# Search for hidden-charm pentaquark states in three-body final states

Jia-Ming Xie  School of Physics, Beihang University, Beijing 100191, China
Xi-Zhe Ling  Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  Ming-Zhu Liu<EMAIL_ADDRESS>School of Space and
Environment, Beihang University, Beijing 100191, China  Li-Sheng Geng
<EMAIL_ADDRESS>School of Physics, Beihang University, Beijing
100191, China  Beijing Key Laboratory of Advanced Nuclear Materials and
Physics, Beihang University, Beijing 100191, China  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

The three pentaquark states, $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$,
discovered by the LHCb Collaboration in 2019, are often interpreted as
$\bar{D}^{(\ast)}\Sigma_{c}$ molecules. Together with their four
$\bar{D}^{(*)}\Sigma_{c}^{\ast}$ partners dictated by heavy quark spin
symmetry they represent a complete multiplet of hadronic molecules of
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$. The pentaquark states were observed in
the $J/\psi p$ invariant mass distributions of the $\Lambda_{b}\rightarrow
J/\psi pK$ decay. It is widely recognized that to understand their nature,
other discovery channels play an important role. In this work, we investigate
two three-body decay modes of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
molecules. The tree-level modes proceed via off-shell $\Sigma_{c}^{(\ast)}$
baryons,
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(\ast)}\left(\Sigma_{c}^{(\ast)}\to\Lambda_{c}\pi\right)\to\bar{D}^{(\ast)}\Lambda_{c}\pi$,
while the triangle-loop modes proceed through
$\bar{D}^{\ast}\Sigma_{c}^{(\ast)}\to J/\psi N\pi$, $\eta_{c}N\pi$ via
$\bar{D}\Sigma_{c}^{(\ast)}$ rescattering to $J/\psi N$ and $\eta_{c}N$. Our
results indicate that the decay widths of the $P_{c}(4457)$ and
$\bar{D}^{(\ast)}\Sigma_{c}^{\ast}$ states into
$\bar{D}^{(\ast)}\Lambda_{c}\pi$ are several MeV, as a result can be observed
in the upcoming Run 3 and Run 4 of LHC. The partial decay widths into
$\bar{D}^{(\ast)}\Lambda_{c}\pi$ of the $P_{c}(4312)$ and $P_{c}(4440)$ states
range from tens to hundreds of keV. In addition, the partial decay widths of
$\bar{D}^{\ast}\Sigma_{c}$ molecules into $J/\psi N\pi$ and $\eta_{c}N\pi$ are
several keV and tens of keV, respectively, and the partial decay widths of
$\bar{D}^{\ast}\Sigma_{c}^{\ast}$ molecules into $J/\psi N\pi$ vary from
several keV to tens of keV. In particular, we show that the spin-5/2
$\bar{D}^{\ast}\Sigma_{c}^{\ast}$ state can be searched for in the $J/\psi
N\pi$ and $\bar{D}^{\ast}\Lambda_{c}\pi$ invariant mass distributions, while
the latter one is more favorable. These three-body decay modes of the
pentaquark states are of great value to further observations of the pentaquark
states and to a better understanding of their nature.

##  I Introduction

The existence of hidden-charm pentaquark states was predicted about ten years
ago Wu  _et al._ (2010, 2011); Wang  _et al._ (2011); Yang  _et al._ (2012);
Yuan  _et al._ (2012); Wu  _et al._ (2012); Garcia-Recio  _et al._ (2013);
Xiao  _et al._ (2013); Uchino  _et al._ (2016); Karliner and Rosner (2015). In
2015, the LHCb Collaboration observed two pentaquark states, named as
$P_{c}(4380)$ and $P_{c}(4450)$, in the $J/\psi p$ invariant mass
distributions of the $\Lambda_{b}\to J/\psi pK$ decay Aaij  _et al._ (2015).
In 2019, the LHCb Collaboration updated their analysis with ten times more
data, showing that the original $P_{c}(4450)$ state splits into two states,
$P_{c}(4440)$ and $P_{c}(4457)$, and in addition a new state $P_{c}(4312)$ is
discovered Aaij  _et al._ (2019a). The masses and decay widths of the three
states are

$\displaystyle M_{P_{c}(4312)}$ $\displaystyle=$ $\displaystyle 4311.9\pm
0.7^{+6.8}_{-0.6}~{}\mbox{MeV}\quad\quad\Gamma_{P_{c}(4312)}=9.8\pm
2.7^{+3.7}_{-4.5}~{}\mbox{MeV},$ $\displaystyle M_{P_{c}(4440)}$
$\displaystyle=$ $\displaystyle 4440.3\pm
1.3^{+4.1}_{-4.7}~{}\mbox{MeV}\quad\quad\Gamma_{P_{c}(4440)}=20.6\pm
4.9^{+8.7}_{-10.1}~{}\mbox{MeV},$ (1) $\displaystyle M_{P_{c}(4457)}$
$\displaystyle=$ $\displaystyle 4457.3\pm
0.6^{+4.1}_{-1.7}~{}\mbox{MeV}\quad\quad\Gamma_{P_{c}(4457)}=6.4\pm
2.0^{+5.7}_{-1.9}~{}\mbox{MeV}.$

Two more pentaquark states were reported in the following years, though only
with a significance of about 3$\sigma$. A hidden-charm pentaquark with
strangeness, $P_{cs}(4459)$, is visible in the $J/\psi\Lambda$ invariant mass
spectrum Aaij  _et al._ (2021), and a hidden-charm pentaquark $P_{c}(4337)$
was found in both the $J/\psi p$ and $J/\psi\bar{p}$ invariant mass spectrum
Aaij  _et al._ (2022a). The former has been predicted by many studies as the
$SU(3)$-flavor partner of the pentaquark states, $P_{c}(4312)$, $P_{c}(4380)$,
$P_{c}(4440)$, and $P_{c}(4457)$ Chen  _et al._ (2016, 2017); Shen  _et al._
(2019); Xiao  _et al._ (2019a); Wang  _et al._ (2020a), while the latter is
more difficult to understand. It could be a $\chi_{c0}(1S)p$ bound state Yan
_et al._ (2022), a compact multiquark state Deng (2022), a cusp effect
Nakamura  _et al._ (2021), or a reflection effect Wang  _et al._ (2021).
Therefore, we will leave a study of $P_{cs}(4459)$ and $P_{c}(4337)$ to a
future work.

Trying to understand the nature of the pentaquark states has led to intensive
theoretical studies. In Refs. Liu  _et al._ (2019, 2021) we have employed both
an effective field theory (EFT) and the one-boson-exchange (OBE) model to
describe the three pentaquark states as $\bar{D}^{(\ast)}\Sigma_{c}$ molecules
by reproducing their masses, which is later confirmed by many other
theoretical studies Chen  _et al._ (2019a); He (2019); Chen  _et al._ (2019b);
Xiao  _et al._ (2019b); Yamaguchi  _et al._ (2020); Pavon Valderrama (2019);
Du  _et al._ (2020); He and Chen (2019); Wang  _et al._ (2020b). In addition
to the $\bar{D}^{(\ast)}\Sigma_{c}$ channel, the role of the
$\bar{D}\Lambda_{c1}$ channel has been studied in relation with the
$P_{c}(4457)$ state Burns and Swanson (2019); Peng  _et al._ (2021a); Yalikun
_et al._ (2021). In Refs. Xiao  _et al._ (2019c); Lin and Zou (2019) with the
effective Lagrangian approach the authors reproduced the decay widths of the
pentaquark states in the hadronic molecular picture. With the same approach,
Wu et al. calculated the ratios of the production rates of the pentaquark
states in the $\Lambda_{b}$ decays, and obtained results in agreement with the
LHCb measurements Wu and Chen (2019). Although the molecular interpretation is
the most popular, other explanations are also available, e.g., hadro-charmonia
Eides  _et al._ (2020), compact pentaquark states Ali and Parkhomenko (2019);
Mutuk (2019); Wang (2020); Cheng and Liu (2019); Weng  _et al._ (2019); Zhu
_et al._ (2019); Pimikov  _et al._ (2020); Ruangyoo  _et al._ (2022); Deng
(2022), virtual states Fernández-Ramírez  _et al._ (2019) or double triangle
singularities Nakamura (2021).

To investigate the molecular nature of the pentaquark states, many other
methods have been proposed. Historically, the existence of the $\Omega$ baryon
indeed verified the quark model, where the $SU(3)$-flavor symmetry plays an
important role. Following the same principle, among others, we proposed that,
if future experiments could discover the four heavy quark spin symmetry (HQSS)
partners of the $\bar{D}^{(\ast)}\Sigma_{c}^{\ast}$ molecules, it would
support their molecular nature Liu  _et al._ (2019). Apart from HQSS, their
$SU(3)$-flavor symmetry partners Xiao  _et al._ (2019a); Wang  _et al._
(2020a) and heavy quark flavor symmetry partners Wang  _et al._ (2019); Azizi
_et al._ (2021) have been predicted, the existence of which can also support
the molecular nature of the pentaquark states. In our previous work Pan  _et
al._ (2020), we showed that one can study instead the
$\Xi_{cc}^{(\ast)}\Sigma_{c}^{(\ast)}$ system, because it can be related to
the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ system via heavy antiquark diquark
symmetry (HADS). In addition, lattice QCD simulations can provide valuable
information to understand the pentaquark states, but a complete simulation of
the pentaquark system with all the relevant coupled channels is complicated
Skerbis and Prelovsek (2019); Sugiura  _et al._ (2019). Recently, we predicted
the existence of a three-body hadronic molecule $\Sigma_{c}\bar{D}\bar{K}$ Wu
_et al._ (2021), which can be viewed as an excited state of the $P_{cs}(4459)$
state. If the $\Sigma_{c}\bar{D}\bar{K}$ molecule is discovered in the future,
it will provide a non-trivial check on the molecular nature of the pentaquark
states.

