# Hadronic molecules in B decays

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

###### Abstract

There are eighteen possibly existing $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}D_{s}^{(*)-}$ hadronic molecular states.
We construct their corresponding interpolating currents and calculate their
decay constants using QCD sum rules. Based on these results, we calculate
their relative production rates in $B$ and $B^{*}$ decays through the current
algebra, and calculate their relative branching ratios through the Fierz
rearrangement, as summarized in Table 3. Our results support the
interpretations of the $X(3872)$, $Z_{c}(3900)^{0}$, $Z_{c}(4020)^{0}$, and
$X_{0}(2900)$ as the molecular states $D\bar{D}^{*}$ of $J^{PC}=1^{++}$,
$D\bar{D}^{*}$ of $J^{PC}=1^{+-}$, $D^{*}\bar{D}^{*}$ of $J^{PC}=1^{+-}$, and
$D^{*}\bar{K}^{*}$ of $J^{P}=0^{+}$, respectively. Our results also suggest
that the $Z_{cs}(3985)^{-}$, $Z_{cs}(4000)^{+}$, and $Z_{cs}(4220)^{+}$ are
strange partners of the $X(3872)$, $Z_{c}(3900)^{0}$, and $Z_{c}(4020)^{0}$,
respectively. In the calculations we estimate the lifetime of a weakly-coupled
composite particle $A=|BC\rangle$ to be $1/t_{A}\approx
1/t_{B}+1/t_{C}+\Gamma_{A\to BC}+\cdots$, with $\cdots$ partial widths of
other possible decay channels.

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

##  I Introduction

Since the discovery of the $X(3872)$ in 2003 by Belle Choi:2003ue , a lot of
charmonium-like $XYZ$ states were discovered in the past twenty years pdg .
These structures are good candidates of tetraquark states, which are composed
by two quarks and two antiquarks. Although there is still a long way to fully
understand how the strong interaction binds these quarks and antiquarks
together, this subject has become one of the most intriguing research topics
in particle physics. Their theoretical and experimental studies are
significantly improving our understanding of the strong interaction at the low
energy region Chen:2016qju ; Liu:2019zoy ; Lebed:2016hpi ; Esposito:2016noz ;
Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm ; Karliner:2017qhf ; Guo:2019twa ;
Brambilla:2019esw .

Among all the $XYZ$ states, the $X(3872)$, $X_{0}(2900)$, $Z_{cs}(4000)^{+}$,
and $Z_{cs}(4220)^{+}$ have been observed in $B$ decays Choi:2003ue ;
Aaij:2020hon ; Aaij:2020ypa ; Aaij:2021ivw . Besides, the $Z_{c}(3900)^{0}$
and $Z_{c}(4020)^{0}$ (may) decay into the $J/\psi\pi^{0}$ final states
Ablikim:2013mio ; Liu:2013dau ; Voloshin:2013dpa ; Chen:2019wjd , so they are
also possible to be observed in the $B^{-}\to K^{-}J/\psi\pi^{0}$ decay. In
this paper we shall investigate these six exotic structures as a whole. We
shall study their mass spectra, productions in $B$ and $B^{*}$ decays, and
decay properties. Before doing this, let us briefly review some of their
information, and we refer to the reviews Chen:2016qju ; Liu:2019zoy ;
Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm ;
Karliner:2017qhf ; Guo:2019twa ; Brambilla:2019esw  for detailed discussions.

  * •

The $X(3872)$ is the most puzzling state among all the charmonium-like $XYZ$
states. It is now denoted as the $\chi_{c1}(3872)$ in PDG2020 pdg , but the
mass of $\chi_{c1}(2P)$ was estimated to be $3.95$ GeV Godfrey:1985xj , which
value is significantly larger than the mass of $X(3872)$. Consequently,
various interpretations were proposed to explain it, such as a compact
tetraquark state Maiani:2004vq ; Maiani:2014aja ; Maiani:2020zhr ;
Hogaasen:2005jv ; Ebert:2005nc ; Barnea:2006sd ; Chiu:2006hd ;
Padmanath:2015era , a loosely-bound $D\bar{D}^{*}$ molecular state
Voloshin:1976ap ; Voloshin:2003nt ; Close:2003sg ; Wong:2003xk ;
Braaten:2003he ; Swanson:2003tb ; Tornqvist:2004qy ; Prelovsek:2013cra ;
Braaten:2019ags , and a hybrid charmonium state Close:2003mb ; Li:2004sta ,
etc. The $X(3872)$ was also studied as a conventional $c\bar{c}$ state in
Refs. Barnes:2003vb ; Eichten:2004uh ; Quigg:2004vf ; Kong:2006ni , and was
suggested to be the mixture of a $c\bar{c}$ state with the $D\bar{D}^{*}$
component in Refs. Meng:2005er ; Meng:2014ota ; Suzuki:2005ha .

The quantum number of $X(3872)$ was determined to be $I^{G}J^{PC}=0^{+}1^{++}$
Aaij:2015eva . It has been observed in the $J/\psi\pi\pi$, $J/\psi\pi\pi\pi$,
$\gamma J/\psi$, $\gamma\psi(2S)$, $\chi_{c1}\pi$, and $D^{0}\bar{D}^{*0}$
channels Gokhroo:2006bt ; Aubert:2007rva ; Adachi:2008sua ; Aubert:2008ae ;
Bhardwaj:2011dj ; Aaij:2014ala ; Ablikim:2019soz ; Li:2019kpj . Especially,
the $J/\psi\pi\pi$ and $J/\psi\pi\pi\pi$ channels have comparable branching
ratios Abe:2005ix ; delAmoSanchez:2010jr ; Choi:2011fc ; Ablikim:2019zio :

${\mathcal{B}(X(3872)\xrightarrow{~{}\,\omega\,~{}}J/\psi\pi\pi\pi)\over\mathcal{B}(X(3872)\xrightarrow{~{}~{}\rho~{}~{}}J/\psi\pi\pi)}=1.6^{+0.4}_{-0.3}\pm
0.2\,,$ (1)

which implies a large isospin violation.

  * •

In 2013 BESIII discovered the charged $Z_{c}(3900)^{\pm}$ in the $Y(4260)\to
J/\psi\pi^{+}\pi^{-}$ decay Ablikim:2013mio , which observation was confirmed
by Belle Liu:2013dau  and CLEO Xiao:2013iha . Later BESIII observed the
charged $Z_{c}(4020)^{\pm}$ in the $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ process
Ablikim:2013wzq . Because the $Z_{c}(3900)^{\pm}$ and $Z_{c}(4020)^{\pm}$ both
couple strongly to charmonia and yet they are charged, these two structures
are definitely not conventional $c\bar{c}$ states. There have been various
theoretical models developed to explain them, such as compact tetraquark
states Maiani:2004vq ; Maiani:2014aja ; Qiao:2013dda ; Wang:2016mmg ;
Azizi:2020itk , loosely-bound $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ molecular
states Voloshin:1976ap ; Khemchandani:2013iwa ; Wang:2013cya ; Guo:2013sya ;
Wilbring:2013cha ; He:2014nya ; Prelovsek:2014swa ; Chen:2015igx ;
Dong:2021juy ; Yan:2021tcp , hadro-quarkonia Braaten:2013boa ;
Dubynskiy:2008mq , or due to kinematical threshold effects Chen:2011xk ;
Liu:2013vfa ; Swanson:2014tra , etc.

The $Z_{c}(3900)$ has been observed in the $J/\psi\pi$, $h_{c}\pi$, and
$D\bar{D}^{*}$ channels Ablikim:2013mio ; Liu:2013dau ; Ablikim:2013xfr ;
Ablikim:2015swa , and the quantum number of its neutral one was determined to
be $I^{G}J^{PC}=1^{+}1^{+-}$ Collaboration:2017njt . The $Z_{c}(4020)$ has
been observed in the $h_{c}\pi$ and $D^{*}\bar{D}^{*}$ channels
Ablikim:2013wzq ; Ablikim:2013emm , and the quantum number of its neutral one
may also be $I^{G}J^{PC}=1^{+}1^{+-}$. Evidence for the
$Z_{c}(3900)^{\pm}\rightarrow\eta_{c}\rho^{\pm}$ decay was reported at
$\sqrt{s}=4.226$ GeV with the relative branching ratio Ablikim:2019ipd :

${\mathcal{B}(Z_{c}(3900)^{\pm}\rightarrow\eta_{c}\rho^{\pm})\over\mathcal{B}(Z_{c}(3900)^{\pm}\rightarrow
J/\psi\pi^{\pm})}=2.2\pm 0.9\,.$ (2)

This ratio was suggested in Ref. Esposito:2014hsa  to be useful to
discriminate the compact tetraquark and hadronic molecule scenarios, and it
has been calculated by many theoretical methods in Refs. Dias:2013xfa ;
Agaev:2016dev ; Wang:2017lot ; Goerke:2016hxf ; Ke:2013gia ; Patel:2014zja ;
Li:2014pfa ; Dong:2013iqa ; Faccini:2013lda ; Xiao:2018kfx ; Wang:2018pwi .

  * •

In 2020 BESIII reported their observation of the $Z_{cs}(3985)^{-}$ in the
$K^{+}$ recoil-mass spectra of the $e^{+}e^{-}\to
K^{+}(D^{-}_{s}D^{*0}+D^{*-}_{s}D^{0})$ process Ablikim:2020hsk . Later LHCb
reported their observation of the $Z_{cs}(4000)^{+}$ and $Z_{cs}(4220)^{+}$ in
the $J/\psi K^{+}$ invariant mass spectrum of the $B^{+}\to J/\psi\phi K^{+}$
decay Aaij:2021ivw , and one can naturally deduce the existence of the
$Z_{cs}(4000)^{-}$ and $Z_{cs}(4220)^{-}$ as their antiparticles. These
structures couple strongly to charmonia and yet they are charged, so they are
definitely exotic hadrons/structures. There have been some theoretical methods
developed to explain the $Z_{cs}(3985)^{-}$, $Z_{cs}(4000)^{-}$, and
$Z_{cs}(4220)^{-}$, such as compact tetraquark states Cui:2006mp ; Wang:2009bd
; Azizi:2020zyq ; Wang:2020iqt ; Jin:2020yjn ; Sungu:2020zvk , loosely-bound
$DD_{s}^{*-}$, $D^{*}D_{s}^{-}$, and $D^{*}D_{s}^{*-}$ molecular states
Lee:2008uy ; Meng:2020ihj ; Yang:2020nrt ; Liu:2020nge ; Du:2020vwb ;
Chen:2020yvq ; Cao:2020cfx ; Sun:2020hjw ; Ikeno:2021ptx ; Wang:2020htx ;
Guo:2020vmu , or due to kinematical threshold effects Chen:2013wca ;
Wang:2020dmv ; Wang:2020kej ; Zhu:2021vtd ; Ge:2021sdq , etc.

  * •

In 2020 LHCb reported their observation of the $X_{0}(2900)$ in the
$D^{-}K^{+}$ invariant mass spectrum of the $B^{+}\to D^{+}D^{-}K^{+}$ decay
Aaij:2020hon ; Aaij:2020ypa . This structure has the quark content
$\bar{c}\bar{s}ud$, so it is an exotic hadron with valence quarks of four
different flavors. Various theoretical methods were applied to explain it,
such as a compact tetraquark state Karliner:2020vsi ; He:2020jna ; Lu:2020qmp
; Wang:2020xyc ; Wang:2020prk ; Yang:2021izl , a loosely-bound
$D^{*}\bar{K}^{*}$ molecular state Molina:2010tx ; Molina:2020hde ;
Liu:2020nil ; Huang:2020ptc ; Hu:2020mxp ; Xiao:2020ltm ; Tan:2020cpu ;
Abreu:2020ony ; Qi:2021iyv , or due to triangle singularities Liu:2020orv ,
etc.

Summarizing the above discussions, we quickly notice that the $X(3872)$,
$Z_{c}(3900)^{0}$, $Z_{c}(4020)^{0}$, and $X_{0}(2900)$ can be interpreted in
the hadronic molecular picture as the molecular states $D\bar{D}^{*}$ of
$J^{PC}=1^{++}$, $D\bar{D}^{*}$ of $J^{PC}=1^{+-}$, $D^{*}\bar{D}^{*}$ of
$J^{PC}=1^{+-}$, and $D^{*}\bar{K}^{*}$ of $J^{P}=0^{+}$, respectively; the
$Z_{cs}(3985)^{-}/Z_{cs}(4000)^{-}$ and $Z_{cs}(4220)^{-}$ can be interpreted
in the hadronic molecular picture as the molecular states
$DD_{s}^{*-}/D^{*}D_{s}^{-}$ of $J^{P}=1^{+}$ and $D^{*}D_{s}^{*-}$ of
$J^{PC}=1^{+}$. Besides, there may exist the molecular states $D\bar{D}$ of
$J^{PC}=0^{++}$, $D^{*}\bar{D}^{*}$ of $J^{PC}=0^{++}/2^{++}$, $D\bar{K}$ of
$J^{P}=0^{+}$, $D\bar{K}^{*}/D^{*}\bar{K}$ of $J^{P}=1^{+}$,
$D^{*}\bar{K}^{*}$ of $J^{P}=1^{+}/2^{+}$, $DD_{s}^{-}$ of $J^{P}=0^{+}$, and
$D^{*}D_{s}^{*-}$ of $J^{P}=0^{+}/2^{+}$.

Altogether there are eighteen possibly existing $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}D^{(*)-}$ molecular states. We use the
symbol

$|D^{(*)}\bar{D}^{(*)}/D^{(*)}\bar{K}^{(*)}/D^{(*)}\bar{D}_{s}^{(*)};J^{P(C)}\rangle\,,$
(3)

to denote them, where $\bar{D}_{s}^{(*)}$ is used to denote $D_{s}^{(*)-}$. To
make this paper more understandable, we further simply our notations as
follows:

  * •

We shall not differentiate the charged $Z_{c}(3900)^{\pm}$ and the neutral
$Z_{c}(3900)^{0}$. They are both denoted as the $Z_{c}(3900)$, which is
interpreted as the $D\bar{D}^{*}$ molecular state of $J^{PC}=1^{+-}$ in the
present study. So does the $Z_{c}(4020)$, which is interpreted as the
$D^{*}\bar{D}^{*}$ molecular state of $J^{PC}=1^{+-}$. Note that their
negative charge-conjugation parity $C=-$ is only due to the neutral ones,
while the charged ones are not charge-conjugated.

  * •

We shall also denote the $Z_{cs}(3985)^{-}$, $Z_{cs}(4000)^{\pm}$, and
$Z_{cs}(4220)^{\pm}$ as the $Z_{cs}(3985)$, $Z_{cs}(4000)$, and
$Z_{cs}(4220)$, respectively. Both the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ can
be interpreted as the $DD_{s}^{*-}/D^{*}D_{s}^{-}$ molecular states of
$J^{P}=1^{+}$, and in the present study we shall further consider the mixing
between the $DD_{s}^{*-}$ and $D^{*}D_{s}^{-}$ components, since their
thresholds are quite close to each other. By doing this we can obtain strange
partners of the $|D\bar{D}^{*};1^{++}\rangle$ and
$|D\bar{D}^{*};1^{+-}\rangle$, which are denoted for convenience as
$|D\bar{D}_{s}^{*};1^{++}\rangle$ and $|D\bar{D}_{s}^{*};1^{+-}\rangle$,
respectively. We shall use them to separately explain the $Z_{cs}(3985)$ and
$Z_{cs}(4000)$, but note that they are not charge-conjugated actually. See
Sec. II.3 for detailed discussions.

The above notations can be used to well differentiate the $X(3872)$,
$Z_{c}(3900)$, and $Z_{c}(4020)$ as well as the $Z_{cs}(3985)$,
$Z_{cs}(4000)$, and $Z_{cs}(4220)$. We refer to Refs. Gamermann:2006nm ;
Dai:2015bcc ; Mutuk:2018zxs ; Wu:2019vbk ; Cao:2020dlz ; Dong:2020hxe ;
Prelovsek:2020eiw ; Chen:2020eyu ; Wang:2020dgr ; Burns:2020xne ; Wei:2021usz
; Ozdem:2021yvo ; Yang:2021sue  for some relevant theoretical studies, and to
the reviews Chen:2016qju ; Liu:2019zoy ; Lebed:2016hpi ; Esposito:2016noz ;
Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm ; Karliner:2017qhf ; Guo:2019twa ;
Brambilla:2019esw  again for their detailed discussions.

In this paper we shall systematically study the eighteen possibly existing
$D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$
hadronic molecular states. We shall systematically construct their
corresponding interpolating currents, and apply the method of QCD sum rules to
calculate their decay constants. The obtained results will be used to further
study their production and decay properties. This method has been applied to
systematically investigate $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ hadronic molecular
states in Ref. Chen:2020opr .

We use the isoscalar $D\bar{D}$ molecular state of $J^{PC}=0^{++}$,
$|D\bar{D};0^{++}\rangle$, as an example to briefly show our procedures.
Firstly, we construct its corresponding hidden-charm tetraquark current:

$\eta_{1}(x)=\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma_{5}q_{b}(x)\,,$
(4)

which is a local meson-meson current coupling to $|D\bar{D};0^{++}\rangle$
through

$\langle 0|\eta_{1}|D\bar{D};0^{++}\rangle=f_{|D\bar{D};0^{++}\rangle}\,.$ (5)

In principle, one needs to explicitly use the non-local current in order to
exactly describe the $D\bar{D}$ molecular state, but this can not be done yet
within the present QCD sum rule framework. In the above expressions, $a$ and
$b$ are color indices; $q(x)$ is an $up$ or $down$ quark field, and $c(x)$ is
a $charm$ quark field; the decay constant $f_{|D\bar{D};0^{++}\rangle}$ can be
calculated using QCD sum rules, which is an important input parameter when
studying production and decay properties of $|D\bar{D};0^{++}\rangle$.

Secondly, we investigate the three-body $B^{-}\to K^{-}D^{0}\bar{D}^{0}$
decay, where $|D\bar{D};0^{++}\rangle$ can be produced. The total quark
content of the final states is $\bar{u}c\bar{c}s\bar{u}u$. We apply the Fierz
rearrangement to carefully examine the combination of these six quarks, from
which we select the current $\eta_{1}$, so that the relative production rate
of $|D\bar{D};0^{++}\rangle$ can be estimated.

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

$\eta_{1}\rightarrow-{1\over
12}~{}\bar{q}_{a}\gamma_{5}q_{a}~{}\bar{c}_{b}\gamma_{5}c_{b}+{1\over
12}~{}\bar{q}_{a}\gamma_{\mu}q_{a}~{}\bar{c}_{b}\gamma^{\mu}c_{b}+\cdots\,.$
(6)

Accordingly, $\eta_{1}$ couples to the $\eta_{c}\eta$ and $J/\psi\omega$
channels simultaneously:

$\displaystyle\langle 0|\eta_{1}|\eta_{c}\eta\rangle$ $\displaystyle\approx$
$\displaystyle-{1\over 12}~{}\langle
0|\bar{q}_{a}\gamma_{5}q_{a}|\eta\rangle~{}\langle
0|\bar{c}_{b}\gamma_{5}c_{b}|\eta_{c}\rangle+\cdots\,,$ $\displaystyle\langle
0|\eta_{1}|J/\psi\omega\rangle$ $\displaystyle\approx$ $\displaystyle{1\over
12}~{}\langle 0|\bar{q}_{a}\gamma_{\mu}q_{a}|\omega\rangle~{}\langle
0|\bar{c}_{b}\gamma^{\mu}c_{b}|J/\psi\rangle+\cdots\,.$

We can use these two equations to straightforwardly calculate the relative
branching ratio of the $|D\bar{D};0^{++}\rangle$ decay into $\eta_{c}\eta$ to
its decay into $J/\psi\omega$.

This paper is organized as follows. In Sec. II we systematically construct
hidden-charm tetraquark currents corresponding to the $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular
states. We use them to perform QCD sum rule analyses and calculate their decay
constants. The obtained results are used in Sec. III to study the productions
of the $D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and
$D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular states in $B$ and $B^{*}$ decays
through the current algebra. In Sec. IV we use the Fierz rearrangement of the
Dirac and color indices to study their decay properties, and calculate some of
their relative branching ratios. The obtained results are summarized and
discussed in Sec. V. This paper has a supplementary file “OPE.nb” containing
all the spectral densities.

##  II Interpolating currents

In this section we systematically construct (strange) hidden-charm and open-
charm tetraquark interpolating currents corresponding to
$D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$
hadronic molecular states. We separately construct them in the following
subsections.

