# Correlation of the hidden-charm molecular tetraquarks and the charmoniumlike
structures existing in the $B\to XYZ+K$

Fu-Lai Wang1,2<EMAIL_ADDRESS>Xin-Dian Yang1,2<EMAIL_ADDRESS>Rui
Chen4,5<EMAIL_ADDRESS>Xiang Liu1,2,3111Corresponding author
<EMAIL_ADDRESS>1School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China
4Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Department of Physics and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha
410081, China
5Center of High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

The molecular assignments to the three $P_{c}$ states and the similar
production mechanism between the $\Lambda_{b}\to P_{c}+K$ and $B\to XYZ+K$
convince us the $B$ decaying to a charmonium state plus light mesons could be
the appropriate production process to search for the charmoniumlike molecular
tetraquarks. In this work, we systematically study the interactions between a
charmed (charmed-strange) meson and an anti-charmed (anti-charm-strange)
meson, which include the $D^{(*)}\bar{D}^{(*)}$, $\bar{D}^{(*)}\bar{D}_{1}$,
$D^{(*)}\bar{D}_{2}^{*}$, $D_{s}^{(*)}\bar{D}_{s}^{(*)}$,
${D}_{s}^{(*)}\bar{D}_{s0}^{*}$, $D_{s}^{(*)}\bar{D}_{s1}^{\prime}$,
${D}_{s}^{(*)}\bar{D}_{s1}$, $D_{s}^{(*)}\bar{D}_{s2}^{*}$ systems. After
adopting the one-boson-exchange effective potentials, our numerical results
indicate that, on one hand, there can exist a serial of isoscalar
charmoniumlike $\mathcal{D}\bar{\mathcal{D}}$ and
$\mathcal{D}_{s}\bar{\mathcal{D}}_{s}$ molecular states, on the other hand, we
can fully exclude the charged charmoniumlike states as the isovector
charmoniumlike molecules. Meanwhile, we discuss the two-body hidden-charm
decay channels for the obtained $\mathcal{D}\bar{\mathcal{D}}$ and
$\mathcal{D}_{s}\bar{\mathcal{D}}_{s}$ molecules, especially the
$D^{*}\bar{D}^{*}$ molecular tetraquarks. By analyzing the experimental data
collected from the $B\to XYZ+K$ and the mass spectrum and two-body hidden-
charm decay channels for the obtained $\mathcal{D}\bar{\mathcal{D}}$ and
$\mathcal{D}_{s}\bar{\mathcal{D}}_{s}$ molecules, we find several possible
hints of the existence of the charmoniumlike molecular tetraquarks, i.e., a
peculiar characteristic mass spectrum of the isoscalar $D^{*}\bar{D}^{*}$
molecular systems can be applied to identify the charmoniumlike molecule. We
look forward to the future experiments like the LHCb, Belle II, and BESIII
Collaborations can test our results with more precise experimental data.

##  I Introduction

In 2015, the LHCb Collaboration analyzed the $\Lambda_{b}\to J/\psi pK$ decay
and reported two $P_{c}$ structures ($P_{c}(4380)$ and $P_{c}(4450)$) existing
in the $J/\psi p$ invariant mass spectrum Aaij:2015tga . After four years, the
LHCb revised $\Lambda_{b}\to J/\psi pK$ process with higher precision data and
found that former $P_{c}(4450)$ contains two substructures $P_{c}(4440)$ and
$P_{c}(4457)$ Aaij:2019vzc . Besides, a new enhancement structure
$P_{c}(4312)$ was announced Aaij:2019vzc . In fact, this updated result of
$P_{c}$ states provides a direct evidence to support the existence of the
hidden-charm molecular pentaquark states Li:2014gra ; Wu:2010jy ;
Karliner:2015ina ; Wang:2011rga ; Yang:2011wz ; Wu:2012md ; Chen:2015loa .

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Figure 1: The production mechanisms of the $P_{c}$ states from the
$\Lambda_{b}$ baryon decays and the $XYZ$ states from the $B$ meson decays.

In Fig. 1, we present the quark level description of the $\Lambda_{b}\to
P_{c}K$, which is a typical hadronic weak decay. If replacing $ud$ quarks of
the $\Lambda_{b}$ by an antiquark $\bar{q}$, we may get the $B\to XYZ+K$
process, where $XYZ$ denote the charmoniumlike structures. Obviously, due to
the similar production mechanism between the $\Lambda_{b}\to P_{c}K$ and $B\to
XYZ+K$ (see Fig. 1), we naturally conjecture that the $B\to XYZ+K$ should be
the ideal processes to produce the hidden-charm molecular tetraquark states,
especially with establishing the hidden-charm molecular tetraquark states. As
is well known, the $X(3872)$ is a typical example of hidden-charm molecular
tetraquark state, which was found in the $B^{\pm}\to
J/\psi\pi^{+}\pi^{-}K^{\pm}$ process by the Belle collaboration in 2003
Choi:2003ue . The $X(3872)$ was suggested to be an isoscalar $D\bar{D}^{*}$
molecular state with $J^{PC}=1^{++}$ Wong:2003xk ; Swanson:2003tb ;
Suzuki:2005ha ; Liu:2008fh ; Thomas:2008ja ; Liu:2008tn ; Lee:2009hy .

In fact, the road of identifying the hidden-charm molecular tetraquark states
is getting confusing Chen:2016qju ; Liu:2013waa ; Hosaka:2016pey ; Liu:2019zoy
; Brambilla:2019esw ; Olsen:2017bmm ; Guo:2017jvc . In 2004, the Belle
Collaboration reported the observation of the charmoniumlike state $Y(3940)$
in the $J/\psi\omega$ invariant mass spectrum of the $B\to J/\psi\omega K$
Abe:2004zs . In 2009, the CDF Collaboration observed another charmoniumlike
state $Y(4140)$ in the $J/\psi\phi$ invariant mass spectrum of the $B\to
J/\psi\phi K$ Aaltonen:2009tz . In Ref. Liu:2009ei , Liu and Zhu noticed the
similarity between the $Y(3940)$ and $Y(4140)$, and proposed $S-$wave
$D^{*}\bar{D}^{*}$ and $D_{s}^{*}\bar{D}_{s}^{*}$ molecular states assignment
to the $Y(3940)$ and $Y(4140)$, respectively, where the $J^{PC}$ quantum
numbers for the $Y(3940)$ and $Y(4140)$ should be either $0^{++}$ or $2^{++}$
due to a simple selection rules from the parity and angular momentum
conservation Liu:2009ei . In the subsequent experiments, the LHCb analyzed the
$B\to J/\psi\phi K$ process, where the $Y(4140)$ was confirmed and other three
new structures $X(4274)$, $X(4500)$ and $X(4700)$ were discovered in the
$J/\psi\phi$ invariant mass spectrum Aaij:2016iza . However, the LHCb
announced that the preferred $J^{PC}$ quantum number of the $Y(4140)$ is
$1^{++}$ Aaij:2016nsc , which obviously cannot support the molecular
assignment to the $Y(4140)$ suggested in Ref. Liu:2009ei . If checking the
experimental and theoretical research status of the $Z(4430)$, which was
reported by the Belle in the $B\to\psi(3686)\pi K$ Choi:2007wga , the same
situation again happens. Since the $Z(4430)$ is near the threshold of
$D^{*}\bar{D}_{1}^{(\prime)}$ channel, the authors of Refs. Liu:2007bf ;
Liu:2008xz  suggested that the $Z(4430)$ can be the good candidate of the
$S-$wave $D_{1}\bar{D}^{*}$ molecular state222To be convenient, we use the
shorthand notation $\mathcal{A}\overline{\mathcal{B}}$ to denote the
$\mathcal{A}\overline{\mathcal{B}}+c.c.$ system in the following parts, where
the notations $\mathcal{A}$ and $\mathcal{B}$ represent two different charmed
(charmed-strange) mesons, respectively. with $J^{P}=0^{-},1^{-},2^{-}$ by the
one-boson-exchange (OBE) model calculation. However, the Belle
Chilikin:2013tch  and LHCb Aaij:2014jqa  reanalyzed the $B\to\psi(3686)\pi K$
and indicated that the $Z(4430)$ has quantum number $J^{P}=1^{+}$, which is
contradict with the $S-$wave $D_{1}\bar{D}^{*}$ molecular state assignment
Liu:2007bf ; Liu:2008xz .

Although there were a dozen of charmoniumlike structures reported by
experiments in the $B$ meson decays Brambilla:2019esw , unfortunately we have
not definitely identify one hidden-charm molecular tetraquark state until now
Chen:2016qju ; Liu:2013waa ; Hosaka:2016pey ; Liu:2019zoy ; Brambilla:2019esw
; Olsen:2017bmm ; Guo:2017jvc . It is big challenge to assign the molecular
state explanations to the charmoniumlike structures from the $B\to XYZ+K$
processes.

If we still believe the similar production mechanism between the
$\Lambda_{b}\to P_{c}K$ and $B\to XYZ+K$ (see Fig. 1), we have reason to
believe the existence of the hidden-charm molecular tetraquark state in the
$B\to XYZ+K$ processes. Facing this situation mentioned above, we should try
to find possible solutions existing in the reported experimental data of the
$B\to XYZ+K$. In fact, the lesson of observation of the $P_{c}$ states in 2015
Aaij:2015tga  and 2019 Aaij:2019vzc  may inspire us. In 2015, the LHCb
measured the $J^{P}$ quantum numbers of the $P_{c}(4380)$ and $P_{c}(4450)$,
gave that their preferred $J^{P}$ quantum numbers are of opposite parity
Aaij:2015tga , which is a challenge to the hadronic molecular assignment
Chen:2015loa . This situation was dramatically changed with further LHCb
experiment in 2019, where the $P_{c}(4450)$ is composed of two substructures
$P_{c}(4440)$ and $P_{c}(4457)$ Aaij:2019vzc , which means that the
measurement of spin-parity quantum number of the observed $P_{c}(4450)$ can be
ignored Aaij:2015tga . To some extent, this fact reflects the importance of
higher precision to the study of hadron spectroscopy.

With the running of the Belle II Kou:2018nap  and the accumulation of Run II
and Run III data at the LHCb Bediaga:2018lhg , obviously investigation of the
charmoniumlike $XYZ$ states must enter a new era. Thus, we should
systematically reexamine the correlation of the hidden-charm molecular states
and the charmoniumlike structures existing in the $B\to XYZ+K$ processes
Chen:2016qju ; Liu:2013waa ; Hosaka:2016pey ; Liu:2019zoy ; Brambilla:2019esw
; Olsen:2017bmm ; Guo:2017jvc , where the constraint from $J^{PC}$ quantum
numbers and so-called resonance parameters of depicting these observed
charmoniumlike $XYZ$ structures should be more cautious in the interpretation
of these resonances as the molecular states.

Along this line, we systematically restudy the $S-$wave interactions between a
charmed (charmed-strange) meson and an anticharmed (anticharmed-strange) meson
in the framework of the one-boson-exchange (OBE) model in this work
Chen:2016qju ; Liu:2019zoy . In concrete calculations, both the $S-$$D$ wave
mixing effect and the coupled channel effect are taken into account.
Additionally, we discuss the two-body hidden-charm decay channels for the
obtained $\mathcal{D}\bar{\mathcal{D}}$ and
$\mathcal{D}_{s}\bar{\mathcal{D}}_{s}$ molecules, especially the
$D^{*}\bar{D}^{*}$ molecular tetraquarks. Based on above discussion, we find a
series of possible hints of the hidden-charm molecular tetraquarks existing in
the released experimental data of the $B$ meson decays, which will be a main
task of the present work.

The remainder of this paper is organized as follows. In Sec. II, a comparison
of the relevant experimental data and the corresponding thresholds will be
given. In Sec. III, the mass spectrum and the two-body hidden-charm decay
channels of these discussed hidden-charm molecular tetraquark systems will be
given, and a series of possible hints of the hidden-charm molecular
tetraquarks existing in the reported experimental data of the $B$ meson decays
will be presented. Finally, a brief summary will be given in Sec. IV.

##  II A comparison of the experimental data and the corresponding thresholds

In this work, the main task is to find possible hints of the hidden-charm
molecular tetraquarks existing in the reported experimental data of the $B\to
XYZ+K$ Brambilla:2019esw , and the hidden-charm molecular tetraquarks are
composed of a charmed (charmed-strange) meson and an anticharmed (anticharmed-
strange) meson. Thus, we need to make comparison of the involved experimental
data and the corresponding thresholds of charmed meson pairs or charmed-
strange meson pairs.

Table 1: The thresholds of charmed (charmed-strange) meson pairs (in unit of
MeV). Without hidden-strange quantum number
---
$DD$ | $DD^{*}$ | $D^{*}D^{*}$ | $DD_{0}^{*}$ | $DD_{1}$ | $DD_{1}^{\prime}$ | $DD_{2}^{*}$
3734.48 | 3875.80 | 4017.12 | 4191.74 | 4289.24 | 4294.24 | 4330.29
$D^{*}D_{0}^{*}$ | $D^{*}D_{1}$ | $D^{*}D_{1}^{\prime}$ | $D^{*}D_{2}^{*}$ | $D_{0}^{*}D_{0}^{*}$ | $D_{0}^{*}D_{1}$ | $D_{0}^{*}D_{1}^{\prime}$
4333.06 | 4430.56 | 4435.56 | 4471.61 | 4649.00 | 4746.50 | 4751.50
$D_{0}^{*}D_{2}^{*}$ | $D_{1}D_{1}$ | $D_{1}^{\prime}D_{1}$ | $D_{1}^{\prime}D_{1}^{\prime}$ | $D_{1}D_{2}^{*}$ | $D_{1}^{\prime}D_{2}^{*}$ | $D_{2}^{*}D_{2}^{*}$
4787.55 | 4844.00 | 4849.00 | 4854.00 | 4885.05 | 4890.05 | 4926.10
With hidden-strange quantum number
$D_{s}D_{s}$ | $D_{s}D_{s}^{*}$ | $D_{s}^{*}D_{s}^{*}$ | $D_{s}D_{s0}^{*}$ | $D_{s}D_{s1}^{\prime}$ | $D_{s}^{*}D_{s0}^{*}$ | $D_{s}D_{s1}$
3936.68 | 4080.54 | 4224.40 | 4286.14 | 4427.84 | 4430 | 4503.45
$D_{s}D_{s2}^{*}$ | $D_{s}^{*}D_{s1}^{\prime}$ | $D_{s0}^{*}D_{s0}^{*}$ | $D_{s}^{*}D_{s1}$ | $D_{s}^{*}D_{s2}^{*}$ | $D_{s0}^{*}D_{s1}^{\prime}$ | $D_{s0}^{*}D_{s1}$
4537.44 | 4571.70 | 4635.60 | 4647.31 | 4681.30 | 4777.30 | 4852.91
$D_{s0}^{*}D_{s2}^{*}$ | $D_{s1}^{\prime}D_{s1}^{\prime}$ | $D_{s1}^{\prime}D_{s1}$ | $D_{s1}D_{s1}$ | $D_{s1}^{\prime}D_{s2}^{*}$ | $D_{s1}D_{s2}^{*}$ | $D_{s2}^{*}D_{s2}^{*}$
4886.90 | 4919.00 | 4994.61 | 5028.60 | 5070.22 | 5104.20 | 5138.20

Usually, the $S$-wave and $P$-wave charmed mesons can be grouped into three
doublets $H=[D(0^{-}),D^{\ast}(1^{-})]$,
$S=[D_{0}^{\ast}(0^{+}),D^{\prime}_{1}(1^{+})]$, and
$T=[D_{1}(1^{+}),D^{\ast}_{2}(2^{+})]$ according to the heavy quark spin
symmetry Wise:1992hn . Similarly, there also exist three doublets for the
$S-$wave and $P$-wave charmed-strange mesons, i.e., $H=[D_{s},D_{s}^{\ast}]$,
$S=[D_{s0}^{\ast},D_{s1}^{\prime}]$, and $T=[D_{s1},D_{s2}^{\ast}]$
Wise:1992hn . In Table 1, we list these thresholds of charmed meson pairs or
charmed-strange meson pairs Zyla:2020zbs , which are distributed over a very
wide energy range $3.7\sim 5.2$ GeV. However, if only considering the kinetics
of the $B\to XYZ+K$ decays, the maximum mass of the involved $XYZ$ structures
should be $M(XYZ)_{max}=M(B)-M(K)=4783~{}{\rm MeV}$, which makes us select
these thresholds with mass lower than 4783 MeV when making further analysis.
Additionally, the charmed mesons in $S-$doublet have broad widths around
several hundred MeV Zyla:2020zbs , which may be a obstacle for the formation
of the hadronic molecular states Hanhart:2010wh ; Filin:2010se . Different
from the charmed mesons in $S-$doublet, these charmed mesons in the
$H$-doublet and $T$-doublet have narrow widths Zyla:2020zbs , which can be
regarded as the suitable components to form the hadronic molecular states
Liu:2008xz ; Hanhart:2010wh ; Filin:2010se . Just considering the above fact,
we consider the cases of $H\bar{H}$ and $H\bar{T}$ for charmed meson pairs and
the cases of $H\bar{H}$, $H\bar{S}$, and $H\bar{T}$ for charmed-strange meson
pairs when performing a comparison of these thresholds with the released
experimental data.

At present, the experimental information of the $B\to XYZ+K$ are very abundant
Chen:2016qju ; Liu:2013waa ; Hosaka:2016pey ; Liu:2019zoy ; Brambilla:2019esw
; Olsen:2017bmm ; Guo:2017jvc . Here, the $XYZ$ data are from the
$J/\psi\pi\pi$ invariant mass spectrum of the $B\to J/\psi\pi\pi K$
Choi:2003ue , the $J/\psi\omega$ invariant mass spectrum of the $B\to
J/\psi\omega K$ Abe:2004zs ; Aubert:2007vj ; delAmoSanchez:2010jr , the
$J/\psi\eta$ invariant mass spectrum of the $B\to J/\psi\eta K$ Aubert:2004fc
, the $J/\psi\phi$ invariant mass spectrum of the $B\to J/\psi\phi K$
Aaltonen:2009tz ; Chatrchyan:2013dma ; Aaij:2016iza ; Aaij:2012pz ;
Abazov:2013xda ; Lees:2014lra ; Aaltonen:2011at ; Aaij:2016nsc ; Aaij:2021ivw
, the $\eta_{c}\pi$ invariant mass spectrum of the $B\to\eta_{c}\pi K$
Aaij:2018bla , the $J/\psi\pi$ invariant mass spectrum of the $B\to J/\psi\pi
K$ Chilikin:2014bkk ; Abazov:2018cyu ; Aaij:2019ipm , the $\psi(3686)\pi$
invariant mass spectrum of the $B\to\psi(3686)\pi K$ Choi:2007wga ;
Mizuk:2009da ; Chilikin:2013tch ; Aaij:2014jqa , the $\chi_{c1}\pi$ invariant
mass spectrum of the $B\to\chi_{c1}\pi K$ Mizuk:2008me ; Lees:2011ik ;
Bhardwaj:2015rju ; Bhardwaj:2019spn , and the $\chi_{c2}\pi$ invariant mass
spectrum of the $B\to\chi_{c2}\pi K$ Bhardwaj:2015rju . For clearly discussing
the present issue, we categorized these $XYZ$ data into three groups:

  * •

Isoscalar $XYZ$ data without hidden-strange quantum number are involved in the
$B\to J/\psi\omega K$ and $B\to J/\psi\eta K$;

  * •

Isoscalar $XYZ$ data with hidden-strange quantum number are relevant to the
$B\to J/\psi\phi K$;

  * •

Isovector $XYZ$ data without hidden-strange quantum number have relation to
five decay processes, i.e., the $B\to\eta_{c}\pi K$, $B\to J/\psi\pi K$,
$B\to\psi(3686)\pi K$, $B\to\chi_{c1}\pi K$, and $B\to\chi_{c2}\pi K$.