The decay and production mechanisms of the pentaquark states have also
attracted considerable attention, which can offer valuable means to reveal
their nature. In Ref. Sakai  _et al._ (2019), assuming the pentaquark states
as hadronic molecules, Sakai et al. found that the decay width of
$P_{c}(4312)\to\eta_{c}p$ is about three times larger than that of
$P_{c}(4312)\to J/\psi p$, in agreement with the results of the quark
interchange model Wang  _et al._ (2020b). However, if $P_{c}(4312)$ is
regarded as a compact pentaquark state, it will dominantly decay into
$\bar{D}^{\ast}\Lambda_{c}$ rather than $\eta_{c}p$ Weng  _et al._ (2019). In
Ref.Guo  _et al._ (2019), Guo et al. estimated that the branching ratio
Br$(P_{c}(4457)\to J/\psi\Delta)$/Br$(P_{c}(4457)\to J/\psi p)$ ranges from a
few percent to about $30\%$ in the molecular picture, which shows a larger
isospin breaking in comparison with the decays of ordinary hadrons. As a
result, experimental measurements of such branching ratios are of great value
to reveal the internal structure of the pentaquark states. In Ref. Chen  _et
al._ (2022), Chen et al. estimated the production rates of $P_{c}(4312)$,
$P_{c}(4440)$ and $P_{c}(4457)$ in $pp$ collisions using the constrained
phasespace coalescence model and the parton and hadron cascade (PACIAE) model.
Their results show that the production rates of the pentaquark states, the
nucleuslike states, and the hadronic molecular states are of the same order.
In Ref. Yang and Guo (2021), Yang et al. estimated the production rates of
$P_{c}$’s in lepton-proton processes, where the $P_{c}$’s are treated as
hadronic molecules.

To further explore the decay mechanism of the pentaquark states in the
molecular picture, in this work we study the three-body decays of the seven
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ pentaquark states. The decay mechanism
of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules is similar to that of
the $T_{cc}$ state recently discovered by the LHCb Collaboration Aaij  _et
al._ (2022b, c), where the decay of $T_{cc}$ in the molecular picture proceeds
via the off-shell $D^{\ast}$ meson Meng  _et al._ (2021); Ling  _et al._
(2022); Chen  _et al._ (2021); Feijoo  _et al._ (2021); Yan and Valderrama
(2022); Fleming  _et al._ (2021); Albaladejo (2022); Du  _et al._ (2022);
Mikhasenko (2022). As for the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules,
we will consider two three-body decay modes. The tree-level modes proceed via
off-shell $\Sigma_{c}^{(\ast)}$ baryons, i.e.
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(\ast)}\left(\Sigma_{c}^{(\ast)}\to\Lambda_{c}\pi\right)\to\bar{D}^{(\ast)}\Lambda_{c}\pi$,
while the triangle-loop modes proceed through
$\bar{D}^{\ast}\Sigma_{c}^{(\ast)}\to J/\psi N\pi$, $\eta_{c}N\pi$ via
$\bar{D}\Sigma_{c}^{(\ast)}$ rescattering to $J/\psi N$ or $\eta_{c}N$. We use
the effective Lagrangian approach to estimate the decay widths of these two
modes in this work. We hope that these decay modes can offer further insights
into the nature of the pentaquark states.

This paper is organized as follows. In Sec. II, we introduce the coupled-
channel contact-range EFT of $\eta_{c}N-J/\psi
N-\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ for the pentaquark system as well as
present the decay amplitudes of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
molecules via tree-level and triangle-loop diagrams using the effective
Lagrangian approach. In Sec. III, we present the mass spectrum of
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$, their relevant couplings and the decay
widths of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules into
$\bar{D}^{(\ast)}\Lambda_{c}\pi$ and $J/\psi(\eta_{c})N\pi$. Finally, this
paper ends with a short summary in Sec. IV.

##  II Theoretical framework

The three-body decay modes of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
molecules can proceed via two types of Feynman diagrams as shown in Fig. 1 and
Fig. 2: tree-level diagrams and triangle-loop diagrams. The binding energies
of the $\bar{D}^{\ast}\Sigma_{c}^{(\ast)}$ molecules relative to their mass
thresholds are around 4-20 MeV Liu  _et al._ (2019); Zyla  _et al._ (2020a),
while the mass thresholds of $\bar{D}\pi\Sigma_{c}^{(\ast)}$ are 5 MeV less
than those of $\bar{D}^{\ast}\Sigma_{c}^{(\ast)}$, which implies that the
$\bar{D}^{\ast}\Sigma_{c}^{(\ast)}$ molecules decaying into
$\bar{D}\pi\Sigma_{c}^{(\ast)}$ via off-shell $\bar{D}^{\ast}$ meson are
heavily suppressed or forbidden. The phase space of
$\Sigma_{c}^{(\ast)}\to\Lambda_{c}\pi$ is more than 30 MeV, so that the tree-
level decays of
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}\to\bar{D}^{(\ast)}\left(\Sigma_{c}^{(\ast)}\to\Lambda_{c}\pi\right)\to\bar{D}^{(\ast)}\Lambda_{c}\pi$
are allowed, as shown in Fig. 1. The rescattering between final states
$\bar{D}^{(\ast)}\Lambda_{c}\pi$ can form the decay modes of the triangle-loop
diagrams, which are small due to the fact that the loop diagrams are
suppressed with respect to the tree-level diagrams Yan and Valderrama (2022),
satisfying the power counting of EFT. Although the decay mode of
$\bar{D}^{\ast}\to\bar{D}\pi$ is not allowed at tree level, their final states
$\bar{D}\Sigma_{c}^{(\ast)}$ would couple to the $J/\psi N$ and $\eta_{c}N$
channels according to HQSS Sakai  _et al._ (2019). Therefore, considering the
final-state interactions, it would lead to the decay modes via the triangle-
loop mechanism as shown in Fig. 2.

