# The low-lying hidden- and double-charm tetraquark states in a constituent
quark model with Instanton-induced Interaction

Jin-Bao Wang  School of Physical Science and Technology, Southwest University,
Chongqing 400715, China  Gang Li<EMAIL_ADDRESS>College of Physics and
Engineering, Qufu Normal University, Qufu 273165, China  Chun-Sheng An
<EMAIL_ADDRESS>School of Physical Science and Technology, Southwest
University, Chongqing 400715, China  Cheng-Rong Deng  School of Physical
Science and Technology, Southwest University, Chongqing 400715, China  Ju-Jun
Xie  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 101408, China  School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, Henan 450001, China  Lanzhou Center for
Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,
Lanzhou University, Lanzhou 730000, China

###### Abstract

Spectrum of the low-lying hidden- and double-charm tetraquark states are
investigated in a nonrelativistic quark potential model, where the Instanton-
induced interaction is taken as the residual spin-dependent hyperfine
interaction between quarks. The model parameters are fixed by fitting the
spectrum of the ground hadron states. Our numerical results show that masses
of several presently studied tetraquark states are close to those of the
experimentally observed candidates of exotic meson, which indicates that the
corresponding compact tetraquark components may take considerable
probabilities in those observed exotic states.

##  I Introduction

More and more exotic hadron candidates, which exhibit different properties
from mesons and baryons in the traditional quark model, have been observed
experimentally PDG  since the discovery of $X(3872)$ Belle 2003 . The typical
mesonic exotic states, such as the charmonium-like state $Z_{c}(3900)^{\pm}$
observed in the $\pi^{\pm}J/\psi$ invariant mass spectra of the process
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ BESIII 2013  and the
$Z_{cs}(3985)$ state observed in the $D_{s}^{-}D^{*0}$ and $D_{s}^{*-}D^{0}$
final states of the $e^{+}e^{-}\rightarrow
K^{+}D_{s}^{-}D^{*0}(D_{s}^{*-}D^{0})$ reaction BESIII 2021 , may contain at
least four $c\bar{c}q\bar{q}$ quarks. Theoretically, the exotic meson states
can be explained as compact tetraquark states in quark potential model
Yang:2009zzp ; Deng:2016rus ; Richard:2017una ; Luo:2017eub ; JinX 2021 ;
YangG 2021 ; Deng 2021 , QCD sum rule Chen:2013pya ; Chen:2015ata ;
Wang:2017dtg ; Wang:2019got , and diquark-diquark picture D.Ebert 2007 ;
Esposito:2013fma ; Shi:2021jyr . While, they could be also described as the
molecular states in effective field theory Liu:2009qhy ; WangP 2013 ; HeJ 2013
; Aceti:2014kja ; ChenR 2021 ; Feijoo:2021ppq . See also Refs. Chen:2016qju ;
Lebed:2016hpi ; Guo:2017jvc ; Liu:2019zoy ; Brambilla:2019esw  for recent
reviews.

Very recently, an exotic state $T_{cc}^{+}$ was reported in the
$D^{0}D^{0}\pi^{+}$ mass spectrum by LHCb collaboration LHCbTcc1 ; LHCbTcc2 ,
which has a mass very close to the $D^{0}D^{*+}$ threshold and its width is
extremely narrow. In fact, taking into account the heavy quark symmetry
between $\Xi_{cc}$ and $T_{cc}$, the existence of tetraquark states which
contain two heavy quarks has been predicted theoretically in Refs. Zhu 2013 ;
Kar.M 2017 , after the discovery of double-charm baryon $\Xi_{cc}^{++}$ Xicc .
Especially, the binding energy of $DD^{*}$ molecule predicted by one boson
exchange model is in agreement with the experimental value. Within the
$DD^{*}$ molecule nature, the strong and radiative decay widths of
$T^{+}_{cc}$ are studied in Ref. Feijoo:2021ppq ; Ling:2021bir . The
successful prediction and the discovery of $T_{cc}^{+}$ are enormously helpful
to our understanding of the hadron structure.

The constituent quark model (CQM), in which different versions of the explicit
hyperfine interactions between quarks have been proposed, is one of the most
successful phenomenological method to study the hadron structure. For
instance, in the Godfrey-Isgur model Isgur , all the effects of one gluon
exchange interactions (OGE) are included, and it gives a good description for
meson spectra from $\pi$ to $\Upsilon$ meson. The chiral constituent quark
model Glozman:1995fu ; Glozman:1995xy ; Vijande:2004he ; Deng:2014gqa ;
Yang:2015bmv ; Deng:2017xlb  is another widely used version to investigate
hadron spectrum, which usually contains one boson exchange (OBE) interactions,
characterizing chiral symmetry spontaneously broken of QCD at low energy.
Tentatively, different versions of CQM have been compared in the estimations
on spectrum of baryon excitations in a multiquark picture recently Yuan:2012wz
; Yuan:2012zs ; An:2013zoa ; An:2014lga ; An:2017lwg ; Wang:2021rjk .

In addition to the widely employed OGE- and OBE-CQM, the instanton-induced
interaction between quarks has also been used in Refs. Yuan:2012wz ;
Yuan:2012zs ; An:2013zoa  to investigate the spectrum of baryon excitations as
pentaquark sates. This kind of interaction was firstly introduced by ’t Hooft
to solve the $U(1)$-problem tHooft , and the instanton vacuum of QCD could
provide a mechanism of spontaneous chiral symmetry breaking of QCD
Diakonov:1983hh ; Diakonov:1985eg . Phenomenologically, it has been shown that
one can successfully reproduce the spectrum of light hadrons Shuryak:1988bf ;
Blask:1990ez ; Brau:1998sxe ; Semay:2001th  and the single charm baryons
Migura:2006ep  employing the instanton-induced interaction. Furthermore, the
instanton-induced interaction has also been employed to study the tetraquark
states Metch ; tHooft:2008rus . Consequently, here we use it to investigate
the spectra of low-lying hidden- and double-charm tetraquark states.

The present manuscript is organized as follows. In Sec. II, we present the
formalism of the CQM with instanton-induced interaction between quarks,
including effective Hamiltonian, wave functions for tetraquark configurations
and model parameters. The masses of hidden- and double-charm tetraquark states
in our model and discussions are given in Sec. III. Finally, Sec. IV is a
brief summary of present manuscript.

##  II Framework

###  II.1 Effective Hamiltonian

In present work, a non-relativistic quark potential model is employed for
calculations on the spectra of hidden- and double-charm tetraquark states. The
dynamics of the model are governed by an effective Hamiltonian as follow:

$H_{eff.}=\sum_{i=1}^{4}\left(m_{i}+T_{i}\right)-T_{C.M.}+V_{Conf.}+V_{Ins.}\,,$
(1)

where $m_{i}$ and $T_{i}$ are the constituent mass and kinetic energy of
$i$-th quark, $T_{C.M.}$ denotes center of mass kinetic energy. $V_{Conf.}$
stands for the quark confinement potential, which is taken to be the widely
accepted Cornell potential in present work Cornell :

$V_{Conf.}=\sum_{i<j}-\frac{3}{16}\,\left(\vec{\lambda}^{c}_{i}\cdot\vec{\lambda}^{c}_{j}\right)\,\left(b\,r_{ij}-\frac{4}{3}\frac{\alpha_{ij}}{r_{ij}}+C_{0}\right)\,,$
(2)

where $\vec{\lambda}^{c}_{i(j)}$ is Gell-Mann matrix in $SU(3)$ color space
acting on the $i(j)$-th quark, $b$, $\alpha_{ij}$ and $C_{0}$ are strength of
quark confinement, QCD effective coupling constant between two quarks and zero
point energy, respectively.

For the residual spin-dependent interaction $V_{Ins.}$ in Eq. (1), we employ a
phenomenologically extended version of ’t Hooft’s instanton-induced
interaction Migura:2006ep :

$V_{Ins.}=V^{qq}_{Ins.}+V^{q\bar{q}}_{Ins.}\,,$ (3)

with

$\displaystyle V^{qq}_{Ins.}$ $\displaystyle=$
$\displaystyle\sum_{i<j}-\hat{g}^{qq}_{ij}\Big{(}P_{ij}^{S=1}P_{ij}^{C,\bf{6}}+2P_{ij}^{S=0}P_{ij}^{C,\bf{\bar{3}}}\Big{)}\delta^{3}(\vec{r}_{ij})\,,$
$\displaystyle V^{q\bar{q}}_{Ins.}$ $\displaystyle=$
$\displaystyle\sum_{i<j}\hat{g}^{q\bar{q}}_{ij}\Bigg{[}\frac{3}{2}P_{ij}^{S=1}P_{ij}^{C,\bf{8}}+P_{ij}^{S=0}\Big{(}\frac{1}{2}P_{ij}^{C,\bf{8}}$
(4)
$\displaystyle+8P_{ij}^{C,\bf{1}}\Big{)}\Bigg{]}\delta^{3}(\vec{r}_{ij})\,,$

where $V^{qq}_{Ins.}$ is for the interaction between a quark-quark or
antiquark-antiquark pair, and $V^{q\bar{q}}_{Ins.}$ is for that between a
quark-antiquark pair. $\hat{g}^{qq}_{ij}$ and $\hat{g}^{q\bar{q}}_{ij}$ are
flavor-dependent coupling strength operators, whose explicit matrix elements
are shown in Table 1 and Table 2. $P_{ij}^{S=0}$ and $P_{ij}^{S=1}$ are spin
projector operators onto spin-singlet and spin-triplet states, respectively;
$P_{ij}^{C,\bf{\bar{3}}}$, $P_{ij}^{C,\bf{6}}$, $P_{ij}^{C,\bf{1}}$ and
$P_{ij}^{C,\bf{8}}$ are color projector operators onto color anti-triplet
$\bf{\bar{3}}_{c}$, color sextet $\bf{6}_{c}$, color singlet $\bf{1}_{c}$ and
color octet $\bf{8}_{c}$, respectively.

