# Fully-strange tetraquark $ss\bar{s}\bar{s}$ spectrum and possible
experimental evidence

Feng-Xiao Liu1,4, Ming-Sheng Liu1,4, Xian-Hui Zhong1,4 111E-mail:
<EMAIL_ADDRESS>Qiang Zhao2,3,4 222E-mail<EMAIL_ADDRESS>1)
Department of Physics, Hunan Normal University, and Key Laboratory of Low-
Dimensional Quantum Structures and Quantum Control of Ministry of Education,
Changsha 410081, China  2) Institute of High Energy Physics, Chinese Academy
of Sciences, Beijing 100049, 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

In this work we construct 36 tetraquark configurations for the $1S$-, $1P$-,
and $2S$-wave states, and make a prediction of the mass spectrum for the
tetraquark $ss\bar{s}\bar{s}$ system in the framework of a nonrelativistic
potential quark model without the diquark-antidiquark approximation. The model
parameters are well determined by our previous study of the strangeonium
spectrum. We find that the resonances $f_{0}(2200)$ and $f_{2}(2340)$ may
favor the assignments of ground states $T_{(ss\bar{s}\bar{s})0^{++}}(2218)$
and $T_{(ss\bar{s}\bar{s})2^{++}}(2378)$, respectively, and the newly observed
$X(2500)$ at BESIII may be a candidate of the lowest mass $1P$-wave $0^{-+}$
state $T_{(ss\bar{s}\bar{s})0^{-+}}(2481)$. Signals for the other $0^{++}$
ground state $T_{(ss\bar{s}\bar{s})0^{++}}(2440)$ may also have been observed
in the $\phi\phi$ invariant mass spectrum in $J/\psi\to\gamma\phi\phi$ at
BESIII. The masses of the $J^{PC}=1^{--}$ $T_{ss\bar{s}\bar{s}}$ states are
predicted to be in the range of $\sim 2.44-2.99$ GeV, which indicates that the
$\phi(2170)$ resonance may not be a good candidate of the
$T_{ss\bar{s}\bar{s}}$ state. This study may provide a useful guidance for
searching for the $T_{ss\bar{s}\bar{s}}$ states in experiments.

##  I Introduction

From the Review of Particle Physics (RPP) of Particle Data Group
Tanabashi:2018oca , above the mass range of $2.0$ GeV one can see that there
are several unflavored $q\bar{q}$ isoscaler states, such as $f_{0}(2200)$,
$f_{2}(2150)$, $f_{2}(2300)$, $f_{2}(2340)$ etc., dominantly decaying into
$\phi\phi$, $\eta\eta$, and/or $K\bar{K}$ final states. The decay modes
indicate that these states might be good candidates for conventional
$s\bar{s}$ meson resonances. Recently, we carried out a systematical study of
the mass spectrum and strong decay properties of the $s\bar{s}$ system in Ref.
Li:2020xzs . It shows that these states cannot be easily accommodated by the
conventional $s\bar{s}$ meson spectrum. While they may be candidates for
tetraquark $ss\bar{s}\bar{s}$ ($T_{ss\bar{s}\bar{s}}$) states, it is easy to
understand that they can fall apart into $\phi\phi$ and $\eta\eta$ final
states through quark rearrangements, or easily decay into $K\bar{K}$ final
states through a pair of $s\bar{s}$ annihilations and then a pair of light
quark creations. The mass analysis with the relativistic quark model in Ref.
Ebert:2008id  supports the $f_{0}(2200)$, and $f_{2}(2340)$ to be assigned as
the $T_{ss\bar{s}\bar{s}}$ ground states with $0^{++}$ and $2^{++}$,
respectively. However, a relativized quark model calculation Lu:2019ira  only
favors $f_{2}(2300)$ to be a $T_{ss\bar{s}\bar{s}}$ state.

Some other candidates of the $T_{ss\bar{s}\bar{s}}$ states from experiment are
also suggested in the literature. For example the vector meson resonance
$\phi(2170)$ listed in RPP Tanabashi:2018oca  is suggested to be a $1^{--}$
$T_{ss\bar{s}\bar{s}}$ state based on the mass analysis of QCD sum rules
tetra1 ; tetra2 ; tetra3 ; tetra5 ; tetra7 , and flux-tube model tetra6 . The
newly observed $X(2239)$ resonance in the $e^{+}e^{-}\to K^{+}K^{-}$ process
at BESIII Ablikim:2018iyx  is suggested to be a candidate of the lowest
$1^{--}$ $T_{ss\bar{s}\bar{s}}$ state in a relativized quark model Lu:2019ira
. Moreover, the newly observed resonances $X(2500)$ observed in
$J/\psi\to\gamma\phi\phi$ Ablikim:2016hlu  and $X(2060)$ observed in
$J/\psi\to\phi\eta\eta^{\prime}$ Ablikim:2018xuz  at BESIII are suggested to
be $0^{-+}$ and $1^{+-}$ $T_{ss\bar{s}\bar{s}}$ states, respectively,
according to the QCD sum rule studies Dong:2020okt ; Cui:2019roq . The
assignment of $X(2500)$ is consistent with that in Ref. Lu:2019ira .

With the recent experimental progresses more quantitative studies on the
$T_{ss\bar{s}\bar{s}}$ states can be carried out and their evidences can also
be searched for in experiments. Very recently, the LHCb Collaboration reported
their results on the observations of tetraquark $cc\bar{c}\bar{c}$
($T_{cc\bar{c}\bar{c}}$) states Aaij:2020fnh . A broad structure above the
$J/\psi J/\psi$ threshold ranging from 6.2 to 6.8 GeV and a narrower structure
$T_{cc\bar{c}\bar{c}}(6900)$ are observed with more than 5 $\sigma$ of
significance level. There are also some vague structure around 7.2 GeV to be
confirmed. These observations could be evidences for genuine
$T_{cc\bar{c}\bar{c}}$ states Lu:2020cns ; Chen:2020xwe ; Jin:2020jfc ;
Wang:2020ols ; liu:2020eha .

The observations of the $T_{cc\bar{c}\bar{c}}$ states above the $J/\psi
J/\psi$ threshold at LHCb may provide an important clue for the underlying
dynamics for the ${cc\bar{c}\bar{c}}$ system. In particular, the narrowness of
$T_{cc\bar{c}\bar{c}}(6900)$ suggests that the there should be more profound
mechanism that “slows down” the fall-apart decays of such a tetraquark system.
Although this may be related to the properties of the static potential of
heavy quark systems, more direct evidences are still needed to disentangle the
dynamical features between the heavy and light flavor systems. As an analogy
of the $T_{cc\bar{c}\bar{c}}$ system, there might exist stable
$T_{ss\bar{s}\bar{s}}$ states above the $\phi\phi$ threshold, and can likely
be observed in the di-$\phi$ mass spectrum. On the other hand, flavor mixings
could be important for the light flavor systems and pure ${ss\bar{s}\bar{s}}$
states may not exist. To answer such questions, systematic calculations of the
${ss\bar{s}\bar{s}}$ system should be carried out. The BESIII experiments can
provide a large data sample for the search of the $T_{ss\bar{s}\bar{s}}$
states in $J/\psi$ and $\psi(2S)$ decays. In theory, although there have been
some predictions of the $T_{ss\bar{s}\bar{s}}$ spectrum within the quark model
Ebert:2008id ; Lu:2019ira ; tetra6  and QCD sum rules Lu:2020cns ;
Chen:2020xwe ; Jin:2020jfc ; Wang:2020ols , most of the studies focus on some
special states in a diquark-antidiquark picture. About the status of the
tetraquark states, some recent review works can be referenced Liu:2019zoy ;
Agaev:2020zad . In this study we intend to provide a systematical calculation
of the mass spectrum of the $1S$, $1P$ and $2S$-wave $T_{ss\bar{s}\bar{s}}$
states without the diquark-antidiquark approximation in a nonrelativistic
potential quark model (NRPQM).

The NRPQM is based on the Hamiltonian proposed by the Cornell model
Eichten:1978tg , which contains a linear confinement and a one-gluon-exchange
(OGE) potential for quark-quark and quark-antiquark interactions. With the
NRPQM, we have successfully described the $s\bar{s}$, $c\bar{c}$, and
$b\bar{b}$ meson spectra Li:2020xzs ; Deng:2016stx ; Deng:2016ktl , and $sss$,
$ccc$ and $bbb$ baryon spectra Liu:2019vtx ; Liu:2019wdr . Furthermore, we
adopted the NRPQM for the study of both $1S$ and $1P$-wave all-heavy
tetraquark states with a Gaussian expansion method Liu:2019zuc ; liu:2020eha .
In this work we continue to extend this method to study the
$T_{ss\bar{s}\bar{s}}$ spectrum by constructing the full tetraquark
configurations without the diquark-antidiquark approximation. With the
parameters determined in our study of the $s\bar{s}$ spectrum Li:2020xzs , we
obtain a relatively reliable prediction of the mass spectrum for 36
$T_{ss\bar{s}\bar{s}}$ states, i.e., 4 $1S$-wave ground states, 20 $1P$-wave
orbital excitations, and 12 $2S$-wave radial excitations.

The paper is organized as follows: a brief introduction to the tetraquark
spectrum is given in Sec. II. In Sec. III, the numerical results and
discussions are presented. A short summary is given in Sec. IV.

##  II MASS SPECTRUM

###  II.1 Hamiltonian

We adopt a NRPQM to calculate the mass spectrum of the $ss\bar{s}\bar{s}$
system. In this model the Hamiltonian is given by

$H=(\sum_{i=1}^{4}m_{i}+T_{i})-T_{G}+\sum_{i<j}V_{ij}(r_{ij}),$ (1)

where $m_{i}$ and $T_{i}$ stand for the constituent quark mass and kinetic
energy of the $i$th quark, respectively; $T_{G}$ stands for the center-of-mass
(c.m.) kinetic energy of the tetraquark system;
$r_{ij}\equiv|\mathbf{r}_{i}-\mathbf{r}_{j}|$ is the distance between the
$i$th and $j$th quark; and $V_{ij}(r_{ij})$ stands for the effective potential
between them. In this work the $V_{ij}(r_{ij})$ adopts a widely used form
Eichten:1978tg ; Godfrey:1985xj ; Swanson:2005 ; Godfrey:2015dia ;
Godfrey:2004ya ; Lakhina:2006fy ; Lu:2016bbk ; Li:2010vx ; Deng:2016stx ;
Deng:2016ktl :

$V_{ij}(r_{ij})=V_{ij}^{conf}(r_{ij})+V_{ij}^{sd}(r_{ij}),$ (2)

where the confinement potential adopts the standard form of the Cornell
potential Eichten:1978tg , which includes the spin-independent linear
confinement potential $V_{ij}^{Lin}(r_{ij})\propto r_{ij}$ and Coulomb-like
potential $V_{ij}^{Coul}(r_{ij})\propto 1/r_{ij}$:

$V_{ij}^{conf}(r_{ij})=-\frac{3}{16}({\mbox{\boldmath$\lambda$\unboldmath}}_{i}\cdot{\mbox{\boldmath$\lambda$\unboldmath}}_{j})\left(b_{ij}r_{ij}-\frac{4}{3}\frac{\alpha_{ij}}{r_{ij}}+C_{0}\right).$
(3)

The constant $C_{0}$ stands for the zero point energy. While the spin-
dependent potential $V_{ij}^{sd}(r_{ij})$ is the sum of the spin-spin contact
hyperfine potential $V_{ij}^{SS}$, the spin-orbit potential $V_{ij}^{SS}$, and
the tensor term $V_{ij}^{T}$:

$V^{sd}_{ij}(r_{ij})=V^{SS}_{ij}+V^{T}_{ij}+V^{LS}_{ij}\ ,$ (4)

with

$V^{SS}_{ij}=-\frac{\alpha_{ij}}{4}({\mbox{\boldmath$\lambda$\unboldmath}}_{i}\cdot{\mbox{\boldmath$\lambda$\unboldmath}}_{j})\left\\{\frac{\pi}{2}\cdot\frac{\sigma^{3}_{ij}e^{-\sigma^{2}_{ij}r_{ij}^{2}}}{\pi^{3/2}}\cdot\frac{16}{3m_{i}m_{j}}(\mathbf{S}_{i}\cdot\mathbf{S}_{j})\right\\},$
(5)
$\begin{split}V^{LS}_{ij}=&-\frac{\alpha_{ij}}{16}\frac{{\mbox{\boldmath$\lambda$\unboldmath}}_{i}\cdot{\mbox{\boldmath$\lambda$\unboldmath}}_{j}}{r_{ij}^{3}}\bigg{(}\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4}{m_{i}m_{j}}\bigg{)}\bigg{\\{}\mathbf{L}_{ij}\cdot(\mathbf{S}_{i}+\mathbf{S}_{j})\bigg{\\}}\\\
&-\frac{\alpha_{ij}}{16}\frac{{\mbox{\boldmath$\lambda$\unboldmath}}_{i}\cdot{\mbox{\boldmath$\lambda$\unboldmath}}_{j}}{r_{ij}^{3}}\bigg{(}\frac{1}{m_{i}^{2}}-\frac{1}{m_{j}^{2}}\bigg{)}\bigg{\\{}\mathbf{L}_{ij}\cdot(\mathbf{S}_{i}-\mathbf{S}_{j})\bigg{\\}},\end{split}$
(6)
$V^{T}_{ij}=-\frac{\alpha_{ij}}{4}({\mbox{\boldmath$\lambda$\unboldmath}}_{i}\cdot{\mbox{\boldmath$\lambda$\unboldmath}}_{j})\cdot\frac{1}{m_{i}m_{j}r_{ij}^{3}}\Bigg{\\{}\frac{3(\mathbf{S}_{i}\cdot\mathbf{r}_{ij})(\mathbf{S}_{j}\cdot\mathbf{r}_{ij})}{r_{ij}^{2}}-\mathbf{S}_{i}\cdot\mathbf{S}_{j}\Bigg{\\}}.$
(7)

In the above equations, $\mathbf{S}_{i}$ stands for the spin of the $i$th
quark, and $\mathbf{L}_{ij}$ stands for the relative orbital angular momentum
between the $i$th and $j$th quark. If the interaction occurs between two
quarks or antiquarks, the
$\mbox{\boldmath$\lambda$\unboldmath}_{i}\cdot\mbox{\boldmath$\lambda$\unboldmath}_{j}$
operator is defined as
$\mbox{\boldmath$\lambda$\unboldmath}_{i}\cdot\mbox{\boldmath$\lambda$\unboldmath}_{j}\equiv\sum_{a=1}^{8}\lambda_{i}^{a}\lambda_{j}^{a}$,
while if the interaction occurs between a quark and an antiquark, the
$\mbox{\boldmath$\lambda$\unboldmath}_{i}\cdot\mbox{\boldmath$\lambda$\unboldmath}_{j}$
operator is defined as
$\mbox{\boldmath$\lambda$\unboldmath}_{i}\cdot\mbox{\boldmath$\lambda$\unboldmath}_{j}\equiv\sum_{a=1}^{8}-\lambda_{i}^{a}\lambda_{j}^{a*}$,
where $\lambda^{a*}$ is the complex conjugate of the Gell-Mann matrix
$\lambda^{a}$. The parameters $b_{ij}$ and $\alpha_{ij}$ denote the strength
of the confinement and strong coupling of the one-gluon-exchange potential,
respectively.

The five parameters $m_{s}$, ${\alpha_{ss}}$, ${\sigma_{ss}}$, $b_{ss}$, and
$C_{0}$ have been determined by fitting the mass spectrum of the strangeonium
in our previous work Li:2020xzs . The quark model parameters adopted in this
work are collected in the Table 1.

###  II.2 Configurations classified in the quark model

To calculate the spectroscopy of a $qq\bar{q}\bar{q}$ ($q\in\\{s,c,b\\}$)
system, first we construct the configurations in the product space of flavor,
color, spin, and spatial parts.

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

$\begin{split}|6\bar{6}\rangle_{c}=&\frac{1}{2\sqrt{6}}\bigg{[}(rb+br)(\bar{b\mathstrut}\bar{r\mathstrut}+\bar{r\mathstrut}\bar{b\mathstrut})+(gr+rg)(\bar{g\mathstrut}\bar{r\mathstrut}+\bar{r\mathstrut}\bar{g\mathstrut})\\\
&+(gb+bg)(\bar{b\mathstrut}\bar{g\mathstrut}+\bar{g\mathstrut}\bar{b\mathstrut})\\\
&+2(rr)(\bar{r}\bar{r})+2(gg)(\bar{g}\bar{g})+2(bb)(\bar{b}\bar{b})\bigg{]},\end{split}$
(8) $\begin{split}|\bar{3}3\rangle_{c}=&\frac{1}{2\sqrt{3}}\bigg{[}(br-
rb)(\bar{b\mathstrut}\bar{r\mathstrut}-\bar{r\mathstrut}\bar{b\mathstrut})-(rg-
gr)(\bar{g\mathstrut}\bar{r\mathstrut}-\bar{r\mathstrut}\bar{g\mathstrut})\\\
&+(bg-
gb)(\bar{b\mathstrut}\bar{g\mathstrut}-\bar{g\mathstrut}\bar{b\mathstrut})\bigg{]}.\end{split}$
(9)

Table 1:  Quark model parameters used in this work. $m_{s}$ (GeV) |  0.60
---|---
${\alpha_{ss}}$ |  0.77
${\sigma_{ss}}$ (GeV) |  0.60
${b}$ (GeV 2) |  0.135
${C_{0}}$ (GeV) |  $-0.519$

In the spin space, there are six spin bases, which are denoted by
$\chi_{S}^{S_{12}S_{34}}$. Where $S_{12}$ stands for the spin quantum numbers
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. The spin wave
functions $\chi_{SS_{z}}^{S_{12}S_{34}}$ with a determined $S_{z}$ ($S_{z}$
stands for the third component of the total spin S) can be explicitly
expressed as follows:

$\displaystyle\chi_{00}^{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\uparrow\downarrow\uparrow\downarrow-\uparrow\downarrow\downarrow\uparrow-\downarrow\uparrow\uparrow\downarrow+\downarrow\uparrow\downarrow\uparrow),$
(10) $\displaystyle\chi_{00}^{11}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{12}}(2\uparrow\uparrow\downarrow\downarrow-\uparrow\downarrow\uparrow\downarrow-\uparrow\downarrow\downarrow\uparrow$
(11)
$\displaystyle-\downarrow\uparrow\uparrow\downarrow-\downarrow\uparrow\downarrow\uparrow+2\downarrow\downarrow\uparrow\uparrow),$
$\displaystyle\chi_{11}^{01}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\uparrow\downarrow\uparrow\uparrow-\downarrow\uparrow\uparrow\uparrow),$
(12) $\displaystyle\chi_{11}^{10}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\uparrow\uparrow\uparrow\downarrow-\uparrow\uparrow\downarrow\uparrow),$
(13) $\displaystyle\chi_{11}^{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\uparrow\uparrow\uparrow\downarrow+\uparrow\uparrow\downarrow\uparrow-\uparrow\downarrow\uparrow\uparrow-\downarrow\uparrow\uparrow\uparrow),$
(14) $\displaystyle\chi_{22}^{11}$ $\displaystyle=$
$\displaystyle\uparrow\uparrow\uparrow\uparrow.$ (15)

In the spatial space, we define the relative Jacobi coordinates with the
single-partial coordinates $\mathbf{r_{i}}$ ($i=1,2,3,4$):

$\displaystyle\mbox{\boldmath$\xi$\unboldmath}_{1}$ $\displaystyle\equiv$
$\displaystyle\mathbf{r_{1}}-\mathbf{r_{2}},$ (16)
$\displaystyle\mbox{\boldmath$\xi$\unboldmath}_{2}$ $\displaystyle\equiv$
$\displaystyle\mathbf{r_{3}}-\mathbf{r_{4}},$ (17)
$\displaystyle\mbox{\boldmath$\xi$\unboldmath}_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{m_{1}\mathbf{r_{1}}+m_{2}\mathbf{r_{2}}}{m_{1}+m_{2}}-\frac{m_{3}\mathbf{r_{3}}+m_{4}\mathbf{r_{4}}}{m_{3}+m_{4}},$
(18) $\displaystyle\mathbf{R}$ $\displaystyle\equiv$
$\displaystyle\frac{m_{1}\mathbf{r_{1}}+m_{2}\mathbf{r_{2}}+m_{3}\mathbf{r_{3}}+m_{4}\mathbf{r_{4}}}{m_{1}+m_{2}+m_{3}+m_{4}}.$
(19)

Note that $\mbox{\boldmath$\xi$\unboldmath}_{1}$ and
$\mbox{\boldmath$\xi$\unboldmath}_{2}$ 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
$\mbox{\boldmath$\xi$\unboldmath}_{3}$ 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:2009kj .

In the Jacobi coordinate system, the spatial wave function
$\Psi_{NLM}(\mbox{\boldmath$\xi$\unboldmath}_{1},\mbox{\boldmath$\xi$\unboldmath}_{2},\mbox{\boldmath$\xi$\unboldmath}_{3},\mathbf{R})$
for a $qq\bar{q}\bar{q}$ system with principal quantum number $N$ and orbital
angular momentum quantum numbers $LM$ may be expressed as the linear
combination of
$\Phi(\mathbf{R})\psi_{\alpha_{1}}(\mbox{\boldmath$\xi$\unboldmath}_{1})\psi_{\alpha_{2}}(\mbox{\boldmath$\xi$\unboldmath}_{2})\psi_{\alpha_{3}}(\mbox{\boldmath$\xi$\unboldmath}_{3})$:

$\displaystyle\Psi_{NLM}(\mbox{\boldmath$\xi$\unboldmath}_{1},\mbox{\boldmath$\xi$\unboldmath}_{2},\mbox{\boldmath$\xi$\unboldmath}_{3},\mathbf{R})\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle=\sum_{\alpha_{1},\alpha_{2},\alpha_{3}}C_{\alpha_{1},\alpha_{2},\alpha_{3}}[\Phi(\mathbf{R})\psi_{\alpha_{1}}(\mbox{\boldmath$\xi$\unboldmath}_{1})\psi_{\alpha_{2}}(\mbox{\boldmath$\xi$\unboldmath}_{2})\psi_{\alpha_{3}}(\mbox{\boldmath$\xi$\unboldmath}_{3})]_{NLM},$
(20)

where $C_{\alpha_{1},\alpha_{2},\alpha_{3}}$ stands for the combination
coefficients, $\Phi(\mathbf{R})$ is the center-of-mass (c.m.) motion wave
function. In the quantum number set
$\alpha_{i}\equiv\\{n_{\xi_{i}},l_{\xi_{i}},m_{\xi_{i}}\\}$, $n_{\xi_{i}}$ is
the principal quantum number, $l_{\xi_{i}}$ is the angular momentum, and
$m_{\xi_{i}}$ is its third component projection. The wave functions
$\psi_{\alpha_{i}}(\mbox{\boldmath$\xi$\unboldmath}_{i})$, which account for
the relative motions, can be written as