The isospin quantum number of these molecular states is important. In the
present study,

  * •

We assume that $|D\bar{D};0^{++}\rangle$, $|D^{*}\bar{D}^{*};0^{++}\rangle$,
and $|D^{*}\bar{D}^{*};2^{++}\rangle$ have $I=0$.

  * •

The isospin of the $X(3872)$ as $|D\bar{D}^{*};1^{++}\rangle$ will be
separately investigated in Sec. IV.1.4. Before that we simply assume it also
has $I=0$.

  * •

We assume that $|D\bar{D}^{*};1^{+-}\rangle$ and
$|D^{*}\bar{D}^{*};1^{+-}\rangle$ have $I=1$.

  * •

We assume that all the $D^{(*)}\bar{K}^{(*)}$ molecular states have $I=0$.

  * •

All the $D^{(*)}\bar{D}_{s}^{(*)}$ molecular states have $I=1/2$.

Keeping this in mind, we shall use $q$ to denote either the $up$ or $down$
quark, so that we do not need to explicitly write the isospin contents out.

###  II.1 $D^{(*)}\bar{D}^{(*)}$ currents

In this subsection we use the $\bar{c}$, $c$, $\bar{q}$, and $q$ ($q=u/d$)
quarks to construct hidden-charm tetraquark interpolating currents. We
consider two types of currents, as illustrated in Fig. 1:

$\displaystyle\theta(x)$ $\displaystyle=$
$\displaystyle[\bar{q}_{a}(x)\Gamma^{\theta}_{1}q_{b}(x)]~{}[\bar{c}_{c}(x)\Gamma^{\theta}_{2}c_{d}(x)]\,,$
(8) $\displaystyle\eta(x)$ $\displaystyle=$
$\displaystyle[\bar{q}_{a}(x)\Gamma^{\eta}_{1}c_{b}(x)]~{}[\bar{c}_{c}(x)\Gamma^{\eta}_{2}q_{d}(x)]\,,$
(9)

where $a\cdots d$ are color indices and $\Gamma_{1/2}^{\theta/\eta}$ are Dirac
matrices. We can relate these two configurations through the Fierz
rearrangement in the Lorentz space and the color rearrangement in the color
space:

$\delta_{ab}\delta_{cd}={1\over 3}\delta_{ad}\delta_{cb}+{1\over
2}\lambda^{n}_{ad}\lambda^{n}_{cb}\,.$ (10)

We shall detailedly discuss this rearrangement in Sec. IV.1 when investigating
decay properties of $D^{(*)}\bar{D}^{(*)}$ molecular states.

(a) $\theta(x)$ currents (b) $\eta(x)$ currents

Figure 1: Two types of hidden-charm tetraquark currents, $\theta(x)$ and
$\eta(x)$. Quarks are shown in red/green/blue color, and antiquarks are shown
in cyan/magenta/yellow color.

There can exist altogether six $D^{(*)}\bar{D}^{(*)}$ hadronic molecular
states:

$\displaystyle|D\bar{D};0^{++}\rangle$ $\displaystyle=$
$\displaystyle|D\bar{D}\rangle_{J=0}\,,$ (11)
$\displaystyle\sqrt{2}|D\bar{D}^{*};1^{++}\rangle$ $\displaystyle=$
$\displaystyle|D\bar{D}^{*}\rangle_{J=1}+|D^{*}\bar{D}\rangle_{J=1}\,,$ (12)
$\displaystyle\sqrt{2}|D\bar{D}^{*};1^{+-}\rangle$ $\displaystyle=$
$\displaystyle|D\bar{D}^{*}\rangle_{J=1}-|D^{*}\bar{D}\rangle_{J=1}\,,$ (13)
$\displaystyle|D^{*}\bar{D}^{*};0^{++}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{D}^{*}\rangle_{J=0}\,,$ (14)
$\displaystyle|D^{*}\bar{D}^{*};1^{+-}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{D}^{*}\rangle_{J=1}\,,$ (15)
$\displaystyle|D^{*}\bar{D}^{*};2^{++}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{D}^{*}\rangle_{J=2}\,.$ (16)

Their corresponding currents are:

$\displaystyle\eta_{1}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma_{5}q_{b}(x)\,,$
(17) $\displaystyle\eta_{2}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma^{\alpha}q_{b}(x)$
(18)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\\{\gamma_{5}\leftrightarrow\gamma^{\alpha}\\}\,,$
$\displaystyle\eta_{3}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma^{\alpha}q_{b}(x)$
(19)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\\{\gamma_{5}\leftrightarrow\gamma^{\alpha}\\}\,,$
$\displaystyle\eta_{4}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma^{\mu}c_{a}(x)~{}\bar{c}_{b}(x)\gamma_{\mu}q_{b}(x)\,,$
(20) $\displaystyle\eta_{5}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{\mu}c_{a}(x)~{}\bar{c}_{b}(x)\sigma^{\alpha\mu}\gamma_{5}q_{b}(x)$
(21)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\\{\gamma_{\mu}\leftrightarrow\sigma^{\alpha\mu}\gamma_{5}\\}\,,$
$\displaystyle\eta_{6}^{\alpha\beta}(x)$ $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\mu\nu}\bar{q}_{a}(x)\gamma_{\mu}c_{a}(x)\bar{c}_{b}(x)\gamma_{\nu}q_{b}(x)\,,$
(22)

where $P^{\alpha\beta,\mu\nu}$ is the spin-2 projection operator

$P^{\alpha\beta,\mu\nu}={1\over 2}g^{\alpha\mu}g^{\beta\nu}+{1\over
2}g^{\alpha\nu}g^{\beta\mu}-{1\over 4}g^{\alpha\beta}g^{\mu\nu}\,.$ (23)

In the above expressions we have used the tensor field
$\bar{c}_{b}\sigma_{\mu\nu}\gamma_{5}q_{b}$ ($\mu,\nu=0\cdots 3$) to construct
the current $\eta_{5}^{\alpha}(x)$ of $J^{PC}=1^{+-}$. In principle, this
tensor field contains both $J^{P}=1^{+}$ and $1^{-}$ components, but its
negative-parity component $\bar{c}_{b}\sigma_{ij}\gamma_{5}q_{b}$
($i,j=1,2,3$) gives the dominant contribution to $\eta_{5}^{\alpha}(x)$.
Hence, the tetraquark current $\eta_{5}^{\alpha}(x)$ of $J^{PC}=1^{+-}$
corresponds to $|D^{*}\bar{D}^{*};1^{+-}\rangle$. Beside it, there exists
another current directly corresponding to $|D^{*}\bar{D}^{*};1^{+-}\rangle$:

$\eta_{7}^{\alpha\beta}(x)=\bar{q}_{a}(x)\gamma^{\alpha}c_{a}(x)~{}\bar{c}_{b}(x)\gamma^{\beta}q_{b}(x)-\\{\alpha\leftrightarrow\beta\\}\,,$
(24)

but this current itself contains both positive- and negative-parity
components, so we do not use it in the present study.

We use $X$ to generally denote the $D^{(*)}\bar{D}^{(*)}$ molecular states,
and assume that the currents $\eta_{1\cdots 6}^{\alpha_{1}\cdots\alpha_{J}}$
of spin-$J$ couple to them through

$\langle
0|\eta^{\alpha_{1}\cdots\alpha_{J}}|X\rangle=f_{X}\epsilon^{\alpha_{1}\cdots\alpha_{J}}\,,$
(25)

where $f_{X}$ is the decay constant, and
$\epsilon^{\alpha_{1}\cdots\alpha_{J}}$ is the traceless and symmetric
polarization tensor.

We apply the method of QCD sum rules Shifman:1978bx ; Reinders:1984sr  to
study the $D^{(*)}\bar{D}^{(*)}$ hadronic molecular states through the
currents $\eta_{1\cdots 6}$. We calculate their spectral densities using the
method of operator product expansion (OPE) at the leading order of
$\alpha_{s}$ and up to the $D({\rm imension})=8$ terms. In the calculations we
calculate the perturbative term, the quark condensates
$\langle\bar{q}q\rangle$/$\langle\bar{s}s\rangle$, the gluon condensate
$\langle g_{s}^{2}GG\rangle$, the quark-gluon mixed condensates $\langle
g_{s}\bar{q}\sigma Gq\rangle$/$\langle g_{s}\bar{s}\sigma Gs\rangle$, and
their combinations $\langle\bar{q}q\rangle^{2}$,
$\langle\bar{q}q\rangle\langle\bar{s}s\rangle$, $\langle\bar{q}q\rangle\langle
g_{s}\bar{q}\sigma Gq\rangle$, $\langle\bar{q}q\rangle\langle
g_{s}\bar{s}\sigma Gs\rangle$, and $\langle\bar{s}s\rangle\langle
g_{s}\bar{q}\sigma Gq\rangle$. We take into account the $charm$ and $strange$
quark masses, but ignore the $up$ and $down$ quark masses. We adopt the
factorization assumption of vacuum saturation for higher dimensional
condensates. The $D=3$ quark condensates
$\langle\bar{q}q\rangle$/$\langle\bar{s}s\rangle$ and the $D=5$ mixed
condensates $\langle g_{s}\bar{q}\sigma Gq\rangle$/$\langle g_{s}\bar{s}\sigma
Gs\rangle$ are both multiplied by the charm quark mass $m_{c}$, which are thus
important power corrections.

The obtained spectral densities are too length, so we list them in the
supplementary file “OPE.nb”. In the calculations we use the following values
for various QCD sum rule parameters pdg ; Yang:1993bp ; Ellis:1996xc ;
Eidemuller:2000rc ; Narison:2002pw ; Gimenez:2005nt ; Jamin:2002ev ;
Ioffe:2002be ; Ovchinnikov:1988gk ; colangelo :

$\displaystyle m_{s}$ $\displaystyle=$ $\displaystyle 96^{+8}_{-4}\mbox{
MeV}\,,$ $\displaystyle m_{c}$ $\displaystyle=$ $\displaystyle
1.275^{+0.025}_{-0.035}\mbox{ GeV}\,,$ $\displaystyle\langle\bar{q}q\rangle$
$\displaystyle=$ $\displaystyle-(0.240\pm 0.010)^{3}\mbox{ GeV}^{3}\,,$
$\displaystyle\langle\bar{s}s\rangle$ $\displaystyle=$ $\displaystyle(0.8\pm
0.1)\times\langle\bar{q}q\rangle\,,$ (26) $\displaystyle\langle
g_{s}^{2}GG\rangle$ $\displaystyle=$ $\displaystyle 0.48\pm 0.14\mbox{
GeV}^{4}\,,$ $\displaystyle\langle g_{s}\bar{q}\sigma Gq\rangle$
$\displaystyle=$ $\displaystyle-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$
$\displaystyle\langle g_{s}\bar{s}\sigma Gs\rangle$ $\displaystyle=$
$\displaystyle-M_{0}^{2}\times\langle\bar{s}s\rangle\,,$ $\displaystyle
M_{0}^{2}$ $\displaystyle=$ $\displaystyle 0.8\pm 0.2\mbox{ GeV}^{2}\,,$

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

Based on these spectral densities, we calculate masses and decay constants of
the $D^{(*)}\bar{D}^{(*)}$ hadronic molecular states. The results are
summarized in Table 1, supporting the interpretations of the $X(3872)$,
$Z_{c}(3900)$, and $Z_{c}(4020)$ as the $|D\bar{D}^{*};1^{++}\rangle$,
$|D\bar{D}^{*};1^{+-}\rangle$, and $|D^{*}\bar{D}^{*};1^{+-}\rangle$ molecular
states, respectively. The accuracy of our QCD sum rule results is moderate but
not enough to extract their binding energies. Therefore, our results can only
suggest but not determine: a) whether these $D^{(*)}\bar{D}^{(*)}$ molecular
states exist or not, and b) whether they are bound states or resonance states.
Note that we have not taken into account the radiative corrections in our QCD
sum rule calculations, which lead to even more theoretical uncertainties.
However, in the present study we are more concerned about the ratios, i.e.,
the relative production rates and the relative branching ratios, whose
uncertainties are significantly reduced. We refer to Refs. Navarra:2006nd ;
Matheus:2006xi ; Zhang:2009em ; Chen:2010ze ; Wang:2013vex ; Wang:2013daa ;
Cui:2013yva ; Zhang:2013aoa ; Chen:2015ata ; Chen:2015moa ; Chen:2016otp ;
Azizi:2017ubq ; Ozdem:2017jqh ; Chen:2019osl ; Wang:2019hnw ; Xu:2020qtg ;
Zhang:2020oze ; Chen:2020aos ; Agaev:2020nrc ; Albuquerque:2020ugi ;
Albuquerque:2021tqd ; Wan:2020oxt ; Wang:2020rcx  for more QCD sum rule
studies.

Table 1: Masses and decay constants of $D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular states, extracted from the currents $\eta_{1\cdots 6}$, $\xi_{1\cdots 6}$, and $\zeta_{1\cdots 6}$, respectively. Currents |  Configuration |  $s_{0}^{min}~{}[{\rm GeV}^{2}]$ | Working Regions |  Pole [%] |  Mass [GeV] |  $f_{X}$ [GeV5] |  Candidate
---|---|---|---|---|---|---|---
$s_{0}~{}[{\rm GeV}^{2}]$ |  $M_{B}^{2}~{}[{\rm GeV}^{2}]$
$\eta_{1}$ | $|D\bar{D};0^{++}\rangle$ | 16.6 | $18.0\pm 1.0$ |  $2.77$–$3.08$ |  $40$–$50$ | $3.73^{+0.08}_{-0.08}$ | $\left(1.42^{+0.28}_{-0.25}\right)\times 10^{-2}$ |
$\eta_{2}^{\alpha}$ | $|D\bar{D}^{*};1^{++}\rangle$ | 17.3 | $19.0\pm 1.0$ |  $2.77$–$3.15$ |  $40$–$51$ | $3.87^{+0.08}_{-0.09}$ | $\left(2.16^{+0.40}_{-0.36}\right)\times 10^{-2}$ | $X(3872)$
$\eta_{3}^{\alpha}$ | $|D\bar{D}^{*};1^{+-}\rangle$ | 17.3 | $19.0\pm 1.0$ |  $2.77$–$3.15$ |  $40$–$51$ | $3.87^{+0.08}_{-0.09}$ | $\left(2.16^{+0.40}_{-0.36}\right)\times 10^{-2}$ | $Z_{c}(3900)$
$\eta_{4}$ | $|D^{*}\bar{D}^{*};0^{++}\rangle$ | 18.7 | $20.0\pm 1.0$ |  $2.72$–$3.00$ |  $40$–$49$ | $4.01^{+0.11}_{-0.11}$ | $\left(3.07^{+0.58}_{-0.52}\right)\times 10^{-2}$ |
$\eta_{5}^{\alpha}$ | $|D^{*}\bar{D}^{*};1^{+-}\rangle$ | 19.6 | $20.0\pm 1.0$ |  $3.05$–$3.14$ |  $40$–$42$ | $4.06^{+0.14}_{-0.13}$ | $\left(4.96^{+0.89}_{-0.80}\right)\times 10^{-2}$ | $Z_{c}(4020)$
$\eta_{6}^{\alpha\beta}$ | $|D^{*}\bar{D}^{*};2^{++}\rangle$ | 17.9 | $20.0\pm 1.0$ |  $2.68$–$3.20$ |  $40$–$55$ | $4.04^{+0.13}_{-0.12}$ | $\left(1.71^{+0.30}_{-0.27}\right)\times 10^{-2}$ |
$\xi_{1}$ | $|D\bar{K};0^{+}\rangle$ | 9.2 | $10.0\pm 1.0$ |  $2.08$–$2.26$ |  $40$–$47$ | $2.72^{+0.10}_{-0.10}$ | $\left(0.77^{+0.18}_{-0.17}\right)\times 10^{-2}$ |
$\xi_{2}^{\alpha}$ | $|D\bar{K}^{*};1^{+}\rangle$ | 10.4 | $11.0\pm 1.0$ |  $2.24$–$2.37$ |  $40$–$45$ | $2.89^{+0.10}_{-0.11}$ | $\left(0.86^{+0.19}_{-0.17}\right)\times 10^{-2}$ |
$\xi_{3}^{\alpha}$ | $|D^{*}\bar{K};1^{+}\rangle$ | 10.0 | $11.0\pm 1.0$ |  $2.13$–$2.35$ |  $40$–$49$ | $2.85^{+0.10}_{-0.11}$ | $\left(0.82^{+0.18}_{-0.17}\right)\times 10^{-2}$ |
$\xi_{4}$ | $|D^{*}\bar{K}^{*};0^{+}\rangle$ | 10.6 | $11.5\pm 1.0$ |  $2.05$–$2.23$ |  $40$–$48$ | $2.91^{+0.10}_{-0.11}$ | $\left(1.29^{+0.31}_{-0.28}\right)\times 10^{-2}$ | $X_{0}(2900)$
$\xi_{5}^{\alpha}$ | $|D^{*}\bar{K}^{*};1^{+}\rangle$ | 12.0 | $12.5\pm 1.0$ |  $2.46$–$2.58$ |  $40$–$44$ | $3.13^{+0.15}_{-0.12}$ | $\left(2.96^{+0.62}_{-0.55}\right)\times 10^{-2}$ |
$\xi_{6}^{\alpha\beta}$ | $|D^{*}\bar{K}^{*};2^{+}\rangle$ | 11.8 | $12.5\pm 1.0$ |  $2.40$–$2.56$ |  $40$–$45$ | $3.09^{+0.12}_{-0.11}$ | $\left(1.10^{+0.23}_{-0.21}\right)\times 10^{-2}$ |
$\zeta_{1}$ | $|D\bar{D}_{s};0^{+}\rangle$ | 18.4 | $19.0\pm 1.0$ |  $3.08$–$3.22$ |  $40$–$44$ | $3.86^{+0.07}_{-0.08}$ | $\left(1.74^{+0.32}_{-0.29}\right)\times 10^{-2}$ |
$\zeta_{2}^{\alpha}$ | $|D\bar{D}_{s}^{*};1^{++}\rangle$ | 19.3 | $20.0\pm 1.0$ |  $3.14$–$3.31$ |  $40$–$45$ | $3.99^{+0.08}_{-0.08}$ | $\left(2.65^{+0.46}_{-0.42}\right)\times 10^{-2}$ | $Z_{cs}(3985)$
$\zeta_{3}^{\alpha}$ | $|D\bar{D}_{s}^{*};1^{+-}\rangle$ | 19.3 | $20.0\pm 1.0$ |  $3.14$–$3.31$ |  $40$–$45$ | $3.99^{+0.08}_{-0.08}$ | $\left(2.65^{+0.46}_{-0.42}\right)\times 10^{-2}$ | $Z_{cs}(4000)$
$\zeta_{4}$ | $|D^{*}\bar{D}_{s}^{*};0^{+}\rangle$ | 21.1 | $22.0\pm 1.0$ |  $3.20$–$3.38$ |  $40$–$45$ | $4.20^{+0.08}_{-0.08}$ | $\left(4.44^{+0.77}_{-0.69}\right)\times 10^{-2}$ |
$\zeta_{5}^{\alpha}$ | $|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$ | 22.6 | $22.0\pm 1.0$ | $\sim 3.69$ | $\sim 37$ | $4.22^{+0.09}_{-0.09}$ | $\left(6.96^{+1.15}_{-1.03}\right)\times 10^{-2}$ | $Z_{cs}(4220)$
$\zeta_{6}^{\alpha\beta}$ | $|D^{*}\bar{D}_{s}^{*};2^{+}\rangle$ | 20.0 | $22.0\pm 1.0$ |  $3.16$–$3.62$ |  $40$–$52$ | $4.20^{+0.09}_{-0.10}$ | $\left(2.39^{+0.36}_{-0.34}\right)\times 10^{-2}$ |

###  II.2 $D^{(*)}\bar{K}^{(*)}$ currents

In this subsection we use the $c$, $s$, $\bar{q}$, and $\bar{q}$ ($q=u/d$)
quarks to construct open-charm tetraquark interpolating currents. We consider
the following type of currents, as illustrated in Fig. 2:

$\xi(x)=[\bar{q}_{a}(x)\Gamma^{\xi}_{1}c_{b}(x)]~{}[\bar{q}_{c}(x)\Gamma^{\xi}_{2}s_{d}(x)]\,,$
(27)

where $\Gamma_{1/2}^{\xi}$ are Dirac matrices. We shall use the Fierz
rearrangement to study this configuration in Sec. IV.2.

Figure 2: Open-charm tetraquark currents $\xi(x)$.

