# Investigation of the tetraquark states $Qq\bar{Q}\bar{q}$ in the improved
chromomagnetic interaction model

Tao Guo  School of Mathematics and Physics, Chengdu University of Technology,
Chengdu 610059, China  Jianing Li  Department of Physics, Tsinghua University,
Beijing 100084, China  Jiaxing Zhao  SUBATECH, Université de Nantes, IMT
Atlantique, IN2P3/CNRS, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France
Lianyi He  Department of Physics, Tsinghua University, Beijing 100084, China

###### Abstract

In the framework of the improved chromomagnetic interaction model, we complete
a systematic study of the $S$-wave tetraquark states $Qq\bar{Q}\bar{q}$
($Q=c,b$, and $q=u,d,s$) with different quantum numbers, $J^{PC}=0^{+(+)}$,
$1^{+(\pm)}$, and $2^{+(+)}$. The mass spectra of tetraquark states are
predicted and the possible decay channels are analyzed by considering both the
angular momentum and $\mathcal{C}$-parity conservation. The recently observed
hidden-charm tetraquark states with strangeness, such as $Z_{cs}(3985)^{-}$,
$X(3960)$, and $Z_{cs}(4220)^{+}$, can be well explained in our model.
Besides, based on the wave function of each tetraquark state, we find that the
low-lying states of each $Qq\bar{Q}\bar{q}$ configuration have a large overlap
to the $Q\bar{Q}$ and $q\bar{q}$ meson basis, instead of $Q\bar{q}$ and
$q\bar{Q}$ meson basis. This indicates one can search these tetraquark states
in future experiments via the channel of $Q\bar{Q}$ and $q\bar{q}$ mesons.

##  I Introduction

Exotic hadrons, especially the heavy flavor exotic hadrons, provide a unique
tool to study the nature of the strong force and the low-energy properties of
Quantum Chromodynamics (QCD). Except for the open-charm tetraquark states,
such as $X_{0,1}(2900)^{0}$ Aaij  _et al._ (2020) and $T_{cc}(3875)^{+}$ Aaij
_et al._ (2022a), dozens of hidden-charm (-bottom) exotic states have been
discovered since the observation of the hidden-charm state $X(3872)$ in 2003
by the Belle Collaboration Choi  _et al._ (2003), see reviews Chen  _et al._
(2016); Esposito  _et al._ (2017); Karliner  _et al._ (2018); Guo  _et al._
(2018); Brambilla  _et al._ (2020). The most fascinating and unknown thing is
the inner structure of these exotic hadrons. Theoretically, these exotic
states can be mainly explained as multiquark states, which can be molecule
states or compact tetraquark states Esposito  _et al._ (2017); Wang  _et al._
(2013); Karliner  _et al._ (2018), hybrid states with $c\bar{c}$-$g$
configuration Close and Page (1995); Oncala and Soto (2017), missing
charmonium states, which their masses can be predicted by potential models but
drastically modified by thresholds Carames  _et al._ (2012); Chen  _et al._
(2013); Swanson (2015), a kinematical effect called triangle singularity
Nakamura (2020); Guo  _et al._ (2020); Braaten  _et al._ (2019). A clear probe
to distinguish multiquark states from hybrid states or charmonia is the
charged hidden-charm (-bottom) exotic state Chiu and Hsieh (2006).

In recent years, many hidden-charm (-bottom) exotic states with non-zero
electric change such as $Z_{c}(3900)^{+}$ Ablikim  _et al._ (2013); Liu  _et
al._ (2013), $Z_{c}(4025)^{+}$ Ablikim  _et al._ (2014), $X(4100)^{+}$ Aaij
_et al._ (2018), $Z_{c}(4430)^{+}$ Choi  _et al._ (2008); Aaij  _et al._
(2014), $Z_{b}(10610)^{+}$, and $Z_{b}(10652)^{+}$ Bondar  _et al._ (2012)
have been observed in experiments. From the decay, one can infer their quark
constitutes are $[cu\bar{c}\bar{d}]$ or $[bu\bar{b}\bar{d}]$. In the meantime,
charged hidden-charm tetraquark states with strangeness are also been found
experimentally. In 2020, the BESIII Collaboration report a charged hidden-
charm exotic structure with strangeness based on the processes of
$e^{+}e^{-}\to K^{+}D_{s}^{-}D^{\ast 0}$ and $K^{+}D_{s}^{\ast-}D^{0}$ Ablikim
_et al._ (2021). Experimental analysis shows that the exotic state has a mass
of $(3982.5_{-2.6}^{+1.8}\pm 2.1)$ MeV and a width of $(12.8_{-4.4}^{+5.3}\pm
3.0)$ MeV, which is close to the $D_{s}^{-}D^{\ast 0}$ and
$D_{s}^{\ast-}D^{0}$ thresholds. It is the first observed candidate of the
charged hidden-charm tetraquark with strangeness, $[cs\bar{c}\bar{u}]$, named
$Z_{cs}(3985)^{-}$. Next, the LHCb Collaboration observed an exotic state,
$Z_{cs}(4000)^{+}$, with a mass of $(4003\pm 6^{+4}_{-14})$ MeV and a width of
$(131\pm 15\pm 16)$ MeV in the $J/\psi K^{+}$ invariant-mass spectrum Aaij
_et al._ (2021). Its quark composition is probably $[cu\bar{c}\bar{s}]$. In
addition, three new candidates named $Z_{cs}(4220)^{+}$, $X(4685)$, and
$X(4630)$ are also observed with high significance in the $J/\psi K^{+}$ and
$J/\psi\phi$ final states, respectively Aaij  _et al._ (2021). Very recently,
a near-threshold peaking structure referred to as $X(3960)$, is discovered by
the LHCb Collaboration in the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum
Aaij  _et al._ (2022b). It’s very likely a hidden-charm and hidden-strange
tetraquark state, $[cs\bar{c}\bar{s}]$. The best fit gives the mass and width
of $X(3960)$ are $(3955\pm 6\pm 22)$ MeV and $(48\pm 17\pm 10)$ MeV,
respectively. The quantum number of this state is favored to be
$I(J^{PC})=0(0^{++})$. In addition, a possible structure $X_{0}(4140)$ is
observed in the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum Aaij  _et al._
(2022b).

On the theoretical side, the mass spectra of $Qq\bar{Q}\bar{q}$ states have
been predicted by the potential model Braaten  _et al._ (2014); Vijande  _et
al._ (2007a, b); Fernandez-Carames  _et al._ (2009); Yang  _et al._ (2021);
Tiwari and Rai (2022), QCD sum rule Chen  _et al._ (2017); Xin  _et al._
(2022), lattice QCD Chiu and Hsieh (2006); Bali  _et al._ (2011); Prelovsek
and Leskovec (2013); Sadl and Prelovsek (2021), effective field theory
Hidalgo-Duque  _et al._ (2013); Ke  _et al._ (2013); Cleven  _et al._ (2013);
Dong  _et al._ (2021a); Ge  _et al._ (2021); Chen and Huang (2022), and the
chromomagnetic interaction (CMI) model Wu  _et al._ (2019). Because the CMI
model Silvestre-Brac (1992); Buccella  _et al._ (2007); Hogaasen and Sorba
(2004); Cui  _et al._ (2006); Maiani  _et al._ (2004); Stancu (2010); Guo  _et
al._ (2011); Luo  _et al._ (2017); Wu  _et al._ (2017, 2019); Cheng  _et al._
(2020) only considers the short-range chromomagnetic interaction between
constituent quarks and antiquarks, it is more suitable to describe the tight
bound states. While for the heavy flavor exotic state, which contains more
than one light quark (antiquark) probably has a large size, the chromoelectric
contribution should be included. This comes to improved chromomagnetic
interaction (ICMI) model Høgaasen  _et al._ (2014); Weng  _et al._ (2018); An
_et al._ (2021); Guo  _et al._ (2022a, b); Weng  _et al._ (2022, 2019, 2021);
An  _et al._ (2022); Weng and Zhu (2022); Liu  _et al._ (2022). The ICMI model
has been used to predict the mass spectra of open heavy-flavor tetraquark
states Guo  _et al._ (2022a, b); Weng  _et al._ (2022), open and hidden heavy-
flavor pentaquark states Weng  _et al._ (2019, 2021); An  _et al._ (2022), and
heavy-flavor dibaryons Weng and Zhu (2022); Liu  _et al._ (2022). In this
work, we investigated the masses, possible decay channels, and inner
structures of changed and change-neutral, open- and hidden-strange tetraquark
states $Qq\bar{Q}\bar{q}$ ($Q=c,b$, and $q=u,d,s$) via the ICMI model firstly.

