# Covalent hadronic molecules induced by shared light quarks

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

###### Abstract

After examining the Feynman diagrams corresponding to the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$, $\bar{D}^{(*)}\Lambda_{c}$,
$D^{(*)}\bar{K}^{*}$, and $D^{(*)}\bar{D}^{(*)}$ hadronic molecular states, we
propose a possible binding mechanism induced by shared light quarks. This
mechanism is similar to the covalent bond in chemical molecules induced by
shared electrons. We use the method of QCD sum rules to calculate its
corresponding light-quark-exchange diagrams, and the obtained results indicate
a model-independent hypothesis: the light-quark-exchange interaction is
attractive when the shared light quarks are totally antisymmetric so that obey
the Pauli principle.

hadronic molecule, covalent bond, QCD sum rules

##  I Introduction

Since the discovery of the $X(3872)$ by Belle in 2003 Choi:2003ue , lots of
charmonium-like $XYZ$ states were discovered in the past two decades pdg .
Some of these structures may contain four quarks and are good candidates of
hidden-charm tetraquark states. In recent years the LHCb Collaboration
continuously observed five $P_{c}/P_{cs}$ states Aaij:2015tga ; Aaij:2019vzc ;
Aaij:2020gdg , which contain five quarks and are good candidates of hidden-
charm pentaquark states. Although there is still a long way to fully
understand how the strong interaction binds these quarks and antiquarks
together, the above exotic structures have become one of the most intriguing
research topics in hadron physics. Their theoretical and experimental studies
are significantly improving our understanding of the non-perturbative
behaviors of the strong interaction at the low energy region. We refer to the
reviews Chen:2016qju ; Liu:2019zoy ; Hosaka:2016pey ; Richard:2016eis ;
Lebed:2016hpi ; Esposito:2016noz ; Ali:2017jda ; Guo:2017jvc ; Olsen:2017bmm ;
Karliner:2017qhf ; Albuquerque:2018jkn ; Guo:2019twa ; Brambilla:2019esw ;
Yang:2020atz ; Dong:2021juy  and references therein for detailed discussions.

Some of the $XYZ$ and $P_{c}/P_{cs}$ states can be interpreted as hadronic
molecular states, which consist of two conventional hadrons Weinberg:1965zz ;
Voloshin:1976ap ; DeRujula:1976zlg ; Tornqvist:1993ng ; Voloshin:2003nt ;
Close:2003sg ; Wong:2003xk ; Braaten:2003he ; Swanson:2003tb ;
Tornqvist:2004qy . For example, the $P_{c}$ states were proposed to be the
$\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular states in Refs. Wu:2010jy ;
Wang:2011rga ; Yang:2011wz ; Karliner:2015ina ; Chen:2015loa ; Chen:2015moa ;
Liu:2019tjn  bound by the one-meson-exchange interaction $\Pi_{M}$, as
depicted in Fig. 2(a). Besides, we know from QCD that there can be the double-
gluon-exchange interaction $\Pi_{G}$ between $\bar{D}^{(*)}$ and
$\Sigma_{c}^{(*)}$, as depicted in Fig. 2(b).

In this paper we propose another possible interaction between $\bar{D}^{(*)}$
and $\Sigma_{c}^{(*)}$ induced by the light-quark-exchange term $\Pi_{Q}$, as
depicted in Fig. 2(c). This term indicates that $\bar{D}^{(*)}$ and
$\Sigma_{c}^{(*)}$ are exchanging and so sharing two light $up/down$ quarks,
as depicted in Fig. 1. It can induce an interaction between $\bar{D}^{(*)}$
and $\Sigma_{c}^{(*)}$, either attractive or repulsive. Note that the two
interactions, $\Pi_{M}$ at the hadron level and $\Pi_{Q}$ at the quark-gluon
level, can overlap with each other. The quark-exchange effect has been studied
in Ref. Hoodbhoy:1986fn  by Hoodbhoy and Jaffe to explain the European Muon
Collaboration (EMC) effect in three-nucleon systems, and later used in Refs.
Modarres:1988sb ; Modarres:2006ry  to study some other nuclei. We also refer
to Ref. wang  for some relevant discussions.

Figure 1: Possible binding mechanism induced by shared light quarks, described
by the light-quark-exchange term $\Pi_{Q}$. Here $q$ denotes a light $up/down$
quark.

(a) $\Pi_{M}$ (b) $\Pi_{G}$

(c) $\Pi_{Q}$

Figure 2: Feynman diagrams between $\bar{D}^{(*)}$ and $\Sigma_{c}^{(*)}$
corresponding to : a) the one-meson-exchange interaction $\Pi_{M}$, b) the
double-gluon-exchange interaction $\Pi_{G}$, and c) the light-quark-exchange
interaction $\Pi_{Q}$. Here $q$ denotes a light $up/down$ quark.

In this paper we shall systematically examine the Feynman diagrams
corresponding to the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$,
$\bar{D}^{(*)}\Lambda_{c}$, $D^{(*)}\bar{K}^{*}$, and $D^{(*)}\bar{D}^{(*)}$
hadronic molecular states. We shall apply the method of QCD sum rules to
investigate the light-quark-exchange term $\Pi_{Q}$, and study its
contributions to these states. Based on the obtained results, we shall study
the binding mechanism induced by shared light quarks. This mechanism is
somewhat similar to the covalent bond in chemical molecules induced by shared
electrons, so we call such hadronic molecules “covalent hadronic molecules”.

Based on the obtained results, we shall further propose a model-independent
hypothesis: “the light-quark-exchange interaction is attractive when the
shared light quarks are totally antisymmetric so that obey the Pauli
principle”. We shall apply this hypothesis to predict some possibly-existing
covalent hadronic molecules. Their binding energies can be at the 10 MeV
level, considering the $P_{c}/P_{cs}$ and $X_{0}(2900)$ Aaij:2020hon ;
Aaij:2020ypa  states as possible covalent hadronic molecules.

Take the $X(3872)$ as another example. We shall find that the light-quark-
exchange term $\Pi_{Q}$ does not contribute to the $D^{(*)}\bar{D}^{(*)}$
molecules, suggesting the $D^{(*)}\bar{D}^{(*)}$ covalent molecules not to
exist. However, there can still be the $D^{(*)}\bar{D}^{(*)}$ hadronic
molecules induced by some other binding mechanisms, such as the one-meson-
exchange interaction Weinberg:1965zz ; Voloshin:1976ap ; DeRujula:1976zlg ;
Tornqvist:1993ng ; Voloshin:2003nt ; Close:2003sg ; Wong:2003xk ;
Braaten:2003he ; Swanson:2003tb ; Tornqvist:2004qy . Especially, the $X(3872)$
can be interpreted as such a $D\bar{D}^{*}$ hadronic molecule, while it was
suggested to be a compact tetraquark state in Refs. Maiani:2004vq ;
Maiani:2014aja ; Hogaasen:2005jv ; Ebert:2005nc ; Barnea:2006sd , a
conventional $c\bar{c}$ state in Refs. Barnes:2003vb ; Eichten:2004uh , and
the mixture of a $c\bar{c}$ state with the $D\bar{D}^{*}$ component in Refs.
Meng:2005er ; Meng:2014ota .

This paper is organized as follows. In Sec. II we systematically investigate
correlation functions of the $D^{-}\Sigma_{c}^{++}$,
$\bar{D}^{0}\Sigma_{c}^{+}$, $I=1/2$ $\bar{D}\Sigma_{c}$, and $I=3/2$
$\bar{D}\Sigma_{c}$ hadronic molecules. In Sec. III we apply the method of QCD
sum rules to investigate the light-quark-exchange term $\Pi_{Q}$, and study
its contributions to these molecules. In Sec. IV we follow the same procedures
and study the
$\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}$,
$\bar{D}^{(*)}\Lambda_{c}$, $D^{(*)}\bar{K}^{*}$, and $D^{(*)}\bar{D}^{(*)}$
hadronic molecules. The obtained results are summarized and discussed in Sec.
V, where we propose the above hypothesis and predict more covalent hadronic
molecules.

##  II Correlation functions of
$D^{-}\Sigma_{c}^{++}/\bar{D}^{0}\Sigma_{c}^{+}/\bar{D}\Sigma_{c}$ molecules

In this section we investigate correlation functions of the
$D^{-}\Sigma_{c}^{++}$, $\bar{D}^{0}\Sigma_{c}^{+}$, $I=1/2$
$\bar{D}\Sigma_{c}$, and $I=3/2$ $\bar{D}\Sigma_{c}$ hadronic molecular
states.

###  II.1 $D^{-}\Sigma_{c}^{++}$ correlation function

In this subsection we investigate correlation function of the
$D^{-}\Sigma_{c}^{++}$ molecule. Firstly, we investigate correlation functions
of the $D^{-}$ meson and the $\Sigma_{c}^{++}$ baryon. Their corresponding
interpolating currents are

$\displaystyle J^{D^{-}}(x)$ $\displaystyle=$
$\displaystyle\bar{c}_{a}(x)\gamma_{5}d_{a}(x)\,,$ (1) $\displaystyle
J^{\Sigma_{c}^{++}}(x)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\epsilon^{abc}u^{T}_{a}(x)\mathbb{C}\gamma^{\mu}u_{b}(x)\gamma_{\mu}\gamma_{5}c_{c}(x)\,,$
(2)

where $a\cdots c$ are color indices; $\mathbb{C}=i\gamma_{2}\gamma_{0}$ is the
charge-conjugation operator; the coefficient $1/\sqrt{2}$ is an isospin
factor.

We write down two-point correlation functions of the $D^{-}$ meson and the
$\Sigma_{c}^{++}$ baryon in the coordinate space:

$\displaystyle\Pi^{D^{-}}(x)$ $\displaystyle=$ $\displaystyle\langle
0|\mathbb{T}\left[J^{D^{-}}(x)J^{D^{-},\dagger}(0)\right]|0\rangle$
$\displaystyle=$ $\displaystyle-{\bf Tr}\left[i{\bf
S}_{q}^{aa^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{a^{\prime}a}(-x)\gamma_{5}\right]\,,$
$\displaystyle\Pi^{\Sigma_{c}^{++}}(x)$ $\displaystyle=$ $\displaystyle\langle
0|\mathbb{T}\left[J^{\Sigma_{c}^{++}}(x)\bar{J}^{\Sigma_{c}^{++}}(0)\right]|0\rangle$
$\displaystyle=$
$\displaystyle\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}{\bf
Tr}\left[i{\bf S}_{q}^{bb^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{aa^{\prime}}(x))^{\rm T}\mathbb{C}\gamma^{\mu}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}\,.$

Their corresponding Feynman diagrams are depicted in Fig. 3(a) and Fig. 3(b),
respectively. In the present study we do not differentiate propagators of the
light $up$ and $down$ quarks, and use $i{\bf S}_{q}^{ab}(x)=i{\bf
S}_{u}^{ab}(x)=i{\bf S}_{d}^{ab}(x)$ to denote both of them in the coordinate
space; besides, we use $i{\bf S}_{c}^{ab}(x)$ to denote propagator of the
heavy $charm$ quark in the coordinate space:

$\displaystyle i{\bf S}_{q}^{ab}(x)$ $\displaystyle=$ $\displaystyle\langle
0|\mathbb{T}\left[u^{a}(x)\bar{u}^{b}(0)\right]|0\rangle$ $\displaystyle=$
$\displaystyle\langle
0|\mathbb{T}\left[d^{a}(x)\bar{d}^{b}(0)\right]|0\rangle\,,$ $\displaystyle
i{\bf S}_{c}^{ab}(x)$ $\displaystyle=$ $\displaystyle\langle
0|\mathbb{T}\left[c^{a}(x)\bar{c}^{b}(0)\right]|0\rangle\,.$ (6)

Figure 3: Feynman diagrams corresponding to: a) $\Pi^{D^{-}}(x)$, b)
$\Pi^{\Sigma_{c}^{++}}(x)$, c) $\Pi^{D^{-}\Sigma_{c}^{++}}_{0}(x)$, and d)
$\Pi^{D^{-}\Sigma_{c}^{++}}_{G}(x)$. $\Pi^{D^{-}}(x)$ and
$\Pi^{\Sigma_{c}^{++}}(x)$ are correlation functions of the $D^{-}$ meson and
the $\Sigma_{c}^{++}$ baryon, respectively. The correlation function of the
$D^{-}\Sigma_{c}^{++}$ molecule satisfies
$\Pi^{D^{-}\Sigma_{c}^{++}}(x)=\Pi^{D^{-}\Sigma_{c}^{++}}_{0}(x)+\Pi^{D^{-}\Sigma_{c}^{++}}_{G}(x)$
with
$\Pi^{D^{-}\Sigma_{c}^{++}}_{0}(x)=\Pi^{D^{-}}(x)\times\Pi^{\Sigma_{c}^{++}}(x)$.