In the following, we adopt this line to make comparison of the released
experimental data and the corresponding thresholds.

###  II.1 Isoscalar $XYZ$ data without hidden-strange quantum number

Figure 2: (color online) The $J/\psi\omega$ and $J/\psi\eta$ invariant mass
distributions in the $B\to J/\psi\omega K$ and $B\to J/\psi\eta K$,
respectively, and the comparison with the thresholds of charmed meson pairs.
Here, the experimental data of the $J/\psi\omega$ invariant mass spectrum are
taken from the Belle Abe:2004zs , BaBar Aubert:2007vj , and BABAR
delAmoSanchez:2010jr , which correspond to diagrams (a)-(c), and the
experimental data of the $J/\psi\eta$ invariant mass spectrum is from the
BABAR measurement Aubert:2004fc  (see diagram (d)).

In 2005, the Belle Collaboration analyzed the $J/\psi\omega$ invariant mass
spectrum of the $B\to J/\psi\omega K$ decay, where the charmoniumlike
structure $Y(3940)$ was reported, which can be depicted by resonance
parameters $M=(3943\pm 11\pm 13)~{}{\rm MeV}$ and $\Gamma=(87\pm 22\pm
26)~{}{\rm MeV}$ Abe:2004zs . In 2008, the BABAR Collaboration confirmed this
observation in the same process with a lower mass, named as the $Y(3915)$, and
its mass and width were measured to be $(M,\Gamma)=(3914.6^{+3.8}_{-3.4}\pm
2.0~{}{\rm MeV},34^{+12}_{-8}\pm 5~{}{\rm MeV})$ Aubert:2007vj . In the
subsequent BABAR experiment in 2010 delAmoSanchez:2010jr , the mass and width
of the $Y(3915)$ were measured to be $M=(3919.1^{+3.8}_{-3.5}\pm 2.0)~{}{\rm
MeV}$ and $\Gamma=(31^{+10}_{-8}\pm 5)~{}{\rm MeV}$, respectively.

If checking these data of the $J/\psi\omega$ invariant mass spectrum of the
$B\to J/\psi\omega K$ from the Belle and BABAR Abe:2004zs ; Aubert:2007vj ;
delAmoSanchez:2010jr , we may find the impact of experimental precision on
reflecting the details. In the Belle data Abe:2004zs , there are only four
experimental points sandwiched by the $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$
thresholds. After several years, the number of experimental points in this
energy range reach up to 12 and 14, which correspond to the BABAR data
measured in 2008 Aubert:2007vj  and 2010 delAmoSanchez:2010jr , respectively.
In Fig. 2, we collect all reported data of the $J/\psi\omega$ invariant mass
spectrum from the $B\to J/\psi\omega K$ Abe:2004zs ; Aubert:2007vj ;
delAmoSanchez:2010jr . In fact, the BABAR data released in 2008 Aubert:2007vj
show the possibility of existing two structures around 3.9 GeV below the
$D^{*}\bar{D}^{*}$ threshold in the $J/\psi\omega$ invariant mass spectrum.
Especially, the BABAR measurement in 2010 further enforce this possibility
delAmoSanchez:2010jr .

If $D^{*}$ and $\bar{D}^{*}$ can be bound together to form the hidden-charm
molecular tetraquarks, the $J^{PC}$ quantum numbers of the $S-$wave isoscalar
$D^{*}\bar{D}^{*}$ molecular system must be $0^{++}$, $1^{+-}$, and $2^{++}$
Liu:2009ei . Under this assumption, the behavior of mass spectrum of the
$S-$wave isoscalar $D^{*}\bar{D}^{*}$ molecular states can explain why two
substructures around 3.9 GeV exist in the $J/\psi\omega$ invariant mass
spectrum of the $B\to J/\psi\omega K$ Abe:2004zs ; Aubert:2007vj ;
delAmoSanchez:2010jr . Thus, we strongly encourage our experimental colleagues
to focus on the detail of the structures around 3.9 GeV with more precise
data. If these substructures can be confirmed in future experiments discussed
above, it will provide strong evidence of existing the hidden-charm molecular
tetraquark. Later, we will revisit a dynamics study of the $S-$wave isoscalar
$D^{*}\bar{D}^{*}$ system, and come back to address this point.

Besides the enhancement structures around 3.9 GeV exist in the $J/\psi\omega$
invariant mass spectrum, we may find a very broad structure around 4.3 GeV
exists in the $J/\psi\omega$ invariant mass spectrum of the $B\to J/\psi\omega
K$ Abe:2004zs ; Aubert:2007vj ; delAmoSanchez:2010jr , where there exist four
thresholds ($D\bar{D}_{1}$, $D\bar{D}^{*}_{2}$, $D^{*}\bar{D}_{1}$, and
$D^{*}\bar{D}_{2}^{*}$) in this energy range, which inspire our interest in
exploring whether the isoscalar $D\bar{D}_{1}$, $D\bar{D}^{*}_{2}$,
$D^{*}\bar{D}_{1}$, and $D^{*}\bar{D}_{2}^{*}$ molecular tetraquarks exist in
nature, which will be one of tasks in this work. Obviously, the details of
such broad structure around 4.3 GeV should be given in future experiments with
more precise data accumulation.

For the isoscalar $XYZ$ data without hidden-strange quantum number, we should
mention the measurement of the $J/\psi\eta$ invariant mass spectrum in the
$B\to J/\psi\eta K$ decay. In 2004, the BABAR released the result of the
$J/\psi\eta$ invariant mass spectrum in the $B\to J/\psi\eta K$ Aubert:2004fc
. In Fig. 2, we compare the BABAR data with the several thresholds of charmed
meson pairs, and may find the evidence of one structure below
$D^{*}\bar{D}^{*}$ threshold and possible enhancement structure around 4.3 GeV
which overlaps with the $D\bar{D}_{1}$, $D\bar{D}^{*}_{2}$,
$D^{*}\bar{D}_{1}$, and $D^{*}\bar{D}_{2}^{*}$ thresholds. In fact, this
phenomenon again shows studying the isoscalar $D^{*}\bar{D}^{*}$,
$D\bar{D}_{1}$, $D\bar{D}^{*}_{2}$, $D^{*}\bar{D}_{1}$, and
$D^{*}\bar{D}_{2}^{*}$ hadronic molecular states is an urgent research issue,
which may provide crucial information to find the evidence of the existence of
the hidden-charm molecular tetraquark.

###  II.2 Isoscalar $XYZ$ data with hidden-strange quantum number

Up to now, the $B\to J/\psi\phi K$ decay have been analyzed by the different
experiment collaborations Aaltonen:2009tz ; Chatrchyan:2013dma ; Aaij:2016iza
; Aaij:2012pz ; Abazov:2013xda ; Lees:2014lra ; Aaltonen:2011at ; Aaij:2016nsc
; Aaij:2021ivw , where we are mainly interested in the experimental data of
the $J/\psi\phi$ invariant mass spectrum from the CDF Aaltonen:2009tz , CMS
Chatrchyan:2013dma , and LHCb Aaij:2016iza  in the following discussion. Since
the quark components of the $\phi$ and $J/\psi$ are the $s\bar{s}$ and
$c\bar{c}$, respectively, we present the thresholds of charmed-strange meson
pairs in Fig. 3 when making comparison with these released experimental data.

Figure 3: (color online) The measured $J/\psi\phi$ invariant mass distribution
of the $B\to J/\psi\phi K$ and the thresholds of charmed-strange meson pairs.
Here, the experimental data are taken from the CDF Aaltonen:2009tz , CMS
Chatrchyan:2013dma , and LHCb Aaij:2016iza , which correspond to diagrams
(a)-(c).

First, we should brief introduce the experimental status of the obtained
$J/\psi\phi$ invariant mass spectrum in the $B\to J/\psi\phi K$ decay
Aaltonen:2009tz ; Chatrchyan:2013dma ; Aaij:2016iza . As shown in Fig. 3 (a),
the CDF Collaboration reported the charmoniumlike structure $Y(4140)$ in the
$J/\psi\phi$ mass spectrum of the $B\to J/\psi\phi K$ decay in 2009
Aaltonen:2009tz . When depicting this enhancement structure, the resonance
parameters can be obtained, i.e., the mass and width of the $Y(4140)$ were
measured to be $M=(4143.0\pm 2.9\pm 1.2)~{}{\rm MeV}$ and
$\Gamma=(11.7^{+8.3}_{-5.0}\pm 3.7)~{}{\rm MeV}$ Aaltonen:2009tz ,
respectively. Besides the $Y(4140)$ structure, the CDF also reported another
enhancement structure, named as the $Y(4274)$, in the $J/\psi\phi$ invariant
mass spectrum, where its mass and width are
$(M,\Gamma)=(4274.4^{+8.4}_{-6.7}\pm 1.9~{}{\rm MeV},32.3^{+21.9}_{-15.3}\pm
7.6~{}{\rm MeV})$ Aaltonen:2009tz , respectively. Later, the CMS confirmed
these two structures $Y(4140)$ and $Y(4274)$ in the same decay process in
2014, where the resonance parameters of the $Y(4140)$ and $Y(4274)$ were
measured to be $(M,\Gamma)_{Y(4140)}=(4148.0\pm 2.4\pm 6.3~{}{\rm
MeV},28^{+15}_{-11}\pm 19~{}{\rm MeV})$ and $(M,\Gamma)_{Y(4274)}=(4313.8\pm
5.3\pm 7.3~{}{\rm MeV},38^{+30}_{-15}\pm 16~{}{\rm MeV})$ Chatrchyan:2013dma ,
respectively. Surprisingly, the LHCb Collaboration announced four
charmoniumlike resonances in the $J/\psi\phi$ mass spectrum of the $B\to
J/\psi\phi K$ decay in 2017 Aaij:2016iza . Here, we collect their resonance
parameters, i.e., $(M,\Gamma)_{Y(4140)}=(4146.5\pm 4.5^{+4.6}_{-2.8}~{}{\rm
MeV},83\pm 21^{+21}_{-14}~{}{\rm MeV})$, $(M,\Gamma)_{Y(4274)}=(4273.3\pm
8.3^{+17.2}_{-3.6}~{}{\rm MeV},56.2\pm 10.9^{+8.4}_{-11.1}~{}{\rm MeV})$,
$(M,\Gamma)_{X(4500)}=(4506\pm 11^{+12}_{-15}~{}{\rm MeV},92\pm
21^{+21}_{-20}~{}{\rm MeV})$, and $(M,\Gamma)_{X(4700)}=(4704\pm
10^{+14}_{-24}~{}{\rm MeV},120\pm 31^{+42}_{-33}~{}{\rm MeV})$ Aaij:2016iza .

If comparing three experimental data listed in Fig. 3, we find that the
precision of the LHCb data Aaij:2016iza  is higher than the CDF and CMS data
Aaltonen:2009tz ; Chatrchyan:2013dma , where there are 12 experimental points
in the energy range sandwiched by the $D_{s}\bar{D}_{s}^{*}$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ thresholds for the LHCb data Aaij:2016iza , which
reflect some abundant details difference from former CDF and CMS
Aaltonen:2009tz ; Chatrchyan:2013dma . When only adopting a Breit-Wigner
formula to describe this resonance structure around 4140 MeV, the width given
by the LHCb Aaij:2016iza  becomes wider than that from the CDF and CMS
Aaltonen:2009tz ; Chatrchyan:2013dma . This phenomenon is puzzling for us
since the LHCb released this result Aaij:2016iza , especially the recent LHCb
result of the $Y(4140)$ Aaij:2021ivw .

We notice an interesting fact that this structure around 4140 MeV reported by
the CDF is below the $D_{s}^{*}\bar{D}_{s}^{*}$ threshold Aaltonen:2009tz ,
and Liu and Zhu proposed that the $Y(4140)$ observed by the CDF can be
regarded as partner of the $Y(3940)$ due to the similarity between the
$Y(3940)$ and $Y(4140)$ in 2009 Liu:2009ei . Thus, the
$D_{s}^{*}\bar{D}_{s}^{*}$ hadronic molecular state explanation to the
$Y(4140)$ was given Liu:2009ei . Along this line, there may exist $S-$wave
$D_{s}^{*}\bar{D}_{s}^{*}$ molecular states, which is similar to the situation
of the $S-$wave isoscalar $D^{*}\bar{D}^{*}$ molecular system discussed in
Sec. II.1. Facing such abundant details given by the LHCb in 2017 Aaij:2016iza
, we may conjecture that the $Y(4140)$ structure may be contain at least two
substructures, which can be tested by future experiments based on more precise
data. If introducing this proposal, the conclusion of the spin-parity quantum
number $J^{PC}=1^{++}$ for the $Y(4140)$ given by the LHCb seems
unreliable333For the $S-$wave $D_{s}^{*}\bar{D}_{s}^{*}$ molecular system, its
$J^{PC}$ quantum numbers are either $0^{++}$ or $2^{++}$, which is due to a
selection rule for the quantum numbers Liu:2009ei . However, the LHCb
measurement suggested $J^{PC}=1^{++}$ for the $Y(4140)$ Aaij:2016iza , which
results in the difficulty to understand the $Y(4140)$ under the hadronic
molecular state assignment. Aaij:2016iza . Here, we need to mention that the
recent LHCb Collaboration updated the amplitude analysis of the $B^{+}\to
J/\psi\phi K^{+}$ decay with higher precision data and reported the $X(4140)$
with $J^{PC}=1^{++}$ and $X(4150)$ with $J^{PC}=2^{-+}$ Aaij:2021ivw , which
imply the very complicated structures around the $D_{s}^{*}\bar{D}_{s}^{*}$
threshold. In the following, the investigation of the $S-$wave
$D_{s}^{*}\bar{D}_{s}^{*}$ hadronic molecular system will become a main point
of this work, which will be further discussed in the next section.

Besides the structure around 4140 MeV existing in the $J/\psi\phi$ invariant
mass spectrum, we should focus on the $Y(4274)$ Aaltonen:2009tz ;
Chatrchyan:2013dma ; Aaij:2016iza , which is just near the
$D_{s}\bar{D}_{s0}^{*}$ threshold. Thus, the study of the
$D_{s}\bar{D}_{s0}^{*}$ molecular system will be paid attention in this work.
Although the LHCb reported $X(4500)$ and $X(4700)$ in 2017 Aaij:2016iza , we
can find more abundant structures above 4250 MeV existing in the $J/\psi\phi$
invariant mass spectrum if carefully checking the LHCb data Aaij:2016iza .
Indeed, the recent LHCb Collaboration announced the observation of other new
resonance structures existing in this energy range with more precise data,
i.e., the $X(4630)$ with $J^{PC}=1^{-+}$ and $X(4685)$ with $J^{PC}=1^{++}$
Aaij:2021ivw . We also notice that there are abundant thresholds of charmed-
strange meson pairs in this interesting energy range. Thus, we will
investigate the hidden-charm and hidden-strange molecular tetraquarks involved
these thresholds in this work.

###  II.3 Isovector $XYZ$ data without hidden-strange quantum number

Figure 4: (color online) The measured $\psi(2S)\pi$, $\eta_{c}\pi$,
$\chi_{c1}\pi$ and $\chi_{c2}\pi$ invariant mass distributions in the $B\to
XYZ+K$ decays, and the comparison with the thresholds of charmed meson pairs.
Here, the experimental data of the $\psi(2S)\pi$ are taken from (a) the Belle
Choi:2007wga , (b) the Belle Mizuk:2009da , (c) the Belle Chilikin:2013tch ,
and (d) the LHCb Aaij:2014jqa , while the experimental data of $\eta_{c}\pi$
is taken from the LHCb Aaij:2018bla  (see diagram (e)). In addition, the
experimental data of $\chi_{c1}\pi$ are taken from (f) the Belle Mizuk:2008me
, (g) the BABAR Lees:2011ik , (h) the Belle Bhardwaj:2015rju , and (i) the
Belle Bhardwaj:2019spn , while the experimental result of the $\chi_{c2}\pi$
invariant mass spectrum is given by the Belle Bhardwaj:2015rju  (see diagram
(j)). Figure 5: (color online) The $J/\psi\pi$ invariant mass distribution in
the $B\to J/\psi\pi K$ and the relevant thresholds of charmed meson pairs.
Here, the experimental datas are taken from the Belle Chilikin:2014bkk  ((a)
for $1.20~{}{\rm GeV}^{2}<m^{2}(K\pi)<2.05~{}{\rm GeV}^{2}$ and (b) for
$2.05~{}{\rm GeV}^{2}<m^{2}(K\pi)<3.20~{}{\rm GeV}^{2}$), the DØ
Abazov:2018cyu  ((c) for $4.25~{}{\rm GeV}<m(J/\psi\pi^{+}\pi^{-})<4.30~{}{\rm
GeV}$ and (d) for $4.30~{}{\rm GeV}<m(J/\psi\pi^{+}\pi^{-})<4.40~{}{\rm
GeV}$), and (e) the LHCb Aaij:2019ipm , respectively.

In the past 18 years, the isovector $XYZ$ data from the $B\to XYZ+K$ were
accumulated Chen:2016qju ; Liu:2013waa ; Hosaka:2016pey ; Liu:2019zoy ;
Brambilla:2019esw ; Olsen:2017bmm ; Guo:2017jvc . In Figs. 4-5, we collected
all relevant experimental data and make a comparison with several typical
thresholds of charmed meson pairs. For different decay processes listed in
Figs. 4-5, we briefly review the experimental information:

  1. 1.

The $B\to\psi(2S)\pi K$ decay: As a super star among these reported charged
charmoniumlike structures, the $Z^{+}(4430)$ structure was first observed by
the Belle Collaboration in the $\psi(2S)\pi$ invariant mass distribution of
the $B\to\psi(2S)\pi K$ decay in 2008 Choi:2007wga . Here, its mass and width
were measured to be $M=(4433\pm 4\pm 2)~{}{\rm MeV}$ and
$\Gamma=(45^{+18}_{-13}{}^{+30}_{-13})~{}{\rm MeV}$, respectively Choi:2007wga
. In the following years, the Belle Collaboration continued their studies on
the $Z^{+}(4430)$, and the measured mass of the $Z^{+}(4430)$ structure are
$M=(4443^{+15}_{-12}{}^{+19}_{-13})~{}{\rm MeV}$ Mizuk:2009da  and
$M=(4485^{+22}_{-22}{}^{+28}_{-11})~{}{\rm MeV}$ Chilikin:2013tch ,
respectively. After ten years, the LHCb Collaboration confirmed the existence
of the $Z^{+}(4430)$ structure Aaij:2014jqa , and the resonance parameters are
$M=(4475\pm 7^{+15}_{-25})~{}{\rm MeV}$ and $\Gamma=(172\pm
13^{+37}_{-34})~{}{\rm MeV}$, respectively. But, the width from the LHCb
Aaij:2014jqa  is more broad than that from the Belle Choi:2007wga ;
Mizuk:2009da ; Chilikin:2013tch . Additionally, the LHCb found a new structure
$Z^{+}(4240)$ in the $\psi(2S)\pi$ invariant mass distribution of the
$B\to\psi(2S)\pi K$, which can be depicted by resonance parameters $M=(4239\pm
18^{+45}_{-10})~{}{\rm MeV}$ and $\Gamma=(220\pm 47^{+108}_{-74})~{}{\rm
MeV}$, respectively Aaij:2014jqa . Comparing with the Belle data Choi:2007wga
; Mizuk:2009da ; Chilikin:2013tch , the LHCb data have smaller error bars
Aaij:2014jqa . Since the $Z^{+}(4430)$ structure is near several thresholds of
charmed meson pair, there were extensive discussion of the hidden-charm
molecular tetraquark explanation to the $Z^{+}(4430)$ Liu:2007bf ; Liu:2008xz
; Close:2009ag . In this work, we will still dedicate this topic, which is the
study of the isovector hidden-charm molecular tetraquark involved in this
energy range.

  2. 2.