One should note that instanton-induced interaction is a pure contact
interaction, and $\delta^{3}\left(\vec{r}_{ij}\right)$ will lead to an unbound
Hamiltonian if we don’t treat it perturbatively. Here we regularize it as in
Refs. Isgur ; Vijande:2004he ; Metch ,

$\delta^{3}\left(\vec{r}_{ij}\right)\,\rightarrow\,\left(\frac{\sigma}{\sqrt{\pi}}\right)^{3}\mathrm{exp}\left(-\sigma^{2}\,r_{ij}^{2}\right)\,.$
(5)

The regularization parameter $\sigma$ should correlate to the finite size of
constituent quarks and therefore flavor dependent Vijande:2004he , but here we
just make $\sigma$ to be a unified parameter for different quark flavors and
absorb the flavor dependence into $\hat{g}^{qq}_{ij}$ and
$\hat{g}^{q\bar{q}}_{ij}$.

Table 1: Matrix elements of flavor-dependent coupling strength operators $\hat{g}^{qq}_{ij}$. $g_{f_{1}f_{2}}$ is flavor-dependent model parameter, $n=u,d$ stands for light flavor. | $ud$ | $du$ | $us$ | $ds$ | $sd$ | $su$ | $uc$ | $dc$ | $cd$ | $cu$ | $sc$ | $cs$ | $uu$ | $dd$ | $ss$ | $cc$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$ud$ | $g_{nn}$ | $-g_{nn}$ | 0 | 0 | 0 | 0
$du$ | $-g_{nn}$ | $g_{nn}$
$us$ | 0 | $g_{ns}$ | $0$ | $0$ | $-g_{ns}$ | 0 | 0 | 0
$ds$ | $0$ | $g_{ns}$ | $-g_{ns}$ | $0$
$sd$ | $0$ | $-g_{ns}$ | $g_{ns}$ | $0$
$su$ | $-g_{ns}$ | $0$ | $0$ | $g_{ns}$
$uc$ | 0 | 0 | $g_{nc}$ | 0 | 0 | $-g_{nc}$ | 0 | 0
$dc$ | 0 | $g_{nc}$ | $-g_{nc}$ | 0
$cd$ | 0 | $-g_{nc}$ | $g_{nc}$ | 0
$cu$ | $-g_{nc}$ | 0 | 0 | $g_{nc}$
$sc$ | 0 | 0 | 0 | $g_{sc}$ | $-g_{sc}$ | 0
$cs$ | $-g_{sc}$ | $g_{sc}$
$uu$ | 0 | 0 | 0 | 0 | 0
$dd$
$ss$
$cc$
Table 2: Matrix elements of flavor-dependent coupling strength operator $\hat{g}^{q\bar{q}}_{ij}$. $g_{f_{1}f_{2}}$ is flavor-dependent model parameter, $n=u,d$ stands for light flavor. | $u\bar{d}$ | $d\bar{u}$ | $u\bar{s}$ | $d\bar{s}$ | $s\bar{d}$ | $s\bar{u}$ | $u\bar{c}$ | $d\bar{c}$ | $c\bar{d}$ | $c\bar{u}$ | $s\bar{c}$ | $c\bar{s}$ | $u\bar{u}$ | $d\bar{d}$ | $s\bar{s}$ | $c\bar{c}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$u\bar{d}$ | $-g_{nn}$ | $0$ | 0 | 0 | 0 | 0
$d\bar{u}$ | $0$ | $-g_{nn}$
$u\bar{s}$ | 0 | $-g_{ns}$ | 0 | 0 | 0 | 0 | 0 | 0
$d\bar{s}$ | 0 | $-g_{ns}$ | 0 | 0
$s\bar{d}$ | 0 | $0$ | $-g_{ns}$ | 0
$s\bar{u}$ | 0 | $0$ | $0$ | $-g_{ns}$
$u\bar{c}$ | 0 | 0 | $-g_{nc}$ | 0 | 0 | 0 | 0 | 0
$d\bar{c}$ | $0$ | $-g_{nc}$ | 0 | 0
$c\bar{d}$ | $0$ | $0$ | $-g_{nc}$ | 0
$c\bar{u}$ | $0$ | $0$ | $0$ | $-g_{nc}$
$s\bar{c}$ | 0 | 0 | 0 | $-g_{sc}$ | 0 | 0
$c\bar{s}$ | $0$ | $-g_{sc}$
$u\bar{u}$ | 0 | 0 | 0 | 0 | 0 | $g_{nn}$ | $g_{ns}$ | $g_{nc}$
$d\bar{d}$ | $g_{nn}$ | $0$ | $g_{ns}$ | $g_{nc}$
$s\bar{s}$ | $g_{ns}$ | $g_{ns}$ | $0$ | $g_{sc}$
$c\bar{c}$ | $g_{nc}$ | $g_{nc}$ | $g_{sc}$ | $0$

###  II.2 Configurations of four quark systems

Considering the Pauli principle for two-quark (-antiquark) subsystems, there
are eight possible flavor-spin-color symmetry configurations for
$qq\bar{q}\bar{q}$ systems, as follows:

$\displaystyle\hskip 28.45274pt|1\rangle$
$\displaystyle=\\{qq\\}_{{\bf{6}}_{c}}\\{\bar{q}\bar{q}\\}_{\bar{\bf{6}}_{c}}\,,$
$\displaystyle|2\rangle$
$\displaystyle=[qq]_{\bar{\bf{3}}_{c}}[\bar{q}\bar{q}]_{{\bf{3}}_{c}}\,,\hskip
28.45274pt$ $\displaystyle|3\rangle$
$\displaystyle=\\{qq\\}^{*}_{\bar{\bf{3}}_{c}}[\bar{q}\bar{q}]_{{\bf{3}}_{c}}\,,$
$\displaystyle|4\rangle$
$\displaystyle=[qq]_{\bar{\bf{3}}_{c}}\\{\bar{q}\bar{q}\\}^{*}_{{\bf{3}}_{c}}\,,$
$\displaystyle|5\rangle$
$\displaystyle=[qq]^{*}_{{\bf{6}}_{c}}\\{\bar{q}\bar{q}\\}_{\bar{\bf{6}}_{c}}\,,$
$\displaystyle|6\rangle$
$\displaystyle=\\{qq\\}_{{\bf{6}}_{c}}[\bar{q}\bar{q}]^{*}_{\bar{\bf{6}}_{c}}\,,$
$\displaystyle|7\rangle$
$\displaystyle=\\{qq\\}^{*}_{\bar{\bf{3}}_{c}}\\{\bar{q}\bar{q}\\}^{*}_{{\bf{3}}_{c}}\,,$
$\displaystyle|8\rangle$
$\displaystyle=[qq]^{*}_{{\bf{6}}_{c}}[\bar{q}\bar{q}]^{*}_{\bar{\bf{6}}_{c}}\,,$
(6)

where $\\{\dots\\}$ denotes a permutation symmetric flavor wave function and
$[\dots]$ denotes a permutation antisymmetric flavor wave function of two-
quark (-antiquark) subsystem. The superscript $*$ means diquark or antidiquark
forms a spin-triplet, while the configurations without superscript are spin-
singlet. The blackened number in the subscript denotes color wave function of
two-quark (-antiquark) subsystem. As we can see, there are two possible color
neutral structures of tetraquark systems, namely,
$\bf{6}_{c}\otimes\bf{\bar{6}}_{c}$ and $\bf{\bar{3}}_{c}\otimes\bf{3}_{c}$.
Note that here we only consider the $S$-wave tetraquark systems, so the
spatial wave functions are always completely permutation symmetric.

Explicit flavor wave functions of the studied hidden- and double-charm
tetraquark systems are presented in the following subsections, respectively.

####  II.2.1 Hidden-Charm Systems

There are nine flavor configurations for $q\bar{q}c\bar{c}$ with $q$ the light
quarks $u$, $d$, and $s$, which are: $uc\bar{u}\bar{c}$, $uc\bar{d}\bar{c}$,
$uc\bar{s}\bar{c}$, $dc\bar{d}\bar{c}$, $dc\bar{s}\bar{c}$,
$sc\bar{s}\bar{c}$, $dc\bar{u}\bar{c}$, $sc\bar{u}\bar{c}$, and
$sc\bar{d}\bar{c}$. Compositions of these configurations could lead to the
tetraquark states with quantum numbers isospin $I$ and strangeness $S$, as
below:

  * •

$I=0,\,S=0.$

$\displaystyle X_{n\bar{n}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(uc\bar{u}\bar{c}+dc\bar{d}\bar{c}\right),$
(7) $\displaystyle X_{s\bar{s}}$ $\displaystyle=sc\bar{s}\bar{c}.$ (8)

  * •

$I=\frac{1}{2},\,S=\pm 1.$

$\displaystyle Z_{cs}^{+}=uc\bar{s}\bar{c},$
$\displaystyle\bar{Z}_{cs}^{0}=-sc\bar{d}\bar{c},$ (9) $\displaystyle
Z_{cs}^{0}=dc\bar{s}\bar{c}.$ $\displaystyle Z_{cs}^{-}=sc\bar{u}\bar{c}.$
(10)

  * •

$I=1,\,S=0.$

$\displaystyle Z_{c}^{+}$ $\displaystyle=-uc\bar{d}\bar{c},$ (11)
$\displaystyle Z_{c}^{0}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(uc\bar{u}\bar{c}-dc\bar{d}\bar{c}\right),$
(12) $\displaystyle Z_{c}^{-}$ $\displaystyle=dc\bar{u}\bar{c}.$ (13)

$X$ and $Z_{c}^{0}$ are pure neutral systems, thus they could have $C$-parity.
Note that $\bar{Z}_{cs}^{0}$ and $Z_{cs}^{-}$ are the conjugate states of
$Z_{cs}^{0}$ and $Z_{cs}^{+}$, respectively. Thus, we only consider the latter
set in following, since a state should share the same energy with its
conjugate state in present model.