There can exist altogether six $D^{(*)}\bar{K}^{(*)}$ hadronic molecular
states:

$\displaystyle|D\bar{K};0^{+}\rangle$ $\displaystyle=$
$\displaystyle|D\bar{K}\rangle_{J=0}\,,$ (28)
$\displaystyle|D\bar{K}^{*};1^{+}\rangle$ $\displaystyle=$
$\displaystyle|D\bar{K}^{*}\rangle_{J=1}\,,$ (29)
$\displaystyle|D^{*}\bar{K};1^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{K}\rangle_{J=1}\,,$ (30)
$\displaystyle|D^{*}\bar{K}^{*};0^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{K}^{*}\rangle_{J=0}\,,$ (31)
$\displaystyle|D^{*}\bar{K}^{*};1^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{K}^{*}\rangle_{J=1}\,,$ (32)
$\displaystyle|D^{*}\bar{K}^{*};2^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}\bar{K}^{*}\rangle_{J=2}\,.$ (33)

Their corresponding currents are:

$\displaystyle\xi_{1}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{q}_{b}(x)\gamma_{5}s_{b}(x)\,,$
(34) $\displaystyle\xi_{2}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{q}_{b}(x)\gamma^{\alpha}s_{b}(x)\,,$
(35) $\displaystyle\xi_{3}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma^{\alpha}c_{a}(x)~{}\bar{q}_{b}(x)\gamma_{5}s_{b}(x)\,,$
(36) $\displaystyle\xi_{4}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma^{\mu}c_{a}(x)~{}\bar{q}_{b}(x)\gamma_{\mu}s_{b}(x)\,,$
(37) $\displaystyle\xi_{5}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{\mu}c_{a}(x)~{}\bar{q}_{b}(x)\sigma^{\alpha\mu}\gamma_{5}s_{b}(x)$
(38)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\\{\gamma_{\mu}\leftrightarrow\sigma^{\alpha\mu}\gamma_{5}\\}\,,$
$\displaystyle\xi_{6}^{\alpha\beta}(x)$ $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\mu\nu}\bar{q}_{a}(x)\gamma_{\mu}c_{a}(x)\bar{q}_{b}(x)\gamma_{\nu}s_{b}(x)\,.$
(39)

We use the currents $\xi_{1\cdots 6}$ to perform numerical analyses, and
calculate masses and decay constants of $D^{(*)}\bar{K}^{(*)}$ molecular
states. The obtained results are summarized in Table 1. Our results support
the interpretation of the $X_{0}(2900)$ as the
$|D^{*}\bar{K}^{*};0^{+}\rangle$ molecular state. However, masses of
$|D\bar{K};0^{+}\rangle$ and $|D^{*}\bar{K};1^{+}\rangle$ are significantly
larger than the $D\bar{K}$ and $D^{*}\bar{K}$ thresholds at about 2360 MeV and
2500 MeV, respectively. This diversity may be due to the nature of $K$ mesons
as Nambu-Goldstone bosons. Again, the accuracy of our QCD sum rule results is
moderate but not enough to determine: a) whether these $D^{(*)}\bar{K}^{(*)}$
molecular states exist or not, and b) whether they are bound states or
resonance states.

###  II.3 $D^{(*)}\bar{D}_{s}^{(*)}$ currents

In this subsection we use the $c$, $\bar{c}$, $s$, and $\bar{q}$ ($q=u/d$)
quarks to construct strange hidden-charm tetraquark interpolating currents. We
consider the following type of currents:

$\zeta(x)=[\bar{q}_{a}(x)\Gamma^{\zeta}_{1}c_{b}(x)]~{}[\bar{c}_{c}(x)\Gamma^{\zeta}_{2}s_{d}(x)]\,,$
(40)

where $\Gamma_{1/2}^{\zeta}$ are Dirac matrices. Their Fierz rearrangements
are quite similar to those for $\eta(x)$, just with one light $up$/$down$
quark replaced by another $strange$ quark.

There can exist altogether six $D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular
states:

$\displaystyle|D\bar{D}_{s};0^{+}\rangle$ $\displaystyle=$
$\displaystyle|DD_{s}^{-}\rangle_{J=0}\,,$ (41)
$\displaystyle\sqrt{2}|D\bar{D}_{s}^{*};1^{++}\rangle$ $\displaystyle=$
$\displaystyle|DD_{s}^{*-}\rangle_{J=1}+|D^{*}D_{s}^{-}\rangle_{J=1}\,,$ (42)
$\displaystyle\sqrt{2}|D\bar{D}_{s}^{*};1^{+-}\rangle$ $\displaystyle=$
$\displaystyle|DD_{s}^{*-}\rangle_{J=1}-|D^{*}D_{s}^{-}\rangle_{J=1}\,,$ (43)
$\displaystyle|D^{*}\bar{D}_{s}^{*};0^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}D_{s}^{*-}\rangle_{J=0}\,,$ (44)
$\displaystyle|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}D_{s}^{*-}\rangle_{J=1}\,,$ (45)
$\displaystyle|D^{*}\bar{D}_{s}^{*};2^{+}\rangle$ $\displaystyle=$
$\displaystyle|D^{*}D_{s}^{*-}\rangle_{J=2}\,.$ (46)

Their corresponding currents are:

$\displaystyle\zeta_{1}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma_{5}s_{b}(x)\,,$
(47) $\displaystyle\zeta_{2}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma^{\alpha}s_{b}(x)$
(48)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\\{\gamma_{5}\leftrightarrow\gamma^{\alpha}\\}\,,$
$\displaystyle\zeta_{3}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{5}c_{a}(x)~{}\bar{c}_{b}(x)\gamma^{\alpha}s_{b}(x)$
(49)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\\{\gamma_{5}\leftrightarrow\gamma^{\alpha}\\}\,,$
$\displaystyle\zeta_{4}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma^{\mu}c_{a}(x)~{}\bar{c}_{b}(x)\gamma_{\mu}s_{b}(x)\,,$
(50) $\displaystyle\zeta_{5}^{\alpha}(x)$ $\displaystyle=$
$\displaystyle\bar{q}_{a}(x)\gamma_{\mu}c_{a}(x)~{}\bar{c}_{b}(x)\sigma^{\alpha\mu}\gamma_{5}s_{b}(x)$
(51)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\\{\gamma_{\mu}\leftrightarrow\sigma^{\alpha\mu}\gamma_{5}\\}\,,$
$\displaystyle\zeta_{6}^{\alpha\beta}(x)$ $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\mu\nu}\bar{q}_{a}(x)\gamma_{\mu}c_{a}(x)\bar{c}_{b}(x)\gamma_{\nu}s_{b}(x)\,.$
(52)

In the above expressions we have considered the mixing between the
$DD_{s}^{*-}$ and $D^{*}D_{s}^{-}$ components, because their thresholds are
very close to each other. After doing this, $|D\bar{D}_{s}^{*};1^{++}\rangle$
is the strange partner of $|D\bar{D}^{*};1^{++}\rangle$, so we denote its
quantum number as $J^{PC}=1^{++}$ for convenience;
$|D\bar{D}_{s}^{*};1^{+-}\rangle$ is the strange partner of
$|D\bar{D}^{*};1^{+-}\rangle$, so we also denote its quantum number as
$J^{PC}=1^{+-}$.

We use the currents $\zeta_{1\cdots 6}$ to perform numerical analyses, and
calculate masses and decay constants of $D^{(*)}\bar{D}_{s}^{(*)}$ molecular
states. The obtained results are summarized in Table 1. Our results support
the interpretations of the $Z_{cs}(3985)$, $Z_{cs}(4000)$, and $Z_{cs}(4220)$
as the $|D\bar{D}_{s}^{*};1^{++}\rangle$, $|D\bar{D}_{s}^{*};1^{+-}\rangle$,
and $|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$ molecular states, respectively.
However, the accuracy of our QCD sum rule results is not enough to
differentiate the $Z_{cs}(3985)$ and $Z_{cs}(4000)$.

To better understand $D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and
$D^{(*)}\bar{D}_{s}^{(*)}$ molecular states, we shall further study their
production and decay properties in the following sections. Especially, the
decay constants $f_{X}$ calculated in this section are important input
parameters.

##  III Productions through the current algebra

In this section we study productions of $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular
states in $B$ and $B^{*}$ decays through the current algebra. We shall
calculate their relative production rates, such as $\mathcal{B}(B^{-}\to
K^{-}X):\mathcal{B}(B^{-}\to K^{-}X^{\prime})$, with $X$ and $X^{\prime}$
different molecular states. This method has been applied in Ref. Chen:2020opr
to systematically study productions of $\bar{D}^{(*)}\Sigma_{c}^{(*)}$
hadronic molecular states in $\Lambda_{b}^{0}$ decays.

(a) $B^{-}\to K^{-}D^{(*)0}{\bar{D}}^{(*)0}$ (b) $B^{-}\to
D^{-}D^{(*)0}{\bar{K}}^{(*)0}$

(c) $B^{-}\to\phi D^{(*)0}D_{s}^{(*)-}$ (d) $B^{-}\to K^{-}D^{(*)+}{D}^{(*)-}$

(e) $B^{-}\to D^{-}D^{(*)+}{K}^{(*)-}$

Figure 3: Possible production mechanisms of $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular
states in $B^{-}$ decays.

As depicted in Fig. 3, the quark content of the initial state $B^{-}$ is
$\bar{u}b$. When it decays, first the $b$ quark decays into a $c$ quark by
emitting a $W^{-}$ boson, and this $W^{-}$ boson translates into a pair of
$\bar{c}$ and $s$ quarks, both of which are Cabibbo-favored; then they pick up
an isoscalar quark-antiquark pair $\bar{u}u+\bar{d}d+\bar{s}s$ from the
vacuum; finally they hadronize into the $D^{(*)}\bar{D}^{(*)}\bar{K}^{(*)}$
and $D^{(*)}\bar{D}_{s}^{(*)}\phi$ final states:

$\displaystyle B^{-}$ $\displaystyle=$
$\displaystyle\bar{u}b\to\bar{u}c~{}\bar{c}s\to\bar{u}c~{}\bar{c}s~{}(\bar{u}u+\bar{d}d+\bar{s}s)$
(53) $\displaystyle\to$ $\displaystyle
D^{(*)}\bar{D}^{(*)}\bar{K}^{(*)}+D^{(*)}\bar{D}_{s}^{(*)}\phi+\cdots\,.$

Among all the possible final states, $K^{-}D^{(*)0}\bar{D}^{(*)0}$,
$K^{-}D^{(*)+}{D}^{(*)-}$, $D^{-}D^{(*)0}\bar{K}^{(*)0}$, and
$D^{-}D^{(*)+}K^{(*)-}$ can further produce the neutral $D^{(*)}\bar{D}^{(*)}$
and $D^{(*)}\bar{K}^{(*)}$ molecular states, and $\phi D^{(*)0}D_{s}^{(*)-}$
can further produce the charged $D^{(*)}\bar{D}_{s}^{(*)}$ molecular states.

We shall develop three Fierz rearrangements in Sec. III.1 to describe the
production mechanisms depicted in Fig. 3(a,b,c), and use them to perform
numerical analyses separately in Sec. III.2, Sec. III.3, and Sec. III.4.
Productions of $D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and
$D^{(*)}\bar{D}_{s}^{(*)}$ molecular states in $B^{*}$ decays will be
similarly investigated in Sec. III.5.

###  III.1 Fierz rearrangement

To describe the production mechanism depicted in Fig. 3(a), we use the color
rearrangement given in Eq. (10), and apply the Fierz transformation to
interchange the $s_{d}$ and $u_{f}$ quarks:

$\displaystyle B^{-}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle J_{B^{-}}=[\delta^{ab}\bar{u}_{a}\gamma_{5}b_{b}]$ (54)
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
(55) $\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}\times~{}[\delta^{ef}\bar{u}_{e}u_{f}]$
(56) $\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle{\delta^{ab}\delta^{cf}\delta^{ed}\over
3}\times\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}\times\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}\times\bar{u}_{e}u_{f}~{}+~{}\cdots$
(57) $\displaystyle\xlongequal{{\rm Fierz:}s_{d}\leftrightarrow u_{f}}$
$\displaystyle-{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}u_{f}]\times[\delta^{ed}\bar{u}_{e}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
$\displaystyle+{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})u_{f}]\times[\delta^{ed}\bar{u}_{e}\gamma_{5}s_{d}]$
$\displaystyle-{i\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\sigma^{\mu\rho}\gamma_{5}u_{f}]\times[\delta^{ed}\bar{u}_{e}\gamma_{\mu}(1-\gamma_{5})s_{d}]~{}+~{}\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle+{1\over
24}\times\Big{(}\eta_{3}^{\alpha}-\eta_{2}^{\alpha}\Big{)}\times[\bar{u}_{a}\gamma_{\alpha}\gamma_{5}s_{a}]$
$\displaystyle+{1\over 12}\times\eta_{4}\times[\bar{u}_{a}\gamma_{5}s_{a}]$
$\displaystyle+{i\over
24}\times\eta_{5}^{\alpha}\times[\bar{u}_{a}\gamma_{\alpha}\gamma_{5}s_{a}]~{}+~{}\cdots\,.$

Similarly, we describe the production mechanism depicted in Fig. 3(b) as:

$\displaystyle B^{-}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle J_{B^{-}}=[\delta^{ab}\bar{u}_{a}\gamma_{5}b_{b}]$ (60)
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
(61) $\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}\times~{}[\delta^{ef}\bar{d}_{e}d_{f}]$
(62) $\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle{\delta^{ab}\delta^{cf}\delta^{ed}\over
3}\times\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}\times\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}\times\bar{d}_{e}d_{f}~{}+~{}\cdots$
(63) $\displaystyle\xlongequal{{\rm Fierz:}s_{d}\leftrightarrow d_{f}}$
$\displaystyle-{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}d_{f}]\times[\delta^{ed}\bar{d}_{e}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
$\displaystyle+{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})d_{f}]\times[\delta^{ed}\bar{d}_{e}\gamma_{5}s_{d}]$
$\displaystyle-{i\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{\mu}(1-\gamma_{5})d_{f}]\times[\delta^{ed}\bar{d}_{e}\sigma^{\mu\rho}\gamma_{5}s_{d}]~{}+~{}\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle-{1\over
12}\times\xi_{3}^{\alpha}\times[\bar{c}_{a}\gamma_{\alpha}\gamma_{5}d_{a}]$
$\displaystyle-{1\over 12}\times\xi_{4}\times[\bar{c}_{a}\gamma_{5}d_{a}]$
$\displaystyle+{i\over
24}\times\xi_{5}^{\alpha}\times[\bar{c}_{a}\gamma_{\alpha}\gamma_{5}d_{a}]~{}+~{}\cdots\,,$

and the production mechanism depicted in Fig. 3(c) as:

$\displaystyle B^{-}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle J_{B^{-}}=[\delta^{ab}\bar{u}_{a}\gamma_{5}b_{b}]$ (66)
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
(67) $\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}\times~{}[\delta^{ef}\bar{s}_{e}s_{f}]$
(68) $\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle{\delta^{ab}\delta^{cf}\delta^{ed}\over
3}\times\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}\times\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}\times\bar{s}_{e}s_{f}~{}+~{}\cdots$
(69) $\displaystyle\xlongequal{{\rm Fierz:}s_{d}\leftrightarrow s_{f}}$
$\displaystyle-{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}s_{f}]\times[\delta^{ed}\bar{s}_{e}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
$\displaystyle-{i\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{\rho}(1-\gamma_{5})c_{b}]\times[\delta^{cf}\bar{c}_{c}\sigma^{\mu\rho}\gamma_{5}s_{f}]\times[\delta^{ed}\bar{s}_{e}\gamma_{\mu}(1-\gamma_{5})s_{d}]~{}+~{}\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle-{1\over
24}\times\Big{(}\zeta_{3}^{\alpha}-\zeta_{2}^{\alpha}\Big{)}\times[\bar{s}_{a}\gamma_{\alpha}s_{a}]$
$\displaystyle-{i\over
24}\times\zeta_{5}^{\alpha}\times[\bar{s}_{a}\gamma_{\alpha}s_{a}]~{}+~{}\cdots\,.$

The brief explanations to the Fierz rearrangement given in Eq. (54) are as
follows:

  * •

Eq. (55) describes the Cabibbo-favored weak decay process, $b\to c+\bar{c}s$,
via the $V$-$A$ current.

  * •

Eq. (56) describes the production of the $\bar{u}u$ pair from the vacuum via
the ${}^{3}P_{0}$ quark pair creation mechanism.

  * •

In Eq. (57) we apply the color rearrangement given in Eq. (10).

  * •

In Eq. (III.1) we apply the Fierz transformation to interchange the $s_{d}$
and $u_{f}$ quarks.

  * •

In Eq. (III.1) we combine the four quarks $\bar{u}_{a}c_{b}\bar{c}_{c}u_{f}$
together so that $D^{(*)0}\bar{D}^{(*)0}$ molecular states can be produced.

The other two Fierz rearrangements given in Eqs. (60) and (66) can be
similarly investigated.

In the above expression, we only consider $\eta_{1\cdots 6}$, $\xi_{1\cdots
6}$, and $\zeta_{1\cdots 6}$ defined in Eqs. (17–22), Eqs. (34–39), and Eqs.
(47–52). These currents couple to $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ molecular states
through $S$-wave. Besides, there may exist some other currents coupling to
these states through $P$-wave, but we do not take them into account in the
present study. Hence, some other molecular states such as
$|D\bar{D};0^{++}\rangle$ may still be produced in $B^{-}$ decays, and
omissions of these currents cause some theoretical uncertainties.

Beside the three production mechanisms depicted in Fig. 3(a,b,c), there exist
some other possible production mechanisms, such as those depicted in Fig.
3(d,e). However, the mechanism depicted in Fig. 3(d) contains the color factor
${\delta^{ad}\delta^{cf}\delta^{eb}\over
9}\times\bar{u}_{a}c_{b}\bar{c}_{c}s_{d}\bar{d}_{e}d_{f}$, and the one
depicted in Fig. 3(e) contains the color factor
${\delta^{ad}\delta^{cf}\delta^{eb}\over
9}\times\bar{u}_{a}c_{b}\bar{c}_{c}s_{d}\bar{d}_{e}d_{f}$, so both of them are
suppressed.

Very recently, the LHCb Collaboration reported their observations of $X(4630)$
and $X(4685)$ in the $B^{+}\to K^{+}X\to K^{+}J/\psi\phi$ decay process
Aaij:2021ivw . Although these two structures were also observed in $B$ decays,
their production mechanisms may not be (completely) the same as those depicted
in Fig. 3(a,b,c), so we do not investigate them in the present study.

###  III.2 Productions of $D^{(*)}\bar{D}^{(*)}$ molecules

In this subsection we use the Fierz rearrangement given in Eq. (54) to perform
numerical analyses, and calculate relative production rates of
$D^{(*)}\bar{D}^{(*)}$ molecular states in $B^{-}\to K^{-}X$ decays. To do
this we need the following couplings to the $K^{-}$ meson:

$\displaystyle\langle 0|\bar{u}_{a}i\gamma_{5}s_{a}|K^{-}(q)\rangle$
$\displaystyle=$ $\displaystyle\lambda_{K}\,,$ (72) $\displaystyle\langle
0|\bar{u}_{a}\gamma_{\mu}\gamma_{5}s_{a}|K^{-}(q)\rangle$ $\displaystyle=$
$\displaystyle iq_{\mu}f_{K}\,,$

where $f_{K}=155.6$ MeV pdg  and $\lambda_{K}={f_{K}^{2}m_{K}\over
m_{q}+m_{s}}$ with the isospin-averaged current quark mass
$m_{q}={m_{u}+m_{d}\over 2}$.

We extract from Eq. (54) the following decay channels:

  1. 1.

The decay of $B^{-}$ into $|D\bar{D}^{*};1^{++}\rangle K^{-}$ is contributed
by $\eta_{2}^{\alpha}\times[\bar{u}_{a}\gamma_{\alpha}\gamma_{5}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D\bar{D}^{*};1^{++}(\epsilon_{1},q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{i~{}d_{1}\over
24}f_{K}f_{|D\bar{D}^{*};1^{++}\rangle}~{}\epsilon_{1}\cdot q_{2}\,.$

The decay constant $f_{|D\bar{D}^{*};1^{++}\rangle}$ has been calculated in
the previous section and given in Table 1. The overall factor $d_{1}$ is
related to: a) the coupling of $J_{B^{-}}$ to $B^{-}$, b) the weak and
${}^{3}P_{0}$ decay processes described by Eqs. (55) and (56), c) the isospin
factor between $|D^{0}\bar{D}^{*0};1^{++}\rangle$ and
$|D\bar{D}^{*};1^{++}\rangle$, and d) the dynamical process depicted in Fig.
3(a).