The $B\to\eta_{c}\pi K$ decay: In 2018, the LHCb found the evidence of a broad
charmonium-like structure in the $\eta_{c}\pi$ invariant mass spectrum of the
$B\to\eta_{c}\pi K$ decay, which was named as the $X(4100)$ with the mass
$M=(4096\pm 20^{+18}_{-22})~{}{\rm MeV}$ and the width $\Gamma=(152\pm
58^{+60}_{-35})~{}{\rm MeV}$ Aaij:2018bla . In fact, there exists event
accumulation around 4.5 GeV, where several thresholds of charmed meson pairs
can be found (see Fig. 4 (e)).

  3. 3.

The $B\to\chi_{c1}\pi K$ decay: For the $\chi_{c1}\pi$ invariant mass
distribution in the $B\to\chi_{c1}\pi K$ decay, two charged charmoniumlike
structures $Z^{+}(4051)$ and $Z^{+}(4248)$ were announced by the Belle
Collaboration in the exclusive $B\to\chi_{c1}\pi K$ decay Mizuk:2008me , where
their resonance parameters were measured to be
$(M,\Gamma)_{Z^{+}(4051)}=(4051\pm 14^{+20}_{-41}~{}{\rm
MeV},82^{+21}_{-17}{}^{+47}_{-22}~{}{\rm MeV})$ and
$(M,\Gamma)_{Z^{+}(4248)}=(4248^{+44}_{-29}{}^{+180}_{-35}~{}{\rm
MeV},177^{+54}_{-39}{}^{+316}_{-61}~{}{\rm MeV})$ Mizuk:2008me , respectively.
However, these two charged charmoniumlike structures were not seen in the
following BABAR experiment Lees:2011ik . In 2016, the Belle Collaboration
carried out a new measurement of the $B\to\chi_{c1}\pi K$, where the
distribution of the $\chi_{c1}\pi$ invariant mass spectrum was given
Bhardwaj:2015rju . In fact, the line shape of the $\chi_{c1}\pi$ invariant
mass spectrum is complicated. Additionally, for searching for the $X(3872)$
and $X(3915)$ decaying to the final state $\chi_{c1}\pi$ in the $B$ meson
decay, the Belle measured the $\chi_{c1}\pi$ invariant mass spectrum from the
$B$ meson decay, by which they did not find the significant signals of the
$X(3872)$ and $X(3915)$ Bhardwaj:2019spn .

  4. 4.

The $B\to\chi_{c2}\pi K$ decay: In 2016, the Belle Collaboration provided the
measured $\chi_{c2}\pi$ invariant mass spectrum with low precision
Bhardwaj:2015rju . Here, the comparison of data and thresholds is shown.

  5. 5.

The $B\to J/\psi\pi K$ decay: As shown in Fig. 5 (a)-(b), the Belle444Besides
announcing the $Z_{c}(4200)^{+}$ structure, the Belle also gave the evidence
for the $Z(4430)^{+}$ structure Chilikin:2014bkk . Collaboration presented the
results of an amplitude analysis of the $B\to J/\psi\pi K$ decay in 2014, and
reported a very broad charged charmoniumlike structure $Z_{c}(4200)^{+}$,
where its mass and width were measured to be
$(M,\Gamma)=(4196^{+31}_{-29}{}^{+17}_{-13}~{}{\rm
MeV},370^{+70}_{-70}{}^{+70}_{-132}~{}{\rm MeV})$ Chilikin:2014bkk ,
respectively. We may find such broad structure overlaps with many thresholds
(see Fig. 5 (a)-(b) for more details). The correlation of this broad structure
with the corresponding isovector hidden-charm molecular tetraquarks should be
investigated, which will become an important issue in this work. In 2018, the
DØ Collaboration analyzed the data of the $J/\psi\pi$ invariant mass spectrum
from the $B\to J/\psi\pi K$ Abazov:2018cyu , where they only focused on
$3600<m_{J/\psi\pi}<4200$ MeV range inspired by the observed
$Z_{c}(3900)^{\pm}$ from the BESIIII Ablikim:2013mio  and Belle Liu:2013dau .
And the DØ data show the evidence of charged charmoniumlike structure similar
to the $Z_{c}(3900)^{\pm}$, where the measured mass of the $Z_{c}(3900)^{\pm}$
is $3895.0\pm 5.2^{+4.0}_{-2.7}~{}{\rm MeV}$. Since the width information of
such structure is still absent in the DØ measurement, it is hard to conclude
that the DØ evidence truly corresponds to the $Z_{c}(3900)^{\pm}$
Ablikim:2013mio ; Liu:2013dau . Additionally, the $J/\psi\pi$ invariant mass
spectrum of the $B\to J/\psi\pi K$ decay has been studied by the LHCb
Collaboration in 2019 Aaij:2019ipm , and two enhancement structures are
visible at 4200 MeV and 4600 MeV.

As introduced above, the isovector $XYZ$ data of the $B\to XYZ+K$ decays
stimulate our interest in exploring the isovector hidden-charm molecular
systems, which will be illustrated in Sec. III. By this study, we want to
answer whether these isovector $XYZ$ structures have close relation to the
isovector hidden-charm molecular tetraquarks.

##  III Mass spectrum of the charmoniumlike molecular tetraquark systems

In this section, we restudy the mass spectrum behaviors from a pair of charmed
(charm-strange) meson and anticharmed (anti-charm-strange) meson interactions,
these charmed or anticharmed mesons are in the $H$, $S$, and $T$ doublets.
Here, we still adopt the OBE model and consider the $S-D$ wave mixing effect
and the coupled channel effect. As is well known, since Yukawa firstly
proposed the nucleon-nucleon interaction is mediated through the pion-exchange
in 1935 Yukawa:1935xg , the OBE model obtains great promotion. On one hand,
theorists consider the scalar meson ($\sigma$) and vector meson
$(\rho/\omega)$ exchanges interactions to depict the intermediate and short
range interactions, respectively. On the other hand, several various
corrections are introduced to discuss the fine properties of the hadron-hadron
interactions, like the isospin breaking effects, the coupled-channel effects,
the spin-orbit force, and the recoil corrections. Up to now, the OBE model
have been frequently adopted to study the hadron-hadron interactions in the
heavy flavor sector Chen:2016qju ; Liu:2019zoy .

###  III.1 OBE effective potentials

For deducing the OBE effective potentials for the hadron-hadron interactions
at the hadronic level quantitatively, we usually adopt the effective
Lagrangian approach. The general procedures include three typical steps
Wang:2020dya ; Wang:2019nwt ; Wang:2019aoc ; Wang:2020bjt ; Wang:2021hql ;
Chen:2018pzd ; Yang:2021sue . First, we can write out the scattering amplitude
$\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})$ for the relevant scattering process
$h_{1}h_{2}\to h_{3}h_{4}$ according to the effective Lagrangians. And then,
the effective potentials in the momentum space $\mathcal{V}^{h_{1}h_{2}\to
h_{3}h_{4}}_{E}(\bm{q})$ can be related to the corresponding scattering
amplitudes $\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})$ by using the Breit
approximation Breit:1929zz ; Breit:1930zza , i.e.,

$\displaystyle\mathcal{V}^{h_{1}h_{2}\to
h_{3}h_{4}}_{E}(\bm{q})=-\frac{\mathcal{M}(h_{1}h_{2}\to
h_{3}h_{4})}{\sqrt{\prod_{i}2m_{i}\prod_{f}2m_{f}}},$ (3.1)

where $m_{i}\,(i=h_{1},\,h_{2})$ and $m_{f}\,(f=h_{3},\,h_{4})$ denote the
masses of the initial and final states, respectively. Finally, the effective
potentials in the coordinate space $\mathcal{V}^{h_{1}h_{2}\to
h_{3}h_{4}}_{E}(\bm{r})$ can be obtained by performing the Fourier
transformation, i.e.,

$\displaystyle\mathcal{V}^{h_{1}h_{2}\to
h_{3}h_{4}}_{E}(\bm{r})=\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}^{h_{1}h_{2}\to
h_{3}h_{4}}_{E}(\bm{q})\mathcal{F}^{2}(q^{2},m_{E}^{2}),$ (3.2)

which will be applied to search for the bound state solutions by solving the
coupled channel Schr$\rm\ddot{o}$dinger equation, and we can further extract
the bound state properties from the obtained bound state solutions. Because
the discussed hadrons are not pointlike particles, we introduce the monopole
type form factor in each interaction vertex Tornqvist:1993ng ;
Tornqvist:1993vu , i.e.,

$\displaystyle\mathcal{F}(q^{2},m_{E}^{2})=\frac{\Lambda^{2}-m_{E}^{2}}{\Lambda^{2}-q^{2}},$
(3.3)

which reflects the finite size effect of the discussed hadrons and compensate
the off-shell effect of the exchanged light mesons Wang:2020dya . Here,
$\Lambda$, $m_{E}$, and $q$ are the cutoff parameter, the mass, and the four
momentum of the exchanged light mesons, respectively.

Subsequently, let us construct the relevant effective Lagrangians. According
to the heavy quark limit Wise:1992hn , the relevant super-fields
$H^{(Q)}_{a}$, $H^{(\overline{Q})}_{a}$, $S^{(Q)}_{a}$,
$S^{(\overline{Q})}_{a}$, $T^{(Q)\mu}_{a}$, and $T^{(\overline{Q})\mu}_{a}$
can be defined as as Ding:2008gr

$\displaystyle H^{(Q)}_{a}$ $\displaystyle=$
$\displaystyle{\mathcal{P}}_{+}\left(D^{*(Q)\mu}_{a}\gamma_{\mu}-D^{(Q)}_{a}\gamma_{5}\right),$
$\displaystyle H^{(\overline{Q})}_{a}$ $\displaystyle=$
$\displaystyle\left(\bar{D}^{*(\overline{Q})\mu}_{a}\gamma_{\mu}-\bar{D}^{(\overline{Q})}_{a}\gamma_{5}\right){\mathcal{P}}_{-},$
$\displaystyle S^{(Q)}_{a}$ $\displaystyle=$
$\displaystyle{\mathcal{P}}_{+}\left(D^{\prime(Q)\mu}_{1a}\gamma_{\mu}\gamma_{5}-D^{*(Q)}_{0a}\right),$
$\displaystyle S^{(\overline{Q})}_{a}$ $\displaystyle=$
$\displaystyle\left(D^{\prime(\overline{Q})\mu}_{1a}\gamma_{\mu}\gamma_{5}-D^{*(\overline{Q})}_{0a}\right){\mathcal{P}}_{-},$
$\displaystyle T^{(Q)\mu}_{a}$ $\displaystyle=$
$\displaystyle{\mathcal{P}}_{+}\left[D^{*(Q)\mu\nu}_{2a}\gamma_{\nu}-\sqrt{\frac{3}{2}}D^{(Q)}_{1a\nu}\gamma_{5}\left(g^{\mu\nu}-\frac{1}{3}\gamma^{\nu}\left(\gamma^{\mu}-v^{\mu}\right)\right)\right],$
$\displaystyle T^{(\overline{Q})\mu}_{a}$ $\displaystyle=$
$\displaystyle\left[\bar{D}^{*(\overline{Q})\mu\nu}_{2a}\gamma_{\nu}-\sqrt{\frac{3}{2}}\bar{D}^{(\overline{Q})}_{1a\nu}\gamma_{5}\left(g^{\mu\nu}-\frac{1}{3}\gamma^{\nu}\left(\gamma^{\mu}-v^{\mu}\right)\right)\right]{\mathcal{P}}_{-},$

respectively. Here, ${\mathcal{P}}_{\pm}=(1\pm{v}\\!\\!\\!/)/2$ are the
projection operators, and $v^{\mu}=(1,\bf{0})$ denotes the four velocity in
the nonrelativistic approximation. Their conjugate fields read as
$\overline{X}=\gamma_{0}X^{\dagger}\gamma_{0}$ with $X=H^{(Q)}_{a}$,
$H^{(\overline{Q})}_{a}$, $S^{(Q)}_{a}$, $S^{(\overline{Q})}_{a}$,
$T^{(Q)\mu}_{a}$, and $T^{(\overline{Q})\mu}_{a}$.

According to the heavy quark symmetry, the chiral symmetry, and the hidden
local symmetry Casalbuoni:1992gi ; Casalbuoni:1996pg ; Yan:1992gz ;
Harada:2003jx ; Bando:1987br , one can construct the effective Lagrangians
describing the interactions between the (anti)charmed mesons in the
$H/S/T$-doublet and the light scalar, pseudoscalar, and vector mesons
Ding:2008gr ,

$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle
g_{\sigma}\left\langle
H^{(Q)}_{a}\sigma\overline{H}^{(Q)}_{a}\right\rangle+g_{\sigma}\left\langle\overline{H}^{(\overline{Q})}_{a}\sigma
H^{(\overline{Q})}_{a}\right\rangle$
$\displaystyle+g^{\prime}_{\sigma}\left\langle
S^{(Q)}_{a}\sigma\overline{S}^{\,(Q)}_{a}\right\rangle+g^{\prime}_{\sigma}\left\langle\overline{S}^{\,(\overline{Q})}_{a}\sigma
S^{(\overline{Q})}_{a}\right\rangle$
$\displaystyle+g^{\prime\prime}_{\sigma}\left\langle
T^{(Q)\mu}_{a}\sigma\overline{T}^{(Q)}_{a\mu}\right\rangle+g^{\prime\prime}_{\sigma}\left\langle\overline{T}^{(\overline{Q})\mu}_{a}\sigma
T^{(\overline{Q})}_{a\mu}\right\rangle$
$\displaystyle+\frac{h_{\sigma}}{f_{\pi}}\left[\left\langle
S^{(Q)}_{a}\gamma^{\mu}\partial_{\mu}\sigma\overline{H}^{\,({Q})}_{a}\right\rangle-\left\langle\overline{H}^{\,(\overline{Q})}_{a}\gamma^{\mu}\partial_{\mu}\sigma
S^{(\overline{Q})}_{a}\right\rangle+H.c.\right]$
$\displaystyle+\frac{h^{\prime}_{\sigma}}{f_{\pi}}\left[\left\langle
T^{(Q)\mu}_{a}\partial_{\mu}\sigma\overline{H}^{(Q)}_{b}\right\rangle+\left\langle\overline{H}^{(\overline{Q})}_{a}\partial_{\mu}\sigma
T^{(\overline{Q})\mu}_{b}\right\rangle+H.c.\right]$
$\displaystyle+ig\left\langle
H^{(Q)}_{b}{\mathcal{A}}\\!\\!\\!/_{ba}\gamma_{5}\overline{H}^{\,({Q})}_{a}\right\rangle+ig\left\langle\overline{H}^{(\overline{Q})}_{a}{\mathcal{A}}\\!\\!\\!/_{ab}\gamma_{5}H^{\,(\overline{Q})}_{b}\right\rangle$
$\displaystyle+i\tilde{k}\left\langle S^{(Q)}_{b}{\cal
A}\\!\\!\\!/_{ba}\gamma_{5}\overline{S}^{\,(Q)}_{a}\right\rangle+i\tilde{k}\left\langle\overline{S}^{\,(\overline{Q})}_{a}{\cal
A}\\!\\!\\!/_{ab}\gamma_{5}S^{(\overline{Q})}_{b}\right\rangle$
$\displaystyle+ik\left\langle
T^{\,(Q)\mu}_{b}{\mathcal{A}}\\!\\!\\!/_{ba}\gamma_{5}\overline{T}^{(Q)}_{a\mu}\right\rangle+ik\left\langle\overline{T}^{\,(\overline{Q})\mu}_{a}{\mathcal{A}}\\!\\!\\!/_{ab}\gamma_{5}T^{(\overline{Q})}_{b\mu}\right\rangle$
$\displaystyle+\left[ih\left\langle S^{(Q)}_{b}{\cal
A}\\!\\!\\!/_{ba}\gamma_{5}\overline{H}^{\,(Q)}_{a}\right\rangle+ih\left\langle\overline{H}^{\,(\overline{Q})}_{a}{\cal
A}\\!\\!\\!/_{ab}\gamma_{5}S^{(\overline{Q})}_{b}\right\rangle+H.c.\right]$
$\displaystyle+\left[i\left\langle
T^{(Q)\mu}_{b}\left(\frac{h_{1}}{\Lambda_{\chi}}D_{\mu}{\mathcal{A}}\\!\\!\\!/+\frac{h_{2}}{\Lambda_{\chi}}D\\!\\!\\!\\!/{\mathcal{A}}_{\mu}\right)_{ba}\gamma_{5}\overline{H}^{\,(Q)}_{a}\right\rangle+H.c.\right]$
$\displaystyle+\left[i\left\langle\overline{H}^{\,(\overline{Q})}_{a}\left(\frac{h_{1}}{\Lambda_{\chi}}{\mathcal{A}}\\!\\!\\!/\stackrel{{\scriptstyle\leftarrow}}{{D_{\mu}^{\prime}}}+\frac{h_{2}}{\Lambda_{\chi}}{\mathcal{A}}_{\mu}\stackrel{{\scriptstyle\leftarrow}}{{D\\!\\!\\!/^{\prime}}}\right)_{ab}\gamma_{5}T^{(\overline{Q})\mu}_{b}\right\rangle+H.c.\right]$
$\displaystyle+\left\langle iH^{(Q)}_{b}\left(\beta
v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})+\lambda\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ba}\overline{H}^{\,(Q)}_{a}\right\rangle$
$\displaystyle-\left\langle i\overline{H}^{(\overline{Q})}_{a}\left(\beta
v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})-\lambda\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ab}H^{\,(\overline{Q})}_{b}\right\rangle$
$\displaystyle+\left\langle
iS^{(Q)}_{b}\left(\beta^{\prime}v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})+\lambda^{\prime}\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ba}\overline{S}^{\,(Q)}_{a}\right\rangle$
$\displaystyle-\left\langle
i\overline{S}^{(\overline{Q})}_{a}\left(\beta^{\prime}v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})-\lambda^{\prime}\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ab}S^{\,(\overline{Q})}_{b}\right\rangle$
$\displaystyle+\left\langle
iT^{\,(Q)}_{b\lambda}\left(\beta^{\prime\prime}v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})+\lambda^{\prime\prime}\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ba}\overline{T}^{(Q)\lambda}_{a}\right\rangle$
$\displaystyle-\left\langle
i\overline{T}^{\,(\overline{Q})}_{a\lambda}\left(\beta^{\prime\prime}v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})-\lambda^{\prime\prime}\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ab}T^{(\overline{Q})\lambda}_{b}\right\rangle$
$\displaystyle+\left[\left\langle H^{(Q)}_{b}(i\zeta\gamma^{\mu}({\cal
V}_{\mu}-\rho_{\mu})+i\mu\sigma^{\lambda\nu}F_{\lambda\nu}(\rho))_{ba}\overline{S}^{\,(Q)}_{a}\right\rangle+H.c.\right]$
$\displaystyle+\left[\left\langle\overline{S}^{\,(\overline{Q})}_{a}(i\zeta\gamma^{\mu}({\cal
V}_{\mu}-\rho_{\mu})+i\mu\sigma^{\lambda\nu}F_{\lambda\nu}(\rho))_{ab}H^{(\overline{Q})}_{b}\right\rangle+H.c.\right]$
$\displaystyle+\left[\left\langle
T^{(Q)\mu}_{b}\left(i\zeta_{1}({\mathcal{V}}_{\mu}-\rho_{\mu})+\mu_{1}\gamma^{\nu}F_{\mu\nu}(\rho)\right)_{ba}\overline{H}^{\,(Q)}_{a}\right\rangle+H.c.\right]$
$\displaystyle-\left[\left\langle\overline{H}^{\,(\overline{Q})}_{a}\left(i\zeta_{1}({\mathcal{V}}_{\mu}-\rho_{\mu})-\mu_{1}\gamma^{\nu}F_{\mu\nu}(\rho)\right)_{ab}T^{(\overline{Q})\mu}_{b}\right\rangle+H.c.\right],$

