# Hidden and double charm-strange tetraquarks and their decays in a potential
quark model

Feng-Xiao Liu1,4 111E-mail<EMAIL_ADDRESS>Ru-Hui Ni2,4, Xian-Hui Zhong2,4
222E-mail<EMAIL_ADDRESS>Qiang Zhao1,3 333E-mail<EMAIL_ADDRESS>1)
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  2) Department of Physics, Hunan Normal University, and Key Laboratory
of Low-Dimensional Quantum Structures and Quantum Control of Ministry of
Education, Changsha 410081, China  3) University of Chinese Academy of
Sciences, Beijing 100049, China  4) Synergetic Innovation Center for Quantum
Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081,
China

###### Abstract

We carry out a systematic study of the $1S$-wave hidden and double charm-
strange tetraquarks $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ in a
nonrelativistic potential quark model framework with the explicitly correlated
Gaussian method, and the mass spectra, color-spin configurations and possible
decay modes are obtained. We find that although these states are all above
their open flavor thresholds, their rearrangement decay widths are rather
narrow which can be understood by the mismatching of the wave functions
between the initial and final states. It implies that the tetraquarks of
$cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ may have a good chance to exist as
genuine tetraquark states. It also shows that the color-spin configurations of
the $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ systems are quite different. We
find that for a physical state of $cs\bar{c}\bar{s}$ its color configurations
can be dominated by either the $|11\rangle_{c}$ or $|88\rangle_{c}$ ones. It
suggests that some hidden charm-strange tetraquark states may strongly couple
to two color-singlet hadrons if the kinematics and dynamics allow. In
contrast, we find that the color configurations $|11\rangle_{c}$ and
$|88\rangle_{c}$ in a double charm-strange $cc\bar{s}\bar{s}$ state are rather
compatible. It may suggest that an overall color-singlet tetraquark (i.e. a
genuine color-singlet) should always play a role in the $T_{cc\bar{s}\bar{s}}$
states. Discussions taking into account some experimental candidates are
presented, and suggestions on further experimental searches are also made.

##  I Introduction

Since the $X(3872)$ was first observed at Belle in 2003 Choi  _et al._ (2003),
many candidates of exotic hadrons have been reported by experiments such as
_BABAR_ , Belle, LHCb, and BESIII, etc. (e.g. see recent reviews Chen  _et
al._ (2016); Esposito  _et al._ (2017); Karliner  _et al._ (2018); Guo  _et
al._ (2018); Brambilla  _et al._ (2020)). The masses of these exotic states
are generally located near particular thresholds of two conventional open-
heavy-flavor mesons. Some of these states found in the $J/\psi\phi$ or
$D_{s}^{+}D_{s}^{-}$ invariant mass spectrum may contain both a $c\bar{c}$
pair and an $s\bar{s}$ pair, which implies that these structures are exotic
candidates consisting of $cs\bar{c}\bar{s}$ constituent, such as multi-quark
state, hadronic molecule, or hybrid.

Recently, the LHCb Collaboration reported evidence of a near-threshold peaking
structure referred to as $X\left(3960\right)$ and another structure
$X_{0}\left(4140\right)$ observed in the $D_{s}^{+}D_{s}^{-}$ invariant mass
spectrum in the decay $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ Aaij  _et al._
(2023). Before the observation of $X\left(3960\right)$, a plethora of exotic
structures, that are good candidates of $cs\bar{c}\bar{s}$ tetraquarks, have
been found, such as $X\left(4140\right)$ Aaltonen  _et al._ (2009); Chatrchyan
_et al._ (2014); Abazov  _et al._ (2014); Aaij  _et al._ (2017),
$X\left(4274\right)$ Aaltonen  _et al._ (2017); Aaij  _et al._ (2017),
$X\left(4350\right)$ Shen  _et al._ (2010), $X\left(4500\right)$ Aaij  _et
al._ (2017), $X\left(4630\right)$ Aaij  _et al._ (2021), $X\left(4685\right)$
Aaij  _et al._ (2021), and $X\left(4700\right)$ Aaij  _et al._ (2017).

These signals are peculiarly interesting mainly because their probable
constituent quark contents can be $cs\bar{c}\bar{s}$ in the hidden charm
sector where the relatively heavier mass of the strange quark may provide
quite different information concerning the dynamics describing the $D_{s}^{+}$
and $D_{s}^{-}$ interactions at the hadronic level or describing the
constituent quark interactions at the quark level if we compare such a system
with exotic candidates, such as $X(3872)$.

On the theory side, the mass spectrum of the $cs\bar{c}\bar{s}$ system has
been studied by various approaches, such as the potential quark model Braaten
_et al._ (2014); Vijande  _et al._ (2007a, b); Fernandez-Carames  _et al._
(2009); Yang  _et al._ (2021); Tiwari and Rai (2023), QCD sum rule Chen  _et
al._ (2017a); Xin  _et al._ (2022), lattice QCD (LQCD) 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 (2023), and the chromomagnetic interaction (CMI) model Wu  _et
al._ (2019).

About the nature of the charmonium-like states, there are many explanations in
the literature. To know about the experimental and theoretical status of these
charmonium-like states, one can refer to the recent reviews Dong  _et al._
(2021b); Ali  _et al._ (2017); Esposito  _et al._ (2017); Olsen  _et al._
(2018); Lebed  _et al._ (2017); Liu  _et al._ (2019a); Chen  _et al._ (2016);
Brambilla  _et al._ (2020); Chen  _et al._ (2023a); Guo  _et al._ (2018).

Recently, the LHCb Collaboration reported the first observation of a double-
charm tetraquark $T_{cc}^{+}(3875)$ ($cc\bar{u}\bar{d}$) in the
$D^{0}D^{0}\pi^{+}$ invariant mass distribution Aaij  _et al._ (2022a, b). Its
quantum numbers favor $IJ^{P}=01^{+}$. The double-charm tetraquark systems
were studied in the literature before the observation of $T_{cc}^{+}(3875)$
and the theoretical predictions are quite different Zouzou  _et al._ (1986);
Lipkin (1986); Heller and Tjon (1987); Carlson  _et al._ (1988); Silvestre-
Brac and Semay (1993); Pepin  _et al._ (1997); Brink and Stancu (1998); Janc
and Rosina (2004); Zhang  _et al._ (2008); Barnea  _et al._ (2006); Vijande
_et al._ (2007a); Navarra  _et al._ (2007); Manohar and Wise (1993); Carames
_et al._ (2011); Ohkoda  _et al._ (2012). With the observation of
$T_{cc}^{+}(3875)$, a lot of interpretations and studies have been done based
on different scenarios, such as compact states Kim  _et al._ (2022); Chen  _et
al._ (2023b), hadronic molecules Feijoo  _et al._ (2021); Dong  _et al._
(2021b); Dai  _et al._ (2022); Albaladejo (2022); Du  _et al._ (2022); Ke  _et
al._ (2022); Abreu (2022); Xin and Wang (2022); Peng  _et al._ (2023),
triangle singularity (TS) mechanism Braaten  _et al._ (2022), etc. It was also
investigated by LQCD calculations Padmanath and Prelovsek (2022); Chen  _et
al._ (2022); Lyu  _et al._ (2023); Du  _et al._ (2023), and in its production
and decays Hu  _et al._ (2021); Ling  _et al._ (2022); Meng  _et al._ (2021);
Yan and Valderrama (2022). In Refs. Qiu  _et al._ (2024) combined analyses of
the $X(3872)$, $Z_{c}(3900)$ and $T_{cc}^{+}(3875)$ are presented, which
reveal peculiar dynamics arising from the $D^{(*)}\bar{D}^{(*)}$ and
$D^{(*)}D^{(*)}$ threshold interactions.

In this work, we investigate the mass spectra, possible decay modes, and inner
structures of tetraquark states $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ in
the nonrelativistic potential quark model (NRPQM). Our attention is paid to
cases where strong couplings to the nearby $S$-wave decay channels can be
identified. Taking into account the experimental fact that some of those
exotic candidates are located in the vicinity of an $S$-wave threshold and the
possibility of multi-quark states in QCD cannot be eliminated, we are
interested in any clues for connecting the quark model multi-quark
configurations and the open channel dynamics with each other. Such information
should be helpful for unifying the quark model prescription of the multi-quark
spectra with the threshold dynamics based on hadron-hadron interactions. As
the first step forward, we will focus on the $1S$-wave spectra of the
$cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ tetraquarks. We mention in advance
that because of the configuration mixings the $1S$-wave spectra have become
rich enough for raising a lot of interesting questions.

As follows, we first give an introduction of the framework of the NRPQM in
Sec. II. Then, the mass spectra and strong decay properties of the $1S$-wave
$cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ states will be presented in Sec. III
with discussions. A summary will be given in Sec. IV.

##  II Model and method

###  II.1 Mass spectrum

####  II.1.1 Hamiltonian

The mass spectra of the tetraquarks are calculated within the NRPQM, which has
been widely adopted to deal with the mesons and baryon spectra. In this model,
the Hamiltonian is given by

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left(\underset{i=1}{\overset{4}{\sum}}m_{i}+T_{i}\right)-T_{cm}+\underset{i<j}{\sum}V_{ij}\left(r_{ij}\right),$
(1)

where $m_{i}$ and $T_{i}$ stand for the constituent quark mass and kinetic
energy of the $i$-th quark, respectively; $T_{cm}$ stands for the center-of-
mass (c.m.) kinetic energy of the tetraquark system;
$\boldsymbol{r}_{ij}\equiv|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|$ is the
distance between the $i$-th and $j$-th quarks; and $V_{ij}\left(r_{ij}\right)$
stands for the effective potential between them. In this work the
$V_{ij}\left(r_{ij}\right)$ adopts a widely used form Eichten  _et al._
(1978); Godfrey and Isgur (1985); Barnes  _et al._ (2005); Godfrey and Moats
(2015); Godfrey (2004); Lakhina and Swanson (2007); Lü  _et al._ (2016); Li
_et al._ (2011); Deng  _et al._ (2017a, b); Liu  _et al._ (2021a, 2019b, b):

$\displaystyle V_{ij}\left(r_{ij}\right)$ $\displaystyle=$
$\displaystyle-\frac{3}{16}\left(\boldsymbol{\lambda}_{i}\cdot\boldsymbol{\lambda}_{j}\right)\left(b_{ij}r_{ij}-\frac{4}{3}\frac{\alpha_{ij}}{r_{ij}}+c_{ij}\right)$
$\displaystyle+$
$\displaystyle-\frac{\alpha_{ij}}{4}\left(\boldsymbol{\lambda}_{i}\cdot\boldsymbol{\lambda}_{j}\right)\left\\{\frac{\pi}{2}\frac{\sigma_{ij}^{3}e^{-\sigma_{ij}^{2}r_{ij}^{2}}}{\pi^{3/2}}\frac{16}{3m_{i}m_{j}}\left(\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j}\right)\right\\},$

where $\boldsymbol{\lambda}_{i,j}$ are the color operators acting on the
$i,j$-th quarks, $\boldsymbol{S}_{i,j}$ represent the spin operators of the
$i,j$-th quarks. The parameters $b_{ij}$ and $c_{ij}$ denote the confinement
potential strength and the zero point energy, respectively. And $\alpha_{ij}$
denote the the strong coupling for the OGE potential. It should be mentioned
that the tensor and spin-orbit potential do not contribute to the $1S$-wave
tetraquarks considered here.