$\psi_{\alpha_{i}}(\mbox{\boldmath$\xi$\unboldmath}_{i})=R_{n_{\xi_{i}}l_{\xi_{i}}}(\xi_{i})Y_{l_{\xi_{i}}m_{\xi_{i}}}(\hat{\mbox{\boldmath$\xi$\unboldmath}_{i}}),$
(21)

where $Y_{l_{\xi_{i}}m_{\xi_{i}}}(\hat{\mbox{\boldmath$\xi$\unboldmath}_{i}})$
is the spherical harmonic function, and $R_{n_{\xi_{i}}l_{\xi_{i}}}(\xi_{i})$
is the radial part. It is seen that for an excited state, there are three
spatial excitation modes corresponding to three independent internal wave
functions $\psi_{\alpha_{i}}(\mbox{\boldmath$\xi$\unboldmath}_{i})$
($i=1,2,3$), which are denoted as $\xi_{1}$, $\xi_{2}$, and $\xi_{3}$,
respectively, in the present work. One point should be emphasized that
considering the fact that the $ss\bar{s}\bar{s}$ system is composed of equal
mass constituent quarks and antiquarks, we adopt a single set of Jacobi
coordinates in this study as an approximation. In fact, the four-body wave
function describing a scalar $ss\bar{s}\bar{s}$ state contains a small
contribution of internal angular momentum. This contribution is neglected in
our calculations. To precisely treat an $N$-body system, one can involve
several different sets of Jacobi coordinates as those done in Refs.
Hiyama:2003cu ; Hiyama:2018ivm ; Meng:2020knc ; Hiyama:2005cf ; Hiyama:2018ukv
; Meng:2019fan ; or adopt a single set of Jacobi coordinates
$X=(\mbox{\boldmath$\xi$\unboldmath}_{i},\mbox{\boldmath$\xi$\unboldmath}_{2},...,\mbox{\boldmath$\xi$\unboldmath}_{N-1})$
with non diagonal Gaussians $e^{-XAX^{T}}$ as those done in Refs. Varga:1996zz
; Varga:1996jr ; Varga:1997xga ; Brink:1998as , where $A$ is a symmetric
matrix.

Taking into account the Pauli principle and color confinement for the four-
quark system $qq\bar{q}\bar{q}$, we have 4 configurations for $1S$-wave ground
states, 20 configurations for the $1P$-wave orbital excitations, and 12
configurations for the $2S$-wave radial excitations. The spin-parity quantum
numbers, notations, and wave functions for these configurations are presented
in Table 2. With the wave functions for all the configurations, the mass
matrix elements of the Hamiltonian can be worked out.

To work out the matrix elements in the coordinate space, we expand the radial
part $R_{n_{\xi_{i}}l_{\xi_{i}}}(\xi_{i})$ with a series of harmonic
oscillator functions Liu:2019vtx ; liu:2020eha :

$R_{n_{\xi_{i}}l_{\xi_{i}}}(\xi_{i})=\sum_{\ell=1}^{n}\mathcal{C}_{\xi_{i}\ell}~{}\phi_{n_{\xi_{i}}l_{\xi_{i}}}(\omega_{\xi_{i}\ell},\xi_{i}),$
(22)

with

$\displaystyle\phi_{n_{\xi_{i}}l_{\xi_{i}}}(\omega_{\xi_{i}\ell},\xi_{i})$
$\displaystyle=\left(\mu_{\xi_{i}}\omega_{\xi_{i}\ell}\right)^{\frac{3}{4}}\Bigg{[}\frac{2^{l_{\xi}+2-n_{\xi}}(2l_{\xi_{i}}+2n_{\xi_{i}}+1)!!}{\sqrt{\pi}n_{\xi_{i}}![(2l_{\xi_{i}}+1)!!]^{2}}\Bigg{]}^{\frac{1}{2}}\left(\sqrt{\mu_{\xi_{i}}\omega_{{\xi_{i}}\ell}}\xi_{i}\right)^{l_{\xi_{i}}}$
(23) $\displaystyle\times
e^{-\frac{1}{2}\mu_{\xi_{i}}\omega_{\xi_{i}\ell}{\xi_{i}}^{2}}F\left(-n_{\xi_{i}},l_{\xi_{i}}+\frac{3}{2},\mu_{\xi_{i}}\omega_{\xi_{i}\ell}{\xi_{i}}^{2}\right),$

where
$F\left(-n_{\xi_{i}},l_{\xi_{i}}+\frac{3}{2},\mu_{\xi_{i}}\omega_{\xi_{i}\ell}{\xi_{i}}^{2}\right)$
is the confluent hypergeometric function. It should be pointed out that if
there are no radial excitations, the expansion method with harmonic oscillator
wave functions are just the same as the Gaussian expansion method adopted in
the literature Hiyama:2003cu ; Hiyama:2018ivm .

For an $ss\bar{s}\bar{s}$ system, if we ensure that the spatial wave function
with Jacobi coordinates can transform into the single particle coordinate
system, the harmonic oscillator frequencies $\omega_{\xi_{i}\ell}$ ($i=1,2,3$)
can be related to the harmonic oscillator stiffness factor $K_{\ell}$ with
$\omega_{\xi_{1}\ell}=\sqrt{2K_{\ell}/\mu_{\xi_{1}}}$,
$\omega_{\xi_{2}\ell}=\sqrt{2K_{\ell}/\mu_{\xi_{2}}}$, and
$\omega_{\xi_{3}\ell}=\sqrt{4K_{\ell}/\mu_{\xi_{3}}}$. Considering the reduced
masses $\mu_{\xi_{1}}=\mu_{\xi_{2}}=m_{s}/2$, $\mu_{\xi_{3}}=m_{s}$ for
$T_{(ss\bar{s}\bar{s})}$, one has
$\omega_{\xi_{1}\ell}=\omega_{\xi_{2}\ell}=\omega_{\xi_{3}\ell}=\sqrt{4K_{\ell}/m_{s}}$.
It indicates that the harmonic oscillator frequencies $\omega_{\xi_{i}\ell}$
for $T_{(ss\bar{s}\bar{s})}$ are not independent. According to the relation
$\omega_{\xi_{1}\ell}=\omega_{\xi_{2}\ell}=\omega_{\xi_{3}\ell}=\omega_{\ell}$,
the expansion of $\prod_{i=1}^{3}R_{n_{\xi_{i}}l_{\xi_{i}}}({\xi_{i}})$ can be
simplified as

$\begin{split}\prod_{i=1}^{3}R_{n_{\xi_{i}}l_{\xi_{i}}}({\xi_{i}})=&\sum_{\ell}^{n}\sum_{\ell^{\prime}}^{n}\sum_{\ell^{\prime\prime}}^{n}\mathcal{C}_{{\xi_{1}}\ell}\mathcal{C}_{{\xi_{2}}\ell^{\prime}}\mathcal{C}_{{\xi_{3}}\ell^{\prime\prime}}~{}\phi_{n_{\xi_{1}}l_{\xi_{1}}}(\omega_{{\xi_{1}}\ell},{\xi_{1}})~{}\phi_{n_{\xi_{2}}l_{\xi_{2}}}(\omega_{{\xi_{2}}\ell^{\prime}},{\xi_{2}})~{}\phi_{n_{\xi_{3}}l_{\xi_{3}}}(\omega_{{\xi_{3}}\ell^{\prime\prime}},{\xi_{3}})~{}\delta_{\ell\ell^{\prime}}\delta_{\ell\ell^{\prime\prime}}\\\
=&\sum_{\ell}^{n}\mathcal{C}_{\ell}~{}\phi_{n_{\xi_{1}}l_{\xi_{1}}}(\omega_{\ell},{\xi_{1}})~{}\phi_{n_{\xi_{2}}l_{\xi_{2}}}(\omega_{\ell},{\xi_{2}})~{}\phi_{n_{\xi_{3}}l_{\xi_{3}}}(\omega_{\ell},{\xi_{3}}).\end{split}$
(24)

Then we introduce oscillator length parameters $d_{\ell}$ that can be related
to the harmonic oscillator frequencies $\omega_{\ell}$ with
$1/d^{2}_{\ell}=m_{s}\omega_{\ell}$. Following the method of Refs.
Hiyama:2003cu ; Hiyama:2018ivm , we let the $d_{\ell}$ parameters form a
geometric progression

$d_{\ell}=d_{1}a^{\ell-1}\ \ \ (\ell=1,...,n),$ (25)

where $n$ is the number of harmonic oscillator functions, and $a$ is the ratio
coefficient. There are three parameters $\\{d_{1},d_{n},n\\}$ to be determined
through the variation method. It is found that with the parameter set {0.085
fm, 3.399 fm, 15} for the $ss\bar{s}\bar{s}$ system, we can obtain stable
solutions. The numerical results should be independent of the parameter
$d_{1}$. To confirm this point, as done in the literature Hiyama:2005cf ;
Hiyama:2018ukv ; Meng:2019fan  we scale the parameter $d_{1}$ of the basis
functions as $d_{1}\to\alpha d_{1}$. The mass of a $T_{ss\bar{s}\bar{s}}$
state should be stable at a resonance energy insensitive to the scaling
parameter $\alpha$. As an example, we plot the masses of 12 $2S$-wave
$T_{ss\bar{s}\bar{s}}$ 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 each 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:2019vtx ; Liu:2019zuc . Finally, the physical states can be
obtained by diagonalizing the mass matrix of different configurations with the
same $J^{PC}$ numbers.