Here, the covariant derivatives are written as
$D_{\mu}=\partial_{\mu}+\mathcal{V}_{\mu}$ and
$D^{\prime}_{\mu}=\partial_{\mu}-\mathcal{V}_{\mu}$. And the axial current
$\mathcal{A}_{\mu}$, the vector current ${\cal V}_{\mu}$, the vector meson
field $\rho_{\mu}$, and the vector meson strength tensor $F_{\mu\nu}(\rho)$
are defined as

$\displaystyle{\mathcal{A}}_{\mu}=\frac{1}{2}\left(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger}\right)_{\mu},$
$\displaystyle{\mathcal{V}}_{\mu}=\frac{1}{2}\left(\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger}\right)_{\mu},$
$\displaystyle\xi=\exp(i\mathbb{P}/f_{\pi}),\quad\rho_{\mu}=\frac{i{g_{V}}}{{\sqrt{2}}}\mathbb{V}_{\mu},$
$\displaystyle
F_{\mu\nu}(\rho)=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+\left[\rho_{\mu},\rho_{\nu}\right].$
(3.6)

respectively. The light pseudoscalar meson matrix $\mathbb{P}$ and the light
vector meson matrix $\mathbb{V}_{\mu}$ have the conventional form Wang:2020bjt
; Wang:2021hql ; Chen:2018pzd , which can be expressed as

$\displaystyle\left.\begin{array}[]{l}{\mathbb{P}}={\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right)},\\\
{\mathbb{V}}_{\mu}={\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right)}_{\mu},\end{array}\right.$ (3.15)

respectively. After expanding the compact effective Lagrangians in Eq. (III.1)
to the leading order of the pseudo-Goldstone field $\xi$, we can further
obtain the concrete effective Lagrangians (see Refs. Wang:2019nwt ;
Wang:2019aoc ; Wang:2020dya ; Shen:2010ky ; Hu:2010fg  for more information).
The normalized relations for these discussed charmed mesons are written as

$\displaystyle\left.\begin{array}[]{ll}\langle
0|D|c\bar{q}(0^{-})\rangle=\sqrt{m_{D}},&\langle
0|D^{*\mu}|c\bar{q}(1^{-})\rangle=\epsilon^{\mu}\sqrt{m_{D^{*}}},\\\ \langle
0|D_{0}^{*}|c\bar{q}(0^{+})\rangle=\sqrt{m_{D_{0}^{*}}},&\langle
0|D_{1}^{\prime\mu}|c\bar{q}(1^{+})\rangle=\epsilon^{\mu}\sqrt{m_{D_{1}^{\prime}}},\\\
\langle
0|D_{1}^{\mu}|c\bar{q}(1^{+})\rangle=\epsilon^{\mu}\sqrt{m_{D_{1}}},&\langle
0|D_{2}^{*\mu\nu}|c\bar{q}(2^{+})\rangle=\zeta^{\mu\nu}\sqrt{m_{D_{2}^{*}}},\\\
\end{array}\right.$ (3.19)

respectively. Here, $\epsilon^{\mu}_{m}\,(m=0,\pm 1)$ and
$\zeta^{\mu\nu}_{m}(m=0,\pm 1,\pm 2)$ correspond to the polarization vector
and tensor, respectively. In the static limit, they have the form of
$\epsilon_{0}^{\mu}=\left(0,0,0,-1\right)$,
$\epsilon_{\pm}^{\mu}=\left(0,\,\pm 1,\,i,\,0\right)/\sqrt{2}$, and
$\zeta^{\mu\nu}_{m}=\sum_{m1,m2}\langle
1,m_{1};1,m_{2}|2,m\rangle\epsilon^{\mu}_{m_{1}}\epsilon^{\nu}_{m_{2}}$
Cheng:2010yd .

In order to obtain the concrete effective potentials, one need to further
construct the wave functions for the investigated systems. They include the
color part, the flavor part, the spin-orbit part, and the spatial wave
functions. For the systems composed by colorless hadrons, the color wave
functions are simply $\bm{1}$. In Table 2, we summarize the flavor wave
functions $|I,I_{3}\rangle$ for the $S$-wave
$\mathcal{A}\overline{\mathcal{A}}$ and $\mathcal{A}\overline{\mathcal{B}}$
systems, here, notations $\mathcal{A}$ and $\mathcal{B}$ stand for the
different charmed mesons, and $J$, $J_{1}$, and $J_{2}$ correspond to the
total angular momentum quantum numbers of the discussed charmoniumlike systems
$\mathcal{A}\overline{\mathcal{B}}$, the charmed (charmed-strange) mesons
$\mathcal{A}$, and the charmed (charmed-strange) mesons $\mathcal{B}$,
respectively. In particular, we need to emphasize the $C$ parity for the
discussed systems is determined by $C=cx_{1}x_{2}(-1)^{J-J_{1}-J_{2}}$ with
$c=\pm 1$, where the charge conjugate transformation conventions satisfy
$\mathcal{A}\leftrightarrow x_{1}\overline{\mathcal{A}}$ and
$\mathcal{B}\leftrightarrow x_{2}\overline{\mathcal{B}}$ Liu:2007bf ;
Liu:2008xz ; Liu:2008fh ; Liu:2008tn ; Sun:2012sy ; Wang:2020dya ; Li:2015exa
; Li:2013bca ; Hu:2010fg ; Shen:2010ky ; Dong:2021juy ; Liu:2013rxa ;
Chen:2015add .

Table 2: Flavor wave functions $|I,I_{3}\rangle$ for these discussed $S$-wave $\mathcal{A}\overline{\mathcal{A}}$ and $\mathcal{A}\overline{\mathcal{B}}$ systems. Here, the notations $\mathcal{A}$ and $\mathcal{B}$ stand for different charmed mesons, and $I$ and $I_{3}$ represent their isospin and the third component of these discussed charmoniumlike systems, respectively. Systems | $|I,I_{3}\rangle$ | Flavor wave functions
---|---|---
$\mathcal{A}\overline{\mathcal{A}}$ | $|1,1\rangle$ | $\mathcal{A}^{+}\overline{\mathcal{A}}^{0}$
$|1,0\rangle$ | $\dfrac{1}{\sqrt{2}}\left(\mathcal{A}^{0}\overline{\mathcal{A}}^{0}-\mathcal{A}^{+}\mathcal{A}^{-}\right)$
$|1,-1\rangle$ | ${\mathcal{A}}^{0}\mathcal{A}^{-}$
$|0,0\rangle$ | $\dfrac{1}{\sqrt{2}}\left(\mathcal{A}^{0}\overline{\mathcal{A}}^{0}+\mathcal{A}^{+}\mathcal{A}^{-}\right)$
$\mathcal{A}\overline{\mathcal{B}}$ | $|1,1\rangle$ | $\dfrac{1}{\sqrt{2}}\left(\mathcal{A}^{+}\overline{\mathcal{B}}^{0}+c\mathcal{B}^{+}\overline{\mathcal{A}}^{0}\right)$
$|1,0\rangle$ | $\dfrac{1}{2}\left[\left(\mathcal{A}^{0}\overline{\mathcal{B}}^{0}-\mathcal{A}^{+}{\mathcal{B}}^{-}\right)+c\left(\mathcal{B}^{0}\overline{\mathcal{A}}^{0}-\mathcal{B^{+}}{\mathcal{A}}^{-}\right)\right]$
$|1,-1\rangle$ | $\dfrac{1}{\sqrt{2}}\left({\mathcal{A}}^{0}\mathcal{B}^{-}+c{\mathcal{B}}^{0}\mathcal{A}^{-}\right)$
$|0,0\rangle$ | $\dfrac{1}{2}\left[\left(\mathcal{A}^{0}\overline{\mathcal{B}}^{0}+\mathcal{A}^{+}{\mathcal{B}}^{-}\right)+c\left(\mathcal{B}^{0}\overline{\mathcal{A}}^{0}+\mathcal{B^{+}}{\mathcal{A}}^{-}\right)\right]$

The spin-orbit wave functions $|^{2S+1}L_{J}\rangle$ for the
$|{\mathcal{D}}_{J_{1}}{\overline{\mathcal{D}}}_{J_{2}}\rangle$ systems can be
constructed as

$\displaystyle|{\mathcal{D}}_{0}{\overline{\mathcal{D}}}_{1}\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m_{L}}C^{J,M}_{1m,Lm_{L}}\epsilon_{m}^{\mu}|Y_{L,m_{L}}\rangle,$
(3.20) $\displaystyle|{\mathcal{D}}_{0}{\overline{\mathcal{D}}}_{2}\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m_{L}}C^{J,M}_{2m,Lm_{L}}\zeta_{m}^{\mu\nu}|Y_{L,m_{L}}\rangle,$
(3.21) $\displaystyle|{\mathcal{D}}_{1}{\overline{\mathcal{D}}}_{1}\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{1m,1m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\epsilon_{m}^{\mu}\epsilon_{m^{\prime}}^{\nu}|Y_{L,m_{L}}\rangle,$
(3.22) $\displaystyle|{\mathcal{D}}_{1}{\overline{\mathcal{D}}}_{2}\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{1m,2m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\epsilon_{m}^{\lambda}\zeta_{m^{\prime}}^{\mu\nu}|Y_{L,m_{L}}\rangle.$
(3.23)

In the above expressions, the notations ${\mathcal{D}}_{0}$,
${\mathcal{D}}_{1}$, and ${\mathcal{D}}_{2}$ denote the charmed (charm-
strange) mesons with the total angular momentum quantum numbers $J=0,1$, and
$2$, respectively. $C^{e,f}_{ab,cd}$ is the Clebsch-Gordan coefficient, and
$|Y_{L,m_{L}}\rangle$ is the spherical harmonics function. In Table 3, we
summary the relevant spin-orbit wave functions $|{}^{2S+1}L_{J}\rangle$ and
the discussed channels under considering the coupled channel effect.

Table 3: The relevant quantum numbers $J^{P}$ and possible channels $|{}^{2S+1}L_{J}\rangle$ involved in this work. Here, $...$ means that the $S-$wave components for the corresponding channels do not exist. $J^{P}$ | ${\mathcal{D}}_{0}\overline{{\mathcal{D}}}_{0}$ | ${\mathcal{D}}_{0}\overline{{\mathcal{D}}}_{1}$ | ${\mathcal{D}}_{0}\overline{{\mathcal{D}}}_{2}$ | ${\mathcal{D}}_{1}\overline{{\mathcal{D}}}_{1}$ | ${\mathcal{D}}_{1}\overline{{\mathcal{D}}}_{2}$
---|---|---|---|---|---
$0^{\pm}$ | $|{}^{1}\mathbb{S}_{0}\rangle$ | $...$ | $...$ | $|{}^{1}\mathbb{S}_{0}\rangle/|{}^{5}\mathbb{D}_{0}\rangle$ | $...$
$1^{\pm}$ | $...$ | $|{}^{3}\mathbb{S}_{1}\rangle/|{}^{3}\mathbb{D}_{1}\rangle$ | $...$ | $|{}^{3}\mathbb{S}_{1}\rangle/|{}^{3,5}\mathbb{D}_{1}\rangle$ | $|{}^{3}\mathbb{S}_{1}\rangle/|{}^{3,5,7}\mathbb{D}_{1}\rangle$
$2^{\pm}$ | $...$ | $...$ | $|{}^{5}\mathbb{S}_{2}\rangle/|{}^{5}\mathbb{D}_{2}\rangle$ | $|{}^{5}\mathbb{S}_{2}\rangle/|{}^{1,3,5}\mathbb{D}_{2}\rangle$ | $|{}^{5}\mathbb{S}_{2}\rangle/|{}^{3,5,7}\mathbb{D}_{2}\rangle$
$3^{\pm}$ | $...$ | $...$ | $...$ | $...$ | $|{}^{7}\mathbb{S}_{3}\rangle/|{}^{3,5,7}\mathbb{D}_{3}\rangle$

For the
$\mathcal{A}\overline{\mathcal{B}}\to\mathcal{A}\overline{\mathcal{B}}$
processes, there exist the direct channel and cross channel Feynman diagrams
Liu:2007bf ; Wang:2020dya , where $\mathcal{A}$ and $\mathcal{B}$ stand for
two different charmed (charm-strange) mesons. The total effective potentials
can be written as Liu:2008fh ; Liu:2008tn ; Sun:2012sy ; Wang:2020dya

$\displaystyle\mathcal{V}_{Total}(\bm{r})=\mathcal{V}_{D}(\bm{r})+c\,\mathcal{V}_{C}(\bm{r}).$
(3.24)

In Fig. 6, we present the direct channel and cross channel Feynman diagrams.
For the $\mathcal{A}\overline{\mathcal{A}}$ systems, there exist the direct
channel contribution.

---
Figure 6: The direct channel and cross channel Feynman diagrams for the
$\mathcal{A}\overline{\mathcal{B}}\to\mathcal{A}\overline{\mathcal{B}}$
processes. Here, the notations $\mathcal{A}$ and $\mathcal{B}$ represent two
different charmed (charmed-strange) mesons.

With the standard procedures of the OBE model Wang:2020dya ; Wang:2019nwt ;
Wang:2019aoc ; Wang:2020bjt ; Wang:2021hql ; Chen:2018pzd , we can finally
obtain the effective potentials in the coordinate space. In Appendix A, we
collect the concrete expressions for all the OBE effective potentials. We
estimate the coupling constants by fitting the reported experimental data and
using several theoretical models Liu:2008xz ; Casalbuoni:1996pg ; Wang:2020dya
; Falk:1992cx ; Isola:2003fh ; Cleven:2016qbn ; Dong:2019ofp ; He:2019csk ;
Wang:2019nwt ; Wang:2019aoc ; Wang:2020lua ; Riska:2000gd ; Hu:2010fg ;
Bardeen:2003kt ; Cheng:2004ru ; He:2016pfa ; Liu:2010hf ; He:2013oma ;
He:2012zd . In particular, we fix the phases between these coupling constants
by the quark model Riska:2000gd . In Table 4, we collect their values. In
addition, we need to introduce the following parameters of the hadron masses,
$m_{\sigma}=600.00~{}\rm{MeV}$, $m_{\pi}=137.27~{}\rm{MeV}$,
$m_{\eta}=547.86~{}\rm{MeV}$, $m_{\rho}=775.49~{}\rm{MeV}$,
$m_{\omega}=782.65~{}\rm{MeV}$, $m_{\phi}=1019.46~{}\rm{MeV}$,
$m_{D}=1867.24~{}\rm{MeV}$, $m_{D^{\ast}}=2008.56~{}\rm{MeV}$,
$m_{D_{0}^{*}}=2324.50~{}\rm{MeV}$, $m_{D_{1}^{\prime}}=2427.00~{}\rm{MeV}$,
$m_{D_{1}}=2422.00~{}\rm{MeV}$, $m_{D^{\ast}_{2}}=2463.05~{}\rm{MeV}$,
$m_{D_{s}}=1968.34~{}\rm{MeV}$, $m_{D_{s}^{\ast}}=2112.20~{}\rm{MeV}$,
$m_{D_{s0}^{*}}=2317.80~{}\rm{MeV}$, $m_{D_{s1}^{\prime}}=2459.50~{}\rm{MeV}$,
$m_{D_{s1}}=2535.11~{}\rm{MeV}$, and $m_{D^{\ast}_{s2}}=2569.10~{}\rm{MeV}$
Zyla:2020zbs .

Table 4: A summary of the coupling constants adopted in our calculations. Units of the coupling constants $h^{\prime}=(h_{1}+h_{2})/\Lambda_{\chi}$, $\lambda$, $\lambda^{\prime}$, $\lambda^{\prime\prime}$, $\mu$, and $\mu_{1}$ are $\rm{GeV}^{-1}$, and the coupling constant $f_{\pi}$ is given in unit of GeV. $g_{\sigma}$ | $g^{\prime}_{\sigma}$ | $g^{\prime\prime}_{\sigma}$ | $|h_{\sigma}|$ | $|h^{\prime}_{\sigma}|$ | $g$
---|---|---|---|---|---
$-0.76$ | $0.76$ | $0.76$ | $0.32$ | $0.35$ | $0.59$
$\tilde{k}$ | $k$ | $|h|$ | $|h^{\prime}|$ | $f_{\pi}$ | $\beta$
$0.59$ | $0.59$ | $0.56$ | $0.55$ | $0.132$ | $-0.90$
$\beta^{\prime}$ | $\beta^{\prime\prime}$ | $\lambda$ | $\lambda^{\prime}$ | $\lambda^{\prime\prime}$ | $|\zeta|$
$0.90$ | $0.90$ | $-0.56$ | $0.56$ | $0.56$ | $0.727$
$|\mu|$ | $|\zeta_{1}|$ | $\mu_{1}$ | $g_{V}$ |  |
$0.364$ | $0.20$ | $0$ | $5.83$ |  |

###  III.2 Numerical results and discussions

In this section, we attempt to find the loosely bound state solutions for the
charmed (charm-strange) meson and anticharmed (anti-charm-strange) meson
systems by solving the coupled channel Schr$\ddot{\rm{o}}$dinger equation. As
the only one free parameter here, we vary the cutoff parameters in the range
of $0.8$-$3.0~{}\rm{GeV}$. The loosely bound state with cutoff value around
$1.0\rm~{}{GeV}$ can be the prime hadronic molecular candidate, since this
value range is widely accepted as a reasonable input parameter based on the
experience of studying the deuteron Tornqvist:1993ng ; Tornqvist:1993vu ;
Wang:2019nwt . As is well known, a reasonable loosely bound hadronic molecule
should satisfy its binding energy is around several to several tens MeV, and
its typical size should be larger than the size of all the component hadrons
Chen:2016qju ; Chen:2017xat .

Here, we need to emphasis that we mainly focus on the mass spectrum for the
hidden-charm molecular tetraquark systems in this work, which is inspired by
the abundant experimental data. In addition, we also give rough estimations of
the branching ratios of the two-body hidden-charm decay behaviors for the
$D^{*}\bar{D}^{*}$ molecular tetraquarks within the heavy quark symmetry
analysis, which is due to possible peculiar characteristic mass spectrum of
the isoscalar $D^{*}\bar{D}^{*}$ molecular systems existing in the reported
experimental data of the $B$ meson decays (see Fig. 2). For other obtained
$\mathcal{D}\bar{\mathcal{D}}$ and $\mathcal{D}_{s}\bar{\mathcal{D}}_{s}$
molecules, we only simply list the two-body hidden-charm decay channels. Their
decay behaviors will be further discussed in the future work. Our results will
be categorized into three corresponding cases:

  1. 1.

The isoscalar charmoniumlike molecular systems without hidden-strange quantum
number,

  2. 2.

The charmoniumlike molecular systems with hidden-strange quantum number,

  3. 3.

The isovector hidden-charm molecular tetraquark systems.

####  III.2.1 Isoscalar charmoniumlike molecular systems without hidden-
strange quantum number

##### The isoscalar $D^{*}\bar{D}^{*}$ system.

In Table 5, we present the corresponding bound state properties for the
$S-$wave isoscalar $D^{*}\bar{D}^{*}$ system. When cutoff values vary from 0.8
to 3.0 GeV, we can obtain bound state solutions for the $S-$wave isoscalar
$D^{*}\bar{D}^{*}$ states with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$. And we
can obtain the relation of
$\Lambda[0(0^{++})]<\Lambda[0(1^{+-})]<\Lambda[0(2^{++})]$. Suppose bound
states with a smaller cutoff binds deeper when we set the same binding energy.
We can find the isoscalar $D^{*}\bar{D}^{*}$ interaction with
$I(J^{PC})=0(0^{++})$ is strongest attractive, followed by the states with
$I(J^{PC})=0(1^{+-})$ and $0(2^{++})$. Thus, we can conclude these three
states are possible isoscalar hidden-charm molecular tetraquark candidates,
and their masses satisfy $M[0(0^{++})]<M[0(1^{+-})]<M[0(2^{++})]$. Here, our
results are also consistent with the conclusions in Refs. Liu:2008tn ;
Liu:2008mi ; Liu:2009ei ; Sun:2011uh ; Sun:2012zzd ; Zhao:2015mga ;
Liu:2017mrh ; Dong:2021juy ; Liu:2016kqx ; Dai:2018nmw ; Ding:2020dio ;
Yang:2017prf ; Zhang:2006ix ; Tornqvist:1993ng ; DeRujula:1976zlg .

