# The explanation of some exotic states in the $cs\bar{c}\bar{s}$ tetraquark
system

Xuejie Liu1<EMAIL_ADDRESS>Hongxia Huang2<EMAIL_ADDRESS>(Corresponding author) Jialun Ping2<EMAIL_ADDRESS>(Corresponding author)
Dianyong Chen1<EMAIL_ADDRESS>Xinmei Zhu3<EMAIL_ADDRESS>1School of
Physics, Southeast University, Nanjing 210094, P. R. China  2Department of
Physics, Nanjing Normal University, Nanjing 210023, P.R. China  3Department of
Physics, Yangzhou University, Yangzhou 225009, P. R. China

###### Abstract

Inspired by the recent observation of $\chi_{c0}(3930)$, $X(4685)$ and
$X(4630)$ by the LHCb Collaboration and some exotic resonances such as
$X(4350)$, $X(4500)$, etc. by several experiment collaborations, the
$cs\bar{c}\bar{s}$ tetraquark systems with $IJ^{P}=00^{+}$, $01^{+}$ and
$02^{+}$ are systematically investigated in the framework of the quark
delocalization color screening model(QDCSM). Two structures, the meson-meson
and diquark-antidiquark structures, as well as the channel-coupling of all
channels of these two configurations are considered in this work. The
numerical results indicate that the molecular bound state $\bar{D}_{s}D_{s}$
with $IJ^{P}=00^{+}$ can be supposed to explain the $\chi_{c0}(3930)$.
Besides, by using the stabilization method, several resonant states are
obtained. There are four $IJ^{P}=00^{+}$ states around the resonance mass 4035
MeV, 4385 MeV, 4524 MeV, and 4632 MeV, respectively; one $IJ^{P}=01^{+}$ state
around the resonance mass 4327 MeV; and two $IJ^{P}=02^{+}$ states around the
resonance mass 4419 MeV and 4526 MeV, respectively. All of them are compact
tetraquarks. Among these states, $X(4350)$, $X(4500)$ and $X(4700)$ can be
explained as the compact tetraquark state with $IJ^{P}=00^{+}$, and the
$X(4274)$ is possible to be a candidate of the compact tetraquark state with
$IJ^{P}=01^{+}$. More experimental tests are expected to check the existence
of all these possible resonance states.

###### pacs:

13.75.Cs, 12.39.Pn, 12.39.Jh

##  I Introduction

The underlying theory of strong interaction is quantum chromodynamics (QCD),
which is nonperturbative at low energy. Therefore, it is very difficult to be
solved by the hadron spectrum model independently. Although the traditional
quark model can well describe the hadron spectrum, which is classified as
mesons, composed of $q\bar{q}$ and baryons, composed of $qqq$ tr_model1 ;
tr_model2 , the emergence of plenty of states or resonant structures in
experiments can not fit the hadron spectrum predicted by the naive quark model
in last two decades. Those states are called exotic states. A lot of effort
has been made to understand the nature of these states, but there is still
controversy about their nature.

Actually, most of these states are in the proximity of di-hadron thresholds.
To name a few famous examples, the observed $X(3872)$ Bell-X(3872)  and
$Z_{c}(3900)$ Zc(3900)1 ; Zc(3900)2 ; Zc(3900)3  are around the $D\bar{D}^{*}$
threshold and the $Z_{c}(4020)$ Zc(4020)1 ; Zc(4020)2  are near the
$D^{*}\bar{D}^{*}$ threshold, etc. Recently, more representative states were
searched for experimentally, such as $Z_{c}(3985)$ Zc(3985) , which are near
the $\bar{D}_{s}D^{*}$ and $\bar{D}^{*}_{s}D$ thresholds.