Accordingly, the symmetry configurations of hidden-charm systems in Eq. (6)
could be classified by quantum numbers $J^{P(C)}$, which are listed in Table
3.

Table 3: The symmetry configurations of hidden-charm systems. For $Z_{cs}$ states | For $X$ and $Z_{c}$ states
---|---
$J^{P}$ | Configuration | $J^{PC}$ | Configuration
$0^{+}$ | $|1\rangle$ | $0^{++}$ | $|1\rangle$
| $|2\rangle$ |  | $|2\rangle$
| $|7\rangle$ |  | $|7\rangle$
| $|8\rangle$ |  | $|8\rangle$
$1^{+}$ | $|3\rangle$ | $1^{++}$ | $|3^{\prime}\rangle=\frac{1}{\sqrt{2}}\left(|3\rangle+|4\rangle\right)$
| $|4\rangle$ |  | $|5^{\prime}\rangle=\frac{1}{\sqrt{2}}\left(|5\rangle+|6\rangle\right)$
| $|5\rangle$ | $1^{+-}$ | $|4^{\prime}\rangle=\frac{1}{\sqrt{2}}\left(|3\rangle-|4\rangle\right)$
| $|6\rangle$ |  | $|6^{\prime}\rangle=\frac{1}{\sqrt{2}}\left(|5\rangle-|6\rangle\right)$
| $|7\rangle$ |  | $|7\rangle$
| $|8\rangle$ |  | $|8\rangle$
$2^{+}$ | $|7\rangle$ | $2^{++}$ | $|7\rangle$
| $|8\rangle$ |  | $|8\rangle$

####  II.2.2 Double-Charm Systems

Similarly as in the previous section, the flavor wave functions of
$cc\bar{q}\bar{q}$ systems with proper quantum numbers can be decomposed as
follows:

  * •

$I=0,\,S=0\,.$

$\left(T_{cc}^{0,0}\right)^{+}=\frac{1}{\sqrt{2}}cc\left(-\bar{d}\bar{u}+\bar{u}\bar{d}\right)\,,$
(14)

  * •

$I=0,\,S=2\,.$

$\left(T_{cc}^{0,2}\right)^{++}=cc\bar{s}\bar{s}\,,$ (15)

  * •

$I=\frac{1}{2},\,S=1\,.$

$\displaystyle\left(T_{cc}^{\frac{1}{2},1}\right)^{+}$
$\displaystyle=-\frac{1}{\sqrt{2}}cc\left(\bar{d}\bar{s}\pm\bar{s}\bar{d}\right)\,,$
(16) $\displaystyle\left(T_{cc}^{\frac{1}{2},1}\right)^{0}$
$\displaystyle=\frac{1}{\sqrt{2}}cc\left(\bar{u}\bar{s}\pm\bar{s}\bar{u}\right)\,,$
(17)

  * •

$I=1,\,S=0\,.$

$\displaystyle\left(T_{cc}^{1,0}\right)^{++}=cc\bar{d}\bar{d}\,,$ (18)
$\displaystyle\left(T_{cc}^{1,0}\right)^{+}=-\frac{1}{\sqrt{2}}cc\left(\bar{d}\bar{u}+\bar{u}\bar{d}\right)\,,$
(19) $\displaystyle\left(T_{cc}^{1,0}\right)^{0}=cc\bar{u}\bar{u}\,,$ (20)

here we use $\left(T_{cc}^{I,S}\right)^{Q}$ to denote $cc\bar{q}\bar{q}$
systems, where $I$, $S$ and $Q$ are isospin, strangeness, and electric charge
of the state, respectively.

For the double-charm states, the flavor wave function of diquark must be
symmetric since it is $cc$, and the symmetry of antidiquark flavor wave
functions is determined by Eqs. (14–20). All the configurations of double-
charm tetraquark systems and their corresponding quantum numbers are listed in
Table 4.

Table 4: The symmetry configuraitons of double-charm systems. $T_{cc}^{I,S}$ | $J^{P}$ | Configuration
---|---|---
$T_{cc}^{0,0}$ | $1^{+}$ | $|3\rangle$
|  | $|6\rangle$
$T_{cc}^{0,2}$ | $0^{+}$ | $|1\rangle$
|  | $|7\rangle$
| $1^{+}$ | $|7\rangle$
| $2^{+}$ | $|7\rangle$
$T_{cc}^{\frac{1}{2},1}$ | $0^{+}$ | $|1\rangle$
|  | $|7\rangle$
| $1^{+}$ | $|3\rangle$
|  | $|6\rangle$
|  | $|7\rangle$
| $2^{+}$ | $|7\rangle$
$T_{cc}^{1,0}$ | $0^{+}$ | $|1\rangle$
|  | $|7\rangle$
| $1^{+}$ | $|7\rangle$
| $2^{+}$ | $|7\rangle$

###  II.3 Wave functions

For the calculation of Hamiltonian matrices, a Gaussian functions expansion
method Zhang:2007mu ; Zhang:2005jz ; Liu:2019zuc ; GEM  is used here to solve
the Schrödinger equation of Hamiltonian Eq. (1), where the orbital wave
function of the ground $S$-wave tetraquark system can be expanded by a series
of Gaussian functions,

$\Psi(\\{\vec{r}_{i}\\})=\prod_{i=1}^{4}\sum_{\mathcal{l}}^{n}C_{i\mathcal{l}}\left(\frac{1}{\pi
b_{i\mathcal{l}}^{2}}\right)^{3/4}\mathrm{exp}\left[-\frac{1}{2b_{i\mathcal{l}}^{2}}r_{i}^{2}\right]\,,$
(21)

here $\\{b_{i\mathcal{l}}\\}$ are harmonic oscillator length parameters and
can be related to the frequencies $\\{\omega_{\mathcal{l}}\\}$ with
$1/b_{i\mathcal{l}}^{2}=m_{i}\omega_{\mathcal{l}}$, where we have taken the
ansatz that $\omega_{\mathcal{l}}$ is independent to the quark mass as in Ref.
Liu:2019zuc . Then the orbital wave function can be simplified to be

$\displaystyle\Psi(\\{\vec{r}_{i}\\})=$
$\displaystyle\sum_{\mathcal{l}}^{n}C_{\mathcal{l}}\prod_{i=1}^{4}\left(\frac{m_{i}\omega_{\mathcal{l}}}{\pi}\right)^{3/4}\mathrm{exp}\left[-\frac{m_{i}\omega_{\mathcal{l}}}{2}r_{i}^{2}\right]$
$\displaystyle=$
$\displaystyle\sum_{\mathcal{l}}^{n}C_{\mathcal{l}}\,\psi\left(\omega_{\mathcal{l}},\\{\vec{r}_{i}\\}\right)\,,$
(22)

which is often adopted for the calculations of multiquark systems Zhang:2007mu
; Zhang:2005jz .

On the other hand, here we define the Jacobi coordinates by
$\\{\vec{r}_{i}\\}$ as

$\displaystyle\vec{\xi}_{1}$ $\displaystyle=$
$\displaystyle\vec{r}_{1}-\vec{r}_{2}\,,$ (23) $\displaystyle\vec{\xi}_{2}$
$\displaystyle=$ $\displaystyle\vec{r}_{3}-\vec{r}_{4}\,,$ (24)
$\displaystyle\vec{\xi}_{3}$ $\displaystyle=$
$\displaystyle\frac{m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2}}{m_{1}+m_{2}}-\frac{m_{3}\vec{r}_{3}+m_{4}\vec{r}_{4}}{m_{3}+m_{4}}\,,$
(25) $\displaystyle\vec{R}$ $\displaystyle=$
$\displaystyle\frac{m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2}+m_{3}\vec{r}_{3}+m_{4}\vec{r}_{4}}{m_{1}+m_{2}+m_{3}+m_{4}}\,,$
(26)

to remove contributions of the motion of the center of mass directly. Where we
have enumerated the two quarks by $1,2$ and the two antiquarks by $3,4$,
respectively.

In the calculations we define $1/b_{\mathcal{l}}^{2}\equiv
m_{q}\omega_{\mathcal{l}}$ ($m_{q}$ denotes the constituent mass of the
corresponding quark). Following the work of Ref. GEM , the parameters
$\\{b_{\mathcal{l}}\\}$ are set to be geometric series,

$b_{\mathcal{l}}=b_{1}a^{\mathcal{l}-1}\hskip
22.76228pt\left(\mathcal{l}=1,2,...,n\right)\,,$ (27)

where $n$ is the number of Gaussian functions and $a$ the ratio coefficient,
then there are only thee parameters $\\{b_{1},b_{n},n\\}$ need to be
determined.