Table 5: Bound state properties for the $S-$wave isoscalar $D^{\ast}\bar{D}^{\ast}$ system. Cutoff $\Lambda$, binding energy $E$, and root-mean-square (RMS) radius $r_{RMS}$ are in units of GeV, MeV, and fm, respectively. Here, we label the major probability for the corresponding channels in a bold manner. Effect | Single channel |  $S-$$D$ wave mixing effect
---|---|---
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{1}\mathbb{S}_{0}/{}^{5}\mathbb{D}_{0})$
$0^{++}$ | 0.92 | $-0.56$ | 3.98 | 0.91 | $-0.61$ | 3.89 |  99.56/0.44
0.99 | $-11.81$ | 1.09 | 0.98 | $-10.80$ | 1.15 |  99.42/0.58
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$
$1^{+-}$ | 1.07 | $-0.38$ | 4.54 | 1.05 | $-0.35$ | 4.67 |  99.49/0.51
1.16 | $-12.01$ | 1.09 | 1.15 | $-12.35$ | 1.10 |  99.13/0.87
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{5}\mathbb{S}_{2}/{}^{1}\mathbb{D}_{2}/{}^{5}\mathbb{D}_{2})$
$2^{++}$ | 2.06 | $-0.28$ | 5.18 | 1.82 | $-0.33$ | 5.05 |  99.03/0.07/0.90
3.00 | $-12.35$ | 1.26 | 2.81 | $-12.45$ | 1.28 |  98.00/0.15/1.85

After that, we give rough estimations of the branching ratios of the two-body
hidden-charm decay behaviors for these possible $S-$wave isoscalar
$D^{\ast}\bar{D}^{\ast}$ molecular candidates by using the heavy quark
symmetry. As an approximate symmetry, the heavy quark symmetry is often
applied to study the structures of the hadrons which contain the heavy quarks.
In order to perform the heavy quark symmetry analysis for the two-body hidden-
charm decay behaviors, we should first expand the spin wave functions of the
heavy hadrons systems $|\ell_{1}s_{1}j_{1},\ell_{2}s_{2}j_{2},JM\big{\rangle}$
in terms of the heavy quark basis
$|\ell_{1}\ell_{2}L,s_{1}s_{2}S,JM\big{\rangle}$, i.e.,

$\displaystyle\left|\ell_{1}s_{1}j_{1},\ell_{2}s_{2}j_{2},JM\right\rangle$
$\displaystyle=\sum_{S,L}\hat{S}\hat{L}\hat{j_{1}}\hat{j_{2}}\left\\{\begin{array}[]{ccc}\ell_{1}&\ell_{2}&L\\\
s_{1}&s_{2}&S\\\
j_{1}&j_{2}&J\end{array}\right\\}|\ell_{1}\ell_{2}L,s_{1}s_{2}S,JM\big{\rangle}$
(3.28)

with $\hat{A}=\sqrt{2A+1}$. Here, the 9-$j$ symbol is used to relate two bases
$|\ell_{1}s_{1}j_{1},\ell_{2}s_{2}j_{2},JM\big{\rangle}$ and
$|\ell_{1}\ell_{2}L,s_{1}s_{2}S,JM\big{\rangle}$ with the investigated system
coupled in a different way Ozpineci:2013zas .

For these possible $S-$wave isoscalar $D^{\ast}\bar{D}^{\ast}$ molecular
candidates, the two-body hidden-charm decay channels include the
$\eta_{c}\eta$, $\eta_{c}\eta^{\prime}$, $\eta_{c}\omega$, $J/\psi\eta$, and
$J/\psi\omega$ channels. We can expand their spin wave functions in the heavy
quark spin symmetry basis, i.e.,

$\displaystyle\left|0^{++}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\left|0_{q\bar{q}}^{-+},0_{c\bar{c}}^{-+},0^{++}\right\rangle-\frac{1}{2}\left|1_{q\bar{q}}^{--},1_{c\bar{c}}^{--},0^{++}\right\rangle,$
(3.29) $\displaystyle\left|1^{+-}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|1_{q\bar{q}}^{--},0_{c\bar{c}}^{-+},1^{+-}\right\rangle+\frac{1}{\sqrt{2}}\left|0_{q\bar{q}}^{-+},1_{c\bar{c}}^{--},1^{+-}\right\rangle,$
(3.30) $\displaystyle\left|2^{++}\right\rangle$ $\displaystyle=$
$\displaystyle\left|1_{q\bar{q}}^{--},1_{c\bar{c}}^{--},2^{++}\right\rangle,$
(3.31)

where $L_{q\bar{q}}^{P_{q},C_{q}}$and $S_{c\bar{c}}^{P_{Q},C_{Q}}$ stand for
the spin parities for the light-flavor meson and charmonium state,
respectively. In the heavy quark symmetry, we can estimate that

  * •

The isoscalar $D^{*}\bar{D}^{*}$ molecular state with $J^{PC}=0^{++}$ can
decay into the $\eta_{c}\eta^{(\prime)}$ and $J/\psi\omega$ channels through
the $S-$wave interaction, the relative decay ratio
$\mathcal{B}_{0}=\Gamma_{0}[J/\psi\omega]/\Gamma_{0}[\eta_{c}\eta^{(\prime)}]$
is roughly $1:3$.

  * •

For the isoscalar $D^{*}\bar{D}^{*}$ molecular state with $J^{PC}=1^{+-}$, it
can decay into the $J/\psi\eta^{(\prime)}$ and $\eta_{c}\omega$ via the
$S-$wave coupling. The relative decay branching ratio for the
$J/\psi\eta^{(\prime)}$ and $\eta_{c}\omega$ channels is
$\mathcal{B}_{1}=\Gamma_{1}[J/\psi\eta]/\Gamma_{1}[\eta_{c}\omega]=1:1$. Since
the phase space for the $D^{*}\bar{D}^{*}\to J/\psi\eta$ is larger than that
in the $\eta_{c}\omega$ final state around 100 MeV, the partial decay widths
for these two hidden-charm decay processes satisfy
$\Gamma_{1}[J/\psi\eta]>\Gamma_{1}[\eta_{c}\omega]$, it leads to the
$J/\psi\eta$ channel is the prime decay channel to search for the isoscalar
$D^{*}\bar{D}^{*}$ molecular state with $J^{PC}=1^{+-}$.

  * •

For the isoscalar $D^{*}\bar{D}^{*}$ molecular state with $J^{PC}=2^{++}$, the
$J/\psi\omega$ is the only one two-body hidden-charm decay mode by the
$S-$wave interaction.

In addition, the isoscalar $D^{*}\bar{D}^{*}$ molecular state with
$J^{PC}=0^{++}$ can strongly couple to the $D\bar{D}$ channel via the $S-$wave
coupling, and the $D^{*}\bar{D}^{*}[0(2^{++})]$ can decay into the
$D\bar{D}^{*}$ and $D\bar{D}$ channels through the $D$-wave interactions,
which indicate that the two-body open-charm decay widths satisfy
$\Gamma_{\text{Open}}[0(0^{++})]>\Gamma_{\text{Open}}[0(2^{++})]$. Thus, we
can estimate that the strong decay width for the $D^{\ast}\bar{D}^{\ast}$
molecule with $I(J^{PC})=0(0^{++})$ is larger than that for the
$D^{\ast}\bar{D}^{\ast}$ molecule with $I(J^{PC})=0(2^{++})$ 555Here, we
neglect the other decay modes with very small contribution, like the three-
body decay modes, the two-body decay modes via the $D$-wave interaction.. To
summarize, if both the $D^{\ast}\bar{D}^{\ast}$ states with
$I(J^{PC})=0(0^{++})$ and $0(2^{++})$ are the possible charmoniumlike
molecular candidates, their mass and decay width should satisfy
$M[0(0^{++})]<M[0(2^{++})]$ and $\Gamma[0(0^{++})]>\Gamma[0(2^{++})]$,
respectively. These important characters on their mass spectrum and two-body
strong decay behaviors provided here can help us to search and further
identify the $D^{\ast}\bar{D}^{\ast}$ charmoniumlike molecules.

Figure 7: The $J/\psi\omega$ and $J/\psi\eta$ invariant mass spectrum around
the $D^{*}\bar{D}^{*}$ threshold in the $B\to J/\psi\omega K$ Aubert:2007vj ;
delAmoSanchez:2010jr  and $B\to J/\psi\eta K$ Aubert:2004fc , respectively.

According to the above analysis, the $J/\psi\omega$ final state has the
potential to observe the possible $D^{\ast}\bar{D}^{\ast}$ charmoniumlike
molecules with $I(J^{PC})=0(0^{++})$ and $0(2^{++})$. Recalling the BABAR data
presented in Fig. 2, we can find possible evidence of the existence of two
enhancement structures below the $D^{*}\bar{D}^{*}$ threshold by analyzing the
$J/\psi\omega$ invariant mass spectrum of the $B\to J/\psi\omega K$
Aubert:2007vj ; delAmoSanchez:2010jr . In Fig. 7 (a), we label the possible
positions of the $D^{\ast}\bar{D}^{\ast}$ charmoniumlike molecules with
$I(J^{PC})=0(0^{++})$ and $0(2^{++})$ in the $J/\psi\omega$ invariant mass
spectrum in the $B\to J/\psi\omega K$ Aubert:2007vj ; delAmoSanchez:2010jr .
We look forward the future experiments with higher precision data can test our
theoretical predictions.

Meanwhile, it is interesting to note that the $S-$wave $D^{*}\bar{D}^{*}$
state with $I(J^{PC})=0(1^{+-})$ is favored to be the possible charmoniumlike
molecular candidate, and the $J/\psi\eta$ channel is the important two-body
hidden-charm decay mode. Experimental, we may find an enhancement structure
around 3.9 GeV in the $J/\psi\eta$ invariant mass spectrum of the $B\to
J/\psi\eta K$ Aubert:2004fc , which may correspond to the $S-$wave isoscalar
$D^{*}\bar{D}^{*}$ molecular state with $J^{PC}=1^{+-}$ (see Fig. 7 (b)). In
addition, we notice that the BESIII Collaboration Ablikim:2012ht  and the
Belle Collaboration Iwashita:2013wnn  respectively analyzed the $J/\psi\eta$
invariant mass spectrum in the $e^{+}e^{-}$ annihilation process and the $B\to
J/\psi\eta K$ process, unfortunately, no significant signal around 3.9 GeV was
found. This could be because there are only several dozen events or less in
the $J/\psi\eta$ invariant mass spectrum Aubert:2004fc ; Ablikim:2012ht ;
Iwashita:2013wnn , in comparison with the observations of the $J/\psi\omega$
invariant mass spectrum Abe:2004zs ; Aubert:2007vj ; delAmoSanchez:2010jr , we
expect more precision experimental data to further check the structure around
3.9 GeV in the $J/\psi\eta$ invariant mass spectrum.

In short, due to the lack of the sufficiently accurate experimental results,
the present data sample from the $B$ meson decays was not large enough to
analyze these possible charmoniumlike molecular tetraquark structures in the
$J/\psi\omega$ and $J/\psi\eta$ invariant mass spectrum Abe:2004zs ;
Aubert:2007vj ; delAmoSanchez:2010jr ; Aubert:2004fc ; Iwashita:2013wnn , and
the study of these possible charmoniumlike molecular tetraquark candidates
will become an important research field at the precision frontier in future
experiments. Thus, we strongly expect experimental colleagues to focus on the
detailed structures around 3.9 GeV in the $J/\psi\omega$ and $J/\psi\eta$
invariant mass spectrum with more precise experimental data, like the LHCb,
Belle II, and BESIII, it will provide strong evidence of existing
charmoniumlike molecular tetraquark states if these possible enhancement
structures can be confirmed in future experiments.

##### The others isoscalar $\mathcal{D}\bar{\mathcal{D}}$ systems.

In this section, we mainly try to understand very broad structure around 4.3
GeV in the $J/\psi\omega$ invariant mass spectrum of the $B\to J/\psi\omega K$
process Abe:2004zs ; Aubert:2007vj ; delAmoSanchez:2010jr . The mass
thresholds of the $D^{\ast}\bar{D}_{1}$, $D^{\ast}\bar{D}^{\ast}_{2}$,
$D\bar{D}^{\ast}_{2}$, and $D\bar{D}_{1}$ systems locate around this energy
region. In general, the pion exchange interaction usually plays a crucial role
in forming the hadronic molecular states Chen:2016qju . In the following, we
discuss these $S-$wave isoscalar charmoniumlike molecular tetraquark systems
with three different groups, i.e., the $S-$wave isoscalar charmoniumlike
molecular tetraquark systems with the pion exchange contribution occurring in
the direct channel effective potentials, the $S-$wave isoscalar charmoniumlike
molecular tetraquark systems with the pion exchange contribution occurring in
the cross channel effective potentials, and the $S-$wave isoscalar
charmoniumlike molecular tetraquark systems without the pion exchange process.
For the sake of completeness, we also discuss the mass spectrum and the two-
body hidden-charm decay channels for the $S-$wave isoscalar $D\bar{D}$ and
$D\bar{D}^{*}$ systems.

(i).

The $S-$wave isoscalar $D^{\ast}\bar{D}_{1}$ and $D^{\ast}\bar{D}^{\ast}_{2}$
systems.

By performing numerical calculations, we can obtain the loosely bound state
solutions for the $S-$wave isoscalar $D^{\ast}\bar{D}_{1}$ and
$D^{\ast}\bar{D}^{\ast}_{2}$ systems when the cutoff values are tuned from 0.8
to 3.0 GeV. In Table 6, we present the corresponding bound state solutions.
Compared to the high spin states, we find that the low spin states can be
easier to bind as charmoniumlike molecular candidates for the $S-$wave
isoscalar $D^{\ast}\bar{D}_{1}$ and $D^{\ast}\bar{D}^{\ast}_{2}$ systems.
Here, we also consider the coupled channel effect, and find the coupled
channel effect plays a minor role in the above discussed systems. In fact,
there are several papers on the predictions of the possible $S-$wave isoscalar
$D^{\ast}\bar{D}_{1}$ and $D^{\ast}\bar{D}^{\ast}_{2}$ charmoniumlike
molecular tetraquark states Close:2010wq ; Li:2015exa ; Close:2009ag ;
He:2017mbh ; Li:2013bca ; Zhu:2013sca .

Table 6: Bound state solutions for the $S-$wave isoscalar $D^{\ast}\bar{D}_{1}$ and $D^{\ast}\bar{D}^{\ast}_{2}$ systems. Conventions are the same as Table 5. Effect | Single channel |  $S$-$D$ wave mixing effect
---|---|---
$D^{*}\bar{D}_{1}$
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{1}\mathbb{S}_{0}/{}^{5}\mathbb{D}_{0})$
$0^{--}$ | 0.96 | $-0.59$ | 3.73 | 0.95 | $-0.52$ | 3.92 |  99.70/0.30
1.03 | $-11.12$ | 1.07 | 1.03 | $-12.40$ | 1.03 |  99.59/0.41
$0^{-+}$ | 0.92 | $-0.56$ | 3.91 | 0.91 | $-0.55$ | 3.96 |  99.53/0.47
0.99 | $-11.42$ | 1.08 | 0.99 | $-12.82$ | 1.05 |  99.41/0.59
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1}/{}^{5}\mathbb{D}_{1})$
$1^{--}$ | 1.10 | $-0.48$ | 4.11 | 1.08 | $-0.33$ | 4.61 |  99.64/0.35/0.01
1.20 | $-12.57$ | 1.03 | 1.19 | $-12.74$ | 1.03 |  99.38/0.61/0.01
$1^{-+}$ | 1.06 | $-0.45$ | 4.25 | 1.04 | $-0.34$ | 4.68 |  99.47/0.52/0.01
1.15 | $-11.80$ | 1.07 | 1.14 | $-11.75$ | 1.09 |  99.13/0.86/0.01
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{5}\mathbb{S}_{2}/{}^{1}\mathbb{D}_{2}/{}^{3}\mathbb{D}_{2}/{}^{5}\mathbb{D}_{2})$
$2^{--}$ | 2.56 | $-0.32$ | 4.89 | 2.60 | $-0.33$ | 4.90 |  98.90/0.05/$o(0)$/1.05
2.58 | $-9.86$ | 1.16 | 2.74 | $-8.65$ | 1.33 |  93.06/4.18/$o(0)$/2.75
$2^{-+}$ | 1.73 | $-0.81$ | 3.68 | 1.59 | $-0.34$ | 4.90 |  99.30/0.08/$o(0)$/0.62
2.14 | $-12.16$ | 1.24 | 2.07 | $-12.74$ | 1.24 |  98.77/0.27/$o(0)$/0.93
$D^{*}\bar{D}_{2}^{*}$
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1}/{}^{5}\mathbb{D}_{1}/{}^{7}\mathbb{D}_{1})$
$1^{--}$ | 0.94 | $-0.28$ | 4.78 | 0.94 | $-0.56$ | 3.90 |  99.77/0.07/$o(0)$/0.16
1.02 | $-12.82$ | 1.01 | 1.02 | $-13.86$ | 0.99 |  99.69/0.05/$o(0)$/0.26
$1^{-+}$ | 0.97 | $-0.27$ | 4.80 | 0.97 | $-0.51$ | 4.00 |  99.81/0.09/$o(0)$/0.10
1.05 | $-10.92$ | 1.09 | 1.05 | $-11.81$ | 1.06 |  99.71/0.19/$o(0)$/0.09
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{5}\mathbb{S}_{2}/{}^{3}\mathbb{D}_{2}/{}^{5}\mathbb{D}_{2}/{}^{7}\mathbb{D}_{2})$
$2^{--}$ | 1.21 | $-0.67$ | 3.68 | 1.19 | $-0.28$ | 4.84 |  99.67/$o(0)$/0.33/$o(0)$
1.36 | $-12.76$ | 1.04 | 1.35 | $-12.82$ | 1.05 |  99.38/0.01/0.60/0.01
$2^{-+}$ | 1.11 | $-0.31$ | 4.77 | 1.10 | $-0.28$ | 4.84 |  99.78/$o(0)$/0.22/$o(0)$
1.21 | $-11.55$ | 1.08 | 1.21 | $-12.66$ | 1.04 |  99.59/0.01/0.40/$o(0)$
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{7}\mathbb{S}_{3}/{}^{3}\mathbb{D}_{3}/{}^{5}\mathbb{D}_{3}/{}^{7}\mathbb{D}_{3})$
$3^{--}$ | 1.90 | $-0.32$ | 4.94 | 1.73 | $-0.31$ | 4.98 |  99.21/0.07/$o(0)$/0.72
2.87 | $-12.26$ | 1.23 | 2.64 | $-12.24$ | 1.25 |  98.36/0.39/0.03/1.22
$3^{-+}$ | 1.97 | $-0.32$ | 4.94 | 1.77 | $-0.33$ | 4.91 |  99.03/0.04/$o(0)$/0.93
2.74 | $-12.49$ | 1.21 | 2.51 | $-12.41$ | 1.24 |  97.70/0.19/0.03/2.07

In Fig. 8, we present the cutoff parameter $\Lambda$ dependence of the binding
energy $E$ for the $S$-wave $D^{\ast}\bar{D}_{1}$ state with
$I(J^{PC})=0(0^{--})$, and there exist the loosely bound state solutions when
the cutoff parameter is larger than 0.96 GeV, where the binding energy
increases with the cutoff value monotonically. For simplicity, we only take
$\Lambda$=0.96 and 1.03 GeV to present the loosely bound state solutions for
the $S$-wave $D^{\ast}\bar{D}_{1}$ state with $I(J^{PC})=0(0^{--})$ in Table
6.