Parameters adopted in this work are collected in the Table 1, which are
extracted by fitting the mass spectra of the mesons Zyla  _et al._ (2020) as
also shown in Table 2. The same quark model parameters are adopted as in Refs.
Deng  _et al._ (2017a); Li  _et al._ (2021, 2020).

Table 1: Quark model parameters used in this work. $m_{u/d}$ (GeV) | 0. | 350 |  |
---|---|---|---|---
$m_{s}$ (GeV) | 0. | 600 |  |
$m_{c}$ (GeV) | 1. | 483 |  |
$m_{b}$ (GeV) | 4. | 852 |  |
|  $ss$ |  $cc$ |  $cs$
$b_{qq}$ $(\mathrm{GeV}^{2})$ | 0. | 1350 | 0. | 1425 | 0. | 1230
$c_{qq}$ (GeV) | -0. | 5190 | 0. | 0 | -0. | 1745
$\sigma_{qq}$ (GeV) | 0. | 6000 | 1. | 1384 | 0. | 8000
$\alpha_{qq}$ | 0. | 7700 | 0. | 5461 | 0. | 6800
Table 2: The theoretical mass spectra of $s\bar{s}$, $c\bar{c}$, and $c\bar{s}$ compared with the observations (labeled with Obs.) from the PDG Workman  _et al._ (2022). The unit of mass is MeV.  |  $s\bar{s}$ |  $c\bar{c}$ |  $c\bar{s}$
---|---|---|---
$n^{2S+1}L_{J}$ | Obs. | Mass | Obs. | Mass | Obs. | Mass
$1^{1}S_{0}$ |  | 797 |  $\eta_{c}$(2984) | 2984 | $D_{s}(1968)$ | 1969
$1^{3}S_{1}$ | $\phi\left(1020\right)$ | 1017 |  $J/\psi$(3097) | 3097 | $D_{s}^{*}(2112)$ | 2112
$2^{1}S_{0}$ | $\eta\left(1475\right)$ | 1619 |  $\eta_{c}\left(2S\right)$(3638) | 3635 | $\cdot\cdot\cdot$ | 2655
$2^{3}S_{1}$ | $\phi\left(1680\right)$ | 1699 | $\psi\left(3686\right)$ | 3679 | $D_{s1}^{*}\left(2700\right)$ | 2721
$1^{3}P_{0}$ | $f_{0}\left(1370\right)$ | 1373 | $\chi_{c0}(3415)$ | 3417 | $\cdot\cdot\cdot$ | 2419
$1^{1}P_{1}\left(1P_{1}\right)$ | $h_{1}\left(1415\right)$ | 1462 | $h_{c}(3525)$ | 3522 | $\cdot\cdot\cdot$ | 2518
$1^{3}P_{1}\left(1P^{\prime}_{1}\right)$ | $f_{1}\left(1420\right)$ | 1492 | $\chi_{c1}(3511)$ | 3516 | $D_{s1}\left(2536\right)$ | 2534
$1^{3}P_{2}$ | $f_{2}^{\prime}\left(1525\right)$ | 1513 | $\chi_{c2}(3556)$ | 3552 | $D_{s2}^{*}\left(2573\right)$ | 2550
$1^{1}D_{2}\left(1D_{2}\right)$ | $\cdot\cdot\cdot$ | 1825 | $\cdot\cdot\cdot$ | 3806 | $\cdot\cdot\cdot$ | 2812
$1^{3}D_{1}$ | $\cdot\cdot\cdot$ | 1808 | $\psi\left(3770\right)$ | 3787 | $D_{s1}^{*}\left(2860\right)$ | 2836
$1^{3}D_{2}\left(1D^{\prime}_{2}\right)$ | $\cdot\cdot\cdot$ | 1840 | $\psi_{2}\left(3823\right)$ | 3807 | $\cdot\cdot\cdot$ | 2865
$1^{3}D_{3}$ | $\cdot\cdot\cdot$ | 1822 | $\psi_{3}\left(3842\right)$ | 3812 | $D_{s3}^{*}\left(2860\right)$ | 2822

####  II.1.2 Tetraquark configurations

The wave function for a tetraquark system can be constructed as a product of
the flavor, color, spin, and spatial configurations.

In the color space, there are two color-singlet bases
$|6_{12}\bar{6}_{34}\rangle_{c}$ and $|\bar{3}_{12}3_{34}\rangle_{c}$, their
wave functions are given by

$\displaystyle\left|6_{12}\bar{6}_{34}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{6}}\left[\left(rb+br\right)\left(\bar{r}\bar{b}+\bar{b}\bar{r}\right)+\left(bg+gb\right)\left(\bar{b}\bar{g}+\bar{g}\bar{b}\right)\right.$
(3)
$\displaystyle+\left(gr+rg\right)\left(\bar{g}\bar{r}+\bar{r}\bar{g}\right)$
$\displaystyle\left.+2\left(rr\right)\left(\bar{r}\bar{r}\right)+2\left(bb\right)\left(\bar{b}\bar{b}\right)+2\left(gg\right)\left(\bar{g}\bar{g}\right)\right],$
$\displaystyle\left|\bar{3}_{12}3_{34}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}\left[\left(rb-
br\right)\left(\bar{r}\bar{b}-\bar{b}\bar{r}\right)+\left(bg-
gb\right)\left(\bar{b}\bar{g}-\bar{g}\bar{b}\right)\right.$ (4)
$\displaystyle\left.+\left(gr-
gr\right)\left(\bar{g}\bar{r}-\bar{r}\bar{g}\right)\right]\ ,$

where $r\ (\bar{r})$, $b\ (\bar{b})$ and $g\ (\bar{g})$ denote the color wave
function of a single quark (antiquark).

In the spin space, there are six spin bases, which are denoted by
$\chi_{SS_{z}}^{S_{12}S_{34}}$. Where $S_{12}$ stands for the spin quantum
number for the diquark ($q_{1}q_{2}$) or antidiquark
($\bar{q}_{1}\bar{q}_{2}$), while $S_{34}$ stands for the spin quantum number
for the antidiquark ($\bar{q}_{3}\bar{q}_{4}$) or diquark ($q_{3}q_{4}$). $S$
is the total spin quantum number of the tetraquark $qq\bar{q}\bar{q}$ system,
while $S_{z}$ stands for the third component of the total spin
$\boldsymbol{S}$. The spin wave functions $\chi_{SS_{z}}^{S_{12}S_{34}}$ with
a determined $S_{z}$ can be explicitly expressed as follows:

$\displaystyle\chi_{00}^{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\uparrow\downarrow\uparrow\downarrow-\downarrow\uparrow\uparrow\downarrow-\uparrow\downarrow\downarrow\uparrow+\downarrow\uparrow\downarrow\uparrow\right),$
(5) $\displaystyle\chi_{00}^{11}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{12}}(2\uparrow\uparrow\downarrow\downarrow-\uparrow\downarrow\uparrow\downarrow-\downarrow\uparrow\uparrow\downarrow-\uparrow\downarrow\downarrow\uparrow$
(6)
$\displaystyle-\downarrow\uparrow\downarrow\uparrow+2\downarrow\downarrow\uparrow\uparrow),$
$\displaystyle\chi_{11}^{01}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\uparrow\downarrow\uparrow\uparrow-\downarrow\uparrow\uparrow\uparrow\right),$
(7) $\displaystyle\chi_{11}^{10}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\uparrow\uparrow\uparrow\downarrow-\uparrow\uparrow\downarrow\uparrow\right),$
(8) $\displaystyle\chi_{11}^{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\uparrow\uparrow\uparrow\downarrow+\uparrow\uparrow\downarrow\uparrow-\uparrow\downarrow\uparrow\uparrow-\downarrow\uparrow\uparrow\uparrow\right),$
(9) $\displaystyle\chi_{22}^{11}$ $\displaystyle=$
$\displaystyle\uparrow\uparrow\uparrow\uparrow.$ (10)

In the spatial space, the relative Jacobi coordinates with the single-partial
coordinates $\boldsymbol{r_{i}}$ ($i=1,2,3,4$) are defined by

$\displaystyle\boldsymbol{\xi}_{1}$ $\displaystyle\equiv$
$\displaystyle\boldsymbol{r_{1}}-\boldsymbol{r_{2}},$ (11)
$\displaystyle\boldsymbol{\xi}_{2}$ $\displaystyle\equiv$
$\displaystyle\boldsymbol{r_{3}}-\boldsymbol{r_{4}},$ (12)
$\displaystyle\boldsymbol{\xi}_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{m_{1}\boldsymbol{r_{1}}+m_{2}\boldsymbol{r_{2}}}{m_{1}+m_{2}}-\frac{m_{3}\boldsymbol{r_{3}}+m_{4}\boldsymbol{r_{4}}}{m_{3}+m_{4}},$
(13) $\displaystyle\boldsymbol{R}$ $\displaystyle\equiv$
$\displaystyle\frac{m_{1}\boldsymbol{r_{1}}+m_{2}\boldsymbol{r_{2}}+m_{3}\boldsymbol{r_{3}}+m_{4}\boldsymbol{r_{4}}}{m_{1}+m_{2}+m_{3}+m_{4}}.$
(14)

Note that $\boldsymbol{\xi}_{1}$ and $\boldsymbol{\xi}_{2}$ stand for the
relative Jacobi coordinates between two quarks $q_{1}$ and $q_{2}$ (or
antiquarks $\bar{q}_{1}$ and $\bar{q}_{2}$), and two antiquarks $\bar{q}_{3}$
and $\bar{q}_{4}$ (or quarks $q_{3}$ and $q_{4}$), respectively. While
$\boldsymbol{\xi}_{3}$ stands for the relative Jacobi coordinate between
diquark $qq$ and anti-diquark $\bar{q}\bar{q}$. Using the above Jacobi
coordinates, it is easy to obtain basis functions that have well-defined
symmetry under permutations of the pairs ($12$) and ($34$) Vijande  _et al._
(2009).

Considering the Pauli principle and color confinement for the
$cs\bar{c}\bar{s}$ system, we have 12 $1S$ configurations, while for the
$cc\bar{s}\bar{s}$ we have 4 $1S$ configurations. The spin-parity quantum
numbers, notations, and wave functions for these configurations are presented
in Table 3 and Table 4.