Table 2:  Configurations for the tetraquark $qq\bar{q}\bar{q}$ system up to the $2S$-wave states. $J^{P(C)}$ | Configuration |  Wave Function
---|---|---
$J^{PC}$=$0^{++}$ | $1^{1}S_{0^{++}(6\bar{6})_{c}}$ | $\psi_{000}^{1S}\chi_{00}^{00}$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$0^{++}$ | $1^{1}S_{0^{++}(\bar{3}3)_{c}}$ | $\psi_{000}^{1S}\chi_{00}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{+-}$ | $1^{3}S_{1^{+-}(\bar{3}3)_{c}}$ | $\psi_{000}^{1S}\chi_{11}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{++}$ | $1^{5}S_{2^{++}(\bar{3}3)_{c}}$ | $\psi_{000}^{1S}\chi_{22}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$0^{--}$ | $1^{3}P_{0^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{6}}\left(\psi_{011}^{\xi_{1}}\chi_{1-1}^{10}-\psi_{010}^{\xi_{1}}\chi_{10}^{10}+\psi_{01-1}^{\xi_{1}}\chi_{11}^{10}-\psi_{011}^{\xi_{2}}\chi_{1-1}^{01}+\psi_{010}^{\xi_{2}}\chi_{10}^{01}-\psi_{01-1}^{\xi_{2}}\chi_{11}^{01}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$0^{--}$ | $1^{3}P_{0^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{6}}\left(\psi_{011}^{\xi_{1}}\chi_{1-1}^{01}-\psi_{010}^{\xi_{1}}\chi_{10}^{01}+\psi_{01-1}^{\xi_{1}}\chi_{11}^{01}-\psi_{011}^{\xi_{2}}\chi_{1-1}^{10}+\psi_{010}^{\xi_{2}}\chi_{10}^{10}-\psi_{01-1}^{\xi_{2}}\chi_{11}^{10}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$0^{-+}$ | $1^{3}P_{0^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{6}}\left(\psi_{011}^{\xi_{1}}\chi_{1-1}^{10}-\psi_{010}^{\xi_{1}}\chi_{10}^{10}+\psi_{01-1}^{\xi_{1}}\chi_{11}^{10}+\psi_{011}^{\xi_{2}}\chi_{1-1}^{01}-\psi_{010}^{\xi_{2}}\chi_{10}^{01}+\psi_{01-1}^{\xi_{2}}\chi_{11}^{01}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$0^{-+}$ | $1^{3}P_{0^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{6}}\left(\psi_{011}^{\xi_{1}}\chi_{1-1}^{01}-\psi_{010}^{\xi_{1}}\chi_{10}^{01}+\psi_{01-1}^{\xi_{1}}\chi_{11}^{01}+\psi_{011}^{\xi_{2}}\chi_{1-1}^{10}-\psi_{010}^{\xi_{2}}\chi_{10}^{10}+\psi_{01-1}^{\xi_{2}}\chi_{11}^{10}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$0^{-+}$ | $1^{3}P_{0^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\sqrt{\frac{1}{3}}\left(\psi_{011}^{\xi_{3}}\chi_{1-1}^{11}-\psi_{010}^{\xi_{3}}\chi_{10}^{11}+\psi_{01-1}^{\xi_{3}}\chi_{11}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{--}$ | $1^{3}P_{1^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\frac{1}{2}\left(\psi_{011}^{\xi_{1}}\chi_{10}^{10}-\psi_{010}^{\xi_{1}}\chi_{11}^{10}-\psi_{011}^{\xi_{2}}\chi_{10}^{01}+\psi_{010}^{\xi_{2}}\chi_{11}^{01}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$1^{--}$ | $1^{3}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\frac{1}{2}\left(\psi_{011}^{\xi_{1}}\chi_{10}^{01}-\psi_{010}^{\xi_{1}}\chi_{11}^{01}-\psi_{011}^{\xi_{2}}\chi_{10}^{10}+\psi_{010}^{\xi_{2}}\chi_{11}^{10}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{--}$ | $1^{5}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\sqrt{\frac{1}{10}}\psi_{011}^{\xi_{3}}\chi_{20}^{11}-\sqrt{\frac{3}{10}}\psi_{010}^{\xi_{3}}\chi_{21}^{11}+\sqrt{\frac{3}{5}}\psi_{01-1}^{\xi_{3}}\chi_{22}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{--}$ | $1^{1}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\psi_{011}^{\xi_{3}}\chi_{00}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{--}$ | $1^{1}P_{1^{--}(6\bar{6})_{c}\left(\xi_{3}\right)}$ | $\psi_{011}^{\xi_{3}}\chi_{00}^{00}$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$1^{-+}$ | $1^{3}P_{1^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\frac{1}{2}\left(\psi_{011}^{\xi_{1}}\chi_{10}^{10}-\psi_{010}^{\xi_{1}}\chi_{11}^{10}+\psi_{011}^{\xi_{2}}\chi_{10}^{01}-\psi_{010}^{\xi_{2}}\chi_{11}^{01}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$1^{-+}$ | $1^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\frac{1}{2}\left(\psi_{011}^{\xi_{1}}\chi_{10}^{01}-\psi_{010}^{\xi_{1}}\chi_{11}^{01}+\psi_{011}^{\xi_{2}}\chi_{10}^{10}-\psi_{010}^{\xi_{2}}\chi_{11}^{10}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{-+}$ | $1^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{011}^{\xi_{3}}\chi_{10}^{11}-\psi_{010}^{\xi_{3}}\chi_{11}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{--}$ | $1^{3}P_{2^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{011}^{\xi_{1}}\chi_{11}^{10}-\psi_{011}^{\xi_{2}}\chi_{11}^{01}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$2^{--}$ | $1^{3}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{011}^{\xi_{1}}\chi_{11}^{01}-\psi_{011}^{\xi_{2}}\chi_{11}^{10}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{--}$ | $1^{5}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\sqrt{\frac{1}{3}}\psi_{011}^{\xi_{3}}\chi_{21}^{11}-\sqrt{\frac{2}{3}}\psi_{010}^{\xi_{3}}\chi_{22}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{-+}$ | $1^{3}P_{2^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{011}^{\xi_{1}}\chi_{11}^{10}+\psi_{011}^{\xi_{2}}\chi_{11}^{01}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$2^{-+}$ | $1^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{011}^{\xi_{1}}\chi_{11}^{01}+\psi_{011}^{\xi_{2}}\chi_{11}^{10}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{-+}$ | $1^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\psi_{011}^{\xi_{3}}\chi_{11}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$3^{--}$ | $1^{5}P_{3^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\psi_{011}^{\xi_{3}}\chi_{22}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$0^{+-}$ | $2^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{00}^{00}-\psi_{100}^{\xi_{2}}\chi_{00}^{00}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$0^{+-}$ | $2^{1}S_{0^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{00}^{11}-\psi_{100}^{\xi_{2}}\chi_{00}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$0^{++}$ | $2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{00}^{00}+\psi_{100}^{\xi_{2}}\chi_{00}^{00}\right)$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$0^{++}$ | $2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{00}^{11}+\psi_{100}^{\xi_{2}}\chi_{00}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$0^{++}$ | $2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{3}\right)}$ | $\psi_{100}^{\xi_{3}}\chi_{00}^{00}$ | $|6\bar{6}\rangle^{c}$
$J^{PC}$=$0^{++}$ | $2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\psi_{100}^{\xi_{3}}\chi_{00}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{+-}$ | $2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{11}^{11}+\psi_{100}^{\xi_{2}}\chi_{11}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{+-}$ | $2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\psi_{100}^{\xi_{3}}\chi_{11}^{11}$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$1^{++}$ | $\begin{array}[]{c}2^{3}S_{1^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{11}^{11}-\psi_{100}^{\xi_{2}}\chi_{11}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{+-}$ | $\begin{array}[]{c}2^{5}S_{2^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{22}^{11}-\psi_{100}^{\xi_{2}}\chi_{22}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{++}$ | $2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | $\sqrt{\frac{1}{2}}\left(\psi_{100}^{\xi_{1}}\chi_{22}^{11}+\psi_{100}^{\xi_{2}}\chi_{22}^{11}\right)$ | $|\bar{3}3\rangle^{c}$
$J^{PC}$=$2^{++}$ | $2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | $\psi_{100}^{\xi_{3}}\chi_{22}^{11}$ | $|\bar{3}3\rangle^{c}$
Table 3:  Predicted mass spectrum for the $ss\bar{s}\bar{s}$ system with the HOEM. $J^{P(C)}$ | Configuration |  $\langle H\rangle$ (MeV) | Mass (MeV) | Eigenvector
---|---|---|---|---
$0^{++}$ | $\begin{array}[]{l}1^{1}S_{0^{++}(6\bar{6})_{c}}\\\ 1^{1}S_{0^{++}(\bar{3}3)_{c}}\end{array}$ | $\left(\begin{array}[]{cc}2365&-105\\\ -105&2293\end{array}\right)$ | $\left(\begin{array}[]{c}2218\\\ 2440\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{cc}-0.58&-0.81\end{array}\right)\\\ \left(\begin{array}[]{cc}-0.81&0.58\end{array}\right)\end{array}\right)$
$1^{+-}$ | $\begin{array}[]{l}1^{3}S_{1^{+-}(\bar{3}3)_{c}}\end{array}$ | $\left(2323\right)$ | $2323$ | $1$
$2^{++}$ | $\begin{array}[]{l}1^{5}S_{2^{++}(\bar{3}3)_{c}}\end{array}$ | $\left(2378\right)$ | $2378$ | $1$
$0^{--}$ | $\begin{array}[]{l}{}^{3}P_{0^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{0^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2635&154\\\ 154&2694\end{array}\right)$ | $\left(\begin{array}[]{c}2507\\\ 2821\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{cc}-0.77&0.64\end{array}\right)\\\ \left(\begin{array}[]{cc}0.64&0.77\end{array}\right)\end{array}\right)$
$0^{-+}$ | $\begin{array}[]{l}{}^{3}P_{0^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{0^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{0^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{ccc}2616&-35&-111\\\ -35&2685&56\\\ -111&56&2576\end{array}\right)$ | $\left(\begin{array}[]{c}2481\\\ 2635\\\ 2761\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{ccc}0.61&-0.11&0.78\end{array}\right)\\\ \left(\begin{array}[]{ccc}-0.56&-0.76&0.34\end{array}\right)\\\ \left(\begin{array}[]{ccc}0.56&-0.64&-0.53\end{array}\right)\end{array}\right)$
$1^{--}$ | $\begin{array}[]{l}{}^{3}P_{1^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{5}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}\\\ {}^{1}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}\\\ {}^{1}P_{1^{--}(6\bar{6})_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{ccccc}2585&-154&-89&-46&90\\\ -154&2694&42&22&-76\\\ -89&42&2584&-8&29\\\ -46&22&-8&2636&-51\\\ 90&-76&29&-51&2889\end{array}\right)$ | $\left(\begin{array}[]{c}2445\\\ 2567\\\ 2627\\\ 2766\\\ 2984\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{ccccc}-0.80&-0.38&-0.43&-0.14&0.11\end{array}\right)\\\ \left(\begin{array}[]{ccccc}0.18&0.57&-0.78&-0.05&0.15\end{array}\right)\\\ \left(\begin{array}[]{ccccc}0.03&0.11&0.11&-0.97&-0.18\end{array}\right)\\\ \left(\begin{array}[]{ccccc}-0.42&0.57&0.43&0.00&0.56\end{array}\right)\\\ \left(\begin{array}[]{ccccc}0.38&-0.43&-0.07&-0.19&0.79\end{array}\right)\end{array}\right)$
$1^{-+}$ | $\begin{array}[]{l}{}^{3}P_{1^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{ccc}2628&95&25\\\ 95&2712&12\\\ 25&12&2633\end{array}\right)$ | $\left(\begin{array}[]{c}2564\\\ 2632\\\ 2778\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{ccc}0.83&-0.51&-0.21\end{array}\right)\\\ \left(\begin{array}[]{ccc}-0.09&0.25&-0.96\end{array}\right)\\\ \left(\begin{array}[]{ccc}0.55&0.82&0.17\end{array}\right)\end{array}\right)$
$2^{--}$ | $\begin{array}[]{l}{}^{3}P_{2^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{5}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{ccc}2620&-217&-50\\\ -217&2725&24\\\ -50&24&2665\end{array}\right)$ | $\left(\begin{array}[]{c}2446\\\ 2657\\\ 2907\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{ccc}0.79&0.60&0.12\end{array}\right)\\\ \left(\begin{array}[]{ccc}-0.03&0.23&-0.97\end{array}\right)\\\ \left(\begin{array}[]{ccc}-0.61&0.76&0.20\end{array}\right)\end{array}\right)$
$2^{-+}$ | $\begin{array}[]{l}{}^{3}P_{2^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ {}^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{ccc}2638&138&-33\\\ 138&2733&-16\\\ -33&-16&2673\end{array}\right)$ | $\left(\begin{array}[]{c}2537\\\ 2669\\\ 2837\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{ccc}0.82&-0.56&0.13\end{array}\right)\\\ \left(\begin{array}[]{ccc}0.00&0.23&0.97\end{array}\right)\\\ \left(\begin{array}[]{ccc}-0.58&-0.79&0.19\end{array}\right)\end{array}\right)$
$3^{--}$ | $\begin{array}[]{c}{}^{5}P_{3^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{c}2719\end{array}\right)$ | $2719$ | $1$
$0^{+-}$ | $\begin{array}[]{l}2^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{1}S_{0^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2848&-27\\\ -27&2942\end{array}\right)$ | $\left(\begin{array}[]{c}2841\\\ 2949\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{cc}-0.97&-0.26\end{array}\right)\\\ \left(\begin{array}[]{cc}-0.26&0.97\end{array}\right)\end{array}\right)$
$0^{++}$ | $\begin{array}[]{l}2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{3}\right)}\\\ 2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{cccc}2859&-53&-61&-18\\\ -53&2903&-20&-49\\\ -61&-20&3218&-40\\\ -18&-49&-40&2856\end{array}\right)$ | $\left(\begin{array}[]{c}2781\\\ 2876\\\ 2948\\\ 3232\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{cccc}-0.61&-0.52&-0.16&-0.57\end{array}\right)\\\ \left(\begin{array}[]{cccc}0.67&0.02&0.03&-0.74\end{array}\right)\\\ \left(\begin{array}[]{cccc}0.39&-0.85&0.07&0.34\end{array}\right)\\\ \left(\begin{array}[]{cccc}0.15&0.02&-0.98&0.09\end{array}\right)\end{array}\right)$
$1^{+-}$ | $\begin{array}[]{l}2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2920&-44\\\ -44&2867\end{array}\right)$ | $\left(\begin{array}[]{c}2842\\\ 2945\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{cc}-0.49&-0.87\end{array}\right)\\\ \left(\begin{array}[]{cc}-0.87&0.49\end{array}\right)\end{array}\right)$
$1^{++}$ | $\begin{array}[]{l}2^{3}S_{1^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(2954\right)$ | 2954 | 1
$2^{+-}$ | $\begin{array}[]{l}2^{5}S_{2^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(2977\right)$ | 2977 | 1
$2^{++}$ | $\begin{array}[]{l}2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2952&-28\\\ -28&2888\end{array}\right)$ | $\left(\begin{array}[]{c}2878\\\ 2963\end{array}\right)$ | $\left(\begin{array}[]{c}\left(\begin{array}[]{cc}-0.35&-0.94\end{array}\right)\\\ \left(\begin{array}[]{cc}-0.94&0.35\end{array}\right)\end{array}\right)$
Table 4:  The average contributions of each part of the Hamiltonian to the $ss\bar{s}\bar{s}$ configurations with the HOEM. $\langle T\rangle$ stands for the contribution of the kinetic energy term. $\langle V^{Lin}\rangle$ and $\langle V^{Coul}\rangle$ stand for the contributions from the linear confinement potential and Coulomb type potential, respectively. $\langle V^{SS}\rangle$, $\langle V^{T}\rangle$, and $\langle V^{LS}\rangle$ stand for the contributions from the spin-spin interaction term, the tensor potential term, and the spin-orbit interaction term, respectively. The second number in every column is calculated with the GEM.  $J^{P(C)}$ | Configuration | Mass | $\langle T\rangle$ | $\langle V^{Lin}\rangle$ | $\langle V^{Coul}\rangle$ | $\langle V^{SS}\rangle$ | $\langle V^{T}\rangle$ | $\langle V^{LS}\rangle$
---|---|---|---|---|---|---|---|---
$0^{++}$ | $1^{1}S_{0^{++}(6\bar{6})_{c}}$ | 2365 | 807 | 930 | -774 | 40.75 |  |
$1^{1}S_{0^{++}(\bar{3}3)_{c}}$ | 2293 | 884 | 890 | -812 | -30.29 |  |
$1^{+-}$ | $\begin{array}[]{c}1^{3}S_{1^{+-}(\bar{3}3)_{c}}\end{array}$ | 2323 | 851 | 906 | -797 | 0 |  |
$2^{++}$ | $\begin{array}[]{c}1^{5}S_{2^{++}(\bar{3}3)_{c}}\end{array}$ | 2378 | 793 | 937 | -767 | 53.2 |  |
$0^{--}$ | $1^{3}P_{0^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2635 | 827 | 1077 | -660 | 4.42 | 34.94 | -11.65
$1^{3}P_{0^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2694 | 902 | 1093 | -644 | -0.95 | 9.02 | -27.06
$0^{-+}$ | $1^{3}P_{0^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2616 | 863 | 1056 | -675 | 26.94 | -4.2 | -12.6
$1^{3}P_{0^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2685 | 922 | 1083 | -652 | 8.3 | -9.39 | -28.16
$1^{3}P_{0^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2576 | 976 | 1015 | -703 | 9.96 | -20.88 | -62.65
$1^{--}$ | $1^{3}P_{1^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2585 | 895 | 1037 | -688 | 5.06 | -20.12 | -6.71
$1^{3}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2694 | 902 | 1093 | -644 | -0.95 | -4.51 | -13.53
$1^{5}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2584 | 972 | 1018 | -702 | 43.08 | -14.6 | -93.84
$1^{1}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2636 | 877 | 1068 | -665 | -5.24 | 0 | 0
$1^{1}P_{1^{--}(6\bar{6})_{c}\left(\xi_{3}\right)}$ | 2889 | 848 | 1210 | -564 | 33.29 | 0 | 0
$1^{-+}$ | $1^{3}P_{1^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2628 | 845 | 1066 | -668 | 26.34 | 2.03 | -6.08
$1^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2712 | 882 | 1106 | -637 | 8.27 | 4.33 | -13
$1^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2633 | 884 | 1064 | -668 | 9.08 | 8.73 | -26.19
$2^{--}$ | $1^{3}P_{2^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2620 | 845 | 1066 | -667 | 4.59 | 3.63 | 6.05
$1^{3}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2725 | 859 | 1119 | -628 | -0.58 | 0.83 | 12.4
$1^{5}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2665 | 846 | 1087 | -653 | 35.77 | 11.33 | -24.27
$2^{-+}$ | $1^{3}P_{2^{-+}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2638 | 833 | 1074 | -662 | 25.88 | -0.39 | 5.92
$1^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2733 | 854 | 1123 | -626 | 8.23 | -0.82 | 12.29
$1^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2673 | 830 | 1096 | -646 | 8.53 | -1.56 | 23.44
$3^{--}$ | $1^{5}P_{3^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2719 | 780 | 1131 | -625 | 31.94 | -2.80 | 41.99
$0^{+-}$ | $2^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2848/ 2927 | 716/ 844 | 1122/ 1225 | -366/ -529 | 14.48/ 24.74 |  |
$2^{1}S_{0^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2942/ 2931 | 934/ 941 | 1247/ 1245 | -598/ -618 | -4.08/ 0.8 |  |
$0^{++}$ | $2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2858/ 2874 | 839/ 863 | 1181/ 1206 | -548/ -585 | 24.54/ 27.9 |  |
$2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2903/ 2918 | 956/ 954 | 1236/ 1235 | -639/ -620 | -12.68/ -12.4 |  |
$2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{3}\right)}$ | 3216/ 3148 | 898/ 922 | 1394/ 1364 | -465/ -517 | 26.81/ 17.52 |  |
$2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2855/ 2841 | 891/ 899 | 1189/ 1197 | -583/ -622 | -3.65/ 4.74 |  |
$1^{+-}$ | $2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2919/ 2934 | 937/ 926 | 1247/ 1252 | -631/ -610 | 4.36/ 3.51 |  |
$2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2866/ 2851 | 877/ 895 | 1195/ 1201 | -575/ -621 | 7.29/ 14.61 |  |
$1^{++}$ | $2^{3}S_{1^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2954/ 2943 | 919/ 926 | 1256/ 1256 | -591/ -614 | 7.95/ 12.47 |  |
$2^{+-}$ | $2^{5}S_{2^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2976/ 2965 | 890/ 905 | 1272/ 1271 | -577/ -607 | 30.03/ 34.34 |  |
$2^{++}$ | $2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ | 2952/ 2964 | 902/ 898 | 1268/ 1272 | -615/ -601 | 35.6/ 33.36 |  |
$2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ | 2887/ 2871 | 851/ 868 | 1207/ 1220 | -560/ -612 | 27.25/ 33.3 |  |
Figure 1: Predicted masses of 12 $2S$-wave $T_{ss\bar{s}\bar{s}}$
configurations as a function of the scaling factor $\alpha$. Figure 2: Mass
spectrum for the $ss\bar{s}\bar{s}$ system. The solid and dotted lines stand
for the results predicted by the GEM and HOEM, respectively. Table 5:
Predicted mass spectrum for the $2S$-wave $ss\bar{s}\bar{s}$ system with the
GEM.

