# Hidden-charm pentaquark states through the current algebra:
From their productions to decays

Hua-Xing Chen<EMAIL_ADDRESS>School of Physics, Southeast University,
Nanjing 210094, China

###### Abstract

There may exist seven $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular
states. We construct their corresponding interpolating currents, and calculate
their masses and decay constants using QCD sum rules. Based on these results,
we calculate their relative production rates in $\Lambda_{b}^{0}$ decays
through the current algebra, i.e., $\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}K^{-}):\mathcal{B}(\Lambda_{b}^{0}\to P_{c}^{\prime}K^{-})$ with $P_{c}$
and $P_{c}^{\prime}$ two different states. We also study their decay
properties through the Fierz rearrangement, and further calculate these ratios
in the $J/\psi p$ mass spectrum, i.e., $\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}K^{-}\to J/\psi pK^{-}):\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{\prime}K^{-}\to J/\psi pK^{-})$. Our results suggest that the
$\bar{D}^{*}\Sigma_{c}^{*}$ molecular states of $J^{P}=1/2^{-}$ and $3/2^{-}$
are possible to be observed in future experiments.

exotic hadron, pentaquark state, hadronic molecule, interpolating current, QCD
sum rules, current algebra, Fierz rearrangement

##  I Introduction

Since the discovery of the $X(3872)$ by Belle in 2003 Choi:2003ue , many
charmonium-like $XYZ$ states were discovered in the past two decades pdg .
Some of these structures may contain four quarks, $\bar{c}c\bar{q}q$
($q=u/d$), so they are good candidates of hidden-charm tetraquark states.

In recent years the LHCb Collaboration continuously observed as many as five
interesting exotic structures:

  * •

In 2015 LHCb observed two structures $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$
in the $J/\psi p$ invariant mass spectrum of the $\Lambda_{b}^{0}\to J/\psi
pK^{-}$ decays Aaij:2015tga :

$\displaystyle P_{c}(4380)^{+}:$ $\displaystyle M=$ $\displaystyle 4380\pm
8\pm 29\mbox{ MeV}\,,$ $\displaystyle\Gamma=$ $\displaystyle 205\pm 18\pm
86\mbox{ MeV}\,,$ $\displaystyle P_{c}(4450)^{+}:$ $\displaystyle M=$
$\displaystyle 4449.8\pm 1.7\pm 2.5\mbox{ MeV}\,,$ $\displaystyle\Gamma=$
$\displaystyle 39\pm 5\pm 19\mbox{ MeV}\,.$

This observation is supported by the later LHCb experiment investigating the
$J/\psi p$ invariant mass spectrum of the $\Lambda_{b}^{0}\to J/\psi p\pi^{-}$
decays Aaij:2016ymb .

  * •

In 2019 LHCb observed a new structure $P_{c}(4312)^{+}$ and further separated
the $P_{c}(4450)^{+}$ into two substructures $P_{c}(4440)^{+}$ and
$P_{c}(4457)^{+}$, still in the $J/\psi p$ invariant mass spectrum of the
$\Lambda_{b}^{0}\to J/\psi pK^{-}$ decays Aaij:2019vzc :

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

  * •

In 2020 LHCb reported the evidence of a hidden-charm pentaquark state with
strangeness, $P_{cs}(4459)^{0}$, in the $J/\psi\Lambda$ invariant mass
spectrum of the $\Xi_{b}^{-}\to J/\psi\Lambda K^{-}$ decays lhcb :

$\displaystyle P_{cs}(4459)^{0}:$ $\displaystyle M=$ $\displaystyle 4458.8\pm
2.9^{+4.7}_{-1.1}\mbox{ MeV}\,,$ $\displaystyle\Gamma=$ $\displaystyle 17.3\pm
6.5^{+8.0}_{-5.7}\mbox{ MeV}\,.$

These structures contain at least five quarks, $\bar{c}cuud$ or $\bar{c}cuds$,
so they are perfect candidates of hidden-charm pentaquark states. The
charmonium-like $XYZ$ and hidden-charm pentaquark states have attracted lots
of attentions, and their studies have significantly improved our understanding
of the non-perturbative behaviors of the strong interaction at the low energy
region Chen:2016qju ; Liu:2019zoy ; Chen:2022asf ; Lebed:2016hpi ;
Esposito:2016noz ; Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm ;
Karliner:2017qhf ; Brambilla:2019esw ; Guo:2019twa ; Meng:2022ozq .

To understand the above $P_{c}$ and $P_{cs}$ states, various theoretical
interpretations were proposed, such as loosely-bound hadronic molecular states
Chen:2019bip ; Chen:2019asm ; Liu:2019tjn ; He:2019ify ; Huang:2019jlf ;
Guo:2019kdc ; Fernandez-Ramirez:2019koa ; Xiao:2019aya ; Meng:2019ilv ;
Wu:2019adv ; Yamaguchi:2019seo ; Valderrama:2019chc ; Liu:2019zvb ;
Burns:2019iih ; Wang:2019ato ; Gutsche:2019mkg ; Du:2019pij ; Xiao:2020frg ;
Ozdem:2018qeh ; Wang:2019hyc ; Zhang:2019xtu ; Azizi:2016dhy , tightly-bound
compact pentaquark states Maiani:2015vwa ; Lebed:2015tna ; Stancu:2019qga ;
Giron:2019bcs ; Ali:2019npk ; Weng:2019ynv ; Eides:2019tgv ; Cheng:2019obk ;
Ali:2019clg ; Wang:2019got ; Wang:2020eep , and kinematical effects
Guo:2015umn ; Liu:2015fea ; Bayar:2016ftu ; Kuang:2020bnk , etc. Especially,
the three narrow states $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$ are just below the $\bar{D}\Sigma_{c}$ and
$\bar{D}^{*}\Sigma_{c}$ thresholds, so it is natural to explain them as the
$\bar{D}^{(*)}\Sigma_{c}$ hadronic molecular states, whose existence had been
predicted in Refs. Wu:2010jy ; Wang:2011rga ; Yang:2011wz ; Karliner:2015ina ;
Wu:2012md  before the LHCb experiment performed in 2015 Aaij:2015tga ; the
other narrow state $P_{cs}(4459)^{0}$ is just below the $\bar{D}^{*}\Xi_{c}$
threshold, so it is natural to explain it as the $\bar{D}^{*}\Xi_{c}$
molecular state Chen:2020uif ; Peng:2020hql .

However, these exotic structures were only observed by LHCb Aaij:2015tga ;
Aaij:2016ymb ; Aaij:2019vzc ; lhcb . It is crucial to search for their partner
states as well as some other potential decay channels, in order to further
understand their nature. There have been some theoretical studies on this
subject, using effective approaches Guo:2019fdo ; Xiao:2019mst ; Cao:2019kst ;
Lin:2019qiv , the quark interchange model Wang:2019spc ; Dong:2020nwk , the
heavy quark symmetry Voloshin:2019aut ; Sakai:2019qph , and QCD sum rules
Xu:2019zme , etc. We refer to reviews Chen:2016qju ; Liu:2019zoy ;
Chen:2022asf ; Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ; Ali:2017jda ;
Olsen:2017bmm ; Karliner:2017qhf ; Brambilla:2019esw ; Guo:2019twa ;
Meng:2022ozq  and references therein for detailed discussions.

In this paper we shall systematically investigate hidden-charm pentaquark
states as $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular states through
their corresponding hidden-charm pentaquark interpolating currents. We shall
systemically construct all the relevant currents, and apply the method of QCD
sum rules to calculate their masses and decay constants. The obtained results
will be used to further study their production and decay properties.

Our strategy is quite straightforward. Firstly, we construct a hidden-charm
pentaquark current, such as

$\displaystyle\sqrt{2}\xi_{1}(x)$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}(x)\gamma_{5}d_{b}(x)]$
$\displaystyle~{}~{}~{}~{}~{}\times[\epsilon^{cde}u_{c}^{T}(x)\mathbb{C}\gamma_{\mu}u_{d}(x)\gamma^{\mu}\gamma_{5}c_{e}(x)]\,,$

where $a\cdots e$ are color indices. It is the current best coupling to the
$D^{-}\Sigma_{c}^{++}$ molecular state of $J^{P}=1/2^{-}$, through

$\langle 0|\xi_{1}|D^{-}\Sigma_{c}^{++};1/2^{-}(q)\rangle=f_{1}u(q)\,,$ (8)

where $u(q)$ is the Dirac spinor of $|D^{-}\Sigma_{c}^{++};1/2^{-}\rangle$.
Its decay constant $f_{1}$ can be calculated using QCD sum rules.

Secondly, we investigate the three-body $\Lambda_{b}^{0}\to J/\psi pK^{-}$
decays. The total quark content of the final states is $udc\bar{c}s\bar{u}u$,
where the intermediate states $D^{(*)-}\Sigma_{c}^{(*)++}K^{-}$ can be
produced. We apply the Fierz rearrangement to carefully examine the
combination of these seven quarks, from which we select the current $\xi_{1}$
and evaluate the relative production rate of
$|D^{-}\Sigma_{c}^{++};1/2^{-}\rangle$.

Thirdly, we apply the Fierz rearrangement of the Dirac and color indices to
transform the current $\xi_{1}$ into

$\sqrt{2}\xi_{1}\rightarrow{1\over
6}~{}[\bar{c}_{a}\gamma_{5}c_{a}]~{}N-{1\over
12}~{}[\bar{c}_{a}\gamma_{\mu}c_{a}]~{}\gamma^{\mu}\gamma_{5}N+\cdots\,,$ (9)

where
$N=\epsilon^{abc}(u_{a}^{T}\mathbb{C}d_{b})\gamma_{5}u_{c}-\epsilon^{abc}(u_{a}^{T}\mathbb{C}\gamma_{5}d_{b})u_{c}$
is the Ioffe’s light baryon field well coupling to the proton Ioffe:1981kw ;
Ioffe:1982ce ; Espriu:1983hu . Accordingly, $\xi_{1}$ couples to the
$\eta_{c}p$ and $J/\psi p$ channels simultaneously:

$\displaystyle\langle 0|\xi_{1}|\eta_{c}p\rangle$ $\displaystyle\approx$
$\displaystyle{\sqrt{2}\over 12}\langle
0|\bar{c}_{a}\gamma_{5}c_{a}|\eta_{c}\rangle~{}\langle 0|N|p\rangle+\cdots\,,$
(10) $\displaystyle\langle 0|\xi_{1}|\psi p\rangle$ $\displaystyle\approx$
$\displaystyle-{\sqrt{2}\over 24}\langle
0|\bar{c}_{a}\gamma_{\mu}c_{a}|\psi\rangle~{}\gamma^{\mu}\gamma_{5}\langle
0|N|p\rangle+\cdots\,.$

We can use these two equations to straightforwardly calculate the relative
branching ratio of the $|D^{-}\Sigma_{c}^{++};1/2^{-}\rangle$ decay into
$\eta_{c}p$ to its decay into $J/\psi p$ Yu:2017zst . We refer to Ref.
Chen:2020pac  for detailed discussions. There we have applied the same method
to study decay properties of $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$ as $\bar{D}^{(*)}\Sigma_{c}$ molecular states, and in this
paper we shall apply it to study decay properties of
$\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular states.

This paper is organized as follows. In Sec. II we systematically construct the
hidden-charm pentaquark currents corresponding to
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular states. We use them to
perform QCD sum rule analyses in Sec. III, and calculate their masses and
decay constants. The obtained results are used in Sec. IV to study their
productions in $\Lambda_{b}^{0}$ decays through the current algebra. In Sec. V
we use the Fierz rearrangement of the Dirac and color indices to study decay
properties of $\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular states, and calculate
some of their relative branching ratios. The obtained results are summarized
and discussed in Sec. VI.

##  II Hidden-charm pentaquark interpolating currents

(a) $\theta(x)$ currents: $[\bar{c}c][uud]$ (b) $\eta(x)$ currents:
$[\bar{c}u][udc]$

(c) $\xi(x)$ currents: $[\bar{c}d][uuc]$

Figure 1: Three types of hidden-charm pentaquark interpolating currents, that
are the $\theta(x)$, $\eta(x)$, and $\xi(x)$ currents. Quarks are shown in
red/green/blue color, and antiquarks are shown in cyan/magenta/yellow color.
Taken from Ref. Chen:2020pac .

In this section we use the $\bar{c}$, $c$, $u$, $u$, and $d$ ($q=u/d$) quarks
to construct hidden-charm pentaquark interpolating currents. We consider the
following three types of currents:

$\displaystyle\theta(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{a}(x)\Gamma^{\theta}_{1}c_{b}(x)]~{}\Big{[}[q^{T}_{c}(x)\mathbb{C}\Gamma^{\theta}_{2}q_{d}(x)]~{}\Gamma^{\theta}_{3}q_{e}(x)\Big{]}\,,$
$\displaystyle\eta(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{a}(x)\Gamma^{\eta}_{1}u_{b}(x)]~{}\Big{[}[u^{T}_{c}(x)\mathbb{C}\Gamma^{\eta}_{2}d_{d}(x)]~{}\Gamma^{\eta}_{3}c_{e}(x)\Big{]}\,,$
$\displaystyle\xi(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{a}(x)\Gamma^{\xi}_{1}d_{b}(x)]~{}\Big{[}[u^{T}_{c}(x)\mathbb{C}\Gamma^{\xi}_{2}u_{d}(x)]~{}\Gamma^{\xi}_{3}c_{e}(x)\Big{]}\,,$

where $a\cdots e$ are color indices, $\Gamma_{1/2/3}^{\theta/\eta/\xi}$ are
Dirac matrices, and $\mathbb{C}=i\gamma_{2}\gamma_{0}$ is the charge-
conjugation operator. We illustrate them in Fig. 1. These three configurations
can be related together by the Fierz rearrangement in the Lorentz space as
well as the color rearrangement:

$\delta^{ab}\epsilon^{cde}=\delta^{ac}\epsilon^{bde}+\delta^{ad}\epsilon^{cbe}+\delta^{ae}\epsilon^{cdb}\,.$
(12)

This will be detailedly discussed in Sec. V. There we shall construct the
$\theta(x)$ currents by combining charmonium operators and light baryon
fields.

In this section we construct the $\eta(x)$ and $\xi(x)$ currents, and further
use them to construct currents corresponding to
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular states. To do this we
combine charmed meson operators and charmed baryon fields. There are five
independent charmed meson operators:

$\displaystyle\bar{c}_{a}q_{a}\,[0^{+}]\,,\,\bar{c}_{a}\gamma_{5}q_{a}\,[0^{-}]\,,$
(13)
$\displaystyle\bar{c}_{a}\gamma_{\mu}q_{a}\,[1^{-}]\,,\,\bar{c}_{a}\gamma_{\mu}\gamma_{5}q_{a}\,[1^{+}]\,,\,\bar{c}_{a}\sigma_{\mu\nu}q_{a}\,[1^{\pm}]\,.$

Besides, there exists another one,
$\bar{c}_{d}\sigma_{\mu\nu}\gamma_{5}q_{d}$, but it is related to
$\bar{c}_{d}\sigma_{\mu\nu}q_{d}$ through

$\sigma_{\mu\nu}\gamma_{5}={i\over
2}\epsilon_{\mu\nu\rho\sigma}\sigma^{\rho\sigma}\,.$ (14)

Especially, we need the $J^{P}=0^{-}$ and $1^{-}$ ones to construct the
$\eta(x)$ and $\xi(x)$ currents, which couple to the ground-state charmed
mesons $\mathcal{D}=D/D^{*}$:

$\displaystyle J_{D}$ $\displaystyle=$
$\displaystyle\bar{c}_{a}\gamma_{5}q_{a}\,,$ (15) $\displaystyle J_{D^{*}}$
$\displaystyle=$ $\displaystyle\bar{c}_{a}\gamma_{\mu}q_{a}\,.$

Charmed baryon fields have been systematically constructed and studied in
Refs. Liu:2007fg ; Chen:2017sci ; Cui:2019dzj ; Dmitrasinovic:2020wye  using
the method of QCD sum rules Shifman:1978bx ; Reinders:1984sr  within the heavy
quark effective theory Grinstein:1990mj ; Eichten:1989zv ; Falk:1990yz . In
this paper we need the following charmed baryon fields $J_{\mathcal{B}}$,
which couple to the ground-state charmed baryons
$\mathcal{B}=\Lambda_{c}/\Sigma_{c}/\Sigma_{c}^{*}$:

$\displaystyle J_{\Lambda_{c}^{+}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}[u_{a}^{T}\mathbb{C}\gamma_{5}d_{b}]c_{c}\,,$
$\displaystyle\sqrt{2}J_{\Sigma_{c}^{++}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}[u_{a}^{T}\mathbb{C}\gamma_{\mu}u_{b}]\gamma^{\mu}\gamma_{5}c_{c}\,,$
$\displaystyle J_{\Sigma_{c}^{+}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}[u_{a}^{T}\mathbb{C}\gamma_{\mu}d_{b}]\gamma^{\mu}\gamma_{5}c_{c}\,,$
$\displaystyle\sqrt{2}J_{\Sigma_{c}^{0}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}[d_{a}^{T}\mathbb{C}\gamma_{\mu}d_{b}]\gamma^{\mu}\gamma_{5}c_{c}\,,$
(16) $\displaystyle\sqrt{2}J^{\alpha}_{\Sigma_{c}^{*++}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}P_{3/2}^{\alpha\mu}[u_{a}^{T}\mathbb{C}\gamma_{\mu}u_{b}]c_{c}\,,$
$\displaystyle J^{\alpha}_{\Sigma_{c}^{*+}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}P_{3/2}^{\alpha\mu}[u_{a}^{T}\mathbb{C}\gamma_{\mu}d_{b}]c_{c}\,,$
$\displaystyle\sqrt{2}J^{\alpha}_{\Sigma_{c}^{*0}}$ $\displaystyle=$
$\displaystyle\epsilon^{abc}P_{3/2}^{\alpha\mu}[d_{a}^{T}\mathbb{C}\gamma_{\mu}d_{b}]c_{c}\,.$

Here $P_{3/2}^{\mu\nu}$ is the spin-3/2 projection operator

$P_{3/2}^{\mu\nu}=g^{\mu\nu}-{1\over 4}\gamma^{\mu}\gamma^{\nu}\,.$ (17)

In the molecular picture the $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$ are usually interpreted as the $\bar{D}\Sigma_{c}$ and
$\bar{D}^{*}\Sigma_{c}$ hadronic molecular states Wu:2012md ; Chen:2019asm ;
Liu:2019tjn . Their relevant currents have been constructed in Ref.
Chen:2020pac . In this paper we further construct the $\bar{D}\Sigma_{c}^{*}$
and $\bar{D}^{*}\Sigma_{c}^{*}$ currents. We summarize all of them here, for
completeness.