Table 3: Configurations of $1S$ states for $T_{cs\bar{c}\bar{s}}$ system. The subscripts and superscripts of the configurations are the spin quantum numbers and representations of the color SU(3) group, respectively. $\psi$, $\chi$, and $\left|6\bar{6}\right\rangle_{c}$/ $\left|\bar{3}3\right\rangle_{c}$ stand for the wave functions in the spatial, spin, and color spaces, respectively.  $J^{PC}$ | Configuration | $\underline{\hskip 14.22636pt\mathrm{Wave~{}function}\hskip 14.22636pt}$
---|---|---
$0^{++}$ | $\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{0}$ | $\psi_{000}^{1S}$ | $\chi_{00}^{00}$ | $\left|6\bar{6}\right\rangle_{c}$
| $\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{0}$ | $\psi_{000}^{1S}$ | $\chi_{00}^{00}$ | $\left|\bar{3}3\right\rangle_{c}$
| $\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{0}$ | $\psi_{000}^{1S}$ | $\chi_{00}^{11}$ | $\left|6\bar{6}\right\rangle_{c}$
| $\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{0}$ | $\psi_{000}^{1S}$ | $\chi_{00}^{11}$ | $\left|\bar{3}3\right\rangle_{c}$
$1^{+-}$ | $\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}$ | $\psi_{000}^{1S}$ | $\chi_{1m}^{11}$ | $\left|6\bar{6}\right\rangle_{c}$
| $\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}$ | $\psi_{000}^{1S}$ | $\chi_{1m}^{11}$ | $\left|\bar{3}3\right\rangle_{c}$
| $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}-\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}\right)$ | $\psi_{000}^{1S}$ | $\frac{1}{\sqrt{2}}\left(\chi_{1m}^{10}-\chi_{1m}^{01}\right)$ | $\left|6\bar{6}\right\rangle_{c}$
| $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}-\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}\right)$ | $\psi_{000}^{1S}$ | $\frac{1}{\sqrt{2}}\left(\chi_{1m}^{10}-\chi_{1m}^{01}\right)$ | $\left|\bar{3}3\right\rangle_{c}$
$1^{++}$ | $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}+\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}\right)$ | $\psi_{000}^{1S}$ | $\frac{1}{\sqrt{2}}\left(\chi_{1m}^{10}+\chi_{1m}^{01}\right)$ | $\left|6\bar{6}\right\rangle_{c}$
| $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}+\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}\right)$ | $\psi_{000}^{1S}$ | $\frac{1}{\sqrt{2}}\left(\chi_{1m}^{10}+\chi_{1m}^{01}\right)$ | $\left|\bar{3}3\right\rangle_{c}$
$2^{++}$ | $\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{2}$ | $\psi_{000}^{1S}$ | $\chi_{2m}^{11}$ | $\left|6\bar{6}\right\rangle_{c}$
| $\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{2}$ | $\psi_{000}^{1S}$ | $\chi_{2m}^{11}$ | $\left|\bar{3}3\right\rangle_{c}$
Table 4: Configurations of $1S$ states for $T_{cc\bar{s}\bar{s}}$ system. The caption is the same as that of Table 3. $J^{P}$ | Configuration |  $\underline{\hskip 5.69046pt\mathrm{Wave~{}function}\hskip 5.69046pt}$
---|---|---
$0^{+}$ | $\left|\left\\{cc\right\\}_{0}^{6}\left\\{\bar{s}\bar{s}\right\\}_{0}^{\bar{6}}\right\rangle_{0}$ | $\psi_{000}^{1S}$ | $\chi_{00}^{00}$ | $\left|6\bar{6}\right\rangle_{c}$
| $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{0}$ | $\psi_{000}^{1S}$ | $\chi_{00}^{11}$ | $\left|\bar{3}3\right\rangle_{c}$
$1^{+}$ | $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{1}$ | $\psi_{000}^{1S}$ | $\chi_{11}^{11}$ | $\left|\bar{3}3\right\rangle_{c}$
$2^{+}$ | $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{2}$ | $\psi_{000}^{1S}$ | $\chi_{22}^{11}$ | $\left|\bar{3}3\right\rangle_{c}$

####  II.1.3 Numerical method

With the wave functions for all the configurations, the mass matrix elements
of the Hamiltonian can be worked out. To solve the four-body problem
accurately, we adopt the explicitly correlated Gaussian method Mitroy  _et
al._ (2013); Varga and Suzuki (1995). It is a well-established variational
method to solve quantum-mechanical few-body problems in molecular, atomic, and
nuclear physics. The trail wave function of the tetraquark states without any
spatial excitations in the coordinate space is expanded in terms of correlated
Gaussian basis. Such a basis function can be written as

$\psi=\exp\left[-\sum_{i<j=1}^{4}\frac{1}{2d_{ij}^{2}}(\boldsymbol{r}_{i}-\boldsymbol{r}_{j})^{2}\right].$
(15)

For a tetraquark system $cs\bar{c}\bar{s}$ with zero angular momentum,
$d_{ij}$ are adjustable parameters. We can take $d_{12}=d_{34}\equiv a$,
$d_{13}\equiv b$, $d_{24}\equiv c$ and $d_{14}=d_{23}\equiv e$. For a
tetraquark system $cc\bar{s}\bar{s}$ with zero angular momentum, we can take
$d_{12}=a$, $d_{34}\equiv b$ and $d_{13}=d_{24}=d_{14}=d_{23}\equiv c$. It is
convenient to use a set of the Jacobi coordinates
$\xi=\left(\xi_{1},\xi_{2},\xi_{3}\right)$, instead of the relative distance
vectors $(\boldsymbol{r}_{i}-\boldsymbol{r}_{j})$. Then the correlated
Gaussian basis function can be rewritten as

$\psi\left(\xi,A\right)=\exp\left(-\sum_{i,j}A_{ij}\boldsymbol{\xi}_{i}\cdot\boldsymbol{\xi}_{j}\right)\equiv\exp\left(-\tilde{\boldsymbol{\xi}}A\boldsymbol{\xi}\right),$
(16)

and $A$ is a $3\times 3$ symmetric positive-definite matrix whose elements are
variational parameters.

The coordinate part of the trial wave function $\psi\left(\xi,A\right)$ can be
formed as a linear combination of the correlated Gaussians

$\displaystyle\psi\left(\xi,A\right)=\sum_{k=1}^{\mathcal{N}}c_{k}G\left(\xi,A_{k}\right).$
(17)

The accuracy of the trial function depends on the length of the expansion
$\mathcal{N}$ and the nonlinear parameters $A_{k}$. In our calculations,
following the method of Ref. Hiyama  _et al._ (2003), we let the variational
parameters form a geometric progression. For example, for a variational
parameter $d$, we take

$\displaystyle d_{n}=d_{1}q^{n-1}(n=1,\cdot\cdot\cdot,n_{d}^{max}).$ (18)

There are three parameters $\\{d_{1},d_{n_{d}^{max}},n^{max}\\}$ to be
determined through the variation method. The length of the expansion
$\mathcal{N}$ is determined to by
$\mathcal{N}=n_{a}^{max}n_{b}^{max}n_{c}^{max}n_{e}^{max}$. In this work,
stable solutions are obtained with
$n_{a}^{max}=n_{b}^{max}=n_{c}^{max}=n_{e}^{max}\equiv n^{max}=5$.

The numerical results should be independent of the parameters
$\\{d_{1},d_{n_{d}^{max}},n_{d}^{max}\\}$. To confirm this point, we scale the
basis number parameter $n^{max}$ as $n^{max}=3\to 6$. As an example, we plot
the masses of 12 $1S$-wave $T_{\left(cs\bar{c}\bar{s}\right)}$ configurations
as a function of $n^{max}$ in Fig. 1. It is found that the numerical results
are nearly independent of $n^{max}$ when it is large enough. On the other
hand, to see the $d_{n}$ independence of the numerical results, as commonly
dealt with in the literature Hiyama  _et al._ (2006, 2018); Meng  _et al._
(2019) we scale the parameter $d_{1}$ of the basis functions as
$d_{1}\to\alpha d_{1}$. The mass of a $T_{cs\bar{c}\bar{s}}$ state should be
stable at a resonance energy insensitive to the scaling parameter $\alpha$. As
an example, we plot the masses of 12 $1S$-wave
$T_{\left(cs\bar{c}\bar{s}\right)}$ configurations as a function of the
scaling factor $\alpha$ in Fig. 1. It is found that the numerical results are
nearly independent of the scaling factor $\alpha$. The stabilization of other
states predicted in this work has also been examined by the same method.

With the mass matrix elements ready for every configuration, the mass of the
tetraquark configuration and its spacial wave function can be determined by
solving a generalized eigenvalue problem. The details can be found in our
previous works Liu  _et al._ (2020, 2019b). The physical states can be
obtained by diagonalizing the mass matrix of different configurations with the
same $J^{PC}$ numbers.

Figure 1: Predicted masses of 12 $1S$-wave $T_{cs\bar{c}\bar{s}}$
configurations as a function of the scaling factor $n^{max}$ and $\alpha$.

###  II.2 Rearrangement decay

With the eigenstates obtained in the previous section, we can calculate the
rearrangement decays of the $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ systems
in a quark-exchange model Barnes  _et al._ (2001). The transition operators
can be extracted from the quark-quark and quark-antiquark interactions
$V_{ij}$ via the quark rearrangement. The decay amplitude $\mathcal{M}(A\to
BC)$ of a tetraquark state is described by

$\displaystyle\mathcal{M}(A\to
BC)=-\sqrt{(2\pi)^{3}}\sqrt{8M_{A}E_{B}E_{C}}\left\langle
BC|\sum_{i<j}V_{ij}|A\right\rangle,$ (19)

where $A$ stands for the initial tetraquark state, $BC$ stands for the final
hadron pair. The potentials $V_{ij}$ between quarks are taken from the
Hamiltonian. $M_{A}$ is the mass of the initial state, while $E_{B}$ and
$E_{C}$ are the energies of the final states $B$ and $C$, respectively, in the
initial-hadron-rest system.

This phenomenological model has been applied to the study of the hidden-charm
decay properties for the multiquark states in the literature Wang  _et al._
(2020); Xiao  _et al._ (2020); Wang  _et al._ (2021); Han and Xiao (2022). For
simplicity, the wave functions of the initial and final state hadrons, i.e.
$A,\ B,\ C$, are adopted in the form of single harmonic oscillator. They are
determined by fitting the wave functions calculated from our potential model.
The partial decay width of $A\to BC$ is given by

$\displaystyle\Gamma=\frac{1}{2J_{A}+1}\frac{|\boldsymbol{q}|}{8\pi
M_{A}^{2}}\left|\mathcal{M}(A\to BC)\right|^{2},$ (20)

where $\boldsymbol{q}$ is the three-vector momentum of the final state $B$ or
$C$ in the initial-hadron-rest frame. We note that the rearrangement decays of
the tetraquarks actually provide a dynamical description of the couplings of
the tetraquarks to two color-singlet hadrons in the final state. It can play a
role as the bare coupling of the elementary component of an exotic state to
the constituent hadrons in a molecular picture if the coupling is via a near-
threshold $S$-wave interaction.