Then we put the $D^{-}$ meson and the $\Sigma_{c}^{++}$ baryon at the same
location, and construct a composite current corresponding to the
$D^{-}\Sigma_{c}^{++}$ molecule,

$\displaystyle J^{D^{-}\Sigma_{c}^{++}}(x)$ $\displaystyle=$ $\displaystyle
J^{D^{-}}(x)\times J^{\Sigma_{c}^{++}}(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{d}(x)\gamma_{5}d_{d}(x)]$ $\displaystyle\times$
$\displaystyle{1\over\sqrt{2}}[\epsilon^{abc}u^{T}_{a}(x)\mathbb{C}\gamma^{\mu}u_{b}(x)\gamma_{\mu}\gamma_{5}c_{c}(x)]\,.$

Its correlation function in the coordinate space is

$\displaystyle\Pi^{D^{-}\Sigma_{c}^{++}}(x)$ (8) $\displaystyle=$
$\displaystyle\langle
0|\mathbb{T}\left[J^{D^{-}\Sigma_{c}^{++}}(x)\bar{J}^{D^{-}\Sigma_{c}^{++}}(0)\right]|0\rangle$
$\displaystyle=$ $\displaystyle-{\bf Tr}\left[i{\bf
S}_{q}^{dd^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{d^{\prime}d}(-x)\gamma_{5}\right]$ $\displaystyle\times$
$\displaystyle\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}{\bf
Tr}\left[i{\bf S}_{q}^{bb^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{aa^{\prime}}(x))^{\rm T}\mathbb{C}\gamma^{\mu}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}$ $\displaystyle=$
$\displaystyle\Pi^{D^{-}\Sigma_{c}^{++}}_{0}(x)+\Pi^{D^{-}\Sigma_{c}^{++}}_{G}(x)\,,$

where

$\Pi^{D^{-}\Sigma_{c}^{++}}_{0}(x)=\Pi^{D^{-}}(x)\times\Pi^{\Sigma_{c}^{++}}(x)\,,$
(9)

is the leading term contributed by non-correlated $D^{-}$ and
$\Sigma_{c}^{++}$, and $\Pi^{D^{-}\Sigma_{c}^{++}}_{G}(x)$ describes the
double-gluon-exchange interaction between them. Their corresponding Feynman
diagrams are depicted in Fig. 3(c) and Fig. 3(d), respectively.

The double-gluon-exchange term $\Pi^{D^{-}\Sigma_{c}^{++}}_{G}(x)$ is at the
$\mathcal{O}(\alpha_{s}^{2})$ order, and so not very large. In the present
study we shall not pay much attention to it, while we shall investigate
another more important term from the next subsection.

###  II.2 $\bar{D}^{0}\Sigma_{c}^{+}$ correlation function

In this subsection we investigate correlation function of the
$\bar{D}^{0}\Sigma_{c}^{+}$ molecule. The interpolating currents corresponding
to the $\bar{D}^{0}$ meson, the $\Sigma_{c}^{+}$ baryon, and the
$\bar{D}^{0}\Sigma_{c}^{+}$ molecule are

$\displaystyle J^{\bar{D}^{0}}(x)$ $\displaystyle=$
$\displaystyle\bar{c}_{a}(x)\gamma_{5}u_{a}(x)\,,$ (10) $\displaystyle
J^{\Sigma_{c}^{+}}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{abc}u^{T}_{a}(x)\mathbb{C}\gamma^{\mu}d_{b}(x)\gamma_{\mu}\gamma_{5}c_{c}(x)\,,$
(11) $\displaystyle J^{\bar{D}^{0}\Sigma_{c}^{+}}(x)$ $\displaystyle=$
$\displaystyle J^{\bar{D}^{0}}(x)\times J^{\Sigma_{c}^{+}}(x)$
$\displaystyle=$ $\displaystyle[\bar{c}_{d}(x)\gamma_{5}u_{d}(x)]$
$\displaystyle\times$
$\displaystyle[\epsilon^{abc}u^{T}_{a}(x)\mathbb{C}\gamma^{\mu}d_{b}(x)\gamma_{\mu}\gamma_{5}c_{c}(x)]\,.$

Their correlation functions in the coordinate space are:

$\displaystyle\Pi^{\bar{D}^{0}}(x)$ $\displaystyle=$ $\displaystyle-{\bf
Tr}\left[i{\bf S}_{q}^{aa^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{a^{\prime}a}(-x)\gamma_{5}\right]\,,$ (13)
$\displaystyle\Pi^{\Sigma_{c}^{+}}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}{\bf
Tr}\left[i{\bf S}_{q}^{bb^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{aa^{\prime}}(x))^{\rm
T}\mathbb{C}\gamma^{\mu}\right]\times\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}\,,$ (14)
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}(x)$ $\displaystyle=$
$\displaystyle-{\bf Tr}\left[i{\bf S}_{q}^{dd^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{d^{\prime}d}(-x)\gamma_{5}\right]\times\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}{\bf
Tr}\left[i{\bf S}_{q}^{bb^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{aa^{\prime}}(x))^{\rm
T}\mathbb{C}\gamma^{\mu}\right]\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}$ (15)
$\displaystyle+{\bf Tr}\left[i{\bf
S}_{q}^{da^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{bb^{\prime}}(x))^{\rm T}\mathbb{C}\gamma^{\mu}i{\bf
S}_{q}^{ad^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{d^{\prime}d}(-x)\gamma_{5}\right]\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)+\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{G}(x)+\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{Q}(x)\,.$

In the above expressions, $\Pi^{\bar{D}^{0}}(x)$ and $\Pi^{\Sigma_{c}^{+}}(x)$
are correlation functions of $\bar{D}^{0}$ and $\Sigma_{c}^{+}$, respective;
$\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)=\Pi^{\bar{D}^{0}}(x)\times\Pi^{\Sigma_{c}^{+}}(x)$
is the leading term contributed by non-correlated $\bar{D}^{0}$ and
$\Sigma_{c}^{+}$, and $\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{G}(x)$ describes the
double-gluon-exchange interaction between them. Their corresponding Feynman
diagrams are depicted in Fig. 4(a-d).

Figure 4: Feynman diagrams corresponding to: a) $\Pi^{\bar{D}^{0}}(x)$, b)
$\Pi^{\Sigma_{c}^{+}}(x)$, c) $\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)$, d)
$\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{G}(x)$, and e)
$\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{Q}(x)$. $\Pi^{\bar{D}^{0}}(x)$ and
$\Pi^{\Sigma_{c}^{+}}(x)$ are correlation functions of the $\bar{D}^{0}$ meson
and the $\Sigma_{c}^{+}$ baryon, respectively. The correlation function of the
$\bar{D}^{0}\Sigma_{c}^{+}$ molecule satisfies
$\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}(x)=\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)+\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{G}(x)+\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{Q}(x)$
with
$\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)=\Pi^{\bar{D}^{0}}(x)\times\Pi^{\Sigma_{c}^{+}}(x)$.

Compared to the case of $D^{-}\Sigma_{c}^{++}$, we find

$\displaystyle\Pi^{\bar{D}^{0}}(x)$ $\displaystyle=$
$\displaystyle\Pi^{D^{-}}(x)\,,$ (16) $\displaystyle\Pi^{\Sigma_{c}^{+}}(x)$
$\displaystyle=$ $\displaystyle\Pi^{\Sigma_{c}^{++}}(x)\,,$ (17)
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)$ $\displaystyle=$
$\displaystyle\Pi^{D^{-}\Sigma_{c}^{++}}_{0}(x)\,,$ (18)
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{G}(x)$ $\displaystyle=$
$\displaystyle\Pi^{D^{-}\Sigma_{c}^{++}}_{G}(x)\,,$ (19)

while there exists an extra term,

$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{Q}(x)$ $\displaystyle=$
$\displaystyle{\bf Tr}\Big{[}i{\bf
S}_{q}^{da^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{bb^{\prime}}(x))^{\rm T}\mathbb{C}\gamma^{\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i{\bf
S}_{q}^{ad^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{d^{\prime}d}(-x)\gamma_{5}\Big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}\,.$

Its corresponding Feynman diagrams are depicted in Fig. 4(e). This term
describes the light-quark-exchange effect between $\bar{D}^{0}$ and
$\Sigma_{c}^{+}$, i.e., $\bar{D}^{0}$ and $\Sigma_{c}^{+}$ are exchanging and
so sharing two light $up$ quarks. It can induce an interaction between
$\bar{D}^{0}$ and $\Sigma_{c}^{+}$, either attractive or repulsive. This term
is color-unconfined, i.e., its contribution decreases as the distance between
$\bar{D}^{0}$ and $\Sigma_{c}^{+}$ increases; besides, the two
exchanged/shared light quarks have the same color. We shall further study it
numerically in Sec. III.3.

###  II.3 $I=1/2$ $\bar{D}\Sigma_{c}$ correlation function

In this subsection we investigate correlation function of the $I=1/2$
$\bar{D}\Sigma_{c}$ molecule. Its corresponding interpolating current is

$J^{\bar{D}\Sigma_{c}}(x)=\sqrt{1\over
3}J^{\bar{D}^{0}\Sigma_{c}^{+}}(x)-\sqrt{2\over
3}J^{D^{-}\Sigma_{c}^{++}}(x)\,,$ (21)

with the correlation function in the coordinate space

$\displaystyle\Pi^{\bar{D}\Sigma_{c}}(x)$ $\displaystyle=$ $\displaystyle-{\bf
Tr}\left[i{\bf S}_{q}^{dd^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{d^{\prime}d}(-x)\gamma_{5}\right]\times\epsilon^{abc}\epsilon^{a^{\prime}b^{\prime}c^{\prime}}{\bf
Tr}\left[i{\bf S}_{q}^{bb^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{aa^{\prime}}(x))^{\rm
T}\mathbb{C}\gamma^{\mu}\right]\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}$ (22)
$\displaystyle-{\bf Tr}\left[i{\bf
S}_{q}^{da^{\prime}}(x)\gamma^{\mu^{\prime}}\mathbb{C}(i{\bf
S}_{q}^{bb^{\prime}}(x))^{\rm T}\mathbb{C}\gamma^{\mu}i{\bf
S}_{q}^{ad^{\prime}}(x)\gamma_{5}i{\bf
S}_{c}^{d^{\prime}d}(-x)\gamma_{5}\right]\gamma_{\mu}\gamma_{5}i{\bf
S}_{c}^{cc^{\prime}}(x)\gamma_{\mu^{\prime}}\gamma_{5}$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{0}(x)+\Pi^{\bar{D}\Sigma_{c}}_{G}(x)+\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)\,.$

Compared to the case of $\bar{D}^{0}\Sigma_{c}^{+}$ given in Eq. (15), only
the terms $\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)$ and
$\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{Q}(x)$ are different/oppisite:

$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{0}(x)$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{0}(x)\,,$ (23)
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{G}(x)$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{G}(x)\,,$ (24)
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)$ $\displaystyle=$
$\displaystyle-\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}_{Q}(x)\,.$ (25)

Accordingly, if the term $\Pi_{G}(x)$ induces a repulsive interaction to the
$\bar{D}^{0}\Sigma_{c}^{+}$ molecule, it would induce an attractive
interaction to the $I=1/2$ $\bar{D}\Sigma_{c}$ molecule; and vice verse. We
shall further study it numerically in Sec. III.1.

###  II.4 $I=3/2$ $\bar{D}\Sigma_{c}$ correlation function

For completeness, we investigate correlation function of the $I=3/2$
$\bar{D}\Sigma_{c}$ molecule in this subsection. Its corresponding
interpolating current is

$J_{I=3/2}^{\bar{D}\Sigma_{c}}(x)=\sqrt{2\over
3}J^{\bar{D}^{0}\Sigma_{c}^{+}}(x)+\sqrt{1\over
3}J^{D^{-}\Sigma_{c}^{++}}(x)\,,$ (26)

with the correlation function in the coordinate space

$\Pi_{I=3/2}^{\bar{D}\Sigma_{c}}(x)=\Pi^{\bar{D}\Sigma_{c}}_{0}(x)+\Pi^{\bar{D}\Sigma_{c}}_{G}(x)-2\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)\,.$
(27)

In the above expression $\Pi^{\bar{D}\Sigma_{c}}_{0}(x)$,
$\Pi^{\bar{D}\Sigma_{c}}_{G}(x)$, and $\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)$ are
taken from the $I=1/2$ $\bar{D}\Sigma_{c}$ molecule.