There can exist altogether seven $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic
molecular states, that are $\bar{D}\Sigma_{c}$ of $J^{P}={1\over 2}^{-}$,
$\bar{D}^{*}\Sigma_{c}$ of $J^{P}={1\over 2}^{-}/{3\over 2}^{-}$,
$\bar{D}\Sigma_{c}^{*}$ of $J^{P}={3\over 2}^{-}$, and
$\bar{D}^{*}\Sigma_{c}^{*}$ of $J^{P}={1\over 2}^{-}/{3\over 2}^{-}/{5\over
2}^{-}$:

$\displaystyle|\bar{D}\Sigma_{c};{1/2}^{-};\theta\rangle$ (18)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{0}\Sigma_{c}^{+};1/2^{-}\rangle+\sin\theta~{}|D^{-}\Sigma_{c}^{++};1/2^{-}\rangle\,,$
$\displaystyle|\bar{D}^{*}\Sigma_{c};{1/2}^{-};\theta\rangle$ (19)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{*0}\Sigma_{c}^{+};1/2^{-}\rangle+\sin\theta~{}|D^{*-}\Sigma_{c}^{++};1/2^{-}\rangle\,,$
$\displaystyle|\bar{D}^{*}\Sigma_{c};{3/2}^{-};\theta\rangle$ (20)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{*0}\Sigma_{c}^{+};3/2^{-}\rangle+\sin\theta~{}|D^{*-}\Sigma_{c}^{++};3/2^{-}\rangle\,,$
$\displaystyle|\bar{D}\Sigma_{c}^{*};{3/2}^{-};\theta\rangle$ (21)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle+\sin\theta~{}|D^{-}\Sigma_{c}^{*++};3/2^{-}\rangle\,,$
$\displaystyle|\bar{D}^{*}\Sigma_{c}^{*};{1/2}^{-};\theta\rangle$ (22)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle+\sin\theta~{}|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\,,$
$\displaystyle|\bar{D}^{*}\Sigma_{c}^{*};{3/2}^{-};\theta\rangle$ (23)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle+\sin\theta~{}|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\,,$
$\displaystyle|\bar{D}^{*}\Sigma_{c}^{*};{5/2}^{-};\theta\rangle$ (24)
$\displaystyle~{}~{}~{}~{}=\cos\theta~{}|\bar{D}^{*0}\Sigma_{c}^{*+};5/2^{-}\rangle+\sin\theta~{}|D^{*-}\Sigma_{c}^{*++};5/2^{-}\rangle\,,$

where $\theta$ is an isospin parameter, satisfying $\theta=-55^{\rm o}$ for
$I=1/2$ and $\theta=35^{\rm o}$ for $I=3/2$. In the present study we shall
concentrate on the former $I=1/2$ states, so that we can simplify the
notations to be

$\displaystyle|\bar{D}^{(*)}\Sigma_{c}^{(*)};J^{P}\rangle$ $\displaystyle=$
$\displaystyle{\sqrt{1/3}}~{}|\bar{D}^{(*)0}\Sigma_{c}^{(*)+};J^{P}\rangle-{\sqrt{2/3}}~{}|D^{(*)-}\Sigma_{c}^{(*)++};J^{P}\rangle.$

Their relevant interpolating currents are:

$J_{i}=\cos\theta~{}\eta_{i}+\sin\theta~{}\xi_{i}\,,$ (26)

where

$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{5}u_{b}]~{}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}d_{d}\gamma^{\mu}\gamma_{5}c_{e}]$
(27) $\displaystyle=$ $\displaystyle\bar{D}^{0}~{}\Sigma_{c}^{+}\,,$
$\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}\gamma^{\nu}\gamma_{5}~{}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}d_{d}\gamma^{\mu}\gamma_{5}c_{e}]$
(28) $\displaystyle=$
$\displaystyle\bar{D}^{*0}_{\nu}~{}\gamma^{\nu}\gamma_{5}~{}\Sigma_{c}^{+}\,,$
$\displaystyle\eta_{3}^{\alpha}$ $\displaystyle=$ $\displaystyle
P_{3/2}^{\alpha\nu}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}d_{d}\gamma^{\mu}\gamma_{5}c_{e}]$
(29) $\displaystyle=$ $\displaystyle
P_{3/2}^{\alpha\nu}~{}\bar{D}^{*0}_{\nu}~{}\Sigma_{c}^{+}\,,$
$\displaystyle\eta_{4}^{\alpha}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{5}u_{b}]~{}P_{3/2}^{\alpha\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}d_{d}c_{e}]$
(30) $\displaystyle=$ $\displaystyle\bar{D}^{0}~{}\Sigma_{c}^{*+;\alpha}\,,$
$\displaystyle\eta_{5}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}P_{3/2}^{\nu\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}d_{d}c_{e}]$
(31) $\displaystyle=$
$\displaystyle\bar{D}^{*0}_{\nu}~{}\Sigma_{c}^{*+;\nu}\,,$
$\displaystyle\eta_{6}^{\alpha}$ $\displaystyle=$
$\displaystyle[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}P_{3/2}^{\alpha\rho}~{}\gamma^{\nu}\gamma_{5}~{}P^{3/2}_{\rho\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma^{\mu}d_{d}c_{e}]$
(32) $\displaystyle=$
$\displaystyle\bar{D}^{*0}_{\nu}~{}P_{3/2}^{\alpha\rho}~{}\gamma^{\nu}\gamma_{5}~{}\Sigma_{c;\rho}^{*+}\,,$
$\displaystyle\eta_{7}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\nu\rho}_{5/2}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}u_{b}]~{}P^{3/2}_{\rho\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma^{\mu}d_{d}c_{e}]$
(33) $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\nu\rho}_{5/2}~{}\bar{D}^{*0}_{\nu}~{}\Sigma_{c;\rho}^{*+}\,,$

and

$\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}[\delta^{ab}\bar{c}_{a}\gamma_{5}d_{b}]~{}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}u_{d}\gamma^{\mu}\gamma_{5}c_{e}]$
(34) $\displaystyle=$ $\displaystyle D^{-}~{}\Sigma_{c}^{++}\,,$
$\displaystyle\xi_{2}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}d_{b}]~{}\gamma^{\nu}\gamma_{5}~{}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}u_{d}\gamma^{\mu}\gamma_{5}c_{e}]$
(35) $\displaystyle=$ $\displaystyle
D^{*-}_{\nu}~{}\gamma^{\nu}\gamma_{5}~{}\Sigma_{c}^{++}\,,$
$\displaystyle\xi_{3}^{\alpha}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}P_{3/2}^{\alpha\nu}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}d_{b}]~{}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}u_{d}\gamma^{\mu}\gamma_{5}c_{e}]$
(36) $\displaystyle=$ $\displaystyle
P_{3/2}^{\alpha\nu}~{}D^{*-}_{\nu}~{}\Sigma_{c}^{++}\,,$
$\displaystyle\xi_{4}^{\alpha}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}[\delta^{ab}\bar{c}_{a}\gamma_{5}d_{b}]~{}P_{3/2}^{\alpha\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}u_{d}c_{e}]$
(37) $\displaystyle=$ $\displaystyle D^{-}~{}\Sigma_{c}^{*++;\alpha}\,,$
$\displaystyle\xi_{5}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}d_{b}]~{}P_{3/2}^{\nu\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma_{\mu}u_{d}c_{e}]$
(38) $\displaystyle=$ $\displaystyle D^{*-}_{\nu}~{}\Sigma_{c}^{*++;\nu}\,,$
$\displaystyle\xi_{6}^{\alpha}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}d_{b}]~{}P_{3/2}^{\alpha\rho}\gamma^{\nu}\gamma_{5}~{}P^{3/2}_{\rho\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma^{\mu}u_{d}c_{e}]$
(39) $\displaystyle=$ $\displaystyle
D^{*-}_{\nu}~{}P_{3/2}^{\alpha\rho}~{}\gamma^{\nu}\gamma_{5}~{}\Sigma_{c;\rho}^{*++}\,,$
$\displaystyle\xi_{7}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}P^{\alpha\beta,\nu\rho}_{5/2}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}d_{b}]~{}P^{3/2}_{\rho\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma^{\mu}u_{d}c_{e}]$
(40) $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\nu\rho}_{5/2}~{}D^{*-}_{\nu}~{}\Sigma_{c;\rho}^{*++}\,.$

In the above expressions, we have used $\mathcal{D}$ and $\mathcal{B}$ to
denote the charmed meson operators $J_{\mathcal{D}}$ and the charmed baryon
fields $J_{\mathcal{B}}$ for simplicity; $P_{5/2}^{\mu\nu,\rho\sigma}$ is the
spin-5/2 projection operator

$\displaystyle P_{5/2}^{\mu\nu,\rho\sigma}$ $\displaystyle=$
$\displaystyle{1\over 2}g^{\mu\rho}g^{\nu\sigma}+{1\over
2}g^{\mu\sigma}g^{\nu\rho}-{1\over 6}g^{\mu\nu}g^{\rho\sigma}$
$\displaystyle-{1\over 12}g^{\mu\rho}\gamma^{\nu}\gamma^{\sigma}-{1\over
12}g^{\mu\sigma}\gamma^{\nu}\gamma^{\rho}$ $\displaystyle-{1\over
12}g^{\nu\sigma}\gamma^{\mu}\gamma^{\rho}-{1\over
12}g^{\nu\rho}\gamma^{\mu}\gamma^{\sigma}\,.$

##  III Masses and decay constants through QCD sum rules

In this section we use the method of QCD sum rules Shifman:1978bx ;
Reinders:1984sr  to study $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states
through the currents $J_{1\cdots 7}$, i.e., $J_{1,2,5}$ of $J^{P}=1/2^{-}$,
$J^{\alpha}_{3,4,6}$ of $J^{P}=3/2^{-}$, and $J_{7}^{\alpha\beta}$ of
$J^{P}=5/2^{-}$. We shall calculate their masses and decay constants, and the
obtained results will be used in the next section to further calculate their
relative production rates. Some of the calculations have been done in Refs.
Chen:2015moa ; Chen:2016otp ; Xiang:2017byz ; Chen:2019bip , and we refer to
Refs. Chen:2020uif ; Wang:2019hyc ; Zhang:2019xtu ; Azizi:2016dhy ;
Wang:2019got  for more relevant QCD sum rule studies.

###  III.1 Correlation functions

We assume that the currents $J_{1\cdots 7}$ couple to the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states $X_{1\cdots 7}$ through

$\displaystyle\langle 0|J_{1,2,5}|X_{1,2,5};1/2^{-}\rangle$ $\displaystyle=$
$\displaystyle f_{X_{1,2,5}}u(p)\,,$ $\displaystyle\langle
0|J_{3,4,6}^{\alpha}|X_{3,4,6};3/2^{-}\rangle$ $\displaystyle=$ $\displaystyle
f_{X_{3,4,6}}u^{\alpha}(p)\,,$ (42) $\displaystyle\langle
0|J_{7}^{\alpha\beta}|X_{7};5/2^{-}\rangle$ $\displaystyle=$ $\displaystyle
f_{X_{7}}u^{\alpha\beta}(p)\,,$

where $u(p)$, $u^{\alpha}(p)$, and $u^{\alpha\beta}(p)$ are spinors of the
$X_{1\cdots 7}$. The two-point correlation functions extracted from these
currents can be written as:

$\displaystyle\Pi_{1,2,5}\left(q^{2}\right)$ $\displaystyle=$ $\displaystyle
i\int d^{4}xe^{iq\cdot x}\langle
0|T\left[J_{1,2,5}(x)\bar{J}_{1,2,5}(0)\right]|0\rangle$ (43) $\displaystyle=$
$\displaystyle(q\\!\\!\\!/~{}+M_{X_{1,2,5}})~{}\Pi_{1,2,5}\left(q^{2}\right)\,,$
$\displaystyle\Pi^{\alpha\alpha^{\prime}}_{3,4,6}\left(q^{2}\right)$
$\displaystyle=$ $\displaystyle i\int d^{4}xe^{iq\cdot x}\langle
0|T\left[J^{\alpha}_{3,4,6}(x)\bar{J}^{\alpha^{\prime}}_{3,4,6}(0)\right]|0\rangle$
(44) $\displaystyle=$
$\displaystyle\mathcal{G}_{3/2}^{\alpha\alpha^{\prime}}(q\\!\\!\\!/~{}+M_{X_{3,4,6}})~{}\Pi_{3,4,6}\left(q^{2}\right)\,,$
$\displaystyle\Pi^{\alpha\beta,\alpha^{\prime}\beta^{\prime}}_{7}\left(q^{2}\right)$
$\displaystyle=$ $\displaystyle i\int d^{4}xe^{iq\cdot x}\langle
0|T\left[J^{\alpha\beta}_{7}(x)\bar{J}^{\alpha^{\prime}\beta^{\prime}}_{7}(0)\right]|0\rangle$
(45) $\displaystyle=$
$\displaystyle\mathcal{G}_{5/2}^{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(q\\!\\!\\!/~{}+M_{X_{7}})~{}\Pi_{7}\left(q^{2}\right)\,,$

where $\mathcal{G}_{3/2}^{\mu\nu}$ and $\mathcal{G}_{5/2}^{\mu\nu,\rho\sigma}$
are coefficients of the spin-3/2 and spin-5/2 propagators, respectively:

$\displaystyle\mathcal{G}_{3/2}^{\mu\nu}(p)$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}-{1\over
3}\gamma^{\mu}\gamma^{\nu}-{p^{\mu}\gamma^{\nu}-p^{\nu}\gamma^{\mu}\over
3m}-{2p^{\mu}p^{\nu}\over 3m^{2}}\,,$
$\displaystyle\mathcal{G}_{5/2}^{\mu\nu,\rho\sigma}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})-\frac{1}{5}g^{\mu\nu}g^{\rho\sigma}$
$\displaystyle-$
$\displaystyle\frac{1}{10}(g^{\mu\rho}\gamma^{\nu}\gamma^{\sigma}+g^{\mu\sigma}\gamma^{\nu}\gamma^{\rho}+g^{\nu\rho}\gamma^{\mu}\gamma^{\sigma}+g^{\nu\sigma}\gamma^{\mu}\gamma^{\rho})$
$\displaystyle+$
$\displaystyle\frac{1}{10m}\Bigl{(}g^{\mu\rho}(p^{\nu}\gamma^{\sigma}-p^{\sigma}\gamma^{\nu})+g^{\mu\sigma}(p^{\nu}\gamma^{\rho}-p^{\rho}\gamma^{\nu})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+g^{\nu\rho}(p^{\mu}\gamma^{\sigma}-p^{\sigma}\gamma^{\mu})+g^{\nu\sigma}(p^{\mu}\gamma^{\rho}-p^{\rho}\gamma^{\mu})\Bigr{)}$
$\displaystyle+$
$\displaystyle\frac{1}{5m^{2}}(g^{\mu\nu}p^{\rho}p^{\sigma}+g^{\rho\sigma}p^{\mu}p^{\nu})$
$\displaystyle-$
$\displaystyle\frac{2}{5m^{2}}(g^{\mu\rho}p^{\nu}p^{\sigma}+g^{\mu\sigma}p^{\nu}p^{\rho}+g^{\nu\rho}p^{\mu}p^{\sigma}+g^{\nu\sigma}p^{\mu}p^{\rho})$
$\displaystyle+$
$\displaystyle\frac{1}{10m^{2}}\Bigl{(}\gamma^{\mu}p^{\nu}(\gamma^{\rho}p^{\sigma}+\gamma^{\sigma}p^{\rho})+\gamma^{\nu}p^{\mu}(\gamma^{\rho}p^{\sigma}+\gamma^{\sigma}p^{\rho})\Bigl{)}$
$\displaystyle+$
$\displaystyle\frac{1}{5m^{3}}\Bigl{(}p^{\rho}p^{\sigma}(\gamma^{\mu}p^{\nu}+\gamma^{\nu}p^{\mu})-p^{\mu}p^{\nu}(\gamma^{\rho}p^{\sigma}+\gamma^{\sigma}p^{\rho})\Bigl{)}$
$\displaystyle+$
$\displaystyle\frac{2}{5m^{4}}p^{\mu}p^{\nu}p^{\rho}p^{\sigma}\,.$

In the above expressions we have assumed that the states $X_{1\cdots 7}$ have
the same spin-parity quantum numbers as the currents $J_{1\cdots 7}$, so that
we can use the “non-$\gamma_{5}$ coupling” in Eqs. (42); while we need to use
the “$\gamma_{5}$ coupling”:

$\langle 0|J_{1\cdots 7}|X_{1\cdots 7}^{\prime}\rangle=f_{X^{\prime}_{1\cdots
7}}\gamma_{5}u(p)\,,$ (48)

if the states $X^{\prime}_{1\cdots 7}$ have the opposite parity from the
currents $J_{1\cdots 7}$; or we can use the partner currents
$\gamma_{5}J_{1\cdots 7}$, which also have the opposite parity:

$\langle 0|\gamma_{5}J_{1\cdots 7}|X_{1\cdots 7}\rangle=f_{X_{1\cdots
7}}\gamma_{5}u(p)\,.$ (49)

From Eqs. (48) and (49) we can derive another “non-$\gamma_{5}$ coupling”
between $\gamma_{5}J_{1\cdots 7}$ and $X^{\prime}_{1\cdots 7}$:

$\langle 0|\gamma_{5}J_{1\cdots 7}|X_{1\cdots
7}^{\prime}\rangle=f_{X^{\prime}_{1\cdots 7}}u(p)\,.$ (50)

We refer to Refs. Chung:1981cc ; Jido:1996ia ; Kondo:2005ur ; Ohtani:2012ps
for detailed discussions.