Table 5: The average contributions of each part of the Hamiltonian to the $T_{cs\bar{c}\bar{s}\ }\left(1S\right)$ configurations. In the table, we define that $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}\right)^{\pm}\equiv\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}\pm\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}\right)$ and $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}\right)^{\pm}\equiv\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}\pm\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}\right)$. $J^{PC}$ | State | Mass | $\left\langle T\right\rangle$ | $\left\langle V^{Lin}\right\rangle$ | $\left\langle V^{Coul}\right\rangle$ | $\left\langle V^{SS}\right\rangle$ | $\left\langle V^{c}\right\rangle$
---|---|---|---|---|---|---|---
$0^{++}$ | $\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{0}$ | 4388 | 773 | 740 | -854 | 18. | 13 | -455
| $\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{0}$ | 4403 | 855 | 701 | -859 | -68. | 53 | -391
| $\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{0}$ | 4243 | 954 | 670 | -951 | -140. | 51 | -455
| $\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{0}$ | 4448 | 797 | 724 | -828 | -19. | 73 | -391
$1^{+-}$ | $\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}$ | 4305 | 874 | 698 | -910 | -67. | 32 | -455
| $\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}$ | 4467 | 774 | 734 | -815 | 0. | 34 | -391
| $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}\right)^{-}$ | 4366 | 798 | 730 | -867 | -5. | 46 | -455
| $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}\right)^{-}$ | 4444 | 801 | 722 | -831 | -22. | 69 | -391
$1^{++}$ | $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}\right)^{+}$ | 4382 | 785 | 735 | -859 | 10. | 93 | -455
| $\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}\right)^{+}$ | 4448 | 798 | 723 | -829 | -18. | 96 | -392
$2^{++}$ | $\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{2}$ | 4414 | 746 | 751 | -837 | 43. | 23 | -455
| $\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{2}$ | 4503 | 732 | 753 | -792 | 35. | 39 | -391
Table 6: The average contributions of each part of the Hamiltonian to the $T_{cc\bar{s}\bar{s}\ }\left(1S\right)$ configurations. $J^{P}$ |  State | Mass | $\left\langle T\right\rangle$ | $\left\langle V^{Lin}\right\rangle$ | $\left\langle V^{Coul}\right\rangle$ | $\left\langle V^{SS}\right\rangle$ | $\left\langle V^{c}\right\rangle$
---|---|---|---|---|---|---|---
$0^{+}$ | $\left|\left\\{cc\right\\}_{0}^{6}\left\\{\bar{s}\bar{s}\right\\}_{0}^{\bar{6}}\right\rangle_{0}$ | 4465 | 735 | 682 | -831 |  19. | 99 | -306
| $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{0}$ | 4432 | 802 | 740 | -831 | -10. | 54 | -434
$1^{+}$ | $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{1}$ | 4449 | 780 | 750 | -820 | 7. | 11 | -434
$2^{+}$ | $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{2}$ | 4481 | 740 | 769 | -799 | 38. | 16 | -434
Figure 2: Predicted mass spectra of $1S$ states for the $cs\bar{c}\bar{s}$ and $T_{cc\bar{s}\bar{s}}$ systems. Table 7: Predicted mass spectra of $1S$ states for the $T_{cs\bar{c}\bar{s}}$ system. $J^{PC}$ | State |  $\left\langle H\right\rangle$ (MeV) | Mass (MeV) | Eigenvector
---|---|---|---|---
$0^{++}$ | $\begin{array}[]{c}\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{0}\\\ \left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{0}\\\ \left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{0}\\\ \left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{0}\end{array}$ | $\left(\begin{array}[]{rrrr}4388&-58&-15&-71\\\ -58&4403&-81&-3\\\ -15&-81&4243&-76\\\ -71&-3&-76&4448\end{array}\right)$ | $\left(\begin{array}[]{r}4174\\\ 4356\\\ 4424\\\ 4526\end{array}\right)$ | $\left(\begin{array}[]{rrrr}0.26&0.37&0.84&0.31\\\ 0.82&0.24&-0.46&0.26\\\ 0.11&-0.80&0.11&0.58\\\ 0.50&-0.40&0.28&-0.71\end{array}\right)$
$1^{+-}$ | $\begin{array}[]{c}\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}\\\ \left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}\\\ \frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}-\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}\right)\\\ \frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}-\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}\right)\end{array}$ | $\left(\begin{array}[]{rrrr}4305&-68&-28&-23\\\ -68&4467&-21&-10\\\ -28&-21&4366&-117\\\ -23&-10&-117&4444\end{array}\right)$ | $\left(\begin{array}[]{r}4239\\\ 4322\\\ 4492\\\ 4529\end{array}\right)$ | $\left(\begin{array}[]{rrrr}0.66&0.27&0.57&0.41\\\ 0.68&0.20&-0.57&-0.41\\\ 0.33&-0.94&-0.03&0.12\\\ 0.04&-0.11&0.58&-0.80\end{array}\right)$
$1^{++}$ | $\begin{array}[]{c}\frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{0}^{\bar{6}}\right\rangle_{1}+\left|\left(cs\right)_{0}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{1}\right)\\\ \frac{1}{\sqrt{2}}\left(\left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{0}^{3}\right\rangle_{1}+\left|\left(cs\right)_{0}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{1}\right)\end{array}$ | $\left(\begin{array}[]{rr}4382&-2\\\ -2&4448\end{array}\right)$ | $\left(\begin{array}[]{r}4382\\\ 4448\end{array}\right)$ | $\left(\begin{array}[]{rr}1.00&0.03\\\ 0.03&-1.00\end{array}\right)$
$2^{++}$ | $\begin{array}[]{c}\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{2}\\\ \left|\left(cs\right)_{1}^{\bar{3}}\left(\bar{c}\bar{s}\right)_{1}^{3}\right\rangle_{2}\end{array}$ | $\left(\begin{array}[]{rr}4414&-54\\\ -54&4503\end{array}\right)$ | $\left(\begin{array}[]{r}4389\\\ 4528\end{array}\right)$ | $\left(\begin{array}[]{rr}0.90&0.43\\\ 0.43&-0.90\end{array}\right)$
Table 8: Predicted mass spectra of $1S$ states for the $T_{cc\bar{s}\bar{s}}$ system. $J^{P}$ | State |  $\left\langle H\right\rangle$ (MeV) | Mass (MeV) | Eigenvector
---|---|---|---|---
$0^{+}$ | $\begin{array}[]{c}\left|\left\\{cc\right\\}_{0}^{6}\left\\{\bar{s}\bar{s}\right\\}_{0}^{\bar{6}}\right\rangle_{0}\\\ \left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{0}\end{array}$ | $\left(\begin{array}[]{rr}4465&-60\\\ -60&4432\end{array}\right)$ | $\left(\begin{array}[]{r}4386\\\ 4510\end{array}\right)$ | $\left(\begin{array}[]{rr}-0.61&-0.80\\\ -0.80&0.61\end{array}\right)$
$1^{+}$ | $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{1}$ | 4449 | 4449 | 1
$2^{+}$ | $\left|\left\\{cc\right\\}_{1}^{\bar{3}}\left\\{\bar{s}\bar{s}\right\\}_{1}^{3}\right\rangle_{2}$ | 4481 | 4481 | 1
Table 9: The proportions of different color configurations and the root mean
square radius for each state of $cs\bar{c}\bar{s}$. And
$\langle\boldsymbol{r}_{12}^{2}\rangle^{\frac{1}{2}}$=$\langle\boldsymbol{r}_{34}^{2}\rangle^{\frac{1}{2}}$,
$\langle\boldsymbol{r}_{14}^{2}\rangle^{\frac{1}{2}}$=$\langle\boldsymbol{r}_{23}^{2}\rangle^{\frac{1}{2}}$.

State | $\left|6\bar{6}\right\rangle$ | $\left|\bar{3}3\right\rangle$ | $\left|11\right\rangle$ | $\left|88\right\rangle$ | $\langle\boldsymbol{r}_{12}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{13}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{24}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{14}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{12-34}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{13-24}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{14-23}^{2}\rangle^{\frac{1}{2}}$
---|---|---|---|---|---|---|---|---|---|---|---
$T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4174\right)$ | 77$\%$ | 23$\%$ | 92$\%$ | 8$\%$ | 0.64 | 0.50 | 0.66 | 0.60 | 0.38 | 0.46 | 0.42
$T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4356\right)$ | 88$\%$ | 12$\%$ | 70$\%$ | 30$\%$ | 0.69 | 0.52 | 0.70 | 0.64 | 0.38 | 0.51 | 0.45
$T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4424\right)$ | 2$\%$ | 98$\%$ | 32$\%$ | 68$\%$ | 0.59 | 0.54 | 0.70 | 0.63 | 0.45 | 0.42 | 0.41
$T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4526\right)$ | 34$\%$ | 66$\%$ | 7$\%$ | 93$\%$ | 0.63 | 0.53 | 0.69 | 0.63 | 0.42 | 0.45 | 0.43
$T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4239\right)$ | 76$\%$ | 24$\%$ | 98$\%$ | 2$\%$ | 0.67 | 0.51 | 0.68 | 0.63 | 0.39 | 0.49 | 0.44
$T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4322\right)$ | 79$\%$ | 21$\%$ | 95$\%$ | 5$\%$ | 0.67 | 0.51 | 0.68 | 0.63 | 0.38 | 0.49 | 0.43
$T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4492\right)$ | 10$\%$ | 90$\%$ | 8$\%$ | 92$\%$ | 0.63 | 0.55 | 0.72 | 0.65 | 0.45 | 0.45 | 0.43
$T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4529\right)$ | 35$\%$ | 65$\%$ | 1$\%$ | 99$\%$ | 0.63 | 0.53 | 0.70 | 0.64 | 0.43 | 0.46 | 0.42
$T_{cs\bar{c}\bar{s}\ 1^{++}}\left(4382\right)$ | 100$\%$ |  | 70$\%$ | 30$\%$ | 0.71 | 0.53 | 0.72 | 0.63 | 0.38 | 0.50 | 0.47
$T_{cs\bar{c}\bar{s}\ 1^{++}}\left(4448\right)$ |  | 100$\%$ | 30$\%$ | 70$\%$ | 0.60 | 0.55 | 0.72 | 0.64 | 0.46 | 0.43 | 0.42
$T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4389\right)$ | 82$\%$ | 18$\%$ | 98$\%$ | 2$\%$ | 0.71 | 0.54 | 0.73 | 0.67 | 0.40 | 0.52 | 0.47
$T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4528\right)$ | 18$\%$ | 82$\%$ | 2$\%$ | 98$\%$ | 0.65 | 0.57 | 0.75 | 0.67 | 0.47 | 0.47 | 0.45

Table 10: The proportions of different color configurations and the root mean
square radius for each state of $cc\bar{s}\bar{s}$. And
$\langle\boldsymbol{r}_{13}^{2}\rangle^{\frac{1}{2}}$=$\langle\boldsymbol{r}_{24}^{2}\rangle^{\frac{1}{2}}$=$\langle\boldsymbol{r}_{14}^{2}\rangle^{\frac{1}{2}}$=$\langle\boldsymbol{r}_{23}^{2}\rangle^{\frac{1}{2}}$,
$\langle\boldsymbol{r}_{13-24}^{2}\rangle^{\frac{1}{2}}$=$\langle\boldsymbol{r}_{14-23}^{2}\rangle^{\frac{1}{2}}$.