$J^{P(C)}$ | Configuration |  $\langle H\rangle$ (MeV) | Mass (MeV) | Eigenvector
---|---|---|---|---
$0^{+-}$ | $\begin{array}[]{l}2^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{1}S_{0^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2927&38\\\ 38&2931\end{array}\right)$ | $\left(\begin{array}[]{c}2891\\\ 2967\end{array}\right)$ | $\left(\begin{array}[]{cc}-0.73&0.69\\\ 0.69&0.73\end{array}\right)$
$0^{++}$ | $\begin{array}[]{l}2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{3}\right)}\\\ 2^{1}S_{0^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{cccc}2874&-48&27&-24\\\ -48&2918&-8&-42\\\ 27&-8&3148&-31\\\ -24&-42&-31&2841\end{array}\right)$ | $\left(\begin{array}[]{c}2798\\\ 2876\\\ 2954\\\ 3155\end{array}\right)$ | $\left(\begin{array}[]{cccc}-0.50&-0.46&-0.04&-0.73\\\ 0.74&0.20&-0.14&-0.62\\\ 0.43&-0.87&-0.06&0.25\\\ -0.11&0.04&-0.99&0.10\end{array}\right)$
$1^{+-}$ | $\begin{array}[]{l}2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2934&-40\\\ -40&2851\end{array}\right)$ | $\left(\begin{array}[]{c}2835\\\ 2950\end{array}\right)$ | $\left(\begin{array}[]{cc}-0.38&-0.93\\\ -0.93&0.38\end{array}\right)$
$1^{++}$ | $\begin{array}[]{l}2^{3}S_{1^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(2943\right)$ | 2943 | 1
$2^{+-}$ | $\begin{array}[]{l}2^{5}S_{2^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\end{array}$ | $\left(2965\right)$ | 2965 | 1
$2^{++}$ | $\begin{array}[]{l}2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}\\\ 2^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}\end{array}$ | $\left(\begin{array}[]{cc}2964&-43\\\ -43&2871\end{array}\right)$ | $\left(\begin{array}[]{c}2854\\\ 2981\end{array}\right)$ | $\left(\begin{array}[]{cc}-0.36&-0.93\\\ -0.93&0.36\end{array}\right)$

##  III Results and discussions

Our predictions of the $T_{ss\bar{s}\bar{s}}$ mass spectrum with the HOEM are
given in Table 3, where the components of different configurations for a
physical state can be seen. For example, the two $0^{++}$ ground states are
mixing states between two different configurations
${}^{1}S_{0^{++}(6\bar{6})_{c}}$ and ${}^{1}S_{0^{++}(\bar{3}3)_{c}}$ due to a
strong contribution of the confinement potential to the non-diagonal elements.
To see the contributions from each part of the Hamiltonian to the mass of
different configurations, we also present our results in Table 4. It is found
that both the kinetic energy term $\langle T\rangle$ and the linear
confinement potential term $\langle V^{Lin}\rangle$ contribute a large
positive value to the mass, while the Coulomb type potential $\langle
V^{Coul}\rangle$ has a large cancelation with these two terms. The spin-spin
interaction term $\langle V^{SS}\rangle$, the tensor potential term $\langle
V^{T}\rangle$, and/or the spin-orbit interaction term $\langle V^{LS}\rangle$
have also sizeable contributions to some configurations. Thus, as a reliable
calculation, both the spin-independent and spin-dependent potentials should be
reasonably included for the $ss\bar{s}\bar{s}$ system. For clarity, our
predicted $T_{ss\bar{s}\bar{s}}$ spectrum is plotted in Fig. 2.