Figure 8: (color online) The cutoff parameter $\Lambda$ dependence of the
binding energy $E$ for the $S$-wave $D^{\ast}\bar{D}_{1}$ state with
$I(J^{PC})=0(0^{--})$.

(ii).

The $S-$wave isoscalar $D\bar{D}^{\ast}_{2}$ and $D\bar{D}^{\ast}$ systems.

For the $D\bar{D}^{\ast}_{2}$ and $D\bar{D}^{\ast}$ systems, the $\pi$
exchange occurs in the $D\bar{D}^{\ast}_{2}\to{D}^{\ast}_{2}\bar{D}$ and
$D\bar{D}^{\ast}\to{D}^{\ast}\bar{D}$ processes, and the interaction Feynman
diagram corresponds to the Cross diagram in Fig. 6. In Table 7, we collect the
bound state solutions for the $S-$wave isoscalar $D\bar{D}^{\ast}_{2}$ and
$D\bar{D}^{\ast}$ systems. It is obvious that these $S-$wave isoscalar
$D\bar{D}^{\ast}_{2}$ and $D\bar{D}^{\ast}$ states can be possible
charmoniumlike molecular candidates as their bound state solutions satisfy the
typical characters for a loosely bound hadronic molecule666When the cutoff
$\Lambda$ is taken around 1 GeV, the binding energy is around several to
several tens MeV, and the size of bound state is larger than the size of its
component.. In fact, the $S-$wave isoscalar $D\bar{D}^{*}$ molecular states
have been extensively studied in Refs. Wong:2003xk ; Swanson:2003tb ;
Suzuki:2005ha ; Liu:2008fh ; Thomas:2008ja ; Liu:2008tn ; Lee:2009hy ;
Zhao:2014gqa ; Li:2012cs ; He:2014nya ; Voloshin:2003nt ; Close:2003sg ;
Tornqvist:2004qy ; Sun:2011uh ; Sun:2012zzd ; Wang:2017dcq ; Ding:2020dio ;
Sun:2017wgf ; Yang:2017prf ; Ding:2009vj ; Zhang:2006ix ; Liu:2017mrh ;
Tornqvist:1993ng ; DeRujula:1976zlg . However, our knowledge of the $S$-wave
isoscalar $D\bar{D}^{\ast}_{2}$ molecular states is still not enough up to now
Li:2015exa ; Li:2013bca .

Table 7: Bound state solutions for the $S-$wave isoscalar $D\bar{D}^{\ast}_{2}$ and $D\bar{D}^{\ast}$ systems. Conventions are the same as Table 5. Effect | Single channel |  $S-$$D$ wave mixing effect
---|---|---
$D\bar{D}_{2}^{*}[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{5}\mathbb{S}_{2}/{}^{5}\mathbb{D}_{2})$
$2^{--}$ | 1.46 | $-0.25$ | 5.08 | 1.46 | $-0.34$ | 4.74 |  99.99/0.01
| 1.85 | $-12.22$ | 1.11 | 1.82 | $-12.40$ | 1.11 |  99.91/0.09
$2^{-+}$ | 1.30 | $-0.36$ | 4.69 | 1.29 | $-0.24$ | 5.17 |  99.99/0.01
| 1.47 | $-12.03$ | 1.11 | 1.47 | $-12.89$ | 1.08 |  99.93/0.07
$D\bar{D}^{*}[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$
$1^{+-}$ | 1.62 | $-0.36$ | 4.70 | 1.36 | $-0.39$ | 4.71 |  97.38/2.62
1.77 | $-12.51$ | 1.07 | 1.49 | $-12.32$ | 1.19 |  92.15/7.85
$1^{++}$ | 1.18 | $-0.27$ | 5.15 | 1.08 | $-0.27$ | 5.22 |  99.05/0.95
1.53 | $-12.39$ | 1.19 | 1.30 | $-12.09$ | 1.23 |  96.99/3.01

(iii).

The $S-$wave isoscalar $D\bar{D}_{1}$ and $D\bar{D}$ systems.

Despite the $\pi$ exchange does not contribute to the effective potentials for
the $S-$wave $D\bar{D}_{1}$ system as the parity forbidden, the scalar and
vector mesons exchange interactions may be strong enough to generate an bound
state Chen:2017vai ; Wang:2019nwt . As shown in Table 8, our numerical results
suggest the isoscalar $D\bar{D}_{1}$ states with
$I(J^{PC})=0[(1^{--}),(1^{-+})]$ and the $S-$wave $D\bar{D}$ state with
$I(J^{PC})=0(0^{++})$ can be possible charmoniumlike molecular candidates. In
fact, the $S-$wave isoscalar $D\bar{D}_{1}$ molecular states were intensively
discussed in Refs. Ding:2008gr ; Dong:2019ofp ; Close:2010wq ; Li:2015exa ;
Close:2009ag ; Li:2013bca ; Wang:2020lua , which may be related to the
$Y(4260)$777In 2017, the BESIII gave more precise data of the $e^{+}e^{-}\to
J/\psi\pi^{+}\pi^{-}$ Ablikim:2016qzw , which shows that the $Y(4260)$
Aubert:2005rm  is split into two resonances $Y(4220)$ and $Y(4320)$.
Aubert:2005rm . In Refs. Liu:2008tn ; Liu:2008mi ; Liu:2017mrh ; Ding:2020dio
; Yang:2017prf ; Zhang:2006ix ; DeRujula:1976zlg , the $S-$wave $D\bar{D}$
bound state with $I(J^{PC})=0(0^{++})$ was estimated.

Table 8: Bound state solutions for the $S-$wave isoscalar $D\bar{D}_{1}$ system. Conventions are the same as Table 5. Effect | Single channel |  $S-$$D$ wave mixing effect
---|---|---
$D\bar{D}[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ |  |  |  |
$0^{++}$ | 1.46 | $-0.29$ | 5.08 |  |  |  |
1.76 | $-12.55$ | 1.15 |  |  |  |
$D\bar{D}_{1}[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$
$1^{--}$ | 1.39 | $-0.36$ | 4.72 | 1.39 | $-0.39$ | 4.60 |  99.98/0.02
| 1.67 | $-12.13$ | 1.14 | 1.67 | $-12.42$ | 1.13 |  99.92/0.08
$1^{-+}$ | 1.38 | $-0.29$ | 4.92 | 1.38 | $-0.32$ | 4.80 |  99.98/0.02
| 1.63 | $-12.63$ | 1.09 | 1.63 | $-12.89$ | 1.08 |  99.93/0.07

Let’s give a short summary, we can predict a serial of possible charmoniumlike
molecules composed by the $S-$wave isoscalar $D^{\ast}\bar{D}_{1}$,
$D^{\ast}\bar{D}^{\ast}_{2}$, $D\bar{D}^{\ast}_{2}$, $D\bar{D}^{\ast}$,
$D\bar{D}_{1}$, and $D\bar{D}$ systems. In Table 9, we summary their two-body
hidden-charm decay information for all the possible $S-$wave isoscalar
$D^{\ast}\bar{D}_{1}$, $D^{\ast}\bar{D}^{\ast}_{2}$, $D\bar{D}^{\ast}_{2}$,
$D\bar{D}^{\ast}$, $D\bar{D}_{1}$, and $D\bar{D}$ charmoniumlike molecules.
For example, the $J/\psi\omega$ channel is the two-body hidden-charm decay
mode for the isoscalar $D^{*}\bar{D}_{1}[(0,1,2)^{-+}]$,
$D^{*}\bar{D}_{2}^{*}[3^{-+}]$, and $D\bar{D}_{1}[1^{-+}]$ bound states,
perhaps, it is possible to observe the experimental signal of these possible
charmoniumlike molecules in the $J/\psi\omega$ final state. When we recall the
experimental data of the $B\to J/\psi\omega K$ Abe:2004zs ; Aubert:2007vj ;
delAmoSanchez:2010jr , our predictions of these possible charmoniumlike
molecules also reflect the complexity of the structures in the $J/\psi\omega$
invariant mass spectrum around 4.3 GeV. At present, it is a little difficult
to definitely identify these possible charmoniumlike molecules, we hope
further experiments can provide more precise measurement of the $B$ meson
decay into a charmonium state plus a light-flavor meson. In addition, the
$\chi_{cJ}(1P)\omega$ with $J=0,1,2$ are the two-body hidden-charm decay
channels for the possible $S-$wave isoscalar $D^{\ast}\bar{D}_{1}$,
$D^{\ast}\bar{D}^{\ast}_{2}$, $D\bar{D}^{\ast}_{2}$, and $D\bar{D}_{1}$
systems of negative $C$-parity in our calculations.

Table 9: A summary of the two-body hidden-charm decay channels for all the possible $S-$wave isoscalar $D^{\ast}\bar{D}_{1}$, $D^{\ast}\bar{D}^{\ast}_{2}$, $D\bar{D}^{\ast}_{2}$, $D\bar{D}^{\ast}$, $D\bar{D}_{1}$, and $D\bar{D}$ charmoniumlike molecules. States | Two-body hidden-charm decay channels
---|---
$D^{*}\bar{D}_{1}[0^{-+}]$ |  ${J/\psi\omega}$, $\chi_{c0}(1P)\eta$, $\chi_{c0}(1P)\eta^{\prime}$
$D^{*}\bar{D}_{1}[0^{--}]$ |  ${\chi_{c1}(1P)\omega}$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$
$D^{*}\bar{D}_{1}[1^{-+}]$ |  ${J/\psi\omega}$, $\chi_{c1}(1P)\eta$, $\eta_{c}\eta$, $\eta_{c}\eta^{\prime}$,
$D^{*}\bar{D}_{1}[1^{--}]$ |  ${\chi_{c2}(1P)\omega}$, $\chi_{c1}(1P)\omega$, $\chi_{c0}(1P)\omega$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$
$D^{*}\bar{D}_{1}[2^{-+}]$ |  ${J/\psi\omega}$, $\chi_{c2}(1P)\eta$
$D^{*}\bar{D}_{1}[2^{--}]$ |  ${\chi_{c2}(1P)\omega}$, $\chi_{c1}(1P)\omega$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$
$D^{*}\bar{D}_{2}^{*}[1^{-+}]$ |  ${\chi_{c1}(1P)\eta}$, $\eta_{c}\eta$, $\eta_{c}\eta^{\prime}$, $J/\psi\omega$, $\chi_{c1}(1P)\eta^{\prime}$
$D^{*}\bar{D}_{2}^{*}[1^{--}]$ |  ${\chi_{c2}(1P)\omega}$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$, $\chi_{c0}(1P)\omega$, $\chi_{c1}(1P)\omega$
$D^{*}\bar{D}_{2}^{*}[2^{-+}]$ |  ${\chi_{c2}(1P)\eta}$, $J/\psi\omega$
$D^{*}\bar{D}_{2}^{*}[2^{--}]$ |  ${\chi_{c2}(1P)\omega}$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$, $\chi_{c1}(1P)\omega$
$D^{*}\bar{D}_{2}^{*}[3^{-+}]$ | ${J/\psi\omega}$
$D^{*}\bar{D}_{2}^{*}[3^{--}]$ | ${\chi_{c2}(1P)\omega}$
$D\bar{D}_{2}^{*}[2^{-+}]$ |  ${\chi_{c2}(1P)\eta}$, $J/\psi\omega$
$D\bar{D}_{2}^{*}[2^{--}]$ |  ${\chi_{c1}(1P)\omega}$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$
$D\bar{D}^{*}[1^{++}]$ | $J/\psi\omega$
$D\bar{D}^{*}[1^{+-}]$ |  $\eta_{c}\omega$, $J/\psi\eta$
$D\bar{D}_{1}[1^{-+}]$ |  ${J/\psi\omega}$, $\eta_{c}\eta^{\prime}$, $\eta_{c}\eta$, $\chi_{c1}(1P)\eta$
$D\bar{D}_{1}[1^{--}]$ |  ${\chi_{c0}(1P)\omega}$, $\eta_{c}\omega$, $J/\psi\eta$, $J/\psi\eta^{\prime}$
$D\bar{D}[0^{++}]$ | $\eta_{c}\eta$

####  III.2.2 Charmoniumlike molecular tetraquark systems with hidden-strange
quantum number

With the help of the $\rm{SU}(3)$ flavor symmetry, we can further study the
interactions between the charm-strange meson and anti-charm-strange meson, in
this section, we analyze the existence probability of the charmoniumlike
molecules with hidden-strange quantum number and give their two-body hidden-
charm decay channels.

##### The $D_{s}^{*}\bar{D}_{s}^{*}$ system.

For the $D_{s}^{*}\bar{D}_{s}^{*}$ system, the $\eta$ and $\phi$ exchanges
contribute to the effective potentials in the OBE model. The relevant
numerical results for the $S-$wave $D_{s}^{\ast}\bar{D}_{s}^{\ast}$ system are
given in Table 10, and the cutoff parameters are taken in the range from 1.0
to 3.0 GeV.

Table 10: Bound state solutions for the $S-$wave $D_{s}^{\ast}\bar{D}_{s}^{\ast}$ system. Conventions are the same as Table 5. Effect | Single channel |  $S-$$D$ wave mixing effect
---|---|---
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{1}\mathbb{S}_{0}/{}^{5}\mathbb{D}_{0})$
$0^{++}$ | 1.59 | $-0.72$ | 3.49 | 1.58 | $-0.23$ | 4.98 |  99.98/0.02
1.65 | $-11.09$ | 0.98 | 1.65 | $-11.34$ | 0.97 |  99.95/0.05
$J^{PC}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$
$1^{+-}$ | 1.89 | $-0.52$ | 4.00 | 1.88 | $-0.26$ | 4.89 |  99.98/0.02
2.00 | $-12.37$ | 0.95 | 2.00 | $-12.52$ | 0.94 |  99.96/0.04

For the $S-$wave $D_{s}^{*}\bar{D}_{s}^{*}$ states with $J^{PC}=0^{++}$ and
$1^{+-}$, we can obtain the loosely bound state solutions for these states
when the cutoff values are taken to be around 1.6 GeV and 1.9 GeV,
respectively. Thus, the $S-$wave $D_{s}^{*}\bar{D}_{s}^{*}$ states with
$J^{PC}=0^{++}$ and $1^{+-}$ can be the possible hidden-charm and hidden-
strange molecular tetraquark candidates, especially the
$D_{s}^{*}\bar{D}_{s}^{*}$ molecular state with $J^{PC}=0^{++}$ Ding:2009vd ;
Meng:2020cbk . If taking a large cutoff $\Lambda>3.0$ GeV, there exist the
loosely bound state solutions for the $D_{s}^{*}\bar{D}_{s}^{*}$ state with
$J^{PC}=2^{++}$ in Ref. Liu:2009ei . However, such cutoff parameter is far
from the usual value around 1.0 GeV Tornqvist:1993ng ; Tornqvist:1993vu ;
Wang:2019nwt , which is consistent with our numerical results.

##### The $D_{s}\bar{D}_{s}$, $D_{s}\bar{D}_{s}^{\ast}$,
$D_{s}\bar{D}_{s0}^{\ast}$, $D_{s}\bar{D}_{s1}^{\prime}$,
$D_{s}^{\ast}\bar{D}_{s0}^{\ast}$, $D_{s}^{\ast}\bar{D}_{s1}^{\prime}$,
${D}_{s}^{(*)}\bar{D}_{s1}$, and $D_{s}^{(*)}\bar{D}_{s2}^{*}$ systems.

Besides the $S-$wave $D_{s}^{*}\bar{D}_{s}^{*}$ system, we also investigate
the bound state properties of the $S-$wave $D_{s}\bar{D}_{s}$,
$D_{s}\bar{D}_{s}^{\ast}$, $D_{s}\bar{D}_{s0}^{\ast}$,
$D_{s}\bar{D}_{s1}^{\prime}$, $D_{s}^{\ast}\bar{D}_{s0}^{\ast}$, and
$D_{s}^{\ast}\bar{D}_{s1}^{\prime}$ systems by tuning the cutoff values
$\Lambda$ from 1.0 to 3.0 GeV, and the corresponding numerical results are
listed in Table 11. If we still adopt the general criterion of the loosely
hadron-hadron molecule Wang:2020dya ; Tornqvist:1993ng ; Tornqvist:1993vu ;
Wang:2019nwt , we can find

  * •

There may exist several possible charmoniumlike molecular tetraquark
candidates with hidden-strange quantum number, such as the
$D_{s}\bar{D}_{s}^{\ast}$ state with $J^{PC}=1^{+-}$ Liu:2017mrh ;
Meng:2020cbk , the $D_{s}\bar{D}_{s0}^{\ast}$ state with $J^{PC}=0^{-\mp}$
Shen:2010ky ; Liu:2010hf ; He:2013oma ; He:2016pfa , the
$D_{s}\bar{D}_{s1}^{\prime}$ state with $J^{PC}=1^{--}$, the
$D_{s}^{\ast}\bar{D}_{s1}^{\prime}$ states with $J^{PC}=0^{-\mp}$ and
$1^{-+}$. In particular, the coupled channel effect plays an important role in
generating these loosely bound states, i.e., the $D_{s}\bar{D}_{s}^{\ast}$
state with $J^{PC}=1^{+-}$, the $D_{s}\bar{D}_{s0}^{\ast}$ state with
$J^{PC}=0^{-\mp}$, and the $D_{s}\bar{D}_{s1}^{\prime}$ state with
$J^{PC}=1^{--}$.

  * •

If we increase the cutoff value around 2.0 GeV, we can obtain loosely bound
state solutions for the $D_{s}\bar{D}_{s}^{\ast}$ state with $J^{PC}=1^{++}$,
the $D_{s}^{\ast}\bar{D}_{s0}^{\ast}$ state with $J^{PC}=1^{--}$, and the
$D_{s}^{\ast}\bar{D}_{s1}^{\prime}$ state with $J^{PC}=1^{--}$. They may be
the possible charmoniumlike molecular candidates with hidden-strange quantum
number.

  * •

In addition, we do not obtain bound state solutions for the $D_{s}\bar{D}_{s}$
state with $J^{PC}=0^{++}$, the $D_{s}\bar{D}_{s1}^{\prime}$ state with
$J^{PC}=1^{-+}$, and the $D_{s}^{\ast}\bar{D}_{s1}^{\prime}$ state with
$J^{PC}=2^{-\mp}$ by tuning cutoff values from $1.0$ to $3.0$ GeV.