State | $\left|6\bar{6}\right\rangle$ | $\left|\bar{3}3\right\rangle$ | $\left|11\right\rangle$ | $\left|88\right\rangle$ | $\langle\boldsymbol{r}_{12}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{34}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{13}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{12-34}^{2}\rangle^{\frac{1}{2}}$ | $\langle\boldsymbol{r}_{13-24}^{2}\rangle^{\frac{1}{2}}$
---|---|---|---|---|---|---|---|---|---
$T_{cc\bar{s}\bar{s}\ 0^{+}}\left(4386\right)$ | 36$\%$ | 64$\%$ | 45$\%$ | 55$\%$ | 0.57 | 0.74 | 0.65 | 0.45 | 0.45
$T_{cc\bar{s}\bar{s}\ 0^{+}}\left(4510\right)$ | 64$\%$ | 36$\%$ | 55$\%$ | 45$\%$ | 0.59 | 0.76 | 0.65 | 0.42 | 0.48
$T_{cc\bar{s}\bar{s}\ 1^{+}}\left(4449\right)$ |  | 100$\%$ | 33$\%$ | 67$\%$ | 0.51 | 0.69 | 0.65 | 0.49 | 0.41
$T_{cc\bar{s}\bar{s}\ 2^{+}}\left(4481\right)$ |  | 100$\%$ | 33$\%$ | 67$\%$ | 0.51 | 0.71 | 0.68 | 0.51 | 0.42

##  III Results and discussions

###  III.1 Mass spectra and quark configurations

The predicted masses of each configuration for the $cs\bar{c}\bar{s}$ and
$cc\bar{s}\bar{s}$ systems have been listed in Table 5 and 6, respectively.
The contributions from each part of the Hamiltonian to these configurations
are further analyzed. The results are listed in Table 5 and 6 as well. It
shows that the averaged kinetic energy $\langle T\rangle$, the linear
confining potential $\langle V^{Lin}\rangle$, and the Coulomb potential
$\langle V^{Coul}\rangle$ have the same order of magnitude. Furthermore, it is
found that the spin-spin interaction plays an important role in the
configuration
$\left|\left(cs\right)_{1}^{6}\left(\bar{c}\bar{s}\right)_{1}^{\bar{6}}\right\rangle_{0}$.
The predicted mass for this configuration is notably ($\sim 60-260$ MeV)
smaller than the other configurations due to the strong attractive spin-spin
interactions $\langle V^{SS}\rangle\simeq-140$ MeV.

After considering configuration mixing, one can obtain the physical states.
The predicted mass spectra for the $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$
systems have been given in Tables 7 and 8 and also shown in Fig. 2. The masses
of the $1S$-wave $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ states are
predicted to be in the range of $\sim 4.1-4.6$ GeV and $\sim 4.3-4.6$ GeV,
respectively. One can see that the physical states are usually mixtures of two
different color configurations $|6\bar{6}\rangle_{c}$ and
$|\bar{3}3\rangle_{c}$. For the physical states with $J^{P}=0^{++}$, $1^{+-}$
and $2^{++}$, the mixing between different color configurations is
particularly evident. The configuration mixing effects can cause notable mass
shifts to the physical states. It shows that our results of the $1S$-wave
$cs\bar{c}\bar{s}$ states are much higher than other theoretical works Li  _et
al._ (2023); Guo  _et al._ (2023); Lü and Dong (2016), but consistent with
Ref. Wu  _et al._ (2016). Meanwhile, our results of the $1S$-wave
$cc\bar{s}\bar{s}$ states are consistent with other theoretical works Lü  _et
al._ (2020); Ebert  _et al._ (2007); Zhang  _et al._ (2008).

###  III.2 Proportions of different color configurations

It should be mentioned that except for the color configurations
$|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$, one can also select the
$|11\rangle_{c}$ and $|88\rangle_{c}$ representations when constructing the
tetraquark wave functions. The two sets of color configurations are equivalent
to each other. The $|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$
configurations can be expressed by $|11\rangle_{c}$ and $|88\rangle_{c}$
through the Fierz transformation Wang  _et al._ (2019). Namely, one can
extract the $|11\rangle_{c}$ and $|88\rangle_{c}$ components in a physical
states expressed with the $|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$
configurations. A large $|11\rangle_{c}$ component may indicate a potentially
large coupling for the initial tetraquark coupling to a nearby $S$-wave
channel.

Due to different symmetries of wave functions, there are slight differences in
the calculation methods between the $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$
systems. Taking the physical state $T_{cs\bar{c}\bar{s}\
2^{++}\left(4389\right)}$ as an example, it shows that there exist
configurations of different colors, but with the same spatial, flavor, and
spin wave functions. For convenience, taking the physical state
$T_{cs\bar{c}\bar{s}\ 2^{++}\left(4389\right)}$ that only includes two
configurations as an example, the wave function for the physical state is
given by:

$\displaystyle T_{cs\bar{c}\bar{s}\ 2^{++}\left(4389\right)}$ $\displaystyle=$
$\displaystyle
c_{1}\chi_{22}^{11}\left|6\bar{6}\right\rangle_{c}+c_{2}\chi_{22}^{11}\left|\bar{3}3\right\rangle_{c}\
,$ (21)

where $c_{1}$ and $c_{2}$ are the elements of the mixing matrix given in Table
7. Expressing the wave function in the basis of
$\left|11\right\rangle_{c}/\left|88\right\rangle_{c}$, we have:

$\displaystyle T_{cs\bar{c}\bar{s}\ 2^{++}\left(4389\right)}$ $\displaystyle=$
$\displaystyle\left(c_{1}\ \sqrt{\frac{2}{3}}+c_{2}\
\sqrt{\frac{1}{3}}\right)\chi_{22}^{11}\left|11\right\rangle_{c}$ (22)
$\displaystyle+$ $\displaystyle\left(c_{1}\ \sqrt{\frac{1}{3}}-c_{2}\
\sqrt{\frac{2}{3}}\right)\chi_{22}^{11}\left|88\right\rangle_{c}\ .$

So the probabilities for $T_{cs\bar{c}\bar{s}\ 2^{++}\left(4389\right)}$
staying in different color configurations can be extracted by

$\displaystyle\left|11\right\rangle_{c}$ $\displaystyle:$
$\displaystyle\left|\sqrt{\frac{2}{3}}c_{1}+\sqrt{\frac{1}{3}}c_{2}\right|^{2},$
$\displaystyle\left|88\right\rangle_{c}$ $\displaystyle:$
$\displaystyle\left|\sqrt{\frac{1}{3}}c_{1}-\sqrt{\frac{2}{3}}c_{2}\right|^{2}.$
(23)

It is interesting to note that for possible mixing elements $c_{1}$ and
$c_{2}$, it might happen that a physical state of $cs\bar{c}\bar{s}$ can be
dominated by either the $|11\rangle_{c}$ or $|88\rangle_{c}$ configurations.
It implies that some tetraquark states may strongly couple to two color-
singlet hadrons if the kinematics and dynamics allow.

Carrying on a similar analysis of the $cc\bar{s}\bar{s}$ system and taking the
physical state $T_{cc\bar{s}\bar{s}\ 0^{++}\left(4386\right)}$ as an example,
the wave function for the physical state is given by $T_{cc\bar{s}\bar{s}\
0^{++}\left(4386\right)}=c_{1}\
\chi_{00}^{00}\left|6\bar{6}\right\rangle_{c}+c_{2}\
\chi_{00}^{11}\left|\bar{3}3\right\rangle_{c}$. It can also be expressed as:

$\displaystyle T_{cc\bar{s}\bar{s}\ 0^{++}\left(4386\right)}$ $\displaystyle=$
$\displaystyle\left(c_{1}\ \chi_{00}^{00}\sqrt{\frac{2}{3}}+c_{2}\
\chi_{00}^{11}\sqrt{\frac{1}{3}}\right)\left|11\right\rangle_{c}$ (24)
$\displaystyle+$ $\displaystyle\left(c_{1}\
\chi_{00}^{00}\sqrt{\frac{1}{3}}-c_{2}\
\chi_{00}^{11}\sqrt{\frac{2}{3}}\right)\left|88\right\rangle_{c}\ ,$

where $c_{1}$ and $c_{2}$ are the elements of the mixing matrix given in Table
8. Note that the spin configurations of $\chi_{00}^{00}$ and $\chi_{00}^{11}$
are linearly independent to each other. They are part of the total wave
function for the $|11\rangle_{c}$ and $|88\rangle_{c}$ configurations. Namely,
to extract the probability amplitudes for the $|11\rangle_{c}$ and
$|88\rangle_{c}$ components one needs to calculate the spin transition matrix
elements first. By taking the orthogonality between $\chi_{00}^{00}$ and
$\chi_{00}^{11}$ in Eq. (24) we can extract the probabilities for
$T_{cc\bar{s}\bar{s}\ 0^{++}\left(4386\right)}$ in the color configurations
$|11\rangle_{c}$ and $|88\rangle_{c}$ as follows, respectively:

$\displaystyle\left|11\right\rangle_{c}$ $\displaystyle:$
$\displaystyle\frac{2}{3}c_{1}^{2}+\frac{1}{3}c_{2}^{2},$
$\displaystyle\left|88\right\rangle_{c}$ $\displaystyle:$
$\displaystyle\frac{1}{3}c_{1}^{2}+\frac{2}{3}c_{2}^{2}.$ (25)

It is interesting to note that for any possible values for the mixing elements
$c_{1}$ and $c_{2}$, the color configurations $|11\rangle_{c}$ and
$|88\rangle_{c}$ in a physical state can be compatible. This may indicate that
an overall color-singlet tetraquark (i.e. $|88\rangle_{c}$ configuration)
should always play a role in the $T_{cc\bar{s}\bar{s}}$ system.

Using the wave function obtained by solving the Schrödinger equation, the
components of different color configurations in the physical
$cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ states can be extracted which are
listed in Tables 10 and 10, respectively. It shows that for the
$cs\bar{c}\bar{s}$ system, some physical states may have a large
$\left|11\right\rangle_{c}$ component, while some have a large
$\left|88\right\rangle_{c}$. In contrast, for the $cc\bar{s}\bar{s}$ system
$\left|11\right\rangle_{c}$ and $\left|88\right\rangle$ occupy a similar
proportion. These numerical values confirms our expectations based on the
mixing patterns presented in Eqs. (III.2) and (III.2). We also note that among
those mixing states with the same quantum numbers for the $cs\bar{c}\bar{s}$
system, the higher mass states seem to have a lower component of
$|11\rangle_{c}$. Phenomenologically, the higher mass state in the mixing
multiplet is more likely to be an overall color-singlet tetraquark.