\begin{overpic}[scale={0.09}]{tree.png} \put(12.0,26.0){$P_{c1}^{+}$}
\par\put(40.0,3.0){$D^{-}$} \par\put(38.0,33.0){$\Sigma_{c}^{++}$}
\put(68.0,52.0){$\Lambda_{c}^{+}$} \put(69.0,20.0){$\pi^{+}$}
\put(45.0,-6.0){(a1)} \end{overpic} \begin{overpic}[scale={0.09}]{tree.png}
\put(12.0,26.0){$P_{c1}^{+}$} \par\put(38.0,3.0){$\bar{D}^{0}$}
\par\put(39.0,33.0){$\Sigma_{c}^{+}$} \put(68.0,52.0){$\Lambda_{c}^{+}$}
\put(69.0,20.0){$\pi^{0}$} \put(45.0,-6.0){(a2)} \end{overpic}
\begin{overpic}[scale={0.09}]{tree.png} \put(12.0,26.0){$P_{c2}^{+}$}
\par\put(40.0,3.0){$D^{-}$} \par\put(35.0,33.0){$\Sigma_{c}^{\ast++}$}
\put(68.0,52.0){$\Lambda_{c}^{+}$} \put(69.0,20.0){$\pi^{+}$}
\put(45.0,-6.0){(b1)} \end{overpic} \begin{overpic}[scale={0.09}]{tree.png}
\put(12.0,26.0){$P_{c2}^{+}$} \par\put(38.0,3.0){$\bar{D}^{0}$}
\par\put(38.0,33.0){$\Sigma_{c}^{\ast+}$} \put(68.0,52.0){$\Lambda_{c}^{+}$}
\put(69.0,20.0){$\pi^{0}$} \put(45.0,-6.0){(b2)} \end{overpic}
---
\begin{overpic}[scale={0.09}]{tree.png} \put(8.0,27.0){$P_{c3|c4}^{+}$}
\par\put(36.0,3.0){$D^{\ast-}$} \par\put(38.0,33.0){$\Sigma_{c}^{++}$}
\put(68.0,52.0){$\Lambda_{c}^{+}$} \put(69.0,20.0){$\pi^{+}$}
\put(45.0,-6.0){(c1)} \end{overpic} \begin{overpic}[scale={0.09}]{tree.png}
\put(8.0,27.0){$P_{c3|c4}^{+}$} \par\put(35.0,3.0){$\bar{D}^{\ast 0}$}
\par\put(39.0,33.0){$\Sigma_{c}^{+}$} \put(68.0,52.0){$\Lambda_{c}^{+}$}
\put(69.0,20.0){$\pi^{0}$} \put(45.0,-6.0){(c2)} \end{overpic}
\begin{overpic}[scale={0.09}]{tree.png} \put(3.0,27.0){$P_{c5|c6|c7}^{+}$}
\par\put(36.0,3.0){$D^{\ast-}$} \par\put(35.0,33.0){$\Sigma_{c}^{\ast++}$}
\put(68.0,52.0){$\Lambda_{c}^{+}$} \put(69.0,20.0){$\pi^{+}$}
\put(45.0,-6.0){(d1)} \end{overpic} \begin{overpic}[scale={0.09}]{tree.png}
\put(3.0,27.0){$P_{c5|c6|c7}^{+}$} \par\put(35.0,3.0){$\bar{D}^{\ast 0}$}
\par\put(38.0,33.0){$\Sigma_{c}^{\ast+}$} \put(68.0,52.0){$\Lambda_{c}^{+}$}
\put(69.0,20.0){$\pi^{0}$} \put(45.0,-6.0){(d2)} \end{overpic}
Figure 1: Tree-level diagrams for the decays of $P_{c1}\to\bar{D}\Sigma_{c}\to\bar{D}\Lambda_{c}\pi$ (a), $P_{c2}\to\bar{D}\Sigma_{c}^{\ast}\to\bar{D}\Lambda_{c}\pi$ (b), $P_{c3|c4}\to\bar{D}^{\ast}\Sigma_{c}\to\bar{D}^{\ast}\Lambda_{c}\pi$ (c) and $P_{c5|c6|c7}\to\bar{D}^{\ast}\Sigma_{c}^{\ast}\to\bar{D}^{\ast}\Lambda_{c}\pi$ (d). \begin{overpic}[scale={0.26}]{tr1.png} \put(11.0,34.0){$P_{c3|c4}^{+}$} \par\put(34.0,14.0){$\Sigma_{c}^{+}$} \par\put(33.0,39.0){$\bar{D}^{\ast 0}$} \put(53.0,28.0){$D^{-}$} \put(67.0,48.0){$\pi^{+}$} \put(60.0,14.0){$J/\psi(\eta_{c})$} \put(55.0,5.0){$n$} \put(45.0,0.0){(a1)} \end{overpic} |  \begin{overpic}[scale={0.26}]{tr1.png} \put(11.0,34.0){$P_{c3|c4}^{+}$} \par\put(34.0,14.0){$\Sigma_{c}^{+}$} \par\put(33.0,39.0){$\bar{D}^{\ast 0}$} \put(53.0,28.0){$\bar{D}^{0}$} \put(67.0,48.0){$\pi^{0}$} \put(60.0,14.0){$J/\psi(\eta_{c})$} \put(55.0,5.0){$p$} \put(45.0,0.0){(a2)} \end{overpic} |  \begin{overpic}[scale={0.26}]{tr1.png} \put(11.0,34.0){$P_{c3|c4}^{+}$} \par\put(32.0,14.0){$\Sigma_{c}^{++}$} \par\put(33.0,39.0){$D^{\ast-}$} \put(53.0,28.0){$D^{-}$} \put(67.0,48.0){$\pi^{0}$} \put(60.0,14.0){$J/\psi(\eta_{c})$} \put(55.0,5.0){$p$} \put(45.0,0.0){(a3)} \end{overpic}
---|---|---
\begin{overpic}[scale={0.26}]{tr1.png} \put(8.0,34.0){$P_{c5|c6|c7}^{+}$} \par\put(33.0,14.0){$\Sigma_{c}^{\ast+}$} \par\put(33.0,39.0){$\bar{D}^{\ast 0}$} \put(53.0,28.0){$D^{-}$} \put(67.0,48.0){$\pi^{+}$} \put(59.0,13.0){$J/\psi$} \put(55.0,5.0){$n$} \put(45.0,0.0){(b1)} \end{overpic} |  \begin{overpic}[scale={0.26}]{tr1.png} \put(8.0,34.0){$P_{c5|c6|c7}^{+}$} \par\put(33.0,14.0){$\Sigma_{c}^{\ast+}$} \par\put(33.0,39.0){$\bar{D}^{\ast 0}$} \put(53.0,28.0){$\bar{D}^{0}$} \put(67.0,48.0){$\pi^{0}$} \put(59.0,13.0){$J/\psi$} \put(55.0,5.0){$p$} \put(45.0,0.0){(b2)} \end{overpic} |  \begin{overpic}[scale={0.26}]{tr1.png} \put(8.0,34.0){$P_{c5|c6|c7}^{+}$} \par\put(30.0,14.0){$\Sigma_{c}^{\ast++}$} \par\put(33.0,39.0){$D^{\ast-}$} \put(53.0,28.0){$D^{-}$} \put(67.0,48.0){$\pi^{0}$} \put(59.0,13.0){$J/\psi$} \put(55.0,5.0){$p$} \put(45.0,0.0){(b3)} \end{overpic}
Figure 2: Triangle-loop diagrams for the decays of
$P_{c3|c4}\to\bar{D}^{\ast}\Sigma_{c}\to J/\psi(\eta_{c})N\pi$ (a) and
$P_{c5|c6|c7}\to\bar{D}^{\ast}\Sigma_{c}^{\ast}\to J/\psi N\pi$ (b) via the
rescattering of final states.

To calculate the Feynman diagrams of Figs. 1 and 2, we have to describe the
interactions related to each vertex, which can be classified into three
categories. The first category involves the interactions between the hadronic
molecules and their constituents. In this work, the seven
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ hadronic molecules are denoted by
$P_{c1}$, $P_{c2}$, $\ldots$, $P_{c7}$, following the order of scenario A of
Table I in Ref. Liu  _et al._ (2019) (see also Table II). Here, we should note
that $P_{c1}$, $P_{c3}$, and $P_{c5}$ have $J=1/2$, $P_{c2}$ and $P_{c4}$ have
$J=3/2$, and $P_{c7}$ has $J=5/2$, respectively. Their interactions with the
corresponding constituents are described by the following Lagrangians Lin and
Zou (2019):

$\displaystyle\mathcal{L}_{P_{c1}\bar{D}\Sigma_{c}}$ $\displaystyle=$
$\displaystyle-ig_{P_{c1}\bar{D}\Sigma_{c}}\bar{\Sigma}_{c}\bar{D}P_{c1},$
$\displaystyle\mathcal{L}_{P_{c2}\bar{D}\Sigma_{c}^{\ast}}$ $\displaystyle=$
$\displaystyle-
ig_{P_{c2}\bar{D}\Sigma_{c}^{\ast}}\bar{\Sigma}_{c\mu}^{\ast}\bar{D}P_{c2}^{\mu},$
$\displaystyle\mathcal{L}_{P_{c3}\bar{D}^{\ast}\Sigma_{c}}$ $\displaystyle=$
$\displaystyle
g_{P_{c3}\bar{D}^{\ast}\Sigma_{c}}\bar{\Sigma}_{c}\gamma^{5}\widetilde{\gamma}^{\mu}\bar{D}^{\ast}_{\mu}P_{c3},$
$\displaystyle\mathcal{L}_{P_{c4}\bar{D}^{\ast}\Sigma_{c}}$ $\displaystyle=$
$\displaystyle
g_{P_{c4}\bar{D}^{\ast}\Sigma_{c}}\bar{\Sigma}_{c}\bar{D}^{\ast}_{\mu}P_{c4}^{\mu},$
(2) $\displaystyle\mathcal{L}_{P_{c5}\bar{D}^{\ast}\Sigma_{c}^{\ast}}$
$\displaystyle=$ $\displaystyle
g_{P_{c5}\bar{D}^{\ast}\Sigma_{c}^{\ast}}\bar{\Sigma}_{c}^{\ast\mu}\bar{D}^{\ast}_{\mu}P_{c5},$
$\displaystyle\mathcal{L}_{P_{c6}\bar{D}^{\ast}\Sigma_{c}^{\ast}}$
$\displaystyle=$ $\displaystyle
g_{P_{c6}\bar{D}^{\ast}\Sigma_{c}^{\ast}}\bar{\Sigma}_{c\mu}^{\ast}\bar{D}^{\ast}_{\nu}\gamma^{5}\widetilde{\gamma}^{\nu}P_{c6}^{\mu},$
$\displaystyle\mathcal{L}_{P_{c7}\bar{D}^{\ast}\Sigma_{c}^{\ast}}$
$\displaystyle=$ $\displaystyle
g_{P_{c7}\bar{D}^{\ast}\Sigma_{c}^{\ast}}\bar{\Sigma}_{c\mu}^{\ast}\bar{D}^{\ast}_{\nu}P_{c7}^{\mu\nu},$

where $g_{P_{c1...c7}\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}}$ are the couplings,
which will be determined by the EFT approach described below.
$\widetilde{\gamma}^{\mu}$ is defined as
$\widetilde{\gamma}^{\mu}=(g^{\mu\nu}-p^{\mu}p^{\nu}/p^{2}){\gamma_{\nu}}$,
where $p$ is the momentum of the initial $P_{c}$ states.