The paper is organized as follows. A brief introduction of the ICMI model and
the wave functions of the tetraquark $Qq\bar{Q}\bar{q}$ systems in the color-
spin space will be given in Sec. II. In Sec. III, by substituting the
effective masses and coupling strengths into the ICMI model, we calculated the
mass spectra and wave functions of the $S$-wave tetraquark state
$Qq\bar{Q}\bar{q}$. In addition, the related analysis of possible decay
channels and the inner structures are presented in this section. We summarize
in Sec. IV.

##  II Model description

Analog to the meson and baryon, in this paper we study the heavy flavor
tetraquark state $Qq\bar{Q}\bar{q}$. The state can be considered as the
composition of a heavy quark $Q$, a heavy antiquark $\bar{Q}$, a light quark
$q$, and a light antiquark $\bar{q}$ in the quark model, where $Q=c,b$, and
$q=u,d,s$. At the leading order, the strong interaction between constituent
quarks (antiquarks) can be estimated by the one-gluon-exchange (OGE)
potential. For $S$-wave tetraquark states, the spin-orbit angular momentum
coupling part disappears. The total potential can be reduced to two parts De
Rujula  _et al._ (1975); Oka and Takeuchi (1989)

$V_{ij}^{\text{OGE}}=V^{\text{cm}}_{ij}+V^{\text{ce}}_{ij},$ (1)

with the chromomagnetic interaction part,

$V^{\text{cm}}_{ij}=-{\alpha_{s}\pi\delta({\bf r}_{ij})\over
6m_{i}m_{j}}{\lambda}_{i}^{c}\cdot{\lambda}_{j}^{c}\mbox{\boldmath{$\sigma$}}_{i}\cdot\mbox{\boldmath{$\sigma$}}_{j},$
(2)

and the chromoelectric interaction part,

$V^{\text{ce}}_{ij}={\alpha_{s}\over
4r_{ij}}{\lambda}_{i}^{c}\cdot{\lambda}_{j}^{c}.$ (3)

The parameter $m_{i}$ is the effective mass of the $i$th constituent quark,
$\alpha_{s}$ is the running coupling constant, $r_{ij}=|{\bf r}_{ij}|=|{\bf
r}_{i}-{\bf r}_{j}|$ is the spatial distance between the $i$-th and $j$-th
quarks. The notation ${\lambda}_{i}^{c}$ ($c=1,...,8$) are the Gell-Mann
matrices acting on the color space of the $i$th quark, and the notation
$\mbox{\boldmath{$\sigma$}}_{i}$ are the Pauli matrices on the spin space of
the $i$th quark. In addition, the $\lambda_{i}^{c}$ should be replaced by
$-\lambda_{i}^{c*}$ if the subscript $i$ (or $j$) denotes an antiquark.
Integrating the spatial wave function part, we can get the ICMI model, which
consists of the total mass of the constituent quarks, chromomagnetic
interaction, and chromoelectric interaction. Therefore, the effective
Hamiltonian of a tetraquark system in the ICMI model reads Høgaasen  _et al._
(2014):

$H=\sum\limits_{i=1}^{4}m_{i}+H_{\text{cm}}+H_{\text{ce}},$ (4)

where the chromomagnetic interaction term can be expressed as

$H_{\text{cm}}=-\sum\limits_{i<j}v_{ij}{\lambda}_{i}^{c}\cdot{\lambda}_{j}^{c}\mbox{\boldmath{$\sigma$}}_{i}\cdot\mbox{\boldmath{$\sigma$}}_{j},$
(5)

and the chromoelectric interaction term can be expressed as

$H_{\text{ce}}=-\sum\limits_{i<j}c_{ij}{\lambda}_{i}^{c}\cdot{\lambda}_{j}^{c}.$
(6)

The model parameters $v_{ij}$ and $c_{ij}$ incorporate the effects of the
effective mass of the constituent quarks, the spatial configuration of the
tetraquark system, and the running coupling constant. Considering the symmetry
in the color-spin space, the chromoelectric interaction term and constituent
quarks mass term can be consolidated into one term Weng  _et al._ (2018,
2019),

$H_{0}\equiv\sum\limits_{i=1}^{4}m_{i}+H_{\text{ce}}=-\frac{3}{16}\sum\limits_{i<j}m_{ij}{\lambda}_{i}^{c}\cdot{\lambda}_{j}^{c},$
(7)

with an introduced parameter $m_{ij}=m_{i}+m_{j}+16c_{ij}/3$, which is related
to the effective mass $m_{i}$ and $m_{j}$ of the constituent quarks and the
coupling strength $c_{ij}$ of the chromoelectric interaction. Then, the
Hamiltonian of the ICMI model can be simplified as,

$H=H_{0}+H_{\text{cm}}.$ (8)

The above parameters, such as $v_{ij}$ and $m_{ij}$ can be obtained by fitting
the conventional hadron spectra.

Aiming to solve the eigen equations with given Hamiltonian (8), we need to
construct the wave function of $Qq\bar{Q}\bar{q}$ system, firstly. For the
tetraquark states $Q_{1}q_{2}\bar{Q}_{3}\bar{q}_{4}$ where $Q=c,b$ and
$q=u,d,s$, there are two types of decomposition of the wave function in color
space based on the SU$(3)$ group theory. They physically correspond to two
different configurations in color space, the diquark-antidiquark configuration
labeled as $|(Q_{1}q_{2})(\bar{Q}_{3}\bar{q}_{4})\rangle$ and the meson-meson
configuration labeled as $|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ (or
$|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$). Taking into account the
symmetry characteristics, these two configurations can be properly connected
by a linear transformation. It is convenient to see the total spin of the
$S$-wave tetraquark states can be 0, 1, and 2, so the possible quantum numbers
are $J^{P}=0^{+}$, $1^{+}$, and $2^{+}$. Now, we can construct the color-spin
wave function for given tetraquark states and we only show the results with
the $|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ basis.

For the scalar tetraquark states with $J^{P}=0^{+}$, the color-spin basis wave
functions $|\alpha_{i}\rangle$ ($i=1,2,3,4$) can be built as,

$\displaystyle|\alpha_{1}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{1}_{0}\otimes(q_{2}\bar{q}_{4})^{1}_{0}\rangle_{0}^{1},$
$\displaystyle|\alpha_{2}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{1}_{1}\otimes(q_{2}\bar{q}_{4})^{1}_{1}\rangle_{0}^{1},$
$\displaystyle|\alpha_{3}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{8}_{0}\otimes(q_{2}\bar{q}_{4})^{8}_{0}\rangle_{0}^{1},$
$\displaystyle|\alpha_{4}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{8}_{1}\otimes(q_{2}\bar{q}_{4})^{8}_{1}\rangle_{0}^{1},$
(9)

where the superscripts and subscripts denote the total color and spin of
$Q_{1}\bar{Q}_{3}$, $q_{2}\bar{q}_{4}$ subsystems, and the tetraquark
$Q_{1}q_{2}\bar{Q}_{3}\bar{q}_{4}$ systems, respectively. We know the charge-
neutral system has definite $\mathcal{C}$-parity. So, if $Q_{1}$ and $Q_{3}$,
as well as $q_{2}$ and $q_{4}$ are of the same flavor. All the above color-
spin bases have a positive charge conjugation, $J^{PC}=0^{++}$.