Using the orbital wave function in Eq. (22), we compute the diagonal elements
of the Hamiltonian matrices by solving the generalized matrix eigenvalue
problem:

$\sum_{\mathcal{l}}^{n}\sum_{\mathcal{l}^{\prime}}^{n}C^{i}_{\mathcal{l}}\left(H^{d}_{\mathcal{l}\mathcal{l}^{\prime}}-E_{i}^{d}N_{\mathcal{l}\mathcal{l}^{\prime}}\right)C^{i}_{\mathcal{l}^{\prime}}=0\,,$
(28)

where $i=1\text{--}n$ and

$\displaystyle H^{d}_{\mathcal{l}\mathcal{l}^{\prime}}=$
$\displaystyle\langle\psi\left(\omega_{\mathcal{l}}\right)(FSC)|H_{eff.}|\psi\left(\omega_{\mathcal{l}^{\prime}}\right)(FSC)\rangle\,,$
(29) $\displaystyle N_{\mathcal{l}\mathcal{l}^{\prime}}=$
$\displaystyle\langle\psi\left(\omega_{\mathcal{l}}\right)(FSC)|\psi\left(\omega_{\mathcal{l}^{\prime}}\right)(FSC)\rangle\,,$
(30)

here $(FSC)$ stands for flavor$\,\otimes\,$spin$\,\otimes\,$color wave
function. One can choose a set of $\\{b_{1},b_{n},n\\}$ tentatively, then
extend and densify the harmonic oscillator length parameters to get a minimum
energy $E^{d}_{m}$, which should be correlative to the energy of physical
state according to the Rayleigh-Ritz variational principle. In present work we
take $\\{b_{1},b_{n},n\\}=\\{0.02\,\text{fm},6\,\text{fm},40\\}$ for the light
quarks, and one can directly obtain the ranges for the strange and charm
quarks by replacing the light quark mass by strange and charm quark masses,
respectively. Besides, the nondiagonal elements of Hamiltonian matrices can be
easily obtained using $\\{C_{\mathcal{l}}^{m}\\}$.

###  II.4 Model Parameters

There are sixteen parameters in our model, namely, constituent quark masses
$m_{n}$, $m_{s}$ and $m_{c}$; quark confinement strength $b$, QCD effective
coupling constants $\alpha_{ij}$ ($i$ and $j$ are quark flavor), and zero
point energy $C_{0}$ for Cornell potential; $g_{nn}$, $g_{ns}$, $g_{nc}$,
$g_{sc}$, and $\sigma$ for instanton-induced interaction. All these above
model parameters are collected in Table 5.

Table 5: Parameters used in this work. $m_{n}=340\,\text{MeV}$ | $m_{s}=511\,\text{MeV}$
---|---
$m_{c}=1674\,\text{MeV}$ |
$b=0.39\,\text{GeV}\cdot\text{fm}^{-1}$ | $C_{0}=-296\,\text{MeV}$
$\alpha_{nn}=0.600$ | $\alpha_{ns}=0.600$
$\alpha_{ss}=0.562$ | $\alpha_{nc}=0.509$
$\alpha_{sc}=0.480$ | $\alpha_{cc}=0.450$
$g_{nn}=0.169\times 10^{-4}\,\text{MeV}^{-2}$ | $g_{ns}=0.107\times 10^{-4}\,\text{MeV}^{-2}$
$g_{nc}=0.040\times 10^{-4}\,\text{MeV}^{-2}$ | $g_{sc}=0.031\times 10^{-4}\,\text{MeV}^{-2}$
$\sigma=485\,\text{MeV}$ |

Since none of the exotic states is identified as a pure tetraquark state, one
cannot take the masses of these exotic states as input to evaluate the model
parameters. In our model, the interactions between quarks and antiquarks in
tetraquark states are simple superposition of two-body confinements and tow-
body residual interactions, which is the same as the description of
traditional baryons and mesons in CQM. Therefore, we just extract model
parameters from traditional hadron spectra. In this work, eleven mesons and
eight baryons are taken as inputs to determine these model parameters.

It is worthy to mention that, if the pseudoscalar mesons whose dominant
contents are considered to be pure $q_{f_{1}}\bar{q}_{f_{2}}$ with
$f_{1}=f_{2}$, as shown in Table 2, the instanton-induced interaction effects
will vanish, this should lead to the mass relation $m_{\eta_{c}}=m_{J/\psi}$,
it’s obviously unreasonable. In fact, as discussed in tHooft:1999cta , the
instanton-induced interaction should cause mixing between this kind of states
with quantum number $0^{-+}$ naturally, although the mixing between the
$c\bar{c}$ pair with the light quark-antiquark pairs should be very small, it
will obviously influence on the mass of $\eta_{c}$. Consequently, to get the
correct mass relations, one has to consider the flavor-configuration mixing
for $\eta$, $\eta^{\prime}$ and $\eta_{c}$. However, the mixing angles are
unknown and difficult to determine tHooft:1999cta . Thus, these sates $\eta$,
$\eta^{\prime}$ and $\eta_{c}$ are not taken into account for determining the
model parameters shown in table 5. In addition, the effective strong coupling
constant between two charm quarks is determined by fitting the mass of
$J/\psi$.

With these model parameters shown in Table 5, the model calculations for the
hadron spectra are shown in Table 6.

Table 6: Ground state hadron spectrum. The columns denoted by “M” are model calculations while the columns denoted by “PDG” are experimental data for the masses of these states in PDG PDG . The units are in MeV. States | M | PDG | States | M | PDG
---|---|---|---|---|---
$\pi$ | $139$ | $138$ | $J/\psi$ | $3097$ | $3097$
$\rho$ | $775$ | $775$ | $\Sigma_{c}$ | $2447$ | $2454$
$\omega$ | $775$ | $783$ | $\Sigma_{c}^{*}$ | $2518$ | $2518$
$\phi$ | $1019$ | $1019$ | $\Xi_{c}$ | $2507$ | $2469$
$K$ | $499$ | $496$ | $\Xi_{c}^{\prime}$ | $2554$ | $2577$
$K^{*}$ | $894$ | $895$ | $\Xi_{c}^{*}$ | $2627$ | $2646$
$D$ | $1866$ | $1867$ | $\Omega_{c}$ | $2664$ | $2695$
$D^{*}$ | $2009$ | $2009$ | $\Omega_{c}^{*}$ | $2737$ | $2766$
$D_{s}$ | $1969$ | $1968$ | $\Xi_{cc}$ | $3607$ | $3621$
$D_{s}^{*}$ | $2111$ | $2112$ |  |  |

##  III Results and Discussions

With the model parameters in Table 5, one can calculate the mass spectra of
the ground $S$-wave hidden- and double-charm tetraquark systems. The numerical
results are shown in Tables 7-13.

As discussed in the Sec. II, there are two possible color structures of
tetraquark configurations, namely, $\bf{6}_{c}\otimes\bf{\bar{6}}_{c}$ and
$\bf{\bar{3}}_{c}\otimes\bf{3}_{c}$. The results of color factor
$\langle\vec{\lambda}^{c}_{i}\cdot\vec{\lambda}^{c}_{j}\rangle$ show that, the
Cornell potentials are attractive between each of quarks and antiquarks in
tetraquark states with $\bf{\bar{3}}_{c}\otimes\bf{3}_{c}$ color structure
(configurations $|2\rangle,|3\rangle,|4\rangle,|7\rangle$). However, they are
repulsive for those states with $\bf{6}_{c}\otimes\bf{\bar{6}}_{c}$ color
structure (configurations $|1\rangle,|5\rangle,|6\rangle,|8\rangle$).
Intuitively, one might think that the tetraquark states with
$\bf{\bar{3}}_{c}\otimes\bf{3}_{c}$ structure could be more bound than those
states with the $\bf{6}_{c}\otimes\bf{\bar{6}}_{c}$, and thus have lower
energies.

However, this is not always true. Because, on one hand, a tighter binding is
accompanied by a higher quark kinetic energy, on the other hand a color
dependent residual interaction, Instanton-induced interaction, is considered
in the model, which has a much more complicated interaction structure in
tetraquark states compared with traditional meson states.

Consequently, a physical tetraquark state should be composed of the two kinds
of components with $\bf{\bar{3}}_{c}\otimes\bf{3}_{c}$ and
$\bf{6}_{c}\otimes\bf{\bar{6}}_{c}$ color structures. In Tables 7, 9, 10 and
13, the numerical results including mixing of the two kinds of components are
presented in the last two columns. Obviously, effects of configurations mixing
are important.

In the following subsections, we discuss the results of $X$, $Z_{c}$, $Z_{cs}$
and double-charm tetraquark states, respectively.