The two-point correlation functions derived from Eqs. (48) and (49) are
similar to Eqs. (43)–(45), but just with $(q\\!\\!\\!/~{}+M_{X})$ replaced by
$(-q\\!\\!\\!/~{}+M_{X})$. Based on this feature, we can extract parities of
the $X_{1\cdots 7}$: we use the terms proportional to $\bf 1$ to evaluate
masses of the $X_{1\cdots 7}$, which are then compared with those terms
proportional to $q\\!\\!\\!/~{}$ to extract their parities.

In QCD sum rule studies we need to calculate the two-point correlation
function $\Pi\left(q^{2}\right)$ at both hadron and quark-gluon levels. At the
hadron level, we use the dispersion relation to write it as

$\Pi(q^{2})={\frac{1}{\pi}}\int^{\infty}_{s_{<}}\frac{{\rm
Im}\Pi(s)}{s-q^{2}-i\varepsilon}ds\,,$ (51)

with $s_{<}$ the physical threshold. We define the imaginary part of the
correlation function as the spectral density $\rho(s)$, which can be evaluated
at the hadron level by inserting intermediate hadron states
$\sum_{n}|n\rangle\langle n|$:

$\displaystyle\rho_{\rm phen}(s)$ $\displaystyle\equiv$ $\displaystyle{\rm
Im}\Pi(s)/\pi$ $\displaystyle=$
$\displaystyle\sum_{n}\delta(s-M^{2}_{n})\langle 0|\eta|n\rangle\langle
n|{\eta^{\dagger}}|0\rangle$ $\displaystyle=$ $\displaystyle
f_{X}^{2}\delta(s-m_{X}^{2})+\mbox{continuum}.$

In the last step we adopt the usual parametrization of one-pole dominance for
the ground state $X$ together with a continuum contribution.

At the quark-gluon level we calculate $\Pi\left(q^{2}\right)$ using the method
of operator product expansion (OPE), and extract its corresponding spectral
density $\rho_{\rm OPE}(s)$. After performing the Borel transformation at both
hadron and quark-gluon levels, we approximate the continuum using the spectral
density above a threshold value $s_{0}$ (quark-hadron duality), and arrive at
the sum rule equation

$\Pi(s_{0},M_{B}^{2})\equiv
f^{2}_{X}e^{-M_{X}^{2}/M_{B}^{2}}=\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{\rm
OPE}(s)ds\,.$ (53)

It can be used to further calculate $M_{X}$ and $f_{X}$ through

$\displaystyle M^{2}_{X}(s_{0},M_{B})$ $\displaystyle=$
$\displaystyle\frac{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}s\rho_{\rm
OPE}(s)ds}{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{\rm OPE}(s)ds}\,,$ (54)
$\displaystyle f_{X}^{2}(s_{0},M_{B})$ $\displaystyle=$ $\displaystyle
e^{M_{X}^{2}(s_{0},M_{B})\over
M_{B}^{2}}\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{\rm OPE}(s)ds\,.$

In the present study we have calculated OPEs at the leading order of
$\alpha_{s}$ and up to the $D({\rm imension})=10$ terms, including the
perturbative term, the charm quark mass, the quark condensate
$\langle\bar{q}q\rangle$, the gluon condensate $\langle g_{s}^{2}GG\rangle$,
the quark-gluon mixed condensate $\langle g_{s}\bar{q}\sigma Gq\rangle$, and
their combinations $\langle\bar{q}q\rangle^{2}$,
$\langle\bar{q}q\rangle\langle g_{s}\bar{q}\sigma Gq\rangle$,
$\langle\bar{q}q\rangle^{3}$, and $\langle g_{s}\bar{q}\sigma Gq\rangle^{2}$.
We summarized the obtained spectral densities $\rho_{1\cdots 7}(s)$ in
Appendix A, which are extracted from the currents $J_{1\cdots 7}$,
respectively.

In the calculations we have ignored the chirally suppressed terms with the
light quark mass, and adopt the factorization assumption of vacuum saturation
for higher dimensional condensates, i.e.,
$\langle(\bar{q}q)^{2}\rangle=\langle\bar{q}q\rangle^{2}$,
$\langle(\bar{q}q)(g_{s}\bar{q}\sigma Gq)\rangle=\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma Gq\rangle$,
$\langle(\bar{q}q)^{3}\rangle=\langle\bar{q}q\rangle^{3}$, and
$\langle(g_{s}\bar{q}\sigma Gq)^{2}\rangle=\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}$. We find that the $D=3$ quark condensate
$\langle\bar{q}q\rangle$ and the $D=5$ mixed condensate $\langle
g_{s}\bar{q}\sigma Gq\rangle$ are both multiplied by the charm quark mass
$m_{c}$, which are thus important power corrections.

In the next subsection we shall use the spectral densities $\rho_{1\cdots
7}(s)$ to perform numerical analyses, and calculate masses and decay constants
of the $X_{1\cdots 7}$. Before doing this, let us investigate the current
$J_{1}$ as an example. It has the quantum number $J^{P}=1/2^{-}$ and couples
to the $\bar{D}\Sigma_{c}$ molecular state $X_{1}$. Its spectral density
$\rho_{1}(s)$ is given in Eqs. (LABEL:ope:J1). We find that the terms timed by
$m_{c}$ are almost positively proportional to those terms timed by
$q\\!\\!\\!/$. Hence, the parity of $X_{1}$ is extracted to be negative, that
is the same as $J_{1}$; in other words, $J_{1}$ mainly couples to a negative-
parity state. Similarly, all the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular
states defined in Eqs. (18-24) are determined to have the negative parity.

###  III.2 Mass analyses

In this subsection we use the spectral densities $\rho_{1\cdots 7}(s)$
extracted from the currents $J_{1\cdots 7}$ to perform numerical analyses, and
calculate masses and decay constants of the $X_{1\cdots 7}$. As discussed in
the previous subsection, we only use the terms proportional to $m_{c}$ to do
this.

We use the current $J_{1}$ as an example, whose spectral density $\rho_{1}(s)$
can be found in Eq. (LABEL:ope:J1). We use the following values for various
QCD sum rule parameters Yang:1993bp ; Eidemuller:2000rc ; Narison:2002pw ;
Gimenez:2005nt ; Jamin:2002ev ; Ioffe:2002be ; Ovchinnikov:1988gk ;
Ellis:1996xc ; colangelo :

$\displaystyle m_{c}$ $\displaystyle=$ $\displaystyle
1.275^{+0.025}_{-0.035}\mbox{ GeV}\,,$ $\displaystyle\langle\bar{q}q\rangle$
$\displaystyle=$ $\displaystyle-(0.24\pm 0.01)^{3}\mbox{ GeV}^{3}\,,$
$\displaystyle\langle g_{s}^{2}GG\rangle$ $\displaystyle=$
$\displaystyle(0.48\pm 0.14)\mbox{ GeV}^{4}\,,$ (56) $\displaystyle\langle
g_{s}\bar{q}\sigma Gq\rangle$ $\displaystyle=$ $\displaystyle
M_{0}^{2}\times\langle\bar{q}q\rangle\,,$ $\displaystyle M_{0}^{2}$
$\displaystyle=$ $\displaystyle(0.8\pm 0.2)\mbox{ GeV}^{2}\,,$

where the running mass in the $\overline{MS}$ scheme is used for the charm
quark.

Figure 2: Convergence (solid curve, defined in Eq. (57)) and Pole-Contribution
(dashed curve, defined in Eq. (58)) as functions of the Borel mass $M_{B}$.
These curves are obtained using the current $J_{1}$ when setting $s_{0}=24$
GeV2.

There are two free parameters in Eqs. (54) and (LABEL:eq:fx): the Borel mass
$M_{B}$ and the threshold value $s_{0}$. We use two criteria to constrain the
Borel mass $M_{B}$ for a fixed $s_{0}$. The first criterion is to insure the
convergence of OPE series. It is done by requiring the $D=10$ terms
($m_{c}\langle\bar{q}q\rangle^{3}$ and $\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}$) to be less than 10%, so that the lower limit of $M_{B}$ can be
determined:

$\mbox{Convergence}\equiv\left|\frac{\Pi^{D=10}(\infty,M_{B})}{\Pi(\infty,M_{B})}\right|\leq
10\%\,.$ (57)

We show this function in Fig. 2 using the solid curve, and find that the OPE
convergence improves with the increase of $M_{B}$. This criterion leads to
$\left(M_{B}^{min}\right)^{2}=3.27$ GeV2, when setting $s_{0}=24$ GeV2.

The second criterion is to insure the validity of one-pole parametrization. It
is done by requiring the pole contribution to be larger than 40%, so that the
upper limit of $M_{B}$ can be determined:

$\mbox{Pole-Contribution}\equiv\frac{\Pi(s_{0},M_{B})}{\Pi(\infty,M_{B})}\geq
40\%\,.$ (58)

We show this function in Fig. 2 using the dashed curve, and find that it
decreases with the increase of $M_{B}$. This criterion leads to
$\left(M_{B}^{max}\right)^{2}=3.52$ GeV2, when setting $s_{0}=24$ GeV2.

Figure 3: Variations of the mass $M_{X}$ (left) and the decay constant $f_{X}$
(right) with respect to the Borel mass $M_{B}$, calculated using the current
$J_{1}$. In both panels, the short-dashed, solid and long-dashed curves are
obtained by setting $s_{0}=23$, $24$ and $25$ GeV2, respectively.

Altogether we extract the working region of Borel mass to be $3.27$
GeV${}^{2}<M_{B}^{2}<3.52$ GeV2 for the current $J_{1}$ with the threshold
value $s_{0}=24$ GeV2. We show variations of $M_{X_{1}}$ and $f_{X_{1}}$ with
respect to the Borel mass $M_{B}$ in Fig. 3. They are shown in a broader
region $3.0$ GeV${}^{2}\leq M_{B}^{2}\leq 4.0$ GeV2, and they are more stable
inside the above Borel window.

Redoing the same procedures by changing $s_{0}$, we find that there exist non-
vanishing Borel windows as long as $s_{0}\geq s_{0}^{min}=22.4$ GeV2.
Accordingly, we choose $s_{0}$ to be slightly larger with the uncertainty $\pm
1.0$ GeV, that is $s_{0}=24.0\pm 1.0$ GeV2. Altogether our working regions for
the current $J_{1}$ are determined to be $23.0$ GeV${}^{2}\leq s_{0}\leq 25.0$
GeV2 and $3.27$ GeV${}^{2}\leq M_{B}^{2}\leq 3.52$ GeV2, where we calculate
the mass and decay constant of $X_{1}$ to be:

$\displaystyle M_{X_{1}}$ $\displaystyle=$ $\displaystyle
4.30^{+0.10}_{-0.10}\mbox{ GeV}\,,$ (59) $\displaystyle f_{X_{1}}$
$\displaystyle=$ $\displaystyle\left(1.19^{+0.19}_{-0.18}\right)\times
10^{-3}\mbox{ GeV}^{6}\,.$

Here the central values correspond to $M_{B}^{2}=3.40$ GeV2 and $s_{0}=24.0$
GeV2. Their uncertainties come from the threshold value $s_{0}$, the Borel
mass $M_{B}$, the charm quark mass $m_{c}$, and various QCD sum rule
parameters listed in Eqs. (56). This mass value is consistent with the
experimental mass of the $P_{c}(4312)^{+}$ Aaij:2019vzc , supporting it to be
the $I=1/2$ $\bar{D}\Sigma_{c}$ molecular state of $J^{P}=1/2^{-}$.

Table 1: Masses and decay constants of the $X_{1\cdots 7}$, extracted from the currents $J_{1\cdots 7}$. Currents | Configuration | $s_{0}^{min}$ | Working Regions | Pole [%] | Mass [GeV] |  $f_{X}$ [GeV6] | Candidate
---|---|---|---|---|---|---|---
[${\rm GeV}^{2}$] | $s_{0}~{}[{\rm GeV}^{2}]$ | $M_{B}^{2}~{}[{\rm GeV}^{2}]$
$J_{1}$ | $|\bar{D}\Sigma_{c};1/2^{-}\rangle$ | 22.4 | $24.0\pm 1.0$ |  $3.27$–$3.52$ |  $40$–$48$ | $4.30^{+0.10}_{-0.10}$ | $\left(1.19^{+0.19}_{-0.18}\right)\times 10^{-3}$ | $P_{c}(4312)^{+}$
$J_{2}$ | $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$ | 25.5 | $27.0\pm 1.0$ |  $3.78$–$3.99$ |  $40$–$46$ | $4.48^{+0.10}_{-0.10}$ | $\left(2.24^{+0.34}_{-0.30}\right)\times 10^{-3}$ | $P_{c}(4457)^{+}$
$J_{3}$ | $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$ | 24.6 | $26.0\pm 1.0$ |  $3.51$–$3.72$ |  $40$–$46$ | $4.46^{+0.11}_{-0.10}$ | $\left(1.15^{+0.18}_{-0.16}\right)\times 10^{-3}$ | $P_{c}(4440)^{+}$
$J_{4}$ | $|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle$ | 24.2 | $25.0\pm 1.0$ |  $3.33$–$3.45$ |  $40$–$44$ | $4.43^{+0.10}_{-0.10}$ | $\left(0.65^{+0.11}_{-0.10}\right)\times 10^{-3}$ |
$J_{5}$ | $|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle$ | 26.0 | $27.0\pm 1.0$ |  $3.43$–$3.56$ |  $40$–$44$ | $4.51^{+0.10}_{-0.11}$ | $\left(1.12^{+0.19}_{-0.17}\right)\times 10^{-3}$ |
$J_{6}$ | $|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle$ | 25.3 | $27.0\pm 1.0$ |  $3.69$–$3.98$ |  $40$–$48$ | $4.52^{+0.11}_{-0.11}$ | $\left(0.85^{+0.14}_{-0.13}\right)\times 10^{-3}$ |
$J_{7}$ | $|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle$ | 24.7 | $26.0\pm 1.0$ |  $3.22$–$3.42$ |  $40$–$46$ | $4.55^{+0.15}_{-0.13}$ | $\left(0.65^{+0.11}_{-0.10}\right)\times 10^{-3}$ |

Similarly, we use the spectral densities $\rho_{2\cdots 7}(s)$ extracted from
the currents $J_{2\cdots 7}$ to perform numerical analyses, and calculate
masses and decay constants of the $X_{2\cdots 7}$. Especially, the sum rule
results extracted from the currents $J_{6}^{\alpha}$ and $J_{7}^{\alpha\beta}$
are

$\displaystyle M_{X_{6}}$ $\displaystyle=$ $\displaystyle
4.64^{+0.10}_{-0.10}\mbox{ GeV}\,,$ $\displaystyle f_{X_{6}}$ $\displaystyle=$
$\displaystyle\left(1.01^{+0.15}_{-0.14}\right)\times 10^{-3}\mbox{
GeV}^{6}\,,$ (60) $\displaystyle M_{X_{7}}$ $\displaystyle=$ $\displaystyle
4.64^{+0.14}_{-0.12}\mbox{ GeV}\,,$ $\displaystyle f_{X_{7}}$ $\displaystyle=$
$\displaystyle\left(0.77^{+0.12}_{-0.11}\right)\times 10^{-3}\mbox{
GeV}^{6}\,.$

These two mass values are both not far from the $\bar{D}^{*}\Sigma_{c}^{*}$
threshold at $M_{D^{*}}+M_{\Sigma_{c}^{*}}=4527$ MeV, but a bit larger than
that. To get a better description of $\bar{D}^{*}\Sigma_{c}^{*}$ molecular
states that may lie just below the $\bar{D}^{*}\Sigma_{c}^{*}$ threshold, we
slightly release the criterion given in Eq. (57) to be

$\mbox{Convergence}\equiv\left|\frac{\Pi^{D=10}(\infty,M_{B})}{\Pi(\infty,M_{B})}\right|\leq
15\%\,.$ (61)

Now the masses and decay constants extracted from the currents
$J_{6}^{\alpha}$ and $J_{7}^{\alpha\beta}$ are modified to be

$\displaystyle M^{\prime}_{X_{6}}$ $\displaystyle=$ $\displaystyle
4.52^{+0.11}_{-0.11}\mbox{ GeV}\,,$ $\displaystyle f^{\prime}_{X_{6}}$
$\displaystyle=$ $\displaystyle\left(0.85^{+0.14}_{-0.13}\right)\times
10^{-3}\mbox{ GeV}^{6}\,,$ (62) $\displaystyle M^{\prime}_{X_{7}}$
$\displaystyle=$ $\displaystyle 4.55^{+0.15}_{-0.13}\mbox{ GeV}\,,$
$\displaystyle f^{\prime}_{X_{7}}$ $\displaystyle=$
$\displaystyle\left(0.65^{+0.11}_{-0.10}\right)\times 10^{-3}\mbox{
GeV}^{6}\,.$

Besides, the mass of $|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle$ is calculated to
be $4.43^{+0.10}_{-0.10}$ GeV, which is consistent with, but also a bit larger
than, the $\bar{D}\Sigma_{c}^{*}$ threshold at $M_{D}+M_{\Sigma_{c}^{*}}=4385$
MeV. All these divergences indicate that the accuracy of our QCD sum rule
results is moderate but not good enough to extract the binding energies of the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states. Therefore, our results can
only suggest but not determine: a) whether these
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states exist or not, and b) whether
they are bound states or resonance states. However, in the present study we
are more concerned about the ratios, i.e., the relative production rates and
the relative branching ratios, whose uncertainties can be significantly
reduced. Accordingly, the decay constants $f_{X}$ calculated in this section
are input parameters that are more important than the masses $M_{X}$. Note
that the decay constants $f_{X}$ can also be used within the QCD sum rule
method to directly calculate the partial decay widths through the three-point
correlation functions, but we shall not do this in the present study.