For the axial vector tetraquark states with quantum number $J^{P}=1^{+}$, the
color-spin basis wave functions $|\beta_{i}\rangle$ ($i=1,2,...,6$) can be
built as,

$\displaystyle|\beta_{1}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{1}_{0}\otimes(q_{2}\bar{q}_{4})^{1}_{1}\rangle_{1}^{1},$
$\displaystyle|\beta_{2}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{1}_{1}\otimes(q_{2}\bar{q}_{4})^{1}_{0}\rangle_{1}^{1},$
$\displaystyle|\beta_{3}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{1}_{1}\otimes(q_{2}\bar{q}_{4})^{1}_{1}\rangle_{1}^{1},$
$\displaystyle|\beta_{4}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{8}_{0}\otimes(q_{2}\bar{q}_{4})^{8}_{1}\rangle_{1}^{1},$
$\displaystyle|\beta_{5}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{8}_{1}\otimes(q_{2}\bar{q}_{4})^{8}_{0}\rangle_{1}^{1},$
$\displaystyle|\beta_{6}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{8}_{1}\otimes(q_{2}\bar{q}_{4})^{8}_{1}\rangle_{1}^{1}.$
(10)

Also, if $Q_{1}$ and $Q_{3}$, as well as $q_{2}$ and $q_{4}$ in the tetraquark
$Q_{1}q_{2}\bar{Q}_{3}\bar{q}_{4}$ systems are of the same flavor, the
tetraquark state has a definite $\mathcal{C}$-parity. The bases
$|\beta_{3}\rangle$ and $|\beta_{6}\rangle$ do not change under the symmetry
operation of charge conjugation, which gives $J^{PC}=1^{++}$, while
$|\beta_{1}\rangle$, $|\beta_{2}\rangle$, $|\beta_{4}\rangle$, and
$|\beta_{5}\rangle$ change the sign under the operation of charge conjugation,
which gives $J^{PC}=1^{+-}$.

For $J^{P}=2^{+}$ states, the color-spin basis wave functions
$|\gamma_{i}\rangle$ ($i=1,2$) are given by

$\displaystyle|\gamma_{1}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{1}_{1}\otimes(q_{2}\bar{q}_{4})^{1}_{1}\rangle_{2}^{1},$
$\displaystyle|\gamma_{2}\rangle\equiv|(Q_{1}\bar{Q}_{3})^{8}_{1}\otimes(q_{2}\bar{q}_{4})^{8}_{1}\rangle_{2}^{1}.$
(11)

Similarly, each basis introduced above has a definite charge conjugation if
$Q_{1}$ and $Q_{3}$, as well as $q_{2}$ and $q_{4}$ are of the same flavor.
All the above color-spin bases of the tetraquark systems have a positive
$\mathcal{C}$-parity.

The wave function $\Psi$ of the tetraquark state $Qq\bar{Q}\bar{q}$ with a
given quantum number $J^{P}$ can be expressed as the superposition of the
bases shown above,

$\Psi=\sum_{i=1}^{N_{cs}}c_{i}|\kappa_{i}\rangle,$ (12)

where $N_{cs}$ represents the number of the color-spin basis and $c_{i}$
represents the amplitude for various color-spin basis, which satisfies the
normalization condition $\sum_{i=1}^{N_{cs}}|c_{i}|^{2}=1$. Here,
$|\kappa_{i}\rangle=|\alpha_{i}\rangle$, or $|\beta_{i}\rangle$, or
$|\gamma_{i}\rangle$ depends on the quantum numbers. With this wave function,
we can obtain the matrix form of the Hamiltonian (8),
$\langle\Psi|H|\Psi\rangle$ Guo  _et al._ (2022a). The mass spectra of
tetraquark states can be obtained by diagonalizing this matrix. And the
probability $|c_{i}|^{2}$ can be used to analyze the possible decay channels
of the tetraquark states.

##  III Results and discussion

The parameters, such as $m_{ij}$ and $v_{ij}$, in the ICMI model can be
extracted by fitting the masses, especially the low-lying conventional
hadrons, which have been observed in experiments. In this work, we adopt the
parameters obtained in Ref. Guo  _et al._ (2022a). Now, we have calculated the
mass spectra and wave functions of the tetraquark $Qq\bar{Q}\bar{q}$ ($Q=c,b$,
and $q=u,d,s$) systems with various quantum numbers $J^{P}=0^{+(+)}$,
$1^{+(\pm)}$, and $2^{+(+)}$. Tetraquark state has definite
$\mathcal{C}$-parity as long as $Q(q)$ and $\bar{Q}(\bar{q})$ are the same
flavor. The $\mathcal{C}$-parity of the tetraquark state is determined by the
basis as discussed in the previous section. The tetraquark states can strongly
decay into a pair of mesons. The $\mathcal{C}$-parity is conserved in these
processes. For the $D\bar{D}$ ($D^{\ast}\bar{D}^{\ast}$) pair, its
$\mathcal{C}$-parity of can be estimated by $(-1)^{L+S}$, where $S$ is the
total spin and $L$ is relative angular momentum. While for other meson pairs,
such as $D\bar{D}^{\ast}$, $D_{s}\bar{D}_{s}^{\ast}$, the $\mathcal{C}$-parity
can be either positive or negative. The mass spectra of the tetraquark states
are shown in Figs. 1-3. Also for comparison, we plot all possible meson-meson
thresholds in each figure. The superposition amplitudes $\\{c_{i}\\}$ of the
corresponding color-spin wave functions for each tetraquark state are listed
in Tables 1-3.

Figure 1:  Mass spectra of $S$-wave tetraquark states (a) $cn\bar{c}\bar{n}$
($n=u,d$), (b) $cs\bar{c}\bar{n}$, and (c) $cs\bar{c}\bar{s}$ with different
quantum numbers. The black dashed and red dot-dashed lines demonstrate all
possible meson-meson thresholds.

Figure 2:  Mass spectra of $S$-wave tetraquark states (a) $bn\bar{b}\bar{n}$
($n=u,d$), (b) $bs\bar{b}\bar{n}$, and (c) $bs\bar{b}\bar{s}$ with different
quantum numbers. The black dashed and red dot-dashed lines demonstrate all
possible meson-meson thresholds.