Compared to Eq. (22), if the term $\Pi_{Q}(x)$ induces an attractive
interaction to the $I=1/2$ $\bar{D}\Sigma_{c}$ molecule, it would induce a
repulsive interaction to the $I=3/2$ $\bar{D}\Sigma_{c}$ molecule; and vice
verse.

##  III QCD sum rule studies of
$D^{-}\Sigma_{c}^{++}/\bar{D}^{0}\Sigma_{c}^{+}/\bar{D}\Sigma_{c}$ molecules

In this section we apply the method of QCD sum rules Shifman:1978bx ;
Reinders:1984sr  to investigate the light-quark-exchange term $\Pi_{Q}(x)$,
and study its contributions to the $D^{-}\Sigma_{c}^{++}$,
$\bar{D}^{0}\Sigma_{c}^{+}$, $I=1/2$ $\bar{D}\Sigma_{c}$, and $I=3/2$
$\bar{D}\Sigma_{c}$ hadronic molecular states.

In QCD sum rule analyses we consider the two-point correlation function in the
momentum space:

$\displaystyle\Pi(q^{2})$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iqx}\langle 0|\mathbb{T}\left[J(x)\bar{J}(0)\right]|0\rangle$ (28)
$\displaystyle=$ $\displaystyle i\int d^{4}xe^{iqx}~{}\Pi(x)\,,$

where $J(x)$ is an interpolating current. We generally assume it to be a
composite current, coupling to the molecular state $X\equiv|YZ\rangle$ through

$\langle 0|J|X\rangle=f_{X}u_{X}\,.$ (29)

We write $\Pi(q^{2})$ in the form of dispersion relation as

$\Pi(q^{2})=\int^{\infty}_{s_{<}}\frac{\rho(s)}{s-q^{2}-i\varepsilon}ds\,,$
(30)

where $s_{<}$ is the physical threshold and $\rho(s)\equiv{\rm Im}\Pi(s)/\pi$
is the spectral density.

At the quark-gluon level we calculate $\Pi(q^{2})$ using the method of
operator product expansion (OPE) up to certain order. According to Eqs. (8),
(15), (22), and (27), we further separate it into

$\Pi(q^{2})\approx\Pi_{0}(q^{2})+\Pi_{Q}(q^{2})\,,$ (31)

and define their imaginary parts to be $\rho_{0}(s)$ and $\rho_{Q}(s)$. Here
we have omitted the other term $\Pi_{G}(q^{2})$, since the light-quark-
exchange term $\Pi_{Q}(q^{2})$ is much larger.

At the hadron level we evaluate the spectral density by inserting intermediate
hadron states $\sum_{n}|n\rangle\langle n|$:

$\displaystyle\rho(s)$ $\displaystyle=$
$\displaystyle\sum_{n}\delta(s-M^{2}_{n})\langle 0|J|n\rangle\langle
n|\bar{J}|0\rangle$ (32) $\displaystyle=$ $\displaystyle
f^{2}_{X}\delta(s-M^{2}_{X})+\rm{continuum}\,,$

where we have adopted a parametrization of one pole dominance for the ground
state $X$ together with a continuum contribution.

Given $X\equiv|YZ\rangle$ to be a molecular state, its mass $M_{X}$ can be
expanded as

$\displaystyle M_{X}$ $\displaystyle=$ $\displaystyle M_{Y}+M_{Z}+\Delta M$
$\displaystyle\equiv$ $\displaystyle M_{0}+\Delta M\,.$

Then we insert Eq. (32) into Eq. (30), and expand it as

$\displaystyle\Pi(q^{2})$ $\displaystyle=$ $\displaystyle{f_{X}^{2}\over
M_{X}^{2}-q^{2}}+\cdots$ $\displaystyle\approx$ $\displaystyle{f_{X}^{2}\over
M_{0}^{2}-q^{2}}-{2M_{0}f_{X}^{2}\over\left(M_{0}^{2}-q^{2}\right)^{2}}\Delta
M+\cdots\,.$

The former term is contributed by the non-correlated $Y$ and $Z$, and the
latter term is contributed by their interactions. Compared to Eq. (31), we
further obtain:

$\displaystyle\Pi_{0}(q^{2})$ $\displaystyle=$ $\displaystyle{f_{X}^{2}\over
M_{0}^{2}-q^{2}}+\cdots\,,$ (35) $\displaystyle\Pi_{Q}(q^{2})$
$\displaystyle=$
$\displaystyle-{2M_{0}f_{X}^{2}\over\left(M_{0}^{2}-q^{2}\right)^{2}}\Delta
M+\cdots\,.$ (36)

We perform the Borel transformation to the above correlation functions at both
hadron and quark-gluon levels. After assuming contributions from the continuum
to be approximated by the OPE spectral densities $\rho_{0}(s)$ and
$\rho_{Q}(s)$ above a threshold value $s_{0}$, we arrive at two sum rule
equations:

$\displaystyle f^{2}_{X}e^{-M_{0}^{2}/M_{B}^{2}}$ $\displaystyle=$
$\displaystyle\Pi_{0}(M_{B}^{2},s_{0})=\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{0}(s)ds\,,$
(37) $\displaystyle-{2M_{0}f_{X}^{2}\over M_{B}^{2}}\Delta
Me^{-M_{0}^{2}/M_{B}^{2}}$ $\displaystyle=$
$\displaystyle\Pi_{Q}(M_{B}^{2},s_{0})=\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{Q}(s)ds\,.$
(38)

There are two free parameters in Eqs. (37) and (38): the Borel mass $M_{B}$
and the threshold value $s_{0}$. Differentiating Eq. (37) with respect to
$1/M_{B}^{2}$, we obtain

$M^{2}_{0}=\frac{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}s\rho(s)ds}{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho(s)ds}\,.$
(39)

This equation can be used to relate $M_{B}$ and $s_{0}$, so that there is only
one free parameter left.

Dividing Eq. (38) by Eq. (37), we obtain

$-{2M_{0}\over M_{B}^{2}}\Delta
M={\Pi_{Q}\over\Pi_{0}}=\frac{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{Q}(s)ds}{\int^{s_{0}}_{s_{<}}e^{-s/M_{B}^{2}}\rho_{0}(s)ds}\,.$
(40)

This equation can be used to calculate $\Delta M$.

We shall use Eq. (40) to study contributions of the term $\Pi_{Q}(x)$ to the
$D^{-}\Sigma_{c}^{++}$, $\bar{D}^{0}\Sigma_{c}^{+}$, $I=1/2$
$\bar{D}\Sigma_{c}$, and $I=3/2$ $\bar{D}\Sigma_{c}$ hadronic molecular
states, separately in the following subsections. Before doing this, we note
that $\Delta M$ is actually not the binding energy, but relates to some
potential $V(r)$ between $Y$ and $Z$ induced by exchanged/shared light quarks:

  * •

Because we are using local currents in QCD sum rule analyses,

$V(r=0)=\Delta M\,.$ (41)

  * •

Because the term $\Pi_{Q}(x)$ is color-unconfined, its contribution decreases
as $r$ increases:

$V(r\rightarrow\infty)\rightarrow 0\,.$ (42)

We may build a model and use the light-quark-exchange potential $V(r)$ to
derive the binding energy of $X$, but this will not be done in the present
study. Other than this, we shall calculate $\Delta M$ and qualitatively study
several hadronic molecules possibly bound by this potential, through which we
shall propose a model-independent hypothesis for such molecules.

The binding mechanism induced by the light-quark-exchange potential $V(r)$ is
somewhat similar to the covalent bond in chemical molecules induced by shared
electrons, so we call such hadronic molecules “covalent hadronic molecules”.

###  III.1 $I=1/2$ $\bar{D}\Sigma_{c}$ sum rules

In this subsection we apply QCD sum rules to study the $I=1/2$
$\bar{D}\Sigma_{c}$ molecule. We have calculated its correlation function at
the leading order of $\alpha_{s}$ and up to the $D({\rm imension})=10$ terms,
including the perturbative term, 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 $\langle g_{s}\bar{q}\sigma Gq\rangle^{2}$.
The extracted spectral density

$\rho^{\bar{D}\Sigma_{c}}(s)=\rho_{0}^{\bar{D}\Sigma_{c}}(s)+\rho_{Q}^{\bar{D}\Sigma_{c}}(s)\,,$
(43)

is given in Appendix A.

To perform numerical analyses, we use the following values for various QCD sum
rule parameters in the present study pdg ; Yang:1993bp ; Ellis:1996xc ;
Eidemuller:2000rc ; Narison:2002pw ; Gimenez:2005nt ; Jamin:2002ev ;
Ioffe:2002be ; Ovchinnikov:1988gk ; colangelo :

$\displaystyle m_{s}$ $\displaystyle=$ $\displaystyle 96^{+8}_{-4}\mbox{
MeV}\,,$ $\displaystyle m_{c}$ $\displaystyle=$ $\displaystyle
1.275^{+0.025}_{-0.035}\mbox{ GeV}\,,$ $\displaystyle\langle\bar{q}q\rangle$
$\displaystyle=$ $\displaystyle-(0.240\pm 0.010)^{3}\mbox{ GeV}^{3}\,,$
$\displaystyle\langle\bar{s}s\rangle$ $\displaystyle=$ $\displaystyle(0.8\pm
0.1)\times\langle\bar{q}q\rangle\,,$ (44) $\displaystyle\langle
g_{s}^{2}GG\rangle$ $\displaystyle=$ $\displaystyle 0.48\pm 0.14\mbox{
GeV}^{4}\,,$ $\displaystyle\langle g_{s}\bar{q}\sigma Gq\rangle$
$\displaystyle=$ $\displaystyle-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$
$\displaystyle\langle g_{s}\bar{s}\sigma Gs\rangle$ $\displaystyle=$
$\displaystyle-M_{0}^{2}\times\langle\bar{s}s\rangle\,,$ $\displaystyle
M_{0}^{2}$ $\displaystyle=$ $\displaystyle 0.8\pm 0.2\mbox{ GeV}^{2}\,,$

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

There are two free parameters in Eqs. (37) and (38): the Borel mass $M_{B}$
and the threshold value $s_{0}$. We use Eq. (39) to constrain them by setting
pdg :

$\displaystyle M_{0}^{\bar{D}\Sigma_{c}}$ $\displaystyle=$
$\displaystyle{1\over
3}\times\left(M_{\bar{D}^{0}}+M_{\Sigma_{c}^{+}}\right)+{2\over
3}\times\left(M_{D^{-}}+M_{\Sigma_{c}^{++}}\right)$ (45) $\displaystyle=$
$\displaystyle 4321.66~{}{\rm MeV}\,.$

The derived relation between $M_{B}$ and $s_{0}$ is depicted in Fig. 5, which
will be used in the following calculations.

Figure 5: Relation between the Borel mass $M_{B}$ and the threshold value
$s_{0}$, constrained by Eq. (39).

There are two criteria to constrain the Borel mass $M_{B}$. The first
criterion is to insure the convergence of OPE series, 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 15%:

$\mbox{Convergence}\equiv\left|\frac{\Pi^{D=10}(\infty,M_{B})}{\Pi(\infty,M_{B})}\right|\leq
15\%\,.$ (46)

This criterion determines the lower limit of $M_{B}$. As shown in Fig. 6 using
the solid curve, we find it to be $\left(M_{B}^{min}\right)^{2}=3.11$ GeV2.

Figure 6: Convergence (solid) and Pole-Contribution (dashed) as functions of
the Borel mass $M_{B}$.

The second criterion is to insure the validity of one-pole parametrization, by
requiring the pole contribution to be larger than 40%:

$\mbox{Pole-
Contribution}\equiv\frac{\Pi(s_{0}(M_{B}),M_{B})}{\Pi(\infty,M_{B})}\geq
40\%\,.$ (47)

This determines the upper limit of $M_{B}$. As shown in Fig. 6 using the
dashed curve, we find it to be $\left(M_{B}^{max}\right)^{2}=3.48$ GeV2.