  2. 2.

The decay of $B^{-}$ into $|D\bar{D}^{*};1^{+-}\rangle K^{-}$ is contributed
by $\eta_{3}^{\alpha}\times[\bar{u}_{a}\gamma_{\alpha}\gamma_{5}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D\bar{D}^{*};1^{+-}(\epsilon_{1},q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle{i~{}d_{1}\over
24}f_{K}f_{|D\bar{D}^{*};1^{+-}\rangle}~{}\epsilon_{1}\cdot q_{2}\,.$

  3. 3.

The decay of $B^{-}$ into $|D^{*}\bar{D}^{*};0^{++}\rangle K^{-}$ is
contributed by $\eta_{4}\times[\bar{u}_{a}\gamma_{5}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D^{*}\bar{D}^{*};0^{++}(q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle{d_{1}\over
12}\lambda_{K}f_{|D^{*}\bar{D}^{*};0^{++}\rangle}\,.$

  4. 4.

The decay of $B^{-}$ into $|D^{*}\bar{D}^{*};1^{+-}\rangle K^{-}$ is
contributed by
$\eta_{5}^{\alpha}\times[\bar{u}_{a}\gamma_{\alpha}\gamma_{5}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D^{*}\bar{D}^{*};1^{+-}(\epsilon_{1},q_{1})~{}K^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{d_{1}\over
24}f_{K}f_{|D^{*}\bar{D}^{*};1^{+-}\rangle}~{}\epsilon_{1}\cdot q_{2}\,.$

In the present study we adopt the possible interpretations of the $X(3872)$,
$Z_{c}(3900)$, and $Z_{c}(4020)$ as the $|D\bar{D}^{*};1^{++}\rangle$,
$|D\bar{D}^{*};1^{+-}\rangle$, and $|D^{*}\bar{D}^{*};1^{+-}\rangle$ molecular
states, respectively. As summarized in Table 1, the masses of the
$D^{(*)}\bar{D}^{(*)}$ molecular states calculated through the QCD sum rule
method have considerable uncertainties, based on which we are not able to
study their production and decay properties, e.g., we are not able to
calculate the partial decay width of the $|D\bar{D}^{*};1^{+-}\rangle$
molecular state into the $D\bar{D}^{*}$ channel due to the ambiguous phase
space, as will be studied in Sec. IV.1.5. Hence, we assume the masses of the
$D^{(*)}\bar{D}^{(*)}$ molecular states to be either the realistic masses or
at the lowest $D^{(*)}\bar{D}^{(*)}$ thresholds pdg :

$\displaystyle M_{|D\bar{D};0^{++}\rangle}$ $\displaystyle\approx$
$\displaystyle M_{D^{0}}+M_{\bar{D}^{0}}=3730~{}{\rm MeV}\,,$ $\displaystyle
M_{|D\bar{D}^{*};1^{++}\rangle}$ $\displaystyle=$ $\displaystyle
M_{X(3872)}=3871.69~{}{\rm MeV}\,,$ $\displaystyle
M_{|D\bar{D}^{*};1^{+-}\rangle}$ $\displaystyle=$ $\displaystyle
M_{Z_{c}(3900)}=3888.4~{}{\rm MeV}\,,$ (77) $\displaystyle
M_{|D^{*}\bar{D}^{*};0^{++}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{*0}}+M_{\bar{D}^{*0}}=4014~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{D}^{*};1^{+-}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{Z_{c}(4020)}=4024.1~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{D}^{*};2^{++}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{*0}}+M_{\bar{D}^{*0}}=4014~{}{\rm MeV}\,.$

Then we can evaluate the above production amplitudes to obtain the following
partial decay widths:

$\displaystyle\Gamma(B^{-}\to K^{-}|D\bar{D};0^{++}\rangle)$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle\Gamma(B^{-}\to
K^{-}|D\bar{D}^{*};1^{++}\rangle)$ $\displaystyle=$ $\displaystyle
d_{1}^{2}~{}0.40\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{-}\to K^{-}|D\bar{D}^{*};1^{+-}\rangle)$
$\displaystyle=$ $\displaystyle d_{1}^{2}~{}0.38\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{-}\to
K^{-}|D^{*}\bar{D}^{*};0^{++}\rangle)$ $\displaystyle=$ $\displaystyle
d_{1}^{2}~{}4.21\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{-}\to K^{-}|D^{*}\bar{D}^{*};1^{+-}\rangle)$
$\displaystyle=$ $\displaystyle d_{1}^{2}~{}1.35\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{-}\to
K^{-}|D^{*}\bar{D}^{*};2^{++}\rangle)$ $\displaystyle=$ $\displaystyle 0\,.$
(78)

###  III.3 Productions of $D^{(*)}\bar{K}^{(*)}$ molecules

Similarly, we use the Fierz rearrangement given in Eq. (60) to perform
numerical analyses, and calculate relative production rates of
$D^{(*)}\bar{K}^{(*)}$ molecular states in $B^{-}\to D^{-}X$ decays:

  1. 5.

The decay of $B^{-}$ into $|D^{*}\bar{K};1^{+}\rangle D^{-}$ is contributed by
$\xi_{3}^{\alpha}\times[\bar{c}_{a}\gamma_{\alpha}\gamma_{5}d_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D^{*}\bar{K};1^{+}(\epsilon_{1},q_{1})~{}D^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{i~{}d_{2}\over
12}f_{D}f_{|D^{*}\bar{K};1^{+}\rangle}~{}\epsilon_{1}\cdot q_{2}\,.$

The overall factor $d_{2}$ is related to: a) the coupling of $J_{B^{-}}$ to
$B^{-}$, b) the weak and ${}^{3}P_{0}$ decay processes described by Eqs. (61)
and (62), c) the isospin factor between $|D^{*0}\bar{K}^{0};1^{+}\rangle$ and
$|D^{*}\bar{K};1^{+}\rangle$, and d) the dynamical process depicted in Fig.
3(b).

  2. 6.

The decay of $B^{-}$ into $|D^{*}\bar{K}^{*};0^{+}\rangle D^{-}$ is
contributed by $\xi_{4}\times[\bar{c}_{a}\gamma_{5}d_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D^{*}\bar{K}^{*};0^{+}(q_{1})~{}D^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{d_{2}\over
12}\lambda_{D}f_{|D^{*}\bar{K}^{*};0^{+}\rangle}\,.$

  3. 7.

The decay of $B^{-}$ into $|D^{*}\bar{K}^{*};1^{+}\rangle D^{-}$ is
contributed by
$\xi_{5}^{\alpha}\times[\bar{c}_{a}\gamma_{\alpha}\gamma_{5}d_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D^{*}\bar{K}^{*};1^{+}(\epsilon_{1},q_{1})~{}D^{-}(q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{d_{2}\over
24}f_{D}f_{|D^{*}\bar{K}^{*};1^{+}\rangle}~{}\epsilon_{1}\cdot q_{2}\,.$

In the above expressions $f_{D}$ and $\lambda_{D}$ are decay constants of the
$D^{-}$ meson, whose definitions and values can be found in Table 2.

In the present study we adopt the possible interpretation of the $X_{0}(2900)$
as the $|D^{*}\bar{K}^{*};0^{+}\rangle$ molecular state. Accordingly, we
assume masses of $D^{(*)}\bar{K}^{(*)}$ molecular states to be either the
realistic mass or at the lowest $D^{(*)}\bar{K}^{(*)}$ thresholds pdg :

$\displaystyle M_{|D\bar{K};0^{+}\rangle}$ $\displaystyle\approx$
$\displaystyle M_{D^{0}}+M_{K^{+}}=2359~{}{\rm MeV}\,,$ $\displaystyle
M_{|D\bar{K}^{*};1^{+}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{0}}+M_{K^{*+}}=2756~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{K};1^{+}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{*0}}+M_{K^{+}}=2501~{}{\rm MeV}\,,$ (82) $\displaystyle
M_{|D^{*}\bar{K}^{*};0^{+}\rangle}$ $\displaystyle=$ $\displaystyle
M_{X_{0}(2900)}=2866~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{K}^{*};1^{+}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{*0}}+M_{K^{*+}}=2899~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{K}^{*};2^{+}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{*0}}+M_{K^{*+}}=2899~{}{\rm MeV}\,.$

Then we can evaluate the above production amplitudes to obtain the following
partial decay widths:

$\displaystyle\Gamma(B^{-}\to D^{-}|D\bar{K};0^{+}\rangle)$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle\Gamma(B^{-}\to
D^{-}|D\bar{K}^{*};1^{+}\rangle)$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\Gamma(B^{-}\to D^{-}|D^{*}\bar{K};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle d_{2}^{2}~{}2.13\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{-}\to
D^{-}|D^{*}\bar{K}^{*};0^{+}\rangle)$ $\displaystyle=$ $\displaystyle
d_{2}^{2}~{}3.18\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{-}\to D^{-}|D^{*}\bar{K}^{*};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle d_{2}^{2}~{}2.23\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{-}\to
D^{-}|D^{*}\bar{K}^{*};2^{+}\rangle)$ $\displaystyle=$ $\displaystyle 0\,.$
(83)

###  III.4 Productions of $D^{(*)}\bar{D}_{s}^{(*)}$ molecules

Similarly, we use the Fierz rearrangement given in Eq. (66) to perform
numerical analyses, and calculate relative production rates of
$D^{(*)}\bar{D}_{s}^{(*)}$ molecular states in $B^{-}\to\phi X$ decays:

  1. 8.

The decay of $B^{-}$ into $|D\bar{D}_{s}^{*};1^{++}\rangle\phi$ is contributed
by $\zeta_{2}^{\alpha}\times[\bar{s}_{a}\gamma_{\alpha}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D\bar{D}_{s}^{*};1^{++}(\epsilon_{1},q_{1})~{}\phi(\epsilon_{2},q_{2})\rangle$
$\displaystyle=$ $\displaystyle{d_{3}\over
24}m_{\phi}f_{\phi}f_{|D\bar{D}_{s}^{*};1^{++}\rangle}~{}\epsilon_{1}\cdot\epsilon_{2}\,.$

The overall factor $d_{3}$ is related to: a) the coupling of $J_{B^{-}}$ to
$B^{-}$, b) the weak and ${}^{3}P_{0}$ decay processes described by Eqs. (67)
and (68), and c) the dynamical process depicted in Fig. 3(c). In this case
there does not exist the isospin factor.

  2. 9.

The decay of $B^{-}$ into $|D\bar{D}_{s}^{*};1^{+-}\rangle\phi$ is contributed
by $\zeta_{3}^{\alpha}\times[\bar{s}_{a}\gamma_{\alpha}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D\bar{D}_{s}^{*};1^{+-}(\epsilon_{1},q_{1})~{}\phi(\epsilon_{2},q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{d_{3}\over
24}m_{\phi}f_{\phi}f_{|D\bar{D}_{s}^{*};1^{+-}\rangle}~{}\epsilon_{1}\cdot\epsilon_{2}\,.$

  3. 10.

The decay of $B^{-}$ into $|D^{*}\bar{D}_{s}^{*};1^{+}\rangle\phi$ is
contributed by $\zeta_{5}^{\alpha}\times[\bar{s}_{a}\gamma_{\alpha}s_{a}]$:

$\displaystyle\langle
B^{-}(q)~{}|~{}D^{*}\bar{D}_{s}^{*};1^{+}(\epsilon_{1},q_{1})~{}\phi(\epsilon_{2},q_{2})\rangle$
$\displaystyle=$ $\displaystyle-{i~{}d_{3}\over
24}m_{\phi}f_{\phi}f_{|D^{*}\bar{D}_{s}^{*};1^{+}\rangle}~{}\epsilon_{1}\cdot\epsilon_{2}\,.$

In the above expressions $f_{\phi}$ is decay constant of the $\phi$ meson.
Besides, we also need $f_{\phi}^{T}$, and their definitions are:

$\displaystyle\langle 0|\bar{s}_{a}\gamma_{\mu}s_{a}|\phi(\epsilon,q)\rangle$
$\displaystyle=$ $\displaystyle m_{\phi}f_{\phi}\epsilon_{\mu}\,,$ (87)
$\displaystyle\langle
0|\bar{s}_{a}\sigma_{\mu\nu}s_{a}|\phi(\epsilon,q)\rangle$ $\displaystyle=$
$\displaystyle if^{T}_{\phi}(q_{\mu}\epsilon_{\nu}-q_{\nu}\epsilon_{\mu})\,,$

where $f_{\phi}=233$ MeV pdg  and $f_{\phi}^{T}=177$ MeV Becirevic:2004qv .

In the present study we adopt the possible interpretations of the
$Z_{cs}(3985)$, $Z_{cs}(4000)$, and $Z_{cs}(4220)$ as the
$|D\bar{D}_{s}^{*};1^{++}\rangle$, $|D\bar{D}_{s}^{*};1^{+-}\rangle$, and
$|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$ molecular states, respectively.
Accordingly, we assume masses of $D^{(*)}\bar{D}_{s}^{(*)}$ molecular states
to be either the realistic masses or at the lowest $D^{(*)}\bar{D}_{s}^{(*)}$
thresholds pdg :

$\displaystyle M_{|D\bar{D}_{s};0^{+}\rangle}$ $\displaystyle\approx$
$\displaystyle M_{D^{0}}+M_{D_{s}^{-}}=3833~{}{\rm MeV}\,,$ $\displaystyle
M_{|D\bar{D}_{s}^{*};1^{++}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{Z_{cs}(3985)}=3982.5~{}{\rm MeV}\,,$ $\displaystyle
M_{|D\bar{D}_{s}^{*};1^{+-}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{Z_{cs}(4000)}=4003~{}{\rm MeV}\,,$ (88) $\displaystyle
M_{|D^{*}\bar{D}_{s}^{*};0^{+}\rangle}$ $\displaystyle=$ $\displaystyle
M_{D^{*0}}+M_{D_{s}^{*-}}=4119~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{D}_{s}^{*};1^{+}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{Z_{cs}(4220)}=4216~{}{\rm MeV}\,,$ $\displaystyle
M_{|D^{*}\bar{D}_{s}^{*};2^{+}\rangle}$ $\displaystyle\approx$ $\displaystyle
M_{D^{*0}}+M_{D_{s}^{*-}}=4119~{}{\rm MeV}\,.$

Then we can evaluate the above production amplitudes to obtain the following
partial decay widths:

$\displaystyle\Gamma(B^{-}\to\phi|D\bar{D}_{s};0^{+}\rangle)$ $\displaystyle=$
$\displaystyle 0\,,$
$\displaystyle\Gamma(B^{-}\to\phi|D\bar{D}_{s}^{*};1^{++}\rangle)$
$\displaystyle=$ $\displaystyle d_{3}^{2}~{}2.53\times 10^{-10}~{}{\rm
GeV}^{13}\,,$
$\displaystyle\Gamma(B^{-}\to\phi|D\bar{D}_{s}^{*};1^{+-}\rangle)$
$\displaystyle=$ $\displaystyle d_{3}^{2}~{}2.38\times 10^{-10}~{}{\rm
GeV}^{13}\,,$
$\displaystyle\Gamma(B^{-}\to\phi|D^{*}\bar{D}_{s}^{*};0^{+}\rangle)$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\Gamma(B^{-}\to\phi|D^{*}\bar{D}_{s}^{*};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle d_{3}^{2}~{}5.77\times 10^{-10}~{}{\rm
GeV}^{13}\,,$
$\displaystyle\Gamma(B^{-}\to\phi|D^{*}\bar{D}_{s}^{*};2^{+}\rangle)$
$\displaystyle=$ $\displaystyle 0\,.$ (89)

From Eqs. (78), (83), and (89), we can derive the relative production rate
$\mathcal{R}_{1}\equiv{\mathcal{B}\left(B^{-}\rightarrow K^{-}X/D^{-}X/\phi
X\right)\over\mathcal{B}\left(B^{-}\rightarrow
D^{-}|D^{*}\bar{K}^{*};0^{+}\rangle\right)/d_{2}}$. We summarize the obtained
results in Table 3, which will be further discussed in Sec. V. The difference
among the three overall factors $d_{1,2,3}$ is partly caused by the different
dynamical processes described by Eqs. (III.1), (III.1), and (III.1), i.e., the
attraction between $D^{(*)}$ and $\bar{D}^{(*)}$, the attraction between
$D^{(*)}$ and $\bar{K}^{(*)}$, and the attraction between $D^{(*)}$ and
$\bar{D}_{s}^{(*)}$ can be different. However, we are not able to estimate
this in the present study.

###  III.5 Productions in $B^{*}$ decays

In this subsection we follow the same procedures to study productions of
$D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$
molecular states in $B^{*}$ decays.

Similar to Eq. (54), we apply the Fierz transformation to obtain:

$\displaystyle B^{*-}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle
J^{\alpha}_{B^{*-}}=[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}b_{b}]$
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}\times~{}[\delta^{ef}\bar{u}_{e}u_{f}]$
$\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle{\delta^{ab}\delta^{cf}\delta^{ed}\over
3}\times\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}\times\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}\times\bar{u}_{e}u_{f}~{}+~{}\cdots$
$\displaystyle\xlongequal{{\rm Fierz:}s_{d}\leftrightarrow u_{f}}$
$\displaystyle+{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}u_{f}]\times[\delta^{ed}\bar{u}_{e}\gamma^{\alpha}(1-\gamma_{5})s_{d}]$
$\displaystyle-{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma^{\alpha}(1-\gamma_{5})u_{f}]\times[\delta^{ed}\bar{u}_{e}\gamma_{5}s_{d}]$
$\displaystyle+{1\over
12}\times[\delta^{ab}\bar{u}_{a}i\sigma^{\alpha\rho}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})u_{f}]\times[\delta^{ed}\bar{u}_{e}\gamma_{5}s_{d}]~{}+~{}\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle-{1\over
12}\times\eta_{1}\times[\bar{u}_{a}\gamma^{\alpha}\gamma_{5}s_{a}]$
$\displaystyle-{1\over
24}\times\Big{(}\eta_{2}^{\alpha}+\eta_{3}^{\alpha}\Big{)}\times[\bar{u}_{a}\gamma_{5}s_{a}]$
$\displaystyle-{i\over
24}\times\eta_{5}^{\alpha}\times[\bar{u}_{a}\gamma_{5}s_{a}]~{}+~{}\cdots\,.$

Similar to Eq. (60), we apply the Fierz transformation to obtain:

$\displaystyle B^{*-}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle
J^{\alpha}_{B^{*-}}=[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}b_{b}]$
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}\times~{}[\delta^{ef}\bar{d}_{e}d_{f}]$
$\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle{\delta^{ab}\delta^{cf}\delta^{ed}\over
3}\times\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}\times\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}\times\bar{d}_{e}d_{f}~{}+~{}\cdots$
$\displaystyle\xlongequal{{\rm Fierz:}s_{d}\leftrightarrow u_{f}}$
$\displaystyle+{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}d_{f}]\times[\delta^{ed}\bar{d}_{e}\gamma^{\alpha}(1-\gamma_{5})s_{d}]$
$\displaystyle-{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma^{\alpha}(1-\gamma_{5})d_{f}]\times[\delta^{ed}\bar{d}_{e}\gamma_{5}s_{d}]$
$\displaystyle-{1\over
12}\times[\delta^{ab}\bar{u}_{a}i\sigma^{\alpha\rho}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}d_{f}]\times[\delta^{ed}\bar{d}_{e}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}+~{}\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle+{1\over
12}\times\xi_{1}\times[\bar{c}_{a}\gamma^{\alpha}\gamma_{5}d_{a}]$
$\displaystyle+{1\over
12}\times\xi_{2}^{\alpha}\times[\bar{c}_{a}\gamma_{5}d_{a}]$
$\displaystyle+{i\over
24}\times\xi_{5}^{\alpha}\times[\bar{c}_{a}\gamma_{5}d_{a}]~{}+~{}\cdots\,.$

Similar to Eq. (66), we apply the Fierz transformation to obtain:

$\displaystyle B^{*-}$
$\displaystyle\xrightarrow{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle
J^{\alpha}_{B^{*-}}=[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}b_{b}]$
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm weak~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]$
$\displaystyle\xrightarrow{~{}~{}~{}~{}\rm QPC~{}~{}~{}~{}}$
$\displaystyle[\delta^{ab}\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}]~{}\times~{}[\delta^{cd}\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}]~{}\times~{}[\delta^{ef}\bar{s}_{e}s_{f}]$
$\displaystyle\xlongequal{~{}~{}~{}~{}\rm color~{}~{}~{}~{}}$
$\displaystyle{\delta^{ab}\delta^{cf}\delta^{ed}\over
3}\times\bar{u}_{a}\gamma^{\alpha}\gamma_{\rho}(1-\gamma_{5})c_{b}\times\bar{c}_{c}\gamma^{\rho}(1-\gamma_{5})s_{d}\times\bar{s}_{e}s_{f}~{}+~{}\cdots$
$\displaystyle\xlongequal{{\rm Fierz:}s_{d}\leftrightarrow s_{f}}$
$\displaystyle+{1\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{5}s_{f}]\times[\delta^{ed}\bar{s}_{e}\gamma^{\alpha}(1-\gamma_{5})s_{d}]$
$\displaystyle-{i\over
12}\times[\delta^{ab}\bar{u}_{a}\gamma_{5}c_{b}]\times[\delta^{cf}\bar{c}_{c}\gamma_{\mu}(1-\gamma_{5})s_{f}]\times[\delta^{ed}\bar{s}_{e}\sigma^{\alpha\mu}\gamma_{5}s_{d}]~{}+~{}\cdots$
$\displaystyle\xlongequal{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$
$\displaystyle+{1\over
12}\times\zeta_{1}\times[\bar{s}_{a}\gamma^{\alpha}s_{a}]$
$\displaystyle-{i\over
24}\times\Big{(}\zeta_{2\mu}+\zeta_{3\mu}\Big{)}\times[\bar{s}_{a}\sigma^{\alpha\mu}\gamma_{5}s_{a}]~{}+~{}\cdots\,.$

From Eq. (III.5), we obtain the following partial decay widths

$\displaystyle\Gamma(B^{*-}\to K^{-}|D\bar{D};0^{++}\rangle)$ $\displaystyle=$
$\displaystyle d_{4}^{2}~{}0.16\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to K^{-}|D\bar{D}^{*};1^{++}\rangle)$
$\displaystyle=$ $\displaystyle d_{4}^{2}~{}0.62\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{*-}\to
K^{-}|D\bar{D}^{*};1^{+-}\rangle)$ $\displaystyle=$ $\displaystyle
d_{4}^{2}~{}0.61\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to K^{-}|D^{*}\bar{D}^{*};0^{++}\rangle)$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle\Gamma(B^{*-}\to
K^{-}|D^{*}\bar{D}^{*};1^{+-}\rangle)$ $\displaystyle=$ $\displaystyle
d_{4}^{2}~{}2.84\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to K^{-}|D^{*}\bar{D}^{*};2^{++}\rangle)$
$\displaystyle=$ $\displaystyle 0\,.$ (93)

From Eq. (III.5), we obtain the following partial decay widths

$\displaystyle\Gamma(B^{*-}\to D^{-}|D\bar{K};0^{+}\rangle)$ $\displaystyle=$
$\displaystyle d_{5}^{2}~{}0.18\times 10^{-10}~{}{\rm GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to D^{-}|D\bar{K}^{*};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle d_{5}^{2}~{}1.69\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{*-}\to D^{-}|D^{*}\bar{K};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle\Gamma(B^{*-}\to
D^{-}|D^{*}\bar{K}^{*};0^{+}\rangle)$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\Gamma(B^{*-}\to D^{-}|D^{*}\bar{K}^{*};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle d_{5}^{2}~{}4.40\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{*-}\to
D^{-}|D^{*}\bar{K}^{*};2^{+}\rangle)$ $\displaystyle=$ $\displaystyle 0\,.$
(94)

From Eq. (III.5), we obtain the following partial decay widths

$\displaystyle\Gamma(B^{*-}\to\phi|D\bar{D}_{s};0^{+}\rangle)$
$\displaystyle=$ $\displaystyle d_{6}^{2}~{}1.78\times 10^{-10}~{}{\rm
GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to\phi|D\bar{D}_{s}^{*};1^{++}\rangle)$
$\displaystyle=$ $\displaystyle d_{6}^{2}~{}1.09\times 10^{-10}~{}{\rm
GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to\phi|D\bar{D}_{s}^{*};1^{+-}\rangle)$
$\displaystyle=$ $\displaystyle d_{6}^{2}~{}1.02\times 10^{-10}~{}{\rm
GeV}^{13}\,,$
$\displaystyle\Gamma(B^{*-}\to\phi|D^{*}\bar{D}_{s}^{*};0^{+}\rangle)$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\Gamma(B^{*-}\to\phi|D^{*}\bar{D}_{s}^{*};1^{+}\rangle)$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\Gamma(B^{*-}\to\phi|D^{*}\bar{D}_{s}^{*};2^{+}\rangle)$
$\displaystyle=$ $\displaystyle 0\,.$ (95)

In the above expressions $d_{4,5,6}$ are three overall factors.

From Eqs. (93–95), we can derive the relative production rate
$\mathcal{R}_{2}\equiv{\mathcal{B}\left(B^{*-}\rightarrow K^{-}X/D^{-}X/\phi
X\right)\over\mathcal{B}\left(B^{*-}\rightarrow
K^{-}|D\bar{D};0^{++}\rangle\right)/d_{4}}$. We summarize the obtained results
in Table 3, which will be further discussed in Sec. V. Again, the difference
among the three overall factors $d_{4,5,6}$ is partly due to that the
attraction between $D^{(*)}$ and $\bar{D}^{(*)}$, the attraction between
$D^{(*)}$ and $\bar{K}^{(*)}$, and the attraction between $D^{(*)}$ and
$\bar{D}_{s}^{(*)}$ can be different.

##  IV Decay properties through the Fierz rearrangement

The Fierz rearrangement fierz  of the Dirac and color indices has been applied
in Refs. Chen:2019wjd ; Chen:2019eeq  to study strong decay properties of the
$X(3872)$ and $Z_{c}(3900)$ as $D\bar{D}^{*}$ molecular states. In this
section we follow the same procedures to study decay properties of other
$D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$
hadronic molecular states, separately in Sec. IV.1, Sec. IV.2, and Sec. IV.3.
We shall study their two-body and three-body hadronic decays as well as the
four-body decay process $X\to J/\psi\omega\to J/\psi\pi\pi\pi$.

This method has been systematically applied to study light baryon and
tetraquark currents in Refs. Chen:2006hy ; Chen:2006zh ; Chen:2007xr ;
Chen:2008ej ; Chen:2008ne ; Chen:2012ut ; Chen:2008qv ; Chen:2009sf ;
Chen:2012ex , and applied to study strong decay properties of the $P_{c}$ and
$X(6900)$ states in Refs. Chen:2020pac ; Chen:2020xwe . A similar arrangement
of the spin and color indices in the nonrelativistic case can be found in
Refs. Voloshin:2013dpa ; Maiani:2017kyi ; Voloshin:2018pqn ; Voloshin:2019aut
; Wang:2019spc ; Xiao:2019spy ; Cheng:2020nho , which was applied to study
decay properties of the $P_{c}$ and $XYZ$ states.

Before doing this, we note that the Fierz rearrangement in the Lorentz space
is a matrix identity, valid if each quark/antiquark field in the initial and
final currents is at the same location. For example, we can apply the Fierz
rearrangement to transform a non-local current
$\eta=[\bar{q}(x)c(x)]~{}[\bar{c}(y)q(y)]$ into the combination of many non-
local currents $\theta=[\bar{q}(x)q(y)]~{}[\bar{c}(y)c(x)]$, with all the
quark fields remaining at the same locations. We shall keep this in mind, and
omit the coordinates in this section.

Table 2: Couplings of meson operators to meson states, where color indices are omitted for simplicity. All the isovector meson operators have the quark content $\bar{q}\Gamma q=\left(\bar{u}\Gamma u-\bar{d}\Gamma d\right)/\sqrt{2}$, and all the isoscalar light meson operators have the quark content $\bar{q}\Gamma q=\left(\bar{u}\Gamma u+\bar{d}\Gamma d\right)/\sqrt{2}$. Operators–$J_{(\mu\nu)}$ |  $I^{G}J^{PC}$ |  Mesons |  $I^{G}J^{PC}$ |  Couplings |  Decay Constants
---|---|---|---|---|---
$\bar{c}c$ | $0^{+}0^{++}$ | $\chi_{c0}(1P)$ | $0^{+}0^{++}$ | $\langle 0|J|\chi_{c0}\rangle=m_{\chi_{c0}}f_{\chi_{c0}}$ |  $f_{\chi_{c0}}=343$ MeV Veliev:2010gb
$\bar{c}i\gamma_{5}c$ | $0^{+}0^{-+}$ | $\eta_{c}$ | $0^{+}0^{-+}$ | $\langle 0|J|\eta_{c}\rangle=\lambda_{\eta_{c}}$ | $\lambda_{\eta_{c}}={f_{\eta_{c}}m_{\eta_{c}}^{2}\over 2m_{c}}$
$\bar{c}\gamma_{\mu}c$ | $0^{-}1^{--}$ | $J/\psi$ | $0^{-}1^{--}$ | $\langle 0|J_{\mu}|J/\psi\rangle=m_{J/\psi}f_{J/\psi}\epsilon_{\mu}$ |  $f_{J/\psi}=418$ MeV Becirevic:2013bsa
$\bar{c}\gamma_{\mu}\gamma_{5}c$ | $0^{+}1^{++}$ | $\eta_{c}$ | $0^{+}0^{-+}$ | $\langle 0|J_{\mu}|\eta_{c}\rangle=ip_{\mu}f_{\eta_{c}}$ |  $f_{\eta_{c}}=387$ MeV Becirevic:2013bsa
$\chi_{c1}(1P)$ | $0^{+}1^{++}$ | $\langle 0|J_{\mu}|\chi_{c1}\rangle=m_{\chi_{c1}}f_{\chi_{c1}}\epsilon_{\mu}$ |  $f_{\chi_{c1}}=335$ MeV Novikov:1977dq
$\bar{c}\sigma_{\mu\nu}c$ | $0^{-}1^{\pm-}$ | $J/\psi$ | $0^{-}1^{--}$ | $\langle 0|J_{\mu\nu}|J/\psi\rangle=if^{T}_{J/\psi}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f_{J/\psi}^{T}=410$ MeV Becirevic:2013bsa
$h_{c}(1P)$ | $0^{-}1^{+-}$ | $\langle 0|J_{\mu\nu}|h_{c}\rangle=if^{T}_{h_{c}}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}$ |  $f_{h_{c}}^{T}=235$ MeV Becirevic:2013bsa
$\bar{c}q$ | $0^{+}$ | $\bar{D}_{0}^{*}$ | $0^{+}$ | $\langle 0|J|\bar{D}_{0}^{*}\rangle=m_{D_{0}^{*}}f_{D_{0}^{*}}$ |  $f_{D_{0}^{*}}=410$ MeV Narison:2015nxh
$\bar{c}i\gamma_{5}q$ | $0^{-}$ | $\bar{D}$ | $0^{-}$ | $\langle 0|J|\bar{D}\rangle=\lambda_{D}$ | $\lambda_{D}={f_{D}m_{D}^{2}\over{m_{c}+m_{d}}}$
$\bar{c}\gamma_{\mu}q$ | $1^{-}$ | $\bar{D}^{*}$ | $1^{-}$ | $\langle 0|J_{\mu}|\bar{D}^{*}\rangle=m_{D^{*}}f_{D^{*}}\epsilon_{\mu}$ |  $f_{D^{*}}=253$ MeV Chang:2018aut
$\bar{c}\gamma_{\mu}\gamma_{5}q$ | $1^{+}$ | $\bar{D}$ | $0^{-}$ | $\langle 0|J_{\mu}|\bar{D}\rangle=ip_{\mu}f_{D}$ |  $f_{D}=211.9$ MeV pdg
$\bar{D}_{1}$ | $1^{+}$ | $\langle 0|J_{\mu}|\bar{D}_{1}\rangle=m_{D_{1}}f_{D_{1}}\epsilon_{\mu}$ |  $f_{D_{1}}=356$ MeV Narison:2015nxh
$\bar{c}\sigma_{\mu\nu}q$ | $1^{\pm}$ | $\bar{D}^{*}$ | $1^{-}$ | $\langle 0|J_{\mu\nu}|\bar{D}^{*}\rangle=if_{D^{*}}^{T}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f_{D^{*}}^{T}\approx 220$ MeV
– | $1^{+}$ | – | –
$\bar{c}s$ | $0^{+}$ | – | $0^{+}$ | – | –
$\bar{c}i\gamma_{5}s$ | $0^{-}$ | $\bar{D}_{s}$ | $0^{-}$ | $\langle 0|J|\bar{D}_{s}\rangle=\lambda_{D_{s}}$ | $\lambda_{D_{s}}={f_{D_{s}}m_{D_{s}}^{2}\over{m_{c}+m_{s}}}$
$\bar{c}\gamma_{\mu}s$ | $1^{-}$ | $\bar{D}_{s}^{*}$ | $1^{-}$ | $\langle 0|J_{\mu}|\bar{D}^{*}_{s}\rangle=m_{D_{s}^{*}}f_{D_{s}^{*}}\epsilon_{\mu}$ |  $f_{D_{s}^{*}}=301$ MeV pdg
$\bar{c}\gamma_{\mu}\gamma_{5}s$ | $1^{+}$ | $\bar{D}_{s}$ | $0^{-}$ | $\langle 0|J_{\mu}|\bar{D}\rangle=ip_{\mu}f_{D_{s}}$ |  $f_{D_{s}}=257$ MeV pdg
– | $1^{+}$ | – |
$\bar{c}\sigma_{\mu\nu}s$ | $1^{\pm}$ | $\bar{D}^{*}_{s}$ | $1^{-}$ | $\langle 0|J_{\mu\nu}|\bar{D}_{s}^{*}\rangle=if_{D_{s}^{*}}^{T}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ | $f_{D_{s}^{*}}^{T}\approx f_{D_{s}^{*}}\times{f_{D^{*}}^{T}\over f_{D^{*}}}$
– | $1^{+}$ | – | –
$\bar{q}q$ | $0^{+}0^{++}$ | $f_{0}(500)$ | $0^{+}0^{++}$ | $\langle 0|J|f_{0}\rangle=m_{f_{0}}f_{f_{0}}$ |  $f_{f_{0}}\sim 380$ MeV Chen:2019eeq
$\bar{q}i\gamma_{5}q$ | $0^{+}0^{-+}$ | $\eta$ | $0^{+}0^{-+}$ | – | –
$\bar{q}\gamma_{\mu}q$ | $0^{-}1^{--}$ | $\omega$ | $0^{-}1^{--}$ | $\langle 0|J_{\mu}|\omega\rangle=m_{\omega}f_{\omega}\epsilon_{\mu}$ |  $f_{\omega}\approx f_{\rho}=216$ MeV Jansen:2009hr
$\bar{q}\gamma_{\mu}\gamma_{5}q$ | $0^{+}1^{++}$ | $\eta$ | $0^{+}0^{-+}$ | $\langle 0|J_{\mu}|\eta\rangle=ip_{\mu}f_{\eta}$ |  $f_{\eta}=97$ MeV Ottnad:2017bjt ; Guo:2015xva
$f_{1}(1285)$ | $0^{+}1^{++}$ | – | –
$\bar{q}\sigma_{\mu\nu}q$ | $0^{-}1^{\pm-}$ | $\omega$ | $0^{-}1^{--}$ | $\langle 0|J_{\mu\nu}|\omega\rangle=if^{T}_{\omega}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f^{T}_{\omega}\approx f_{\rho}^{T}=159$ MeV Jansen:2009hr
$h_{1}(1170)$ | $0^{-}1^{+-}$ | $\langle 0|J_{\mu\nu}|h_{1}\rangle=if^{T}_{h_{1}}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}$ |  $f_{h_{1}}^{T}\approx f_{b_{1}}^{T}=180$ MeV Ball:1996tb
$\bar{q}q$ | $1^{-}0^{++}$ | – | $1^{-}0^{++}$ | – | –
$\bar{q}i\gamma_{5}q$ | $1^{-}0^{-+}$ | $\pi^{0}$ | $1^{-}0^{-+}$ | $\langle 0|J|\pi^{0}\rangle=\lambda_{\pi}$ | $\lambda_{\pi}={f_{\pi}m_{\pi}^{2}\over m_{u}+m_{d}}$
$\bar{q}\gamma_{\mu}q$ | $1^{+}1^{--}$ | $\rho^{0}$ | $1^{+}1^{--}$ | $\langle 0|J_{\mu}|\rho^{0}\rangle=m_{\rho}f_{\rho}\epsilon_{\mu}$ |  $f_{\rho}=216$ MeV Jansen:2009hr
$\bar{q}\gamma_{\mu}\gamma_{5}q$ | $1^{-}1^{++}$ | $\pi^{0}$ | $1^{-}0^{-+}$ | $\langle 0|J_{\mu}|\pi^{0}\rangle=ip_{\mu}f_{\pi}$ |  $f_{\pi}=130.2$ MeV pdg
$a_{1}(1260)$ | $1^{-}1^{++}$ | $\langle 0|J_{\mu}|a_{1}\rangle=m_{a_{1}}f_{a_{1}}\epsilon_{\mu}$ |  $f_{a_{1}}=254$ MeV Wingate:1995hy
$\bar{q}\sigma_{\mu\nu}q$ | $1^{+}1^{\pm-}$ | $\rho^{0}$ | $1^{+}1^{--}$ | $\langle 0|J_{\mu\nu}|\rho^{0}\rangle=if^{T}_{\rho}(p_{\mu}\epsilon_{\nu}-p_{\nu}\epsilon_{\mu})$ |  $f_{\rho}^{T}=159$ MeV Jansen:2009hr
$b_{1}(1235)$ | $1^{+}1^{+-}$ | $\langle 0|J_{\mu\nu}|b_{1}\rangle=if^{T}_{b_{1}}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}$ |  $f_{b_{1}}^{T}=180$ MeV Ball:1996tb

In the calculations we need couplings of charmonium operators to charmonium
states, which are listed in Table 2. We also need couplings of charmed/light
meson operators to charmed/light meson states, which are also listed in Table
2. We refer to Refs. Chen:2019wjd ; Chen:2019eeq  for detailed discussions of
these couplings.

###  IV.1 Decay properties of $D^{(*)}\bar{D}^{(*)}$ molecules

In this subsection we perform the Fierz rearrangement for the currents
$\eta_{1,4,5,6}$, and use the obtained results to study strong decay
properties of $D^{(*)}\bar{D}^{(*)}$ molecular states. The results obtained in
Refs. Chen:2019wjd ; Chen:2019eeq  using $\eta_{2,3}^{\alpha}$ are also
summarized here for completeness.

We adopt the possible interpretations of the $X(3872)$, $Z_{c}(3900)$, and
$Z_{c}(4020)$ as the $|D\bar{D}^{*};1^{++}\rangle$,
$|D\bar{D}^{*};1^{+-}\rangle$, and $|D^{*}\bar{D}^{*};1^{+-}\rangle$ molecular
states, respectively. Masses of $D^{(*)}\bar{D}^{(*)}$ molecular states are
accordingly chosen, as given in Eqs. (77). Besides, we assume that
$|D\bar{D}^{*};1^{+-}\rangle$ and $|D^{*}\bar{D}^{*};1^{+-}\rangle$ have
$I=1$; $|D\bar{D};0^{++}\rangle$, $|D^{*}\bar{D}^{*};0^{++}\rangle$, and
$|D^{*}\bar{D}^{*};2^{++}\rangle$ have $I=0$; the isospin of the $X(3872)$ as
$|D\bar{D}^{*};1^{++}\rangle$ will be separately investigated in Sec. IV.1.4.

(a) Fierz rearrangement: $\eta\rightarrow\theta$ (b) direct fall-apart process

Figure 4: Fall-apart decay processes of $D^{(*)}\bar{D}^{(*)}$ molecular
states, investigated through the $\eta(x)$ currents.

####  IV.1.1 $|D\bar{D};0^{++}\rangle$

The current $\eta_{1}$ corresponds to the $|D\bar{D};0^{++}\rangle$ molecular
state. We can apply the Fierz rearrangement to interchange the $c_{a}$ and
$q_{b}$ quarks, and transform it into:

$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}c_{a}~{}\bar{c}_{b}\gamma_{5}q_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
12}~{}\bar{q}_{a}q_{a}~{}\bar{c}_{b}c_{b}-{1\over
12}~{}\bar{q}_{a}\gamma_{5}q_{a}~{}\bar{c}_{b}\gamma_{5}c_{b}$
$\displaystyle+{1\over
12}~{}\bar{q}_{a}\gamma_{\mu}q_{a}~{}\bar{c}_{b}\gamma^{\mu}c_{b}-{1\over
12}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}~{}\bar{c}_{b}\gamma^{\mu}\gamma_{5}c_{b}$
$\displaystyle-{1\over
24}~{}\bar{q}_{a}\sigma_{\mu\nu}q_{a}~{}\bar{c}_{b}\sigma^{\mu\nu}c_{b}+\cdots\,,$

where we have only kept the leading-order fall-apart decays described by
color-singlet-color-singlet meson-meson currents, but neglect the
$\mathcal{O}(\alpha_{s})$ corrections described by color-octet-color-octet
meson-meson currents.