We summarized all the above sum rule results in Table 1. Our results are
consistent with those of Ref. Wang:2022ltr , where the authors applied the
same QCD sum rule method to study both the $I=1/2$
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states as well as the $I=3/2$ ones.
Our results support the interpretations of the $P_{c}(4440)^{+}$ and
$P_{c}(4457)^{+}$ Aaij:2019vzc  as the $I=1/2$ $\bar{D}^{*}\Sigma_{c}$
molecular states of $J^{P}=1/2^{-}$ and $3/2^{-}$. Again, the accuracy of our
sum rule results is not good enough to distinguish/identify them. To better
understand them, we move on to study their production and decay properties in
the following sections, where we shall find that the $P_{c}(4440)^{+}$ and
$P_{c}(4457)^{+}$ can be better interpreted in our framework as
$|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$ and
$|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$, respectively.

##  IV Productions through the current algebra

In this section we study productions of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
molecular states in $\Lambda_{b}^{0}$ decays through the current algebra. We
shall calculate their relative production rates, i.e.,
$\mathcal{B}(\Lambda_{b}^{0}\to P_{c}K^{-}):\mathcal{B}(\Lambda_{b}^{0}\to
P_{c}^{\prime}K^{-})$ with $P_{c}$ and $P_{c}^{\prime}$ two different states.
We refer to Refs. Cheng:2015cca ; Chen:2015sxa  for more relevant studies.

(a) $\Lambda_{b}^{0}\to D^{(*)-}\Sigma_{c}^{(*)++}K^{-}$ (b)
$\Lambda_{b}^{0}\to\bar{D}^{(*)0}\Sigma_{c}^{(*)+}K^{-}$

Figure 4: Production mechanisms of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular
states in $\Lambda_{b}^{0}$ decays.

The $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$ were observed
by LHCb in the $J/\psi p$ invariant mass spectrum of the $\Lambda_{b}^{0}\to
J/\psi pK^{-}$ decays. The quark content of the initial state
$\Lambda_{b}^{0}$ is $udb$. In this three-body decay process, first the $b$
quark decays into a $c$ quark by emitting a $W^{-}$ boson, and the $W^{-}$
translates into a pair of $\bar{c}$ and $s$ quarks, both of which are Cabibbo-
favored; then they pick up a pair of $\bar{u}$ and $u$ quarks from the vacuum;
finally they hadronize into the three final states $J/\psi pK^{-}$:

$\Lambda_{b}^{0}=udb\to udc~{}\bar{c}s\to udc~{}\bar{c}s~{}\bar{u}u\to J/\psi
pK^{-}\,.$ (63)

Hence, the total quark content of the final states is $udc\bar{c}s\bar{u}u$,
where the intermediate states $D^{(*)-}\Sigma_{c}^{(*)++}K^{-}$ and
$\bar{D}^{(*)0}\Sigma_{c}^{(*)+}K^{-}$ can also be produced.

In the present study we study productions of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
molecular states by investigating the mechanisms depicted in Fig. 4. Note that
the $u$ quark from the vacuum needs to exchange with either the $u$ or $d$
quark of the $\Lambda_{b}^{0}$, because the $ud$ pair of the $\Lambda_{b}^{0}$
is in a state of $I=0$, while the $\Sigma_{c}$ and $\Sigma_{c}^{*}$ both have
$I=1$.

As depicted in Fig. 4, the weak interaction only involves the initial $b$
quark and the final $c\bar{c}s$ quarks. Hence, considering the quark pair
produced from the vacuum to be $\bar{u}u+\bar{d}d$ of $I=0$, the isospin of
the whole process is also conserved to be $I=0$:

$\displaystyle\Lambda_{b}^{0}$ $\displaystyle\to$ $\displaystyle
udc~{}\bar{c}s~{}(\bar{u}u+\bar{d}d)$ $\displaystyle\to$
$\displaystyle\sqrt{1\over 3}D^{(*)-}\Sigma_{c}^{(*)++}K^{-}+\sqrt{1\over
3}\bar{D}^{(*)0}\Sigma_{c}^{(*)0}\bar{K}^{0}$ $\displaystyle-\sqrt{1\over
6}D^{(*)-}\Sigma_{c}^{(*)+}\bar{K}^{0}-\sqrt{1\over
6}\bar{D}^{(*)0}\Sigma_{c}^{(*)+}K^{-}\,.$

The four fixed isospin factors allow us to consider only the
$D^{(*)-}\Sigma_{c}^{(*)++}K^{-}$ final state, since the results derived from
the $\bar{D}^{(*)0}\Sigma_{c}^{(*)+}K^{-}$ final state are the same.
Accordingly, we only need to consider the exchange of the $u$ quark from the
vacuum and the $d$ quark from the $\Lambda_{b}^{0}$, which is depicted in Fig.
4(a).

Summarizing the above discussions, in this section we shall calculate relative
production rates of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states in
$\Lambda_{b}^{0}$ decays, by investigating the three-body $\Lambda_{b}^{0}\to
D^{(*)-}\Sigma_{c}^{(*)++}K^{-}$ decays, whose mechanism is depicted in Fig.
4(a). We shall develop a Fierz rearrangement to describe this process in Sec.
IV.1, and use it to perform numerical analyses in Sec. IV.2.

###  IV.1 Fierz rearrangement

To describe the production mechanism depicted in Fig. 4(a), we use the color
rearrangement given in Eq. (12) twice to obtain

$\displaystyle\epsilon^{abc}\delta^{de}\delta^{fg}$ $\displaystyle=$
$\displaystyle\left(\epsilon^{ebc}\delta^{da}+\epsilon^{aec}\delta^{db}+\epsilon^{abe}\delta^{dc}\right)\times\delta^{fg}$
$\displaystyle=$
$\displaystyle\epsilon^{gbc}\delta^{da}\delta^{fe}+\epsilon^{egc}\delta^{da}\delta^{fb}+\epsilon^{ebg}\delta^{da}\delta^{fc}$
$\displaystyle+$
$\displaystyle\epsilon^{gec}\delta^{db}\delta^{fa}+\epsilon^{agc}\delta^{db}\delta^{fe}+\epsilon^{aeg}\delta^{db}\delta^{fc}$
$\displaystyle+$
$\displaystyle\epsilon^{gbe}\delta^{dc}\delta^{fa}+\epsilon^{age}\delta^{dc}\delta^{fb}+\epsilon^{abg}\delta^{dc}\delta^{fe}\,.$

Given the initial color structure to be
$[\epsilon^{abc}u_{a}d_{b}c_{c}][\delta^{de}\bar{c}_{d}s_{e}][\delta^{fg}\bar{u}_{f}u_{g}]$,
we need to fifth one
$[\epsilon^{agc}u_{a}u_{g}c_{c}][\delta^{db}\bar{c}_{d}d_{b}][\delta^{fe}\bar{u}_{f}s_{e}]$,
which corresponds to the $D^{(*)-}\Sigma_{c}^{(*)++}K^{-}$ final state.

Besides, we also need to apply the Fierz transformation twice: a) to
interchange the $d_{b}$ and $u_{g}$ quarks, and b) to interchange the $d_{b}$
and $s_{e}$ quarks. Note that the Fierz rearrangement in the Lorentz space is
actually a matrix identity. It is valid if each quark field in the initial and
final currents is at the same location.

The key formula is as follow:

$\displaystyle\Lambda_{b}^{0}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle
J_{\Lambda_{b}^{0}}=[\epsilon^{abc}u_{a}^{T}\mathbb{C}\gamma_{5}d_{b}b_{c}]$
(66) $\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\epsilon^{abc}u_{a}^{T}\mathbb{C}\gamma_{5}d_{b}\gamma_{\rho}(1-\gamma_{5})c_{c}]~{}\times~{}[\delta^{de}\bar{c}_{d}\gamma^{\rho}(1-\gamma_{5})s_{e}]$
(67) $\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\epsilon^{abc}u_{a}^{T}\mathbb{C}\gamma_{5}d_{b}\gamma_{\rho}(1-\gamma_{5})c_{c}]~{}\times~{}[\delta^{de}\bar{c}_{d}\gamma^{\rho}(1-\gamma_{5})s_{e}]~{}\times~{}[\delta^{fg}\bar{u}_{f}u_{g}]$
(68) $\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle\epsilon^{agc}\delta^{db}\delta^{fe}\times
u_{a}^{T}\mathbb{C}\gamma_{5}d_{b}\gamma_{\rho}(1-\gamma_{5})c_{c}\times\bar{c}_{d}\gamma^{\rho}(1-\gamma_{5})s_{e}\times\bar{u}_{f}u_{g}+\cdots$
(69) $\displaystyle\xlongequal{{\rm Fierz:}d_{b}\leftrightarrow u_{g}}$
$\displaystyle-{\delta^{db}\delta^{fe}\over
4}\times[\epsilon^{agc}u_{a}^{T}\mathbb{C}\gamma_{\mu}u_{g}\gamma_{\rho}(1-\gamma_{5})c_{c}]\times\bar{c}_{d}\gamma^{\rho}(1-\gamma_{5})s_{e}\times\bar{u}_{f}\gamma^{\mu}\gamma_{5}d_{b}+\cdots$
(70) $\displaystyle\xlongequal{{\rm Fierz:}d_{b}\leftrightarrow s_{e}}$
$\displaystyle+{1+\gamma_{5}\over
16}\times[\epsilon^{agc}u_{a}^{T}\mathbb{C}\gamma_{\mu}u_{g}\gamma^{\mu}\gamma_{5}c_{c}]\times[\delta^{db}\bar{c}_{d}\gamma_{5}d_{b}]\times[\delta^{fe}\bar{u}_{f}\gamma_{5}s_{e}]$
$\displaystyle+{(1+\gamma_{5})(g^{\nu\rho}-i\sigma^{\nu\rho})\over
32}\times[\epsilon^{agc}u_{a}^{T}\mathbb{C}\gamma_{\mu}u_{g}\gamma^{\mu}\gamma_{5}c_{c}]\times[\delta^{db}\bar{c}_{d}\gamma_{\nu}d_{b}]\times[\delta^{fe}\bar{u}_{f}\gamma_{\rho}\gamma_{5}s_{e}]$
$\displaystyle+{(1+\gamma_{5})(g^{\alpha\nu}\gamma^{\rho}+g^{\alpha\rho}\gamma^{\nu})\over
16}\times[P_{\alpha\mu}^{3/2}\epsilon^{agc}u_{a}^{T}\mathbb{C}\gamma^{\mu}u_{g}c_{c}]\times[\delta^{db}\bar{c}_{d}\gamma_{\nu}d_{b}]\times[\delta^{fe}\bar{u}_{f}\gamma_{\rho}\gamma_{5}s_{e}]+\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle+{1+\gamma_{5}\over
8\sqrt{2}}\times\xi_{1}\times[\bar{u}_{a}\gamma_{5}s_{a}]$
$\displaystyle+{(1+\gamma_{5})(g_{\nu\rho}-i\sigma_{\nu\rho})\over
16\sqrt{2}}\left(\xi_{3}^{\nu}-{1\over
4}\gamma^{\nu}\gamma_{5}\xi_{2}\right)[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$
$\displaystyle+{(1+\gamma_{5})(g_{\alpha\nu}\gamma_{\rho}+g_{\alpha\rho}\gamma_{\nu})\over
8\sqrt{2}}\left(\xi_{7}^{\alpha\nu}-{1\over
9}\gamma^{\alpha}\gamma_{5}\xi_{6}^{\nu}-{1\over
9}\gamma^{\nu}\gamma_{5}\xi_{6}^{\alpha}+{2\over
9}g^{\alpha\nu}\xi_{5}\right)[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$
$\displaystyle+\cdots\,.$

Its brief explanations are as follows:

  * •

Eq. (67) describes the Cabibbo-favored weak decay of $b\to c+\bar{c}s$ via the
$V$-$A$ current.

  * •

Eq. (68) describes the production of the $\bar{u}$ and $u$ quark pair from the
vacuum via the ${}^{3}P_{0}$ quark pair creation mechanism.

  * •

In Eq. (69) we apply the double-color rearrangement given in Eq.
(LABEL:eq:colortwice).

  * •

In Eq. (70) we apply the Fierz transformation to interchange the $d_{b}$ and
$u_{g}$ quarks.

  * •

In Eq. (IV.1) we apply the Fierz transformation to interchange the $d_{b}$ and
$s_{e}$ quarks.

  * •

In Eq. (IV.1) we combine the five $u_{a}u_{g}c_{c}\bar{c}_{d}d_{b}$ quarks
together so that $D^{(*)-}\Sigma_{c}^{(*)++}$ molecular states can be
produced.

In the above expression, we only consider $\xi_{1\cdots 7}$ defined in Eqs.
(34-40), which couple to $D^{(*)-}\Sigma_{c}^{(*)++}$ molecular states through
$S$-wave. Actually, there may exist some other currents coupling to these
states through $P$-wave, which are not included in the present study, such as

$\displaystyle\xi_{6}^{\prime\alpha\beta}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}~{}P^{\alpha\beta,\nu\rho}_{3/2}~{}[\delta^{ab}\bar{c}_{a}\gamma_{\nu}d_{b}]~{}P^{3/2}_{\rho\mu}[\epsilon^{cde}u_{c}^{T}\mathbb{C}\gamma^{\mu}u_{d}c_{e}]$
(73) $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\nu\rho}_{3/2}~{}D^{*-}_{\nu}~{}~{}\Sigma_{c;\rho}^{*++}\,,$

where $P_{3/2}^{\mu\nu,\rho\sigma}$ is the spin-3/2 projection operator with
two antisymmetric Lorentz indices,

$\displaystyle P_{3/2}^{\mu\nu,\rho\sigma}$ $\displaystyle=$
$\displaystyle{1\over 2}g^{\mu\rho}g^{\nu\sigma}-{1\over
2}g^{\mu\sigma}g^{\nu\rho}+{1\over 6}\sigma^{\mu\nu}\sigma^{\rho\sigma}$
$\displaystyle-{1\over 4}g^{\mu\rho}\gamma^{\nu}\gamma^{\sigma}+{1\over
4}g^{\mu\sigma}\gamma^{\nu}\gamma^{\rho}$ $\displaystyle-{1\over
4}g^{\nu\sigma}\gamma^{\mu}\gamma^{\rho}+{1\over
4}g^{\nu\rho}\gamma^{\mu}\gamma^{\sigma}\,.$

The current $\eta_{6}^{\prime\alpha\beta}$ couples to
$|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle$ through

$\langle
0|\eta_{6}^{\prime\alpha\beta}|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle=if^{T}_{6^{\prime}}(p^{\alpha}u^{\beta}-p^{\beta}u^{\alpha})\,,$
(75)

where $u_{\alpha}$ is the spinor of $|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle$.
Besides, it can couple to another state of $J^{P}=3/2^{+}$.

Consequently, $|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle$ may still be produced in
$\Lambda_{b}^{0}$ decays, although its directly corresponding current
$\xi_{4}^{\alpha}$ (and so $J_{4}^{\alpha}$) does not appear in Eq. (IV.1).
Besides, omissions of the “some other possible currents” produce some
theoretical uncertainties.

###  IV.2 Production analyses

In this subsection we use the Fierz rearrangement given in Eq. (IV.1) to
perform numerical analyses. We shall take into account the isospin factors of
Eqs. (II) and (IV), and directly calculate relative production rates of
$I=1/2$ $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states in $\Lambda_{b}^{0}$
decays. To do this we need the following couplings to $K^{-}$:

$\displaystyle\langle 0|\bar{u}_{a}\gamma_{5}s_{a}|K^{-}(q)\rangle$
$\displaystyle=$ $\displaystyle\lambda_{K}\,,$ (76) $\displaystyle\langle
0|\bar{u}_{a}\gamma_{\mu}\gamma_{5}s_{a}|K^{-}(q)\rangle$ $\displaystyle=$
$\displaystyle iq_{\mu}f_{K}\,,$

where $f_{K}=155.6$ MeV pdg  and $\lambda_{K}={f_{K}^{2}m_{K}\over
m_{u}+m_{s}}$.

We extract from Eq. (IV.1) the following decay channels:

  1. 1.

The decay of $\Lambda_{b}^{0}$ into $|\bar{D}\Sigma_{c};1/2^{-}\rangle K^{-}$
is contributed by $\xi_{1}\times[\bar{u}_{a}\gamma_{5}s_{a}]$:

$\displaystyle\langle\Lambda_{b}^{0}(q)~{}|~{}\bar{D}\Sigma_{c};1/2^{-}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle\approx$
$\displaystyle-c~{}i\lambda_{K}f_{|\bar{D}\Sigma_{c};1/2^{-}\rangle}~{}\bar{u}_{\Lambda_{b}^{0}}\left({1+\gamma_{5}\over
16}\right)u\,,$

where $u_{\Lambda_{b}^{0}}$ and $u$ are spinors of $\Lambda_{b}^{0}$ and
$|\bar{D}\Sigma_{c};1/2^{-}\rangle$, respectively. The decay constant
$f_{|\bar{D}\Sigma_{c};1/2^{-}\rangle}$ has been calculated in the previous
section and given in Table 1. The overall factor $c$ is related to: a) the
coupling of $J_{\Lambda_{b}^{0}}$ to $\Lambda_{b}^{0}$, b) the weak and
${}^{3}P_{0}$ decay processes described by Eqs. (67) and (68), and c) the
isospin factors of Eqs. (II) and (IV). We shall use the same factor $c$ for
all the seven $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states, which can
cause a significant theoretical uncertainty not taken into account in the
present study.