Altogether we extract the working region of $M_{B}$ to be $3.11$
GeV${}^{2}<M_{B}^{2}<3.48$ GeV2, where we use Eq. (40) and calculate the mass
correction to be

$\Delta M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}=-95~{}{\rm MeV}\,.$ (48)

We show its variation in Fig. 7 with respect to the Borel mass $M_{B}$. It is
shown in a broader region $3.0$ GeV${}^{2}\leq M_{B}^{2}\leq 4.0$ GeV2, and we
find it quite stable inside the above Borel window.

Figure 7: The mass correction $\Delta M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}$ as
a function of the Borel mass $M_{B}$.

The mass correction $\Delta M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}$ given in Eq.
(48) is negative, suggesting the light-quark-exchange potential
$V^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}(r)$ to be attractive, so there can be the
$\bar{D}\Sigma_{c}$ covalent molecule of $I=1/2$ and $J=1/2$.

###  III.2 $D^{-}\Sigma_{c}^{++}$ sum rules

As shown in Eq. (8), the light-quark-exchange term $\Pi_{Q}(x)$ does not
contribute to the correlation function of the $D^{-}\Sigma_{c}^{++}$ molecule,
so the $D^{-}\Sigma_{c}^{++}$ covalent molecule does not exist.

###  III.3 $\bar{D}^{0}\Sigma_{c}^{+}$ sum rules

We follow Sec. III.1 to study the $\bar{D}^{0}\Sigma_{c}^{+}$ molecule, and
calculate its mass correction to be

$\Delta M^{\bar{D}^{0}\Sigma_{c}^{+}}_{J=1/2}\approx-\Delta
M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}=95~{}{\rm MeV}\,.$ (49)

This result suggests $V^{\bar{D}^{0}\Sigma_{c}^{+}}_{J=1/2}(r)$ to be
repulsive, so the $\bar{D}^{0}\Sigma_{c}^{+}$ covalent molecule does not
exist.

###  III.4 $I=3/2$ $\bar{D}\Sigma_{c}$ sum rules

We follow Sec. III.1 to study the $I=3/2$ $\bar{D}\Sigma_{c}$ molecule, and
calculate its mass correction to be

$\Delta M^{\bar{D}\Sigma_{c}}_{I=3/2,J=1/2}\approx-2\Delta
M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}=190~{}{\rm MeV}\,.$ (50)

This result suggests $V^{\bar{D}\Sigma_{c}}_{I=3/2,J=1/2}(r)$ to be repulsive,
so the $\bar{D}\Sigma_{c}$ covalent molecule of $I=3/2$ and $J=1/2$ does not
exist.

##  IV More hadronic molecules

In this section we follow the procedures used in Sec. III.1 and study more
possibly-existing covalent hadronic molecules. We shall investigate the
$\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}$
molecules, the $\bar{D}^{(*)}\Lambda_{c}$ molecules, the $D^{(*)}\bar{K}^{*}$
molecules, and the $D^{(*)}\bar{D}^{(*)}$ molecules, separately in the
following subsections.

###  IV.1
$\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}$
molecules

In this subsection we investigate the light-quark-exchange term $\Pi_{Q}(x)$
and study its contributions to the
$\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}$
molecules.

The current corresponding to the $\bar{D}^{*}\Sigma_{c}$ molecule is

$\displaystyle
J_{\alpha}^{\bar{D}^{*}\Sigma_{c}}(x)=J_{\alpha}^{\bar{D}^{*}}(x)\times
J^{\Sigma_{c}}(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{d}(x)\gamma_{\alpha}q_{d}(x)]\times[\epsilon^{abc}q^{T}_{a}(x)\mathbb{C}\gamma^{\mu}q_{b}(x)\gamma_{\mu}\gamma_{5}c_{c}(x)]\,.$

Its correlation function is

$\displaystyle\Pi_{\alpha\beta}^{\bar{D}^{*}\Sigma_{c}}(q^{2})$
$\displaystyle=$ $\displaystyle i\int d^{4}xe^{iqx}\langle
0|\mathbb{T}\left[J_{\alpha}^{\bar{D}^{*}\Sigma_{c}}(x)\bar{J}_{\beta}^{\bar{D}^{*}\Sigma_{c}}(0)\right]|0\rangle$
$\displaystyle=$
$\displaystyle\mathcal{G}^{3/2}_{\alpha\beta}(q^{2})~{}\Pi^{\bar{D}^{*}\Sigma_{c}}_{J=3/2}\left(q^{2}\right)+\mathcal{G}^{1/2}_{\alpha\beta}(q^{2})~{}\Pi^{\bar{D}^{*}\Sigma_{c}}_{J=1/2}\left(q^{2}\right)\,,$

where $\Pi^{\bar{D}^{*}\Sigma_{c}}_{J=3/2}\left(q^{2}\right)$ and
$\Pi^{\bar{D}^{*}\Sigma_{c}}_{J=1/2}\left(q^{2}\right)$ are contributed by the
spin-3/2 and spin-1/2 components, respectively.
$\mathcal{G}^{3/2}_{\alpha\beta}\left(q^{2}\right)$ and
$\mathcal{G}^{1/2}_{\alpha\beta}\left(q^{2}\right)$ are coefficients of the
spin-3/2 and spin-1/2 propagators, respectively:

$\displaystyle\mathcal{G}^{3/2}_{\alpha\beta}(q^{2})$ $\displaystyle=$
$\displaystyle\big{(}g_{\alpha\beta}-{\gamma_{\alpha}\gamma_{\beta}\over
3}-{q_{\alpha}\gamma_{\beta}-q_{\beta}\gamma_{\alpha}\over
3M}-{2q_{\alpha}q_{\beta}\over 3M^{2}}\big{)}$ (53)
$\displaystyle\times\,(q\\!\\!\\!/~{}+M)\,,$
$\displaystyle\mathcal{G}^{1/2}_{\alpha\beta}(q)$ $\displaystyle=$
$\displaystyle q_{\alpha}q_{\beta}\times\,(q\\!\\!\\!/~{}+M)\,.$ (54)

In the present study we have only calculated the terms proportional to the
Lorentz coefficient $g_{\alpha\beta}$, so we can only extract the mass
correction to the spin-3/2 $\bar{D}^{*}\Sigma_{c}$ molecule. We find it
negative for the $I=1/2$ one:

$\Delta M^{\bar{D}^{*}\Sigma_{c}}_{I=1/2,J=3/2}=-89~{}{\rm MeV}\,.$ (55)

This result suggests $V_{I=1/2,J=3/2}^{\bar{D}^{*}\Sigma_{c}}(r)$ to be
attractive, so there can be the $\bar{D}^{*}\Sigma_{c}$ covalent molecule of
$I=1/2$ and $J=3/2$.

Similarly, we use the current

$\displaystyle J_{\alpha}^{\bar{D}\Sigma_{c}^{*}}(x)=J^{\bar{D}}(x)\times
J_{\alpha}^{\Sigma_{c}^{*}}(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{d}(x)\gamma_{5}q_{d}(x)]\times[\epsilon^{abc}q^{T}_{a}(x)\mathbb{C}\gamma^{\mu}q_{b}(x)P^{3/2}_{\alpha\mu}c_{c}(x)]\,,$

to perform QCD sum rule analyses and study the $\bar{D}\Sigma_{c}^{*}$
molecule. Here $P^{3/2}_{\alpha\mu}$ is the spin-3/2 projection operator

$P^{3/2}_{\alpha\mu}=g_{\alpha\mu}-{1\over 4}\gamma_{\alpha}\gamma_{\mu}\,.$
(57)

We calculate its mass correction to be

$\Delta M^{\bar{D}\Sigma_{c}^{*}}_{I=1/2,J=3/2}=-86~{}{\rm MeV}\,.$ (58)

This result suggests $V_{I=1/2,J=3/2}^{\bar{D}\Sigma_{c}^{*}}(r)$ to be
attractive, so there can be the $\bar{D}\Sigma_{c}^{*}$ covalent molecule of
$I=1/2$ and $J=3/2$.

The current corresponding to the $\bar{D}^{*}\Sigma_{c}^{*}$ molecule is

$\displaystyle
J_{\alpha_{1}\alpha_{2}}^{\bar{D}^{*}\Sigma_{c}^{*}}(x)=J_{\alpha_{1}}^{\bar{D}^{*}}(x)\times
J_{\alpha_{2}}^{\Sigma_{c}^{*}}(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{d}(x)\gamma_{\alpha_{1}}q_{d}(x)]\times[\epsilon^{abc}q^{T}_{a}(x)\mathbb{C}\gamma^{\mu}q_{b}(x)P^{3/2}_{\alpha_{2}\mu}c_{c}(x)]\,.$

Its correlation function is

$\displaystyle\Pi_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}^{\bar{D}^{*}\Sigma_{c}^{*}}(q^{2})$
$\displaystyle=$ $\displaystyle i\int d^{4}xe^{iqx}\langle
0|\mathbb{T}\left[J_{\alpha_{1}\alpha_{2}}^{\bar{D}^{*}\Sigma_{c}^{*}}(x)\bar{J}_{\beta_{1}\beta_{2}}^{\bar{D}^{*}\Sigma_{c}^{*}}(0)\right]|0\rangle$
$\displaystyle=$
$\displaystyle\mathcal{G}^{5/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})~{}\Pi^{\bar{D}^{*}\Sigma_{c}^{*}}_{J=5/2}\left(q^{2}\right)$
$\displaystyle+$
$\displaystyle\mathcal{G}^{3/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})~{}\Pi^{\bar{D}^{*}\Sigma_{c}^{*}}_{J=3/2}\left(q^{2}\right)$
$\displaystyle+$
$\displaystyle\mathcal{G}^{1/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})~{}\Pi^{\bar{D}^{*}\Sigma_{c}^{*}}_{J=1/2}\left(q^{2}\right)+\cdots\,,$

where $\Pi^{\bar{D}^{*}\Sigma_{c}^{*}}_{J=5/2}\left(q^{2}\right)$,
$\Pi^{\bar{D}^{*}\Sigma_{c}^{*}}_{J=3/2}\left(q^{2}\right)$, and
$\Pi^{\bar{D}^{*}\Sigma_{c}^{*}}_{J=1/2}\left(q^{2}\right)$ are contributed by
the spin-5/2, spin-3/2, and spin-1/2 components, respectively.
$\mathcal{G}^{5/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})$,
$\mathcal{G}^{3/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})$, and
$\mathcal{G}^{1/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})$ are
coefficients of the spin-5/2, spin-3/2, and spin-1/2 propagators,
respectively:

$\displaystyle\mathcal{G}^{5/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})=(q\\!\\!\\!/~{}+M)$
(61)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\big{(}{g_{\alpha_{1}\beta_{1}}g_{\alpha_{2}\beta_{2}}\over
2}+{g_{\alpha_{1}\beta_{2}}g_{\alpha_{2}\beta_{1}}\over
2}-{g_{\alpha_{1}\alpha_{2}}g_{\beta_{1}\beta_{2}}\over 4}\big{)}\,,$
$\displaystyle\mathcal{G}^{3/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})=(q\\!\\!\\!/~{}+M)$
(62)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\big{(}{g_{\alpha_{1}\beta_{1}}g_{\alpha_{2}\beta_{2}}\over
2}-{g_{\alpha_{1}\beta_{2}}g_{\alpha_{2}\beta_{1}}\over 2}\big{)}\,,$
$\displaystyle\mathcal{G}^{1/2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})=(q\\!\\!\\!/~{}+M)\times
g_{\alpha_{1}\alpha_{2}}g_{\beta_{1}\beta_{2}}\,.$ (63)

In the present study we have calculated the terms proportional to the three
Lorentz coefficients $g_{\alpha_{1}\beta_{1}}g_{\alpha_{2}\beta_{2}}$,
$g_{\alpha_{1}\beta_{2}}g_{\alpha_{2}\beta_{1}}$, and
$g_{\alpha_{1}\alpha_{2}}g_{\beta_{1}\beta_{2}}$, so we can extract mass
corrections to all the three $I=1/2$ $\bar{D}^{*}\Sigma_{c}^{*}$ molecules of
$J=5/2$, $J=3/2$, and $J=1/2$:

$\displaystyle\Delta M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=5/2}$
$\displaystyle=$ $\displaystyle-107~{}{\rm MeV}\,,$ (64) $\displaystyle\Delta
M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=3/2}$ $\displaystyle=$
$\displaystyle-47~{}{\rm MeV}\,,$ (65) $\displaystyle\Delta
M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=1/2}$ $\displaystyle=$ $\displaystyle
3.5~{}{\rm MeV}\,.$ (66)

These results suggest $V_{I=1/2,J=5/2}^{\bar{D}^{*}\Sigma_{c}^{*}}(r)$ to be
attractive, so there can be the $\bar{D}^{*}\Sigma_{c}^{*}$ covalent molecule
of $I=1/2$ and $J=5/2$.