As an example, we investigate $|D\bar{D};0^{++}\rangle$ through $\eta_{1}(x)$
and the above Fierz rearrangement. As depicted in Fig. 4(a), when the
$\bar{q}_{a}$ and $q_{d}$ quarks meet each other and the $c_{b}$ and
$\bar{c}_{c}$ quarks meet each other at the same time,
$|D\bar{D};0^{++}\rangle$ can decay into one charmonium meson and one light
meson:

$\displaystyle\left[\delta^{ab}\bar{q}_{a}c_{b}\right]~{}\left[\delta^{cd}\bar{c}_{c}q_{d}\right]$
$\displaystyle\xlongequal{~{}\rm color~{}}$ $\displaystyle{1\over
3}\delta^{ad}\delta^{cb}~{}\bar{q}_{a}c_{b}~{}\bar{c}_{c}q_{d}+\cdots$
$\displaystyle\xrightarrow{~{}\rm Fierz~{}}$ $\displaystyle{1\over
3}~{}\left[\delta^{ad}\bar{q}_{a}q_{d}\right]~{}\left[\delta^{cb}c_{b}c_{b}\right]+\cdots.$

This decay process can be described by the Fierz rearrangement given in Eq.
(IV.1.1).

Assuming the isospin of $|D\bar{D};0^{++}\rangle$ to be $I=0$, we extract the
following three decay channels:

  1. 1.

The decay of $|D\bar{D};0^{++}\rangle$ into $\eta_{c}\eta$ is contributed by
both $\bar{q}_{a}\gamma_{5}q_{a}\times\bar{c}_{b}\gamma_{5}c_{b}$ and
$\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}\times\bar{c}_{b}\gamma^{\mu}\gamma_{5}c_{b}$:

$\displaystyle\langle D\bar{D};0^{++}(p)|\eta_{c}(p_{1})~{}\eta(p_{2})\rangle$
$\displaystyle=$ $\displaystyle{a_{1}\over
12}~{}\lambda_{\eta_{c}}\lambda_{\eta}+{a_{1}\over
12}~{}f_{\eta_{c}}f_{\eta}~{}p_{1}\cdot p_{2}\,,$

where $a_{1}$ is an overall factor, related to the coupling of $\eta_{1}$ to
$|D\bar{D};0^{++}\rangle$ as well as the dynamical process depicted in Fig.
4(a).

  2. 2.

The decay of $|D\bar{D};0^{++}\rangle$ into $J/\psi\omega$ is contributed by
both $\bar{q}_{a}\gamma_{\mu}q_{a}\times\bar{c}_{b}\gamma^{\mu}c_{b}$ and
$\bar{q}_{a}\sigma_{\mu\nu}q_{a}\times\bar{c}_{b}\sigma^{\mu\nu}c_{b}$:

$\displaystyle\langle
D\bar{D};0^{++}(p)|J/\psi(\epsilon_{1},p_{1})~{}\omega(\epsilon_{2},p_{2})\rangle$
$\displaystyle=$ $\displaystyle{a_{1}\over
12}~{}m_{J/\psi}f_{J/\psi}m_{\omega}f_{\omega}~{}\epsilon_{1}\cdot\epsilon_{2}$
$\displaystyle+$ $\displaystyle{a_{1}\over
12}~{}f_{J/\psi}^{T}f_{\omega}^{T}\left(\epsilon_{1}\cdot\epsilon_{2}~{}p_{1}\cdot
p_{2}-\epsilon_{1}\cdot p_{2}~{}\epsilon_{2}\cdot p_{1}\right)\,.$

  3. 3.

The decay of $|D\bar{D};0^{++}\rangle$ into $\chi_{c0}f_{0}(500)$ is
contributed by $\bar{q}_{a}q_{a}\times\bar{c}_{b}c_{b}$:

$\displaystyle\langle
D\bar{D};0^{++}(p)|\chi_{c0}(p_{1})~{}f_{0}(p_{2})\rangle$ $\displaystyle=$
$\displaystyle-{a_{1}\over
12}~{}m_{\chi_{c0}}f_{\chi_{c0}}m_{f_{0}}f_{f_{0}}\,.$

Assuming the mass of $|D\bar{D};0^{++}\rangle$ to be about
$M_{D^{0}}+M_{\bar{D}^{0}}=3730$ MeV, we summarize the above amplitudes to
obtain the following partial decay widths:

$\displaystyle\Gamma(|D\bar{D};0^{++}\rangle\to\eta_{c}\eta)=a_{1}^{2}~{}1.7\times
10^{-4}~{}{\rm GeV}^{7}\,,$ $\displaystyle\Gamma(|D\bar{D};0^{++}\rangle\to
J/\psi\omega\to J/\psi\pi\pi\pi)$ (101)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=a_{1}^{2}~{}1.1\times
10^{-10}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D\bar{D};0^{++}\rangle\to\chi_{c0}f_{0}(500)\to\chi_{c0}\pi\pi)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=a_{1}^{2}~{}9.7\times
10^{-10}~{}{\rm GeV}^{7}\,.$

There are two different terms,
$A\equiv\bar{q}_{a}\gamma_{5}q_{a}\times\bar{c}_{b}\gamma_{5}c_{b}$ and
$B\equiv\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}\times\bar{c}_{b}\gamma^{\mu}\gamma_{5}c_{b}$,
both of which can contribute to the decay of $|D\bar{D};0^{++}\rangle$ into
the $\eta_{c}\eta$ final states. There are also two different terms,
$C\equiv\bar{q}_{a}\gamma_{\mu}q_{a}\times\bar{c}_{b}\gamma^{\mu}c_{b}$ and
$D\equiv\bar{q}_{a}\sigma_{\mu\nu}q_{a}\times\bar{c}_{b}\sigma^{\mu\nu}c_{b}$,
both of which can contribute to the decay of $|D\bar{D};0^{++}\rangle$ into
the $J/\psi\omega$ final states. Phase angles among them, such as the phase
angle between the two coupling constants $f_{J/\psi}$ and $f_{J/\psi}^{T}$,
can not be well determined in the present study. This causes some
uncertainties, which will be investigated in Appendix B.

####  IV.1.2 $|D^{*}\bar{D}^{*};0^{++}\rangle$

The current $\eta_{4}$ corresponds to the $|D^{*}\bar{D}^{*};0^{++}\rangle$
molecular state. We can apply the Fierz rearrangement and transform it into:

$\displaystyle\eta_{4}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma^{\mu}c_{a}~{}\bar{c}_{b}\gamma_{\mu}q_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
3}~{}\bar{q}_{a}q_{a}~{}\bar{c}_{b}c_{b}+{1\over
3}~{}\bar{q}_{a}\gamma_{5}q_{a}~{}\bar{c}_{b}\gamma_{5}c_{b}$
$\displaystyle+{1\over
6}~{}\bar{q}_{a}\gamma_{\mu}q_{a}~{}\bar{c}_{b}\gamma^{\mu}c_{b}+{1\over
6}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}~{}\bar{c}_{b}\gamma^{\mu}\gamma_{5}c_{b}+\cdots\
.$

Assuming the isospin of $|D^{*}\bar{D}^{*};0^{++}\rangle$ to be $I=0$ and its
mass to be about $M_{D^{*0}}+M_{\bar{D}^{*0}}=4014$ MeV, we obtain the
following partial decay widths:

$\displaystyle\Gamma(|D^{*}\bar{D}^{*};0^{++}\rangle\to\eta_{c}\eta)$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}4.0\times 10^{-3}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|D^{*}\bar{D}^{*};0^{++}\rangle\to
J/\psi\omega)$ $\displaystyle=$ $\displaystyle a_{4}^{2}~{}4.8\times
10^{-6}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{D}^{*};0^{++}\rangle\to\chi_{c0}f_{0}(500))$
$\displaystyle=$ $\displaystyle a_{4}^{2}~{}4.1\times 10^{-6}~{}{\rm
GeV}^{7}\,,$

where $a_{4}$ is an overall factor.

Besides, $|D^{*}\bar{D}^{*};0^{++}\rangle$ can directly fall apart into the
$D^{*}$ and $\bar{D}^{*}$ mesons, and further decay into the $D^{*}\bar{D}\pi$
and $D\bar{D}^{*}\pi$ final states. This process is depicted in Fig. 4(b). We
estimate its partial width to be

$\displaystyle\Gamma(|D^{*}\bar{D}^{*};0^{++}\rangle\to D^{*}\bar{D}^{*}\to
D^{*}\bar{D}\pi+D\bar{D}^{*}\pi)$ (104)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=a_{4}^{\prime
2}~{}1.7\times 10^{-6}~{}{\rm GeV}^{7}\,,$

where $a_{4}^{\prime}$ is another overall factor. It is probably larger than
$a_{4}$, and we define their ratio to be $t\equiv a_{4}^{\prime}/a_{4}$. This
parameter measures the dynamical difference between the process depicted in
Fig. 4(a) and the one depicted Fig. 4(b).

The uncertainty of Eq. (104) is quite large, because we choose the mass of
$|D^{*}\bar{D}^{*};0^{++}\rangle$ to be just at the $D^{*0}\bar{D}^{*0}$
threshold, so that only the $D^{*0}\bar{D}^{0}\pi^{0}$ and
$D^{0}\bar{D}^{*0}\pi^{0}$ final states are kinematically allowed. However, if
its realistic mass turns out to be slightly larger, some other final states
such as $D^{*0}\bar{D}^{*0}$ and $D^{*0}D^{+}\pi^{-}$ may also become
possible. Moreover, phase spaces of the $D^{*0}\bar{D}^{0}\pi^{0}$ and
$D^{0}\bar{D}^{*0}\pi^{0}$ final states depend significantly on the mass of
$|D^{*}\bar{D}^{*};0^{++}\rangle$.

####  IV.1.3 $|D^{*}\bar{D}^{*};2^{++}\rangle$

The current $\eta_{6}^{\alpha\beta}$ corresponds to the
$|D^{*}\bar{D}^{*};2^{++}\rangle$ molecular state. We can apply the Fierz
rearrangement and transform it into:

$\displaystyle\eta_{6}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\mu\nu}~{}\bar{q}_{a}\gamma_{\mu}c_{a}~{}\bar{c}_{b}\gamma_{\nu}q_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
6}~{}P^{\alpha\beta,\mu\nu}\bar{q}_{a}\gamma_{\mu}q_{a}~{}\bar{c}_{b}\gamma_{\nu}c_{b}$
$\displaystyle-{1\over
6}~{}P^{\alpha\beta,\mu\nu}\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}~{}\bar{c}_{b}\gamma_{\nu}\gamma_{5}c_{b}$
$\displaystyle+{1\over
6}~{}P^{\alpha\beta,\mu\nu}\bar{q}_{a}\sigma_{\mu\rho}q_{a}~{}\bar{c}_{b}\sigma_{\nu\rho}c_{b}+\cdots\,.$

Assuming the isospin of $|D^{*}\bar{D}^{*};2^{++}\rangle$ to be $I=0$ and its
mass to be about $M_{D^{*0}}+M_{\bar{D}^{*0}}=4014$ MeV, we obtain the
following partial decay widths:

$\displaystyle\Gamma(|D^{*}\bar{D}^{*};2^{++}\rangle\to\eta_{c}\eta)$
$\displaystyle=$ $\displaystyle a_{6}^{2}~{}5.1\times 10^{-7}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|D^{*}\bar{D}^{*};2^{++}\rangle\to
J/\psi\omega)$ $\displaystyle=$ $\displaystyle a_{6}^{2}~{}3.7\times
10^{-5}~{}{\rm GeV}^{7}\,,$

where $a_{6}$ is an overall factor.

Besides, $|D^{*}\bar{D}^{*};2^{++}\rangle$ can directly fall apart into the
$D^{*}$ and $\bar{D}^{*}$ mesons, and further decay into the $D^{*}\bar{D}\pi$
and $D\bar{D}^{*}\pi$ final states:

$\displaystyle\Gamma(|D^{*}\bar{D}^{*};2^{++}\rangle\to D^{*}\bar{D}^{*}\to
D^{*}\bar{D}\pi+D\bar{D}^{*}\pi)$ (107)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=a_{6}^{\prime
2}~{}4.9\times 10^{-6}~{}{\rm GeV}^{7}\,,$

where $a_{6}^{\prime}\approx t\times a_{6}$ is another overall factor.

####  IV.1.4 $|D\bar{D}^{*};1^{++}\rangle$

The current $\eta_{2}^{\alpha}$ corresponds to the
$|D\bar{D}^{*};1^{++}\rangle$ molecular state. Based on this current, we have
systematically studied decay properties of the $X(3872)$ as
$|D\bar{D}^{*};1^{++}\rangle$ in Ref. Chen:2019eeq . We summarize some of
those results here.

The isospin breaking effect of the $X(3872)$ is significant and important to
understand its nature. In Ref. Chen:2019eeq  we assumed it to be the
combination of both $I=0$ and $I=1$ $|D\bar{D}^{*};1^{++}\rangle$ states:

$X(3872)=\cos\theta~{}|D\bar{D}^{*};0^{+}1^{++}\rangle+\sin\theta~{}|D\bar{D}^{*};1^{-}1^{++}\rangle\,.$
(108)

After fine-tuning the isospin-breaking angle to be $\theta=\pm 15^{\circ}$,
the BESIII measurement Ablikim:2019zio ,

${\mathcal{B}(X(3872)\to J/\psi\omega\to
J/\psi\pi\pi\pi)\over\mathcal{B}(X(3872)\rightarrow J/\psi\rho\rightarrow
J/\psi\pi\pi)}=1.6^{+0.4}_{-0.3}\pm 0.2\,,$ (109)

can be explained.

In the present study we further select the positive angle
$\theta=+15^{\circ}$, so that the $D^{0}\bar{D}^{*0}$ component is more than
the $D^{+}D^{*-}$ one:

$\displaystyle X(3872)$ $\displaystyle=$ $\displaystyle\cos
15^{\circ}~{}|D\bar{D}^{*};0^{+}1^{++}\rangle+\sin
15^{\circ}~{}|D\bar{D}^{*};1^{-}1^{++}\rangle$ $\displaystyle=$ $\displaystyle
0.8684~{}|D^{0}\bar{D}^{*0};1^{++}\rangle+0.4959~{}|D^{+}D^{*-};1^{++}\rangle\,.$

Using this angle, decay properties of the $X(3872)$ as
$|D\bar{D}^{*};1^{++}\rangle$ were systematically studied in Ref. Chen:2019eeq
, and the results are summarized in Eq. (133) of Sec. IV.3.

Besides, productions of the $X(3872)$ as $|D\bar{D}^{*};1^{++}\rangle$ in $B$
and $B^{*}$ decays are increased from Eqs. (78) and (93) to be

$\displaystyle\Gamma(B^{-}\to K^{-}|D\bar{D}^{*};1^{++}\rangle)$
$\displaystyle=$ $\displaystyle d_{1}^{2}~{}0.60\times 10^{-10}~{}{\rm
GeV}^{13}\,,$ $\displaystyle\Gamma(B^{*-}\to
K^{-}|D\bar{D}^{*};1^{++}\rangle)$ $\displaystyle=$ $\displaystyle
d_{3}^{2}~{}0.93\times 10^{-10}~{}{\rm GeV}^{13}\,.$

The two ratios $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ given in Table 3 need
to be accordingly modified.

####  IV.1.5 $|D\bar{D}^{*};1^{+-}\rangle$ and
$|D^{*}\bar{D}^{*};1^{+-}\rangle$

The currents $\eta_{3}^{\alpha}$ and $\eta_{5}^{\alpha}$ correspond to the
$|D\bar{D}^{*};1^{+-}\rangle$ and $|D^{*}\bar{D}^{*};1^{+-}\rangle$ molecular
states, respectively. Based on these two currents, we have systematically
studied decay properties of the $Z_{c}(3900)$ as $|D\bar{D}^{*};1^{+-}\rangle$
in Ref. Chen:2019wjd . We summarize some of those results here. Especially, in
Ref. Chen:2019wjd  we have considered the mixing between
$|D\bar{D}^{*};1^{+-}\rangle$ and $|D^{*}\bar{D}^{*};1^{+-}\rangle$:

$\displaystyle|D\bar{D}^{*};1^{+-}\rangle^{\prime}$ $\displaystyle=$
$\displaystyle+\cos\phi~{}|D\bar{D}^{*}\rangle_{1^{+-}}+\sin\phi~{}|D^{*}\bar{D}^{*}\rangle_{1^{+-}}\,,$
$\displaystyle|D^{*}\bar{D}^{*};1^{+-}\rangle^{\prime}$ $\displaystyle=$
$\displaystyle-\sin\phi~{}|D\bar{D}^{*}\rangle_{1^{+-}}+\cos\phi~{}|D^{*}\bar{D}^{*}\rangle_{1^{+-}}\,,$

in order to explain the BESIII measurement Ablikim:2019ipd :

${\mathcal{B}(Z_{c}(3900)^{\pm}\rightarrow\eta_{c}\rho^{\pm})\over\mathcal{B}(Z_{c}(3900)^{\pm}\rightarrow
J/\psi\pi^{\pm})}=2.2\pm 0.9\,.$ (113)

In the present study we do not consider this mixing any more. We just use the
single current $\eta_{3}^{\alpha}$ to study decay properties of the
$Z_{c}(3900)$ as $|D\bar{D}^{*};1^{+-}\rangle$, and use the single current
$\eta_{5}^{\alpha}$ to study the $Z_{c}(4020)$ as
$|D^{*}\bar{D}^{*};1^{+-}\rangle$. The results are summarized in Eqs. (IV.3)
and (135) of Sec. IV.3. Actually, as shown in Table 4 of Appendix B, the above
BESIII measurement given in Eq. (113) can also be explained if we take into
account those unknown phase angles among different coupling constants.

###  IV.2 Decay properties of $D^{(*)}\bar{K}^{(*)}$ molecules

In this subsection we perform the Fierz rearrangement for $\xi_{1\cdots 6}$,
and use the obtained results to study strong decay properties of
$D^{(*)}\bar{K}^{(*)}$ molecular states. Again, we adopt the possible
interpretation of the $X_{0}(2900)$ as the $|D^{*}\bar{K}^{*};0^{+}\rangle$
molecular state. Masses of $D^{(*)}\bar{K}^{(*)}$ molecular states are
accordingly chosen, as given in Eqs. (82).

####  IV.2.1 $|D\bar{K};0^{+}\rangle$ and $|D^{*}\bar{K}^{*};0^{+}\rangle$

The currents $\xi_{1}$ and $\xi_{4}$ correspond to the
$|D\bar{K};0^{+}\rangle$ and $|D^{*}\bar{K}^{*};0^{+}\rangle$ molecular
states, respectively. We can apply the Fierz rearrangement and transform them
into:

$\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}c_{a}~{}\bar{q}_{b}\gamma_{5}s_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
12}~{}\bar{q}_{a}s_{a}~{}\bar{q}_{b}c_{b}-{1\over
12}~{}\bar{q}_{a}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma_{5}c_{b}$
$\displaystyle+{1\over
12}~{}\bar{q}_{a}\gamma_{\mu}s_{a}~{}\bar{q}_{b}\gamma^{\mu}c_{b}-{1\over
12}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma^{\mu}\gamma_{5}c_{b}$
$\displaystyle-{1\over
24}~{}\bar{q}_{a}\sigma_{\mu\nu}s_{a}~{}\bar{q}_{b}\sigma^{\mu\nu}c_{b}+\cdots\,,$
$\displaystyle\xi_{4}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma^{\mu}c_{a}~{}\bar{q}_{b}\gamma_{\mu}s_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
3}~{}\bar{q}_{a}s_{a}~{}\bar{q}_{b}c_{b}+{1\over
3}~{}\bar{q}_{a}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma_{5}c_{b}$
$\displaystyle+{1\over
6}~{}\bar{q}_{a}\gamma_{\mu}s_{a}~{}\bar{q}_{b}\gamma^{\mu}c_{b}+{1\over
6}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma^{\mu}\gamma_{5}c_{b}+\cdots\,.$

Assuming the mass of $|D\bar{K};0^{+}\rangle$ to be about
$M_{D^{0}}+M_{K^{+}}=2359$ MeV, all its strong decay channels turn out to be
kinematically forbidden.