Figure 3:  Mass spectra of $S$-wave tetraquark states (a) $bn\bar{c}\bar{n}$ ($n=u,d$), (b) $bs\bar{c}\bar{n}$, and (c) $bs\bar{c}\bar{s}$ with different quantum numbers. The black dashed and red dot-dashed lines demonstrate all possible meson-meson thresholds. System | $J^{PC}$ | Mass |  $\\{c_{i}\\}$ for $|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ basis |  $\\{c_{i}\\}$ for $|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$ basis
---|---|---|---|---
$cn\bar{c}\bar{n}$ | $0^{+(+)}$ | 4146.5 | (0.06, -0.16, -0.96, -0.23) | (-0.21, 0.90, 0.25, 0.29)
(a) |  | 3970.9 | (0.06, -0.72, -0.05, 0.69) | (-0.37, -0.19, 0.82, 0.39)
|  | 3748.2 | (0.11, 0.68, -0.27, 0.67) | (-0.86, -0.24, -0.26, -0.38)
|  | 3109.3 | (-0.99, 0.02, -0.09, 0.10) | (-0.30, 0.31, -0.44, 0.79)
| $1^{+(-)}$ | 4125.2 | (-0.01, -0.06, 0, -0.45, 0.89, 0) | (0.20, 0.20, -0.88, -0.11, -0.11, 0.35)
|  | 4036.0 | (-0.30, -0.08, 0, 0.85, 0.42, 0) | (0.54, 0.54, 0.24, -0.39, -0.39, -0.25)
|  | 3731.7 | (-0.95, 0.04, 0, -0.27, -0.13, 0) | (-0.34, -0.34, -0.32, -0.36, -0.36, -0.63)
|  | 3226.8 | (-0.02, -0.99, 0, -0.06, -0.09, 0) | (-0.24, -0.24, -0.26, -0.45, -0.45, 0.64)
| $1^{+(+)}$ | 4002.1 | (0, 0, -0.56, 0, 0, 0.83) | (0.42, -0.42, 0, -0.57, 0.57, 0)
|  | 3813.1 | (0, 0, -0.83, 0, 0, -0.56) | (-0.57, 0.57, 0, -0.42, 0.42, 0)
| $2^{+(+)}$ | 4107.5 | (-0.28, 0.96) | (-0.81, 0.58)
|  | 3852.2 | (-0.96, -0.28) | (-0.58, -0.81)
$cs\bar{c}\bar{n}$ | $0^{+}$ | 4220.8 | (0.08, -0.19, -0.95, -0.24) | (0.19, -0.90, -0.29, 0.29)
(b) |  | 4078.5 | (0.08, -0.73, -0.01, 0.68) | (0.33, 0.21, -0.83, -0.39)
|  | 3857.2 | (-0.19, -0.65, 0.29, -0.68) | (-0.85, -0.25, -0.20, -0.43)
|  | 3455.1 | (-0.98, 0.05, -0.13, 0.17) | (-0.38, 0.30, -0.43, 0.77)
| $1^{+}$ | 4203.2 | (-0.03, 0.07, 0.01, 0.56, -0.83, -0.01) | (-0.13, -0.12, 0.90, 0.07, 0.06, -0.39)
|  | 4133.2 | (0.32, 0.14, -0.01, -0.78, -0.52, 0.01) | (0.55, 0.53, 0.13, -0.44, -0.43, -0.18)
|  | 4105.5 | (-0.001, -0.002, -0.60, 0.02, 0.001, 0.80) | (0.39, -0.40, 0.01, -0.58, 0.59, -0.003)
|  | 3923.8 | (0.02, -0.002, 0.80, -0.001, 0.003, 0.60) | (0.59, -0.58, 0.01, 0.40, -0.39, 0.02)
|  | 3846.6 | (-0.95, 0.09, 0.01, -0.27, -0.15, 0.01) | (0.33, 0.35, 0.33, 0.33, 0.34, 0.66)
|  | 3575.6 | (0.04, 0.99, 0.00, 0.09, 0.14, 0.001) | (-0.28, -0.28, 0.25, -0.45, -0.45, 0.62)
| $2^{+}$ | 4209.2 | (-0.29, 0.96) | (-0.80, -0.60)
|  | 3967.2 | (-0.96, -0.29) | (0.60, -0.80)
$cs\bar{c}\bar{s}$ | $0^{++}$ | 4302.7 | (0.11, -0.24, -0.94, -0.22) | (-0.17, 0.88, 0.34, -0.28)
(c) |  | 4195.1 | (0.10, -0.75, 0.05, 0.65) | (0.27, 0.25, -0.84, -0.40)
|  | 3976.5 | (-0.35, -0.60, 0.27, -0.67) | (-0.79, -0.29, -0.09, -0.54)
|  | 3739.3 | (-0.93, 0.11, -0.21, 0.29) | (-0.53, 0.28, -0.41, 0.69)
| $1^{+-}$ | 4289.2 | (0.07, -0.08, 0, -0.65, 0.75, 0) | (0.05, 0.05, -0.90, -0.02, -0.02, 0.43)
|  | 4232.3 | (-0.34, -0.22, 0, 0.69, 0.60, 0) | (0.51, 0.51, 0.03, -0.48, -0.48, -0.10)
|  | 3975.2 | (0.92, -0.25, 0, 0.27, 0.12, 0) | (0.30, 0.30, 0.37, 0.25, 0.25, 0.75)
|  | 3864.9 | (-0.16, -0.94, 0, -0.18, -0.24, 0) | (-0.38, -0.38, 0.22, -0.45, -0.45, 0.50)
| $1^{++}$ | 4215.2 | (0, 0, -0.66, 0, 0, 0.75) | (0.34, -0.34, 0, -0.62, 0.62, 0)
|  | 4039.1 | (0, 0, -0.75, 0, 0, -0.66) | (-0.62, 0.62, 0, -0.34, 0.34, 0)
| $2^{++}$ | 4309.7 | (-0.33, 0.94) | (-0.78, -0.63)
|  | 4092.6 | (-0.94, -0.33) | (0.63, -0.78)
Table 1:  Masses (in MeV) and color-spin wave functions (represented by the superposition amplitudes $\\{c_{i}\\}$) of the $S$-wave hidden-charm tetraquark states with quantum numbers $J^{PC}=0^{+(+)}$, $1^{+(\pm)}$, and $2^{+(+)}$. The label (a), (b), and (c) are corresponding to Fig. 1. System | $J^{PC}$ | Mass |  $\\{c_{i}\\}$ for $|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ basis |  $\\{c_{i}\\}$ for $|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$ basis
---|---|---|---|---
$bn\bar{b}\bar{n}$ | $0^{+(+)}$ | 10821.9 | (-0.05, 0.01, 0.98, 0.18) | (0.30, -0.88, -0.15, 0.35)
(a) |  | 10688.5 | (0.03, -0.17, -0.18, 0.97) | (-0.82, -0.29, 0.47, 0.17)
|  | 10221.4 | (0.01, 0.98, -0.04, 0.17) | (-0.44, -0.21, -0.74, -0.46)
|  | 9533.9 | (0.99, -0.01, 0.06, -0.02) | (0.21, -0.32, 0.46, -0.80)
| $1^{+(-)}$ | 10812.2 | (0.002, -0.06, 0, -0.20, 0.98, 0) | (0.36, 0.36, -0.77, -0.16, -0.16, 0.32)
|  | 10739.2 | (-0.13, -0.02, 0, 0.97, 0.20, 0) | (0.53, 0.53, 0.49, -0.27, -0.27, -0.25)
|  | 10164.9 | (0.99, -0.01, 0, 0.12, 0.02, 0) | (0.23, 0.23, 0.30, 0.44, 0.44, 0.64)
|  | 9595.6 | (-0.003, -0.998, 0, -0.01, -0.06, 0) | (-0.20, -0.20, 0.27, -0.46, -0.46, 0.65)
| $1^{+(+)}$ | 10713.7 | (0, 0, -0.15, 0, 0, 0.99) | (0.62, -0.62, 0, -0.33, 0.33, 0)
|  | 10224.2 | (0, 0, -0.99, 0, 0, -0.15) | (-0.33, 0.33, 0, -0.62, 0.62, 0)
| $2^{+(+)}$ | 10757.2 | (-0.14, 0.99) | (-0.89, -0.45)
|  | 10226.9 | (-0.99, -0.14) | (0.45, -0.89)
$bs\bar{b}\bar{n}$ | $0^{+}$ | 10883.6 | (0.07, -0.01, -0.97, -0.23) | (-0.26, 0.88, 0.13, -0.37)
(b) |  | 10776.5 | (0.04, -0.19, -0.22, 0.96) | (-0.82, -0.25, 0.49, 0.15)
|  | 10335.7 | (0.02, 0.98, -0.05, 0.19) | (-0.46, -0.22, -0.73, -0.46)
|  | 9886.2 | (0.99, -0.01, 0.08, -0.03) | (0.23, -0.33, 0.46, -0.79)
| $1^{+}$ | 10874.7 | (0.02, -0.07, -0.001, -0.31, 0.95, 0.01) | (-0.30, -0.28, 0.82, 0.13, 0.13, -0.36)
|  | 10827.2 | (0.14, 0.04, 0.01, -0.94, -0.31, -0.04) | (0.59, 0.54, 0.40, -0.30, -0.28, -0.21)