  2. 2.

The decay of $\Lambda_{b}^{0}$ into $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle
K^{-}$ is contributed by
$\xi_{2}\times[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$:

$\displaystyle\langle\Lambda_{b}^{0}(q)~{}|~{}\bar{D}^{*}\Sigma_{c};1/2^{-}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle\approx$ $\displaystyle
c~{}if_{K}f_{|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle}q_{2}^{\rho}\times$
$\displaystyle\bar{u}_{\Lambda_{b}^{0}}\left({(1+\gamma_{5})(g_{\nu\rho}-i\sigma_{\nu\rho})\over
32}\cdot\left(-{1\over 4}\gamma^{\nu}\gamma_{5}\right)\right)u\,,$

where $u$ and $f_{|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle}$ are the spinor and
decay constant of $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$, respectively.

  3. 3.

The decay of $\Lambda_{b}^{0}$ into $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle
K^{-}$ is contributed by
$\xi_{3}^{\nu}\times[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$:

$\displaystyle\langle\Lambda_{b}^{0}(q)~{}|~{}\bar{D}^{*}\Sigma_{c};3/2^{-}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle\approx$ $\displaystyle
c~{}if_{K}f_{|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle}q_{2}^{\rho}\times$
$\displaystyle\bar{u}_{\Lambda_{b}^{0}}\left({(1+\gamma_{5})(g_{\nu\rho}-i\sigma_{\nu\rho})\over
32}\right)u^{\nu}\,,$

where $u^{\nu}$ and $f_{|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle}$ are the spinor
and decay constant of $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$, respectively.

  4. 4.

The decay of $\Lambda_{b}^{0}$ into $|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle
K^{-}$ is contributed by
$\xi_{5}\times[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$:

$\displaystyle\langle\Lambda_{b}^{0}(q)~{}|~{}\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle\approx$ $\displaystyle
c~{}if_{K}f_{|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle}q_{2}^{\rho}\times$
$\displaystyle\bar{u}_{\Lambda_{b}^{0}}\left({(1+\gamma_{5})(g_{\alpha\nu}\gamma_{\rho}+g_{\alpha\rho}\gamma_{\nu})\over
16}\cdot{2\over 9}g^{\alpha\nu}\right)u\,,$

where $u$ and $f_{|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle}$ are the spinor
and decay constant of $|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle$,
respectively.

  5. 5.

The decay of $\Lambda_{b}^{0}$ into $|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle
K^{-}$ is contributed by
$\xi_{6}^{\beta}\times[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$:

$\displaystyle\langle\Lambda_{b}^{0}(q)~{}|~{}\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle\approx$ $\displaystyle
c~{}if_{K}f_{|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle}q_{2}^{\rho}\times$
$\displaystyle\bar{u}_{\Lambda_{b}^{0}}\Bigg{(}{(1+\gamma_{5})(g_{\alpha\nu}\gamma_{\rho}+g_{\alpha\rho}\gamma_{\nu})\over
16}$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot\left(-{1\over
9}\gamma^{\alpha}\gamma_{5}g^{\nu\beta}-{1\over
9}\gamma^{\nu}\gamma_{5}g^{\alpha\beta}\right)\Bigg{)}u_{\beta}\,,$

where $u_{\beta}$ and $f_{|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle}$ are the
spinor and decay constant of $|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle$,
respectively.

  6. 6.

The decay of $\Lambda_{b}^{0}$ into $|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle
K^{-}$ is contributed by
$\xi_{7}^{\alpha\nu}\times[\bar{u}_{a}\gamma^{\rho}\gamma_{5}s_{a}]$:

$\displaystyle\langle\Lambda_{b}^{0}(q)~{}|~{}\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle\approx$ $\displaystyle
c~{}if_{K}f_{|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle}q_{2}^{\rho}\times$
$\displaystyle\bar{u}_{\Lambda_{b}^{0}}\left({(1+\gamma_{5})(g_{\alpha\nu}\gamma_{\rho}+g_{\alpha\rho}\gamma_{\nu})\over
16}\right)u^{\alpha\nu}\,,$

where $u^{\alpha\nu}$ and $f_{|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle}$ are
the spinor and decay constant of $|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle$,
respectively.

We shall find that the $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$ can be well interpreted in our framework as
$|\bar{D}\Sigma_{c};1/2^{-}\rangle$, $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$,
and $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$, respectively. Accordingly, we
assume masses of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states to be:

$\displaystyle M_{|\bar{D}\Sigma_{c};1/2^{-}\rangle}$ $\displaystyle=$
$\displaystyle M_{P_{c}(4312)^{+}}=4311.9~{}{\rm MeV}\,,$ $\displaystyle
M_{|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle}$ $\displaystyle=$ $\displaystyle
M_{P_{c}(4457)^{+}}=4457.3~{}{\rm MeV}\,,$ $\displaystyle
M_{|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle}$ $\displaystyle=$ $\displaystyle
M_{P_{c}(4440)^{+}}=4440.3~{}{\rm MeV}\,,$ $\displaystyle
M_{|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle}$ $\displaystyle\approx$
$\displaystyle M_{D}+M_{\Sigma_{c}^{*}}=4385~{}{\rm MeV}\,,$ (83)
$\displaystyle M_{|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle}$
$\displaystyle\approx$ $\displaystyle M_{D^{*}}+M_{\Sigma_{c}^{*}}=4527~{}{\rm
MeV}\,,$ $\displaystyle M_{|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle}$
$\displaystyle\approx$ $\displaystyle M_{D^{*}}+M_{\Sigma_{c}^{*}}=4527~{}{\rm
MeV}\,,$ $\displaystyle M_{|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle}$
$\displaystyle\approx$ $\displaystyle M_{D^{*}}+M_{\Sigma_{c}^{*}}=4527~{}{\rm
MeV}\,.$

Now we can summarize the above production amplitudes to obtain the following
partial decay widths:

$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}\Sigma_{c}\rangle_{1/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle c^{2}~{}6.15\times 10^{-11}~{}{\rm
GeV}^{17}\,,$
$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}\rangle_{1/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle c^{2}~{}8.76\times 10^{-12}~{}{\rm
GeV}^{17}\,,$
$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle c^{2}~{}7.52\times 10^{-12}~{}{\rm
GeV}^{17}\,,$
$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}\Sigma_{c}^{*}\rangle_{3/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{1/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle c^{2}~{}3.57\times 10^{-11}~{}{\rm
GeV}^{17}\,,$
$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{3/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle c^{2}~{}1.38\times 10^{-12}~{}{\rm
GeV}^{17}\,,$
$\displaystyle\Gamma(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{5/2^{-}}K^{-})$
$\displaystyle=$ $\displaystyle 0\,.$ (84)

From these values, we derive the following relative production rates
$\mathcal{R}_{1}(P_{c})\equiv{\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow
P_{c}K^{-}\right)/\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-}\right)}$,
which are summarized in Table 3 and will be further discussed in Sec. VI:

$\displaystyle{\mathcal{B}\Bigg{(}\Lambda_{b}^{0}\rightarrow
K^{-}\Big{(}|\bar{D}\Sigma_{c}\rangle_{1/2^{-}}:|\bar{D}^{*}\Sigma_{c}\rangle_{1/2^{-}}:|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}:|\bar{D}\Sigma_{c}^{*}\rangle_{3/2^{-}}:|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{1/2^{-}}:|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{3/2^{-}}:|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{5/2^{-}}\Big{)}\Bigg{)}\over\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-}\right)}$
(85) $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}8.2~{}~{}~{}~{}:~{}~{}~{}~{}~{}\,1.2\,~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}{\bf
1}~{}~{}~{}~{}~{}~{}\,:~{}~{}~{}~{}~{}~{}0~{}~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}\,4.8\,~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}0.18~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}0~{}\,.$

##  V Decay properties through the Fierz rearrangement

We have applied the Fierz rearrangement fierz  of the Dirac and color indices
to study decay properties of the $P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and
$P_{c}(4457)^{+}$ as $\bar{D}^{(*)}\Sigma_{c}$ molecular states, based on the
currents $J_{1\cdots 3}$ Chen:2020pac . In this section we follow the same
procedures to study decay properties of $\bar{D}^{(*)}\Sigma_{c}^{*}$
molecular states using the currents $J_{4\cdots 7}$. We shall study their
decays into charmonium mesons and spin-1/2 light baryons as well as charmed
mesons and spin-1/2 charmed baryons, such as $J/\psi p$ and
$\bar{D}\Lambda_{c}$, etc.

We refer to Ref. Chen:2020pac  for detailed discussions. This method has been
applied to study strong decay properties of the $Z_{c}(3900)$, $X(3872)$, and
$X(6900)$ in Refs. Chen:2019wjd ; Chen:2019eeq ; Chen:2020xwe , and a similar
arrangement of the spin and color indices in the nonrelativistic case has been
applied to study decay properties of $XYZ$ and $P_{c}$ states in Refs.
Voloshin:2013dpa ; Maiani:2017kyi ; Voloshin:2018pqn ; Voloshin:2019aut ;
Wang:2019spc ; Xiao:2019spy ; Cheng:2020nho .

###  V.1 Input parameters

To study decays of $\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular states into
charmonium mesons and light baryons, we need to use the $\theta(x)$ currents.
We can construct them by combining charmonium operators and light baryon
fields, which has been done in Ref. Chen:2020pac . In the present study we
need couplings of charmonium operators to charmonium states, which are listed
in Table 2. We also need the Ioffe’s light baryon field Ioffe:1981kw ;
Ioffe:1982ce ; Espriu:1983hu ; Chen:2008qv ; Chen:2009sf ; Chen:2010ba ;
Chen:2011rh ; Dmitrasinovic:2016hup :

$\displaystyle N$ $\displaystyle=$ $\displaystyle N_{1}-N_{2}$
$\displaystyle=$
$\displaystyle\epsilon^{abc}(u_{a}^{T}\mathbb{C}d_{b})\gamma_{5}u_{c}-\epsilon^{abc}(u_{a}^{T}\mathbb{C}\gamma_{5}d_{b})u_{c}\,.$

It couples to the proton through

$\langle 0|N|p\rangle=f_{p}u_{p}\,,$ (87)

with $u_{p}$ the Dirac spinor of the proton. The decay constant $f_{p}$ has
been calculated in Ref. Chen:2012ex  to be

$f_{p}=0.011{\rm~{}GeV}^{3}\,.$ (88)

Table 2: Couplings of meson operators to meson states, where color indices are omitted for simplicity. Taken from Ref. Chen:2019wjd . Operators |  $I^{G}J^{PC}$ |  Mesons |  $I^{G}J^{PC}$ |  Couplings |  Decay Constants
---|---|---|---|---|---
$I^{S}=\bar{c}c$ | $0^{+}0^{++}$ | $\chi_{c0}(1P)$ | $0^{+}0^{++}$ | $\langle 0|I^{S}|\chi_{c0}\rangle=m_{\chi_{c0}}f_{\chi_{c0}}$ |  $f_{\chi_{c0}}=343$ MeV Veliev:2010gb
$I^{P}=\bar{c}i\gamma_{5}c$ | $0^{+}0^{-+}$ | $\eta_{c}$ | $0^{+}0^{-+}$ | $\langle 0|I^{P}|\eta_{c}\rangle=\lambda_{\eta_{c}}$ | $\lambda_{\eta_{c}}={f_{\eta_{c}}m_{\eta_{c}}^{2}\over 2m_{c}}$
$I^{V}_{\mu}=\bar{c}\gamma_{\mu}c$ | $0^{-}1^{--}$ | $J/\psi$ | $0^{-}1^{--}$ | $\langle 0|I^{V}_{\mu}|J/\psi\rangle=m_{J/\psi}f_{J/\psi}\epsilon_{\mu}$ |  $f_{J/\psi}=418$ MeV Becirevic:2013bsa
$I^{A}_{\mu}=\bar{c}\gamma_{\mu}\gamma_{5}c$ | $0^{+}1^{++}$ | $\eta_{c}$ | $0^{+}0^{-+}$ | $\langle 0|I^{A}_{\mu}|\eta_{c}\rangle=ip_{\mu}f_{\eta_{c}}$ |  $f_{\eta_{c}}=387$ MeV Becirevic:2013bsa
$\chi_{c1}(1P)$ | $0^{+}1^{++}$ | $\langle 0|I^{A}_{\mu}|\chi_{c1}\rangle=m_{\chi_{c1}}f_{\chi_{c1}}\epsilon_{\mu}$ |  $f_{\chi_{c1}}=335$ MeV Novikov:1977dq
$I^{T}_{\mu\nu}=\bar{c}\sigma_{\mu\nu}c$ | $0^{-}1^{\pm-}$ | $J/\psi$ | $0^{-}1^{--}$ | $\langle 0|I^{T}_{\mu\nu}|J/\psi\rangle=if^{T}_{J/\psi}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f_{J/\psi}^{T}=410$ MeV Becirevic:2013bsa
$h_{c}(1P)$ | $0^{-}1^{+-}$ | $\langle 0|I^{T}_{\mu\nu}|h_{c}\rangle=if^{T}_{h_{c}}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}$ |  $f_{h_{c}}^{T}=235$ MeV Becirevic:2013bsa
$O^{S}=\bar{c}q$ | $0^{+}$ | $\bar{D}_{0}^{*}$ | $0^{+}$ | $\langle 0|O^{S}|\bar{D}_{0}^{*}\rangle=m_{D_{0}^{*}}f_{D_{0}^{*}}$ |  $f_{D_{0}^{*}}=410$ MeV Narison:2015nxh
$O^{P}=\bar{c}i\gamma_{5}q$ | $0^{-}$ | $\bar{D}$ | $0^{-}$ | $\langle 0|O^{P}|\bar{D}\rangle=\lambda_{D}$ | $\lambda_{D}={f_{D}m_{D}^{2}\over{m_{c}+m_{d}}}$
$O^{V}_{\mu}=\bar{c}\gamma_{\mu}q$ | $1^{-}$ | $\bar{D}^{*}$ | $1^{-}$ | $\langle 0|O^{V}_{\mu}|\bar{D}^{*}\rangle=m_{D^{*}}f_{D^{*}}\epsilon_{\mu}$ |  $f_{D^{*}}=253$ MeV Chang:2018aut
$O^{A}_{\mu}=\bar{c}\gamma_{\mu}\gamma_{5}q$ | $1^{+}$ | $\bar{D}$ | $0^{-}$ | $\langle 0|O^{A}_{\mu}|\bar{D}\rangle=ip_{\mu}f_{D}$ |  $f_{D}=211.9$ MeV pdg
$\bar{D}_{1}$ | $1^{+}$ | $\langle 0|O^{A}_{\mu}|\bar{D}_{1}\rangle=m_{D_{1}}f_{D_{1}}\epsilon_{\mu}$ |  $f_{D_{1}}=356$ MeV Narison:2015nxh
$O^{T}_{\mu\nu}=\bar{c}\sigma_{\mu\nu}q$ | $1^{\pm}$ | $\bar{D}^{*}$ | $1^{-}$ | $\langle 0|O^{T}_{\mu\nu}|\bar{D}^{*}\rangle=if_{D^{*}}^{T}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f_{D^{*}}^{T}\approx 220$ MeV
– | $1^{+}$ | – | –

To study decays of $\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular states into charmed
mesons and charmed baryons, we need to use the $\eta(x)$ and $\xi(x)$
currents. They have been constructed in Sec. II by combining charmed meson
operators and charmed baryon fields. In the present study we need couplings of
charmed meson operators to charmed meson states, which are also listed in
Table 2. Besides, we need couplings of the charmed baryon fields,
$J_{\mathcal{B}}$ defined in Eqs. (16), to the ground-state charmed baryons
$\mathcal{B}=\Lambda_{c}/\Sigma_{c}$:

$\langle
0|J_{\mathcal{B}}|\mathcal{B}\rangle=f_{\mathcal{B}}u_{\mathcal{B}}\,.$ (89)

Note that we do not investigate decays of
$|\bar{D}^{(*)}\Sigma_{c}^{*};J^{P}\rangle$ into the
$\bar{D}^{(*)}\Sigma_{c}^{*}$ final states in the present study, because some
of the $J=3/2$ charmed baryon fields still remain unclear Chen:2020pac . The
decay constants $f_{\mathcal{B}}$ have been calculated in Refs. Liu:2007fg ;
Chen:2017sci ; Cui:2019dzj  to be

$\displaystyle f_{\Lambda_{c}}$ $\displaystyle=$ $\displaystyle
0.015{\rm~{}GeV}^{3}\,,$ (90) $\displaystyle f_{\Sigma_{c}}$ $\displaystyle=$
$\displaystyle 0.036{\rm~{}GeV}^{3}\,.$

These values are evaluated using the method of QCD sum rules Shifman:1978bx ;
Reinders:1984sr  within the heavy quark effective theory Grinstein:1990mj ;
Eichten:1989zv ; Falk:1990yz , while the full QCD decay constant $f_{p}$ for
the proton has been given in Eq. (88). These two different schemes cause some,
but not large, theoretical uncertainties.

###  V.2 Fierz rearrangement

In this subsection we perform the Fierz rearrangement separately for
$\eta_{4\cdots 7}$ and $\xi_{4\cdots 7}$. The obtained results will be used
later to study strong decay properties of $\bar{D}\Sigma_{c}^{*}$ and
$\bar{D}^{*}\Sigma_{c}^{*}$ molecular states.

Before doing this, we note again that the Fierz rearrangement in the Lorentz
space is actually a matrix identity. It is valid if each quark field in the
initial and final currents is at the same location, e.g., we can apply the
Fierz rearrangement to transform a non-local current
$\eta=[\bar{c}(x)u(x)]~{}[u(y)d(y)c(y)]$ into the combination of many non-
local currents $\theta=[\bar{c}(x)c(y)]~{}[u(y)d(y)u(x)]$, with all the quark
fields remaining at same locations. Keeping this in mind, we shall omit the
coordinates in this subsection.

