (89) J.-J. Wu, R. Molina, E. Oset and B.S. Zou, _Prediction of narrow $N^{*}$ and $\Lambda^{*}$ resonances with hidden charm above 4 GeV_, _Phys. Rev. Lett._ 105 (2010) 232001 [1007.0573].
  * (90) T. Hyodo and D. Jido, _The nature of the Lambda(1405) resonance in chiral dynamics_ , _Prog. Part. Nucl. Phys._ 67 (2012) 55 [1104.4474].
  * (91) V.R. Debastiani, J.M. Dias, W.H. Liang and E. Oset, _Molecular $\Omega_{c}$ states generated from coupled meson-baryon channels_, _Phys. Rev. D_ 97 (2018) 094035 [1710.04231].
  * (92) J.A. Oller and E. Oset, _Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the $\sigma$, f0(980), a0(980) scalar mesons_, _Nucl. Phys._ A620 (1997) 438 [hep-ph/9702314].
  * (93) L. Roca, E. Oset and J. Singh, _Low lying axial-vector mesons as dynamically generated resonances_ , _Phys. Rev. D_ 72 (2005) 014002 [hep-ph/0503273].
  * (94) Q.X. Yu, R. Pavao, V.R. Debastiani and E. Oset, _Description of the $\Xi_{c}$ and $\Xi_{b}$ states as molecular states_, _Eur. Phys. J. C_ 79 (2019) 167 [1811.11738].
  * (95) F.-Z. Peng, M.-Z. Liu, M. Sánchez Sánchez and M. Pavon Valderrama, _Heavy-hadron molecules from light-meson-exchange saturation_ , _Phys. Rev. D_ 102 (2020) 114020 [2004.05658].
  * (96) F.-Z. Peng, M. Sánchez Sánchez, M.-J. Yan and M. Pavon Valderrama, _Heavy-hadron molecular spectrum from light-meson exchange saturation_ , _Phys. Rev. D_ 105 (2022) 034028 [2101.07213].
  * (97) X.-K. Dong, F.-K. Guo and B.-S. Zou, _A survey of heavy–heavy hadronic molecules_ , _Commun. Theor. Phys._ 73 (2021) 125201 [2108.02673].
  * (98) Z.-C. Yang, Z.-F. Sun, J. He, X. Liu and S.-L. Zhu, _The possible hidden-charm molecular baryons composed of anti-charmed meson and charmed baryon_ , _Chin. Phys. C_ 36 (2012) 6 [1105.2901].
  * (99) M.-X. Duan, L. Qiu, X.-Z. Ling and Q. Zhao, _Predictions for feed-down enhancements at the $\Lambda_{c}\bar{D}$ and $\Lambda_{c}\bar{D}^{*}$ thresholds via the triangle and box singularities_, 2303.13329.
  * (100) Y.-W. Pan, M.-Z. Liu, F.-Z. Peng, M. Sánchez Sánchez, L.-S. Geng and M. Pavon Valderrama, _Model independent determination of the spins of the $P_{c}$(4440) and $P_{c}$(4457) from the spectroscopy of the triply charmed dibaryons_, _Phys. Rev. D_ 102 (2020) 011504 [1907.11220].
  * (101) H. Xing, J. Liang, L. Liu, P. Sun and Y.-B. Yang, _First observation of the hidden-charm pentaquarks on lattice_ , 2210.08555.
  * (102) C.-J. Xiao, Y. Huang, Y.-B. Dong, L.-S. Geng and D.-Y. Chen, _Exploring the molecular scenario of Pc(4312) , Pc(4440) , and Pc(4457)_ , _Phys. Rev. D_ 100 (2019) 014022 [1904.00872].
  * (103) C.W. Xiao, J.X. Lu, J.J. Wu and L.S. Geng, _How to reveal the nature of three or more pentaquark states_ , _Phys. Rev. D_ 102 (2020) 056018 [2007.12106].
  * (104) Y. Yamaguchi, Y. Abe, K. Fukukawa and A. Hosaka, _$\pi J/\psi-D\bar{D}^{*}$ potential described by the quark exchange diagram_, _EPJ Web Conf._ 204 (2019) 01007.
  * (105) N. Isgur and M.B. Wise, _Heavy quark symmetry_ , _Adv. Ser. Direct. High Energy Phys._ 10 (1992) 234.
  * (106) J.M. Flynn and N. Isgur, _Heavy quark symmetry: Ideas and applications_ , _J. Phys. G_ 18 (1992) 1627 [hep-ph/9207223].
  * (107) U. Skerbis and S. Prelovsek, _Nucleon- $J/\psi$ and nucleon-$\eta_{c}$ scattering in $P_{c}$ pentaquark channels from LQCD_, _Phys. Rev._ D99 (2019) 094505 [1811.02285].
  * (108) Y.-R. Liu and M. Oka, _$\Lambda_{c}N$ bound states revisited_, _Phys. Rev. D_ 85 (2012) 014015 [1103.4624].