|  | 10803.1 | (0.01, 0.002, -0.17, -0.03, -0.02, 0.99) | (0.59, -0.64, -0.01, -0.33, 0.36, 0.01)
|  | 10338.9 | (0.004, 0, 0.99, 0, -0.001, 0.16) | (0.34, -0.34, 0.001, 0.62, -0.62, 0.003)
|  | 10280.1 | (0.99, -0.01, -0.003, 0.13, 0.03, -0.002) | (-0.24, -0.24, -0.31, -0.43, -0.44, -0.64)
|  | 9947.9 | (-0.01, -0.99, 0, -0.02, -0.08, 0) | (-0.21, -0.21, 0.28, -0.46, -0.46, 0.65)
| $2^{+}$ | 10847.8 | (-0.14, 0.99) | (-0.89, -0.46)
|  | 10342.2 | (-0.99, -0.14) | (0.46, -0.89)
$bs\bar{b}\bar{s}$ | $0^{++}$ | 10952.6 | (0.09, 0.001, -0.96, -0.27) | (-0.22, 0.89, 0.12, -0.39)
(c) |  | 10864.3 | (0.06, -0.22, -0.26, 0.94) | (0.82, 0.21, -0.52, -0.13)
|  | 10460.5 | (0.03, 0.98, -0.06, 0.21) | (-0.47, -0.23, -0.71, -0.47)
|  | 10185.2 | (0.99, -0.02, 0.10, -0.04) | (-0.25, 0.35, -0.46, 0.78)
| $1^{+-}$ | 10945.9 | (0.05, -0.09, 0, -0.48, 0.87, 0) | (0.18, 0.18, -0.88, -0.08, -0.08, 0.41)
|  | 10913.0 | (-0.15, -0.07, 0, 0.86, 0.48, 0) | (0.60, 0.60, 0.24, -0.33, -0.33, -0.14)
|  | 10406.3 | (0.99, -0.03, 0, 0.15, 0.03, 0) | (0.24, 0.24, 0.32, 0.43, 0.43, 0.65)
|  | 10246.7 | (-0.02, -0.99, 0, -0.03, -0.11, 0) | (-0.23, -0.23, 0.29, -0.46, -0.46, 0.63)
| $1^{++}$ | 10891.9 | (0, 0, -0.19, 0, 0, 0.98) | (0.61, -0.61, 0, -0.36, 0.36, 0)
|  | 10464.4 | (0, 0, -0.98, 0, 0, -0.19) | (-0.36, 0.36, 0, -0.61, 0.61, 0)
| $2^{++}$ | 10937.3 | (-0.16, 0.98) | (-0.88, 0.48)
|  | 10468.3 | (-0.98, -0.16) | (-0.48, -0.88)
Table 2:  Masses (in MeV) and color-spin wave functions (represented by the superposition amplitudes $\\{c_{i}\\}$) of the $S$-wave hidden-bottom tetraquark states with quantum numbers $J^{PC}=0^{+(+)}$, $1^{+(\pm)}$, and $2^{+(+)}$. The label (a), (b), and (c) are corresponding to Fig. 2. System |  $J^{P}$ |  Mass |  $\\{c_{i}\\}$ for $|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ basis |  $\\{c_{i}\\}$ for $|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$ basis
---|---|---|---|---
$bn\bar{c}\bar{n}$ |  $0^{+}$ |  7471.9 |  (-0.06, 0.07, 0.97, 0.23) |  (0.24, -0.90, -0.18, 0.33)
(a) |  |  7303.8 |  (0.06, -0.46, -0.17, 0.87) |  (-0.65, -0.21, 0.68, 0.26)
|  |  7069.5 |  (0.05, 0.89, -0.16, 0.43) |  (-0.68, -0.23, -0.55, -0.43)
|  |  6406.2 |  (0.99, -0.02, 0.08, -0.06) |  (0.25, -0.32, 0.45, -0.80)
|  $1^{+}$ |  7461.8 |  (-0.01, -0.06, 0.02, -0.27, 0.95, 0.14) |  (0.41, 0.21, -0.80, -0.17, -0.13, 0.32)
|  |  7401.3 |  (-0.18, -0.03, -0.12, 0.83, 0.16, 0.49) |  (0.73, 0.13, 0.41, -0.46, -0.07, -0.26)
|  |  7322.6 |  (0.15, 0.05, -0.35, -0.43, -0.23, 0.79) |  (0.17, -0.72, -0.11, -0.21, 0.62, 0.12)
|  |  7088.7 |  (0.14, 0.02, 0.93, -0.03, -0.07, 0.34) |  (0.43, -0.47, 0.06, 0.63, -0.44, 0.07)
|  |  7030.0 |  (0.96, -0.03, -0.10, 0.23, 0.09, -0.08) |  (0.23, 0.38, 0.33, 0.34, 0.44, 0.63)
|  |  6477.1 |  (0.01, 0.99, -0.01, 0.04, 0.08, -0.03) |  (0.20, 0.24, -0.26, 0.46, 0.45, -0.65)
|  $2^{+}$ |  7424.9 |  (-0.22, 0.98) |  (-0.85, -0.53)
|  |  7105.1 |  (-0.98, -0.22) |  (0.53, -0.85)
$bs\bar{c}\bar{n}$ |  $0^{+}$ |  7544.2 |  (-0.08, 0.07, 0.96, 0.26) |  (0.21, -0.90, -0.18, 0.35)
(b) |  |  7404.5 |  (0.08, -0.49, -0.19, 0.85) |  (-0.63, -0.19, 0.72, 0.25)
|  |  7182.0 |  (0.09, 0.87, -0.18, 0.45) |  (-0.69, -0.23, -0.51, -0.46)
|  |  6756.7 |  (0.99, -0.03, 0.10, -0.09) |  (0.29, -0.32, 0.45, -0.78)
|  $1^{+}$ |  7535.8 |  (-0.01, 0.08, -0.02, 0.38, -0.91, -0.13) |  (-0.33, -0.15, 0.84, 0.14, 0.10, -0.36)
|  |  7499.5 |  (-0.19, -0.05, -0.12, 0.79, 0.25, 0.52) |  (0.77, 0.14, 0.32, -0.49, -0.08, -0.22)
|  |  7421.1 |  (0.17, 0.08, -0.37, -0.42, -0.27, 0.76) |  (0.13, -0.71, -0.08, -0.19, 0.66, 0.09)
|  |  7203.3 |  (0.16, 0.04, 0.91, -0.03, -0.08, 0.36) |  (0.44, -0.47, 0.07, 0.64, -0.41, 0.07)
|  |  7145.4 |  (0.96, -0.05, -0.12, 0.23, 0.10, -0.09) |  (0.22, 0.40, 0.33, 0.32, 0.43, 0.64)
|  |  6827.9 |  (0.02, 0.99, -0.02, 0.05, 0.11, -0.05) |  (0.21, 0.28, -0.27, 0.45, 0.45, -0.63)
|  $2^{+}$ |  7524.4 |  (-0.23, 0.97) |  (-0.84, -0.54)
|  |  7221.2 |  (-0.97, -0.23) |  (0.54, -0.84)
$bs\bar{c}\bar{s}$ |  $0^{+}$ |  7615.1 |  (-0.11, 0.08, 0.95, 0.27) |  (0.19, -0.89, -0.20, 0.37)
(c) |  |  7507.2 |  (0.12, -0.56, -0.17, 0.80) |  (-0.55, -0.18, 0.77, 0.26)
|  |  7297.1 |  (0.17, 0.82, -0.20, 0.51) |  (-0.72, -0.27, -0.41, -0.50)
|  |  7049.8 |  (0.97, -0.07, 0.16, -0.16) |  (0.39, -0.33, 0.44, -0.74)
|  $1^{+}$ |  7609.7 |  (-0.07, 0.10, -0.05, 0.56, -0.82, -0.03) |  (-0.15, -0.08, 0.88, 0.04, 0.08, -0.44)
|  |  7588.7 |  (-0.21, -0.09, -0.15, 0.66, 0.44, 0.54) |  (0.80, 0.15, 0.12, -0.55, -0.10, -0.13)
|  |  7519.9 |  (0.19, 0.12, -0.43, -0.41, -0.29, 0.72) |  (0.10, -0.65, -0.07, -0.19, 0.72, 0.07)
|  |  7322.4 |  (0.19, 0.08, 0.88, -0.02, -0.08, 0.42) |  (0.49, -0.49, 0.07, 0.63, -0.34, 0.06)
|  |  7269.2 |  (0.94, -0.13, -0.13, 0.26, 0.10, -0.11) |  (0.20, 0.41, 0.36, 0.26, 0.38, 0.67)
|  |  7121.2 |  (0.07, 0.97, -0.05, 0.10, 0.18, -0.09) |  (0.24, 0.38, -0.27, 0.44, 0.46, -0.58)
|  $2^{+}$ |  7617.1 |  (-0.27, 0.96) |  (-0.82, -0.57)
|  |  7344.9 |  (-0.96, -0.27) |  (0.57, -0.82)
Table 3:  Masses (in MeV) and color-spin wave functions (represented by the
superposition amplitudes $\\{c_{i}\\}$) of the $S$-wave mixed-charm-bottom
tetraquark states with quantum numbers $J^{P}=0^{+}$, $1^{+}$, and $2^{+}$.
The label (a), (b), and (c) are corresponding to Fig. 3.