###  III.1 $X$ states

Table 7: The numerical results of ground $S$-wave $X$ states with $I^{G}=0^{+}$. First and second columns are the quark content of each state and the corresponding $J^{PC}$ quantum numbers, the fourth column are the results of single configuration calculations, fifth and sixth columns are the model results after considering configuration mixing, but not considering $cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ mixing. |  | Single configuration | Configurations mixing
---|---|---|---
Quark content | $J^{PC}$ | Config. | Energies (MeV) | Energies (MeV) | Mixing coefficients
$c\bar{c}n\bar{n}$ | $0^{++}$ | $|1\rangle$ | $4161.55$ | $4039.34$ | $(-0.64,\hskip 4.26773pt-0.43,\hskip 4.26773pt0.00,\hskip 4.26773pt-0.64)$
$|2\rangle$ | $4131.90$ | $4115.93$ | $(0.00,\hskip 4.26773pt-0.75,\hskip 4.26773pt-0.43,\hskip 4.26773pt0.50)$
$|7\rangle$ | $4212.69$ | $4231.32$ | $(0.28,\hskip 4.26773pt-0.46,\hskip 4.26773pt0.84,\hskip 4.26773pt0.02)$
$|8\rangle$ | $4131.28$ | $4250.83$ | $(0.71,\hskip 4.26773pt-0.20,\hskip 4.26773pt-0.34,\hskip 4.26773pt-0.59)$
$c\bar{c}n\bar{n}$ | $1^{++}$ | $|3^{\prime}\rangle$ | $4151.06$ | $4052.78$ | $(0.51,\hskip 4.26773pt-0.86)$
$|5^{\prime}\rangle$ | $4088.23$ | $4186.52$ | $(-0.86,\hskip 4.26773pt-0.51)$
$c\bar{c}n\bar{n}$ | $1^{+-}$ | $|4^{\prime}\rangle$ | $4193.21$ | $4053.08$ | $(-0.46,\hskip 4.26773pt0.51,\hskip 4.26773pt-0.25,\hskip 4.26773pt0.68)$
$|6^{\prime}\rangle$ | $4190.93$ | $4186.87$ | $(-0.26,\hskip 4.26773pt0.41,\hskip 4.26773pt-0.55,\hskip 4.26773pt-0.68)$
$|7\rangle$ | $4211.40$ | $4228.55$ | $(0.75,\hskip 4.26773pt0.05,\hskip 4.26773pt-0.61,\hskip 4.26773pt0.24)$
$|8\rangle$ | $4127.50$ | $4254.52$ | $(0.40,\hskip 4.26773pt0.76,\hskip 4.26773pt0.51,\hskip 4.26773pt-0.11)$
$c\bar{c}n\bar{n}$ | $2^{++}$ | $|7\rangle$ | $4208.78$ | $4055.73$ | $(-0.54,\hskip 4.26773pt-0.84)$
$|8\rangle$ | $4119.53$ | $4272.57$ | $(-0.84,\hskip 4.26773pt0.54)$
$c\bar{c}s\bar{s}$ | $0^{++}$ | $|1\rangle$ | $4341.92$ | $4203.76$ | $(-0.24,\hskip 4.26773pt-0.37,\hskip 4.26773pt-0.14,\hskip 4.26773pt0.89)$
$|2\rangle$ | $4293.15$ | $4270.15$ | $(0.61,\hskip 4.26773pt-0.77,\hskip 4.26773pt0.10,\hskip 4.26773pt-0.14)$
$|7\rangle$ | $4361.49$ | $4340.17$ | $(0.34,\hskip 4.26773pt0.32,\hskip 4.26773pt0.81,\hskip 4.26773pt0.35)$
$|8\rangle$ | $4237.31$ | $4419.79$ | $(-0.67,\hskip 4.26773pt-0.41,\hskip 4.26773pt0.56,\hskip 4.26773pt-0.27)$
$c\bar{c}s\bar{s}$ | $1^{++}$ | $|3^{\prime}\rangle$ | $4328.42$ | $4284.16$ | $(-0.39,\hskip 4.26773pt-0.92)$
$|5^{\prime}\rangle$ | $4292.14$ | $4336.39$ | $(-0.92,\hskip 4.26773pt0.39)$
$c\bar{c}s\bar{s}$ | $1^{+-}$ | $|4^{\prime}\rangle$ | $4350.70$ | $4265.06$ | $(0.00,\hskip 4.26773pt0.00,\hskip 4.26773pt-0.17,\hskip 4.26773pt0.98)$
$|6^{\prime}\rangle$ | $4347.16$ | $4283.64$ | $(0.70,\hskip 4.26773pt-0.72,\hskip 4.26773pt0.00,\hskip 4.26773pt0.00)$
$|7\rangle$ | $4372.05$ | $4375.42$ | $(0.00,\hskip 4.26773pt0.00,\hskip 4.26773pt0.98,\hskip 4.26773pt0.17)$
$|8\rangle$ | $4268.42$ | $4414.23$ | $(0.72,\hskip 4.26773pt0.70,\hskip 4.26773pt0.00,\hskip 4.26773pt0.00)$
$c\bar{c}s\bar{s}$ | $2^{++}$ | $|7\rangle$ | $4392.52$ | $4286.55$ | $(0.52,\hskip 4.26773pt-0.85)$
$|8\rangle$ | $4325.86$ | $4431.83$ | $(-0.85,\hskip 4.26773pt-0.52)$

In this subsection, we present the numerical results for spectrum of the
ground $S$-wave $cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ states, those we
call $X$ states.

Similar to $\eta\text{--}\eta^{\prime}$ mixing, there are transitions between
$cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ quark configurations, which can be
naturally caused by the instanton-induced interaction. Consequently, we will
discuss the results within two models: one takes into account only the single
configurations $cn\bar{c}\bar{n}$ or $cs\bar{c}\bar{s}$, respectively; while
the other one includes the mixing of these configurations. The numerical
results of energies for the $I^{G}=0^{+}$ $X$ states are given in Tables 7 and
8, respectively, where the masses and configurations mixing coefficients for
the obtained tetraquark states are shown explicitly.

Meanwhile, we depict the obtained mass spectrum of $X$ states in Fig. 1
comparing with the experimental measurements, where red solid lines denote
$cn\bar{c}\bar{n}$ system and blue lines denote $cs\bar{c}\bar{s}$ system,
hollow circles with crosses denote results after considering
$cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ mixing, gray dotted lines are
corresponding $S$-wave charm meson pair thresholds, finally, the green
rectangles represent experimental masses of these $X$ states taken from PDG
PDG , the rectangle widths stand their masses uncertainties.

Figure 1: Mass spectrum of ground $S$-wave $X$ states. Red solid lines denote
$cn\bar{c}\bar{n}$ system and blue lines denote $cs\bar{c}\bar{s}$ system,
hollow circles with crosses denote results after considering
$cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ mixing, gray dotted lines are
corresponding $S$-wave charm meson pair thresholds, the green rectangles
represent $X$ states experimental masses taken from PDG PDG , the rectangle
width stand their masses uncertainties.

Experimentally, in 2003, the $X(3872)$ state was observed by Belle
collaboration Belle 2003  and its quantum number was determined to be
$J^{PC}=1^{++}$ LHCb 2013 . Explicitly, its mass and width are: $M=3871.65\pm
0.06$ and $\Gamma=1.19\pm 0.01\,\text{MeV}$ PDG . Its mass is extremely close
to the $D^{0}\bar{D}^{*0}$ mass threshold, which indicates that $X(3872)$ may
be explained as a $D^{0}\bar{D}^{*0}$ hadronic molecule. In present work, the
lowest energy of $cn\bar{c}\bar{n}$ states with quantum numbers
$I^{G}(J^{PC})=0^{+}(1^{++})$ is about $4053\,\text{MeV}$, which is much
higher than the mass of $X(3872)$. Consequently, our result may support that
the $X(3872)$ is mostly dominated by a hadronic molecular state.

The $X(3930)$ and $X(3915)$ states are observed through two photon fusion
processes by Belle in 2005 Belle 2006  and 2010 Belle 2010 , respectively.
They were suggested to be good candidates of the $2^{3}P_{0}$ and $2^{3}P_{2}$
charmonium states, $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$, respectively Belle
2006 ; Belle 2010 ; BaBar 2012 . From our results shown in Table 7 and Fig. 1,
it is found that our results of the states with quantum numbers
$I^{G}(J^{PC})=0^{+}(0^{++})$ and $I^{G}(J^{PC})=0^{+}(2^{++})$ are all higher
than the masses of $X(3930)$ and $X(3915)$. Thus $X(3930)$ and $X(3915)$
cannot be described by the compact tetraquark states.

The CDF collaboration reported the discoveries of $X(4140)$ and $X(4274)$ in
the $J/\psi\phi$ invariant mass distribution of the process $B^{+}\rightarrow
J/\psi\phi K^{+}$ in 2009 CDF 2009  and 2011 CDF 2011 , respectively. Later,
LHCb collaboration confirmed their existences and determined their spin-parity
quantum numbers to be $J^{P}=1^{+}$ LHCb 2017 ; LHCb PRD 2017 ; LHCb 2021 .
Masses and widths of $X(4140)$ and $X(4274)$ are PDG

$\displaystyle(M_{X(4140)}=4146.8\pm
2.4\,,\Gamma_{X(4140)}=22^{+8}_{-7})\,\text{MeV}\,,$
$\displaystyle(M_{X(4274)}=4274^{+8}_{-6}\,,\Gamma_{X(4274)}=49\pm
12)\,\text{MeV}\,,$

respectively. The next lowest energy of the $cn\bar{c}\bar{n}$ states with
quantum number $I^{G}(J^{PC})=0^{+}(1^{++})$ in present model is around
$4187\,\text{MeV}$, very close to the mass of $X(4140)$ state, while the
lowest energy of the $cs\bar{c}\bar{s}$ states with quantum number
$I^{G}(J^{PC})=0^{+}(1^{++})$ in present model is about $4284\,\text{MeV}$,
very close to the mass of $X(4274)$ state. Thus $X(4140)$ and $X(4274)$ can be
interpreted as compact $c\bar{c}n\bar{n}$ and $c\bar{c}s\bar{s}$ tetraquark
states with quantum number $I^{G}(J^{PC})=0^{+}(1^{++})$, respectively.

The results by considering $cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ mixing
are also given in Fig. 1, which are denoted by hollow circles with crosses. It
can be found that configurations mixing effects in the tetraquark states with
quantum numbers $J^{PC}=0^{++}$ are significant, while those in the
$J^{PC}=1^{++}$ tetraquark states are tiny. The results of four of the
obtained $X$ states with quantum number $J^{PC}=1^{++}$, considering
$cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ mixing, are shown in Table 8 with
explicit probabilities of the admixtures. One can find that every state is
dominated by a single configuration.

Table 8: The results of the $X$ states with quantum numbers $J^{PC}=1^{++}$ by considering $cn\bar{c}\bar{n}$ and $cs\bar{c}\bar{s}$ mixing. Model values | Experimental values
---|---
M (MeV) | $c\bar{c}n\bar{n}$ | $c\bar{c}s\bar{s}$ | States | M (MeV)
$4052.72$ | $99.98\%$ | $0.02\%$ | — | —
$4176.07$ | $93.70\%$ | $6.30\%$ | $X(4140)$ | $4146.8\pm 2.4$
$4284.76$ | $0.46\%$ | $99.54\%$ | $X(4274)$ | $4274^{+8}_{-6}$
$4346.29$ | $5.86\%$ | $94.14\%$ | — | —

Finally, as shown in Fig. 1, one may note that most of the energies for the
obtained $X$ states in present work are near but higher than thresholds of
corresponding $S-$wave meson-meson channels, with only a few exceptions
including the states shown in Table 8. This may indicate that most of the
presently obtained $X$ states with compact tetraquark structure cannot form
considerable components in the physical meson exotics.