Table 11: Bound state solutions for the $S-$wave $D_{s}\bar{D}_{s}^{\ast}$, $D_{s}\bar{D}_{s0}^{\ast}$, $D_{s}\bar{D}_{s1}^{\prime}$, $D_{s}^{\ast}\bar{D}_{s0}^{\ast}$, and $D_{s}^{\ast}\bar{D}_{s1}^{\prime}$ systems. Conventions are the same as Table 5. Effect | Single channel |  $S-$$D$ wave mixing effect | Coupled channel effect
---|---|---|---
States$[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P(D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*})$
$D_{s}\bar{D}_{s}^{*}[1^{+-}]$ | 2.65 | $-0.26$ | 4.96 | 2.27 | $-0.43$ | 4.39 |  99.25/0.75 | 1.63 | $-0.13$ | 5.41 |  96.64/3.36
2.85 | $-11.95$ | 0.98 | 2.43 | $-11.85$ | 1.03 |  96.56/3.44 | 1.66 | $-8.92$ | 0.99 |  78.85/21.15
$D_{s}\bar{D}_{s}^{*}[1^{++}]$ | $\times$ | $\times$ | $\times$ | 2.71 | $-0.27$ | 4.99 |  99.46/0.54 |  |  |  |
$\times$ | $\times$ | $\times$ | 3.00 | $-5.69$ | 1.49 |  97.69/2.31 |  |  |  |
States$[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ |  |  |  |  | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P(D_{s}\bar{D}_{s0}^{*}/D_{s}^{*}\bar{D}_{s1}^{\prime})$
$D_{s}\bar{D}_{s0}^{*}[0^{--}]$ | $\times$ | $\times$ | $\times$ |  |  |  |  | 1.87 | $-2.52$ | 1.77 |  85.11/14.89
$\times$ | $\times$ | $\times$ |  |  |  |  | 1.88 | $-8.21$ | 0.94 |  75.72/24.48
$D_{s}\bar{D}_{s0}^{*}[0^{-+}]$ | 2.83 | $-0.28$ | 4.93 |  |  |  |  | 1.49 | $-0.43$ | 4.27 |  96.97/3.03
3.00 | $-0.92$ | 3.32 |  |  |  |  | 1.52 | $-10.05$ | 1.00 |  81.81/18.19
State$[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P(D_{s}\bar{D}_{s1}^{\prime}/D_{s}^{*}\bar{D}_{s0}^{*}/D_{s}^{*}\bar{D}_{s1}^{\prime})$
$D_{s}\bar{D}_{s1}^{\prime}[1^{--}]$ | 2.17 | $-0.33$ | 4.65 | 2.15 | $-0.29$ | 4.79 |  99.98/0.02 | 1.87 | $-0.31$ | 4.69 |  98.76/1.24/$o(0)$
2.55 | $-12.60$ | 1.00 | 2.51 | $-12.35$ | 1.00 |  99.82/0.18 | 1.99 | $-12.64$ | 0.92 |  87.08/12.92/$o(0)$
State$[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1})$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P(D_{s}^{*}\bar{D}_{s0}^{*}/D_{s}^{*}\bar{D}_{s1}^{\prime})$
$D_{s}^{*}\bar{D}_{s0}^{*}[1^{--}]$ | 2.17 | $-0.31$ | 4.73 | 2.17 | $-0.31$ | 4.72 |  100.00/$o(0)$ | 2.17 | $-0.31$ | 4.72 |  100.00/$o(0)$
2.53 | $-12.42$ | 1.00 | 2.53 | $-12.42$ | 1.00 |  100.00/$o(0)$ | 2.53 | $-12.42$ | 1.00 |  100.00/$o(0)$
States$[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{1}\mathbb{S}_{0}/{}^{5}\mathbb{D}_{0})$ |  |  |  |
$D_{s}^{*}\bar{D}_{s1}^{\prime}[0^{--}]$ | 1.37 | $-0.41$ | 4.33 | 1.37 | $-0.52$ | 3.99 |  99.96/0.04 |  |  |  |
1.45 | $-12.34$ | 0.98 | 1.45 | $-12.82$ | 0.97 |  99.90/0.10 |  |  |  |
$D_{s}^{*}\bar{D}_{s1}^{\prime}[0^{-+}]$ | 1.87 | $-0.33$ | 4.24 | 1.87 | $-0.36$ | 4.13 |  99.99/0.01 |  |  |  |
1.91 | $-10.27$ | 0.85 | 1.91 | $-10.40$ | 0.85 |  99.99/0.01 |  |  |  |
States$[J^{PC}]$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $\Lambda$ | $E$ | $r_{\rm RMS}$ | $P({}^{3}\mathbb{S}_{1}/{}^{3}\mathbb{D}_{1}/{}^{5}\mathbb{D}_{1})$ |  |  |  |
$D_{s}^{*}\bar{D}_{s1}^{\prime}[1^{--}]$ | 2.29 | $-0.59$ | 3.53 | 2.28 | $-0.47$ | 3.87 |  99.97/0.03/$o(0)$ |  |  |  |
2.37 | $-13.38$ | 0.78 | 2.36 | $-13.47$ | 0.79 |  99.88/0.12/$o(0)$ |  |  |  |
$D_{s}^{*}\bar{D}_{s1}^{\prime}[1^{-+}]$ | 1.55 | $-0.22$ | 5.10 | 1.55 | $-0.36$ | 4.52 |  99.95/0.05/$o(0)$ |  |  |  |
1.68 | $-11.38$ | 1.02 | 1.68 | $-12.17$ | 1.00 |  99.84/0.16/$o(0)$ |  |  |  |

In our previous work Wang:2020dya , we systematic study the interactions
between a pair of charm-strange meson and anti-charm-strange meson in the
$H$-doublet or $T$-doublet by using the OBE model and considering the $S-$$D$
wave mixing and the coupled channel effect, and we can predict several
possible $H\bar{T}$-type charmoniumlike molecular states, i.e., the
$D^{\ast}_{s}\bar{D}_{s1}$ molecular states with $J^{PC}=0^{-\pm}/1^{-\pm}$
and the $D^{\ast}_{s}\bar{D}^{\ast}_{s2}$ molecular states with
$J^{PC}=1^{-\pm}/2^{-\pm}$.

In Table 12, we summary their bound properties and two-body hidden-charm decay
channels for all the investigated possible $S-$wave charmoniumlike molecules
with hidden-strange quantum number.

Table 12: A summary of the bound state properties and two-body hidden-charm decay channels for all the investigated possible $S-$wave charmoniumlike molecules with hidden-strange quantum number. Here, notations $\surd$ and $\surd\\!\\!\\!\setminus$ are marked the charmoniumlike molecular candidates with their bound state solutions with the cutoff $\Lambda$ around 1 to 2 GeV and around 2 to 3 GeV, respectively. States | Bound state properties | Two-body hidden-charm decay channels
---|---|---
$D_{s}^{*}\bar{D}_{s}^{*}[0^{++}]$ | $\surd$ |  ${\eta_{c}\eta^{\prime}}$, $J/\psi\phi$, ${\eta_{c}\eta}$, $\chi_{c1}(1P)\eta$
$D_{s}^{*}\bar{D}_{s}^{*}[1^{+-}]$ | $\surd$ |  ${J/\psi\eta^{\prime}}$, $\eta_{c}\phi$, $J/\psi\eta$
$D_{s}\bar{D}_{s}^{*}[1^{++}]$ | $\surd\\!\\!\\!\setminus$ |  ${\chi_{c0}(1P)\eta}$, ${\chi_{c1}(1P)\eta}$
$D_{s}\bar{D}_{s}^{*}[1^{+-}]$ | $\surd$ |  ${\eta_{c}\phi}$, $J/\psi\eta$, $J/\psi\eta^{\prime}$
$D_{s}\bar{D}_{s0}^{*}[0^{-+}]$ | $\surd$ |  ${J/\psi\phi}$, $\chi_{c0}(1P)\eta$
$D_{s}\bar{D}_{s0}^{*}[0^{--}]$ | $\surd$ |  ${\eta_{c}\phi}$, $J/\psi\eta^{\prime}$, $J/\psi\eta$
$D_{s}\bar{D}_{s1}^{\prime}[1^{--}]$ | $\surd$ |  $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$
$D_{s}^{*}\bar{D}_{s0}^{*}[1^{--}]$ | $\surd\\!\\!\\!\setminus$ |  ${\eta_{c}\phi}$, $J/\psi\eta^{\prime}$, $J/\psi\eta$
$D_{s}^{*}\bar{D}_{s1}^{\prime}[0^{-+}]$ | $\surd$ |  ${J/\psi\phi}$, $\chi_{c0}(1P)\eta^{\prime}$, $\chi_{c0}(1P)\eta$
$D_{s}^{*}\bar{D}_{s1}^{\prime}[0^{--}]$ | $\surd$ |  ${\chi_{c1}(1P)\phi}$, $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$
$D_{s}^{*}\bar{D}_{s1}^{\prime}[1^{-+}]$ | $\surd$ |  ${\chi_{c1}(1P)\eta^{\prime}}$, $J/\psi\phi$, $\chi_{c1}(1P)\eta$, ${\eta_{c}\eta}$, ${\eta_{c}\eta^{\prime}}$
$D_{s}^{*}\bar{D}_{s1}^{\prime}[1^{--}]$ | $\surd\\!\\!\\!\setminus$ |  ${\chi_{c0}(1P)\phi}$, $\chi_{c1}(1P)\phi$, $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$
$D^{\ast}_{s}\bar{D}_{s1}[0^{-+}]$ | $\surd$ |  ${\chi_{c0}(1P)\eta^{\prime}}$, $\chi_{c0}(1P)\eta$, $J/\psi\phi$
$D^{\ast}_{s}\bar{D}_{s1}[0^{--}]$ | $\surd$ |  ${\chi_{c1}(1P)\phi}$, $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$
$D^{\ast}_{s}\bar{D}_{s1}[1^{-+}]$ | $\surd$ |  ${J/\psi\phi}$, ${\chi_{c1}(1P)\eta^{\prime}}$, $\chi_{c1}(1P)\eta$, ${\eta_{c}\eta}$, ${\eta_{c}\eta^{\prime}}$
$D^{\ast}_{s}\bar{D}_{s1}[1^{--}]$ | $\surd$ |  ${\chi_{c0}(1P)\phi}$, $\chi_{c2}(1P)\phi$, $\chi_{c1}(1P)\phi$, $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$
$D^{\ast}_{s}\bar{D}^{\ast}_{s2}[1^{-+}]$ | $\surd$ |  ${\chi_{c1}(1P)\eta^{\prime}}$, $J/\psi\phi$, $\chi_{c1}(1P)\eta$, ${\eta_{c}\eta}$, ${\eta_{c}\eta^{\prime}}$
$D^{\ast}_{s}\bar{D}^{\ast}_{s2}[1^{--}]$ | $\surd$ |  ${\chi_{c2}(1P)\phi}$, $\chi_{c1}(1P)\phi$, $\chi_{c0}(1P)\phi$, $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$
$D^{\ast}_{s}\bar{D}^{\ast}_{s2}[2^{-+}]$ | $\surd$ |  ${\chi_{c2}(1P)\eta^{\prime}}$, $\chi_{c2}(1P)\eta$, ${J/\psi\phi}$
$D^{\ast}_{s}\bar{D}^{\ast}_{s2}[2^{--}]$ | $\surd\\!\\!\\!\setminus$ |  ${\chi_{c2}(1P)\phi}$, $\chi_{c1}(1P)\phi$, $J/\psi\eta^{\prime}$, $J/\psi\eta$, ${\eta_{c}\phi}$

The obtained results are theoretic reference to search for possible
charmoniumlike molecular tetraquark with hidden-strange quantum number. So
far, many charmoniumlike structures have been observed in the $J/\psi\phi$
invariant mass spectrum. In particular, the recent LHCb Collaboration updated
the amplitude analysis of the $B^{+}\to J/\psi\phi K^{+}$ decay with the
combined dataset collected in Run I plus Run II Aaij:2021ivw , and they
reported several enhancement structures, some of them locate close to the mass
thresholds of our investigated charmoniumlike molecular tetraquark systems
with hidden-strange quantum number, our study for these charmoniumlike
molecules with hidden-strange quantum number may provide hints to understand
the nature of the reported charmoniumlike states, i.e.,

  1. 1.

The masses of the $X(4140)$ with $J^{PC}=1^{++}$ and the $X(4150)$ with
$J^{PC}=2^{-+}$ Aaij:2021ivw  are close to the $D_{s}^{*}\bar{D}_{s}^{*}$
threshold. It is obvious that their quantum configurations do not fit into the
$S-$wave $D_{s}^{*}\bar{D}_{s}^{*}$ molecules. If their resonance parameters
and quantum numbers can be further confirmed in the future, the assignment of
these complicated structures as the $S-$wave $D_{s}^{*}\bar{D}_{s}^{*}$
molecular states will be facing great challenge.

  2. 2.

When we recall the other charmoniumlike structure $Y(4274)$ Aaltonen:2009tz ;
Chatrchyan:2013dma ; Aaij:2016iza  (see Fig. 3 for more details), its mass are
close to the $D_{s}\bar{D}_{s0}^{\ast}$ threshold, its spin-parity is favored
as $J^{PC}=1^{++}$ by the LHCb Collaboration Aaij:2016iza ; Aaij:2021ivw , the
similar situation happens, our results do not support the $Y(4274)$ with
$J^{PC}=1^{++}$ as the $S-$wave $D_{s}\bar{D}_{s0}^{\ast}$ molecular state.

  3. 3.

In the mass region from 4.5 GeV to 4.7 GeV, the recent LHCb Collaboration
reported the four enhancement structures $X(4500)$, $X(4630)$, $X(4685)$, and
$X(4700)$ in the $J/\psi\phi$ mass spectrum from the $B\to J/\psi\phi K$ decay
Aaij:2021ivw . Our results indicate there can exist six possible
charmoniumlike molecular tetraquark candidates with hidden-strange quantum
number with positive $C$-parity. Some of them can decay to the $J/\psi\phi$
final state. In our former work Yang:2021sue , the $X(4630)$ can be assigned
as the $D_{s}^{*}\bar{D}_{s1}$ charmoniumlike molecule with $J^{PC}=1^{-+}$.
If these six predicted charomoniumlike molecules really exist, it is a big
challenge to search and further identify separately by using the data from the
$B\to J/\psi\phi K$ process.

  4. 4.

In addition, our calculations can predict several possible charmoniumlike
molecular candidates with negative $C$-parity, which deserve to attract the
experiment’s attentions. Because their two-body hidden-charm decay channels
include the $\chi_{c0,1,2}(1P)\phi$, the $B\to\chi_{c0,1,2}(1P)\phi+K\to
J/\psi\gamma\phi+K$ may be the prime decay process to search for these
possible charmoniumlike molecules.

In fact, the phenomena in high mass region are very complicated, beside these
possible charmoniumlike molecular structures, there can exist several
traditional charmonium states with their masses around these discussed
thresholds. For example, the Lanzhou group indicated that the masses for the
$\chi_{c0}(3{}^{3}P_{0})$, $\chi_{c1}(3{}^{3}P_{1})$, and
$\chi_{c2}(3{}^{3}P_{2})$ states are around 4177 MeV, 4197 MeV, and 4213 MeV
Wang:2019mhs ; Wang:2020prx , which is close to the $D_{s}^{*}\bar{D}_{s}^{*}$
thresholds. Both the theoretical side and the experimental side will make many
efforts to clarify this puzzling phenomenon in the future.

By this study, we find a series of possible isoscalar
$\mathcal{D}\bar{\mathcal{D}}$-type hidden-charm molecular tetraquarks. Some
of the isoscalar charmoniumlike $XYZ$ states can be explained as the isoscalar
hidden-charm molecules as their masses and quantum number configurations match
the calculations of the isoscalar $\mathcal{D}\bar{\mathcal{D}}$-type loosely
bound states. Even so, it is difficult to make a definitely decision if they
are pure hidden-charm molecular tetraquarks or not. They are possible to be
mixtures of the conventional charmonium states and the isoscalar hidden-charm
molecular tetraquarks. However, the charged charmoniumlike states are genuine
tetraquark states, if they are the real particles. In the following, we will
search for possible isovector hidden-charm molecular tetraquarks, and check if
the charged $Z$ isovector states have a close relation to these isovector
hidden-charm molecular tetraquarks.

####  III.2.3 Isovector charmoniumlike molecular tetraquark systems

In this section, let us discuss the final proposal whether there exist
isovector charmoniumlike molecules from a charmed meson and an anticharmed
meson interactions, where the charmed (anticharmed) mesons are in the $H/T$
doublets. In comparison with the isoscalar $\mathcal{D}\bar{\mathcal{D}}$
systems, the OBE effective potentials for the $S-$wave isovector
$\mathcal{D}\bar{\mathcal{D}}$ systems are very different. Since the $\pi$ and
$\rho$ exchanges couple to the isospin charge, their properties in the
isovector $\mathcal{D}\bar{\mathcal{D}}$ systems are contrary to those in the
isoscalar $\mathcal{D}\bar{\mathcal{D}}$ systems, and the corresponding
effective potentials are three times weaker (see Eqs. (1.3)-(1.6) for
details). Therefore, the OBE effective potentials for the $S-$wave isovector
$\mathcal{D}\bar{\mathcal{D}}$ systems may be not strong enough to bind
isovector charmoniumlike molecular states.

When we input the corresponding OBE effective potentials, only the $S-$wave
isovector $D^{\ast}\bar{D}_{1}$ state with $J^{PC}=2^{-+}$ can generate the
loosely bound state with the cutoff parameter $\Lambda$ around 3.0 GeV. Even
though we consider the $S-$$D$ wave mixing and the coupled channel effect, the
cutoff $\Lambda$ is still much larger than 1.0 GeV Tornqvist:1993ng ;
Tornqvist:1993vu . If we take the cutoff value from the deuteron as the
reasonable input, the interactions from a pair of charmed meson and
anticharmed meson are not strong enough to bind $S-$wave isovector
charmoniumlike molecular tetraquark states.

Experimentally, there have accumulated abundant experimental observations of
the charged charmoniumlike states from the $B$ meson decays as introduced in
the Sec. II.3. If they are real particles, they cannot be traditional mesons
but the multiquark matters. As shown in Fig. 9, their masses are all above the
mass thresholds of the corresponding $\mathcal{D}\bar{\mathcal{D}}$ systems.
Obviously, they cannot be assigned as the isovector charmoniumlike molecular
states. In short, our results exclude the charged charmoniumlike states as the
isovector meson-meson molecular states, which perfectly match the experimental
observations. In fact, the other groups also support our conclusions
Chen:2015add ; Liu:2008tn ; Sun:2012zzd ; Zhao:2014gqa ; Liu:2008mi ;
Ding:2008gr ; He:2014nya ; Liu:2007bf ; Liu:2009wb ; Liu:2019zoy ;
Uchino:2015uha ; Liu:2017mrh ; Meng:2021rdg ; Wang:2020htx ; Meng:2020ihj ;
Wang:2020dko .

Figure 9: The masses comparison between the charged charmoniumlike states
Chen:2016qju ; Liu:2013waa ; Hosaka:2016pey ; Liu:2019zoy ; Brambilla:2019esw
; Olsen:2017bmm ; Guo:2017jvc  and the $\mathcal{D}\bar{\mathcal{D}}$
thresholds.

Before closing our discussion on the isovector charmoniumlike systems, we
would like to emphasize that we only analyze whether the charged
charmoniumlike states can be the isovector charmoniumlike molecular states
composed of a pair of charmed meson and anticharmed meson. Furthermore, the
interaction of the off-diagonal couplings may be important to reproduce these
charged charmoniumlike states by the threshold cusp HALQCD:2016ofq ;
Yamada:2021bzb ; Dong:2020hxe ; Ortega:2018cnm ; He:2017lhy , such as the $\pi
J/\psi$-$DD$ and $\rho\eta_{c}$-$DD$ couplings, it is an interesting topic to
consider the more complex coupled channels calculation to deal with these
charged charmoniumlike states. In Ref. Chen:2013coa , Lanzhou group once
reproduced the $Z_{c}(3900)$ structure through the initial-single-pion-
emission mechanism. In fact, many theoretical groups propose other different
explanations on the nature of the charged charmoniumlike states, like the
compact tetraquark configurations, the meson-meson scattering states, and the
kinematical effect (including the coupled channel cusp effect, the reflection
mechanism, the interference effect, the initial single pion emission
mechanism, the triangle singularities, the rescattering effect, and so on),
people can refer articles Chen:2016qju ; Liu:2019zoy ; Guo:2017jvc ;
Wang:2020dmv ; Wang:2020axi ; Wang:2020kej ; Chen:2010nv ; Chen:2011kc ;
Chen:2015bft ; Chen:2017uof ; Chen:2011xk ; Chen:2011pv ; Chen:2013coa ;
Chen:2012yr ; Liu:2015isc ; Chen:2015eia ; Huang:2019agb ; Guo:2019twa  for
more details.

##  IV Summary

Exploration of the hadronic molecular states is an interesting and important
research topics in the hadron physics. Since 2003, serials of charmoniumlike
structures have been reported by different experiments in different beams,
different reactions, and different energy regions, some of them are very close
to a pair of charmed meson and anticharmed meson thresholds, which inspired
theorists to explain them in the hadronic molecular scenario. Although the
charmoniumlike hadronic molecular states are not forbidden by the quantum
chromodynamics (QCD), one cannot give a precisely conclusion whether the
charmoniumlike hadronic molecular states exist or not up to now.