The second category involves the decays of the molecular constituents, i.e.,
$\Sigma_{c}^{(\ast)}\to\Lambda_{c}\pi$ and $\bar{D}^{\ast}\to\bar{D}\pi$. The
Lagrangians describing $\Sigma_{c}^{(\ast)}\to\Lambda_{c}\pi$ read

$\displaystyle\mathcal{L}_{\pi\Lambda_{c}{\Sigma}_{c}}$ $\displaystyle=$
$\displaystyle-i\frac{g_{\pi\Lambda_{c}{\Sigma}_{c}}}{f_{\pi}}~{}\bar{\Lambda}_{c}\gamma^{\mu}\gamma_{5}\partial_{\mu}\vec{\phi}_{\pi}\cdot\vec{\tau}{\Sigma}_{c},$
(3) $\displaystyle\mathcal{L}_{\pi\Lambda_{c}{\Sigma}_{c}^{*}}$
$\displaystyle=$
$\displaystyle\frac{g_{\pi\Lambda_{c}{\Sigma}_{c}^{*}}}{f_{\pi}}~{}\bar{\Lambda}_{c}\partial^{\mu}\vec{\phi}_{\pi}\cdot\vec{\tau}{\Sigma}_{c\mu}^{*},$

where the pion decay constant $f_{\pi}=132$ MeV. The couplings
$g_{\pi\Lambda_{c}{\Sigma}_{c}}$ and $g_{\pi\Lambda_{c}{\Sigma}_{c}^{*}}$ can
be determined by reproducing the corresponding experimental data. With the
decay widths of $\Sigma_{c}^{++,+,0}\to\Lambda_{c}^{+}\pi^{+,0,-}$ and
$\Sigma_{c}^{\ast++,+,0}\to\Lambda_{c}^{+}\pi^{+,0,-}$ being 1.89, 2.3, 1.83
MeV and 14.78, 17.2, 15.3 MeV, respectively Zyla  _et al._ (2020b), we obtain
the corresponding couplings $g_{\pi\Lambda_{c}{\Sigma}_{c}}=0.538,0.554,0.532$
and $g_{\pi\Lambda_{c}{\Sigma}_{c}^{*}}=0.984,1.043,1.001$, consistent with
other works Liu and Oka (2012); Cheng and Chua (2015). In addition, we find
that the two kinds of couplings approximately satisfy the relationship
$g_{\pi\Lambda_{c}{\Sigma}_{c}^{*}}=\sqrt{3}g_{\pi\Lambda_{c}{\Sigma}_{c}}$,
which can be derived from the quark model Liu and Oka (2012). The Lagrangian
describing the $D^{\ast}$ decay into $D\pi$ is

$\displaystyle\mathcal{L}_{DD^{\ast}\pi}$ $\displaystyle=$ $\displaystyle-
ig_{DD^{\ast}\pi}(D\partial^{\mu}\pi
D^{\ast{\dagger}}_{\mu}-D_{\mu}^{*}\partial^{\mu}\pi D^{{\dagger}}),$ (4)

where the coupling is determined to be $g_{D^{\ast+}D^{0}\pi^{+}}=16.818$ by
reproducing the decay width of $D^{\ast+}\to D^{0}\pi^{+}$ Zyla  _et al._
(2020b). Experimentally, there exists only an upper limit $\Gamma<2.1$ MeV for
the $D^{\ast 0}$ width. Thus we turn to the quark model Rosner (2013), where
the strong decay width of $D^{\ast 0}$ is estimated to be $\Gamma_{D^{\ast
0}\to D^{0}\pi^{0}}=34.658$ keV. 111We note that the lattice QCD simulation
Becirevic and Sanfilippo (2013) gave a relatively larger value, i.e.,
$\Gamma_{D^{\ast 0}\to D^{0}\pi^{0}}=53\pm 9$ keV. From isospin symmetry, we
expect that the $D^{*0}$ strong decay width is smaller than the $D^{*+}$
strong decay width because the $D^{*0}\to D^{+}\pi^{-}$ decay mode is
kinematically forbidden. As a result, we do not adopt the lattice QCD result.
With these numbers, we obtain the coupling $g_{D^{\ast
0}D^{0}\pi^{0}}=11.688$. It is clear that the strong couplings approximately
satisfy isospin symmetry.

The last category involves the rescattering of final states, which is
described by the scattering amplitude $T$ and is responsible for the dynamical
generation of the pentaquark states. It can be obtained by solving the
Lippmann-Schwinger equation

$\displaystyle T=(1-VG)^{-1}V,$ (5)

where $V$ is the coupled-channel potential determined by the contact EFT
approach described below, and $G$ is the two-body propagator. Here to avoid
the ultraviolet divergence induced by evaluating the loop function $G$ and
keep the unitarity of the $T$ matrix 222The loop function can also be
regularized by other methods such as momentum cut off scheme and dimensional
regularization scheme Oset and Ramos (1998); Jido  _et al._ (2003); Wu  _et
al._ (2010); Hyodo and Jido (2012); Debastiani  _et al._ (2018). , we include
a regulator of Gaussian form
$F(q^{2},k)=e^{-2q^{2}/\Lambda^{2}}/e^{-2k^{2}/\Lambda^{2}}$ in the integral
as

$\displaystyle
G(\sqrt{s})=2m_{1}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{\omega_{1}+\omega_{2}}{2~{}\omega_{1}\omega_{2}}\frac{F(q^{2},k)}{(\sqrt{s})^{2}-(\omega_{1}+\omega_{2})^{2}+i\varepsilon},$
(6)

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, $\Lambda$ is the momentum cutoff, and the c.m. momentum $k$ is ,

$\displaystyle
k=\frac{\sqrt{{s}-(m_{1}+m_{2})^{2}}\sqrt{s-(m_{1}-m_{2})^{2}}}{2{\sqrt{s}}}.$
(7)

The dynamically generated pentaquark states correspond to poles in the
unphysical sheet, which are below the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
mass thresholds and above the mass thresholds of $J/\psi N$ and $\eta_{c}N$.
The effect of scattering amplitude $T$ passing $\sqrt{s}$ to unphysical sheet
has consequences only for the loop functions. We denote the loop function in
physical sheet and unphysical sheet as $G_{I}(\sqrt{s})$ and
$G_{II}(\sqrt{s})$, respectively, which are related by
$G_{II}(\sqrt{s})=G_{I}(\sqrt{s})-2i\mathrm{Im}~{}G_{I}(\sqrt{s})$ Oller and
Oset (1997); Roca  _et al._ (2005). The imaginary part of Eq. (6) can be
derived via Plemelj-Sokhotski formula, i.g.,
$\mathrm{Im}~{}G(\sqrt{s})=-\frac{2m_{1}}{8\pi}\frac{k}{\sqrt{s}}$, and then
the Eq. (6) in the unphysical sheet is written as

$\displaystyle
G_{II}(\sqrt{s})=G_{I}(\sqrt{s}){+~{}i\frac{2m_{1}}{4\pi}\frac{k}{\sqrt{s}}}.$
(8)

To study the decays of the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecules, we first
extend our single-channel contact-range EFT Liu  _et al._ (2019) to include
the $J/\psi N$ and $\eta_{c}N$ channels. This can be easily achieved by
utilizing HQSS in the following way. 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 muck Isgur and Wise (1992); Flynn and Isgur (1992))
$s_{1L}$ and $s_{2L}$, where 1 and 2 denote $\bar{D}^{(*)}$ and
$\Sigma_{c}^{(*)}$, respectively, via the following spin coupling formula,

$\displaystyle|s_{1l},s_{1h},j_{1};s_{2l},s_{2h},j_{2};J\rangle=$ (9)
$\displaystyle\sqrt{(2j_{1}+1)(2j_{2}+1)(2s_{L}+1)(2s_{H}+1)}\left(\begin{matrix}s_{1l}&s_{2l}&s_{L}\\\
s_{1h}&s_{2h}&s_{H}\\\
j_{1}&j_{2}&J\end{matrix}\right)|s_{1l},s_{2l},s_{L};s_{1h},s_{2h},s_{H};J\rangle.$

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

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

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

$\displaystyle V_{\Sigma_{c}\bar{D}}(1/2^{-})$ $\displaystyle=$
$\displaystyle\frac{1}{3}F_{1/2L}+\frac{2}{3}F_{3/2L},$ (11) $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}}(3/2^{-})$ $\displaystyle=$
$\displaystyle\frac{1}{3}F_{1/2L}+\frac{2}{3}F_{3/2L},$ $\displaystyle
V_{\Sigma_{c}\bar{D}^{\ast}}(1/2^{-})$ $\displaystyle=$
$\displaystyle\frac{7}{9}F_{1/2L}+\frac{2}{9}F_{3/2L},$ $\displaystyle
V_{\Sigma_{c}\bar{D}^{\ast}}(3/2^{-})$ $\displaystyle=$
$\displaystyle\frac{1}{9}F_{1/2L}+\frac{8}{9}F_{3/2L},$ $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}^{\ast}}(1/2^{-})$ $\displaystyle=$
$\displaystyle\frac{8}{9}F_{1/2L}+\frac{1}{9}F_{3/2L},$ $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}^{\ast}}(3/2^{-})$ $\displaystyle=$
$\displaystyle\frac{5}{9}F_{1/2L}+\frac{4}{9}F_{3/2L},$ $\displaystyle
V_{\Sigma_{c}^{\ast}\bar{D}^{\ast}}(5/2^{-})$ $\displaystyle=$ $\displaystyle
F_{3/2L},$

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  _et al._ (2019). For the
inelastic potentials, one can see that there are three possible channels,
$\eta_{c}N$, $J/\psi N$ and $J/\psi\Delta$. However, the $J/\psi\Delta$
channel is suppressed due to isospin symmetry breaking. From HQSS, the
potentials between $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ and $J/\psi
N,\eta_{c}N$ are only related to the spin of the light quark $1/2$, denoted by
one coupling: $g=\langle\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}|1_{H}\otimes
1/2_{L}\rangle=\langle\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}|0_{H}\otimes
1/2_{L}\rangle$. 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 and Prelovsek (2019). In this work, we take the potentials of
$V_{J/\psi(\eta_{c})N\to J/\psi(\eta_{c})N}=0$. In the following, we
explicitly show the potentials for all the seven states.