###  III.3 Root mean square radii

To investigate the inner structure of the tetraquark, we also calculate the
root mean square radius between any two particles. The
$\langle\boldsymbol{r}_{12}^{2}\rangle$ is defined as follows:

$\langle\boldsymbol{r}_{12}^{2}\rangle=\int
r_{12}^{2}\left|\Psi\right|^{2}d\boldsymbol{r}_{12}d\boldsymbol{r}_{34}d\boldsymbol{r}_{12-34}.$
(26)

Our results are also listed in Tables 10 and 10. The
$\langle\boldsymbol{r}_{ij}^{2}\rangle^{\frac{1}{2}}$ represents the average
distance between a quark (an antiquark) and an antiquark (a quark). It can be
found from the Table 10 and 10 that the distances between any two particles
share the same size, around 0.6 fm, which is mainly determined by the
constituent quark mass. Our results of
$\langle\boldsymbol{r}_{ij}^{2}\rangle^{\frac{1}{2}}$ are consistent with
those in Ref. Deng  _et al._ (2020). The quantity
$\langle\boldsymbol{r}_{12-34}^{2}\rangle^{\frac{1}{2}}$ stands for the
averaged distance between the quark pair ($qq$) and antiquark pair
($\bar{q}\bar{q}$). The quantity
$\langle\boldsymbol{r}_{13-24}^{2}\rangle^{\frac{1}{2}}$ and
$\langle\boldsymbol{r}_{14-22}^{2}\rangle^{\frac{1}{2}}$ stand for the
averaged distances between two quark-antiquark pairs ($q\bar{q}$). The
calculated results of $\langle\boldsymbol{r}_{ij-kl}^{2}\rangle^{\frac{1}{2}}$
are about 0.5 fm, which are higher than those calculated by Ref. Deng  _et
al._ (2020). The averaged distances between two subclusters are less than the
averaged distances between two particles ($\langle\boldsymbol{r}_{ij-
kl}^{2}\rangle^{\frac{1}{2}}<\langle\boldsymbol{r}_{ij}^{2}\rangle^{\frac{1}{2}}$),
which indicates that the overlap of the two subclusters is extremely strong.
In such a case, interpretations of the internal structures based on the
hadronic molecules would be problematic. Or at least, one has to include long-
ranged dynamics to obtain a more realistic view of the physical states.

###  III.4 Rearrangement decays

The stability of the tetraquark states can be identified by comparing the mass
of tetraquark states and threshold of tetraquark states meson-meson thresholds
$T_{M_{1}M_{2}}=M_{1}\left(c\bar{s}\right)+M_{2}\left(\bar{c}s\right)$ and
$T_{M^{\prime}_{1}M^{\prime}_{2}}=M_{1}\left(c\bar{c}\right)+M_{2}\left(s\bar{s}\right)$.
Because of the contributions of kinetic energy $\langle T\rangle$ and linear
confinement potential $\langle V^{Lin}\rangle$, the mass of all of these
states are higher than the corresponding threshold $T_{M_{1}M_{2}}$.
Tetraquark states with $E>T_{M_{1}M_{2}}$ and
$E>T_{M^{\prime}_{1}M^{\prime}_{2}}$ are unstable and can decay into two color
singlet mesons through the quark rearrangement. It can be found that all the
states of $cs\bar{c}\bar{s}$ lie above the thresholds of $\eta_{c}\eta$ and
$D_{s}\bar{D_{s}}$. However, it is interesting to find that the state
$T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4174\right)$ lies below the thresholds of
$D_{s}^{*}\bar{D_{s}^{*}}$. So this state cannot decay into
$D_{s}^{*}\bar{D_{s}^{*}}$ through strong interactions in the NRPQM. While for
the $cc\bar{s}\bar{s}$ system, all the states lie above the thresholds of
$D_{s}^{*}D_{s}^{*}$. So these states can decay into $D_{s}D_{s}$,
$D_{s}D_{s}^{*}$, or $D_{s}^{*}D_{s}^{*}$.

In addition to calculating the mass spectra, the results of the fall-apart
decays via the quark rearrangement of the $1S$-wave $cs\bar{c}\bar{s}$ and
$cc\bar{s}\bar{s}$ states are given in Tables 11 and 13. It shows that all the
states of the $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ systems, have narrow
decay widths within the range of $(0,16)$ MeV. There are two main reasons for
it. Firstly, since the initial states are in an $S$ wave, the rearrangement
decays are mainly driven by the spin-spin interactions. In contrast, the decay
amplitude caused by the confinement potential part
$-\frac{3}{16}\left(\boldsymbol{\lambda}_{i}\cdot\boldsymbol{\lambda}_{j}\right)\left(b_{ij}r_{ij}-\frac{4}{3}\frac{\alpha_{ij}}{r_{ij}}+c_{ij}\right)$
is negligibly small. Secondly, one notices that the two terms of the decay
amplitude, $\left\langle B_{13}C_{24}|V_{12}+V_{34}|A\right\rangle$ and
$\left\langle B_{13}C_{24}|V_{14}+V_{23}|A\right\rangle$ (or $\left\langle
B_{14}C_{23}|V_{12}+V_{34}|A\right\rangle$ and $\left\langle
B_{14}C_{23}|V_{13}+V_{24}|A\right\rangle$), almost completely cancel out each
other.

Table 11: The predicted decay widths $\Gamma$ (MeV) of the rearrangement decay processes of the ground $cs\bar{c}\bar{s}$ system.  State | $\Gamma_{T\rightarrow\eta_{c}\eta}$ | $\Gamma_{T\rightarrow J/\psi\phi}$ | $\Gamma_{T\rightarrow J/\psi\eta}$ | $\Gamma_{T\rightarrow\eta_{c}\phi}$ | $\Gamma_{T\rightarrow D_{s}\bar{D_{s}}}$ | $\Gamma_{T\rightarrow D_{s}^{*}\bar{D_{s}^{*}}}$ | $\Gamma_{T\rightarrow\frac{1}{\sqrt{2}}\left(D_{s}^{*}\bar{D_{s}}-D_{s}\bar{D_{s}^{*}}\right)}$ | $\Gamma_{T\rightarrow\frac{1}{\sqrt{2}}\left(D_{s}^{*}\bar{D_{s}}+D_{s}\bar{D_{s}^{*}}\right)}$ | $\Gamma_{sum}$
---|---|---|---|---|---|---|---|---|---
$T_{0^{++}}\left(4174\right)$ | 2.0 | 0.07 | ✘ | ✘ | 15 | ✘ | ✘ | ✘ | 18
$T_{0^{++}}\left(4356\right)$ | 1.09 | 3.4 | ✘ | ✘ | 0.07 | 0.16 | ✘ | ✘ | 4.7
$T_{0^{++}}\left(4424\right)$ | 4.8 | 0.85 | ✘ | ✘ | 0.00 | 8.6 | ✘ | ✘ | 14
$T_{0^{++}}\left(4526\right)$ | 0.19 | 7.0 | ✘ | ✘ | 1.3 | 0.03 | ✘ | ✘ | 8.5
$T_{1^{++}}\left(4382\right)$ | ✘ | 0.72 | ✘ | ✘ | ✘ | ✘ | 1.0 | ✘ | 1.7
$T_{1^{++}}\left(4448\right)$ | ✘ | 0.61 | ✘ | ✘ | ✘ | ✘ | 0.42 | ✘ | 1.0
$T_{2^{++}}\left(4389\right)$ | ✘ | 0.80 | ✘ | ✘ | ✘ | 6.0 | ✘ | ✘ | 6.8
$T_{2^{++}}\left(4528\right)$ | ✘ | 0.07 | ✘ | ✘ | ✘ | 0.06 | ✘ | ✘ | 0.13
$T_{1^{+-}}\left(4239\right)$ | ✘ | ✘ | 0.91 | 0.02 | ✘ | 2.9 | ✘ | 12 | 16
$T_{1^{+-}}\left(4322\right)$ | ✘ | ✘ | 0.03 | 1.18 | ✘ | 1.6 | ✘ | 4.6 | 7.4
$T_{1^{+-}}\left(4492\right)$ | ✘ | ✘ | 1.1 | 0.52 | ✘ | 0.24 | ✘ | 1.4 | 3.3
$T_{1^{+-}}\left(4529\right)$ | ✘ | ✘ | 1.1 | 1.6 | ✘ | 0.53 | ✘ | 0.14 | 3.4
Table 12: Candidates for the $cs\bar{c}\bar{s}$. State |  | Mass(MeV) | Width(MeV) | Decay | $J^{P\left(C\right)}$ | Ref.
---|---|---|---|---|---|---
$X\left(3960\right)$ | $3960<T_{0^{++}}\left(4174,4526\right)$ | $3956\pm 5\pm 10$ | $43\pm 13\pm 8$ | $B^{+}\to{\color[rgb]{1,0,0}D_{s}^{-}D_{s}^{+}}K^{+}$ | $0^{++}$ | Aaij  _et al._ (2023)
$X_{0}\left(4140\right)$ | $\Gamma_{T_{0^{++}}\left(4174\right)\to D_{s}\bar{D_{s}}}=15.44$ | $4133\pm 6\pm 6$ | $67\pm 17\pm 7$ | $B^{+}\to{\color[rgb]{1,0,0}D_{s}^{-}D_{s}^{+}}K^{+}$ | $0^{++}$ | Aaij  _et al._ (2023)
$X\left(4500\right)$ | $\Gamma_{T_{0^{++}}\left(4526\right)\rightarrow J/\psi\ \phi\left(1020\right)}=7.01$ | $4506\,_{-\,19}^{+\,16}$ | $92_{-21}^{+30}$ | $B^{+}\to{\color[rgb]{0,0,1}J/\psi\phi}K^{+}$ | $0^{++}$ | Aaij  _et al._ (2017)
$X\left(4700\right)$ | $4700>T_{0^{++}}\left(4174,4526\right)$ | $4704\,_{-\,26}^{+\,17}$ | $120_{-45}^{+52}$ | $B^{+}\to{\color[rgb]{0,0,1}J/\psi\phi}K^{+}$ | $0^{++}$ | Aaij  _et al._ (2017)
$X\left(4350\right)$ |  $\Gamma_{T_{0^{++}}\left(4356\right)\rightarrow J/\psi\ \phi\left(1020\right)}=3.38$, | $4350.6_{-5.1}^{+4.6}\pm 0.7$ | $13.3_{-9}^{+18}\pm 4$ | $\gamma\gamma\to{\color[rgb]{0,0,1}J/\psi\phi}$ | $0/2^{++}$ | Shen  _et al._ (2010)
| $\Gamma_{T_{2^{++}}\left(4389\right)\rightarrow J/\psi\ \phi\left(1020\right)}=0.80$ |  |  |  |  |
$X\left(4140\right)$ | $4140<T_{1^{++}}\left(4382,4448\right)$ | $4143.0\pm 2.9\pm 1.2$ | $11.7_{-5.0}^{+8.3}\pm 3.7$ | $B^{+}\to{\color[rgb]{0,0,1}J/\psi\phi}K^{+}$ | $1^{++}$ | Aaltonen  _et al._ (2009); Chatrchyan  _et al._ (2014); Abazov  _et al._ (2014); Aaij  _et al._ (2017)
$X\left(4274\right)$ | $4274<T_{1^{++}}\left(4382,4448\right)$ | $4274_{-6.7}^{+8.4}\pm 1.9$ | $\Gamma=32.3_{-15.3}^{+21.9}\pm 7.6$ | $B\to{\color[rgb]{0,0,1}J/\psi\phi}K$ | $1^{++}$ | Aaltonen  _et al._ (2017); Aaij  _et al._ (2017)
$X\left(4685\right)$ | $4685>T_{1^{++}}\left(4382,4448\right)$ | $4684\,_{-\,17}^{+\,15}$ | $126\pm 15\,_{-41}^{+37}$ | $B^{+}\to{\color[rgb]{0,0,1}J/\psi\phi}K^{+}$ | $1^{+}$ | Aaij  _et al._ (2021)

###  III.5 Discussions

Based on the tetraquark picture, the $1S$ states for the
$T_{cs\bar{c}\bar{s}}$ and $T_{cs\bar{c}\bar{s}}$ have relatively simple
spectra in the NRPQM. In this Subsection we discuss the experimental
observations and their possible assignments to the calculated spectra.