###  III.2 $Z_{c}$ and $Z_{cs}$ states

The numerical results for $S$-wave $cn\bar{c}\bar{n}$ and $cn\bar{c}\bar{s}$
systems are collected in Table. 9 and Table. 10, respectively. In these
tables, the first and second columns give the quark content of each state and
their corresponding quantum numbers, the fourth column shows the numerical
results considering only single configurations, while numbers in the fifth
column are the results obtained by including the configurations mixing
effects, and the explicit probability amplitudes for the mixed configurations
are shown in the last column.

Table 9: As in table 7 but for the case of $Z_{c}$ states. |  | Single configuration | Configurations mixing
---|---|---|---
Quark content | $I^{G}(J^{PC})$ | Config. | Energies (MeV) | Energies (MeV) | Mixing coefficients
$c\bar{c}n\bar{n}$ | $1^{-}(0^{++})$ | $|1\rangle$ | $4079.46$ | $3777.23$ | $(0.23,\hskip 4.26773pt0.42,\hskip 4.26773pt0.30,\hskip 4.26773pt0.82)$
$|2\rangle$ | $4048.33$ | $4018.55$ | $(-0.76,\hskip 4.26773pt-0.17,\hskip 4.26773pt-0.42,\hskip 4.26773pt0.46)$
$|7\rangle$ | $4093.05$ | $4076.61$ | $(-0.04,\hskip 4.26773pt-0.78,\hskip 4.26773pt0.60,\hskip 4.26773pt0.19)$
$|8\rangle$ | $3872.81$ | $4221.26$ | $(-0.60,\hskip 4.26773pt0.43,\hskip 4.26773pt0.61,\hskip 4.26773pt-0.28)$
$c\bar{c}n\bar{n}$ | $1^{-}(1^{++})$ | $|3^{\prime}\rangle$ | $4099.70$ | $4052.82$ | $(-0.40,\hskip 4.26773pt-0.91)$
$|5^{\prime}\rangle$ | $4061.95$ | $4108.84$ | $(-0.91,\hskip 4.26773pt0.40)$
$c\bar{c}n\bar{n}$ | $1^{+}(1^{+-})$ | $|4^{\prime}\rangle$ | $4095.30$ | $3814.57$ | $(-0.43,\hskip 4.26773pt-0.53,\hskip 4.26773pt-0.31,\hskip 4.26773pt0.66)$
$|6^{\prime}\rangle$ | $4050.89$ | $4050.16$ | $(0.48,\hskip 4.26773pt0.54,\hskip 4.26773pt-0.14,\hskip 4.26773pt0.68)$
$|7\rangle$ | $4118.37$ | $4149.29$ | $(-0.28,\hskip 4.26773pt0.09,\hskip 4.26773pt0.90,\hskip 4.26773pt0.32)$
$|8\rangle$ | $3959.10$ | $4209.65$ | $(0.71,\hskip 4.26773pt-0.65,\hskip 4.26773pt0.26,\hskip 4.26773pt0.07)$
$c\bar{c}n\bar{n}$ | $1^{-}(2^{++})$ | $|7\rangle$ | $4164.70$ | $4052.98$ | $(-0.52,\hskip 4.26773pt-0.85)$
$|8\rangle$ | $4095.25$ | $4206.97$ | $(-0.85,\hskip 4.26773pt0.52)$
Table 10: As in table 7 but for the case of $Z_{cs}$ states. |  | Single configuration | Configurations mixing
---|---|---|---
Quark content | $I(J^{P})$ | Config. | Energies (MeV) | Energies (MeV) | Mixing coefficients
$c\bar{c}n\bar{s}$ | $\frac{1}{2}(0^{+})$ | $|1\rangle$ | $4198.04$ | $3951.72$ | $(0.17,\hskip 4.26773pt0.45,\hskip 4.26773pt-0.24,\hskip 4.26773pt0.84)$
$|2\rangle$ | $4158.90$ | $4129.62$ | $(-0.76,\hskip 4.26773pt-0.23,\hskip 4.26773pt0.45,\hskip 4.26773pt0.41)$
$|7\rangle$ | $4210.36$ | $4186.38$ | $(0.00,\hskip 4.26773pt-0.76,\hskip 4.26773pt-0.61,\hskip 4.26773pt0.23)$
$|8\rangle$ | $4019.17$ | $4318.75$ | $(-0.62,\hskip 4.26773pt0.41,\hskip 4.26773pt-0.61,\hskip 4.26773pt-0.26)$
$c\bar{c}n\bar{s}$ | $\frac{1}{2}(1^{+})$ | $|3\rangle$ | $4205.97$ | $3988.72$ | $(-0.31,\hskip 4.26773pt-0.31,\hskip 4.26773pt0.37,\hskip 4.26773pt-0.37,\hskip 4.26773pt0.29,\hskip 4.26773pt0.68)$
$|4\rangle$ | $4209.70$ | $4159.09$ | $(0.34,\hskip 4.26773pt0.34,\hskip 4.26773pt-0.28,\hskip 4.26773pt0.48,\hskip 4.26773pt0.10,\hskip 4.26773pt0.67)$
$|5\rangle$ | $4176.11$ | $4165.97$ | $(0.31,\hskip 4.26773pt-0.21,\hskip 4.26773pt-0.70,\hskip 4.26773pt-0.60,\hskip 4.26773pt0.03,\hskip 4.26773pt0.09)$
$|6\rangle$ | $4174.25$ | $4212.73$ | $(0.64,\hskip 4.26773pt-0.67,\hskip 4.26773pt0.28,\hskip 4.26773pt0.23,\hskip 4.26773pt-0.05,\hskip 4.26773pt-0.02)$
$|7\rangle$ | $4231.97$ | $4253.47$ | $(-0.20,\hskip 4.26773pt-0.16,\hskip 4.26773pt-0.07,\hskip 4.26773pt0.01,\hskip 4.26773pt-0.92,\hskip 4.26773pt0.28)$
$|8\rangle$ | $4089.37$ | $4307.39$ | $(-0.49,\hskip 4.26773pt-0.52,\hskip 4.26773pt-0.47,\hskip 4.26773pt0.47,\hskip 4.26773pt0.22,\hskip 4.26773pt-0.05)$
$c\bar{c}n\bar{s}$ | $\frac{1}{2}(2^{+})$ | $|7\rangle$ | $4272.18$ | $4167.21$ | $(0.52,\hskip 4.26773pt-0.85)$
$|8\rangle$ | $4205.90$ | $4310.87$ | $(-0.85,\hskip 4.26773pt-0.52)$

In addition, we also depict the mass spectra of $S$-wave $cn\bar{c}\bar{n}$
and $cn\bar{c}\bar{s}$ tetraquark states in Figs. 2 and 3 comparing with the
experimental measurements PDG ; LHCb 2021 ; BESIII 2021 , where the red solid
lines are results obtained in present work, while the green and yellow
rectangles represent the states with and without definite quantum numbers in
experiments, respectively, with the rectangle widths indicating their masses
uncertainties, and the gray dotted lines denote the corresponding $S$-wave
charm meson pair thresholds.

Figure 2: Mass spectra of ground $S$-wave $Z_{c}$ states. Red solid lines
denote the model results. Gray dotted lines are corresponding $S$-wave charm
meson pair thresholds. The green rectangle represents experimental mass of
$Z_{c}(3900)$ with definite quantum number, the rectangle width is its mass
uncertainty. The yellow rectangles represents the electric charmonium-like
states, observed in experiments near 4.1 GeV, whose spin-parity quantum
numbers have not been pinned down yet, the corresponding quantum numbers in
figure are suggested by model calculation, the rectangle width stand their
masses uncertainties, while the deep yellow solid lines in rectangles are
central values of their experimental masses. All the experimental values shown
in this figure are taken from PDG PDG . Figure 3: Mass spectra of ground
$S$-wave $Z_{cs}$ states. Red solid lines denote the model results. Gray
dotted lines are corresponding $S$-wave charm meson pair thresholds. The green
rectangles represent the experimental masses of $Z_{cs}(4000)$ and
$Z_{cs}(4220)$ with definite quantum numbers LHCb 2021 , the rectangle width
stand their masses uncertainties and the deep green solid lines in the
rectangles are central values of their masses. The yellow rectangles represent
the experimental mass of $Z_{cs}(3985)$, the rectangle width stand its mass
uncertainty. Spin-parity quantum numbers of $Z_{cs}(3985)$ in figure are
suggested by model calculation.

In 2013, BESIII collaboration reported the electric charmoniumlike state
$Z_{c}(3900)^{\pm}$ in the $\pi^{\pm}J/\psi$ invariant mass spectrum of the
process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ BESIII 2013 . And the same
structure was also observed in the same process by Belle collaboration Belle
2013 . Later on, $Z_{c}(3885)^{\pm}$ was discovered in the
$(D\bar{D}^{*})^{\pm}$ invariant mass spectrum of the process
$e^{+}e^{-}\rightarrow\pi^{\mp}(D\bar{D}^{*})^{\pm}$ by BESIII collaboration,
which should have a very similar mass with the $Z_{c}(3900)^{\pm}$ state
BESIII 2014 . Since they have the same spin–parity $J^{P}=1^{+}$, similar mass
and width, $Z_{c}(3900)$ and $Z_{c}(3885)$ are probably the same state, and
mainly couple dominantly to the $D\bar{D}^{*}$ channel BESIII 2014 . Thus we
will not distinguish the $Z_{c}(3900)$ and $Z_{c}(3885)$ in the following
discussions.