Figure 8: Mass corrections $\Delta M^{\bar{D}^{(*)}\Sigma_{c}^{(*)}}$ as
functions of the Borel mass $M_{B}$. The curves from top to bottom correspond
to $\Delta M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=1/2}$, $\Delta
M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=3/2}$, $\Delta
M^{\bar{D}^{*}\Sigma_{c}}_{I=1/2,J=3/2}$, $\Delta
M^{\bar{D}\Sigma_{c}^{*}}_{I=1/2,J=3/2}$, $\Delta
M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}$, and $\Delta
M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=5/2}$, respectively.

For completeness, we show variations of $\Delta
M^{\bar{D}^{(*)}\Sigma_{c}^{(*)}}$ in Fig. 8 with respect to the Borel mass
$M_{B}$. Besides, we study the $I=3/2$
$\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}$
molecules and obtained

$\Delta
M^{\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}}_{I=3/2}\approx-2\Delta
M^{\bar{D}^{*}\Sigma_{c}/\bar{D}\Sigma_{c}^{*}/\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2}\,.$
(67)

###  IV.2 $\bar{D}^{(*)}\Lambda_{c}$ molecules

In this subsection we investigate the light-quark-exchange term $\Pi_{Q}(x)$
and study its contributions to the $\bar{D}^{(*)}\Lambda_{c}$ molecules.

The currents corresponding to the $\bar{D}\Lambda_{c}$ and
$\bar{D}^{*}\Lambda_{c}$ molecules are

$\displaystyle J^{\bar{D}\Lambda_{c}}(x)=J^{\bar{D}}(x)\times
J^{\Lambda_{c}}(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{d}(x)\gamma_{5}q_{d}(x)]\times[\epsilon^{abc}q^{T}_{a}(x)\mathbb{C}\gamma_{5}q_{b}(x)c_{c}(x)]\,,$
$\displaystyle
J_{\alpha}^{\bar{D}^{*}\Lambda_{c}}(x)=J_{\alpha}^{\bar{D}^{*}}(x)\times
J^{\Lambda_{c}}(x)$ $\displaystyle=$
$\displaystyle[\bar{c}_{d}(x)\gamma_{\alpha}q_{d}(x)]\times[\epsilon^{abc}q^{T}_{a}(x)\mathbb{C}\gamma_{5}q_{b}(x)c_{c}(x)]\,.$

We use them to perform QCD sum rule analyses, and calculate mass corrections
to the $\bar{D}\Lambda_{c}$ molecule of $J=1/2$ and the
$\bar{D}^{*}\Lambda_{c}$ molecule of $J=3/2$. We find both of them to be
positive. However, we do not obtain the mass correction to the
$\bar{D}^{*}\Lambda_{c}$ molecule of $J=1/2$, just like the case of
$\bar{D}^{*}\Sigma_{c}$. These results suggest
$V_{I=1/2,J=1/2}^{\bar{D}\Lambda_{c}}(r)$ and
$V_{I=1/2,J=3/2}^{\bar{D}^{*}\Lambda_{c}}(r)$ to be both repulsive, so the
$J=1/2$ $\bar{D}\Lambda_{c}$ and $J=3/2$ $\bar{D}^{*}\Lambda_{c}$ covalent
molecules do not exist.

###  IV.3 $D^{(*)}\bar{K}^{*}$ molecules

In this subsection we investigate the light-quark-exchange term $\Pi_{Q}(x)$
and study its contributions to the $D^{(*)}\bar{K}^{*}$ molecules. We do not
investigate the $D^{(*)}\bar{K}$ molecules in the present study, because their
bare masses $M_{D^{(*)}}+M_{\bar{K}}$ can not be easily arrived at within the
present QCD sum rule approach, due to the nature of $K$ mesons as Nambu-
Goldstone bosons.

The current corresponding to the $D\bar{K}^{*}$ molecule is

$\displaystyle J_{\alpha}^{D\bar{K}^{*}}(x)$ $\displaystyle=$ $\displaystyle
J^{\bar{D}}(x)\times J_{\alpha}^{\bar{K}^{*}}(x)$ $\displaystyle=$
$\displaystyle[\bar{q}_{a}(x)\gamma_{5}c_{a}(x)]\times[\bar{q}_{b}(x)\gamma_{\alpha}s_{b}(x)]\,.$

Its correlation function is

$\displaystyle\Pi_{\alpha\beta}^{D\bar{K}^{*}}(q^{2})$ $\displaystyle=$
$\displaystyle i\int d^{4}xe^{iqx}\langle
0|\mathbb{T}\left[J_{\alpha}^{D\bar{K}^{*}}(x)J_{\beta}^{D\bar{K}^{*},\dagger}(0)\right]|0\rangle$
$\displaystyle=$ $\displaystyle\left(g_{\alpha\beta}-{q_{\alpha}q_{\beta}\over
q^{2}}\right)~{}\Pi^{D\bar{K}^{*}}\left(q^{2}\right)+\cdots\,.$

We use $J_{\alpha}^{D\bar{K}^{*}}(x)$ to perform QCD sum rule analyses, and
calculate its mass correction to be

$\Delta M^{D\bar{K}^{*}}_{I=0,J=1}=-180~{}{\rm MeV}\,.$ (72)

This suggests $V^{D\bar{K}^{*}}_{I=0,J=1}(r)$ to be attractive, so there can
be the $D\bar{K}^{*}$ covalent molecule of $I=0$ and $J=1$.

The current corresponding to the $D^{*}\bar{K}^{*}$ molecule is

$\displaystyle J_{\alpha_{1}\alpha_{2}}^{D^{*}\bar{K}^{*}}(x)$
$\displaystyle=$ $\displaystyle J_{\alpha_{1}}^{\bar{D}^{*}}(x)\times
J_{\alpha_{2}}^{\bar{K}^{*}}(x)$ $\displaystyle=$
$\displaystyle[\bar{q}_{a}(x)\gamma_{\alpha_{1}}c_{a}(x)]\times[\bar{q}_{b}(x)\gamma_{\alpha_{2}}s_{b}(x)]\,.$

Its correlation function is

$\displaystyle\Pi_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}^{D^{*}\bar{K}^{*}}(q^{2})$
$\displaystyle=$ $\displaystyle i\int d^{4}xe^{iqx}\langle
0|\mathbb{T}\left[J_{\alpha_{1}\alpha_{2}}^{D^{*}\bar{K}^{*}}(x)J_{\beta_{1}\beta_{2}}^{D^{*}\bar{K}^{*},\dagger}(0)\right]|0\rangle$
$\displaystyle=$
$\displaystyle\mathcal{G}^{2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})~{}\Pi^{D^{*}\bar{K}^{*}}_{J=2}\left(q^{2}\right)$
$\displaystyle+$
$\displaystyle\mathcal{G}^{1}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})~{}\Pi^{D^{*}\bar{K}^{*}}_{J=1}\left(q^{2}\right)$
$\displaystyle+$
$\displaystyle\mathcal{G}^{0}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})~{}\Pi^{D^{*}\bar{K}^{*}}_{J=0}\left(q^{2}\right)+\cdots\,,$

where $\Pi^{D^{*}\bar{K}^{*}}_{J=2}\left(q^{2}\right)$,
$\Pi^{D^{*}\bar{K}^{*}}_{J=1}\left(q^{2}\right)$, and
$\Pi^{D^{*}\bar{K}^{*}}_{J=0}\left(q^{2}\right)$ are contributed by the
spin-2, spin-1, and spin-0 components, respectively.
$\mathcal{G}^{2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})$,
$\mathcal{G}^{1}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})$, and
$\mathcal{G}^{0}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})$ are
coefficients of the spin-2, spin-1, and spin-0 propagators, respectively:

$\displaystyle\mathcal{G}^{2}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})=$
(75)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}{g_{\alpha_{1}\beta_{1}}g_{\alpha_{2}\beta_{2}}\over
2}+{g_{\alpha_{1}\beta_{2}}g_{\alpha_{2}\beta_{1}}\over
2}-{g_{\alpha_{1}\alpha_{2}}g_{\beta_{1}\beta_{2}}\over 4}\,,$
$\displaystyle\mathcal{G}^{1}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})={g_{\alpha_{1}\beta_{1}}g_{\alpha_{2}\beta_{2}}\over
2}-{g_{\alpha_{1}\beta_{2}}g_{\alpha_{2}\beta_{1}}\over 2}\,,$ (76)
$\displaystyle\mathcal{G}^{0}_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}(q^{2})=g_{\alpha_{1}\alpha_{2}}g_{\beta_{1}\beta_{2}}\,.$
(77)

We use $J_{\alpha_{1}\alpha_{2}}^{D^{*}\bar{K}^{*}}(x)$ to perform QCD sum
rule analyses, and extract mass corrections to the three isoscalar
$D^{*}\bar{K}^{*}$ molecules of $J=2$, $J=1$, and $J=0$:

$\displaystyle\Delta M^{D^{*}\bar{K}^{*}}_{I=0,J=2}$ $\displaystyle=$
$\displaystyle-119~{}{\rm MeV}\,,$ (78) $\displaystyle\Delta
M^{D^{*}\bar{K}^{*}}_{I=0,J=1}$ $\displaystyle=$ $\displaystyle-46~{}{\rm
MeV}\,,$ (79) $\displaystyle\Delta M^{D^{*}\bar{K}^{*}}_{I=0,J=0}$
$\displaystyle=$ $\displaystyle-4.5~{}{\rm MeV}\,.$ (80)

These results suggest $V_{I=0,J=2}^{D^{*}\bar{K}^{*}}(r)$ to be attractive, so
there can be the $D^{*}\bar{K}^{*}$ covalent molecule of $I=0$ and $J=2$.

Figure 9: Mass corrections $\Delta M^{D^{(*)}\bar{K}^{*}}$ as functions of the
Borel mass $M_{B}$. The curves from top to bottom correspond to $\Delta
M^{D^{*}\bar{K}^{*}}_{I=0,J=0}$ (short-dashed), $\Delta
M^{D^{*}\bar{K}^{*}}_{I=0,J=1}$ (middle-dashed), $\Delta
M^{D^{*}\bar{K}^{*}}_{I=0,J=2}$ (long-dashed), and $\Delta
M^{D\bar{K}^{*}}_{I=0,J=1}$ (solid), respectively.

We show variations of $\Delta M^{D^{(*)}\bar{K}^{*}}$ in Fig. 9 with respect
to the Borel mass $M_{B}$. Especially, $V_{I=0,J=0}^{D^{*}\bar{K}^{*}}(r)$
depends significantly on the Borel mass $M_{B}$ and so also on the threshold
value $s_{0}$, for which we refer to Sec. V for more model-independent
discussions.

For completeness, we study the isovector $D^{(*)}\bar{K}^{*}$ molecules and
obtained

$\Delta M^{D\bar{K}^{*}/D^{*}\bar{K}^{*}}_{I=1}\approx-\Delta
M^{D\bar{K}^{*}/D^{*}\bar{K}^{*}}_{I=0}\,.$ (81)

###  IV.4 $D^{(*)}\bar{D}^{(*)}$ molecules

The light-quark-exchange term $\Pi_{Q}(x)$ does not contribute to correlation
functions of the $D^{(*)}\bar{D}^{(*)}$ molecules, suggesting the
$D^{(*)}\bar{D}^{(*)}$ covalent molecules not to exist. However, there can
still be the $D^{(*)}\bar{D}^{(*)}$ hadronic molecular states induced by some
other binding mechanisms, such as the one-meson-exchange interaction
Weinberg:1965zz ; Voloshin:1976ap ; DeRujula:1976zlg ; Tornqvist:1993ng ;
Voloshin:2003nt ; Close:2003sg ; Wong:2003xk ; Braaten:2003he ; Swanson:2003tb
; Tornqvist:2004qy .