Assuming the mass of $|D^{*}\bar{K}^{*};0^{+}\rangle$ to be
$M_{X_{0}(2900)}=2866$ MeV, we obtain the following partial decay widths:

$\displaystyle\Gamma(|D^{*}\bar{K}^{*};0^{+}\rangle\to
D\bar{K})=b_{4}^{2}~{}1.6\times 10^{-5}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{K}^{*};0^{+}\rangle\to D^{*}\bar{K}^{*}\to
D^{*}\bar{K}\pi)$ (116)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{4}^{2}~{}5.4\times
10^{-8}~{}{\rm GeV}^{7}\,,$

where $b_{4}$ is an overall factor.

Besides, $|D^{*}\bar{K}^{*};0^{+}\rangle$ can directly fall apart into the
$D^{*}$ and $\bar{K}^{*}$ mesons, and further decay into the $D^{*}\bar{K}\pi$
final states:

$\displaystyle\Gamma(|D^{*}\bar{K}^{*};0^{+}\rangle\to D^{*}\bar{K}^{*}\to
D^{*}\bar{K}\pi)$ (117)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{4}^{\prime
2}~{}1.9\times 10^{-6}~{}{\rm GeV}^{7}\,,$

where $b_{4}^{\prime}\approx t\times b_{4}$ is another overall factor.

####  IV.2.2 $|D\bar{K}^{*};1^{+}\rangle$, $|D^{*}\bar{K};1^{+}\rangle$, and
$|D^{*}\bar{K}^{*};1^{+}\rangle$

The currents $\xi_{2}^{\alpha}$, $\xi_{3}^{\alpha}$, and $\xi_{5}^{\alpha}$
correspond to the $|D\bar{K}^{*};1^{+}\rangle$, $|D^{*}\bar{K};1^{+}\rangle$,
and $|D^{*}\bar{K}^{*};1^{+}\rangle$ molecular states, respectively. We can
apply the Fierz rearrangement and transform them into:

$\displaystyle\xi_{2}^{\alpha}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}c_{a}~{}\bar{q}_{b}\gamma^{\alpha}s_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
12}~{}\bar{q}_{a}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma^{\alpha}c_{b}-{1\over
12}~{}\bar{q}_{a}\gamma^{\alpha}s_{a}~{}\bar{q}_{b}\gamma_{5}c_{b}$
$\displaystyle+{1\over
12}~{}\bar{q}_{a}\gamma^{\alpha}\gamma_{5}s_{a}~{}\bar{q}_{b}c_{b}-{1\over
12}~{}\bar{q}_{a}s_{a}~{}\bar{q}_{b}\gamma^{\alpha}\gamma_{5}c_{b}$
$\displaystyle-{i\over
12}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\sigma^{\alpha\mu}c_{b}-{i\over
12}~{}\bar{q}_{a}\sigma^{\alpha\mu}s_{a}~{}\bar{q}_{b}\gamma_{\mu}\gamma_{5}c_{b}$
$\displaystyle+{i\over
12}~{}\bar{q}_{a}\gamma_{\mu}s_{a}~{}\bar{q}_{b}\sigma^{\alpha\mu}\gamma_{5}c_{b}-{i\over
12}~{}\bar{q}_{a}\sigma^{\alpha\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma_{\mu}c_{b}$
$\displaystyle+\cdots\,,$ $\displaystyle\xi_{3}^{\alpha}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma^{\alpha}c_{a}~{}\bar{q}_{b}\gamma_{5}s_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
12}~{}\bar{q}_{a}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma^{\alpha}c_{b}-{1\over
12}~{}\bar{q}_{a}\gamma^{\alpha}s_{a}~{}\bar{q}_{b}\gamma_{5}c_{b}$
$\displaystyle-{1\over
12}~{}\bar{q}_{a}\gamma^{\alpha}\gamma_{5}s_{a}~{}\bar{q}_{b}c_{b}+{1\over
12}~{}\bar{q}_{a}s_{a}~{}\bar{q}_{b}\gamma^{\alpha}\gamma_{5}c_{b}$
$\displaystyle-{i\over
12}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\sigma^{\alpha\mu}c_{b}-{i\over
12}~{}\bar{q}_{a}\sigma^{\alpha\mu}s_{a}~{}\bar{q}_{b}\gamma_{\mu}\gamma_{5}c_{b}$
$\displaystyle-{i\over
12}~{}\bar{q}_{a}\gamma_{\mu}s_{a}~{}\bar{q}_{b}\sigma^{\alpha\mu}\gamma_{5}c_{b}+{i\over
12}~{}\bar{q}_{a}\sigma^{\alpha\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma_{\mu}c_{b}$
$\displaystyle+\cdots\,,$ $\displaystyle\xi_{5}^{\alpha}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{\mu}c_{a}~{}\bar{q}_{b}\sigma^{\alpha\mu}\gamma_{5}s_{b}-\\{\gamma_{\mu}\leftrightarrow\sigma^{\alpha\mu}\gamma_{5}\\}$
$\displaystyle\rightarrow$ $\displaystyle-{i\over
2}~{}\bar{q}_{a}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma^{\alpha}c_{b}+{i\over
2}~{}\bar{q}_{a}\gamma^{\alpha}s_{a}~{}\bar{q}_{b}\gamma_{5}c_{b}$
$\displaystyle+{1\over
6}~{}\bar{q}_{a}\gamma_{\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\sigma^{\alpha\mu}c_{b}-{1\over
6}~{}\bar{q}_{a}\sigma^{\alpha\mu}s_{a}~{}\bar{q}_{b}\gamma_{\mu}\gamma_{5}c_{b}$
$\displaystyle+\cdots\,.$

Assuming the mass of $|D\bar{K}^{*};1^{+}\rangle$ to be about
$M_{D^{0}}+M_{K^{*+}}=2756$ MeV, we obtain the following partial decay widths:

$\displaystyle\Gamma(|D\bar{K}^{*};1^{+}\rangle\to
D^{*}\bar{K})=b_{2}^{2}~{}1.7\times 10^{-7}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D\bar{K}^{*};1^{+}\rangle\to D\bar{K}^{*}\to
D\bar{K}\pi)$ (121)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{2}^{2}~{}4.7\times
10^{-9}~{}{\rm GeV}^{7}\,,$ $\displaystyle\Gamma(|D\bar{K}^{*};1^{+}\rangle\to
D^{*}\bar{K}^{*}\to D^{*}\bar{K}\pi)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{2}^{2}~{}7.0\times
10^{-10}~{}{\rm GeV}^{7}\,,$

where $b_{2}$ is an overall factor.

Besides, $|D\bar{K}^{*};1^{+}\rangle$ can directly fall apart into the $D$ and
$\bar{K}^{*}$ mesons, and further decay into the $D\bar{K}\pi$ final states:

$\displaystyle\Gamma(|D\bar{K}^{*};1^{+}\rangle\to D\bar{K}^{*}\to
D\bar{K}\pi)$ (122)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{2}^{\prime
2}~{}2.9\times 10^{-6}~{}{\rm GeV}^{7}\,,$

where $b_{2}^{\prime}\approx t\times b_{2}$ is another overall factor.

Assuming the mass of $|D^{*}\bar{K};1^{+}\rangle$ to be about
$M_{D^{*0}}+M_{K^{+}}=2501$ MeV, we obtain the following partial decay widths:

$\displaystyle\Gamma(|D^{*}\bar{K};1^{+}\rangle\to D\bar{K}^{*}\to
D\bar{K}\pi)$ (123)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{3}^{2}~{}5.4\times
10^{-16}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{K};1^{+}\rangle\to D^{*}\bar{K}\to
D\pi\bar{K})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{3}^{2}~{}1.3\times
10^{-12}~{}{\rm GeV}^{7}\,,$

where $b_{3}$ is an overall factor.

Besides, $|D^{*}\bar{K};1^{+}\rangle$ can directly fall apart into the $D^{*}$
and $\bar{K}$ mesons, and further decay into the $D\pi\bar{K}$ final states:

$\displaystyle\Gamma(|D^{*}\bar{K};1^{+}\rangle\to D^{*}\bar{K}\to
D\pi\bar{K})$ (124)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{3}^{\prime
2}~{}3.2\times 10^{-10}~{}{\rm GeV}^{7}\,,$

where $b_{3}^{\prime}\approx t\times b_{3}$ is another overall factor.

Assuming the mass of $|D^{*}\bar{K}^{*};1^{+}\rangle$ to be about
$M_{D^{*0}}+M_{K^{*+}}=2899$ MeV, we obtain the following partial decay
widths:

$\displaystyle\Gamma(|D^{*}\bar{K}^{*};1^{+}\rangle\to D\bar{K}^{*})$
$\displaystyle=$ $\displaystyle b_{5}^{2}~{}3.8\times 10^{-6}~{}{\rm
GeV}^{7}\,,$ $\displaystyle\Gamma(|D^{*}\bar{K}^{*};1^{+}\rangle\to
D^{*}\bar{K})$ $\displaystyle=$ $\displaystyle b_{5}^{2}~{}1.2\times
10^{-5}~{}{\rm GeV}^{7}\,,$

where $b_{5}$ is an overall factor.

Besides, $|D^{*}\bar{K}^{*};1^{+}\rangle$ can directly fall apart into the
$D^{*}$ and $\bar{K}^{*}$ mesons, and further decay into the $D^{*}\bar{K}\pi$
and $D\pi\bar{K}^{*}$ final states:

$\displaystyle\Gamma(|D^{*}\bar{K}^{*};1^{+}\rangle\to D^{*}\bar{K}^{*}\to
D^{*}\bar{K}\pi)$ (126)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{5}^{\prime
2}~{}1.1\times 10^{-5}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{K}^{*};1^{+}\rangle\to D^{*}\bar{K}^{*}\to
D\pi\bar{K}^{*})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{5}^{\prime
2}~{}1.1\times 10^{-9}~{}{\rm GeV}^{7}\,,$

where $b_{5}^{\prime}\approx t\times b_{5}$ is another overall factor.

####  IV.2.3 $|D^{*}\bar{K}^{*};2^{+}\rangle$

The current $\xi_{6}^{\alpha\beta}$ corresponds to the
$|D^{*}\bar{K}^{*};2^{+}\rangle$ molecular state. We can apply the Fierz
rearrangement and transform it into:

$\displaystyle\xi_{6}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
P^{\alpha\beta,\mu\nu}~{}\bar{q}_{a}\gamma_{\mu}c_{a}~{}\bar{q}_{b}\gamma_{\nu}s_{b}$
$\displaystyle\rightarrow$ $\displaystyle-{1\over
6}~{}P^{\alpha\beta,\mu\nu}\bar{q}_{a}\gamma_{\mu}s_{a}~{}\bar{q}_{b}\gamma_{\nu}c_{b}$
$\displaystyle-{1\over
6}~{}P^{\alpha\beta,\mu\nu}\bar{q}_{a}\gamma_{\mu}\gamma_{5}s_{a}~{}\bar{q}_{b}\gamma_{\nu}\gamma_{5}c_{b}$
$\displaystyle+{1\over
6}~{}P^{\alpha\beta,\mu\nu}\bar{q}_{a}\sigma_{\mu\rho}s_{a}~{}\bar{q}_{b}\sigma_{\nu\rho}c_{b}+\cdots\,.$

Assuming the mass of $|D^{*}\bar{K}^{*};2^{+}\rangle$ to be about
$M_{D^{*0}}+M_{K^{*+}}=2899$ MeV, we obtain the following partial decay
widths:

$\displaystyle\Gamma(|D^{*}\bar{K}^{*};2^{+}\rangle\to
D\bar{K})=b_{6}^{2}~{}3.0\times 10^{-7}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{K}^{*};2^{+}\rangle\to D^{*}\bar{K}^{*}\to
D^{*}\bar{K}\pi)$ (128)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{6}^{2}~{}1.3\times
10^{-6}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{K}^{*};2^{+}\rangle\to D^{*}\bar{K}^{*}\to
D\pi\bar{K}^{*})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{6}^{2}~{}7.4\times
10^{-11}~{}{\rm GeV}^{7}\,,$

where $b_{6}$ is an overall factor.

Besides, $|D^{*}\bar{K}^{*};2^{+}\rangle$ can directly fall apart into the
$D^{*}$ and $\bar{K}^{*}$ mesons, and further decay into the $D^{*}\bar{K}\pi$
and $D\pi\bar{K}^{*}$ final states:

$\displaystyle\Gamma(|D^{*}\bar{K}^{*};2^{+}\rangle\to D^{*}\bar{K}^{*}\to
D^{*}\bar{K}\pi)$ (129)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{6}^{\prime
2}~{}1.7\times 10^{-5}~{}{\rm GeV}^{7}\,,$
$\displaystyle\Gamma(|D^{*}\bar{K}^{*};2^{+}\rangle\to D^{*}\bar{K}^{*}\to
D\pi\bar{K}^{*})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=b_{6}^{\prime
2}~{}9.9\times 10^{-10}~{}{\rm GeV}^{7}\,,$

where $b_{6}^{\prime}\approx t\times b_{6}$ is another overall factor.

###  IV.3 Summary of decay properties

We summarize the relative branching ratios obtained in Sec. IV.1 and Sec. IV.2
here. The Fierz rearrangements for $\zeta_{1\cdots 6}$ are quite similar to
those for $\eta_{1\cdots 6}$, just with one $up$/$down$ quark replaced by
another $strange$ quark. Hence, decay properties of $D^{(*)}\bar{D}_{s}^{(*)}$
molecular states can be similarly investigated, and we also summarize their
results here. We use the parameter
$t\approx{a_{i}^{\prime}/a_{i}}\approx{b_{i}^{\prime}/b_{i}}$ ($i=1\cdots 6$)
to measure the dynamical difference between the process depicted in Fig. 4(a)
and the one depicted in Fig. 4(b).

We obtain the following relative branching ratios for
$|D\bar{D};0^{++}\rangle$, $|D^{*}\bar{D}^{*};0^{++}\rangle$, and
$|D^{*}\bar{D}^{*};2^{++}\rangle$, all of which are assumed to have $I=0$:

$\displaystyle\mathcal{B}\Big{(}{|D\bar{D};0^{++}\rangle}\to~{}\eta_{c}\eta~{}:~{}J/\psi\omega(\to\pi\pi\pi)~{}:~{}\chi_{c0}f_{0}(500)(\to\pi\pi)~{}\Big{)}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}:~{}~{}~{}~{}~{}~{}~{}\,10^{-6}\,~{}~{}~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}10^{-5}~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{D}^{*};0^{++}\rangle}\to~{}\eta_{c}\eta~{}:~{}~{}J/\psi\omega~{}~{}:~{}\chi_{c0}f_{0}(500)~{}:~{}D^{*}\bar{D}^{*}(\to\bar{D}\pi)~{}\Big{)}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}:~{}~{}\,0.001~{}~{}:~{}~{}~{}~{}\,0.001\,~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}10^{-4}t~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{D}^{*};2^{++}\rangle}\to~{}\eta_{c}\eta~{}:~{}~{}J/\psi\omega~{}~{}:~{}D^{*}\bar{D}^{*}(\to\bar{D}\pi)~{}\Big{)}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}:~{}~{}~{}~{}73~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}9.5t~{}\,.$
(132)

Decay properties of the $X(3872)$ as $|D\bar{D}^{*};1^{++}\rangle$ have been
systematically studied in Ref. Chen:2019eeq , where we obtain

$\displaystyle{\mathcal{B}\Big{(}|D\bar{D}^{*};1^{++}\rangle\rightarrow
J/\psi\omega(\rightarrow\pi\pi\pi):J/\psi\rho(\rightarrow\pi\pi):~{}\chi_{c0}\pi~{}:\eta_{c}f_{0}(\rightarrow\pi\pi):\chi_{c1}f_{0}(\rightarrow\pi\pi):D\bar{D}^{*}(\rightarrow\bar{D}\pi)~{}\Big{)}}$
(133) $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}~{}~{}~{}~{}:\,0.63~{}({\rm
input})\,:~{}0.015\,:~{}~{}~{}~{}\,0.091~{}~{}~{}~{}:~{}~{}~{}~{}~{}0.086~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}7.4t~{}\,.$

In the calculations we have assumed the $X(3872)$ to be the combination of
both $I=0$ and $I=1$ $|D\bar{D}^{*};1^{++}\rangle$ molecular states.

Decay properties of the $Z_{c}(3900)$ as $|D\bar{D}^{*};1^{+-}\rangle$ have
been systematically studied in Ref. Chen:2019wjd . In the present study we
further study the $Z_{c}(4020)$ as $|D^{*}\bar{D}^{*};1^{+-}\rangle$.
Altogether, we obtain

$\displaystyle{\mathcal{B}\Big{(}|D\bar{D}^{*};1^{+-}\rangle\rightarrow~{}~{}J/\psi\pi~{}~{}:~{}~{}~{}\eta_{c}\rho~{}~{}~{}:~{}~{}~{}h_{c}\pi~{}~{}~{}:~{}\chi_{c1}\rho(\rightarrow\pi\pi)~{}:~{}~{}D\bar{D}^{*}(\to\bar{D}\pi)~{}\Big{)}}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1\,~{}~{}~{}:~{}~{}0.092\,~{}:~{}~{}0.011~{}~{}:~{}~{}~{}~{}~{}~{}10^{-6}~{}~{}~{}~{}~{}\,:~{}~{}~{}~{}~{}~{}~{}74t~{}\,,$
$\displaystyle{\mathcal{B}\Big{(}|D^{*}\bar{D}^{*};1^{+-}\rangle\rightarrow~{}~{}J/\psi\pi~{}~{}:~{}~{}\eta_{c}\rho~{}~{}:~{}~{}h_{c}\pi~{}~{}:~{}\chi_{c1}\rho(\rightarrow\pi\pi)~{}:~{}D^{*}\bar{D}^{*}(\to\bar{D}\pi)~{}\Big{)}}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1\,~{}~{}~{}:~{}\,0.33\,~{}:~{}0.002~{}:~{}~{}~{}~{}~{}~{}10^{-5}\,~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}11t~{}\,.$
(135)

In the calculations we have assumed that the $Z_{c}(3900)$ and $Z_{c}(4020)$
both have $I=1$.

Assuming the mass of $|D\bar{K};0^{+}\rangle$ to be about
$M_{D^{0}}+M_{K^{+}}=2359$ MeV, all its strong decay channels turn out to be
kinematically forbidden.