###  III.1 Discussions of the numerical method

Herein we discuss the differences of numerical results between the expansion
method with the harmonic oscillator wave functions (HOEM) used in present work
and the Gaussian expansion method (GEM) often adopted in the literature. For
the $1S$-, $1P$-wave $T_{(ss\bar{s}\bar{s})}$ states, etc., there are no
radial excitations. Thus, the GEM is the same as the HOEM. For the first
radial excited $2S$-wave $T_{(ss\bar{s}\bar{s})}$ states, the HOEM is
different from the GEM because the trail harmonic oscillator wave functions
are different from the Gaussian functions.

To see the differences between the two expansion methods we also give our
predictions of the $2S$-wave $T_{(ss\bar{s}\bar{s})}$ states based on the GEM.
It should be mentioned that by fully expanding
$\prod_{i=1}^{3}R_{n_{\xi_{i}}l_{\xi_{i}}}({\xi_{i}})$ with the GEM, one
cannot distinguish the $\xi_{1}$ and $\xi_{2}$ excited modes which are defined
for the $2S$ configurations presented in Table 2. Then we cannot numerically
work out the masses for the following states of $0^{+-}$
($2^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ and
$2^{1}S_{0^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$), $1^{++}$
($2^{3}S_{1^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$), and $2^{+-}$
($2^{5}S_{2^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$) listed in Table
2. To overcome this problem, the spatial wave functions containing the radial
excitations are expanded with the Gaussian functions, while the spatial wave
functions containing no excitations are adopted the single Gaussian function
as an approximation. We have tested the single Gaussian approximation in the
calculations of the ground $1S$ $T_{(ss\bar{s}\bar{s})}$ states, the numerical
values are reasonably consistent with those calculated with a series of
Gaussian functions. The differences of the numerical results between these two
methods are about 10 MeV.

Our numerical results for the $2S$-wave $T_{(ss\bar{s}\bar{s})}$ states with
the GEM are listed in Table 4 and Table 5. From Table 4, it is found that the
numerical values for the $0^{+-}$ configuration
$2^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ and $0^{++}$
configuration $2^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{3}\right)}$ calculated
with the HOEM are significantly different from those obtained with the GEM.
For these two configurations, the predicted mass differences by the HOEM and
GEM can reach up to $\sim 70$ MeV. However, for the other $2S$-wave
$T_{(ss\bar{s}\bar{s})}$ configurations the numerical values of these two
methods are comparable with each other. The differences of the predicted
masses between these two methods are about $10-20$ MeV. It should be mentioned
that the Coulomb type potential $\langle V^{Coul}\rangle$ for the $2S$-wave
states seems to be sensitive to the numerical methods as shown in Table 4.

In brief, most of the predictions are consistent with each other between the
HOEM and GEM. The uncertainties from the numerical methods do not change our
main predictions of the $T_{(ss\bar{s}\bar{s})}$ spectrum, Although some
numerical results for the $J^{PC}=0^{+-}$ and $0^{++}$ $2S$ states show a
significant numerical method dependence (See Fig. 2), the GEM may give a
slightly more accurate numerical result based on our tests of the charmonium
spectrum. In the following, our discussions of the $2S$ states are based on
the GEM calculations.

###  III.2 $S$-wave states

There are four $1S$-wave $T_{ss\bar{s}\bar{s}}$ states with $J^{PC}=0^{++},\
1^{+-},\ 2^{++}$ in the quark model. Their masses are predicted to be in the
range of $2.21-2.44$ GeV. In contrast, the $2S$ wave includes twelve states.
Except for the highest mass state $T_{(ss\bar{s}\bar{s})0^{++}}(3155)$, their
masses lie in a relative narrow range of $2.78-2.98$ GeV. Apart from the
conventional quantum numbers, i.e., $J^{PC}=0^{++},\ 1^{++},\ 1^{+-},\
2^{++}$, the $2S$-wave can access exotic quantum numbers, i.e.,
$J^{PC}=0^{+-},\ 2^{+-}$.

####  III.2.1 $0^{++}$ states

In the $1S$-wave mutiplets, the two $0^{++}$ ground states include
$T_{(ss\bar{s}\bar{s})0^{++}}(2218)$ and $T_{(ss\bar{s}\bar{s})0^{++}}(2440)$.
Their mass splitting reaches up to about $200$ MeV. These two states have a
strong mixing between the two color structures $|6\bar{6}\rangle_{c}$ and
$|\bar{3}3\rangle_{c}$. Their masses are much larger than the mass threshold
of $\phi\phi$. Thus, they may easily decay into $\phi\phi$ pair through quark
rearrangements. The mass of the lowest $0^{++}$ $T_{ss\bar{s}\bar{s}}$ in our
model is close to the prediction of $2203$ MeV in the relativistic diquark-
antidiquark model Ebert:2008id . However, it turns out to be much higher than
the predicted value $1716$ MeV by the relativized quark model with a diquark-
antidiquark approximation Lu:2019ira . There might be some crucial dynamics
missing in the diquark-antidiquark approximation. As a test of the diquark-
antidiquark approximation we adopt the approximation as done in Ref.
Lu:2019ira  and calculate the mass of the $0^{++}$ $T_{ss\bar{s}\bar{s}}$
state with the same potential model parameters. We obtain a mass of $1758$
MeV, which is comparable with the prediction of Ref. Lu:2019ira , but is
obviously smaller than the results without the diquark-antidiquark
approximation.

In the $2S$-wave sector, there are four $0^{++}$ states,
$T_{(ss\bar{s}\bar{s})0^{++}}(2798)$, $T_{(ss\bar{s}\bar{s})0^{++}}(2876)$,
$T_{(ss\bar{s}\bar{s})0^{++}}(2954)$, and
$T_{(ss\bar{s}\bar{s})0^{++}}(3155)$, predicted in the NRPQM with GEM. A
strong mixing between the two color structures $|6\bar{6}\rangle_{c}$ and
$|\bar{3}3\rangle_{c}$ is also found among these states. In particular, the
radial excitation modes $(\xi_{1},\xi_{2})$ and $(\xi_{3})$ strongly mix with
each other. The highest state $T_{(ss\bar{s}\bar{s})0^{++}}(3155)$ is nearly a
pure configuration of ${}^{1}S_{0^{++}(6\bar{6})_{c}\left(\xi_{3}\right)}$,
with the color structure $|6\bar{6}\rangle_{c}$ and the radial excitation
between diquark $(ss)$ and anti-diquark $(\bar{s}\bar{s})$. The special color
structure of $T_{(ss\bar{s}\bar{s})0^{++}}(3155)$ leads to a rather large mass
gap $\Delta\simeq 201$ MeV from the nearby
$T_{(ss\bar{s}\bar{s})0^{++}}(2954)$. These $2S$-wave $0^{++}$
$ss\bar{s}\bar{s}$ states may easily decay into $\phi\phi$, $\phi\phi(1680)$
final states through quark rearrangements. They may also easily decay into
$K_{s}^{0}K_{s}^{0}$ and $K^{+}K^{-}$ final states through the $s\bar{s}$
annihilation and a pair of nonstrange $q\bar{q}$ creation. One also notices
that these states may directly decay into $\Xi\bar{\Xi}$ baryon pair with a
light $q\bar{q}$ pair creation.

Some evidences for $T_{(ss\bar{s}\bar{s})0^{++}}(2218)$ and
$T_{(ss\bar{s}\bar{s})0^{++}}(2440)$ may have been seen in the previous
experiments. Recently, Kozhevnikov carried out a dynamical analysis of the
resonance contributions to $J/\psi\to\gamma X\to\gamma\phi\phi$
Kozhevnikov:2019lmy  with the data from BESIII Ablikim:2016hlu . Two $0^{++}$
resonances with masses at $\sim 2.2$ GeV and $\sim 2.4$ GeV, were extracted
from the data. Evidence for a scalar around $2.2$ GeV in the $\phi\phi$ mass
spectra in $B_{s}^{0}\to J/\psi\phi\phi$ Aaij:2016qim  was also reported by
Ref. Kozhevnikov:2017nlr . Considering the mass and decay mode, these two
scalar structures may be good candidates for
$T_{(ss\bar{s}\bar{s})0^{++}}(2218)$ and $T_{(ss\bar{s}\bar{s})0^{++}}(2440)$.

It should be mentioned that $f_{0}(2200)$ is listed in RPP Tanabashi:2018oca
as a well-established state. It has been seen in the $K_{s}^{0}K_{s}^{0}$,
$K^{+}K^{-}$ and $\eta\eta$, and may be assigned to
$T_{(ss\bar{s}\bar{s})0^{++}}(2218)$. Some qualitative features can be
expected: (i) The $0^{++}$ $T_{ss\bar{s}\bar{s}}$ state can decay into
$\eta\eta$, $\eta^{\prime}\eta^{\prime}$, and $\eta\eta^{\prime}$ through
quark rearrangements via the $s\bar{s}$ component in the $\eta$ and
$\eta^{\prime}$ mesons. An approximate branching ratio fraction can be
examined:
$BR(\eta\eta):BR(\eta^{\prime}\eta^{\prime}):BR(\eta\eta^{\prime})\simeq\sin^{4}\alpha_{P}:\cos^{4}\alpha_{P}:2\sin^{2}\alpha_{P}\cos^{2}\alpha_{P}\simeq
0.24:0.25:0.50$, with $\alpha_{P}\equiv\arctan\sqrt{2}+\theta_{P}\simeq
44.7^{\circ}$ and without including the phase space factors. (ii) The $0^{++}$
states may also easily decay into $K_{s}^{0}K_{s}^{0}$ and $K^{+}K^{-}$ final
states through annihilating a pair of $s\bar{s}$ and creating a pair of light
$q\bar{q}$. (iii) It is interesting to note that no conventional $0^{++}$
$s\bar{s}$ states are predicted around 2.2 GeV in most literatures Li:2020xzs
.

To establish the $0^{++}$ ground states $T_{(ss\bar{s}\bar{s})0^{++}}(2218)$
and $T_{(ss\bar{s}\bar{s})0^{++}}(2440)$, a combined study of decay channels,
such as $\phi\phi$, $K_{s}^{0}K_{s}^{0}$, $K^{+}K^{-}$, $\eta\eta$,
$\eta^{\prime}\eta^{\prime}$, and $\eta\eta^{\prime}$, should be necessary.
The $2S$-wave $0^{++}$ states can be probed in these meson pair decay channels
including higher channels such as $\phi\phi(1680)$, and some baryon pair decay
channels such as $\Xi\bar{\Xi}$.

####  III.2.2 $2^{++}$ states

There is only one $2^{++}$ state $T_{(ss\bar{s}\bar{s})2^{++}}(2378)$ in the
$1S$-wave states. This state lies between the two $0^{++}$ ground states, and
has a pure $|\bar{3}3\rangle_{c}$ color structure.
$T_{(ss\bar{s}\bar{s})2^{++}}(2378)$ may have large decay rates into the
$\phi\phi$, $\eta\eta$ and $\eta^{\prime}\eta^{\prime}$ final states through
quark rearrangements, and/or into $K^{(*)}\bar{K}^{(*)}$ final states through
the annihilation of $s\bar{s}$ and creation of a pair of nonstrange
$q\bar{q}$. It should be mentioned that with the diquark-antidiquark
approximation, the mass of the $2^{++}$ state is predicted to be 2192 MeV,
which is about 200 MeV lower than the four-body calculation results.