For the $\eta_{c}N-J/\psi N-\bar{D}\Sigma_{c}$ and $\eta_{c}N-J/\psi
N-\bar{D}\Sigma_{c}^{\ast}$ coupled channels, the contact-range potentials $V$
in matrix form read

$V_{\eta_{c}N-J/\psi
N-\bar{D}\Sigma_{c}}^{J=1/2}=\begin{pmatrix}0&0&\frac{1}{2}g\\\
0&0&\frac{1}{2\sqrt{3}}g\\\
\frac{1}{2}g&\frac{1}{2\sqrt{3}}g&C_{a}\end{pmatrix},\quad V_{\eta_{c}N-J/\psi
N-\bar{D}\Sigma_{c}^{\ast}}^{J=3/2}=\begin{pmatrix}0&0&0\\\
0&0&\frac{1}{\sqrt{3}}g\\\ 0&\frac{1}{\sqrt{3}}g&C_{a}\end{pmatrix}.$ (12)

For the $\eta_{c}N-J/\psi N-\bar{D}^{*}\Sigma_{c}$ coupled channels, the
contact-range potentials for $J=1/2$ and $J=3/2$ read

$V_{\eta_{c}N-J/\psi
N-\bar{D}^{*}\Sigma_{c}}^{J=1/2}=\begin{pmatrix}0&0&\frac{1}{2\sqrt{3}}g\\\
0&0&\frac{5}{6}g\\\
\frac{1}{2\sqrt{3}}g&\frac{5}{6}g&C_{a}-\frac{4}{3}C_{b}\end{pmatrix},\quad
V_{\eta_{c}N-J/\psi N-\bar{D}^{*}\Sigma_{c}}^{J=3/2}=\begin{pmatrix}0&0&0\\\
0&0&-\frac{1}{3}g\\\ 0&-\frac{1}{3}g&C_{a}+\frac{2}{3}C_{b}\end{pmatrix}.$
(13)

Similarly for the $\eta_{c}N-J/\psi N-\bar{D}^{*}\Sigma_{c}^{*}$ coupled
channels, they read

$V_{\eta_{c}N-J/\psi
N-\bar{D}^{*}\Sigma_{c}^{*}}^{J=1/2}=\begin{pmatrix}0&0&\sqrt{\frac{2}{3}}g\\\
0&0&-\frac{\sqrt{2}}{3}g\\\
\sqrt{\frac{2}{3}}g&-\frac{\sqrt{2}}{3}g&C_{a}-\frac{5}{3}C_{b}\end{pmatrix},\quad
V_{\eta_{c}N-J/\psi
N-\bar{D}^{*}\Sigma_{c}^{*}}^{J=3/2}=\begin{pmatrix}0&0&0\\\
0&0&\frac{\sqrt{5}}{3}g\\\
0&\frac{\sqrt{5}}{3}g&C_{a}-\frac{2}{3}C_{b}\end{pmatrix},$ (14)

The $J=5/2$ case contains only one channel , i.e., $\bar{D}^{*}\Sigma_{c}^{*}$
(as we have neglected the $J/\psi\Delta$ channel), and the corresponding $V$
reads

$V_{\eta_{c}N-J/\psi
N-\bar{D}^{*}\Sigma_{c}^{*}}^{J=5/2}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&C_{a}+C_{b}\end{pmatrix}.$ (15)

Using the potentials above we can search for poles generated by the coupled-
channel interactions, and determine the couplings between the molecular states
and their constituents from the residues of the corresponding poles,

$\displaystyle
g_{i}g_{j}=\lim_{{\sqrt{s}}\to{\sqrt{s_{0}}}}\left({\sqrt{s}}-{\sqrt{s_{0}}}\right)T_{ij}(\sqrt{s}),$
(16)

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

With the above Lagrangians we can obtain the amplitudes of the two decay
modes, which are explicitly given in Appendix A. The partial decay widths of
$P_{c}\to\bar{D}^{(\ast)}\Lambda_{c}\pi$ and $P_{c}\to J/\psi(\eta_{c})N\pi$
as a function of $m_{12}^{2}$ and $m_{23}^{2}$ Zyla  _et al._ (2020b) read

$d\Gamma=\frac{1}{(2\pi)^{3}}\frac{1}{2J+1}\frac{\overline{|\mathcal{M}|^{2}}}{32m_{P_{c}}^{3}}dm_{12}^{2}dm_{23}^{2},$
(17)

with $m_{12}$ the invariant mass of $\Lambda_{c}\pi$ or $J/\psi(\eta_{c})N$,
and $m_{23}$ the invariant mass of $\bar{D}^{(\ast)}\pi$ or
$J/\psi(\eta_{c})\pi$ for the $P_{c}\to\bar{D}^{(\ast)}\Lambda_{c}\pi$ or
$P_{c}\to J/\psi(\eta_{c})N\pi$ decay, respectively.

##  III Results and discussions

Table 1: Masses and quantum numbers of relevant hadrons used in this work Zyla  _et al._ (2020a).  Hadron | $I(J^{P})$ | M (MeV) | Hadron | $I(J^{P})$ | M (MeV) | Hadron | $I(J^{P})$ | M (MeV)
---|---|---|---|---|---|---|---|---
$\Sigma_{c}^{++}$ | $1(1/2^{+})$ | $2453.97$ | $\Sigma_{c}^{+}$ | $1(1/2^{+})$ | $2452.65$ | $\Sigma_{c}^{0}$ | $1(1/2^{+})$ | $2453.75$
$\Sigma_{c}^{\ast++}$ | $1(3/2^{+})$ | $2518.41$ | $\Sigma_{c}^{\ast+}$ | $1(3/2^{+})$ | $2517.4$ | $\Sigma_{c}^{\ast 0}$ | $1(3/2^{+})$ | $2518.48$
$\pi^{\pm}$ | $1(0^{-})$ | $139.57$ | $\pi^{0}$ | $1(0^{-})$ | $134.98$ | $\Lambda_{c}^{+}$ | $0(1/2^{+})$ | $2286.46$
$\bar{D}^{0}$ | $\frac{1}{2}(0^{-})$ | $1864.84$ | $D^{-}$ | $\frac{1}{2}(0^{-})$ | $1869.66$ | $p$ | $\frac{1}{2}(1/2^{+})$ | $938.27$
$\bar{D}^{\ast 0}$ | $\frac{1}{2}(1^{-})$ | $2006.85$ | $D^{\ast-}$ | $\frac{1}{2}(1^{-})$ | $2010.26$ | $n$ | $\frac{1}{2}(1/2^{+})$ | $939.57$
$J/\psi$ | $0(1^{-})$ | $3096.90$ | $\eta_{c}$ | $0(0^{-})$ | $2983.90$ |  |  |

In this work, we assume that the pentaquark states are generated by the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$, $\eta_{c}N$ and $J/\psi N$ coupled
channels, and neglect the $\bar{D}^{(\ast)}\Lambda_{c}$ contribution, which is
shown to be rather small with respect to the other three channels in the
chiral unitary approach Xiao  _et al._ (2019b, 2020). As a result, there are
three unknown parameters in the contact-range potentials. Following our
previous work Liu  _et al._ (2019), we study two scenarios A and B. In
Scenario A, the spins of $P_{c}(4440)$ and $P_{c}(4457)$ are $1/2$ and $3/2$,
respectively, while in Scenario B they are $3/2$ and $1/2$. For such coupled-
channel systems, the mass splittings between these channels can be up to 600
MeV. Therefore, we choose the value of the cutoff in the Gaussian regulator as
$\Lambda=1.5$ GeV Peng  _et al._ (2021b). 333We have adopted a larger value
$\Lambda=3$ GeV to check the uncertainties induced by the cutoff, and found
that it affects little the pole couplings to $\eta_{c}N$ and $J/\psi N$.  We
tabulate the masses and quantum numbers of relevant particles in Table 1.

###  III.1 Pole position and couplings

Table 2: Couplings of the contact-range potentials (in units of GeV-1) for Scenario A and B obtained with a cutoff $\Lambda=1.5$ GeV.  Scenario |  $\Lambda$ (GeV) |  $C_{a}$ |  $C_{b}$ |  $g_{1}$ |  $g_{3}$ |  $g_{4}$
---|---|---|---|---|---|---
A |  1.5 |  -52.750 |  5.625 |  7.650 |  6.760 |  12.350
B |  1.5 |  -56.447 |  -5.480 |  7.350 |  4.610 |  18.000