####  V.2.1 $\eta\rightarrow\theta$ and $\xi\rightarrow\theta$

Using the color rearrangement Chen:2020pac

$\delta^{ab}\epsilon^{cde}={1\over 3}~{}\delta^{ae}\epsilon^{bcd}-{1\over
2}~{}\lambda^{ae}_{n}\epsilon^{bcf}\lambda^{fd}_{n}+{1\over
2}~{}\lambda^{ae}_{n}\epsilon^{bdf}\lambda^{fc}_{n}\,,$ (91)

together with the Fierz rearrangement to interchange the $u_{b}$ and $c_{e}$
quark fields, we can transform an $\eta$ current into the combination of many
$\theta$ currents:

$\displaystyle\eta_{4}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle[\bar{c}_{a}\gamma_{\mu}c_{a}]\left(-{1\over
32}g^{\alpha\mu}-{i\over 96}\sigma^{\alpha\mu}\right)N$ (92)
$\displaystyle+~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]\left(-{1\over
32}g^{\alpha\mu}\gamma_{5}-{i\over 96}\sigma^{\alpha\mu}\gamma_{5}\right)N$
$\displaystyle+~{}[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]\left({i\over
48}g^{\alpha\mu}\gamma^{\nu}+{1\over
96}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)N$
$\displaystyle+~{}\cdots\,,$ $\displaystyle\eta_{5}$
$\displaystyle\rightarrow$ $\displaystyle+{1\over
8}~{}[\bar{c}_{a}c_{a}]~{}\gamma_{5}N+{1\over
8}~{}[\bar{c}_{a}\gamma_{5}c_{a}]~{}N$ (93) $\displaystyle+{1\over
16}~{}[\bar{c}_{a}\gamma_{\mu}c_{a}]~{}\gamma^{\mu}\gamma_{5}N-{1\over
16}~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]~{}\gamma^{\mu}N$
$\displaystyle+{1\over
48}~{}[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]~{}\sigma^{\mu\nu}\gamma_{5}N+\cdots\,,$
$\displaystyle\eta_{6}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle[\bar{c}_{a}\gamma_{\mu}c_{a}]\left({3\over
32}g^{\alpha\mu}+{i\over 32}\sigma^{\alpha\mu}\right)N$ (94)
$\displaystyle+~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]\left(-{3\over
32}g^{\alpha\mu}\gamma_{5}-{i\over 32}\sigma^{\alpha\mu}\gamma_{5}\right)N$
$\displaystyle+~{}\cdots\,,$ $\displaystyle\eta_{7}^{\alpha\beta}$
$\displaystyle\rightarrow$ $\displaystyle\Big{(}{i\over
144}\sigma^{\alpha\rho}\epsilon^{\beta\mu\nu\rho}+{1\over
72}g^{\alpha\mu}\sigma^{\beta\nu}\gamma_{5}-{1\over
144}g^{\alpha\beta}\sigma^{\mu\nu}\gamma_{5}\Big{)}$ (95)
$\displaystyle\times~{}[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]~{}N+\cdots\,.$

In the above expressions, we have kept all the color-singlet-color-singlet
meson-baryon terms depending on the $J=1/2$ light baryon fields, but omitted
that: a) the color-octet-color-octet meson-baryon terms, such as
$[\lambda^{ae}_{n}\bar{c}_{a}c_{e}][\epsilon^{bcf}\lambda^{fd}_{n}u_{b}u_{c}d_{d}]$,
and b) terms depending on the $J=3/2$ light baryon fields.

Similarly, we can use Eq. (91) together with the Fierz rearrangement to
interchange the $d_{b}$ and $c_{e}$ quark fields, and transform a $\xi$
current into the combination of many $\theta$ currents:

$\displaystyle\sqrt{2}\xi_{4}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle[\bar{c}_{a}\gamma_{\mu}c_{a}]\left({1\over
16}g^{\alpha\mu}+{i\over 48}\sigma^{\alpha\mu}\right)N$ (96)
$\displaystyle+~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]\left({1\over
16}g^{\alpha\mu}\gamma_{5}+{i\over 48}\sigma^{\alpha\mu}\gamma_{5}\right)N$
$\displaystyle+~{}[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]\left(-{i\over
24}g^{\alpha\mu}\gamma^{\nu}-{1\over
48}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)N$
$\displaystyle+~{}\cdots\,,$ $\displaystyle\sqrt{2}\xi_{5}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
4}~{}[\bar{c}_{a}c_{a}]~{}\gamma_{5}N-{1\over
4}~{}[\bar{c}_{a}\gamma_{5}c_{a}]~{}N$ (97) $\displaystyle-{1\over
8}~{}[\bar{c}_{a}\gamma_{\mu}c_{a}]~{}\gamma^{\mu}\gamma_{5}N+{1\over
8}~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]~{}\gamma^{\mu}N$
$\displaystyle-{1\over
24}~{}[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]~{}\sigma^{\mu\nu}\gamma_{5}N+\cdots\,,$
$\displaystyle\sqrt{2}\xi_{6}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle[\bar{c}_{a}\gamma_{\mu}c_{a}]\left(-{3\over
16}g^{\alpha\mu}-{i\over 16}\sigma^{\alpha\mu}\right)N$ (98)
$\displaystyle+~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]\left({3\over
16}g^{\alpha\mu}\gamma_{5}+{i\over 16}\sigma^{\alpha\mu}\gamma_{5}\right)N$
$\displaystyle+~{}\cdots\,,$ $\displaystyle\sqrt{2}\xi_{7}^{\alpha\beta}$
$\displaystyle\rightarrow$ $\displaystyle\Big{(}-{i\over
72}\sigma^{\alpha\rho}\epsilon^{\beta\mu\nu\rho}-{1\over
36}g^{\alpha\mu}\sigma^{\beta\nu}\gamma_{5}$ (99) $\displaystyle~{}~{}+{1\over
72}g^{\alpha\beta}\sigma^{\mu\nu}\gamma_{5}\Big{)}~{}[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]~{}N+\cdots\,.$

####  V.2.2 $\eta\rightarrow\eta$ and $\eta\rightarrow\xi$

Using the color rearrangement

$\delta^{ab}\epsilon^{cde}={1\over 3}~{}\delta^{ac}\epsilon^{bde}-{1\over
2}~{}\lambda^{ac}_{n}\epsilon^{bdf}\lambda^{fe}_{n}+{1\over
2}~{}\lambda^{ac}_{n}\epsilon^{bef}\lambda^{fd}_{n}\,,$ (100)

together with the Fierz rearrangement to interchange the $u_{b}$ and $u_{c}$
quark fields, we can transform an $\eta$ current into the combination of many
$\eta$ currents.

Using another color rearrangement:

$\delta^{ab}\epsilon^{cde}={1\over 3}~{}\delta^{ad}\epsilon^{cbe}+{1\over
2}~{}\lambda^{ad}_{n}\epsilon^{bcf}\lambda^{fe}_{n}-{1\over
2}~{}\lambda^{ad}_{n}\epsilon^{bef}\lambda^{fc}_{n}\,,$ (101)

together with the Fierz rearrangement to interchange the $u_{b}$ and $d_{d}$
quark fields, we can transform an $\eta$ current into the combination of many
$\xi$ currents.

Altogether, we obtain:

$\displaystyle\eta_{4}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle\left({1\over 16}g^{\alpha\mu}+{i\over
48}\sigma^{\alpha\mu}\right)~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\Lambda_{c}^{+}$
(102) $\displaystyle+$ $\displaystyle\left({i\over
384}\sigma^{\alpha\sigma}\epsilon^{\mu\nu\rho\sigma}-{1\over
128}\epsilon^{\alpha\mu\nu\rho}\right)[\bar{c}_{a}\sigma_{\mu\nu}u_{a}]\gamma_{\rho}\gamma_{5}\Sigma_{c}^{+}$
$\displaystyle+$ $\displaystyle\left({i\sqrt{2}\over
384}\sigma^{\alpha\sigma}\epsilon^{\mu\nu\rho\sigma}-{\sqrt{2}\over
128}\epsilon^{\alpha\mu\nu\rho}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\sigma_{\mu\nu}d_{a}]~{}\gamma_{\rho}\gamma_{5}\Sigma_{c}^{++}+\cdots\,,$
$\displaystyle\eta_{5}$ $\displaystyle\rightarrow$ $\displaystyle-{1\over
4}~{}[\bar{c}_{a}\gamma_{5}u_{a}]~{}\Lambda_{c}^{+}-{1\over
48}~{}[\bar{c}_{a}\sigma_{\mu\nu}u_{a}]~{}\sigma^{\mu\nu}\gamma_{5}\Lambda_{c}^{+}$
(103) $\displaystyle-{1\over
32}~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\gamma^{\mu}\gamma_{5}\Sigma_{c}^{+}+{1\over
32}~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}u_{a}]~{}\gamma^{\mu}\Sigma_{c}^{+}$
$\displaystyle-{\sqrt{2}\over
32}[\bar{c}_{a}\gamma_{\mu}d_{a}]\gamma^{\mu}\gamma_{5}\Sigma_{c}^{++}+{\sqrt{2}\over
32}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}d_{a}]\gamma^{\mu}\Sigma_{c}^{++}$
$\displaystyle+~{}\cdots\,,$ $\displaystyle\eta_{6}^{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle\left({i\over
16}g^{\alpha\mu}\gamma^{\nu}+{1\over
32}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)~{}[\bar{c}_{a}\sigma_{\mu\nu}u_{a}]~{}\Lambda_{c}^{+}$
(104) $\displaystyle+$ $\displaystyle\left({1\over
96}g^{\alpha\mu}\gamma^{\nu}\gamma_{5}+{1\over
96}g^{\alpha\nu}\gamma^{\mu}\gamma_{5}-{1\over
192}g^{\mu\nu}\gamma^{\alpha}\gamma_{5}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{+}$
$\displaystyle+$ $\displaystyle\left({1\over
64}g^{\alpha\mu}\gamma^{\nu}-{1\over 64}g^{\alpha\nu}\gamma^{\mu}-{i\over
64}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}u_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{+}$
$\displaystyle+$ $\displaystyle\left({\sqrt{2}\over
96}g^{\alpha\mu}\gamma^{\nu}\gamma_{5}+{\sqrt{2}\over
96}g^{\alpha\nu}\gamma^{\mu}\gamma_{5}-{\sqrt{2}\over
192}g^{\mu\nu}\gamma^{\alpha}\gamma_{5}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\gamma_{\mu}d_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{++}$
$\displaystyle+$ $\displaystyle\left({\sqrt{2}\over
64}g^{\alpha\mu}\gamma^{\nu}-{\sqrt{2}\over
64}g^{\alpha\nu}\gamma^{\mu}-{i\sqrt{2}\over
64}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}d_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{++}+\cdots\,,$
$\displaystyle\eta_{7}^{\alpha\beta}$ $\displaystyle\rightarrow$
$\displaystyle\Big{(}{1\over 36}g^{\alpha\mu}g^{\beta\nu}-{1\over
144}g^{\alpha\beta}g^{\mu\nu}+{i\over 144}g^{\alpha\mu}\sigma^{\beta\nu}$
(105) $\displaystyle+{i\over
144}g^{\alpha\nu}\sigma^{\beta\mu}\Big{)}~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{+}$
$\displaystyle+$ $\displaystyle\Big{(}{\sqrt{2}\over
36}g^{\alpha\mu}g^{\beta\nu}-{\sqrt{2}\over
144}g^{\alpha\beta}g^{\mu\nu}+{i\sqrt{2}\over
144}g^{\alpha\mu}\sigma^{\beta\nu}$ $\displaystyle+{i\sqrt{2}\over
144}g^{\alpha\nu}\sigma^{\beta\mu}\Big{)}~{}[\bar{c}_{a}\gamma_{\mu}d_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{++}+\cdots\,.$

In the above expressions, we have kept all the color-singlet-color-singlet
meson-baryon terms depending on the $J^{P}=1/2^{+}$ charmed baryon fields,
i.e., $J_{\Lambda_{c}^{+}}$ and $J_{\Sigma_{c}^{+/++}}$ defined in Eqs. (16).
We have omitted that: a) the color-octet-color-octet meson-baryon terms, and
b) terms depending on the $J=3/2$ charmed baryon fields.

####  V.2.3 $\xi\rightarrow\eta$

Using Eqs. (100) and (101) together with the Fierz rearrangement in the
Lorentz space, we can transform a $\xi$ current into the combination of many
$\eta$ currents (but without $\xi$ currents):

$\displaystyle\sqrt{2}\xi_{4}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle\left(-{1\over 8}g^{\alpha\mu}-{i\over
24}\sigma^{\alpha\mu}\right)~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\Lambda_{c}^{+}$
(106) $\displaystyle+$ $\displaystyle\left({i\over
192}\sigma^{\alpha\sigma}\epsilon^{\mu\nu\rho\sigma}-{1\over
64}\epsilon^{\alpha\mu\nu\rho}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\sigma_{\mu\nu}u_{a}]~{}\gamma_{\rho}\gamma_{5}\Sigma_{c}^{+}+\cdots\,,$
$\displaystyle\sqrt{2}\xi_{5}$ $\displaystyle\rightarrow$
$\displaystyle+{1\over
2}~{}[\bar{c}_{a}\gamma_{5}u_{a}]~{}\Lambda_{c}^{+}+{1\over
24}~{}[\bar{c}_{a}\sigma_{\mu\nu}u_{a}]~{}\sigma^{\mu\nu}\gamma_{5}\Lambda_{c}^{+}$
(107) $\displaystyle-{1\over
16}[\bar{c}_{a}\gamma_{\mu}u_{a}]\gamma^{\mu}\gamma_{5}\Sigma_{c}^{+}+{1\over
16}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}u_{a}]\gamma^{\mu}\Sigma_{c}^{+}$
$\displaystyle+~{}\cdots\,,$ $\displaystyle\sqrt{2}\xi_{6}^{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle\left(-{i\over
8}g^{\alpha\mu}\gamma^{\nu}-{1\over
16}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)~{}[\bar{c}_{a}\sigma_{\mu\nu}u_{a}]~{}\Lambda_{c}^{+}$
(108) $\displaystyle+$ $\displaystyle\left({1\over
48}g^{\alpha\mu}\gamma^{\nu}\gamma_{5}+{1\over
48}g^{\alpha\nu}\gamma^{\mu}\gamma_{5}-{1\over
96}g^{\mu\nu}\gamma^{\alpha}\gamma_{5}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{+}$
$\displaystyle+$ $\displaystyle\left({1\over
32}g^{\alpha\mu}\gamma^{\nu}-{1\over 32}g^{\alpha\nu}\gamma^{\mu}-{i\over
32}\epsilon^{\alpha\mu\nu\rho}\gamma_{\rho}\gamma_{5}\right)$
$\displaystyle\times~{}[\bar{c}_{a}\gamma_{\mu}\gamma_{5}u_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{+}+\cdots\,,$
$\displaystyle\sqrt{2}\xi_{7}^{\alpha\beta}$ $\displaystyle\rightarrow$
$\displaystyle\Big{(}{1\over 18}g^{\alpha\mu}g^{\beta\nu}-{1\over
72}g^{\alpha\beta}g^{\mu\nu}+{i\over 72}g^{\alpha\mu}\sigma^{\beta\nu}$ (109)
$\displaystyle+{i\over
72}g^{\alpha\nu}\sigma^{\beta\mu}\Big{)}~{}[\bar{c}_{a}\gamma_{\mu}u_{a}]~{}\gamma_{\nu}\gamma_{5}\Sigma_{c}^{+}+\cdots\,.$

###  V.3 Decay analyses

Based on the Fierz rearrangements derived in the previous subsection, we study
strong decay properties of $\bar{D}^{(*)0}\Sigma_{c}^{*+}$ and
$D^{(*)-}\Sigma_{c}^{*++}$ molecular states in this subsection. As an example,
we shall first investigate $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$
through the $\eta_{4}$ current and the Fierz rearrangements given in Eqs. (92)
and (102). Others will be similarly investigated. The obtained results will be
combined in Sec. V.4 to further study $\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular
states of $I=1/2$.

####  V.3.1 $\eta_{4}\rightarrow\theta/\eta/\xi$

(a) $\eta\rightarrow\theta$ (b) $\eta\rightarrow\eta$

(c) $\eta\rightarrow\xi$

Figure 5: Fall-apart decays of $\bar{D}^{(*)0}\Sigma_{c}^{(*)+}$ molecular
states, investigated through the $\eta$ currents. There are three possible
decay processes: a) $\eta\rightarrow\theta$, b) $\eta\rightarrow\eta$, and c)
$\eta\rightarrow\xi$. Their probabilities are the same (33%), if only
considering the color degree of freedom. Taken from Ref. Chen:2020pac .

As an example, we investigate $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$
through the $\eta_{4}$ current and the Fierz rearrangements given in Eqs. (92)
and (102).

First we study Eq. (92). As depicted in Fig. 5(a), when the $\bar{c}_{a}$ and
$c_{e}$ quarks meet each other and the other three quarks meet together at the
same time, $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ can decay into one
charmonium meson and one light baryon:

$\displaystyle\left[\delta^{ab}\bar{c}_{a}u_{b}\right]~{}\left[\epsilon^{cde}u_{c}d_{d}c_{e}\right]$
$\displaystyle\xlongequal{\rm color}$ $\displaystyle{1\over
3}\delta^{ae}\epsilon^{bcd}~{}\bar{c}_{a}u_{b}~{}u_{c}d_{d}c_{e}+\cdots$
$\displaystyle\xlongequal{\rm Fierz}$ $\displaystyle{1\over
3}~{}\left[\delta^{ae}\bar{c}_{a}c_{e}\right]~{}\left[\epsilon^{bcd}u_{c}d_{d}u_{b}\right]+\cdots.$

Especially, we need to apply the Fierz rearrangement in the first and third
steps to interchange both the color and Dirac indices of the $u_{b}$ and
$c_{e}$ quark fields.