The $f_{2}(2340)$ resonance listed in RPP Tanabashi:2018oca  may be assigned
to $T_{(ss\bar{s}\bar{s})2^{++}}(2378)$. Besides the measured mass
$2345^{+50}_{-40}$ MeV, the observed decay modes $\phi\phi$ and $\eta\eta$ are
consistent with the expectation of the tetraquark scenario. On the other hand,
as a conventional $s\bar{s}$ state the $f_{2}(2340)$ cannot be easily
accommodated by the quark model expectation Li:2020xzs . The relativistic
quark model calculation of Ref. Ebert:2008id  also supports the $f_{2}(2340)$
to be assigned as the $T_{ss\bar{s}\bar{s}}$ ground state with $2^{++}$. To
confirm this assignment, the other main decay modes of
$T_{(ss\bar{s}\bar{s})2^{++}}(2378)$ such as $\eta\eta^{\prime}$,
$\eta^{\prime}\eta^{\prime}$, $K^{(*)}\bar{K}^{(*)}$ should be investigated in
experiment.

For the $2S$-wave sector, there are two $2^{++}$ $ss\bar{s}\bar{s}$ states
$T_{(ss\bar{s}\bar{s})2^{++}}(2854)$ and $T_{(ss\bar{s}\bar{s})2^{++}}(2981)$
predicted in our model, which are dominated by the
${}^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ and
${}^{5}S_{2^{++}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ configurations,
respectively. Their masses are predicted to be above the thresholds of
$\phi\phi$, $\phi\phi(1680)$ and $\Xi(1530)\bar{\Xi}$. Therefore, experimental
search for their signals in these decay channels should be helpful for
understanding these tensor tetraquarks.

####  III.2.3 $1^{+-}$ states

In the $1S$-wave multiplets $T_{(ss\bar{s}\bar{s})1^{+-}}(2323)$ is the only
state with $C=-1$, and has a pure $|\bar{3}3\rangle_{c}$ color structure. Its
mass is about 100 MeV larger than the lowest $1S$-wave state
$T_{(ss\bar{s}\bar{s})0^{++}}(2218)$. Its mass is about $200\sim 300$ MeV
larger than that predicted by the QCD sum rules Cui:2019roq  and the
relativized quark model Lu:2019ira  in the diquark picture. The mass of the
$1^{+-}$ state may be notably underestimated in the diquark picture. As a
comparison we calculate the mass of the $1^{+-}$ state in the diquark picture
using the same potential model parameter set adopted in present work, and
obtain a mass of $1936$ MeV, which is 300 MeV smaller than the NRQPM
prediction 2236 MeV.

$T_{(ss\bar{s}\bar{s})1^{+-}}(2323)$ may easily decay into $\eta\phi$ and
$\eta^{\prime}\phi$ through the quark rearrangements. The decay of
$\psi^{\prime}/J/\psi\to\phi\eta\eta^{\prime}$ can access this state in
$\eta\phi$ and $\eta^{\prime}\phi$ channels. It should be mentioned that some
hints of $T_{(ss\bar{s}\bar{s})1^{+-}}(2323)$ may have been found in the
$\eta^{\prime}\phi$ invariant mass spectrum around $2.3-2.4$ GeV by observing
the $J/\psi\to\phi\eta\eta^{\prime}$ reaction at BESIII recently
Ablikim:2018xuz .

For the $2S$-wave sector, there are two states, i.e. $1^{+-}$
$ss\bar{s}\bar{s}$ $T_{(ss\bar{s}\bar{s})1^{+-}}(2835)$ and
$T_{(ss\bar{s}\bar{s})1^{+-}}(2950)$ predicted in the quark model. There are
sizeable configuration mixings in these two states. The
$T_{(ss\bar{s}\bar{s})1^{+-}}(2835)$ is dominated by the
$2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ configuration, which has a
$|\bar{3}3\rangle_{c}$ color structure, and the radial excitation occurs
between diquark $(ss)$ and anti-diquark $(\bar{s}\bar{s})$ (i.e., the
$\xi_{3}$ mode). The $T_{(ss\bar{s}\bar{s})1^{+-}}(2950)$ is dominated by the
$2^{3}S_{1^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$ configuration,
whose radial excitation occurs in the diquark $(ss)$ and anti-diquark
$(\bar{s}\bar{s})$. Apart from the $\eta\phi$ and $\eta^{\prime}\phi$ decay
channels it may favor decays into a pseudoscalar plus a radially excited
vector (i.e., $\eta\phi(1680)$ and $\eta^{\prime}\phi(1680)$), or a radially
excited pseudoscalar plus a vector (i.e., $\eta(1295)\phi$ and
$\eta(1405)\phi$), through the quark rearrangements.

It should be mentioned that in Refs. Cui:2019roq ; tetra7  the authors suggest
that the new structure $X(2063)$ observed in the
$J/\psi\to\phi\eta\eta^{\prime}$ at BESIII Ablikim:2018xuz  could be a
$1^{+-}$ $T_{ss\bar{s}\bar{s}}$ candidate according to the QCD sum rule
calculation. However, the observed mass of $X(2063)$ is too small to be
comparable with our quark model predictions.

####  III.2.4 $0^{+-}$ and $2^{+-}$ states

In the $2S$-wave multiplets, there are two $0^{+-}$ states,
$T_{(ss\bar{s}\bar{s})0^{+-}}(2891)$ and $T_{(ss\bar{s}\bar{s})0^{+-}}(2967)$,
predicted in the NRPQM with GEM. There is a strong configuration mixing
between ${}^{1}S_{0^{+-}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ and
${}^{1}S_{0^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$. There is only
one $2^{+-}$ state $T_{(ss\bar{s}\bar{s})2^{+-}}(2965)$ corresponding to the
configuration ${}^{5}S_{2^{+-}(\bar{3}3)_{c}\left(\xi_{1},\xi_{2}\right)}$.
The $0^{+-}$ and $2^{+-}$ are exotic quantum numbers which cannot be
accommodated by the conventional $q\bar{q}$ scenario. The $P$-wave decays into
the $\eta h_{1}(1P)$ and $\eta^{\prime}h_{1}(1P)$ channels could be useful for
the search for these states in experiments.

###  III.3 $1P$-wave states

There are twenty $1P$-wave $T_{ss\bar{s}\bar{s}}$ states predicted in the
NRPQM. Apart from the conventional quantum numbers, i.e., $J^{PC}=0^{-+},\
1^{--},\ 2^{-+},\ 2^{--},\ 3^{--}$, the $P$-wave can access exotic quantum
numbers, i.e., $J^{PC}=0^{--},\ 1^{-+}$. The masses of the $1P$-wave
$T_{ss\bar{s}\bar{s}}$ states scatter in a wide range of about $2.4-3.0$ GeV.
The masses of the low-lying $1P$-wave states may highly overlap with the
heaviest $1S$-wave state $T_{(ss\bar{s}\bar{s})0^{++}}(2440)$.

####  III.3.1 $0^{-+}$ states

There are three $0^{-+}$ states, $T_{(ss\bar{s}\bar{s})0^{-+}}(2481)$,
$T_{(ss\bar{s}\bar{s})0^{-+}}(2635)$, and
$T_{(ss\bar{s}\bar{s})0^{-+}}(2761)$, predicted in the NRPQM. They are mixed
states with two color structures $|6\bar{6}\rangle_{c}$ and
$|\bar{3}3\rangle_{c}$, and also mixed states between two orbital excitations
$(\xi_{1},\xi_{2})$ and $\xi_{3}$ modes. They can decay into $\phi h_{1}(1P)$
via an $S$ wave, or $\phi\phi$ via a $P$ wave, through the quark
rearrangements.

In 2016, the BESIII Collaboration observed a new resonance $X(2500)$ with a
mass of $2470^{+15}_{-19}$${}^{+101}_{-23}$ MeV and a width of
$230^{+64}_{-35}$${}^{+56}_{-33}$ MeV in $J/\psi\to\gamma\phi\phi$
Ablikim:2016hlu . The preferred spin-parity numbers for $X(2500)$ are
$J^{PC}=0^{-+}$ Ablikim:2016hlu . The $X(2500)$ resonance may be a candidate
for $T_{(ss\bar{s}\bar{s})0^{-+}}(2481)$ in terms of mass, decay modes and
quantum numbers although $X(2500)$ may favor the $4^{1}S_{0}$ $s\bar{s}$ state
as suggested in our previous work Li:2020xzs . In the recent work of Ref.
Dong:2020okt , the authors also suggested $X(2500)$ to be a $0^{-+}$
$T_{ss\bar{s}\bar{s}}$ state according to the QCD sum rule studies. A
measurement of the branching fraction of $\mathcal{B}[X(2500)\to\phi\phi]$
might provide a test of the nature of $X(2500)$. The decay rate of
$T_{(ss\bar{s}\bar{s})0^{-+}}(2481)$ into $\phi\phi$ through the quark
rearrangements should be significantly larger than that via an $s\bar{s}$ pair
production for the $4^{1}S_{0}$ $s\bar{s}$ state.

####  III.3.2 $1^{--}$ states

There are five $1^{--}$ states, $T_{(ss\bar{s}\bar{s})1^{--}}(2445)$,
$T_{(ss\bar{s}\bar{s})1^{--}}(2567)$, $T_{(ss\bar{s}\bar{s})1^{--}}(2627)$,
$T_{(ss\bar{s}\bar{s})1^{--}}(2766)$, and
$T_{(ss\bar{s}\bar{s})1^{--}}(2984)$, predicted in the NRPQM. Their masses
scatter in a rather wide range of about $2.4-3.0$ GeV. From Table 3, it is
found that there are obvious configuration mixings in these tetraquark states
except that $T_{(ss\bar{s}\bar{s})1^{--}}(2627)$ may nearly be a pure
${}^{1}P_{1^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ state. The lowest state
$T_{(ss\bar{s}\bar{s})1^{--}}(2445)$ is dominated by the
${}^{3}P_{1^{--}(6\bar{6})_{c}\left(\xi_{1},\xi_{2}\right)}$ configuration.
Its orbital excitation mainly occurs within the diquark $(ss)$ or anti-diquark
$(\bar{s}\bar{s})$. Meanwhile, the highest state
$T_{(ss\bar{s}\bar{s})1^{--}}(2984)$ is dominated by the
${}^{1}P_{1^{--}(6\bar{6})_{c}\left(\xi_{3}\right)}$ configuration, and the
orbital excitation occurs between the diquark $(ss)$ and anti-diquark
$(\bar{s}\bar{s})$.

The vector meson $\phi(2170)$ in RPP Tanabashi:2018oca  is suggested to be a
$1^{--}$ $T_{ss\bar{s}\bar{s}}$ state in the literature tetra1 ; tetra2 ;
tetra3 ; tetra5 ; tetra6 ; tetra7  since it is hard to be explained as a
conventional meson state according to its measured decay modes Ablikim:2020pgw
; Ablikim:2020coo . Furthermore, the $X(2239)$ resonance, which was observed
in $e^{+}e^{-}\to K^{+}K^{-}$ by the BESIII Collaboration Ablikim:2018iyx ,
was suggested to be a candidate of the lowest $1^{--}$ $T_{ss\bar{s}\bar{s}}$
state by comparing with the mass spectrum from the relativized quark model
Lu:2019ira  and the QCD two-point sum rule method Azizi:2019ecm . However, our
calculations indicate that neither $\phi(2170)$ nor $X(2239)$ can be assigned
to a $1^{--}$ $T_{ss\bar{s}\bar{s}}$ state since their measured masses are
much lower than our predictions. It should be mentioned that in recent studies
the $\phi(2170)$ was considered as a vector tetraquark state with content
$su\bar{s}\bar{u}$ rather than as a state $ss\bar{s}\bar{s}$ Agaev:2019coa ;
Agaev:2020zad .