The average mass and width of $Z_{c}(3900)$ from PDG PDG  are:
$M_{Z_{c}(3900)}=3887.1\pm 2.6$ and $\Gamma_{Z_{c}(3900)}=28.4\pm
2.6\,\text{MeV}$. In present model, the lowest mass of ground $S$-wave
$cn\bar{c}\bar{n}$ states with quantum numbers $I^{G}(J^{PC})=1^{+}(1^{+-})$,
considering configuration mixing, is about $3815\,\text{MeV}$. This value is
lower than, but not far from, the experimental mass of $Z_{c}(3900)$. This may
indicate that the compact $c\bar{c}n\bar{n}$ tetraquark components may take
notable probability in the $Z_{c}(3900)$ state.

The $Z_{c}(4025)$ state was observed in $D\bar{D}^{*}$ final state in Ref.
BESIII 2014 Zc4025  and, meanwhile, a similar structure $Z_{c}(4020)$ was
observed in $\pi h_{c}$ final state by BESIII collaboration BESIII 2013 Zc4020
. At present, the $Z_{c}(4020)$ and $Z_{c}(4025)$ states are denoted as an
identical state $X(4020)$ in PDG PDG . Its average mass and width are
$M_{Z_{c}(4020)}=4024.1\pm 1.9,\,\Gamma_{Z_{c}(4020)}=13\pm 5\,\text{MeV}$,
and its quantum number are probably $J^{PC}=1^{+-}$ HeJ 2013 . In present
model, the next-lowest mass of ground $S$-wave $cn\bar{c}\bar{n}$ state with
quantum number $I^{G}(J^{PC})=1^{+}(1^{+-})$, considering configuration
mixing, is about $4050\,\text{MeV}$, which is very close to the mass of
$X(4020)$. Therefore, one may expect that the $c\bar{c}n\bar{n}$ tetraquark
configuration obtained here could be a component in the $X(4020)$ state.

Table 11: The comparison of model results with experimental masses for $Z_{c}$ states. Fourth, fifth and sixth columns collect the experimental data of $Z_{c}$ states near $4.1\,\text{GeV}$, and first two columns collect the model results in which model masses are close to the experimental masses. Present results | Experimental data
---|---
$I^{G}(J^{PC})$ | M (MeV) | States | $I^{G}(J^{PC})$ | M (MeV)
$1^{+}(1^{+-})$ | $3814.57$ | $Z_{c}(3900)$ | $1^{+}(1^{+-})$ | $3887.1\pm 2.6$
$1^{+}(1^{+-})$ | $4050.16$ | $X(4020)$ | $1^{+}(?^{?-})$ | $4024.1\pm 1.9$
$1^{-}(1^{++})$ | $4052.82$ | $X(4050)$ | $1^{-}(?^{?+})$ | $4051\pm 14^{+20}_{-41}$
$1^{-}(2^{++})$ | $4052.98$
$1^{+}(1^{+-})$ | $4050.16$ | $X(4055)$ | $1^{+}(?^{?-})$ | $4054\pm 3\pm 1$
$1^{-}(0^{++})$ | $4076.61$ | $X(4100)$ | $1^{-}(?^{??})$ | $4096\pm 20^{+18}_{-22}$
$1^{-}(0^{++})$ | $4221.26$ | $X(4250)$ | $1^{-}(?^{?+})$ | $4248^{+44+180}_{-29-35}$
$1^{-}(2^{++})$ | $4206.97$

Besides, there are several other electric charmonium-like particles observed
experimentally, which need to be further confirmed. Here we also compare the
present numerical results with these experimental data, as shown in the Table
11, the last three columns collect the experimental data of $Z_{c}$ states
near $4.1\,\text{GeV}$, and the first two columns are the presently obtained
numerical results those are close to the experimental data.

The $X(4050)$ and $X(4250)$ states were reported by Belle collaboration in the
$\pi^{+}\chi_{c1}$ invariant mass distribution of the process
$\bar{B}^{0}\rightarrow K^{-}\pi^{+}\chi_{c1}$ Belle 2008 . In present model,
we get two states with masses very close two $X(4050)$, whose quantum numbers
are $I^{G}(J^{PC})=1^{-}(1^{++})$ and $I^{G}(J^{PC})=1^{-}(2^{++})$,
respectively. Meanwhile, masses of two obtained states with the quantum
numbers $I^{G}(J^{PC})=1^{-}(0^{++})$ and $I^{G}(J^{PC})=1^{-}(2^{++})$,
respectively, are a little bit lower than $X(4250)$. Finally, two obtained
states, whose quantum numbers are $I^{G}(J^{PC})=1^{+}(1^{+-})$ and
$1^{-}(0^{++})$, respectively, are very close to the $X(4055)^{\pm}$ observed
in $\pi^{\pm}\psi$ final state through process
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(2S)$ by Belle Belle 2015 , and
$X(4100)$ observed in the $\eta_{c}\pi^{-}$ invariant mass spectrum in
$B^{0}\rightarrow\eta_{c}K^{+}\pi^{-}$ decay by LHCb collaboration LHCb 2018 .
We look forward to further examinations on these results by future
experiments.

Now we turn to the $Z_{cs}$ states. The $Z_{cs}(3985)$ was the first observed
charmonium-like state containing a strange quark, it was discovered in the
$D_{s}^{-}D^{*0}(D_{s}^{*-}D^{0})$ channel by BESIII very recently BESIII 2021
. The mass and width of $Z_{cs}(3985)$ are
$M_{Z_{cs}(3985)}=3982.5^{+1.8}_{-2.6}\pm{2.1}$ and
$\Gamma_{Z_{cs}(3985)}=12.8^{+5.3}_{-4.4}\pm 3.0\,\text{MeV}$. Its mass is
very close to the $D_{s}^{-}D^{*0}(D_{s}^{*-}D^{0})$ mass threshold. Recently,
an theoretical analysis shows that the $Z_{cs}(3985)$ cannot be a pure
$D_{s}^{-}D^{*0}(D_{s}^{*-}D^{0})$ molecule ChenR 2021 . And in Ref. Guo 2021
, it has been shown that the two charm-meson molecular components make up
about 40 percent of the physical $Z_{cs}(3985)$ state.

In addition, the $Z_{cs}(3985)$ state has also been studied within constituent
quark models JinX 2021 ; YangG 2021 . For example, in Ref. JinX 2021 , the
diquark-antidiquark $[cs][\bar{c}\bar{n}]$ configurations have been
investigated, and it’s found that there would be resonance states falling in
the mass range of $3916.5\sim 3964.6$ MeV with quantum numbers $J^{P}=0^{+}$,
and $4008.8\sim 4091.2$ MeV with $J^{P}=1^{+}$. Thus, one may conclude that a
compact tetraquark state with $J^{P}=0^{+}$ or $J^{P}=1^{+}$ may take
considerable probability in $Z_{cs}(3985)$. And in a very recent paper
Ikeno:2021mcb , it’s shown that the $Z_{cs}$ state may correspond to a cusp
structure.

In present model, the lowest mass of ground $S$-wave $cn\bar{c}\bar{s}$ states
with quantum number $I(J^{P})=1/2(0^{+})$ and $I(J^{P})=1/2(1^{+})$ are about
$3952\,\text{MeV}$ and $3989\,\text{MeV}$, respectively. Both of them are
close to the experimental mass of $Z_{cs}(3985)$. Therefore, based on the
above discussions, our results also support that the $Z_{cs}(3985)$ may be not
a pure hadronic molecule, and the compact tetraquark component should take
notable probability in it.

Besides, LHCb collaboration performed an analysis of the $B^{+}\rightarrow
J/\psi\phi K^{+}$ dacay and reported two $Z_{cs}$ particles, $Z_{cs}(4000)$
and $Z_{cs}(4220)$ in the invariant $J/\psi K^{+}$ mass distributions LHCb
2021 . The spin-parity of $Z_{cs}(4000)$ is $J^{P}=1^{+}$ and $Z_{cs}(4220)$
is $J^{P}=1^{+}$ or $J^{P}=1^{-}$. Their fitting masses and widths are

$\displaystyle M_{Z_{cs}(4000)}=4003\pm 6^{+4}_{-14}\,\text{MeV}\,,$
$\displaystyle\Gamma_{Z_{cs}(4000)}=131\pm 15\pm 26\,\text{MeV}\,,$
$\displaystyle M_{Z_{cs}(4220)}=4216\pm 24^{+43}_{-30}\,\text{MeV}\,,$
$\displaystyle\Gamma_{Z_{cs}(4220)}=233\pm 52^{+97}_{-73}\,\text{MeV}\,,$

respectively. Although the mass of $Z_{cs}(4000)$ is very close to that of
$Z_{cs}(3985)$, there is no evidence that $Z_{cs}(4000)$ and $Z_{cs}(3985)$
are the same state LHCb 2021 . In present model, the lowest ground $S$-wave
$cn\bar{c}\bar{s}$ state with quantum numbers $I(J^{P})=1/2(1^{+})$ falls at
$\sim 3989\,\text{MeV}$, which thus could be interpreted as component of
$Z_{cs}(4000)$. Yet, in the mass range $4.16\sim 4.26\,\text{GeV}$ for
$I(J^{P})=1/2(1^{+})$, there are three states whose masses are consistent with
the experimental mass of $Z_{cs}(4220)$.

Accordingly, the experimental data of $Z_{cs}$ states, and presently obtained
results those are close to the experimental data, are collected in Table 12.