The above decay process can be described by the Fierz rearrangement given in
Eq. (92), from which we extract the following two decay channels that are
kinematically allowed:

  1. 1.

The decay of $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ into $\eta_{c}p$ is
contributed by $[\bar{c}_{a}\gamma_{\mu}\gamma_{5}c_{a}]N$:

$\displaystyle\langle\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}(q)~{}|~{}\eta_{c}(q_{1})~{}p(q_{2})\rangle$
$\displaystyle\approx$
$\displaystyle{a_{4}}~{}if_{\eta_{c}}f_{p}q_{1}^{\mu}~{}\bar{u}^{\alpha}\left(-{1\over
32}g_{\alpha\mu}\gamma_{5}-{i\over
96}\sigma_{\alpha\mu}\gamma_{5}\right)u_{p}\,,$

where $u_{\alpha}$ and $u_{p}$ are spinors of
$|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ and proton, respectively; $a_{4}$
is an overall factor related to the coupling of $\eta_{4}$ to
$|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ and the dynamical process of Fig.
5(a).

  2. 2.

The decay of $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ into $J/\psi p$ is
contributed by both $[\bar{c}_{a}\gamma_{\mu}c_{a}]N$ and
$[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]N$:

$\displaystyle\langle\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}(q)|J/\psi(q_{1},\epsilon_{1})~{}p(q_{2})\rangle$
$\displaystyle\approx$
$\displaystyle{a_{4}}~{}m_{J/\psi}f_{J/\psi}f_{p}\epsilon_{1}^{\mu}~{}\bar{u}^{\alpha}\left(-{1\over
32}g_{\alpha\mu}-{i\over 96}\sigma_{\alpha\mu}\right)u_{p}$ $\displaystyle+$
$\displaystyle{a_{4}}~{}if^{T}_{J/\psi}f_{p}~{}\left(q_{1}^{\mu}\epsilon_{1}^{\nu}-q_{1}^{\nu}\epsilon_{1}^{\mu}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\bar{u}^{\alpha}\left({i\over
48}g_{\alpha\mu}\gamma_{\nu}+{1\over
96}\epsilon_{\alpha\mu\nu\rho}\gamma^{\rho}\gamma_{5}\right)u_{p}\,.$

Then we study Eq. (102). As depicted in Fig. 5(b), when the $\bar{c}_{a}$ and
$u_{c}$ quarks meet each other and the other three quarks meet together at the
same time, $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ can decay into one
charmed meson and one charmed baryon. Similarly, we can study the decay
process depicted in Fig. 5(c). These two processes can be described by the
Fierz rearrangement given in Eq. (102), from which we extract only one decay
channel that is kinematically allowed:

  1. 3.

The decay of $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ into
$\bar{D}^{*0}\Lambda_{c}^{+}$ is contributed by
$[\bar{c}_{a}\gamma_{\mu}u_{a}]\Lambda_{c}^{+}$:

$\displaystyle\langle\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}(q)~{}|~{}\bar{D}^{*0}(q_{1},\epsilon_{1})~{}\Lambda_{c}^{+}(q_{2})\rangle$
$\displaystyle\approx$
$\displaystyle{b_{4}}~{}m_{D^{*}}f_{D^{*}}f_{\Lambda_{c}}\epsilon_{1}^{\mu}~{}\bar{u}^{\alpha}\left({1\over
16}g_{\alpha\mu}+{i\over 48}\sigma_{\alpha\mu}\right)u_{\Lambda_{c}}\,,$

where $u_{\Lambda_{c}}$ is the Dirac spinor of the $\Lambda_{c}^{+}$; $b_{4}$
is an overall factor related to the coupling of $\eta_{4}$ to
$|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ and the dynamical processes of
Fig. 5(b,c).

Simply assuming the mass of $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ to be
about $M_{D}+M_{\Sigma_{c}^{*}}\approx 4385$ MeV, we summarize the above decay
amplitudes to obtain the following partial decay widths:

$\displaystyle\Gamma(|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}42~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle\to J/\psi p)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}60~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{4}^{2}~{}1.5\times 10^{4}~{}{\rm
GeV}^{7}\,.$

There are two different terms, $A\equiv[\bar{c}_{a}\gamma_{\mu}c_{a}]N$ and
$B\equiv[\bar{c}_{a}\sigma_{\mu\nu}c_{a}]N$, both of which can contribute to
the decay of $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$ into $J/\psi p$. Let
us see their individual contributions:

$\displaystyle\Gamma(|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle\to J/\psi
p)\big{|}_{A}$ $\displaystyle=$ $\displaystyle a_{4}^{2}~{}1.0\times
10^{4}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle\to J/\psi
p)\big{|}_{B}$ $\displaystyle=$ $\displaystyle a_{4}^{2}~{}1.1\times
10^{4}~{}{\rm GeV}^{7}\,.$

Hence, their contributions are at the same level, but they almost cancel each
other out, suggesting that their interference is important. However, the phase
angle between them, i.e., the phase angle between the two coupling constants
$f_{J/\psi}$ and $f_{J/\psi}^{T}$, can not be well determined in the present
study. We shall investigate its relevant (theoretical) uncertainty in Appendix
B.

####  V.3.2 $\xi_{4}\rightarrow\theta/\eta$

To study $|D^{-}\Sigma_{c}^{*++};3/2^{-}\rangle$, we use the $\xi_{4}$ current
and the Fierz rearrangements given in Eqs. (96) and (106). Assuming its mass
to be the same as $|\bar{D}^{0}\Sigma_{c}^{*+};3/2^{-}\rangle$, we obtain the
following partial decay widths:

$\displaystyle\Gamma(|D^{-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}84~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{-}\Sigma_{c}^{*++};3/2^{-}\rangle\to J/\psi p)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}120~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{4}^{2}~{}3.0\times 10^{4}~{}{\rm
GeV}^{7}\,.$

Here we have used the same overall factors $a_{4}$ and $b_{4}$ as those for
the $\eta_{4}$ current.

####  V.3.3 $\eta_{5}\rightarrow\theta/\eta/\xi$

To study $|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle$, we use the $\eta_{5}$
current and the Fierz rearrangements given in Eqs. (93) and (103). Assuming
its mass to be about $M_{D^{*}}+M_{\Sigma_{c}^{*}}\approx 4527$ MeV, we obtain
the following partial decay widths:

$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}3.3\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to J/\psi p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}1.0\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}3.2\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\chi_{c1}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}1.1\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to h_{c}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}220~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\bar{D}^{0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}3.5\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}1.6\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}1.4\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to
D^{-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{5}^{2}~{}2.9\times
10^{4}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to\bar{D}^{*0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}3.3\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle\to
D^{*-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{5}^{2}~{}6.6\times
10^{4}~{}{\rm GeV}^{7}\,,$

where $a_{5}$ and $b_{5}$ are two overall factors.

####  V.3.4 $\xi_{5}\rightarrow\theta/\eta$

To study $|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle$, we use the $\xi_{5}$
current and the Fierz rearrangements given in Eqs. (97) and (107). Assuming
its mass to be the same as $|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle$, we
obtain the following partial decay widths:

$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}6.5\times 10^{5}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to
J/\psi p)$ $\displaystyle=$ $\displaystyle a_{5}^{2}~{}2.1\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}6.4\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\chi_{c1}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}2.1\times 10^{3}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to
h_{c}p)$ $\displaystyle=$ $\displaystyle a_{5}^{2}~{}450~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\bar{D}^{0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}7.0\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}3.1\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}2.9\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};1/2^{-}\rangle\to\bar{D}^{*0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}6.6\times 10^{4}~{}{\rm
GeV}^{7}\,.$

####  V.3.5 $\eta_{6}\rightarrow\theta/\eta/\xi$

To study $|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle$, we use the $\eta_{6}$
current and the Fierz rearrangements given in Eqs. (94) and (104). Assuming
its mass to be the same as $|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle$, we
obtain the following partial decay widths:

$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}750~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to J/\psi p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}1.2\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}960~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}4.5\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}36~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to
D^{-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{6}^{2}~{}71~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to\bar{D}^{*0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}4.3\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*0}\Sigma_{c}^{*+};3/2^{-}\rangle\to
D^{*-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{6}^{2}~{}8.7\times
10^{4}~{}{\rm GeV}^{7}\,,$

where $a_{6}$ and $b_{6}$ are two overall factors.

####  V.3.6 $\xi_{6}\rightarrow\theta/\eta$

To study $|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle$, we use the $\xi_{6}$
current and the Fierz rearrangements given in Eqs. (98) and (108). Assuming
its mass to be the same as $|\bar{D}^{*0}\Sigma_{c}^{*+};1/2^{-}\rangle$, we
obtain the following partial decay widths:

$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}1.5\times 10^{3}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\to
J/\psi p)$ $\displaystyle=$ $\displaystyle a_{6}^{2}~{}2.3\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}1.9\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}9.1\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}71~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*-}\Sigma_{c}^{*++};3/2^{-}\rangle\to\bar{D}^{*0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}8.7\times 10^{4}~{}{\rm
GeV}^{7}\,.$

####  V.3.7 $\eta_{7}\rightarrow\theta/\eta/\xi$ and
$\xi_{7}\rightarrow\theta/\eta$

To study $|\bar{D}^{*0}\Sigma_{c}^{*+};5/2^{-}\rangle$, we use the $\eta_{7}$
current and the Fierz rearrangements given in Eqs. (95) and (105), but we do
not obtain any non-zero decay channel. This state probably mainly decays into
the spin-1 mesons and spin-3/2 baryons, such as $J/\psi N^{*}$ and
$D^{*}\Sigma_{c}^{*}$, etc. However, these final states are not investigated
in the present study. The same results are obtained for
$|D^{*-}\Sigma_{c}^{*++};5/2^{-}\rangle$.

###  V.4 Isospin analyses

In this subsection we collect the results calculated in the previous
subsection to further study decay properties of $\bar{D}^{(*)}\Sigma_{c}^{*}$
molecular states with $I=1/2$.

Combining the results of Sec. V.3.1 and Sec. V.3.2, we obtain the following
partial decay widths for $|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle$ of $I=1/2$:

$\displaystyle\Gamma(|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}130~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle\to J/\psi p)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}180~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{4}^{2}~{}4.5\times 10^{4}~{}{\rm
GeV}^{7}\,.$

Combining the results of Sec. V.3.3 and Sec. V.3.4, we obtain the following
partial decay widths for $|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle$ of
$I=1/2$:

$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}9.8\times 10^{5}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to
J/\psi p)$ $\displaystyle=$ $\displaystyle a_{5}^{2}~{}3.1\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}9.5\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\chi_{c1}p)$
$\displaystyle=$ $\displaystyle a_{5}^{2}~{}3.2\times 10^{3}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to
h_{c}p)$ $\displaystyle=$ $\displaystyle a_{5}^{2}~{}670~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\bar{D}^{0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}1.1\times 10^{6}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}4.7\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}4.8\times 10^{3}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to
D^{-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{5}^{2}~{}9.6\times
10^{3}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to\bar{D}^{*0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}1.1\times 10^{4}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle\to
D^{*-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{5}^{2}~{}2.2\times
10^{4}~{}{\rm GeV}^{7}\,.$

Combining the results of Sec. V.3.5 and Sec. V.3.6, we obtain the following
partial decay widths for $|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle$ of
$I=1/2$:

$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}2.2\times 10^{3}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to
J/\psi p)$ $\displaystyle=$ $\displaystyle a_{6}^{2}~{}3.5\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}2.9\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}1.4\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}12~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to
D^{-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{6}^{2}~{}24~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to\bar{D}^{*0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{6}^{2}~{}1.4\times 10^{4}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle\to
D^{*-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{6}^{2}~{}2.9\times
10^{4}~{}{\rm GeV}^{7}\,.$

We do not obtain any non-zero decay channel for
$|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle$ of $I=1/2$. This state probably
mainly decays into spin-1 mesons and spin-3/2 baryons, such as $J/\psi N^{*}$
and $\bar{D}^{*}\Sigma_{c}^{*}$, etc. However, these final states are not
investigated in the present study.

Decay properties of $\bar{D}^{(*)}\Sigma_{c}$ molecular states have been
investigated in Ref. Chen:2020pac , including
$|\bar{D}\Sigma_{c};1/2^{-}\rangle$, $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$,
and $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$. There we used them to explain the
$P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$, respectively.
However, we shall find that the $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ can be
better interpreted in our framework as $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$
and $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$, respectively/inversely.

Accordingly, in this paper we assume masses of
$|\bar{D}\Sigma_{c};1/2^{-}\rangle$, $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$,
and $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$ to be $M_{P_{c}(4312)^{+}}=4311.9$
MeV, $M_{P_{c}(4457)^{+}}=4457.3$ MeV, and $M_{P_{c}(4440)^{+}}=4440.3$ MeV,
respectively. Redoing all the calculations, we summarize the results here, and
note that: a) some errors were detected in the results of Ref. Chen:2020pac
when calculating $\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to J/\psi p)$,
and b) different notations are used here for overall factors.

We extract for $|\bar{D}\Sigma_{c};1/2^{-}\rangle$ of $I=1/2$ that:

$\displaystyle\Gamma(|\bar{D}\Sigma_{c};1/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{1}^{2}~{}3.2\times 10^{5}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}\Sigma_{c};1/2^{-}\rangle\to J/\psi
p)$ $\displaystyle=$ $\displaystyle a_{1}^{2}~{}8.5\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}\Sigma_{c};1/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{1}^{2}~{}5.9\times 10^{4}~{}{\rm
GeV}^{7}\,.$

We extract for $|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$ of $I=1/2$ that:

$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{2}^{2}~{}1.8\times 10^{5}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to
J/\psi p)$ $\displaystyle=$ $\displaystyle a_{2}^{2}~{}5.1\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to\chi_{c0}p)$
$\displaystyle=$ $\displaystyle a_{2}^{2}~{}8.0\times 10^{3}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to\chi_{c1}p)$
$\displaystyle=$ $\displaystyle a_{2}^{2}~{}200~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to\bar{D}^{0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{2}^{2}~{}1.7\times 10^{6}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{2}^{2}~{}6.0\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{2}^{2}~{}5.9\times 10^{4}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle\to
D^{-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{2}^{2}~{}1.2\times
10^{5}~{}{\rm GeV}^{7}\,.$

We extract for $|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$ of $I=1/2$ that:

$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle\to\eta_{c}p)$
$\displaystyle=$ $\displaystyle a_{3}^{2}~{}670~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle\to J/\psi p)$
$\displaystyle=$ $\displaystyle a_{3}^{2}~{}1.4\times 10^{5}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle\to\bar{D}^{*0}\Lambda_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{3}^{2}~{}4.6\times 10^{4}~{}{\rm
GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle\to\bar{D}^{0}\Sigma_{c}^{+})$
$\displaystyle=$ $\displaystyle b_{3}^{2}~{}1.4~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle\to
D^{-}\Sigma_{c}^{++})$ $\displaystyle=$ $\displaystyle b_{3}^{2}~{}2.7~{}{\rm
GeV}^{7}\,.$

We use the above partial decay widths to further derive their corresponding
relative branching ratios. The obtained results are summarized in Table 3,
where a new parameter $t\equiv{b_{i}^{2}/a_{i}^{2}}$ ($i=1\cdots 7$) is
introduced to measure which processes happen more easily, the process depicted
in Fig. 5(a) or the processes depicted in Fig. 5(b,c). We shall discuss these
results in Sec. VI.

##  VI Summary and discussions

Table 3: Relative branching ratios of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular states and their relative production rates in $\Lambda_{b}^{0}$ decays. In the 2rd-12th columns we show branching ratios relative to the $J/\psi p$ channel, such as ${\mathcal{B}(P_{c}\to\eta_{c}p)\over\mathcal{B}(P_{c}\to J/\psi p)}$ in the 3rd column. The parameter $t\equiv{b_{i}^{2}/a_{i}^{2}}$ ($i=1\cdots 7$) is introduced to measure which processes happen more easily, the process depicted in Fig. 5(a) or the processes depicted in Fig. 5(b,c). In the 13th column we show the ratio $\mathcal{R}_{1}(P_{c})\equiv{\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow P_{c}K^{-}\right)\over\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-}\right)}$, and in the 14th column we show the ratio $\mathcal{R}_{2}(P_{c})\equiv{\mathcal{B}(\Lambda_{b}^{0}\to P_{c}K^{-}\to J/\psi pK^{-})\over\mathcal{B}(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-}\to J/\psi pK^{-})}$. In order to calculate $\mathcal{R}_{2}$: a) we have simply assumed $t=1$, and b) we have neglected all the spin-3/2 baryons that $P_{c}$ can decay to, such as the $J/\psi N^{*}$ and $\bar{D}\Sigma_{c}^{*}$ final states, etc. Configuration | Decay Channels | Productions
---|---|---
$J/\psi p$ | $\eta_{c}p$ | $\chi_{c0}p$ | $\chi_{c1}p$ | $h_{c}p$ | $\bar{D}^{0}\Lambda_{c}^{+}$ | $\bar{D}^{*0}\Lambda_{c}^{+}$ | $\bar{D}^{0}\Sigma_{c}^{+}$ | $D^{-}\Sigma_{c}^{++}$ | $\bar{D}^{*0}\Sigma_{c}^{+}$ | $D^{*-}\Sigma_{c}^{++}$ |  $\mathcal{R}_{1}$ |  $\mathcal{R}_{2}$
$|\bar{D}\Sigma_{c};1/2^{-}\rangle$ | $1$ | $3.8$ | – | – | – | – | $0.69t$ | – | – | – | – | $8.2$ | $2.0$
$|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle$ | $1$ | $0.35$ | $0.016$ | $10^{-4}$ | – | $3.4t$ | $1.2t$ | $0.12t$ | $0.23t$ | – | – | $1.2$ | $0.25$
$|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle$ | $1$ | $0.005$ | – | – | – | – | $0.34t$ | $10^{-5}t$ | $10^{-5}t$ | – | – | $\bf 1$ | $\bf 1$
$|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle$ | $1$ | $0.70$ | – | – | – | – | $250t$ | – | – | – | – | – | –
$|\bar{D}^{*}\Sigma_{c}^{*};1/2^{-}\rangle$ | $1$ | $31$ | $0.30$ | $0.10$ | $0.02$ | $34t$ | $1.5t$ | $0.15t$ | $0.30t$ | $0.35t$ | $0.70t$ | $4.8$ | $0.09$
$|\bar{D}^{*}\Sigma_{c}^{*};3/2^{-}\rangle$ | $1$ | $0.006$ | – | $0.008$ | – | – | $0.39t$ | $10^{-5}t$ | $10^{-4}t$ | $0.04t$ | $0.08t$ | $0.18$ | $0.16$
$|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle$ | – | – | – | –

In this paper we systematically investigate the seven possibly existing
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular states of $I=1/2$,
including $\bar{D}\Sigma_{c}$ of $J^{P}={1\over 2}^{-}$,
$\bar{D}^{*}\Sigma_{c}$ of $J^{P}={1\over 2}^{-}/{3\over 2}^{-}$,
$\bar{D}\Sigma_{c}^{*}$ of $J^{P}={3\over 2}^{-}$, and
$\bar{D}^{*}\Sigma_{c}^{*}$ of $J^{P}={1\over 2}^{-}/{3\over 2}^{-}/{5\over
2}^{-}$.