##  V Summary and Discussions

In this paper we systematically examine the Feynman diagrams corresponding to
the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$, $\bar{D}^{(*)}\Lambda_{c}$,
$D^{(*)}\bar{K}^{*}$, and $D^{(*)}\bar{D}^{(*)}$ hadronic molecular states.
Take the $\bar{D}\Sigma_{c}$ molecule as an example, first we calculate
correlation functions of $\bar{D}$ and $\Sigma_{c}$ in the coordinate space to
be $\Pi^{\bar{D}}(x)$ and $\Pi^{\Sigma_{c}}(x)$, respectively. Then we
calculate correlation functions of the $D^{-}\Sigma_{c}^{++}$,
$\bar{D}^{0}\Sigma_{c}^{+}$, $I=1/2$ $\bar{D}\Sigma_{c}$, and $I=3/2$
$\bar{D}\Sigma_{c}$ molecules, separated into:

$\displaystyle\Pi^{D^{-}\Sigma_{c}^{++}}(x)$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{0}(x)+\Pi^{\bar{D}\Sigma_{c}}_{G}(x)\,,$
$\displaystyle\Pi^{\bar{D}^{0}\Sigma_{c}^{+}}(x)$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{0}(x)+\Pi^{\bar{D}\Sigma_{c}}_{G}(x)-\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)\,,$
$\displaystyle\Pi_{I=1/2}^{\bar{D}\Sigma_{c}}(x)$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{0}(x)+\Pi^{\bar{D}\Sigma_{c}}_{G}(x)+\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)\,,$
$\displaystyle\Pi_{I=3/2}^{\bar{D}\Sigma_{c}}(x)$ $\displaystyle=$
$\displaystyle\Pi^{\bar{D}\Sigma_{c}}_{0}(x)+\Pi^{\bar{D}\Sigma_{c}}_{G}(x)-2\Pi^{\bar{D}\Sigma_{c}}_{Q}(x)\,,$

where
$\Pi^{\bar{D}\Sigma_{c}}_{0}(x)=\Pi^{\bar{D}}(x)\times\Pi^{\Sigma_{c}}(x)$ is
the leading term contributed by non-correlated $\bar{D}$ and $\Sigma_{c}$;
$\Pi^{\bar{D}\Sigma_{c}}_{G}$ describes the double-gluon-exchange interaction
between them, but we do not take it into account in the present study, because
the other term $\Pi^{\bar{D}\Sigma_{c}}_{Q}$ is much larger.

The term $\Pi^{\bar{D}\Sigma_{c}}_{Q}$ describes the light-quark-exchange
effect between $\bar{D}$ and $\Sigma_{c}$, i.e., $\bar{D}$ and $\Sigma_{c}$
are exchanging and so sharing light $up/down$ quarks. We systematically study
it using the method of QCD sum rules, and calculate the mass correction
$\Delta M$ induced by this term. This parameter $\Delta M$ is actually not the
binding energy, but relates to some potential $V(r)$ between $\bar{D}$ and
$\Sigma_{c}$ induced by exchanged/shared light quarks, satisfying:

$\displaystyle V(r=0)$ $\displaystyle=$ $\displaystyle\Delta M\,,$
$\displaystyle V(r\rightarrow\infty)$ $\displaystyle\rightarrow$
$\displaystyle 0\,.$

We systematically investigate the light-quark-exchange term $\Pi_{Q}$, and
study its contributions to the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$,
$\bar{D}^{(*)}\Lambda_{c}$, $D^{(*)}\bar{K}^{*}$, and $D^{(*)}\bar{D}^{(*)}$
hadronic molecular states. We calculate their mass corrections, and list some
of them here:

$\displaystyle\Delta M^{\bar{D}\Sigma_{c}}_{I=1/2,J=1/2}$ $\displaystyle=$
$\displaystyle-95~{}{\rm MeV}\,,$ $\displaystyle\Delta
M^{\bar{D}^{*}\Sigma_{c}}_{I=1/2,J=3/2}$ $\displaystyle=$
$\displaystyle-89~{}{\rm MeV}\,,$ $\displaystyle\Delta
M^{\bar{D}\Sigma_{c}^{*}}_{I=1/2,J=3/2}$ $\displaystyle=$
$\displaystyle-86~{}{\rm MeV}\,,$ $\displaystyle\Delta
M^{\bar{D}^{*}\Sigma_{c}^{*}}_{I=1/2,J=5/2}$ $\displaystyle=$
$\displaystyle-107~{}{\rm MeV}\,,$ $\displaystyle\Delta
M^{D\bar{K}^{*}}_{I=0,J=1}$ $\displaystyle=$ $\displaystyle-180~{}{\rm
MeV}\,,$ $\displaystyle\Delta M^{D^{*}\bar{K}^{*}}_{I=0,J=2}$ $\displaystyle=$
$\displaystyle-119~{}{\rm MeV}\,.$

These results suggest their corresponding light-quark-exchange potentials
$V(r)$ to be attractive, so there can be

  * •

the $\bar{D}\Sigma_{c}$ covalent molecule of $I=1/2$ and $J=1/2$,

  * •

the $\bar{D}^{*}\Sigma_{c}$ covalent molecule of $I=1/2$ and $J=3/2$,

  * •

the $\bar{D}\Sigma_{c}^{*}$ covalent molecule of $I=1/2$ and $J=3/2$,

  * •

the $\bar{D}^{*}\Sigma_{c}^{*}$ covalent molecule of $I=1/2$ and $J=5/2$,

  * •

the $D\bar{K}^{*}$ covalent molecule of $I=0$ and $J=1$,

  * •

the $D^{*}\bar{K}^{*}$ covalent molecule of $I=0$ and $J=2$.

The binding mechanism induced by shared light quarks is somewhat similar to
the covalent bond in chemical molecules induced by shared electrons, so we
call such hadronic molecules “covalent hadronic molecules”.

Recall that the two shared electrons must spin in opposite directions (and so
totally antisymmetric obeying the Pauli principle) in order to form a chemical
covalent bond. Our QCD sum rule results indicate a similar behavior: “the
light-quark-exchange interaction is attractive when the shared light quarks
are totally antisymmetric so that obey the Pauli principle” (these quarks may
spin in the same direction given their flavor structure capable of being
antisymmetric).

The logical chain of the above hypothesis is quite straightforward. We assume
the two light quarks $q_{A}$ inside $Y$ and $q_{B}$ inside $Z$ are totally
antisymmetric. Hence, $q_{A}$ and $q_{B}$ obey the Pauli principle, so that
they can be exchanged and shared. By doing this, wave-functions of $Y$ and $Z$
overlap with each other, so that there can be the covalent hadronic molecule
$X=|YZ\rangle$. This picture has been depicted in Fig. 1. We believe this
hypothesis even better and more important than our QCD sum rule results,
because it is model independent and more easily applicable.

We apply it to study several examples as follows. The two exchanged light
quarks have the same color and so the symmetric color structure; besides, we
assume their orbital structure to be $S$-wave and so also symmetric;
consequently, we only need to consider their spin and flavor structures.

####  $\bar{D}^{(*)0}[\bar{c}_{1}u_{2}]$–$\Lambda_{c}^{+}[u_{3}d_{4}c_{5}]$
covalent molecules.

Let us exchange $u_{2}$ inside $\bar{D}^{(*)0}$ and $u_{3}$ inside
$\Lambda_{c}^{+}$. $u_{3}$ and $d_{4}$ inside $\Lambda_{c}^{+}$ spin in
opposite directions, $u_{2}$ and $d_{4}$ also need to spin in opposite
directions in order to form another $\Lambda_{c}^{+}$, so $u_{2}$ and $u_{3}$
spin in the same direction with the symmetric spin structure. The flavor
structure of $u_{2}$ and $u_{3}$ is also symmetric, so they are totally
symmetric. Accordingly, the $\bar{D}^{(*)}\Lambda_{c}$ covalent molecules do
not exist. This is consistent with our QCD sum rule result obtained in Sec.
IV.2.

####  $\bar{D}^{0}[\bar{c}_{1}u_{2}]$–$\Sigma_{c}^{+}[u_{3}d_{4}c_{5}]$
covalent molecule.

Let us exchange $u_{2}$ inside $\bar{D}^{0}$ and $u_{3}$ inside
$\Sigma_{c}^{+}$. $\bar{c}_{1}$ and $u_{2}$ inside $\bar{D}^{0}$ spin in
opposite directions, $\bar{c}_{1}$ and $u_{3}$ also need to spin in opposite
directions in order to form another $\bar{D}^{0}$, so $u_{2}$ and $u_{3}$ spin
in the same direction with the symmetric spin structure. The flavor structure
of $u_{2}$ and $u_{3}$ is also symmetric, so they are totally symmetric.
Accordingly, the $\bar{D}^{0}\Sigma_{c}^{+}$ covalent molecule does not exist,
consistent with Sec. III.3.

####  $\bar{D}[\bar{c}_{1}q_{2}]$–$\Sigma_{c}[q_{3}q_{4}c_{5}]$ covalent
molecule ($q=u/d$).

After including the isospin symmetry, the exchange can take place between $up$
and $down$ quarks. Let us exchange $q_{2}$ inside $\bar{D}$ and $q_{3}$ inside
$\Sigma_{c}$. As discussed above, they have the symmetric spin structure, so
they can be totally antisymmetric as long as their flavor structure is
antisymmetric. Accordingly, there can be the $I=1/2$ $\bar{D}\Sigma_{c}$
covalent molecule, but not the $I=3/2$ one. This is consistent with our QCD
sum rule result obtained in Sec. III.1 and Sec. III.4. Similarly, we derive
that there can be the $I=1/2$ $\bar{D}\Sigma_{c}^{*}$ and $I=0$ $D\bar{K}^{*}$
covalent molecules, consistent with Sec. IV.1 and Sec. IV.3.

####  $J=5/2$
$\bar{D}^{*}[\bar{c}_{1}q_{2}]$–$\Sigma_{c}^{*}[q_{3}q_{4}c_{5}]$ covalent
molecule.

Let us exchange $q_{2}$ inside $\bar{D}^{*}$ and $q_{3}$ inside
$\Sigma_{c}^{*}$. In the $J_{z}=+5/2$ component, all the quarks/antiquark spin
in the same direction, so $q_{2}$ and $q_{3}$ have the symmetric spin
structure. They can be totally antisymmetric as long as their flavor structure
is antisymmetric. Accordingly, there can be the $I=1/2$
$\bar{D}^{*}\Sigma_{c}^{*}$ covalent molecule of $J=5/2$, but not the $I=3/2$
one. This is consistent with our QCD sum rule result obtained in Sec. IV.1.
Similarly, we derive that there can be the $D^{*}\bar{K}^{*}$ covalent
molecule of $I=0$ and $J=2$, consistent with Sec. IV.3.

####  $J=0$ $D^{*}[c_{1}\bar{q}_{2}]$–$\bar{K}^{*}[s_{3}\bar{q}_{4}]$ covalent
molecule.

Let us exchange $\bar{q}_{2}$ inside $D^{*}$ and $\bar{q}_{4}$ inside
$\bar{K}^{*}$. We perform the spin decomposition:

$\displaystyle|{\bf 1}_{c\bar{q}}\otimes{\bf 1}_{s\bar{q}};J=0\rangle$
$\displaystyle=$ $\displaystyle{\sqrt{3}\over 2}~{}|{\bf 0}_{cs}\otimes{\bf
0}_{\bar{q}\bar{q}};J=0\rangle-{1\over 2}~{}|{\bf 1}_{cs}\otimes{\bf
1}_{\bar{q}\bar{q}};J=0\rangle\,.$

Hence, there exists both ${\bf s}_{\bar{q}_{2}\bar{q}_{4}}={\bf 0}$ (75%) and
${\bf s}_{\bar{q}_{2}\bar{q}_{4}}={\bf 1}$ (25%) components. The former
becomes attractive when $I=1$, and the latter becomes attractive when $I=0$.
Recall that there are both the para-hydrogen and ortho-hydrogen, where the two
protons spin in opposite directions and in the same direction, respectively.
Similarly, there might be two $J=0$ $D^{*}\bar{K}^{*}$ covalent molecules,
i.e., the two components $|{\bf 0}_{cs}\otimes{\bf
0}_{\bar{q}\bar{q}};J=0\rangle$ of $I=1$ and $|{\bf 1}_{cs}\otimes{\bf
1}_{\bar{q}\bar{q}};J=0\rangle$ of $I=0$. However, our QCD sum rule studies
performed in Sec. IV.3 can not differentiate these hyperfine structures,
because there we have summed over the $D^{*}$ and $\bar{K}^{*}$ polarizations.