The three unknown parameters, $C_{a}$, $C_{b}$, and $g$, can in principle be
determined by reproducing the masses and widths of $P_{c}(4440)$ and
$P_{c}(4457)$. However, we find that with only the two inelastic channels
($J/\psi N$ and $\eta_{c}N$), we cannot obtain a satisfactory fit of the two
decay widths with a single $g$, as already noted in Ref. Du  _et al._ (2021).
Therefore, we finetune $g$ for each of the $P_{c}(4312)$, $P_{c}(4440)$, and
$P_{c}(4457)$ states, and the couplings of $g$ are denoted by $g_{1}$, …,
$g_{7}$, following the order of $P_{c1}$, …, $P_{c7}$. For the state $P_{c2}$,
we did not try to reproduce that of the $P_{c}(4380)$ state of the LHCb
Collaboration Aaij  _et al._ (2015), because quite likely they are not the
same state. In Ref. Du  _et al._ (2020), a reanalysis of the 2019 LHCb data
yields a state that corresponds to our $P_{c2}$, whose width is only about 20
MeV. In the present work, we assume $g_{2}=g_{1}$. As shown later this leads
to a total decay width for $P_{c2}$ ranging from 9 MeV to 14 MeV, which is
equal to the sum of the three-body partial decay widths
($P_{c2}\to{D^{\ast-}}\Lambda_{c}^{+}\pi^{+}$ and $P_{c2}\to\bar{D}^{\ast
0}\Lambda_{c}^{+}\pi^{0}$) and the two-body partial width ($P_{c2}\to J/\psi
p$), corresponding to the results obtained in Scenarios B and A, respectively.
We note that the total width is in reasonable agreement with that of Ref. Du
_et al._ (2021). In Table 2, we present the values of $C_{a}$, $C_{b}$, and
$g$ ($g_{1}$, $g_{3}$, and $g_{4}$) in Scenario A and B by reproducing the
masses and widths of $P_{c}(4440)$, and $P_{c}(4457)$ and the width of
$P_{c}(4312)$. One can see that the value of $g_{1}$ is relatively close to
the value of $g_{3}$, but is much smaller than the value of $g_{4}$, which
indicates that HQSS for the inelastic channels are heavily broken 444The other
channels such as $P-$wave $\bar{D}\Lambda_{c1}$ and $\bar{D}\Sigma_{c}^{\ast}$
can also contribute to the decay widths of $P_{c4}$ Burns and Swanson (2019);
Peng  _et al._ (2021a); Burns and Swanson (2022), which could be partially
responsible for the large value of $g_{4}$. . Since the mass of $P_{c2}$ is
close to $P_{c1}$ and the masses of $P_{c5}$, $P_{c6}$, and $P_{c7}$ are close
to $P_{c3}$, the unknown values of $g_{2}$ and $g_{5}$, $g_{6}$, $g_{7}$ in
the present work are taken to be the same as $g_{1}$ and $g_{3}$,
respectively.

Table 3: Masses of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules as well as their couplings to $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$, $J/\psi N$ and $\eta_{c}N$ in Scenario A for a cutoff $\Lambda=1.5$ GeV. The bold figures are the experimental values used as inputs Aaij  _et al._ (2019b). The symbol - denotes that the $P_{c}$ state does not couple to that particular channel.  State |  Molecule |  $J^{P}$ |  Mass (MeV) |  $g_{P_{c}\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}}$ |  $g_{P_{c}J/\psi N}$ |  $g_{P_{c}\eta_{c}N}$
---|---|---|---|---|---|---
$P_{c1}$ |  $\bar{D}\Sigma_{c}$ |  $(1/2)^{-}$ |  4309.3+4.9$i$ |  2.16 |  0.31 |  0.53
$P_{c2}$ |  $\bar{D}\Sigma_{c}^{\ast}$ |  $(3/2)^{-}$ |  4372.2+4.8$i$ |  2.19 |  0.62 |  -
$P_{c3}$ |  $\bar{D}^{\ast}\Sigma_{c}$ |  $(1/2)^{-}$ |  4440.3+10.3$i$ |  2.60 |  0.83 |  0.29
$P_{c4}$ |  $\bar{D}^{\ast}\Sigma_{c}$ |  $(3/2)^{-}$ |  4457.3+3.2$i$ |  1.70 |  0.49 |  -
$P_{c5}$ |  $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ |  $(1/2)^{-}$ |  4502.7+14.0$i$ |  2.68 |  0.48 |  0.83
$P_{c6}$ |  $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ |  $(3/2)^{-}$ |  4510.5+7.2$i$ |  2.31 |  0.71 |  -
$P_{c7}$ |  $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ |  $(5/2)^{-}$ |  4522.8 |  1.49 |  - |  -
Table 4: Masses of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules as well as their couplings to $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$, $J/\psi N$ and $\eta_{c}N$ in Scenario B for a cutoff $\Lambda=1.5$ GeV. The bold figures are the experimental values used as inputs Aaij  _et al._ (2019b). The symbol - denotes that the $P_{c}$ state does not couple to that particular channel.  State |  Molecule |  $J^{P}$ |  Mass (MeV) |  $g_{P_{c}\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}}$ |  $g_{P_{c}J/\psi N}$ |  $g_{P_{c}\eta_{c}N}$
---|---|---|---|---|---|---
$P_{c1}$ |  $\bar{D}\Sigma_{c}$ |  $(1/2)^{-}$ |  4303.1+4.9$i$ |  2.41 |  0.31 |  0.54
$P_{c2}$ |  $\bar{D}\Sigma_{c}^{\ast}$ |  $(3/2)^{-}$ |  4366.3+3.2$i$ |  2.41 |  0.50 |  -
$P_{c3}$ |  $\bar{D}^{\ast}\Sigma_{c}$ |  $(1/2)^{-}$ |  4457.3+3.2$i$ |  1.71 |  0.46 |  0.16
$P_{c4}$ |  $\bar{D}^{\ast}\Sigma_{c}$ |  $(3/2)^{-}$ |  4440.3+10.3$i$ |  2.60 |  0.88 |  -
$P_{c5}$ |  $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ |  $(1/2)^{-}$ |  4523.7+3.7$i$ |  1.56 |  0.25 |  0.43
$P_{c6}$ |  $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ |  $(3/2)^{-}$ |  4515.9+3.0$i$ |  1.99 |  0.45 |  -
$P_{c7}$ |  $\bar{D}^{\ast}\Sigma_{c}^{\ast}$ |  $(5/2)^{-}$ |  4501.3 |  2.59 |  - |  -

With the parameters in Table 2, we search for poles corresponding to the seven
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules and determine their couplings
to the relevant channels. We present the results obtained in Scenario A and B
in Tables 3 and 4, respectively. Compared with the mass spectra of Ref. Liu
_et al._ (2019), the real part of all the poles remains similar, which
reconfirms that the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ channels play the
dominant role in dynamically generating the complete HQSS muiltplet of
hadronic molecules. Furthermore, the couplings of the seven states to
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ are consistent with those of Ref. Sakai
_et al._ (2019). Note that the couplings
$g_{P_{c}\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}}$ obtained in Table 2 are in
isospin basis. From them and assuming isospin symmetry we can derive the
couplings in particle basis, e.g.,
$g_{P_{c}{D}^{-(\ast)}\Sigma_{c}^{++(\ast)}}=\sqrt{\frac{2}{3}}g_{P_{c}\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}}$
and
$g_{P_{c}\bar{D}^{0(\ast)}\Sigma_{c}^{+(\ast)}}=\sqrt{\frac{1}{3}}g_{P_{c}\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}}$.
With the pole positions and couplings to their constituents determined, we can
now study the three-body decay widths of the pentaquark states.

Table 5: Partial decay widths (in units of MeV) of $P_{c}^{+}\to{D^{(\ast)-}}\Lambda_{c}^{+}\pi^{+}$ and $P_{c}^{+}\to\bar{D}^{(\ast)0}\Lambda_{c}^{+}\pi^{0}$ in Scenario A and Scenario B for a cutoff $\Lambda=1.5$ GeV.  Scenario |  A |  A |  B |  B
---|---|---|---|---
Mode |  ${D^{(\ast)-}}\Lambda_{c}^{+}\pi^{+}$ |  $\bar{D}^{(\ast)0}\Lambda_{c}^{+}\pi^{0}$ |  ${D^{(\ast)-}}\Lambda_{c}^{+}\pi^{+}$ |  $\bar{D}^{(\ast)0}\Lambda_{c}^{+}\pi^{0}$
$P_{c1}$ |  0.034 |  0.141 |  0.004 |  0.037
$P_{c2}$ |  2.085 |  2.166 |  1.468 |  1.479
$P_{c3}$ |  0.002 |  0.033 |  0.517 |  1.793
$P_{c4}$ |  0.170 |  0.591 |  0.001 |  0.011
$P_{c5}$ |  2.508 |  2.219 |  6.906 |  6.280
$P_{c6}$ |  3.087 |  2.758 |  3.866 |  3.529
$P_{c7}$ |  2.807 |  2.578 |  1.033 |  0.915

###  III.2 Decay widths of tree-level diagrams

For the decay modes of Fig. 1, we present the corresponding partial decay
widths in Table 5. The decay width of $P_{c}(4312)$ into
$\bar{D}\Lambda_{c}\pi$ is about hundreds and tens of keV in scenarios A and
B, respectively, which accounts for $0.4\%-2\%$ of its total width. The decay
width of $P_{c}(4457)$ into $\bar{D}\Lambda_{c}\pi$ is up to several MeV,
while for $P_{c}(4440)$ it is only tens of keV. The partial width of
$P_{c}(4457)$ accounts for tens of percent of its total width, while for
$P_{c}(4440)$ it accounts for less than one percent, which indicates that the
$\bar{D}\Lambda_{c}\pi$ mode is a good channel to detect $P_{c}(4457)$ and
verify the molecular nature of $P_{c}(4457)$. In addition, we predict the
partial decay widths of the other four molecules into
$\bar{D}^{(\ast)}\Lambda_{c}\pi$, which are about several MeV, some of them
are even up to tens of MeV, in agreement with the results of Ref. Lin and Zou
(2019). Therefore the $P_{c2,5,6,7}$ states can be observed in the
$\bar{D}^{\ast}\Lambda_{c}\pi$ final states in future experiments, especially
$P_{c7}$, which is particularly difficult to be detected in the $J/\psi p$
invariant mass distribution where only the $D-$wave $J/\psi p$ is allowed. One
should note that the partial decay widths of the seven
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules into
$\bar{D}^{(\ast)}\Lambda_{c}\pi$ in Scenario A and B are of the same order of
magnitude, which indicates that the tree-level decay modes cannot discriminate
the spins of $P_{c}(4440)$ and $P_{c}(4457)$.