Table 12: Comparison of model results with experimental masses for $Z_{cs}$ states. Fourth, fifth and sixth columns collect the experimental data of $Z_{cs}$ states, and first two columns collect the model results in which model masses are close to the experimental masses. Model values | Experiment values
---|---
$I(J^{P})$ | M (MeV) | States | $I(J^{P})$ | M (MeV)
$1/2(0^{+})$ | $3951.72$ | $Z_{cs}(3985)$ | $1/2(?^{?})$ | $3982.5^{+1.8}_{-2.6}\pm 2.1$
$1/2(1^{+})$ | $3988.72$
$1/2(1^{+})$ | $3988.72$ | $Z_{cs}(4000)$ | $1/2(1^{+})$ | $4003\pm 6^{+4}_{-14}$
$1/2(1^{+})$ | $4165.97$ | $Z_{cs}(4220)$ | $1/2(1^{?})$ | $4216\pm 24^{+43}_{-30}$
$1/2(1^{+})$ | $4212.73$
$1/2(1^{+})$ | $4253.47$

###  III.3 Double-charm $T_{cc}$ states

Very recently, the first double-charm exotic state $T_{cc}^{+}$ with
$I(J^{P})=0(1^{+})$, whose minimal quark content is $cc\bar{u}\bar{d}$, was
observed by LHCb LHCbTcc1 . Its mass is very close to and below $D^{0}D^{*+}$
threshold, and its decay width is extremely narrow. A further analysis have
been done by LHCb collaboration in Ref. LHCbTcc2 . Theoretically, the
tetraquark states containing two heavy quarks have been predicted in Refs. Zhu
2013 ; Kar.M 2017 .

In present work, the mass spectra of ground $S$-wave $cc\bar{n}\bar{n}$,
$cc\bar{n}\bar{s}$ and $cc\bar{s}\bar{s}$ tetraquark systems are studied. The
numerical results are listed in Table 13, where the first three columns are
the labels of these states in the model, quark content and quantum number of
each state respectively, the fourth, fifth columns show the results of single
configuration calculations, and the last two columns give the results obtained
by considering configurations mixing effects. The model spectrum of $S$-wave
double-charm tetraquark states are also shown in Fig. 4, where red solid lines
denote $T_{cc}^{0,0}$ states, blue solid lines denote $T_{cc}^{1,0}$ states,
pink solid lines denote $T_{cc}^{1/2,1}$ states, green solid line denote
$T_{cc}^{0,2}$ states.

Table 13: The model results of ground $S$-wave double-charm tetraquark states. First three columns are the labels of these states in the model, quark content and quantum number of each state repectively, the fourth, fifth columns give the results of single configuration calculations, and the sixth, seventh columns give the results after considering configuration mixing. |  |  | Single configuration | Configurations mixing
---|---|---|---|---
States | Quark content | $(I,S)(J^{P})$ | Config. | M (MeV) | M (MeV) | Mixing coefficients
$T_{cc}^{0,0}$ | $cc\bar{n}\bar{n}$ | $(0,0)(1^{+})$ | $|3\rangle$ | $3998.90$ | $3982.12$ | $(-0.95,\hskip 4.26773pt-0.30)$
$|6\rangle$ | $4152.40$ | $4169.18$ | $(-0.30,\hskip 4.26773pt0.95)$
$T_{cc}^{1,0}$ | $cc\bar{n}\bar{n}$ | $(1,0)(0^{+})$ | $|1\rangle$ | $4204.41$ | $4032.30$ | $(0.42,\hskip 4.26773pt-0.91)$
$|7\rangle$ | $4069.69$ | $4241.80$ | $(-0.91,\hskip 4.26773pt-0.42)$
$T_{cc}^{1,0}$ | $cc\bar{n}\bar{n}$ | $(1,0)(1^{+})$ | $|7\rangle$ | $4092.34$ | — | —
$T_{cc}^{1,0}$ | $cc\bar{n}\bar{n}$ | $(1,0)(2^{+})$ | $|7\rangle$ | $4134.59$ | — | —
$T_{cc}^{\frac{1}{2},1}$ | $cc\bar{n}\bar{s}$ | $(\frac{1}{2},1)(0^{+})$ | $|1\rangle$ | $4296.64$ | $4132.95$ | $(-0.45,\hskip 4.26773pt-0.89)$
$|7\rangle$ | $4174.01$ | $4337.71$ | $(-0.89,\hskip 4.26773pt0.45)$
$T_{cc}^{\frac{1}{2},1}$ | $cc\bar{n}\bar{s}$ | $(\frac{1}{2},1)(1^{+})$ | $|3\rangle$ | $4134.14$ | $4115.10$ | $(-0.94,\hskip 4.26773pt-0.34,\hskip 4.26773pt0.00)$
$|6\rangle$ | $4257.37$ | $4197.18$ | $(0.00,\hskip 4.26773pt0.00,\hskip 4.26773pt1.00)$
$|7\rangle$ | $4197.18$ | $4276.41$ | $(0.34,\hskip 4.26773pt-0.94,\hskip 4.26773pt0.00)$
$T_{cc}^{\frac{1}{2},1}$ | $cc\bar{n}\bar{s}$ | $(\frac{1}{2},1)(2^{+})$ | $|7\rangle$ | $4240.55$ | — | —
$T_{cc}^{0,2}$ | $cc\bar{s}\bar{s}$ | $(0,2)(0^{+})$ | $|1\rangle$ | $4387.45$ | $4234.84$ | $(0.48,\hskip 4.26773pt-0.88)$
$|7\rangle$ | $4280.69$ | $4433.30$ | $(-0.88,\hskip 4.26773pt-0.48)$
$T_{cc}^{0,2}$ | $cc\bar{s}\bar{s}$ | $(0,2)(1^{+})$ | $|7\rangle$ | $4304.25$ | — | —
$T_{cc}^{0,2}$ | $cc\bar{s}\bar{s}$ | $(0,2)(2^{+})$ | $|7\rangle$ | $4348.45$ | — | —
Figure 4: The model spectrum of ground $S$-wave double-charm tetraquark
states. Red solid lines denote $T_{cc}^{0,0}$ states, blue solid lines denote
$T_{cc}^{1,0}$ states, pink solid lines denote $T_{cc}^{1/2,1}$ states, green
solid line denote $T_{cc}^{0,2}$ states.

As shown in Table 13, all the presently obtained lowest compact $T_{cc}$
states are higher than thresholds of corresponding charmed meson-meson
channels. Therefore, there may be no stable bound states for $S$-wave double-
charm tetraquark state. For instance, the lowest $S$-wave $cc\bar{n}\bar{n}$
states with quantum number $I(J^{P})=0(1^{+})$ in present model falls at $\sim
3982\,\text{MeV}$, which is about $100\,\text{MeV}$ higher than the
experimental data for $T_{cc}^{+}$. Thus, $T_{cc}^{+}$ cannot be interpreted
as a compact tetraquark state in our model.

In Ref. D.Ebert 2007 , the spectrum of tetraquark states containing double
heavy quarks were investigated within a diquark-antidiquark picture employing
the relativistic constituent quark model, the obtained numerical results are
in agreements with the present ones. Very recently, double-heavy tetraquark
states have been also studied using a Chiral quark model in Ref. Deng 2021 ,
it’s shown that both the meson exchange force and the coupled channel effects
play pivotal roles for reproduction of the binding energy of $T_{cc}^{+}$
correlative to $D^{0}D^{*+}$ threshold, revealing the molecular nature of
$T_{cc}^{+}$ Deng 2021 .

##  IV Summary

In present work, we study the ground $S$-wave hidden- and double-charm
tetraquark states employing a nonrelativistic quark potential model, where the
instanton-induced interaction is employed as the spin-dependent residual
interaction between quarks. The model parameters are fixed by fitting the
spectrum of the ground state hadrons. Our numerical results show that energies
for several presently obtained compact tetraquark states are very close to the
masses of the experimentally observed meson exotic states.

Particularly, two $c\bar{c}n\bar{n}$ and $c\bar{c}s\bar{s}$ tetraquark states
with quantum numbers $I^{G}(J^{PC})=0^{+}(1^{++})$ fall at the energies close
to $X(4140)$ and $X(4274)$, respectively. Energies of several
$c\bar{c}n\bar{n}$ tetraquark states with $I^{G}(J^{PC})=1^{+}(1^{+-})$ are
close to $Z_{c}(3900)$, $Z_{c}(4020)$ and $X(4055)$, and another obtained
$c\bar{c}n\bar{n}$ sate with $I^{G}(J^{PC})=1^{-}(0^{++})$ is close to the
$X(4100)$ state. In addition, the $X(4050)$ and $X(4250)$ states, whose
quantum numbers have not been determined, may be interpreted as compact
$c\bar{c}n\bar{n}$ tetraquark states with quantum numbers
$I^{G}(J^{PC})=1^{-}(1^{++})$ or $I^{G}(J^{PC})=1^{-}(2^{++})$, and
$I^{G}(J^{PC})=1^{-}(0^{++})$ or $I^{G}(J^{PC})=1^{-}(2^{++})$, respectively.

For the exotic meson states with one strange quark, we obtain two states very
close to the experimentally observed $Z_{cs}(3985)$ state, which indicates the
quantum number of $Z_{cs}(3985)$ may be $J^{P}=0^{+}$ or $J^{P}=1^{+}$.
Furthermore, several $c\bar{c}n\bar{s}$ tetraquark states with $J^{P}=1^{+}$
are obtained to fall in the energy range close to the $Z_{cs}(4000)$ and
$Z_{cs}(4220)$ states. Accordingly, one may conclude that the presently
considered compact tetraquark states my take significant probabilities in the
above referred $X$, $Z_{c}$ and $Z_{cs}$ states.

###### Acknowledgements.

We would like to thank Prof. X. H. Zhong for very helpful discussions. This
work is partly supported by the Chongqing Natural Science Foundation under
Project No. cstc2021jcyj-msxmX0078, No. cstc2019jcyj-msxmX0409, and the
National Natural Science Foundation of China under Grant Nos. 12075288,
12075133, 11735003, 11961141012 and 11835015. It is also supported by the
Youth Innovation Promotion Association CAS, Taishan Scholar Project of
Shandong Province (Grant No.tsqn202103062), the Higher Educational Youth
Innovation Science and Technology Program Shandong Province (Grant No.
2020KJJ004).

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