###  III.1 Hidden-charm tetraquark states with $cn\bar{c}\bar{n}$,
$cs\bar{c}\bar{n}$, and $cs\bar{c}\bar{s}$ configurations

####  III.1.1 The $cn\bar{c}\bar{n}$ configuration

The mass spectra of the tetraquark states with $cn\bar{c}\bar{n}$
configuration are shown in Fig. 1(a). The superposition amplitudes of the
corresponding wave functions are listed in Table 1(a). It is worth noting that
the notation $n$ represents a $u$ or $d$ quark from now on. And in our ICMI
model, $u$ and $d$ quarks have the same mass.

(1). There are four $S$-wave tetraquark states with $J^{P}=0^{+}$
($J^{PC}=0^{++}$ for charge-neutral states, such as $[cu\bar{c}\bar{u}]$ and
$[cd\bar{c}\bar{d}]$). Their masses are 3109.3 MeV, 3748.2 MeV, 3970.9 MeV,
and 4146.5 MeV, respectively. The lowest state, 3109.3 MeV, lies slightly
below the $\eta_{c}\pi$ threshold and its wave function has a large fraction
on the $\eta_{c}\pi$ basis ($c_{1}=-0.99$). The 3748.2 MeV is above the
$\eta_{c}\pi$ and $D\bar{D}$ thresholds. Considering the angular momentum and
$\mathcal{C}$-parity conservation (for charge-neutral states), this state is
allowed to decay strongly into the mesons $\eta_{c}\pi$ or $D\bar{D}$. In this
paper, we only discuss the $S$-wave decay and neglect the high-order decays
such as $P$ and $D$-wave, which usually give a small contribution Weng  _et
al._ (2019). The 3970.9 MeV is above the $\eta_{c}\pi$, $D\bar{D}$, and
$J/\psi\rho$ thresholds. Therefore, compared with 3748.2 MeV, this state has
one more possible decay mode, $J/\psi\rho$. The remaining state with a large
mass, 4146.5 MeV, can decay into mesons $\eta_{c}\pi$, $D\bar{D}$,
$J/\psi\rho$, and $D^{\ast}\bar{D}^{\ast}$.

(2). There are six $S$-wave tetraquark states with $J^{P}=1^{+}$. For charge-
neutral tetraquark states, they can be separated into two classes based on the
$\mathcal{C}$-parity. There are four $S$-wave tetraquark states with
$J^{PC}=1^{+-}$. The lowest state with a mass of 3226.8 MeV, is slightly lower
than the $J/\psi\pi$ threshold. Its wave function has a large fraction on the
$J/\psi\pi$ basis ($c_{2}=-0.99$). It can decay into $J/\psi$ and $\pi$ easily
due to the large fraction and small mass difference. Similarly, the state with
a mass of 3731.7 MeV is close to the $\eta_{c}\rho$ threshold. The large
fraction $c_{i}$ indicates it is can decay into $\eta_{c}$ and $\rho$ easily.
In the meantime, it can also decay into the mesons $J/\psi\pi$ (decay into
$D\bar{D}$ via $S$-wave breaks the $\mathcal{C}$-parity). The other two
states, 4036.0 and 4125.2 MeV lie above all meson-meson thresholds.
Considering the conservation law in the decay process, the possible decay
channels are $J/\psi\pi$, $\eta_{c}\rho$, and $D\bar{D}^{\ast}$. Another two
$S$-wave tetraquark states with positive $\mathcal{C}$-parity,
$J^{PC}=1^{++}$. Their masses are 3813.1 MeV and 4002.1 MeV. The state with
3813.1 MeV is near the $J/\psi\rho$ and $D\bar{D}^{\ast}$ threshold. Thus the
state can decay into mesons $\pi^{+}\pi^{-}J/\psi$ via the quantum off-shell
decay process $\rho\rightarrow\pi^{+}\pi^{-}$. The tetraquark states
$Qq\bar{Q}\bar{q}$ have no isospin symmetry. However, if considering the
mixing between different tetraquark states, such as $cu\bar{c}\bar{u}$ and
$cu\bar{d}\bar{d}$ can form a state with definite isospin. Especially, the
state $|X\rangle=(|cu\bar{c}\bar{u}\rangle-|cu\bar{d}\bar{d}\rangle)/\sqrt{2}$
with $I=1$ is in good agreement with the experimental results of $X(3872)$
with $I(J^{PC})=0(1^{++})$ Choi  _et al._ (2003). However, our predicted mass
of 3813.1 MeV is almost 60 MeV lower than 3872 MeV. If that is true, $X(3872)$
probably is a mixing state of excited charmonium ($\chi_{c1}(2P)$) and
tetraquark $|X\rangle$ states, which is similar to a mixed molecule-charmonium
state Badalian  _et al._ (2012); Matheus  _et al._ (2009). Our results are
also consistent with the conclusions of some previous theoretical studies
Maiani  _et al._ (2005); Hogaasen  _et al._ (2006). The 4002.1 MeV is slightly
below the $D^{\ast}{\bar{D}}^{\ast}$ threshold. It can decay into
$D^{\ast}\bar{D}^{\ast}$ via the off-shell process. Therefore, all possible
decay channels are $D^{\ast}{\bar{D}}^{\ast}$, $D\bar{D}$, $J/\psi\rho$, and
$\eta_{c}\pi$.

(3). There are two $S$-wave tetraquark states with $J^{P}=2^{+}$
($J^{PC}=2^{++}$ for charge-neutral states), which masses are 3852.2 MeV and
4107.5 MeV, respectively. The 3852.2 MeV is very close to the $J/\psi\rho$
threshold and its wave function has a large component of $J/\psi\rho$ basis
($c_{1}=0.96$). The 4107.5 MeV is above $D^{\ast}\bar{D}^{\ast}$ and
$J/\psi\rho$ thresholds and may be allowed decay into $D^{\ast}\bar{D}^{\ast}$
and $J/\psi\rho$.

####  III.1.2 The $cs\bar{c}\bar{n}$ configuration

The mass spectra of the tetraquark states with $cs\bar{c}\bar{n}$
configuration are shown in Fig. 1(b). For comparison, we also show the
corresponding meson-meson thresholds in this figure. The masses and wave
functions amplitudes $\\{c_{i}\\}$ are listed in Table 1(b). For the
$cs\bar{c}\bar{n}$ state, the $\mathcal{C}$-parity is not a good quantum
number. So, the quantum numbers of the $S$-wave tetraquark state
$cs\bar{c}\bar{n}$ are $J^{P}=0^{+}$, $1^{+}$, and $2^{+}$.

(1). There are four $J^{P}=0^{+}$ states, which masses are 3455.1 MeV, 3857.2
MeV, 4078.5 MeV, and 4220.8 MeV. The state with the smallest mass, 3455.1 MeV,
is close to the $\eta_{c}K$ threshold and its wave function has a large
component of $\eta_{c}K$ basis ($c_{1}=0.98$). So, it can decay into
$\eta_{c}K$ easily. The 3857.2 MeV is above the $\eta_{c}K$ and $D\bar{D}_{s}$
thresholds and can naturally decay into mesons $\eta_{c}K$ and $D\bar{D}_{s}$.
The 4078.5 MeV is below the $D^{\ast}\bar{D}_{s}^{\ast}$ threshold but above
other possible meson-meson thresholds. The possible decay channels are
$\eta_{c}K$, $D\bar{D}_{s}$, and $J/\psi K^{\ast}$. The state with the largest
mass 4220.8 MeV lies above all possible meson-meson thresholds, which can
decay to $\eta_{c}K$, $D\bar{D}_{s}$, $J/\psi K^{\ast}$, and
$D^{\ast}\bar{D}_{s}^{\ast}$.

(2). There are six $S$-wave tetraquark states with $J^{P}=1^{+}$. Their masses
are 3575.6 MeV, 3846.6 MeV, 3923.8 MeV, 4105.5 MeV, 4133.2 MeV, and 4203.2
MeV, respectively. The state with the smallest mass, 3575.6 MeV, which wave
function has a large component of $J/\psi K$ basis ($c_{2}=0.99$). Because its
mass is very close to the $J/\psi K$ threshold, it can decay into $J/\psi K$
with a large probability. Similarly, the 3846.6 MeV can decay into $\eta_{c}K$
pair due to the large fraction $c_{1}=0.95$. The 3923.8 MeV is above the
$J/\psi K$ and $\eta_{c}K^{\ast}$ thresholds and close to the
$D^{\ast}\bar{D}_{s}$ and $D\bar{D}_{s}^{\ast}$ thresholds. We can see its
wave function has a large $|\beta_{3}\rangle$ component ($c_{3}=0.80$),
indicating that the mesons $\eta_{c}K^{\ast}$ component occupies a large
proportion. To study the weights of this state to $D^{\ast}\bar{D}_{s}$ and
$D\bar{D}_{s}^{\ast}$ bases, we convert the tetraquark configuration
$|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ to
$|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$. As shown in Table 1, we can
see the amplitudes of the state (3923.8 MeV) under this set of
$|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$ basis is
$(0.59,-0.58,0.01,0.4,-0.39,0.02)$. It indicates that the first two components
are approximately the same and relatively large, indicating that the
tetraquark state can be allowed to decay into the mesons $D^{\ast}\bar{D}_{s}$
and $D\bar{D}_{s}^{\ast}$. The results nicely explain the nature of exotic
resonant structure $Z_{cs}(3985)^{-}$ Ablikim  _et al._ (2021). However,
another exotic state $Z_{cs}(4000)^{+}$, whose mass is very close to
$Z_{cs}(3985)^{-}$ but with a larger decay width can not be classified in our
calculations. The 4105.5 MeV is only below the $D^{\ast}\bar{D}_{s}^{\ast}$
threshold and thus can naturally decay into the mesons $J/\psi K$,
$\eta_{c}K^{\ast}$, $J/\psi K^{\ast}$, $D^{\ast}\bar{D}_{s}$, and
$D\bar{D}_{s}^{\ast}$. The remaining two states 4133.2 MeV and 4203.2 MeV lie
above all possible meson-meson thresholds so that all decay modes are
possible. In addition, it is worth noting that the mass and decay channel of
the tetraquark state 4203.2 MeV with quark content $cu\bar{c}\bar{s}$ is
probably the experimentally observed $Z_{cs}(4220)^{+}$ Aaij  _et al._ (2021).