###  III.3 Decay widths of triangle-loop diagrams

(a) $P_{c3}$-Scenario A

(b) $P_{c3}$-Scenario B

(c) $P_{c4}$-Scenario A

(d) $P_{c4}$-Scenario B

Figure 3: Partial decay widths of $P_{c3|c4}^{+}\to J/\psi n\pi^{+}$ and
$P_{c3|c4}^{+}\to J/\psi p\pi^{0}$ as a function of the cutoff
$\Lambda^{\prime}$ in the form factors (32). The results are represented by
the blue solid lines and green dot-dashed lines, respectively, which
correspond to the couplings determined for a cutoff $\Lambda=$1.5 GeV.

For the triangle-loop diagrams, we have introduced a form factor in Eq. (32)
to suppress the ultraviolate divergence. The parameter $\Lambda^{\prime}$ is
usually estimated by the relationship
$\Lambda^{\prime}=m_{E}+\alpha\Lambda_{QCD}$, where $m_{E}$ is the mass of the
exchanged particle, $\Lambda_{QCD}\sim 200-300$ MeV is the scale parameter of
quantum chromodynamics (QCD), and $\alpha$ is a dimensionless parameter of
order unity. Following our previous work, we vary $\alpha$ from 0.5 to 1.5
Ling  _et al._ (2021), and then $\Lambda^{\prime}$ varies from 2.0 to 2.2 GeV.

(a) $P_{c3}$-Scenario A

(b) $P_{c3}$-Scenario B

(c) $P_{c4}$-Scenario A

(d) $P_{c4}$-Scenario B

Figure 4: Partial decay widths of $P_{c3|c4}^{+}\to\eta_{c}n\pi^{+}$ and
$P_{c3|c4}^{+}\to\eta_{c}p\pi^{0}$ as a function of the cutoff
$\Lambda^{\prime}$ in the form factors (32). The results are represented by
the blue solid lines and green dot-dashed lines, respectively, which
correspond to the couplings determined for a cutoff $\Lambda=$1.5 GeV.

In Fig. 3, we plot the partial decay widths of $P_{c}(4440)$ and $P_{c}(4457)$
into $J/\psi N\pi$ as a function of the cutoff parameter $\Lambda^{\prime}$.
One can see that the widths are only several keV in both Scenario A and B,
which are weakly dependent on the variations of both the couplings and the
cutoff $\Lambda^{\prime}$. The widths in both scenarios are similar, which
indicates that one cannot discriminate the spins of $P_{c}(4440)$ and
$P_{c}(4457)$ in the $J/\psi N\pi$ channel, consistent with our previous work
Ling  _et al._ (2021). The partial decay width of $P_{c}(4457)$ into $J/\psi
N\pi$ account for only 0.1 percent of its total decay width, which is much
smaller than that decaying into $\bar{D}\Lambda_{c}\pi$.

(a) $P_{c5}$-Scenario A

(b) $P_{c5}$-Scenario B

(c) $P_{c6}$-Scenario A

(d) $P_{c6}$-Scenario B

(e) $P_{c7}$-Scenario A

(f) $P_{c7}$-Scenario B

Figure 5:  Partial decay widths of $P_{c5|c6|c7}^{+}\to J/\psi n\pi^{+}$ and
$P_{c5|c6|c7}^{+}\to J/\psi p\pi^{0}$ as a function of the cutoff
$\Lambda^{\prime}$ of the form factor (32). The results are represented by the
blue solid lines and green dot-dashed lines, respectively, which correspond to
the couplings determined for a cutoff $\Lambda=$1.5 GeV.

Apart from decaying into $J/\psi N\pi$, $P_{c}(4440)$ and $P_{c}(4457)$ can
decay into $\eta_{c}N\pi$ via $\bar{D}\Sigma_{c}$ rescattering to $\eta_{c}N$
if they have $J=1/2$. It is natural to expect that the partial decay widths of
$P_{c}(4440)$ and $P_{c}(4457)$ into $\eta_{c}N\pi$ are larger than those into
$J/\psi N\pi$ because $P_{c}(4312)$ couples more to $\eta_{c}N$ than to
$J/\psi N$. In Fig. 4, we plot the partial decay widths of $P_{c}(4440)$ and
$P_{c}(4457)$ into $\eta_{c}N\pi$ as a function of the cutoff parameter
$\Lambda^{\prime}$. They are tens of keV, which are much larger than those
into $J/\psi N\pi$ in Fig. 3. However, the widths in both Scenario A and B are
similar, which also cannot discriminate the spins of $P_{c}(4440)$ and
$P_{c}(4457)$. On the other hand, the branching ratios of $P_{c}(4440)$ and
$P_{c}(4457)$ decaying into $\eta_{c}N\pi$, $J/\psi N\pi$, and
$\bar{D}\Lambda_{c}\pi$ obtained in this work can be used to test the
molecular nature of $P_{c}(4440)$ and $P_{c}(4457)$ if these decay channels
are detected in future experiments.

We further predict the partial decay widths of $P_{c5}$, $P_{c6}$, and
$P_{c7}$ into $J/\psi N\pi$ in the triangle-loop mechanism. In Fig. 5, we plot
their widths as a function of the cutoff parameter $\Lambda^{\prime}$. The
partial decay widths of $P_{c5}$ and $P_{c6}$ in Scenario B are several keV,
while those in Scenario A are at the order of 1 keV. The partial decay widths
of $P_{c5}$ in Scenario A and B are more different than those of $P_{c6}$ due
to the larger difference in the phase space of $P_{c5}$ into $J/\psi N\pi$ as
shown in Table 3 and Table 4, similar to the case of $P_{c3}$. The partial
widths of $P_{c7}$ decaying into $J/\psi N\pi$ are less than 1 keV, while the
width in Scenario A is larger than that in Scenario B, similar to the case of
$P_{c4}$. The partial widths of $P_{c7}$ decaying into $J/\psi N\pi$ are much
smaller than those of $P_{c5}$ and $P_{c6}$, which are suppressed by the
higher spin of $P_{c7}$, the smaller phase space of $P_{c7}$ decaying into
$J/\psi N\pi$ in Scenario B, and the smaller coupling of $P_{c7}$ to
$\bar{D}^{\ast}\Sigma_{c}^{\ast}$ in Scenario A.

##  IV Summary and Discussion

Inspired by the discovery of $T_{cc}$ by the LHCb Collaboration in the $DD\pi$
three-body final state, we adopted the effective Lagrangian approach to
systematically study seven $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ hadronic
molecules decaying into three-body final states via two modes, tree level and
triangle loop. In the tree-level mode the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules decay via subsequent decays of
$\Sigma_{c}$ and $\Sigma_{c}^{\ast}$ into $\Lambda_{c}\pi$. In the triangle
mode the $\bar{D}^{(\ast)}\Sigma_{c}^{\ast}$ molecules decay into $J/\psi
N\pi$ and $\eta_{c}N\pi$ via a two-step process, i.e., first $\bar{D}^{\ast}$
decays into $\bar{D}\pi$, and then $\bar{D}\Sigma_{c}^{(\ast)}$ rescatters
into $J/\psi N$ and $\eta_{c}N$. The masses and widths of the
$\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ molecules and relevant couplings were
determined in the coupled-channel contact-range EFT approach.

Our results show that the partial decay widths of $P_{c2}$, $P_{c}(4457)$,
$P_{c5}$, $P_{c6}$ and $P_{c7}$ into $\bar{D}^{(\ast)}\Lambda_{c}\pi$, of the
order of several MeV, are much larger than those of $P_{c}(4312)$ and
$P_{c}(4440)$, and therefore are more accessible in future experiments. The
partial decay widths of $P_{c}(4440)$ and $P_{c}(4457)$ into $J/\psi N\pi$ and
$\eta_{c}N\pi$ are only several and tens of keV, respectively, both of which
are similar in scenarios A and B. We predicted the partial decay widths of
$P_{c5}$, $P_{c6}$, and $P_{c7}$ into $J/\psi N\pi$, among which the width of
$P_{c7}\to J/\psi N\pi$ is one order of magnitude smaller than those of
$P_{c5}$ and $P_{c6}$. Our results suggest that one should look for $P_{c7}$
in the $J/\psi N\pi$ and $\bar{D}^{\ast}\Lambda_{c}\pi$ invariant mass
distributions, while the latter is preferable. These three-body decay modes of
the pentaquark states are of great value to further observations of the
pentaquark states. In addition, those partial decay widths are helpful to test
their molecular nature.

It should be noted that although the predicted partial decay widths of
$P_{c2}$, $P_{c5}$, $P_{c6}$, $P_{c7}$ are all dependent on the adopted value
for the coupling $g$, our qualitative conclusions should be relatively robust,
unless HQSS is broken much strongly than naively anticipated. As a result, the
present study of three-body decay modes is expected to stimulate future
experimental searches for the known and predicted
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecules.

##  V Acknowledgments

MZL thank Jun-Xu Lu and Ya-Wen Pan for useful discussions. This work is
supported in part by the National Natural Science Foundation of China under
Grants No.11975041, No.11735003, 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.