The perfect match between experimental observations and the theoretical
predictions on the three $P_{c}$ states provides a strong evidence of the
existence of hidden-charm meson-baryon molecular pentaquark states. If
replacing $ud$ quarks of $\Lambda_{b}$ by an antiquark $\bar{q}$, the
production mechanisms of the $P_{c}$ states from $\Lambda_{b}$ baryon decays
and the $XYZ$ states from $B$ meson decays are very similar. The $B\to XYZ+K$
decay should be the ideal process to produce charmoniumlike molecular states.

In such a situation, we firstly focus on the isoscalar $D^{*}\bar{D}^{*}$
systems with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$. And we find their
interactions are attractive enough to generate bound charmoniumlike molecular
states. After analyzing their two-body hidden-charm decay behaviors, we find
that the $J/\psi\omega$ and $J/\psi\eta$ channels are the essential decay
modes for the isoscalar $D^{*}\bar{D}^{*}$ molecular states with
$J^{PC}=0^{++}/2^{++}$ and $1^{+-}$, respectively. If checking the data from
the $J/\psi\omega$ invariant mass distribution in the $B\to J/\psi\omega K$
and the $J/\psi\eta$ invariant mass distribution in the $B\to J/\psi\eta K$,
one can roughly find double enhancement structures in $J/\psi\omega$ invariant
mass distribution and single structure in $J/\psi\eta$ invariant mass
distribution, their masses are just below the $D^{*}\bar{D}^{*}$ threshold,
which may correspond to the isoscalar $D^{*}\bar{D}^{*}$ charmoniumlike
molecules with $J^{P}=0^{++}/2^{++}$ and $1^{+-}$, respectively. The behavior
of the double enhancement structures around 3.9 GeV below the
$D^{*}\bar{D}^{*}$ threshold from the $B\to J/\psi\omega K$ and the single
structure in the same energy region from the $B\to J/\psi\eta K$ can provide
crucial information to identify the charmoniumlike molecule. Thus, we suggest
to systematically check the correlation of charmoniumlike molecular states and
charmoniumlike structures existing in the $XYZ$ data of the $B\to XYZ+K$
decay.

After that, we promote our study to the interactions between a charmed (charm-
strange) meson and an anticharmed (anti-charm-strange) meson, including the
$D^{(*)}\bar{D}^{(*)}$, $\bar{D}^{(*)}\bar{D}_{1}$, $D^{(*)}\bar{D}_{2}^{*}$,
$D_{s}^{(*)}\bar{D}_{s}^{(*)}$, ${D}_{s}^{(*)}\bar{D}_{s0}^{*}$,
$D_{s}^{(*)}\bar{D}_{s1}^{\prime}$, ${D}_{s}^{(*)}\bar{D}_{s1}$,
$D_{s}^{(*)}\bar{D}_{s2}^{*}$ systems. After input the OBE effective
potentials, we can find a series of promising isoscalar charmoniumlike
molecular tetraquark candidates as summarized in Tables 9. Our results exclude
these discussed $S-$wave isovector charmoniumlike tetraquark states as the
hadronic molecular candidates in the OBE model. Besides analyzing the mass
spectrum, we also give their two-body hidden-charm decay channels of these
possible molecular tetraquark candidates. In Table 9, we collect the two-body
hidden-charm decay channels for these promising charmoniumlike molecular
candidates. These obtained results can be very helpful to search and identify
these discussed molecular tetraquark candidates.

In this work, our results indicate that the underlying phenomena behind the
wide structure around 4.3 GeV in the $B\to J/\psi\omega K$ is very
complicated. There can exist several possible isoscalar charmoniumlike
molecular candidates, and the $J/\psi\omega$ channel is their two-body hidden-
charm decay channel. In addition, we can predict many isoscalar charmoniumlike
molecular states with negative $C$-parity in the same mass region, and their
two-body hidden-charm decay modes are the $\chi_{cJ}(1P)\omega$ with
$J=0,1,2$. The $B\to\chi_{cJ}(1P)\omega+K\to J/\psi\gamma\omega+K$ may be the
good production process to search for these predicted isoscalar charmoniumlike
molecular states with $C=-$.

Meanwhile, we also study the mass spectrum and give two-body hidden-charm
decay channels for the possible charmoniumlike
$\mathcal{D}_{s}\bar{\mathcal{D}_{s}}$ molecular states with hidden-strange
quantum number summarized in Table 12, where $\mathcal{D}_{s}$ stands for the
charm-strange meson. The $J/\psi\phi$ channel is the two-body hidden-charm
decay channel for most of the possible $\mathcal{D}_{s}\bar{\mathcal{D}_{s}}$
molecular states. In order to further distinguish these structures, one need
more precision experimental data, and more analysis on the other decay channel
could be also useful.

Finding the charmoniumlike states has been going on for around 20 years. With
the running of Belle II Kou:2018nap  and the accumulation of Run II and Run
III data at LHCb Bediaga:2018lhg , the study of the charmoniumlike states will
step into a new area, here, we suggest to systematically check the correlation
of charmoniumlike molecular states and charmoniumlike structures existing in
the $XYZ$ data of the $B\to XYZ+K$ decay. The present work provides a good
start point and new insight for identifying charmoniumlike molecule. We expect
more theoretical and experimental efforts in exploring this important and
intriguing research topic in the coming golden decade.

##  Appendix A Relevant subpotentials

Before presenting the effective potentials for these investigated tetraquark
systems, we first define several operators $\mathcal{O}_{k}^{(\prime)}$
involved in this work, which include

$\displaystyle\mathcal{O}_{1}$ $\displaystyle=$
$\displaystyle{\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2}},~{}~{}~{}~{}~{}~{}\mathcal{O}_{2}={\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{2}},~{}~{}~{}~{}~{}~{}\mathcal{O}_{3}=T({\bm{\epsilon}^{\dagger}_{3}},{\bm{\epsilon}_{2}}),$
$\displaystyle\mathcal{O}_{4}$ $\displaystyle=$
$\displaystyle\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{1}}\right)\left({\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2}}\right),~{}~{}~{}~{}~{}~{}\mathcal{O}_{5}=\left({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2}}\right),$
$\displaystyle\mathcal{O}_{6}$ $\displaystyle=$ $\displaystyle
T({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2}}),$
$\displaystyle\mathcal{O}_{7}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4m}}\cdot{\bm{\epsilon}_{2a}}\right)\left({\bm{\epsilon}^{\dagger}_{4n}}\cdot{\bm{\epsilon}_{2b}}\right),$
$\displaystyle\mathcal{O}_{8}$ $\displaystyle=$
$\displaystyle\frac{2}{27}\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{2a}}\right)\left({\bm{\epsilon}^{\dagger}_{3n}}\cdot{\bm{\epsilon}_{2b}}\right),$
$\displaystyle\mathcal{O}_{9}$ $\displaystyle=$
$\displaystyle\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{3m}},{\bm{\epsilon}^{\dagger}_{3n}})T({\bm{\epsilon}_{2a}},{\bm{\epsilon}_{2b}})$
$\displaystyle+\frac{2}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{3m}},{\bm{\epsilon}_{2a}})T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}_{2b}}),$
$\displaystyle\mathcal{O}_{10}$ $\displaystyle=$
$\displaystyle\frac{2}{27}\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{2a}}\right)T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}_{2b}}),$
$\displaystyle\mathcal{O}_{11}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{1}}\right)\left({\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2}}\right)+\frac{1}{3}\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}^{\dagger}_{4}}\right)\left({\bm{\epsilon}_{1}}\cdot{\bm{\epsilon}_{2}}\right),$
$\displaystyle\mathcal{O}_{12}$ $\displaystyle=$
$\displaystyle\frac{2}{3}T({\bm{\epsilon}^{\dagger}_{3}},{\bm{\epsilon}_{1}})T({\bm{\epsilon}^{\dagger}_{4}},{\bm{\epsilon}_{2}})+\frac{1}{3}T({\bm{\epsilon}^{\dagger}_{3}},{\bm{\epsilon}^{\dagger}_{4}})T({\bm{\epsilon}_{1}},{\bm{\epsilon}_{2}}),$
$\displaystyle\mathcal{O}_{13}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}^{\dagger}_{4}}\right)T({\bm{\epsilon}_{1}},{\bm{\epsilon}_{2}})+\frac{1}{6}\left({\bm{\epsilon}_{1}}\cdot{\bm{\epsilon}_{2}}\right)T({\bm{\epsilon}^{\dagger}_{3}},{\bm{\epsilon}^{\dagger}_{4}})$
$\displaystyle-\frac{1}{3}\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{1}}\right)T({\bm{\epsilon}^{\dagger}_{4}},{\bm{\epsilon}_{2}}),$
$\displaystyle\mathcal{O}_{14}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{1}}\right)\left({\bm{\epsilon}^{\dagger}_{4m}}\cdot{\bm{\epsilon}_{2a}}\right)\left({\bm{\epsilon}^{\dagger}_{4n}}\cdot{\bm{\epsilon}_{2b}}\right),$
$\displaystyle\mathcal{O}_{14}^{\prime}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2}}\right)\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{1a}}\right)\left({\bm{\epsilon}^{\dagger}_{3n}}\cdot{\bm{\epsilon}_{1b}}\right),$
$\displaystyle\mathcal{O}_{15}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4m}}\cdot{\bm{\epsilon}_{2a}}\right)\left[\left({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{4n}}\times{\bm{\epsilon}_{2b}}\right)\right],$
$\displaystyle\mathcal{O}_{15}^{\prime}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{1a}}\right)\left[\left({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{3n}}\times{\bm{\epsilon}_{1b}}\right)\right],$
$\displaystyle\mathcal{O}_{16}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4m}}\cdot{\bm{\epsilon}_{2a}}\right)T({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}^{\dagger}_{4n}}\times{\bm{\epsilon}_{2b}}),$
$\displaystyle\mathcal{O}_{16}^{\prime}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{1a}}\right)T({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2}},{\bm{\epsilon}^{\dagger}_{3n}}\times{\bm{\epsilon}_{1b}}),$
$\displaystyle\mathcal{O}_{17}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{1}}\right)\left({\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2a}}\right)\left({\bm{\epsilon}^{\dagger}_{3n}}\cdot{\bm{\epsilon}_{2b}}\right),$
$\displaystyle\mathcal{O}_{17}^{\prime}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4m}}\cdot{\bm{\epsilon}_{2}}\right)\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{1a}}\right)\left({\bm{\epsilon}^{\dagger}_{4n}}\cdot{\bm{\epsilon}_{1b}}\right),$
$\displaystyle\mathcal{O}_{18}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3m}}\cdot{\bm{\epsilon}_{1}}\right)\left({\bm{\epsilon}^{\dagger}_{4}}\cdot{\bm{\epsilon}_{2a}}\right)T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}_{2b}}),$
$\displaystyle\mathcal{O}_{18}^{\prime}$ $\displaystyle=$
$\displaystyle\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4m}}\cdot{\bm{\epsilon}_{2}}\right)\left({\bm{\epsilon}^{\dagger}_{3}}\cdot{\bm{\epsilon}_{1a}}\right)T({\bm{\epsilon}^{\dagger}_{4n}},{\bm{\epsilon}_{1b}}),$
$\displaystyle\mathcal{O}_{19}$ $\displaystyle=$
$\displaystyle\frac{1}{27}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}}\right)\right]\left({\bm{\epsilon}^{\dagger}_{3n}}\cdot{\bm{\epsilon}_{2b}}\right)$
$\displaystyle+\frac{1}{27}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}}\right)\cdot{\bm{\epsilon}_{2b}}\right]\left[{\bm{\epsilon}^{\dagger}_{3n}}\cdot\left({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}}\right)\right],$
$\displaystyle\mathcal{O}_{19}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{27}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}}\right)\right]\left({\bm{\epsilon}^{\dagger}_{4n}}\cdot{\bm{\epsilon}_{1b}}\right)$
$\displaystyle+\frac{1}{27}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}}\right)\cdot{\bm{\epsilon}_{1b}}\right]\left[{\bm{\epsilon}^{\dagger}_{4n}}\cdot\left({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}}\right)\right],$
$\displaystyle\mathcal{O}_{20}$ $\displaystyle=$
$\displaystyle\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}^{\dagger}_{3n}})T({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}},{\bm{\epsilon}_{2b}})$
$\displaystyle+\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}})T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}_{2b}})$
$\displaystyle+\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}_{2b}})T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}}),$
$\displaystyle\mathcal{O}_{20}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}},{\bm{\epsilon}^{\dagger}_{4n}})T({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}},{\bm{\epsilon}_{1b}})$
$\displaystyle+\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}},{\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}})T({\bm{\epsilon}^{\dagger}_{4n}},{\bm{\epsilon}_{1b}})$
$\displaystyle+\frac{1}{27}\mathcal{\sum}T({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}},{\bm{\epsilon}_{1b}})T({\bm{\epsilon}^{\dagger}_{4n}},{\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}}),$
$\displaystyle\mathcal{O}_{21}$ $\displaystyle=$
$\displaystyle\frac{1}{54}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}}\right)\right]T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}_{2b}})$
$\displaystyle+\frac{1}{54}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}}\right)\cdot{\bm{\epsilon}_{2b}}\right]T({\bm{\epsilon}^{\dagger}_{3n}},{\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}})$
$\displaystyle+\frac{1}{54}\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{3n}}\cdot{\bm{\epsilon}_{2b}}\right)T({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}})$
$\displaystyle+\frac{1}{54}\mathcal{\sum}\left[{\bm{\epsilon}^{\dagger}_{3n}}\cdot\left({\bm{\epsilon}^{\dagger}_{4}}\times{\bm{\epsilon}_{2a}}\right)\right]T({\bm{\epsilon}^{\dagger}_{3m}}\times{\bm{\epsilon}_{1}},{\bm{\epsilon}_{2b}}),$
$\displaystyle\mathcal{O}_{21}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{54}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}}\right)\cdot\left({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}}\right)\right]T({\bm{\epsilon}^{\dagger}_{4n}},{\bm{\epsilon}_{1b}})$
(1.1)
$\displaystyle+\frac{1}{54}\mathcal{\sum}\left[\left({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}}\right)\cdot{\bm{\epsilon}_{1b}}\right]T({\bm{\epsilon}^{\dagger}_{4n}},{\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}})$
$\displaystyle+\frac{1}{54}\mathcal{\sum}\left({\bm{\epsilon}^{\dagger}_{4n}}\cdot{\bm{\epsilon}_{1b}}\right)T({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}},{\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}})$
$\displaystyle+\frac{1}{54}\mathcal{\sum}\left[{\bm{\epsilon}^{\dagger}_{4n}}\cdot\left({\bm{\epsilon}^{\dagger}_{3}}\times{\bm{\epsilon}_{1a}}\right)\right]T({\bm{\epsilon}^{\dagger}_{4m}}\times{\bm{\epsilon}_{2}},{\bm{\epsilon}_{1b}}).$

Here, we define
$\mathcal{S}=\sum_{m,n,a,b}C^{2,m+n}_{1m,1n}C^{2,a+b}_{1a,1b}$, and
$T({\bm{x}},{\bm{y}})=3\left(\hat{\bm{r}}\cdot{\bm{x}}\right)\left(\hat{\bm{r}}\cdot{\bm{y}}\right)-{\bm{x}}\cdot{\bm{y}}$
is the tensor force operator. For these operators
$\mathcal{O}_{k}^{(\prime)}$, they should be sandwiched by the relevant spin-
orbital wave functions $|{}^{2S+1}L_{J}\rangle$ for these investigated
tetraquark systems, such as $\langle
D\bar{D}^{\ast}({}^{3}\mathbb{S}_{1})|\mathcal{O}_{1}|D\bar{D}^{\ast}({}^{3}\mathbb{S}_{1})\rangle=1$.
The obtained operator matrix elements $\mathcal{O}_{k}^{(\prime)}[J]$ are
summarized in Tables 13 and 14, which will be used in our calculations.

Table 13: The relevant operator matrix elements $\mathcal{O}_{k}^{(\prime)}[J]=\langle f|\mathcal{O}_{k}^{(\prime)}|i\rangle$. $\begin{array}[]{l}\mathcal{O}_{1}[1]=\rm{diag}(1,1)\\\ \mathcal{O}_{2}[1]=\rm{diag}(1,1)\end{array}$ $\mathcal{O}_{3}[1]=\left(\begin{array}[]{cc}0&-\sqrt{2}\\\ -\sqrt{2}&1\end{array}\right)$ |  $\begin{array}[]{l}\mathcal{O}_{4}[0]=\rm{diag}(1,1)\\\ \mathcal{O}_{5}[0]=\rm{diag}(2,-1)\end{array}$ $\mathcal{O}_{6}[0]=\left(\begin{array}[]{cc}0&\sqrt{2}\\\ \sqrt{2}&2\end{array}\right)$ | $\begin{array}[]{l}\mathcal{O}_{4}[1]=\rm{diag}(1,1,1)\\\ \mathcal{O}_{5}[1]=\rm{diag}(1,1,-1)\end{array}$
---|---|---
$\mathcal{O}_{6}[1]=\left(\begin{array}[]{ccc}0&-\sqrt{2}&0\\\ -\sqrt{2}&1&0\\\ 0&0&1\end{array}\right)$ | $\begin{array}[]{l}\mathcal{O}_{4}[2]=\rm{diag}(1,1,1,1)\\\ \mathcal{O}_{5}[2]=\rm{diag}(-1,2,1,-1)\\\ \mathcal{O}_{7}[2]=\rm{diag}(1,1)\\\ \mathcal{O}_{8}[2]=\rm{diag}(\frac{2}{27},\frac{2}{27})\end{array}$ | $\mathcal{O}_{6}[2]=\left(\begin{array}[]{cccc}0&\frac{\sqrt{2}}{\sqrt{5}}&0&-\frac{\sqrt{14}}{\sqrt{5}}\\\ \frac{\sqrt{2}}{\sqrt{5}}&0&0&-\frac{2}{\sqrt{7}}\\\ 0&0&-1&0\\\ -\frac{\sqrt{14}}{\sqrt{5}}&-\frac{2}{\sqrt{7}}&0&-\frac{3}{7}\end{array}\right)$
$\mathcal{O}_{9}[2]=\left(\begin{array}[]{cc}\frac{8}{135}&-\frac{4\sqrt{2}}{27\sqrt{35}}\\\ -\frac{4\sqrt{2}}{27\sqrt{35}}&\frac{4}{27}\end{array}\right)$ | $\mathcal{O}_{10}[2]=\left(\begin{array}[]{cc}0&-\frac{\sqrt{7}}{27\sqrt{10}}\\\ -\frac{\sqrt{7}}{27\sqrt{10}}&-\frac{1}{126}\end{array}\right)$ | $\begin{array}[]{l}\mathcal{O}_{11}[0]=\rm{diag}(\frac{2}{3},-\frac{1}{3})\\\ \mathcal{O}_{11}[1]=\rm{diag}(-\frac{1}{3},-\frac{1}{3},-\frac{1}{3})\\\ \mathcal{O}_{11}[2]=\rm{diag}(-\frac{1}{3},\frac{2}{3},-\frac{1}{3},-\frac{1}{3})\end{array}$
$\mathcal{O}_{12}[0]=\left(\begin{array}[]{cc}\frac{4}{3}&-\frac{2\sqrt{2}}{3}\\\ -\frac{2\sqrt{2}}{3}&4\end{array}\right)$ | $\mathcal{O}_{13}[0]=\left(\begin{array}[]{cc}0&\frac{\sqrt{2}}{15}\\\ -\frac{8\sqrt{2}}{15}&-\frac{1}{15}\end{array}\right)$ | $\mathcal{O}_{12}[1]=\left(\begin{array}[]{ccc}-\frac{2}{3}&-\frac{2\sqrt{2}}{3}&0\\\ -\frac{2\sqrt{2}}{3}&0&0\\\ 0&0&-\frac{4}{3}\end{array}\right)$