The $1^{--}$ $T_{ss\bar{s}\bar{s}}$ states may have large decay rates into the
$f_{0}(980)\phi$ via an $S$ wave, or into $\eta\phi$ and $\eta^{\prime}\phi$
via a $P$ wave. There are some experimental evidences for structures around
2.4 GeV observed in the $f_{0}(980)\phi$ invariant mass spectrum from _BABAR_
Aubert:2006bu ; Aubert:2007ur , Belle Shen:2009zze , BESII Ablikim:2007ab ,
and BESIII Ablikim:2014pfc , which could be signals of the $1^{--}$
$ss\bar{s}\bar{s}$ tetraquark states Chen:2018kuu . For the heavier states
$T_{(ss\bar{s}\bar{s})1^{--}}(2766)$ and $T_{(ss\bar{s}\bar{s})1^{--}}(2984)$,
they can also decay into $\Xi\bar{\Xi}$ baryon pair through a $q\bar{q}$ pair
production in vacuum. Thus, experimental search for these states in
$e^{+}e^{-}\to\Xi\bar{\Xi}$ should be very interesting.

####  III.3.3 $1^{-+}$ states

There are three $1^{-+}$ states, $T_{(ss\bar{s}\bar{s})1^{-+}}(2564)$,
$T_{(ss\bar{s}\bar{s})1^{-+}}(2632)$, and
$T_{(ss\bar{s}\bar{s})1^{-+}}(2778)$, predicted in the NRPQM. Note that
$1^{-+}$ are exotic quantum numbers which cannot be accommodated by the
conventional $q\bar{q}$ scenario. Both the lowest mass state
$T_{(ss\bar{s}\bar{s})1^{-+}}(2564)$ and highest mass state
$T_{(ss\bar{s}\bar{s})1^{-+}}(2778)$ are mixed states between the two color
structures $|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$, and their
orbital excitations are dominated by the $(\xi_{1},\xi_{2})$ mode. The middle
state $T_{(ss\bar{s}\bar{s})1^{-+}}(2632)$ is dominated by the
${}^{3}P_{1^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ configuration of which
the orbital excitation mainly occurs between the diquark $(ss)$ and anti-
diquark $(\bar{s}\bar{s})$. It should be noted that a corresponding state in
the relativized quark model Lu:2019ira  has a mass of 2581 MeV, which is about
50 MeV smaller than our prediction.

These $1^{-+}$ $T_{ss\bar{s}\bar{s}}$ states may easily decay into $\phi
h_{1}(1P)$, $\eta^{(^{\prime})}f_{1}(1420)$, $\phi\phi$ channels through the
quark rearrangements. They can be searched for in
$\chi_{cJ}(1P)\to\eta\phi\phi,\phi K^{*}K$ with sufficient $\chi_{cJ}(1P)$
data samples at BESIII, although no obvious structures were found in previous
observations Ablikim:2019yjw ; Ablikim:2015lnn .

####  III.3.4 $2^{-+}$ states

There are three $2^{-+}$ states, $T_{(ss\bar{s}\bar{s})2^{-+}}(2537)$,
$T_{(ss\bar{s}\bar{s})2^{-+}}(2669)$, and
$T_{(ss\bar{s}\bar{s})2^{-+}}(2837)$, predicted in the NRPQM. Both the lowest
mass state $T_{(ss\bar{s}\bar{s})2^{-+}}(2537)$ and highest mass state
$T_{(ss\bar{s}\bar{s})2^{-+}}(2837)$ are mixed states between the two color
structures $|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$. Their orbital
excitations are dominated by the $(\xi_{1},\xi_{2})$ mode. The middle state
$T_{(ss\bar{s}\bar{s})2^{-+}}(2669)$ is dominated by the
${}^{3}P_{2^{-+}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ configuration of which
the orbital excitation occurs between the diquark $(ss)$ and anti-diquark
$(\bar{s}\bar{s})$. A corresponding state in the relativized quark model
Lu:2019ira  has a mass of 2619 MeV, which is about 50 MeV smaller than our
prediction.

These $2^{-+}$ $T_{ss\bar{s}\bar{s}}$ states may easily fall apart into $\phi
h_{1}(1P)$ and $\eta f_{2}^{\prime}(1525)$ in an $S$ wave, or into $\phi\phi$
in a $P$ wave through the quark rearrangements. For the high mass state
$T_{(ss\bar{s}\bar{s})2^{-+}}(2837)$, the strong decay mode $\Xi\bar{\Xi}$
also opens. These states can be searched for in $\chi_{c2}(1P)\to\eta
T_{(ss\bar{s}\bar{s})2^{-+}}\to\eta\eta f_{2}^{\prime}(1525)\to\eta\eta
K\bar{K}$ at BESIII with the sufficient $\chi_{c2}(1P)$ data samples.

####  III.3.5 $0^{--}$ states

There are two states with exotic quantum numbers of $0^{--}$,
$T_{(ss\bar{s}\bar{s})0^{--}}(2507)$ and $T_{(ss\bar{s}\bar{s})0^{--}}(2821)$,
predicted in the NRPQM. These two states have a strong mixing between the two
color structures $|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$. The
orbital excitation is the $(\xi_{1},\xi_{2})$ mode, i.e., the excitation
occurs within the diquark $(ss)$ or anti-diquark $(\bar{s}\bar{s})$. These two
states may have large decay rates into $\phi f_{1}(1285)$ and $\phi
f_{1}(1420)$ in an $S$ wave, or into $\eta\phi$ and $\eta^{\prime}\phi$ in a
$P$ wave through the quark rearrangements. These $0^{--}$ exotic states may be
produced by the reactions
$e^{+}e^{-}\to\eta^{(^{\prime})}X\to\eta^{(^{\prime})}\eta^{(^{\prime})}\phi$
or $J/\psi\to\eta^{(^{\prime})}\eta^{(^{\prime})}\phi$.

####  III.3.6 $2^{--}$ states

There are three $2^{--}$ states, $T_{(ss\bar{s}\bar{s})2^{--}}(2446)$,
$T_{(ss\bar{s}\bar{s})2^{--}}(2657)$, and
$T_{(ss\bar{s}\bar{s})2^{--}}(2907)$, predicted in the NRPQM. Both the lowest
mass state $T_{(ss\bar{s}\bar{s})2^{--}}(2446)$ and highest mass state
$T_{(ss\bar{s}\bar{s})2^{--}}(2907)$ are mixed states between the two color
structures, $|6\bar{6}\rangle_{c}$ and $|\bar{3}3\rangle_{c}$, and their
orbital excitations are dominated by the $(\xi_{1},\xi_{2})$ mode. The middle
state $T_{(ss\bar{s}\bar{s})2^{--}}(2657)$ is dominated by the
${}^{3}P_{2^{--}(\bar{3}3)_{c}\left(\xi_{3}\right)}$ configuration, of which
the orbital excitation occurs between the diquark $(ss)$ and anti-diquark
$(\bar{s}\bar{s})$. A corresponding state in the relativized quark model
Lu:2019ira  has a mass of 2622 MeV, which is consistent with our prediction.
These $2^{--}$ states may easily decay into $\phi f_{1}(1285)$, $\phi
f_{1}(1420)$, and $\phi f_{2}^{\prime}(1525)$ in an $S$ wave, or into
$\eta\phi$, $\eta^{\prime}\phi$ in a $P$ wave through the quark
rearrangements. They can also be searched for in
$e^{+}e^{-}\to\eta^{(^{\prime})}X\to\eta^{(^{\prime})}\eta^{(^{\prime})}\phi$
or vector charmonium decays such as $J/\psi\to\eta\eta\phi$.

####  III.3.7 $3^{--}$ state

There is only one $3^{--}$ state $T_{(ss\bar{s}\bar{s})3^{--}}(2719)$
predicted in the NRPQM. This state has a pure color structure
$|\bar{3}3\rangle_{c}$, and also a pure orbital excitation between the diquark
$(ss)$ and anti-diquark $(\bar{s}\bar{s})$. Our predicted mass is about 60 MeV
larger than that predicted by the relativized quark model Lu:2019ira  with a
diquark approximation. The $3^{--}$ states may easily decay into $\phi
f_{2}^{\prime}(1525)$ in an $S$ wave by the quark rearrangements. Since it has
a high spin, it may be produced relatively easier in $p\bar{p}$ or $pp$
collisions.

##  IV Summary

In this work we calculate the mass spectra for the $1S$, $1P$ and $2S$-wave
$T_{ss\bar{s}\bar{s}}$ states in a nonrelativistic potential quark model
without the often-adopted diquark-antidiquark approximation. The $1S$-wave
ground states lie in the mass range of $\sim 2.21-2.44$ GeV, while the $1P$\-
and $2S$-wave states scatter in a rather wide mass range of $\sim 2.44-2.99$
GeV. For the $2S$-wave states, except for the highest state
$T_{(ss\bar{s}\bar{s})0^{++}}(3155)$ all the other states lie in a relatively
narrow range of $\sim 2.78-2.98$ GeV. We find that most of the physical states
are mixed states with different configurations.

For the $ss\bar{s}\bar{s}$ system it shows that both the kinetic energy
$\langle T\rangle$ and the linear confinement potential $\langle
V^{Lin}\rangle$ contribute a large positive value to the mass, while the
Coulomb type potential $\langle V^{Coul}\rangle$ has a large cancellation with
the these two terms. The spin-spin interaction $\langle V^{SS}\rangle$, tensor
potential $\langle V^{T}\rangle$, and/or the spin-orbit interaction term
$\langle V^{LS}\rangle$ also have sizeable contributions to some
configurations.

Some $T_{ss\bar{s}\bar{s}}$ states may have shown hints in experiment. For
instance, the observed decay modes and masses of $f_{0}(2200)$ and
$f_{2}(2340)$ listed in RPP Tanabashi:2018oca  could be good candidates for
the ground states $T_{(ss\bar{s}\bar{s})0^{++}}(2218)$ and
$T_{(ss\bar{s}\bar{s})2^{++}}(2378)$, respectively. The newly observed
$X(2500)$ at BESIII may be a candidate for the lowest mass $1P$-wave $0^{-+}$
state $T_{(ss\bar{s}\bar{s})0^{-+}}(2481)$. Another $0^{++}$ ground state
$T_{(ss\bar{s}\bar{s})0^{++}}(2440)$ may have shown signals in the $\phi\phi$
channel at BESIII Kozhevnikov:2019lmy ; Ablikim:2016hlu . Our calculation
shows that $\phi(2170)$ may not favor a vector state of
$T_{ss\bar{s}\bar{s}}$, because of the much higher mass obtained in our model.

It should be stressed that as a flavor partner of $T_{cc\bar{c}\bar{c}}$, the
$T_{ss\bar{s}\bar{s}}$ system may have very different dynamic features that
need further studies. One crucial point is that the strange quark is rather
light and the light flavor mixing effects could become non-negligible. It
suggests that strong couplings between $T_{ss\bar{s}\bar{s}}$ and open
strangeness channels could be sizeable. As a consequence, mixings between
$T_{ss\bar{s}\bar{s}}$ and $T_{sq\bar{s}\bar{q}}$ would be inevitable. For an
$S$-wave strong coupling, it may also lead to configuration mixings which can
be interpreted as hadronic molecules for a near-threshold structure. In such a
sense, this study can set up a reference on the basis of orthogonal states.
More elaborated dynamics can be investigated by including the hadron
interactions in the Hamiltonian. For states with exotic quantum numbers,
experimental searches for their signals can be carried out at BESIII and
Belle-II.

## Acknowledgement

The authors thank Wen-Biao Yan for useful discussions on the BESIII results.
Helpful discussions with Atsushi Hosaka and Qi-Fang Lü are greatly
appreciated. This work is supported by the National Natural Science Foundation
of China (Grants Nos. 11775078, U1832173, 11705056, 11425525, and 11521505).
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), the Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB34030302), and National Key Basic Research Program of
China under Contract No. 2015CB856700.

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