We obtain the following relative branching ratios for
$|D^{*}\bar{K};1^{+}\rangle$:

$\displaystyle{\mathcal{B}\Big{(}{|D^{*}\bar{K};1^{+}\rangle}\to~{}D\bar{K}^{*}(\to\bar{K}\pi)~{}:~{}\bar{K}D^{*}(\to
D\pi)~{}\Big{)}}$ (136) $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}10^{-6}\,~{}~{}~{}~{}:~{}~{}~{}~{}t+0.004~{}\,.$

The isospin-averaged width of the $\bar{K}^{*}$ meson is about
$\Gamma_{K^{*}}\approx 49.1$ MeV pdg , which value is quite large and can not
be neglected. We refer to Appendix A for more discussions. Taking this into
account, we estimate widths of $|D\bar{K}^{*};1^{+}\rangle$,
$|D^{*}\bar{K}^{*};0^{+}\rangle$, $|D^{*}\bar{K}^{*};1^{+}\rangle$, and
$|D^{*}\bar{K}^{*};2^{+}\rangle$ to be

$\Gamma_{|D^{(*)}\bar{K}^{*};J^{+}\rangle}\approx\Gamma_{\bar{K}^{*}}+\Gamma_{|D^{(*)}\bar{K}^{*};J^{+}\rangle\to
D\bar{K}/D^{*}\bar{K}/D\bar{K}^{*}/D^{*}\bar{K}^{*}/\cdots}\,,$ (137)

with the following relative branching ratios

$\displaystyle\mathcal{B}\Big{(}{|D\bar{K}^{*};1^{+}\rangle}\to~{}D\bar{K}^{*}(\to\bar{K}\pi)~{}:~{}~{}D^{*}\bar{K}~{}~{}:~{}D^{*}\bar{K}^{*}(\to\bar{K}\pi)~{}\Big{)}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t+0.002~{}~{}~{}:~{}~{}0.057~{}~{}:~{}~{}~{}~{}~{}~{}10^{-4}~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{K}^{*};0^{+}\rangle}\to~{}~{}D\bar{K}~{}~{}:~{}D^{*}\bar{K}^{*}(\to\bar{K}\pi)~{}\Big{)}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}8.5~{}~{}~{}:~{}~{}~{}~{}\,t+0.028~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{K}^{*};1^{+}\rangle}\to~{}~{}D\bar{K}^{*}~{}~{}:~{}~{}D^{*}\bar{K}~{}~{}:~{}D^{*}\bar{K}^{*}(\to\bar{K}\pi)~{}:~{}\bar{K}^{*}D^{*}(\to
D\pi)~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,0.34~{}~{}~{}:~{}~{}~{}~{}1.1\,~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t\,~{}~{}~{}~{}~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}10^{-4}t~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{K}^{*};2^{+}\rangle}\to~{}~{}D\bar{K}~{}~{}:~{}D^{*}\bar{K}^{*}(\to\bar{K}\pi)~{}:~{}\bar{K}^{*}D^{*}(\to
D\pi)~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0.018~{}~{}:~{}~{}~{}~{}\,t+0.074\,~{}~{}~{}~{}:~{}~{}~{}~{}10^{-4}t~{}\,.$
(141)

We obtain the following relative branching ratios for
$D^{(*)}\bar{D}_{s}^{(*)}$ molecular states:

$\displaystyle\mathcal{B}\Big{(}{|D\bar{D}_{s};0^{+}\rangle}\to~{}\eta_{c}\bar{K}~{}:~{}J/\psi\bar{K}^{*}(\to\bar{K}\pi)~{}\Big{)}$
$\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,1~{}~{}:~{}~{}~{}~{}~{}~{}~{}0.001~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D\bar{D}_{s}^{*};1^{++}\rangle}\to~{}J/\psi\bar{K}^{*}(\to\bar{K}\pi)~{}:~{}~{}\chi_{c0}\bar{K}~{}~{}:~{}~{}D\bar{D}_{s}^{*}~{}~{}:~{}~{}\bar{D}_{s}D^{*}(\to
D\pi)~{}~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}:~{}~{}0.012~{}~{}:~{}~{}~{}~{}25~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}~{}68~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D\bar{D}_{s}^{*};1^{+-}\rangle}\to~{}J/\psi\bar{K}~{}:~{}~{}\eta_{c}\bar{K}^{*}~{}~{}:~{}~{}D\bar{D}_{s}^{*}~{}~{}:~{}~{}\bar{D}_{s}D^{*}(\to
D\pi)~{}~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,1~{}~{}~{}:~{}~{}0.093~{}~{}:~{}~{}~{}~{}41~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}~{}87~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{D}_{s}^{*};0^{+}\rangle}\to~{}\eta_{c}\bar{K}~{}:~{}J/\psi\bar{K}^{*}~{}:~{}\bar{D}_{s}^{*}D^{*}(\to
D\pi)~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}:~{}~{}\,0.088\,~{}~{}:~{}~{}~{}~{}~{}~{}0.001~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D_{s}\bar{D}_{s}^{*};1^{+}\rangle}\to~{}J/\psi\bar{K}~{}:~{}~{}\eta_{c}\bar{K}^{*}~{}~{}:~{}~{}h_{c}\bar{K}~{}~{}:~{}\chi_{c1}\bar{K}^{*}(\to\bar{K}\pi)~{}:~{}~{}D_{s}^{*}\bar{D}^{*}(\to
D\pi)~{}~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,1~{}~{}~{}:~{}~{}~{}0.36\,~{}~{}:~{}\,0.002\,~{}:~{}~{}~{}~{}~{}~{}~{}\,10^{-7}\,~{}~{}~{}~{}~{}~{}~{}:~{}~{}~{}~{}~{}~{}~{}~{}31~{}\,,$
$\displaystyle\mathcal{B}\Big{(}{|D^{*}\bar{D}_{s}^{*};2^{+}\rangle}\to~{}\eta_{c}\bar{K}~{}:~{}J/\psi\bar{K}^{*}~{}:~{}\bar{D}_{s}^{*}D^{*}(\to
D\pi)~{}\Big{)}$ $\displaystyle\approx$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1~{}~{}:~{}~{}~{}~{}\,27\,~{}~{}~{}~{}:~{}~{}~{}~{}~{}0.055~{}\,.$
(147)

In the calculations we have adopted the possible interpretations of the
$Z_{cs}(3985)$, $Z_{cs}(4000)$, and $Z_{cs}(4220)$ as the
$|D\bar{D}_{s}^{*};1^{++}\rangle$, $|D\bar{D}_{s}^{*};1^{+-}\rangle$, and
$|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$ molecular states, respectively. Masses of
$D^{(*)}\bar{D}_{s}^{(*)}$ molecular states are accordingly chosen, as given
in Eqs. (88).

##  V Summary and discussions

Table 3: Relative branching ratios of $D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ hadronic molecular states and their relative production rates in $B$ and $B^{*}$ decays. In the 2nd and 3rd columns we show their relative production rates $\mathcal{R}_{1}\equiv{\mathcal{B}\left(B^{-}\rightarrow K^{-}X/D^{-}X/\phi X\right)\over\mathcal{B}\left(B^{-}\rightarrow D^{-}|D^{*}\bar{K}^{*};0^{+}\rangle\right)/d_{2}}$ and $\mathcal{R}_{2}\equiv{\mathcal{B}\left(B^{*-}\rightarrow K^{-}X/D^{-}X/\phi X\right)\over\mathcal{B}\left(B^{*-}\rightarrow K^{-}|D\bar{D};0^{++}\rangle\right)/d_{4}}$. In the 4th-16th/9th/13th columns we show their relative branching ratios, where we use the symbol “$AB$” to generally denote the two-body decay $X\to AB$, the three-body decay process $X\to AB\to Ab_{1}b_{2}$, or the four-body decay process $X\to AB\to Ab_{1}b_{2}b_{3}$, depending on which channel is kinematically allowed. The number $0.63$ with $\dagger$ is fixed by the BESIII measurement ${\mathcal{B}(X(3872)\to J/\psi\omega\to J/\psi\pi\pi\pi)\over\mathcal{B}(X(3872)\rightarrow J/\psi\rho\rightarrow J/\psi\pi\pi)}=1.6^{+0.4}_{-0.3}\pm 0.2$ Ablikim:2019zio , which is related to the isospin breaking effect of the $X(3872)$ interpreted as the $|D\bar{D}^{*};1^{++}\rangle$ molecular state. The parameters $t^{\prime}\approx d_{2}/d_{1}\approx d_{5}/d_{4}$ and $t^{\prime\prime}\approx d_{3}/d_{1}\approx d_{6}/d_{4}$ (partly) measure the dynamical difference among Fig. 3(a,b,c), and the parameter $t\approx{a_{i}^{\prime}/a_{i}}\approx{b_{i}^{\prime}/b_{i}}$ ($i=1\cdots 6$) measures the dynamical difference between Fig. 4(a) and Fig. 4(b). Configuration | Productions | Decay Channels | Candidate
---|---|---|---
$\mathcal{R}_{1}$ |  $\mathcal{R}_{2}$ |  $\eta_{c}\eta$ | $J/\psi\omega$ | $\eta_{c}f_{0}$ | $\chi_{c0}f_{0}$ | $\chi_{c1}f_{0}$ | $J/\psi\pi$ | $\chi_{c0}\pi$ | $h_{c}\pi$ | $\eta_{c}\rho$ | $J/\psi\rho$ | $\chi_{c1}\rho$ | $D\bar{D}^{*}$ | $D^{*}\bar{D}^{*}$
$|D\bar{D};0^{++}\rangle$ | – | $d_{4}$ | $1$ | $10^{-6}$ | – | $10^{-5}$ | – | – | – | – | – | – | – | – | – | –
$|D\bar{D}^{*};1^{++}\rangle$ | $0.19d_{1}$ | $5.7d_{4}$ | – | $1$ | $0.091$ | – | $0.086$ | – | $0.015$ | – | – | $0.63^{\dagger}$ | – | $7.4t$ | – | $X(3872)$
$|D\bar{D}^{*};1^{+-}\rangle$ | $0.12d_{1}$ | $3.7d_{4}$ | – | – | – | – | – | $1$ | – | $0.011$ | $0.092$ | – | $10^{-6}$ | $74t$ | – | $Z_{c}(3900)$
$|D^{*}\bar{D}^{*};0^{++}\rangle$ | $1.3d_{1}$ | – | $1$ | $0.001$ | – | $0.001$ | – | – | – | – | – | – | – | – | $10^{-4}t$ | –
$|D^{*}\bar{D}^{*};1^{+-}\rangle$ | $0.42d_{1}$ | $17d_{4}$ | – | – | – | – | – | $1$ | – | $0.002$ | $0.33$ | – | $10^{-5}$ | – | $11t$ | $Z_{c}(4020)$
$|D^{*}\bar{D}^{*};2^{++}\rangle$ | – | – | $1$ | $73$ | – | – | – | – | – | – | – | – | – | – | $9.5t$ | –
Configuration |  $\mathcal{R}_{1}$ |  $\mathcal{R}_{2}$ |  $\Gamma_{\bar{K}^{*}}$ [MeV] | $D\bar{K}$ | $D\bar{K}^{*}$ | $D^{*}\bar{K}$ | $D^{*}\bar{K}^{*}$ | $D^{*}\bar{K}^{*}$ | Candidate
$|D\bar{K};0^{+}\rangle$ | – | $1.1d_{5}$ | – | – | – | – | – | – | –
$|D\bar{K}^{*};1^{+}\rangle$ | – | $10d_{5}$ |  $49.1$ MeV | – | ${t}+0.002$ | $0.057$ | $10^{-4}$ | – | –
$|D^{*}\bar{K};1^{+}\rangle$ | $0.67d_{2}$ | – | – | – | $10^{-6}$ | ${t}+0.004$ | – | – | –
$|D^{*}\bar{K}^{*};0^{+}\rangle$ | $d_{2}$ | – |  $49.1$ MeV | $8.5$ | – | – | ${t}+0.028$ | – | $X_{0}(2900)$
$|D^{*}\bar{K}^{*};1^{+}\rangle$ | $0.70d_{2}$ | $27d_{5}$ |  $49.1$ MeV | – | $0.34$ | $1.1$ | ${t}$ | $10^{-4}t$ | –
$|D^{*}\bar{K}^{*};2^{+}\rangle$ | – | – |  $49.1$ MeV | $0.018$ | – | – | ${t}+0.074$ | $10^{-4}t$ | –
Configuration |  $\mathcal{R}_{1}$ |  $\mathcal{R}_{2}$ | $\eta_{c}\bar{K}$ | $J/\psi\bar{K}$ | $\chi_{c0}\bar{K}$ | $h_{c}\bar{K}$ | $\eta_{c}\bar{K}^{*}$ | $J/\psi\bar{K}^{*}$ | $\chi_{c1}\bar{K}^{*}$ | $D\bar{D}_{s}^{*}$ | $D^{*}\bar{D}_{s}$ | $D^{*}\bar{D}_{s}^{*}$ | Candidate
$|D\bar{D}_{s};0^{+}\rangle$ | – | $11d_{6}$ | $1$ | – | – | – | – | $0.001$ | – | – | – | – | –
$|D\bar{D}_{s}^{*};1^{++}\rangle$ | $0.79d_{3}$ | $6.6d_{6}$ | – | – | $0.012$ | – | – | $1$ | – | $25$ | $68$ | – | $Z_{cs}(3985)$
$|D\bar{D}_{s}^{*};1^{+-}\rangle$ | $0.75d_{3}$ | $6.2d_{6}$ | – | $1$ | – | – | $0.093$ | – | – | $41$ | $87$ | – | $Z_{cs}(4000)$
$|D^{*}\bar{D}_{s}^{*};0^{+}\rangle$ | – | – | $1$ | – | – | – | – | $0.088$ | – | – | – | $0.001$ | –
$|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$ | $1.8d_{3}$ | – | – | $1$ | – | $0.002$ | $0.36$ | – | $10^{-7}$ | – | – | $31$ | $Z_{cs}(4220)$
$|D^{*}\bar{D}_{s}^{*};2^{+}\rangle$ | – | – | $1$ | – | – | – | – | $27$ | – | – | – | $0.055$ | –

In this paper we systematically investigate the eighteen possibly existing
$D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$
($\bar{D}_{s}^{(*)}\equiv D_{s}^{(*)-}$) hadronic molecular states, including
$|D\bar{D};0^{++}\rangle$, $|D\bar{D}^{*};1^{++}/1^{+-}\rangle$,
$|D^{*}\bar{D}^{*};0^{++}/1^{+-}/2^{++}\rangle$, $|D\bar{K};0^{+}\rangle$,
$|D\bar{K}^{*};1^{+}\rangle$, $|D^{*}\bar{K}^{*};0^{+}/1^{+}/2^{+}\rangle$,
$|D\bar{D}_{s};0^{+}\rangle$, $|D\bar{D}_{s}^{*};1^{++}/1^{+-}\rangle$, and
$|D^{*}\bar{D}_{s}^{*};0^{+}/1^{+}/2^{+}\rangle$. Note that the
$D\bar{D}_{s}^{*}$ molecules are not charge-conjugated, but for convenience we
use $|D\bar{D}_{s}^{*};1^{++}\rangle$ and $|D\bar{D}_{s}^{*};1^{+-}\rangle$ to
denote strange partners of $|D\bar{D}^{*};1^{++}\rangle$ and
$|D\bar{D}^{*};1^{+-}\rangle$, respectively. By doing this, we can
differentiate the $X(3872)$, $Z_{c}(3900)$, and $Z_{c}(4020)$ as well as the
$Z_{cs}(3985)$, $Z_{cs}(4000)$, and $Z_{cs}(4220)$.

We systematically construct their corresponding interpolating currents,
through which we study their mass spectra as well as their production and
decay properties. The isospin of these molecular states is important, and in
the present study we assume: a) $|D\bar{D}^{*};1^{+-}\rangle$ and
$|D^{*}\bar{D}^{*};1^{+-}\rangle$ have $I=1$, b) $|D\bar{D};0^{++}\rangle$,
$|D^{*}\bar{D}^{*};0^{++}\rangle$, and $|D^{*}\bar{D}^{*};2^{++}\rangle$ have
$I=0$, c) the isospin of the $X(3872)$ as $|D\bar{D}^{*};1^{++}\rangle$ is
separately investigated in Sec. IV.1.4, d) all the $D^{(*)}\bar{K}^{(*)}$
molecular states have $I=0$, and e) all the $D^{(*)}\bar{D}_{s}^{(*)}$
molecular states have $I=1/2$.

Firstly, we use the method of QCD sum rules to calculate masses and decay
constants of the $D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and
$D^{(*)}\bar{D}_{s}^{(*)}$ molecular states, and the obtained results are
summarized in Table 1. Our results support: a) the interpretations of the
$X(3872)$, $Z_{c}(3900)$, and $Z_{c}(4020)$ as the
$|D\bar{D}^{*};1^{++}\rangle$, $|D\bar{D}^{*};1^{+-}\rangle$, and
$|D^{*}\bar{D}^{*};1^{+-}\rangle$ molecular states, respectively; b) the
interpretation of the $X_{0}(2900)$ as the $|D^{*}\bar{K}^{*};0^{+}\rangle$
molecular state; c) the interpretations of the $Z_{cs}(3985)$, $Z_{cs}(4000)$,
and $Z_{cs}(4220)$ as the $|D\bar{D}_{s}^{*};1^{++}\rangle$,
$|D\bar{D}_{s}^{*};1^{+-}\rangle$, and $|D^{*}\bar{D}_{s}^{*};1^{+}\rangle$
molecular states, respectively. The uncertainty/accuracy is moderate but not
enough to extract the binding energy as well as to differentiate the
$Z_{cs}(3985)$ and $Z_{cs}(4000)$. Hence, our QCD sum rule results can only
suggest but not determine: a) whether these molecular states exist or not, and
b) whether they are bound states or resonance states. To better understand
them, we further study their production and decay properties, and the decay
constants $f_{X}$ extracted from QCD sum rules are important input parameters.

Secondly, we use the current algebra to study productions of
$D^{(*)}\bar{D}^{(*)}$, $D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$
molecular states in $B$ and $B^{*}$ decays. We derive the relative production
rates

$\displaystyle\mathcal{R}_{1}$ $\displaystyle\equiv$
$\displaystyle{\mathcal{B}\left(B^{-}\rightarrow K^{-}X/D^{-}X/\phi
X\right)\over\mathcal{B}\left(B^{-}\rightarrow
D^{-}|D^{*}\bar{K}^{*};0^{+}\rangle\right)/d_{2}}\,,$ (148)
$\displaystyle\mathcal{R}_{2}$ $\displaystyle\equiv$
$\displaystyle{\mathcal{B}\left(B^{*-}\rightarrow K^{-}X/D^{-}X/\phi
X\right)\over\mathcal{B}\left(B^{*-}\rightarrow
K^{-}|D\bar{D};0^{++}\rangle\right)/d_{4}}\,,$ (149)

and the obtained results are summarized in Table 3. In the calculations we
only consider $\eta_{1\cdots 6}$, $\xi_{1\cdots 6}$, and $\zeta_{1\cdots 6}$
defined in Eqs. (17–22), Eqs. (34–39), and Eqs. (47–52), which couple to these
molecular states through $S$-wave. Besides, there may exist some other
currents coupling to them through $P$-wave, which are not taken into account
in the present study. Consequently, some other molecular states such as
$|D\bar{D};0^{++}\rangle$ may still be produced in $B^{-}$ decays, and
omissions of these currents cause some theoretical uncertainties.

Thirdly, we use the Fierz rearrangement of the Dirac and color indices to
study strong decay properties of $D^{(*)}\bar{D}^{(*)}$,
$D^{(*)}\bar{K}^{(*)}$, and $D^{(*)}\bar{D}_{s}^{(*)}$ molecular states. We
calculate some of their relative branching ratios, and the obtained results
are summarized in Table 3. In the calculations we only consider the leading-
order fall-apart decays described by color-singlet-color-singlet meson-meson
currents, but neglect the $\mathcal{O}(\alpha_{s})$ corrections described by
color-octet-color-octet meson-meson currents. Hence, there may be some other
possible decay channels, and omissions of the $\mathcal{O}(\alpha_{s})$
corrections cause some theoretical uncertainties.

Generally speaking, the uncertainty of our QCD sum rule results is moderate,
as given in Table 1, while uncertainties of relative branching ratios and
relative production rates are much larger. In the present study we use local
currents and work under the naive factorization scheme, so our uncertainties
are significantly larger than the well-developed QCD factorization scheme
Beneke:1999br ; Beneke:2000ry ; Beneke:2001ev , whose uncertainty is at the 5%
level Li:2020rcg . On the other hand, we only calculate the ratios, which
significantly reduces our uncertainties Yu:2017zst . Hence, we roughly
estimate uncertainties of relative branching ratios to be at the
$X^{+100\%}_{-~{}50\%}$ level, and uncertainties of relative production rates
to be at the $X^{+200\%}_{-~{}67\%}$ level.

There are three unfixed parameters in Table 3: the parameters
$t^{\prime}\approx d_{2}/d_{1}\approx d_{5}/d_{4}$ and
$t^{\prime\prime}\approx d_{3}/d_{1}\approx d_{6}/d_{4}$ (partly) measure the
dynamical difference among Fig. 3(a,b,c), and the parameter
$t\approx{a_{i}^{\prime}/a_{i}}\approx{b_{i}^{\prime}/b_{i}}$ ($i=1\cdots 6$)
measures the dynamical difference between Fig. 4(a) and Fig. 4(b). Simply
assuming them to be $t\approx t^{\prime}\approx t^{\prime\prime}\approx 1$
(local currents), we use the results given in Table 3 to draw conclusions as
follows:

####  V.0.1 $X_{0}(2900)$ as $|D^{*}\bar{K}^{*};0^{+}\rangle$

Our QCD sum rule analyses support the interpretation of the $X_{0}(2900)$ as
the $|D^{*}\bar{K}^{*};0^{+}\rangle$ molecular state. Given its width to be
about $\Gamma_{X_{0}(2900)}=57.2$ MeV lhcb , we use Eqs. (137) and (IV.3) to
estimate:

$\Gamma_{X_{0}(2900)}\approx\Gamma_{\bar{K}^{*}}+\Gamma_{X\to
D\bar{K}}+\Gamma_{X\to D^{*}\bar{K}^{*}\to D^{*}\bar{K}\pi}\,,$ (150)

where

$\displaystyle\Gamma_{\bar{K}^{*}}$ $\displaystyle\approx$ $\displaystyle
49.1{~{}\rm MeV}\,,$ $\displaystyle\Gamma_{X\to D^{0}\bar{K}^{0}}$
$\displaystyle\approx$ $\displaystyle 3.6{~{}\rm MeV}\,,$
$\displaystyle\Gamma_{X\to D^{+}K^{-}}$ $\displaystyle\approx$ $\displaystyle