Firstly, we systematically construct their corresponding interpolating
currents, and calculate their masses and decay constants using QCD sum rules.
The results are summarized in Table 1, supporting the interpretations of
$P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$ Aaij:2019vzc  as
the $\bar{D}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}$ molecular states. However,
the accuracy of our sum rule results is not good enough to
distinguish/indentify them. To better understand them, we further study their
production and decay properties. The decay constants $f_{X}$ extracted using
QCD sum rules are important input parameters.

Secondly, we use the current algebra to study productions of
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states in $\Lambda_{b}^{0}$ decays.
We derive the relative production rates

$\mathcal{R}_{1}(P_{c})\equiv{\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow
P_{c}K^{-}\right)\over\mathcal{B}\left(\Lambda_{b}^{0}\rightarrow|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-}\right)}\,,$
(127)

and the obtained results are summarized in Table 3.

Thirdly, we use the Fierz rearrangement of the Dirac and color indices to
study decay properties of $\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular states,
including their decays into charmonium mesons and spin-1/2 light baryons as
well as charmed mesons and spin-1/2 charmed baryons, such as $J/\psi p$ and
$\bar{D}\Lambda_{c}$, etc. We calculate their relative branching ratios, and
the obtained results are also summarized in Table 3. The parameter
$t\equiv{b_{i}^{2}/a_{i}^{2}}$ ($i=1\cdots 7$) is introduced to measure which
processes happen more easily, the process depicted in Fig. 5(a) or the
processes depicted in Fig. 5(b,c). Generally speaking, the exchange of one
light quark with another light quark may be easier than its exchange with
another heavy quark Landau , so it can be the case that $t\geq 1$.

In Table 3 we simply assume $t=1$ to further calculate the ratio
$\mathcal{R}_{1}$ in the $J/\psi p$ mass spectrum, that is

$\mathcal{R}_{2}(P_{c})\equiv{\mathcal{B}(\Lambda_{b}^{0}\to P_{c}K^{-}\to
J/\psi
pK^{-})\over\mathcal{B}(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}\rangle_{3/2^{-}}K^{-}\to
J/\psi pK^{-})}\,.$ (128)

In order to calculate this ratio, we have neglected all the spin-3/2 baryons
that $P_{c}$ can decay to, such as the $J/\psi N^{*}$ and
$\bar{D}\Sigma_{c}^{*}$ final states, etc.

Before drawing conclusions, we would like to note:

  * •

When studying masses and decay constants of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
molecular states through QCD sum rules, we calculate two-point correlation
functions at the quark-gluon level as inputs, while masses of charmed mesons
and baryons at the hadron level are not used as input parameters. Accordingly,
the uncertainty/accuracy is moderate but not enough to extract the binding
energy. This means that our sum rule results can only suggest but not
determine: a) whether these $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ molecular states
exist or not, and b) whether they are bound states or resonance states.
Instead, we need to assume their existence, then we can use the extracted
decay constants to further study their production and decay properties.

  * •

When studying relative production rates of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
molecular states in $\Lambda_{b}^{0}$ decays through the current algebra, we
only investigate the hidden-charm pentaquark currents that can couple to these
states through $S$-wave, i.e., $J_{1\cdots 7}$ defined in Eqs. (26-40). There
may exist some other currents coupling to these states through $P$-wave, which
are not taken into account in the present study. Accordingly,
$|\bar{D}\Sigma_{c}^{*};3/2^{-}\rangle$ and
$|\bar{D}^{*}\Sigma_{c}^{*};5/2^{-}\rangle$ may still be produced in
$\Lambda_{b}^{0}$ decays through these “$P$-wave” currents. Besides, their
omissions produce some theoretical uncertainties.

  * •

When studying decay properties of $\bar{D}^{(*)}\Sigma_{c}^{*}$ molecular
states through the Fierz rearrangement, we only consider the leading-order
fall-apart decays described by color-singlet-color-singlet meson-baryon
currents, but neglect the $\mathcal{O}(\alpha_{s})$ corrections described by
color-octet-color-octet meson-baryon currents, so there can be other possible
decay channels. Besides, we do not consider light/charmed baryon fields of
$J=3/2$, so we can not study their decays into the $J/\psi N^{*}$ and
$\bar{D}\Sigma_{c}^{*}$ final states, etc. However, we have kept all the
light/charmed baryon fields that couple to ground-state light/charmed baryons
of $J^{P}=1/2^{+}$, so their decays into these final states are well
investigated in this paper.

Now we can generally discuss about our uncertainties. The uncertainty of our
QCD sum rule results is moderate, while uncertainties of relative branching
ratios as well as the two ratios $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are
much larger. In the present study we work under the naive factorization
scheme, so our uncertainties are significantly larger than the well-developed
QCD factorization scheme Beneke:1999br ; Beneke:2000ry ; Beneke:2001ev , whose
uncertainty is at the 5% level when investigating conventional (heavy) hadrons
Li:2020rcg . On the other hand, in this paper we only calculate the ratios,
which significantly reduces our uncertainties. Accordingly, we roughly
estimate the uncertainty of relative branching ratios to be at the
$X^{+100\%}_{-~{}50\%}$ level. Due to the omission of the “$P$-wave”
pentaquark currents, the uncertainty of the ratio $\mathcal{R}_{1}$ is roughly
estimated to be at the $X^{+200\%}_{-~{}67\%}$ level. We further roughly
estimate the uncertainty of the ratio $\mathcal{R}_{2}$ to be at the
$X^{+300\%}_{-~{}75\%}$ level (or even larger due to the assumption of $t=1$
and the omission of spin-3/2 baryons that $P_{c}$ can decay to).

Finally, we can draw conclusions using the results summarized in Table 3. The
LHCb experiment Aaij:2019vzc  discovered the $P_{c}(4312)^{+}$,
$P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$, and at the same time measured their
relative contributions $\mathcal{R}\equiv\mathcal{B}(\Lambda^{0}_{b}\to
P_{c}^{+}K^{-})\mathcal{B}(P_{c}^{+}\to J/\psi
p)/\mathcal{B}(\Lambda^{0}_{b}\to J/\psi pK^{-})$ to be:

$\displaystyle\mathcal{R}(P_{c}(4312)^{+})$ $\displaystyle=$ $\displaystyle
0.30\pm 0.07^{+0.34}_{-0.09}\%\,,$ $\displaystyle\mathcal{R}(P_{c}(4440)^{+})$
$\displaystyle=$ $\displaystyle 1.11\pm 0.33^{+0.22}_{-0.10}\%\,,$ (129)
$\displaystyle\mathcal{R}(P_{c}(4457)^{+})$ $\displaystyle=$ $\displaystyle
0.53\pm 0.16^{+0.15}_{-0.13}\%\,,$

from which we can derive

$\displaystyle{\mathcal{R}(P_{c}(4312)^{+})\over\mathcal{R}(P_{c}(4440)^{+})}$
$\displaystyle=$ $\displaystyle 0.27^{+0.32}_{-0.14}\,,$ (130)
$\displaystyle{\mathcal{R}(P_{c}(4457)^{+})\over\mathcal{R}(P_{c}(4440)^{+})}$
$\displaystyle=$ $\displaystyle 0.48^{+0.25}_{-0.25}\,.$

These two values are roughly consistent with our results that

$\displaystyle\mathcal{R}_{2}(|\bar{D}\Sigma_{c};1/2^{-}\rangle)$
$\displaystyle=$
$\displaystyle{\mathcal{R}_{2}(|\bar{D}\Sigma_{c};1/2^{-}\rangle)\over\mathcal{R}_{2}(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle)}\approx
2.0\,,$ $\displaystyle\mathcal{R}_{2}(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle)$
$\displaystyle=$
$\displaystyle{\mathcal{R}_{2}(|\bar{D}^{*}\Sigma_{c};1/2^{-}\rangle)\over\mathcal{R}_{2}(|\bar{D}^{*}\Sigma_{c};3/2^{-}\rangle)}\approx
0.25\,,$

given their uncertainties to be roughly at the $X^{+300\%}_{-~{}75\%}$ level.

Therefore, our result supports the interpretations of $P_{c}(4312)^{+}$,
$P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$ as $\bar{D}\Sigma_{c}$ of
$J^{P}={1/2}^{-}$, $\bar{D}^{*}\Sigma_{c}$ of $J^{P}={3/2}^{-}$, and
$\bar{D}^{*}\Sigma_{c}$ of $J^{P}={1/2}^{-}$, respectively. For completeness,
we also investigate the interpretations of $P_{c}(4440)^{+}$ and
$P_{c}(4457)^{+}$ as the $\bar{D}^{*}\Sigma_{c}$ molecular states of
$J^{P}={1/2}^{-}$ and ${3/2}^{-}$ respectively, and the results are given in
Appendix C.

Our results suggest that the $\bar{D}^{*}\Sigma_{c}^{*}$ molecular states of
$J^{P}=1/2^{-}$ and $3/2^{-}$ are also possible to be observed in the $J/\psi
p$ invariant mass spectrum of the $\Lambda_{b}^{0}\to J/\psi pK^{-}$ decays,
and their relative contributions are estimated to be

$\displaystyle{\mathcal{B}(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{1/2^{-}}K^{-}\to
J/\psi pK^{-})\over\mathcal{B}(\Lambda^{0}_{b}\to J/\psi pK^{-})}$
$\displaystyle\approx$ $\displaystyle 0.1\%\,,$
$\displaystyle{\mathcal{B}(\Lambda_{b}^{0}\to|\bar{D}^{*}\Sigma_{c}^{*}\rangle_{3/2^{-}}K^{-}\to
J/\psi pK^{-})\over\mathcal{B}(\Lambda^{0}_{b}\to J/\psi pK^{-})}$
$\displaystyle\approx$ $\displaystyle 0.2\%\,.$

Their relative branching ratios to the $\eta_{c}p$, $\chi_{c0}p$,
$\chi_{c1}p$, $h_{c}p$, $\bar{D}^{0}\Lambda_{c}^{+}$,
$\bar{D}^{*0}\Lambda_{c}^{+}$, $\bar{D}^{0}\Sigma_{c}^{+}$,
$D^{-}\Sigma_{c}^{++}$, $\bar{D}^{*0}\Sigma_{c}^{+}$, and
$D^{*-}\Sigma_{c}^{++}$ final states are also given for future experimental
searches.

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grants No. 11722540 and No. 12075019, the Jiangsu Provincial Double-
Innovation Program under Grant No. JSSCRC2021488, and the Fundamental Research
Funds for the Central Universities.

##  Appendix A Spectral densities

In this appendix we list the spectral densities $\rho_{1\cdots 7}(s)$
extracted for the currents $J_{1\cdots 7}$. In the following expressions,
$\mathcal{F}(s)=\left[(\alpha+\beta)m_{c}^{2}-\alpha\beta s\right]$,
$\mathcal{H}(s)=\left[m_{c}^{2}-\alpha(1-\alpha)s\right]$, and the integration
limits are $\alpha_{min}=\frac{1-\sqrt{1-4m_{c}^{2}/s}}{2}$,
$\alpha_{max}=\frac{1+\sqrt{1-4m_{c}^{2}/s}}{2}$, $\beta_{min}=\frac{\alpha
m_{c}^{2}}{\alpha s-m_{c}^{2}}$, and $\beta_{max}=1-\alpha$.

The spectral density $\rho_{1}(s)$ extracted for the current $J_{1}$ is

$\displaystyle\rho_{1}(s)$ $\displaystyle=$ $\displaystyle
m_{c}\left(\rho^{pert}_{1a}(s)+\rho^{\langle\bar{q}q\rangle}_{1a}(s)+\rho^{\langle
GG\rangle}_{1a}(s)+\rho^{\langle\bar{q}Gq\rangle}_{1a}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{1a}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1a}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{1a}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{1a}(s)\right)$
$\displaystyle+$ $\displaystyle
q\\!\\!\\!/~{}~{}\left(\rho^{pert}_{1b}(s)+\rho^{\langle\bar{q}q\rangle}_{1b}(s)+\rho^{\langle
GG\rangle}_{1b}(s)+\rho^{\langle\bar{q}Gq\rangle}_{1b}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{1b}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1b}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{1b}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{1b}(s)\right)\,,$

where

$\displaystyle\rho^{pert}_{1a}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{13(1-\alpha-\beta)^{3}}{983040\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{1a}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-(1-\alpha-\beta)^{2}}{768\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{1a}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{13(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{1179648\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)\left(80\alpha^{3}+\alpha^{2}(206\beta-79)+\alpha\left(28\beta^{2}-27\beta-1\right)-26(\beta-1)^{2}\beta\right)}{2359296\pi^{8}\alpha^{5}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{1a}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)\left(14\alpha^{2}+2\alpha(15\beta-7)+(\beta-1)\beta\right)}{8192\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{1a}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-29}{1536\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1a}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-6\alpha-29\beta}{3072\pi^{4}\alpha^{2}\beta}\Bigg{\\}}+\mathcal{H}(s)\times\frac{55}{3072\pi^{4}\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{1a}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{52\alpha^{2}-75\alpha+29}{12288\pi^{4}\alpha}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}m_{c}^{2}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)\times\frac{-13}{6144\pi^{4}\alpha}\Bigg{\\}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{1a}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{3}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{13}{288\pi^{2}}\Bigg{\\}}\,,$
$\displaystyle\rho^{pert}_{1b}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{13(1-\alpha-\beta)^{3}}{491520\pi^{8}\alpha^{4}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{1b}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-29(1-\alpha-\beta)^{2}}{12288\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{1b}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{13(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{589824\pi^{8}\alpha^{4}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)\left(167\alpha^{2}+\alpha(223\beta-166)+80\beta^{2}-79\beta-1\right)}{2359296\pi^{8}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{1b}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)\left(110\alpha^{2}+\alpha(217\beta-110)+3(\beta-1)\beta\right)}{32768\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{1b}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-1}{96\pi^{4}\alpha\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1b}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-5\alpha-15\beta}{3072\pi^{4}\alpha\beta}\Bigg{\\}}+\mathcal{H}(s)\times\frac{31}{3072\pi^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{1b}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{30\alpha^{2}-40\alpha+15}{12288\pi^{4}}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}m_{c}^{2}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)\times\frac{-5}{4096\pi^{4}}\Bigg{\\}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{1b}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{3}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{13\alpha}{576\pi^{2}}\Bigg{\\}}\,.$

The spectral density $\rho_{2}(s)$ extracted for the current $J_{2}$ is

$\displaystyle\rho_{2}(s)$ $\displaystyle=$ $\displaystyle
m_{c}\left(\rho^{pert}_{2a}(s)+\rho^{\langle\bar{q}q\rangle}_{2a}(s)+\rho^{\langle
GG\rangle}_{2a}(s)+\rho^{\langle\bar{q}Gq\rangle}_{2a}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{2a}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2a}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{2a}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{2a}(s)\right)$
$\displaystyle+$ $\displaystyle
q\\!\\!\\!/~{}~{}\left(\rho^{pert}_{2b}(s)+\rho^{\langle\bar{q}q\rangle}_{2b}(s)+\rho^{\langle
GG\rangle}_{2b}(s)+\rho^{\langle\bar{q}Gq\rangle}_{2b}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{2b}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2b}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{2b}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{2b}(s)\right)\,,$

where

$\displaystyle\rho^{pert}_{2a}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{49152\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{2a}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-13(1-\alpha-\beta)^{2}}{3072\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{2a}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{5(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{294912\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(1-\alpha-\beta)\left(32\alpha^{3}-\alpha^{2}(16\beta+31)+\alpha\left(-14\beta^{2}+15\beta-1\right)+10(\beta-1)^{2}\beta\right)}{589824\pi^{8}\alpha^{5}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{2a}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)(23\alpha+2\beta-2)}{4096\pi^{6}\alpha^{3}\beta^{2}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{2a}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-5}{192\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2a}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{7\alpha-20\beta}{1536\pi^{4}\alpha^{2}\beta}\Bigg{\\}}+\mathcal{H}(s)\times\frac{11}{512\pi^{4}\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{2a}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{26\alpha^{2}-53\alpha+20}{6144\pi^{4}\alpha}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}m_{c}^{2}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)\times\frac{-13}{6144\pi^{4}\alpha}\Bigg{\\}}\Bigg{\\}}\,,$