####  III.5.1 $T_{cs\bar{c}\bar{s}}$

  * •

$J^{PC}=0^{++}$

In the $J^{PC}=0^{++}$ sector, there are four $1S$ states for the
$T_{cs\bar{c}\bar{s}}$ system. Including configuration mixing effects, their
physical masses are predicted to be in the range of $\left[4174,4526\right]$
MeV (see Table 7), which are much larger than the mass threshold of
$D_{s}\bar{D_{s}}$. Thus, they may easily decay into $\eta_{c}\eta$,
$J/\psi\phi$, or $D_{s}\bar{D_{s}}$ through the quark rearrangements. For the
ones above the $D_{s}^{*}\bar{D_{s}^{*}}$ threshold, they can also decay into
$D_{s}^{*}\bar{D_{s}^{*}}$ via an $S$ wave 444The $D$-wave transition, in
principle, can occur via the spin-orbital tensor coupling. However, this
transition will be highly suppressed at the order of $(p/\mu)^{2}$ with $p$
and $m$ denoting the typical momentum and reduced mass of two interacting
constituent quarks. For the ground state tetraquarks investigated in this work
we have neglected the $D$-wave contributions as a reasonable approximation. .
The calculated partial decay widths are listed in Table 11.

The predicted masses of $T_{cs\bar{c}\bar{s}\ 0^{++}}$ are much higher than
the state $X\left(3960\right)$ observed by LHCb Aaij  _et al._ (2023). Because
of this, it is difficult to accommodate the states $X\left(3960\right)$ as one
of the $1S$ states of $T_{cs\bar{c}\bar{s}}$ with $J^{PC}=0^{++}$ in the
NRPQM. The investigation in an improved chromomagnetic interaction model Guo
_et al._ (2023) and an extended recoupling model Badalian and Simonov (2023)
indicates that $X\left(3960\right)$ may be interpreted as $0^{++}$
$cs\bar{c}\bar{s}$ tetraquark states. In Ref. Li  _et al._ (2023), the
$X\left(3960\right)$ is interpreted as tetraquark $cs\bar{c}\bar{s}$ states by
assuming that the $X\left(4140\right)$ is the lower $1^{++}$
$cs\bar{c}\bar{s}$ tetraquark. Apart from the interpretations which treat
$X\left(3960\right)$ as genuine tetraquark states, there are also other
explanations proposed in the literature. For instance, in Refs. Bayar  _et
al._ (2023); Ji  _et al._ (2022, 2023); Xin  _et al._ (2022); Mutuk (2022);
Chen and Huang (2023) the authors interpreted it as a hadronic molecule.

The lowest energy of $T_{cs\bar{c}\bar{s}\ 0^{++}}$ is quite close to the mass
of the state $X_{0}\left(4140\right)$, and the partial width
$\Gamma_{T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4174\right)\to D_{s}\bar{D_{s}}}$
is predicted to be $\sim 15$ MeV, which implies a possibility that the main
component of the state $X_{0}\left(4140\right)$ may be the state
$cs\bar{c}\bar{s}$ with $J^{PC}=0^{++}$. In Refs. Li  _et al._ (2023); Guo
_et al._ (2023); Badalian and Simonov (2023), the $X_{0}\left(4140\right)$ is
interpreted as tetraquark $cs\bar{c}\bar{s}$ states while the conclusion that
the $X_{0}\left(4140\right)$ is a $D_{s}^{+}D_{s}^{-}$ molecule is drawn in
Ref. Agaev  _et al._ (2023).

In our calculation results, there are two states, $T_{cs\bar{c}\bar{s}\
0^{++}}\left(4356\right)$ and $T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4389\right)$,
close to $4350$ MeV. And the partial widths of $\Gamma_{T_{cs\bar{c}\bar{s}\
0^{++}}\left(4356\right)\rightarrow J/\psi\phi}$ and
$\Gamma_{T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4389\right)\rightarrow J/\psi\phi}$
are predicted to be $3.38$ and $0.80$ MeV, respectively. It implies a
possibility that the main component of the state $X\left(4350\right)$ may be
the state $T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4356\right)$. In Refs. Li  _et
al._ (2023); Yang and Ping (2019); Deng  _et al._ (2020); Wu  _et al._ (2016),
the $X\left(4350\right)$ was explained as an $cs\bar{c}\bar{s}$ tetraquark
state with $J^{PC}=0^{++}$ or $2^{++}$. However, the state cannot be
interpreted as a $cs\bar{c}\bar{s}$ tetraquark with either $0^{++}$ and
$2^{++}$ in the QCD sum rules Chen  _et al._ (2017a). The $X\left(4350\right)$
can also be accommodated by the charmonium spectrum within an unquenched quark
model including coupled-channel effects Deng  _et al._ (2023).

The highest mass state $T_{cs\bar{c}\bar{s}\ 0^{++}}(4526)$ may be a candidate
of $X\left(4500\right)$. The partial width $\Gamma_{T_{cs\bar{c}\bar{s}\
0^{++}}\left(4526\right)\rightarrow J/\psi\phi}$ is predicted to be $\sim 7.$
MeV, which is about 10% of the measured total width $\Gamma_{exp}\simeq
92_{-21}^{+30}$ MeV of $X\left(4500\right)$. The predicted masses for the
$0^{++}$ states are much lower than that of the state $X\left(4700\right)$. In
this way, it is difficult to accommodate the $X\left(4700\right)$ as $1S$
states for $T_{cs\bar{c}\bar{s}}$ with $J^{PC}=0^{++}$ in the NRPQM. However,
we cannot rule out the possibility of interpreting the state
$X\left(4700\right)$ as excited state for $cs\bar{c}\bar{s}$ system. And
because of their high masses, the $X\left(4500\right)$ and
$X\left(4700\right)$ may be interpreted as the orbitally or radially excited
tetraquark or molecular states Chen  _et al._ (2017b); Maiani  _et al._
(2016); Lü and Dong (2016); Deng  _et al._ (2020); Zhu (2016); Wu  _et al._
(2016). In addition to the $cs\bar{c}\bar{s}$ explanation,
$X\left(4500\right)$ and $X\left(4700\right)$ can also be accommodated by the
charmonium spectrum Deng  _et al._ (2023); Ortega  _et al._ (2016).

In brief, the $X_{0}\left(4140\right)$, $X\left(4350\right)$, and
$X\left(4500\right)$ may be interpreted as the $1S$ states for the
$T_{cs\bar{c}\bar{s}}$ system with $J^{PC}=0^{++}$. The second highest
$0^{++}$ state has mass $4424$ MeV. Its partial widths into $\eta_{c}\eta$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ channels are predicted to be
$\Gamma_{T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4424\right)\to\eta_{c}\eta}\simeq
4.8$ MeV and $\Gamma_{T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4424\right)\to
D_{s}^{*}\bar{D_{s}^{*}}}\simeq 8.6$ MeV, respectively. Experimental search
for $T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4424\right)$ in $\eta_{c}\eta$ or
$D_{s}^{*}\bar{D_{s}^{*}}$ is strongly recommended.

  * •

$J^{PC}=1^{++}$

In the $J^{PC}=1^{++}$ sector, there are two $1S$ states for the
$T_{cs\bar{c}\bar{s}}$ system. Including configuration mixing effects, the
physical masses are predicted to be in the range of $\left[4382,4448\right]$
MeV (see Table 7). Although their masses are much larger than the mass
threshold of $J/\psi\phi$, it shows that the two states have rather narrow
decay widths within the range of $\sim 0-1$ MeV. This feature appears in the
hidden-charm decays of the $T_{cs\bar{c}\bar{s}}$ system which can be
explained by the mismatching of the wave function overlap between the heavy-
heavy and light-light clusters.

In our calculation the lowest mass state with $J^{PC}=1^{++}$ is $4382$ MeV,
which is much higher than the two observed states $X\left(4140\right)$ and
$X\left(4270\right)$. This could be an indication that additional dynamics in
addition to the quark potential should be present. Similar to the situation in
the charmonium spectrum that the calculated mass of $\chi_{c1}(2P)$ in the
potential quark model is much higher than the observed $X(3872)$. A broadly
accepted explanation is that the strong coupling between $\chi_{c1}(2P)$ and
the $D\bar{D}^{*}+c.c.$ threshold may provide a source of the short-distance
dynamics for the $D\bar{D}^{*}+c.c.$ interaction, and the unitarization of the
$D\bar{D}^{*}+c.c.$ final state interaction will dynamically generate a pole,
and shift the bare pole mass (i.e. the mass of $\chi_{c1}(2P)$) to the
physical one near threshold (see e.g. the review of Ref. Guo  _et al._
(2018)).

In the studies of QCD sum rules Chen  _et al._ (2017b) and simple color-
magnetic interaction models Wu  _et al._ (2016) the states,
$X\left(4140\right)$ and $X\left(4274\right)$, are interpreted as $S$-wave
tetraquark states with $1^{++}$. In contrast, $X\left(4140\right)$ was the
only state explained as an $cs\bar{c}\bar{s}$ tetraquark state with
$J^{PC}=1^{++}$ in Refs. Li  _et al._ (2023); Wang (2021); Maiani  _et al._
(2016); Lü and Dong (2016), while in Refs. Liu  _et al._ (2021c); Deng  _et
al._ (2020); Yang and Ping (2019), $X\left(4270\right)$ was the only state
explained as an $cs\bar{c}\bar{s}$ tetraquark with $J^{PC}=1^{++}$. In Refs.
Deng  _et al._ (2023); Ortega  _et al._ (2016); Lü and Dong (2016), the
$X\left(4274\right)$ was assigned as a good candidate for the charmonium.

The predicted $1S$ states with $J^{PC}=1^{++}$ in the NRPQM are much lower
than that of $X\left(4685\right)$ observed by LHCb Aaij  _et al._ (2021).
However, we cannot rule out the possibility of $X\left(4685\right)$ and
$X\left(4630\right)$ being the excited states for the $cs\bar{c}\bar{s}$
system.