It is useful to generally discuss how many light quarks at most are there in
the lowest orbit ($q=u/d$):

  1. 1.

In the $\bar{D}^{(*)}[\bar{c}_{1}q_{2}]K^{(*)}[\bar{s}_{3}q_{4}]$ covalent
molecule, the two exchanged light quarks $q_{2}$ and $q_{4}$ have the same
color and so the symmetric color structure; besides, we assume their orbital
structure to be $S$-wave and so also symmetric; consequently, there are two
possible configurations satisfying the Pauli principle (${\bf S}$=symmetric
and ${\bf A}$=antisymmetric):

$\begin{array}[]{cccccc}\hline\cr\hline\cr&{\rm color}&{\rm flavor}&{\rm
spin}&{\rm orbital}&{\rm total}\\\ \hline\cr q_{2}\leftrightarrow q_{4}&{\bf
S}&{\bf A}&{\bf S}&{\bf S}&{\bf A}\\\ \hline\cr q_{2}\leftrightarrow
q_{4}&{\bf S}&{\bf S}&{\bf A}&{\bf S}&{\bf A}\\\
\hline\cr\hline\cr\end{array}$

Hence, two antiquarks can share (at most) two quarks, with the quantum numbers
$(I)J^{P}=(0)1^{+}$ or $(1)0^{+}$.

  2. 2.

In the $\bar{D}^{(*)}[\bar{c}_{1}q_{2}]$–$\Sigma_{c}^{(*)}[q_{3}q_{4}c_{5}]$
covalent molecule, there are three light $up/down$ quarks. We assume the two
exchanged quarks to be $q_{2}$ and $q_{3}$ with the same color. There are also
two possible configurations:

$\begin{array}[]{cccccc}\hline\cr\hline\cr&{\rm color}&{\rm flavor}&{\rm
spin}&{\rm orbital}&{\rm total}\\\ \hline\cr q_{2}\leftrightarrow q_{3}&{\bf
S}&{\bf A}&{\bf S}&{\bf S}&{\bf A}\\\ q_{2}\leftrightarrow q_{4}&{\bf A}&{\bf
S}&{\bf S}&{\bf S}&{\bf A}\\\ q_{3}\leftrightarrow q_{4}&{\bf A}&{\bf S}&{\bf
S}&{\bf S}&{\bf A}\\\ \hline\cr q_{2}\leftrightarrow q_{3}&{\bf S}&{\bf
S}&{\bf A}&{\bf S}&{\bf A}\\\ q_{2}\leftrightarrow q_{4}&{\bf A}&{\bf S}&{\bf
S}&{\bf S}&{\bf A}\\\ q_{3}\leftrightarrow q_{4}&{\bf A}&{\bf S}&{\bf S}&{\bf
S}&{\bf A}\\\ \hline\cr\hline\cr\end{array}$

They satisfy the condition that any two of the three light quarks are totally
antisymmetric so that obey the Pauli principle. Hence, one quark and one
antiquark can share (at most) three quarks, with $(I)J^{P}=({1\over 2}){3\over
2}^{+}$ or $({3\over 2}){1\over 2}^{+}$. However, the
$\bar{D}^{(*)}\Lambda_{c}$ system can not satisfy the above condition.

  3. 3.

In the $\Sigma_{c}^{(*)}[q_{1}q_{2}c_{3}]$–$\Sigma_{b}^{(*)}[q_{4}q_{5}c_{6}]$
covalent molecule, there are four light $up/down$ quarks. We assume that
$q_{1}$ and $q_{4}$ can be exchanged with the same color, and $q_{2}$ and
$q_{5}$ can also be exchanged with the same color. There can be four possible
configurations:

$\begin{array}[]{cccccc}\hline\cr\hline\cr&{\rm color}&{\rm flavor}&{\rm
spin}&{\rm orbital}&{\rm total}\\\ \hline\cr q_{1}\leftrightarrow q_{4}&{\bf
S}&{\bf A}&{\bf S}&{\bf S}&{\bf A}\\\ q_{2}\leftrightarrow q_{5}&{\bf S}&{\bf
A}&{\bf S}&{\bf S}&{\bf A}\\\ q_{1}\leftrightarrow q_{2}&{\bf A}&{\bf S}&{\bf
S}&{\bf S}&{\bf A}\\\ q_{1}\leftrightarrow q_{5}&{\bf A}&{\bf S}&{\bf S}&{\bf
S}&{\bf A}\\\ q_{2}\leftrightarrow q_{4}&{\bf A}&{\bf S}&{\bf S}&{\bf S}&{\bf
A}\\\ q_{4}\leftrightarrow q_{5}&{\bf A}&{\bf S}&{\bf S}&{\bf S}&{\bf A}\\\
\hline\cr q_{1}\leftrightarrow q_{4}&{\bf S}&{\bf S}&{\bf A}&{\bf S}&{\bf
A}\\\ q_{2}\leftrightarrow q_{5}&{\bf S}&{\bf A}&{\bf S}&{\bf S}&{\bf A}\\\
\cdots\\\ \hline\cr q_{1}\leftrightarrow q_{4}&{\bf S}&{\bf A}&{\bf S}&{\bf
S}&{\bf A}\\\ q_{2}\leftrightarrow q_{5}&{\bf S}&{\bf S}&{\bf A}&{\bf S}&{\bf
A}\\\ \cdots\\\ \hline\cr q_{1}\leftrightarrow q_{4}&{\bf S}&{\bf S}&{\bf
A}&{\bf S}&{\bf A}\\\ q_{2}\leftrightarrow q_{5}&{\bf S}&{\bf S}&{\bf A}&{\bf
S}&{\bf A}\\\ \cdots\\\ \hline\cr\hline\cr\end{array}$

They satisfy that any two of the four light quarks are totally antisymmetric
so that obey the Pauli principle. Hence, two quarks can share (at most) four
quarks, with $(I)J^{P}=(0)2^{+}$, $(1)1^{+}$, or $(2)0^{+}$. However, neither
$\Sigma_{c}^{(*)}\Lambda_{b}$ nor $\Lambda_{b}\Lambda_{b}$ can satisfy the
above condition.

We apply the above model-independent hypothesis to predict more covalent
hadronic molecules ($q=u/d$):

  * •

Induced by shared light $up/down$ quarks, there can be the $I=0$
$\bar{D}^{(*)}[\bar{c}q]$–$B^{(*)}[\bar{b}q]$, $I=0$
$\bar{D}^{(*)}[\bar{c}q]$–$\Xi_{c}^{(\prime*)}[csq]$, $I=0$
$\Sigma_{c}^{(*)}[cqq]$–$\Sigma_{b}^{(*)}[bqq]$, and $I=1/2$
$\Sigma_{c}^{(*)}[cqq]$–$\Xi_{b}^{(\prime*)}[bsq]$ covalent molecules, etc.
Their binding energies can be at the 10 MeV level, considering the
$P_{c}/P_{cs}$ and $X_{0}(2900)$ states as possible covalent hadronic
molecules. We list them in Table 1, and note that they are still waiting to be
carefully analysed in our future QCD sum rule studies.

  * •

If the $strange$ quark is also exchangeable, there can be the
$\bar{D}_{s}^{(*)}[\bar{c}s]$–$B^{(*)}[\bar{b}q]$ and
$\bar{D}_{s}^{(*)}[\bar{c}s]$–$\Xi_{c}^{(\prime*)}[csq]$ covalent molecules,
etc. These states are induced by shared light $up/down/strange$ quarks, with
the light $SU(3)$ flavor structure to be antisymmetric.

  * •

If the heavy-quark-exchange interaction is negligible, there might be the
$I=0$ $\bar{D}^{(*)}[\bar{c}q]$–$\bar{D}^{(*)}[\bar{c}q]$, $I=0$
$\Sigma_{c}^{(*)}[cqq]$–$\Sigma_{c}^{(*)}[cqq]$, and $I=1/2$
$\Sigma_{c}^{(*)}[cqq]$–$\Xi_{c}^{(\prime*)}[csq]$ covalent molecules, etc.
These states are still induced by shared light $up/down$ quarks.

  * •

Especially, we propose to search for the
$D^{*-}[\bar{c}d]$–$K^{*0}[\bar{s}d]$, $D^{*-}[\bar{c}d]$–$D^{*-}[\bar{c}d]$,
and $D^{*-}[\bar{c}d]$–$B^{*0}[\bar{b}d]$ covalent molecules of $I=1$ and
$J=0$. These states might exist, when the two light quarks spin in opposite
directions and the two antiquarks also spin in opposite directions, just like
the para-hydrogen.

To end this paper, we note again that the one-meson-exchange interaction
$\Pi_{M}$ at the hadron level and the light-quark-exchange interaction
$\Pi_{Q}$ at the quark-gluon level can overlap with each other. Hence, we may
also attempt to understand the nuclear force based on the above model-
independent hypothesis very roughly: a) the isoscalar $up$-$down$ quark pairs
inside $proton/neutron$ can not be exchanged, forming repulsive cores; b) the
other $up/down$ quarks inside $proton/neutron$ can be freely
exchanged/shared/moved, inducing some interactions among nucleons.

Table 1: Possibly-existing covalent hadronic molecules $|(I)J^{P}\rangle$ induced by shared light $up/down$ quarks, derived from the hypothesis “the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that obey the Pauli principle”. Here, $q$ and $s$ denote the light $up/down$ and $strange$ quarks, respectively; $Q$ and $Q^{\prime}$ denote two different heavy quarks; the symbols $[AB]=AB-BA$ and $\\{AB\\}=AB+BA$ denote the antisymmetric and symmetric light $SU(3)$ flavor structures, respectively. The states with $\checkmark$ have been confirmed in QCD sum rule calculations of the present study, but the states with $?$ and $??$ can not. This is because the latter contain relatively-polarized components, while one needs to sum over polarizations of these components within QCD sum rule method and so can not differentiate these hyperfine structures. Moreover, the states with $??$ contain light $up/down$ quarks with the symmetric flavor/isospin structure, whose masses can be considerably larger than their partners with the antisymmetric flavor/isospin structure. The states without any identification are still waiting to be carefully analysed in our future QCD sum rule studies. | $|\bar{Q}q,{1\over 2}0^{-}\rangle$ | $|\bar{Q}q,{1\over 2}1^{-}\rangle$ | $|Q[qq],0{1\over 2}^{+}\rangle$ | $|Q\\{qq\\},1{1\over 2}^{+}\rangle$ | $|Q\\{qq\\},1{3\over 2}^{+}\rangle$ | $|Q[sq],{1\over 2}{1\over 2}^{+}\rangle$ | $|Q\\{sq\\},{1\over 2}{1\over 2}^{+}\rangle$ | $|Q\\{sq\\},{1\over 2}{3\over 2}^{+}\rangle$
---|---|---|---|---|---|---|---|---
$|\bar{Q}^{\prime}q,{1\over 2}0^{-}\rangle$ | $|(0)0^{+}\rangle$ |  $|(0)1^{+}\rangle$ ($\checkmark$) | – |  $|({1\over 2}){1\over 2}^{-}\rangle$ ($\checkmark$) |  $|({1\over 2}){3\over 2}^{-}\rangle$ ($\checkmark$) | $|(0){1\over 2}^{-}\rangle$ | $|(0){1\over 2}^{-}\rangle$ | $|(0){3\over 2}^{-}\rangle$
$|\bar{Q}^{\prime}q,{1\over 2}1^{-}\rangle$ |  | $\begin{array}[]{c}|(0)0^{+}\rangle~{}(?)\\\ |(1)0^{+}\rangle~{}(??)\\\ |(0)1^{+}\rangle~{}(?)\\\ |(1)1^{+}\rangle~{}(??)\\\ |(0)2^{+}\rangle~{}(\checkmark)\end{array}$ | – | $\begin{array}[]{c}|({1\over 2}){1\over 2}^{-}\rangle~{}(?)\\\ |({3\over 2}){1\over 2}^{-}\rangle~{}(??)\\\ |({1\over 2}){3\over 2}^{-}\rangle~{}(\checkmark)\\\ |({3\over 2}){3\over 2}^{-}\rangle~{}(??)\end{array}$ | $\begin{array}[]{c}|({1\over 2}){1\over 2}^{-}\rangle~{}(?)\\\ |({3\over 2}){1\over 2}^{-}\rangle~{}(??)\\\ |({1\over 2}){3\over 2}^{-}\rangle~{}(?)\\\ |({3\over 2}){3\over 2}^{-}\rangle~{}(??)\\\ |({1\over 2}){5\over 2}^{-}\rangle~{}(\checkmark)\end{array}$ | $\begin{array}[]{c}|(0){1\over 2}^{-}\rangle\\\ |(0){3\over 2}^{-}\rangle\end{array}$ | $\begin{array}[]{c}|(0){1\over 2}^{-}\rangle\\\ |(1){1\over 2}^{-}\rangle\\\ |(0){3\over 2}^{-}\rangle\\\ |(1){3\over 2}^{-}\rangle\end{array}$ | $\begin{array}[]{c}|(0){1\over 2}^{-}\rangle\\\ |(1){1\over 2}^{-}\rangle\\\ |(0){3\over 2}^{-}\rangle\\\ |(1){3\over 2}^{-}\rangle\\\ |(0){5\over 2}^{-}\rangle\end{array}$
$|Q[qq],0{1\over 2}^{+}\rangle$ |  |  | – | – | – | – | – | –
$|Q\\{qq\\},1{1\over 2}^{+}\rangle$ |  |  |  | $\begin{array}[]{c}|(0)1^{+}\rangle\\\ |(1)0/1^{+}\rangle\\\ |(2)0/1^{+}\rangle\end{array}$ | $\begin{array}[]{c}|(0)1/2^{+}\rangle\\\ |(1)1/2^{+}\rangle\\\ |(2)1^{+}\rangle\end{array}$ | $|({1\over 2})0/1^{+}\rangle$ | $\begin{array}[]{c}|({1\over 2})0/1^{+}\rangle\\\ |({3\over 2})0/1^{+}\rangle\end{array}$ | $\begin{array}[]{c}|({1\over 2})1/2^{+}\rangle\\\ |({3\over 2})1/2^{+}\rangle\end{array}$
$|Q\\{qq\\},1{3\over 2}^{+}\rangle$ |  |  |  |  | $\begin{array}[]{c}|(0)1/2/3^{+}\rangle\\\ |(1)0/1/2^{+}\rangle\\\ |(2)0/1^{+}\rangle\end{array}$ | $|({1\over 2})1/2^{+}\rangle$ | $\begin{array}[]{c}|({1\over 2})1/2^{+}\rangle\\\ |({3\over 2})1/2^{+}\rangle\end{array}$ | $\begin{array}[]{c}|({1\over 2})0/1/2/3^{+}\rangle\\\ |({3\over 2})0/1/2^{+}\rangle\end{array}$