(3). For $J^{P}=2^{+}$, masses of two $S$-wave tetraquark states are 3967.2
MeV and 4209.2 MeV. The lower state lies slightly below the $J/\psi K^{\ast}$
threshold and its wave function has a large component of $J/\psi K^{\ast}$
basis ($c_{1}=-0.96$). So, it can be searched in the $J/\psi K^{\ast}$
channel. The higher state lies above the $J/\psi K^{\ast}$ and
$D^{\ast}\bar{D}_{s}^{\ast}$ thresholds and its dominant decay modes are
mesons $J/\psi K^{\ast}$ and $D^{\ast}\bar{D}_{s}^{\ast}$.

####  III.1.3 The $cs\bar{c}\bar{s}$ configuration

For the tetraquark $cs\bar{c}\bar{s}$ configuration, it has definite
$\mathcal{C}$-parity. All possible quantum numbers are $J^{PC}=0^{++}$,
$1^{+-}$, $1^{++}$, and $2^{++}$. The mass spectra of the tetraquark state
with $cs\bar{c}\bar{s}$ configuration are shown in Fig. 1(c) with
corresponding meson-meson thresholds. For the $s\bar{s}$ system, there is no
pure spin singlet state due to the mixing between $u\bar{u}$ and $d\bar{d}$.
For the spin-triplet state, the mixing angle is opportune to form a very
nearly pure $s\bar{s}$ state, which is named $\phi$ meson. So, we only plot
the $\eta_{c}\phi$ and $J/\psi\phi$ thresholds. The superposition amplitudes
$\\{c_{i}\\}$ of the corresponding tetraquark wave functions are listed in
Table 1(c).

(1). There are four $S$-wave tetraquark states with $J^{PC}=0^{++}$, whose
masses are 3739.3 MeV, 3976.5 MeV, 4195.1 MeV, and 4302.7 MeV, respectively.
From the wave function of the lowest state, 3739.3 MeV, we can see it has a
large component based on $\eta_{c}$ and the spin singlet state $s\bar{s}$.
Because $s\bar{s}$ content mixed with $u\bar{u}$ and $d\bar{d}$. So, this
state may be a superposition state of $\eta_{c}\eta$ and
$\eta_{c}\eta^{\prime}$. We will not discuss the decay channels of these mixed
states in this paper. Considering its mass is largely below the meson-meson
thresholds, it would be a stable tetraquark state. The 3976.5 MeV is close to
and above the $D_{s}\bar{D}_{s}$ threshold and thus can naturally decay into
mesons $D_{s}\bar{D}_{s}$. Now, we convert the
$|(Q_{1}\bar{Q}_{3})(q_{2}\bar{q}_{4})\rangle$ configuration to the
$|(Q_{1}\bar{q}_{4})(q_{2}\bar{Q}_{3})\rangle$ configuration. Under this set
of color-spin basis, the superposition amplitude $\\{c_{i}\\}$ of this
tetraquark state is $(-0.79,-0.29,-0.09,-0.54)$. It can be found that the
amplitude $c_{1}$ of the $D_{s}\bar{D}_{s}$ component is -0.79. So, the main
decay channel of this state is mesons $D_{s}\bar{D}_{s}$. These analyses
indicate this state probably is the newly observed exotic hadron state
$X(3960)$ with $J^{PC}=0^{++}$. The 4195.1 MeV is above the $D_{s}\bar{D}_{s}$
and $J/\psi\phi$ thresholds but below the $D_{s}^{\ast}\bar{D}_{s}^{\ast}$
threshold. So, the possible decay channels are $D_{s}\bar{D}_{s}$ and
$J/\psi\phi$. And this state might be the observed $X_{0}(4140)$ at LHCb Aaij
_et al._ (2022b). The last state with a mass of 4302.7 MeV is above all
possible meson-meson thresholds and can be allowed to decay into
$D_{s}\bar{D}_{s}$, $D_{s}^{\ast}\bar{D}_{s}^{\ast}$, and $J/\psi\phi$.

(2). There are four $S$-wave tetraquark states with $J^{PC}=1^{+-}$ and masses
3864.9 MeV, 3975.2 MeV, 4232.3 MeV, and 4289.2 MeV, respectively. Without the
spin singlet $s\bar{s}$ meson, the lowest state with a mass of 3864.9 MeV is
probably a stable tetraquark state. While the state, 3975.2 MeV, with a large
$\eta_{c}\phi$ component, could decay into $\eta_{c}\phi$ via the off-shell
process. The two highest states, 4232.3 MeV and 4289.2 MeV are located above
all possible meson-meson thresholds. Considering the conservation of angular
momentum and $\mathcal{C}$-parity in the decay process, they can decay into
the $D_{s}^{\ast}\bar{D}_{s}^{\ast}$ and $D_{s}\bar{D}_{s}^{\ast}$ via the $S$
wave. Because $D_{s}\bar{D}_{s}^{\ast}$ pair can form a negative
$\mathcal{C}$-parity state via,
$(D_{s}\bar{D}_{s}^{\ast}-{D}_{s}^{\ast}\bar{D}_{s})/\sqrt{2}$. Besides, there
are two $S$-wave tetraquark states with $J^{PC}=1^{++}$ and masses 4039.1 MeV
and 4215.2 MeV, respectively. The state with a mass of 4039.1 MeV has large
fractions on both $J/\psi\phi$ and $D_{s}\bar{D}_{s}^{\ast}$ bases. Satisfying
the $\mathcal{C}$-parity conversion, it can decay into $J/\psi\phi$ and a
mixed state of $D_{s}\bar{D}_{s}^{\ast}$ and its antiparticle,
$(D_{s}\bar{D}_{s}^{\ast}+D_{s}^{\ast}\bar{D}_{s})/\sqrt{2}$. The state,
4215.2 MeV, is very close to the $D_{s}^{\ast}\bar{D}_{s}^{\ast}$ threshold.
It can decay into $J/\psi\phi$, while the $P$ wave decay into the final
$D_{s}^{\ast}\bar{D}_{s}^{\ast}$ pair is allowed.

(3). For $J^{PC}=2^{++}$, we find two $S$-wave tetraquark states with masses
of 4092.6 MeV and 4309.7 MeV, respectively. The wave function of 4092.6 MeV
has a large $J/\psi\phi$ component, it can decay into $J/\psi\phi$. The 4309.7
MeV is above all possible meson-meson thresholds and can be allowed the decay
into mesons $J/\psi\phi$ and $D_{s}^{\ast}\bar{D}_{s}^{\ast}$.

###  III.2 Hidden-bottom tetraquark states with $bn\bar{b}\bar{n}$,
$bs\bar{b}\bar{n}$, and $bs\bar{b}\bar{s}$ configurations

The hidden-bottom tetraquark states can be realized by replacing the charm
quark in the hidden-charm tetraquark states with the bottom quark.
Substituting the corresponding model parameters into Eq. (8), we obtain the
mass spectra and wave functions of the $S$-wave hidden-bottom tetraquark
states. These hidden-bottom tetraquark states can be divided into three
configurations, namely $bn\bar{b}\bar{n}$, $bs\bar{b}\bar{n}$, and
$bs\bar{b}\bar{s}$. Their mass spectra and corresponding meson-meson
thresholds are presented in Fig. 2. The superposition amplitudes $\\{c_{i}\\}$
of the wave functions of these tetraquark states are listed in Table 2. The
analysis method is similar to the previous hidden-charm tetraquark states. So,
we will not discuss them one by one and only focus on two experimental
observed states, $Z_{b}(10610)^{+}$ and $Z_{b}(10652)^{+}$ Bondar  _et al._
(2012). They are observed in the decay channels of $\Upsilon(ns)\pi$ and
$h_{b}(mp)\pi$ with $n=1,2,3$ and $m=1,2$. Their quantum numbers are probably
$J^{P}=1^{+}$ but without the information of the $\mathcal{C}$-parity. In our
model, the state with a mass of 10713.7 MeV and 10739.2 MeV with $J^{P}=1^{+}$
are probably the observed $Z_{b}(10610)^{+}$ and $Z_{b}(10652)^{+}$
respectively, as shown in Fig. 2(a). If so, these two states should have
different $\mathcal{C}$-parities. Their wave function has a very large
fraction on the $\beta_{4}$ and $\beta_{6}$ basis, which indicates these two
states might be the diquark-antiquark bound states.