  * •

$J^{PC}=2^{++}$

In the $J^{PC}=2^{++}$ sector, two $1S$ states are predicted for the
$T_{cs\bar{c}\bar{s}}$ system with configuration mixing included. The physical
masses are predicted to be in the range of $\left[4389,4528\right]$ MeV (see
Table 7). Their masses are sufficient for decaying into $J/\psi\phi$ and
$D_{s}^{*}\bar{D_{s}^{*}}$, and as shown in Table 11 the partial decay widths
turn out to be rather small. It is interesting to see that the partial decay
width $\Gamma_{T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4389\right)\rightarrow
D_{s}^{*}\bar{D_{s}^{*}}}=6.0$ MeV is much larger than
$\Gamma_{T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4528\right)\rightarrow
D_{s}^{*}\bar{D_{s}^{*}}}=0.06$ MeV, although the mass of
$T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4389\right)$ is smaller.

Recall that the observed state $X\left(4350\right)$ in $\gamma\gamma\to
J/\psi\phi$ Shen  _et al._ (2010) may have $J^{PC}=0^{++}$ or $2^{++}$, and
its width is $13.3_{-9}^{+18}\pm 4$ MeV. It seems that it may be assigned as
either $T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4356\right)$ or
$T_{cs\bar{c}\bar{s}\ 2^{++}}\left(4389\right)$ within the error. As shown in
Table 11 that the partial decay width of $\Gamma_{T_{cs\bar{c}\bar{s}\
2^{++}}\left(4389\right)\to D_{s}^{*}\bar{D_{s}^{*}}}=6.0$ MeV is
significantly larger than that of $T_{cs\bar{c}\bar{s}\
0^{++}}\left(4356\right)$. This feature may be useful for identifying these
two states in experiment. At this moment the measured width of
$X\left(4350\right)$ still has large uncertainties. We anticipate that an
improved measurement of its total width may allow us to understand better its
internal structure.

  * •

$J^{PC}=1^{+-}$

In the $J^{PC}=1^{+-}$ sector, there are four $1S$ mixing states in the range
of $\left[4239,4529\right]$ MeV (see Table 7). Again, we see that their
rearrangement decay widths are generally small, although they have
sufficiently large phase spaces for their decays into various channels. In
Table 11 their partial decay widths into $J/\psi\eta$, $\eta_{c}\phi$,
$D_{s}\bar{D_{s}^{*}}/D_{s}^{*}\bar{D_{s}}$, or $D_{s}^{*}\bar{D_{s}^{*}}$
through quark rearrangements are listed. It is interesting to note that the
lightest state $T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4239\right)$ appears to have
the largest partial decay widths into $D_{s}^{*}\bar{D}_{s}^{*}$ and
$D_{s}^{*}\bar{D}_{s}+c.c.$, i.e. $\Gamma_{T_{1^{+-}}\left(4239\right)\to
D_{s}^{*}\bar{D}_{s}^{*}}\simeq 2.9$ MeV and
$\Gamma_{T_{1^{+-}}\left(4239\right)\to D_{s}^{*}\bar{D_{s}}+c.c.}\simeq 12$
MeV, respectively. These large partial decays widths indicate large couplings
for $T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4239\right)$ to the
$D_{s}^{*}\bar{D}_{s}^{*}$ and $D_{s}^{*}\bar{D}_{s}+c.c.$ channels. Taking
into account the closeness of $T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4239\right)$
to the $D_{s}^{*}\bar{D}_{s}^{*}$ threshold, it may imply that
$T_{cs\bar{c}\bar{s}\ 1^{+-}}\left(4239\right)$ could be a good candidate
combing both tetraquark and hadronic molecule dynamics. The tetraquark
configuration can provide a source of the short-distance dynamics while the
unitarized interactions between $D_{s}^{*}\bar{D}_{s}^{*}$ may lead to pole
structures near the $D_{s}^{*}\bar{D}_{s}^{*}$ threshold. Experimental search
for this state in both $D_{s}^{*}\bar{D}_{s}^{*}$ and
$D_{s}^{*}\bar{D}_{s}+c.c.$ channels is strongly recommended.

####  III.5.2 $T_{cc\bar{s}\bar{s}}$

Table 13: The predicted decay widths $\Gamma$ (MeV) of the rearrangement decay processes of the ground $cc\bar{s}\bar{s}$ system.  State | $\Gamma_{T\rightarrow D_{s}\bar{D}_{s}}$ | $\Gamma_{T\rightarrow D_{s}^{*}\bar{D}_{s}^{*}}$ | $\Gamma_{T\rightarrow\bar{D}_{s}D_{s}^{*}+c.c.}$ | $\Gamma_{sum}$
---|---|---|---|---
$T_{0^{+}}\left(4386\right)$ | 0.56 | 0.05 | ✘ | 0.61
$T_{0^{+}}\left(4510\right)$ | 2.85 | 3.55 | ✘ | 6.40
$T_{1^{+}}\left(4449\right)$ | ✘ | ✘ | 0.46 | 0.46
$T_{2^{+}}\left(4481\right)$ | ✘ | 13.85 | ✘ | 13.85

  * •

$J^{P}=0^{+}$

In the $J^{P}=0^{+}$ sector, there are two $1S$ mixing states with
$J^{P}=0^{+}$ in the range of $\left[4386,4510\right]$ MeV (see Table 8). It
shows that their masses are much larger than the mass threshold of
$D_{s}^{*}D_{s}^{*}$. Thus, they may easily decay into $D_{s}\bar{D}_{s}$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ in an $S$ wave. As listed in Table 13, its partial
decay widths for the $D_{s}\bar{D}_{s}$ and $D_{s}^{*}\bar{D}_{s}^{*}$
channels are rather small, which are just 0.56 and 0.05 MeV, respectively. In
contrast, $T_{cc\bar{s}\bar{s}\ 2^{++}}\left(4510\right)$ decays into
$D_{s}\bar{D}_{s}$ with a width of 2.9 MeV, and a significantly large partial
decay width of 3.6 MeV into the $D_{s}^{*}D_{s}^{*}$ channel. This indicates a
strong coupling for $T_{cc\bar{s}\bar{s}\ 2^{++}}\left(4510\right)$ to the
$D_{s}^{*}\bar{D}_{s}^{*}$ channel. Experimental search for these two states
in the $D_{s}\bar{D}_{s}$ and $D_{s}^{*}\bar{D}_{s}^{*}$ channels seems to be
promising.

  * •

$J^{P}=1^{+}$

In the $J^{P}=1^{+}$ sector, there is only one $1S$ state with $J^{P}=1^{+}$
and a mass of $4449$ MeV (see Table 8). Its mass is much larger than the mass
threshold of $\bar{D}_{s}D_{s}^{*}+c.c.$. However, the calculation shows that
its partial decay width $\Gamma_{T_{cc\bar{s}\bar{s}\
1^{+}}\left(4449\right)\rightarrow D_{s}D_{s}^{*}+c.c.}\simeq 0.46$ MeV is a
rather small value (see Table 13). While it may be difficult to observe this
state in experiment based on the present experimental statistics, a search of
its existence may provide a peculiar evidence for the exotic object as a
genuine tetraquark.

  * •

$J^{P}=2^{+}$

In the $J^{P}=2^{+}$ sector, there is only one $1S$ state with $J^{P}=2^{+}$
and a mass of $4481$ MeV (see Table 8). Its mass is much larger than the mass
threshold of $D_{s}^{*}\bar{D}_{s}^{*}$. The calculated partial decay width
$\Gamma_{T_{cc\bar{s}\bar{s}\ 2^{+}}\left(4481\right)\to
D_{s}^{*}\bar{D}_{s}^{*}}\simeq 14$ MeV is listed Table 13. Our calculation
suggests that $T_{cc\bar{s}\bar{s}\ 2^{+}}(4481)$ is also a narrow state. Its
sizeable coupling to the $D_{s}^{*}\bar{D}_{s}^{*}$ channel implies that it
has a good chance to be observed in the $D_{s}^{*}\bar{D}_{s}^{*}$ decay
channel.

##  IV summary

In this work we present a systematic study of the $1S$-wave hidden and double
charm-strange tetraquarks $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ in the
NRPQM. The explicitly correlated Gaussian method Mitroy  _et al._ (2013);
Varga and Suzuki (1995) is adopted to solve the four-body problem numerically,
and the mass spectra, color-spin configurations and possible decay modes are
obtained. It is worth noting that for such four-body systems, the $1S$-wave
spectra have become rich enough for producing many interesting phenomena. We
find that although these states are all above their open flavor thresholds
their rearrangement decay widths are rather narrow which can be understood by
the mismatching of the wave functions between the initial and final states. It
implies that tetraquarks of $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ may have
a better chance to exist as genuine tetraquark states. Meanwhile, by
extracting the color-spin configurations of the $cs\bar{c}\bar{s}$ and
$cc\bar{s}\bar{s}$ systems we find that for a physical state of
$cs\bar{c}\bar{s}$ its color configurations can be dominated by either the
$|11\rangle_{c}$ or $|88\rangle_{c}$ ones. It suggests that some hidden charm-
strange tetraquark states may strongly couple to two color-singlet hadrons if
the kinematics and dynamics allow. In contrast, we find that the color
configurations $|11\rangle_{c}$ and $|88\rangle_{c}$ in a double charm-strange
$cc\bar{s}\bar{s}$ state can be compatible. It may suggest that an overall
color-singlet tetraquark (i.e. $|88\rangle_{c}$ configuration) should always
play a role in the $T_{cc\bar{s}\bar{s}}$ states.

We also make an analysis of the calculated spectra taking into account some
experimental candidates. For the hidden charm-strange system we find that
$X_{0}\left(4140\right)$, $X\left(4350\right)$, and $X\left(4500\right)$ can
be interpreted as the $1S$ $T_{cs\bar{c}\bar{s}}$ states with $J^{PC}=0^{++}$.
While there are four $J^{PC}=0^{++}$ states are expected, the experimental
search for the predicted $T_{cs\bar{c}\bar{s}\ 0^{++}}\left(4424\right)$ in
$\eta_{c}\eta$ or $D_{s}^{*}\bar{D_{s}^{*}}$ is strongly recommended. Our
predictions for the $J^{PC}=1^{++}$ states seem to have higher masses than the
experimental candidates. This is similar to the situation for $X(3872)$ where
the potential quark model predicts a higher mass of the $\chi_{c1}(2P)$ state
than the experimental observation. This may provide an analogue in the hidden
charm-strange sector. Further detailed investigation of the couplings of these
$1^{++}$ states to the $D_{s}\bar{D}_{s}^{*}+c.c.$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ channels will be needed. There still lack of
experimental evidences for the $J^{PC}=1^{+-}$ states in the hidden charm-
strange sector and all the states in the double charm-strange sector. Our
calculations shows that these states mostly turn out to be narrow. Thus, the
experimental search for their signals at LHCb in some preferred decay channels
are strongly recommended.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grants No.12235018, No.12175065, No.E411645Z10). Q.Z. is also supported in
part, by the DFG and NSFC funds to the Sino-German CRC 110 “Symmetries and the
Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG Project-ID
196253076), National Key Basic Research Program of China under Contract No.
2020YFA0406300, and Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB34030302).

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