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grants No. 11722540 and No. 12075019, and the Fundamental Research Funds
for the Central Universities.

##  Appendix A Spectral densities

In this appendix we list spectral densities extracted from the currents
$J^{D^{-}\Sigma_{c}^{++}}$, $J^{\bar{D}^{0}\Sigma_{c}^{+}}$,
$J^{\bar{D}\Sigma_{c}}$ of $I=1/2$, and $J_{I=3/2}^{\bar{D}\Sigma_{c}}$. These
currents are defined in Eqs. (II.1), (II.2), (21), and (26), respectively. 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 $\rho^{\bar{D}\Sigma_{c}}(s)$ extracted from the current
$J^{\bar{D}\Sigma_{c}}$ of $I=1/2$ can be separated into

$\rho^{\bar{D}\Sigma_{c}}(s)=\rho_{0}^{\bar{D}\Sigma_{c}}(s)+\rho_{Q}^{\bar{D}\Sigma_{c}}(s)\,.$
(83)

The leading term $\rho_{0}^{\bar{D}\Sigma_{c}}(s)$ is

$\displaystyle\rho_{0}^{\bar{D}\Sigma_{c}}(s)$ $\displaystyle=$ $\displaystyle
m_{c}~{}\left(\rho^{pert}_{1}(s)+\rho^{\langle\bar{q}q\rangle}_{1}(s)+\rho^{\langle
GG\rangle}_{1}(s)+\rho^{\langle\bar{q}Gq\rangle}_{1}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{1}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{1}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{1}(s)\right)$
$\displaystyle+$ $\displaystyle
q\\!\\!\\!/~{}\left(\rho^{pert}_{2}(s)+\rho^{\langle\bar{q}q\rangle}_{2}(s)+\rho^{\langle
GG\rangle}_{2}(s)+\rho^{\langle\bar{q}Gq\rangle}_{2}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{2}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{2}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{2}(s)\right)\,,$

where

$\displaystyle\rho^{pert}_{1}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{81920\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{1}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-(1-\alpha-\beta)^{2}}{1024\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{1}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{98304\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)\left(3\alpha^{3}+\alpha^{2}(8\beta-3)+\alpha(\beta-1)\beta-(\beta-1)^{2}\beta\right)}{98304\pi^{8}\alpha^{5}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{1}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{3(1-\alpha-\beta)(\alpha+2\beta-1)}{2048\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{1}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-1}{64\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{1}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-1}{128\pi^{4}\alpha^{2}}\Bigg{\\}}+\mathcal{H}(s)\times\frac{1}{64\pi^{4}\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{1}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{(\alpha-1)(2\alpha-1)}{512\pi^{4}\alpha}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}m_{c}^{2}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)\times\frac{-1}{512\pi^{4}\alpha}\Bigg{\\}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{1}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{3}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{1}{24\pi^{2}}\Bigg{\\}}\,,$
$\displaystyle\rho^{pert}_{2}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{40960\pi^{8}\alpha^{4}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{2}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-(1-\alpha-\beta)^{2}}{512\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{2}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{49152\pi^{8}\alpha^{4}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)^{2}(2\alpha+\beta)}{32768\pi^{8}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{2}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{3(1-\alpha-\beta)(\alpha+2\beta-1)}{1024\pi^{6}\alpha\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{2}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-1}{128\pi^{4}\alpha\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{2}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-1}{256\pi^{4}\alpha}\Bigg{\\}}+\mathcal{H}(s)\times\frac{1}{128\pi^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{2}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{(\alpha-1)(2\alpha-1)}{1024\pi^{4}}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}m_{c}^{2}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)\times\frac{-1}{1024\pi^{4}}\Bigg{\\}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{2}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{3}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{\alpha}{48\pi^{2}}\Bigg{\\}}\,.$

The light-quark-exchange term $\rho_{Q}^{\bar{D}\Sigma_{c}}(s)$ is

$\displaystyle\rho_{Q}^{\bar{D}\Sigma_{c}}(s)$ $\displaystyle=$ $\displaystyle
m_{c}~{}\left(\rho^{pert}_{3}(s)+\rho^{\langle\bar{q}q\rangle}_{3}(s)+\rho^{\langle
GG\rangle}_{3}(s)+\rho^{\langle\bar{q}Gq\rangle}_{3}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{3}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{3}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{3}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{3}(s)\right)$
$\displaystyle+$ $\displaystyle
q\\!\\!\\!/~{}\left(\rho^{pert}_{4}(s)+\rho^{\langle\bar{q}q\rangle}_{4}(s)+\rho^{\langle
GG\rangle}_{4}(s)+\rho^{\langle\bar{q}Gq\rangle}_{4}(s)+\rho^{\langle\bar{q}q\rangle^{2}}_{4}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{4}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}_{4}(s)+\rho^{\langle\bar{q}q\rangle^{3}}_{4}(s)\right)\,,$

where

$\displaystyle\rho^{pert}_{3}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{983040\pi^{8}\alpha^{5}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{3}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-(1-\alpha-\beta)^{2}}{3072\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{3}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{1179648\pi^{8}\alpha^{5}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)^{2}\left(8\alpha^{2}+6\alpha\beta+\alpha-2(\beta-1)\beta\right)}{2359296\pi^{8}\alpha^{5}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{3}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)\left(2\alpha^{2}+\alpha(6\beta-2)+(\beta-1)\beta\right)}{8192\pi^{6}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{3}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-5}{1536\pi^{4}\alpha^{2}\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{3}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-6\alpha-5\beta}{3072\pi^{4}\alpha^{2}\beta}\Bigg{\\}}+\mathcal{H}(s)\times\frac{7}{3072\pi^{4}\alpha}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{3}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{4\alpha^{2}-3\alpha+5}{12288\pi^{4}\alpha}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}m_{c}^{2}\delta\left(s-{m_{c}^{2}\over\alpha(1-\alpha)}\right)\times\frac{-1}{6144\pi^{4}\alpha}\Bigg{\\}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{3}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{3}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{1}{288\pi^{2}}\Bigg{\\}}\,,$
$\displaystyle\rho^{pert}_{4}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{5}\times\frac{(1-\alpha-\beta)^{3}}{491520\pi^{8}\alpha^{4}\beta^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle}_{4}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{3}\times\frac{-5(1-\alpha-\beta)^{2}}{12288\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle GG\rangle}_{4}(s)$ $\displaystyle=$
$\displaystyle{\langle
g_{s}^{2}GG\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}m_{c}^{2}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)^{3}\left(\alpha^{3}+\beta^{3}\right)}{589824\pi^{8}\alpha^{4}\beta^{4}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\mathcal{F}(s)^{3}\times\frac{(\alpha+\beta-1)\left(23\alpha^{2}+\alpha(7\beta-22)+8\beta^{2}-7\beta-1\right)}{2359296\pi^{8}\alpha^{3}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle}_{4}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{(1-\alpha-\beta)\left(14\alpha^{2}+\alpha(25\beta-14)+3(\beta-1)\beta\right)}{32768\pi^{6}\alpha^{2}\beta^{3}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}_{4}(s)$ $\displaystyle=$
$\displaystyle{\langle\bar{q}q\rangle^{2}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)^{2}\times\frac{-1}{384\pi^{4}\alpha\beta}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}_{4}(s)$
$\displaystyle=$ $\displaystyle{\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma
Gq\rangle}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\int^{\beta_{max}}_{\beta_{min}}d\beta\Bigg{\\{}\mathcal{F}(s)\times\frac{-5\alpha-3\beta}{3072\pi^{4}\alpha\beta}\Bigg{\\}}+\mathcal{H}(s)\times\frac{7}{3072\pi^{4}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}_{4}(s)$ $\displaystyle=$
$\displaystyle{\langle g_{s}\bar{q}\sigma
Gq\rangle^{2}}\Bigg{\\{}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{6\alpha^{2}-4\alpha+3}{12288\pi^{4}}\Bigg{\\}}+\int^{1}_{0}d\alpha\Bigg{\\{}\frac{-1}{4096\pi^{4}}\Bigg{\\}}\Bigg{\\}}\,,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{3}}_{4}(s)$ $\displaystyle=$
$\displaystyle{m_{c}\langle\bar{q}q\rangle^{3}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\Bigg{\\{}\frac{\alpha}{576\pi^{2}}\Bigg{\\}}\,.$

The spectral densities $\rho^{D^{-}\Sigma_{c}^{++}}(s)$,
$\rho^{\bar{D}^{0}\Sigma_{c}^{+}}(s)$, and
$\rho_{I=3/2}^{\bar{D}\Sigma_{c}}(s)$ extracted from the currents
$J^{D^{-}\Sigma_{c}^{++}}$, $J^{\bar{D}^{0}\Sigma_{c}^{+}}$, and
$J_{I=3/2}^{\bar{D}\Sigma_{c}}$ are related to $\rho^{\bar{D}\Sigma_{c}}(s)$
through

$\displaystyle\rho^{D^{-}\Sigma_{c}^{++}}(s)$ $\displaystyle=$
$\displaystyle\rho_{0}^{\bar{D}\Sigma_{c}}(s)\,,$ (86)
$\displaystyle\rho^{\bar{D}^{0}\Sigma_{c}^{+}}(s)$ $\displaystyle=$
$\displaystyle\rho_{0}^{\bar{D}\Sigma_{c}}(s)-\rho_{Q}^{\bar{D}\Sigma_{c}}(s)\,,$
(87) $\displaystyle\rho_{I=3/2}^{\bar{D}\Sigma_{c}}(s)$ $\displaystyle=$
$\displaystyle\rho_{0}^{\bar{D}\Sigma_{c}}(s)-2\rho_{Q}^{\bar{D}\Sigma_{c}}(s)\,.$
(88)

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