System |  $J^{P}$ |  Mass |  Meson-Meson
---|---|---|---
$cn\bar{c}\bar{n}$ |  $0^{+}$ |  3109.3 |  $\eta_{c}\pi$
$cn\bar{c}\bar{n}$ |  $1^{+}$ |  3226.8 |  $J/\psi\pi$
$cn\bar{c}\bar{n}$ |  $1^{+}$ |  3731.7 |  $\eta_{c}\rho$
$cn\bar{c}\bar{n}$ |  $2^{+}$ |  3852.2 |  $J/\psi\rho$
$cs\bar{c}\bar{n}$ |  $0^{+}$ |  3455.1 |  $\eta_{c}K$
$cs\bar{c}\bar{n}$ |  $1^{+}$ |  3575.6 |  $J/\psi K$
$cs\bar{c}\bar{n}$ |  $1^{+}$ |  3846.6 |  $\eta_{c}K^{\ast}$
$cs\bar{c}\bar{n}$ |  $2^{+}$ |  3967.2 |  $J/\psi K^{\ast}$
$bn\bar{b}\bar{n}$ |  $0^{+}$ |  9533.9 |  $\eta_{b}\pi$
$bn\bar{b}\bar{n}$ |  $1^{+}$ |  9595.6 |  $\Upsilon\pi$
$bn\bar{b}\bar{n}$ |  $1^{+}$ |  10164.9 |  $\eta_{b}\rho$
$bn\bar{b}\bar{n}$ |  $1^{+}$ |  10224.2 |  $\Upsilon\rho$
$bn\bar{b}\bar{n}$ |  $2^{+}$ |  10226.9 |  $\Upsilon\rho$
$bs\bar{b}\bar{n}$ |  $0^{+}$ |  9886.2 |  $\eta_{b}K$
$bs\bar{b}\bar{n}$ |  $1^{+}$ |  9947.9 |  $\Upsilon K$
$bs\bar{b}\bar{n}$ |  $1^{+}$ |  10280.1 |  $\eta_{b}K^{\ast}$
$bs\bar{b}\bar{n}$ |  $1^{+}$ |  10338.9 |  $\Upsilon K^{\ast}$
$bs\bar{b}\bar{n}$ |  $2^{+}$ |  10342.2 |  $\Upsilon K^{\ast}$
$bs\bar{b}\bar{s}$ |  $1^{+}$ |  10406.3 |  $\eta_{b}\phi$
$bs\bar{b}\bar{s}$ |  $1^{+}$ |  10464.4 |  $\Upsilon\phi$
$bs\bar{b}\bar{s}$ |  $2^{+}$ |  10468.3 |  $\Upsilon\phi$
$bn\bar{c}\bar{n}$ |  $0^{+}$ |  6406.2 |  $B_{c}\pi$
$bn\bar{c}\bar{n}$ |  $1^{+}$ |  6477.1 |  $B_{c}^{\ast}\pi$
$bn\bar{c}\bar{n}$ |  $1^{+}$ |  7030.0 |  $B_{c}\rho$
$bn\bar{c}\bar{n}$ |  $2^{+}$ |  7105.1 |  $B_{c}^{\ast}\rho$
$bs\bar{c}\bar{n}$ |  $0^{+}$ |  6756.7 |  $B_{c}K$
$bs\bar{c}\bar{n}$ |  $1^{+}$ |  6827.9 |  $B_{c}^{\ast}K$
$bs\bar{c}\bar{n}$ |  $1^{+}$ |  7145.4 |  $B_{c}K^{\ast}$
$bs\bar{c}\bar{n}$ |  $2^{+}$ |  7221.2 |  $B_{c}^{\ast}K^{\ast}$
$bs\bar{c}\bar{s}$ |  $2^{+}$ |  7344.9 |  $B_{c}^{\ast}\phi$
Table 4:  Tetraquark states, which have a large overlap ($|c_{i}|^{2}>90\%$)
to the $Q\bar{Q}$ and $q\bar{q}$ mesons. Their masses (in MeV), quantum
numbers, and meson-meson contents are listed. $B_{c}^{\ast}$ with
$J^{P}=1^{-}$ has not been found in experiments.

###  III.3 Tetraquark states with $bn\bar{c}\bar{n}$, $bs\bar{c}\bar{n}$, and
$bs\bar{c}\bar{s}$ configurations

The mixed charm-bottom tetraquark state can be obtained by substituting a
charm quark in the hidden-charm tetraquark state with a bottom quark, or by
substituting a bottom quark in the hidden-bottom tetraquark state with a charm
quark. The difference, however, is that these configurations break the charge
conjugation symmetry. Hence, the quantum numbers for the charm-bottom
tetraquark systems, $bn\bar{c}\bar{n}$, $bs\bar{c}\bar{n}$, and
$bs\bar{c}\bar{s}$, are $J^{P}=0^{+}$, $1^{+}$, and $2^{+}$. Similarly, we can
obtain their masses and wave functions by using the ICMI model. Our
theoretical results are shown in Fig. 3 and Table 3. The analysis method is
similar to the previous hidden-charm tetraquark states.

##  IV summary

In this work, we complete a systematic study of the $S$-wave tetraquark states
$Qq\bar{Q}\bar{q}$ ($Q=c,b$ and $q=u,d,s$) via the ICMI model, which includes
both the chromomagnetic and chromoelectric interaction. The parameters in the
ICMI model are obtained by fitting the known hadron spectra and they are taken
directly from the previous work Guo  _et al._ (2022a). The mass spectra,
possible decay channels, and inner structures of the $S$-wave tetraquark
$Qq\bar{Q}\bar{q}$ with quantum numbers $J^{PC}=0^{+(+)}$, $1^{+(\pm)}$, and
$2^{+(+)}$ are presented and analyzed. For the charge-neutral tetraquark
state, the charge conjugation is also considered when analyzing the possible
decay channels.

The recently observed hidden-charm tetraquark states, such as
$Z_{cs}(3985)^{-}$, $X(3960)$, $Z_{cs}(4220)^{+}$, and so on, can be well
explained in our model. The tetraquark state $cs\bar{c}\bar{u}$ with mass
3923.8 MeV and the quantum number $J^{P}=1^{+}$ can be considered as a good
candidate for $Z_{cs}(3985)^{-}$. For the $cs\bar{c}\bar{s}$ configuration, we
find a tetraquark state with a mass of 3976.5 MeV and a quantum number of
$J^{PC}=0^{++}$. The properties of this state are in good agreement with
$X(3960)$. In the meantime, based on the wave functions of each $S$-wave
tetraquark $Qq\bar{Q}\bar{q}$, we find the low-lying states in each
configuration have a significant component of
$|(Q_{1}\bar{Q}_{3})^{1}(q_{2}\bar{q}_{4})^{1}\rangle$ basis (the probability
$|c_{i}|^{2}>90\%$) as shown in Table 4. This indicates they have a large
probability to decay into $Q\bar{Q}$ and $q\bar{q}$ mesons instead of
$Q\bar{q}$ and $q\bar{Q}$ mesons. In some sense, these states are probably the
molecule states of $Q\bar{Q}$ and $q\bar{q}$. Of course, other evidence, such
as the mean radius and hadron level interactions, are needed to determine
whether it is a molecular state Voloshin (2006); Zhang  _et al._ (2006); Liu
_et al._ (2009); Dong  _et al._ (2021b); Chen  _et al._ (2022). Our
predictions on these exotic tetraquark states $Qq\bar{Q}\bar{q}$ can be
examined in future experiments.

###### Acknowledgements.

T. Guo is supported by the Scientific Research Foundation of Chengdu
University of Technology under grant No.10912-KYQD2022-09557; J. Li and L. He
are supported by the National Natural Science Foundation of China under grant
No. 11890712; J. Zhao is supported by the European Union’s Horizon 2020
research and innovation program under grant agreement No 824093 (STRONG-2020).

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