##  Appendix A Invariant amplitudes for the tree-level and triangle-loop
processes

The tree-level amplitudes of the $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$
hadronic molecules decaying into $\bar{D}^{(\ast)}\Lambda_{c}\pi$ read

$\displaystyle\mathcal{M}_{P_{c1}\to\bar{D}\Lambda_{c}\pi}$ $\displaystyle=$
$\displaystyle
i\frac{g_{P_{c1}\bar{D}\Sigma_{c}}g_{\Sigma_{c}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1}){p\\!\\!\\!/}_{2}\gamma_{5}\frac{1}{{k\\!\\!\\!/}_{1}-m_{\Sigma_{c}}}u(k_{0}),$
(18) $\displaystyle\mathcal{M}_{P_{c2}\to\bar{D}\Lambda_{c}\pi}$
$\displaystyle=$ $\displaystyle
i\frac{g_{P_{c2}\bar{D}\Sigma_{c}^{\ast}}g_{\Sigma_{c}^{\ast}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1}){p}_{2\mu}\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}P^{\mu\nu}(k_{1})u_{\nu}(k_{0}),$
(19) $\displaystyle\mathcal{M}_{P_{c3}\to\bar{D}^{\ast}\Lambda_{c}\pi}$
$\displaystyle=$ $\displaystyle
i\frac{g_{P_{c3}\bar{D}^{\ast}\Sigma_{c}}g_{\Sigma_{c}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1}){p\\!\\!\\!/}_{2}\gamma_{5}\frac{1}{{k\\!\\!\\!/}_{1}-m_{\Sigma_{c}}}\widetilde{\gamma}^{\mu}\gamma_{5}\varepsilon_{\mu}(p_{3})u(k_{0}),$
(20) $\displaystyle\mathcal{M}_{P_{c4}\to\bar{D}^{\ast}\Lambda_{c}\pi}$
$\displaystyle=$ $\displaystyle
i\frac{g_{P_{c4}\bar{D}^{\ast}\Sigma_{c}}g_{\Sigma_{c}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1}){p\\!\\!\\!/}_{2}\gamma_{5}\frac{1}{{k\\!\\!\\!/}_{1}-m_{\Sigma_{c}}}\varepsilon_{\mu}(p_{3})u^{\mu}(k_{0}),$
(21) $\displaystyle\mathcal{M}_{P_{c5}\to\bar{D}^{\ast}\Lambda_{c}\pi}$
$\displaystyle=$ $\displaystyle
i\frac{g_{P_{c5}\bar{D}^{\ast}\Sigma_{c}^{\ast}}g_{\Sigma_{c}^{\ast}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1})p_{2\mu}\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}P^{\mu\nu}(k_{1})\varepsilon_{\nu}(p_{3})u(k_{0}),$
(22) $\displaystyle\mathcal{M}_{P_{c6}\to\bar{D}^{\ast}\Lambda_{c}\pi}$
$\displaystyle=$ $\displaystyle
i\frac{g_{P_{c6}\bar{D}^{\ast}\Sigma_{c}^{\ast}}g_{\Sigma_{c}^{\ast}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1})p_{2\mu}\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}P^{\mu\nu}(k_{1})\varepsilon_{\rho}(p_{3})\gamma_{5}\widetilde{\gamma}^{\rho}u_{\nu}(k_{0}),$
(23) $\displaystyle\mathcal{M}_{P_{c7}\to\bar{D}^{\ast}\Lambda_{c}\pi}$
$\displaystyle=$ $\displaystyle
i\frac{g_{P_{c7}\bar{D}^{\ast}\Sigma_{c}^{\ast}}g_{\Sigma_{c}^{\ast}\Lambda_{c}\pi}}{f_{\pi}}~{}\bar{u}(p_{1})p_{2\mu}\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}P^{\mu\nu}(k_{1})\varepsilon_{\rho}(p_{3}){u_{\nu}}^{\rho}(k_{0}),$
(24)

where $k_{0}$ and $k_{1}$ denote the momenta of initial states and
intermediate states, and the momenta of $\Lambda_{c}$, $\pi$, and
$\bar{D}^{(\ast)}$ are represented by $p_{1}$, $p_{2}$, and $p_{3}$,
respectively. $\bar{u}(p_{1})$ and $u(k_{0})$ denote the final and initial
spinor wave functions, respectively.
$P^{\mu\nu}(p)=g^{\mu\nu}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}-\frac{\gamma^{\nu}p^{\mu}-\gamma^{\mu}p^{\nu}}{3m}-\frac{2p^{\mu}p^{\nu}}{3m^{2}}$
denotes the propagator of a massive particle of spin $3/2$.

The amplitudes of the $\bar{D}^{\ast}\Sigma_{c}$ molecules decaying into
$\eta_{c}N\pi$ read

$\displaystyle i\mathcal{M}_{1/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c3}\bar{D}^{\ast}\Sigma_{c}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})T_{\bar{D}\Sigma_{c}\to\eta_{c}N}(\sqrt{s})\frac{1}{{/\\!\\!\\!k}_{1}-m_{\Sigma_{c}}}$
(25)
$\displaystyle\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}\widetilde{\gamma}_{\beta}\gamma_{5}u(k_{0})F(q^{2}),$
$\displaystyle i\mathcal{M}_{3/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c4}\bar{D}^{\ast}\Sigma_{c}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})T_{\bar{D}\Sigma_{c}\to\eta_{c}N}(\sqrt{s})\frac{1}{{/\\!\\!\\!k}_{1}-m_{\Sigma_{c}}}$
(26)
$\displaystyle\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}u_{\beta}(k_{0})F(q^{2}),$

and the amplitudes of the $\bar{D}^{\ast}\Sigma_{c}^{(\ast)}$ molecules
decaying into $J/\psi N\pi$ read

$\displaystyle i\mathcal{M}_{1/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c3}\bar{D}^{\ast}\Sigma_{c}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})\varepsilon_{\mu}(p_{2})\gamma^{\mu}\gamma_{5}T_{\bar{D}\Sigma_{c}\to
J/\psi N}(\sqrt{s})\frac{1}{{/\\!\\!\\!k}_{1}-m_{\Sigma_{c}}}$ (27)
$\displaystyle\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}\widetilde{\gamma}_{\beta}\gamma_{5}u(k_{0})F(q^{2}),$
$\displaystyle i\mathcal{M}_{3/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c4}\bar{D}^{\ast}\Sigma_{c}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})\varepsilon_{\mu}(p_{2})\gamma^{\mu}\gamma_{5}T_{\bar{D}\Sigma_{c}\to
J/\psi N}(\sqrt{s})\frac{1}{{/\\!\\!\\!k}_{1}-m_{\Sigma_{c}}}$ (28)
$\displaystyle\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}u_{\beta}(k_{0})F(q^{2}),$
$\displaystyle i\mathcal{M}_{1/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c5}\bar{D}^{\ast}\Sigma_{c}^{\ast}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})T_{\bar{D}\Sigma_{c}^{\ast}\to
J/\psi N}(\sqrt{s})\varepsilon_{\nu}(p_{2})$ (29)
$\displaystyle\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}{P^{\nu}}_{\beta}(k_{1})\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}u(k_{0})F(q^{2}),$
$\displaystyle i\mathcal{M}_{3/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c6}\bar{D}^{\ast}\Sigma_{c}^{\ast}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})T_{\bar{D}\Sigma_{c}^{\ast}\to
J/\psi N}(\sqrt{s})\varepsilon_{\nu}(p_{2})$ (30)
$\displaystyle\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}P^{\nu\rho}(k_{1})\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}\gamma_{5}\widetilde{\gamma}_{\beta}u_{\rho}(k_{0})F(q^{2}),$
$\displaystyle i\mathcal{M}_{5/2}$ $\displaystyle=$ $\displaystyle
g_{P_{c7}\bar{D}^{\ast}\Sigma_{c}^{\ast}}g_{D^{\ast}D\pi}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{u}(p_{1})T_{\bar{D}\Sigma_{c}^{\ast}\to
J/\psi N}(\sqrt{s})\varepsilon_{\nu}(p_{2})$ (31)
$\displaystyle\frac{{k\\!\\!\\!/}_{1}+m_{\Sigma_{c}^{\ast}}}{k_{1}^{2}-m_{\Sigma_{c}^{\ast}}^{2}+im_{\Sigma_{c}^{\ast}}\Gamma_{\Sigma_{c}^{\ast}}}P^{\nu\rho}(k_{1})\frac{1}{q^{2}-m_{\bar{D}}^{2}}p_{3\alpha}\frac{-g^{\alpha\beta}+\frac{k_{2}^{\alpha}k_{2}^{\beta}}{m_{\bar{D}^{\ast}}^{2}}}{k_{2}^{2}-m_{\bar{D}^{\ast}}^{2}}u_{\beta\rho}(k_{0})F(q^{2}),$

where $q$, $k_{0}$, $k_{1}$, $k_{2}$, $p_{1}$, $p_{2}$ and $p_{3}$ represent
the momenta of $\bar{D}$, $P_{c}$, $\Sigma_{c}^{(\ast)}$, $\bar{D}^{\ast}$,
$N$, $J/\psi(\eta_{c})$ and $\pi$, respectively, and
$T_{\bar{D}\Sigma_{c}^{(\ast)}-J/\psi(\eta_{c})N}$ represent the potentials of
inelastic scattering. To avoid the divergence induced by the loop function, we
have introduced a form factor of the form

$F(q^{2})=\left(\frac{\Lambda^{\prime 2}-m_{\bar{D}}^{2}}{\Lambda^{\prime
2}-q^{2}}\right)^{2}.$ (32)

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