Inspired by the results of recent LHCb experiment about the $X_{c0}(3930)$
which is just below the $D_{s}\bar{D}_{s}$ threshold LHCb_X3930.1 ;
LHCb_X3930.2 , we have a great interest in studying the tetraquark system
composed of $cs\bar{c}\bar{s}$. Besides, several experiment collaborations
have discovered some relevant resonance states. For instance, in 2009, the CDF
Collaboration reported a new state $X(4140)$ in the $J/\psi\phi$ invariant
mass distribution CDF-CCSS  and this structure was then observed by other
collaborations, such as LHCb, CMS, D0, and BABAR in next few years LHCb-CCSS ;
CMS-CCSS ; D0-CCSS ; BABAR-CCSS ; in 2010, a narrow resonance $X(4350)$ was
found in the $\gamma\gamma\longrightarrow J/\psi\phi$ process by the Belle
Collaboration Bell-CCSS , which favored the spin parity $J^{PC}=0^{++}$ or
$2^{++}$; in 2017, the CDF Collaboration observed the resonance state
$X(4274)$ in the $B^{+}\longrightarrow J/\psi\phi K^{+}$ decay with
$3.1\sigma$ significance CDF-CCSS-4274 ; in 2016, in the $B^{+}\longrightarrow
J/\psi\phi K^{+}$ decay, the LHCb Collaboration confirmed the existence of the
$X(4140)$ and $X(4274)$ LHCb-CCSS-2017.1 . Their quantum numbers are measured
to be $J^{PC}=1^{++}$; in the same process, the Collaboration observed two
higher resonances, $X(4500)$ and $X(4700)$ with $J^{PC}=0^{++}$ LHCb-
CCSS-2017.2 ; in 2021, an improved full amplitude analysis of the
$B^{+}\longrightarrow J/\psi\phi K^{+}$ decay is performed by using 6 times
larger signal yield than the previous analysis, the LHCb Collaboration
discovered two new hadron states, which are $X(4685)$ and $X(4630)$ LHCb-
CCSS-2021 .

To identify the internal structure of these resonance states, there are a
large number of theoretical studies about the tetraquark state
$cs\bar{c}\bar{s}$. In Ref Canonical-CCSS , the $X(4140)$ resonance appeared
as a cusp in the $J/\psi\phi$ channel and the $X(4274)$, $X(4500)$, $X(4700)$
were all defined as conventional charmonium states by using a nonrelativistic
constituent quark model, respectively. In the relativized quark model, the
resonance of $X(4140)$ was regarded as the $cs\bar{c}\bar{s}$ tetraquark
ground state, the $X(4700)$ was assigned as the 2S excited tetraquark state,
$X(4500)$ was explained as the tetraquark composed of one 2S scalar diquark
and one scalar antidiquark, and $X(4274)$ was a good candidate of the
conventional $\chi_{c1}$ state relative-CCSS . It was argued that the cusp
effects might explain the structure of the $X(4140)$, but failed to account
for the $X(4274)$ effect-cups-CCSS . In Ref di-antiquark-CCSS , the $X(4140)$
and $X(4270)$ with the tetraquark interpretation was consistent with $X(4350)$
while the interpretation of the $X(4500)$ and $X(4700)$ needed orbital or
radial excitation in the simple color-magnetic interaction model. Maiani et
al. proposed that the $X(4140)$ and the $X(4274)$ could be explained as the
ground state 1S-multiplet of diquark-antiquark tetraquarks while the $X(4500)$
and $X(4700)$ were radially excited 2S states di-antiquark-CCSS1 . Stancu
argued that $X(4140)$ could be the strange partner of X(3872) in a tetraquark
interpretation within a simple quark model with the chromomagnetic interaction
Stancu-CCSS . In Ref QCD-sum-rule-CCSS , based on the diquark-antidiquark
configuration within the framework of QCD sum rules, the $X(4500)$ and the
$X(4700)$ were interpreted as the D-wave $cs\bar{c}\bar{s}$ tetraquark states
of $J^{P}=0^{+}$. According to the calculation with multiquark color flux-tube
model, Deng et al. pointed out that the $X(4500)$ and the $X(4700)$ were
S-wave radial excited states $[cs][\bar{c}\bar{s}]$ color-flux-tube-CCSS .
Moreover, Yang explained the $X(4274)$ as the $cs\bar{c}\bar{s}$ tetraquark
states with $J^{PC}=1^{++}$, $X(4350)$ as a good candidate of the compact
tetraquark state with $J^{PC}=0^{++}$, and the $X(4700)$ as the 2S radial
excited tetraquark state with $J^{PC}=0^{++}$ CHQM-CCSS . In Refs. wang-
diquark-CCSS1 ; wang-diquark-CCSS2 , the $X(4500)$ was observed as the first
radial excited state of the axial-vector-diquark-axial-vector-antidiquark type
scalar $cs\bar{c}\bar{s}$ tetraquark state and the $X(4700)$ was assigned as
the ground state vector-diquark-vector-antidiquark type scalar
$cs\bar{c}\bar{s}$ tetraquark state, but the results disfavored assigning the
$X(4140)$ to the $J^{PC}=1^{++}$ diquark-antiquark type $cs\bar{c}\bar{s}$
tetraquark state. A rescattering mechanism was used in Ref rescattering-CCSS
to understand the nature of $X(4140)$, $X(4350)$, $X(4500)$ and $X(4700)$,
among which the $X(4140)$ and the $X(4700)$ could be simulated due to the
$D^{*}_{s}D_{s}$ rescattering and the $\psi^{{}^{\prime}}\phi$ rescattering.
However, this mechanism failed to generate the $X(4274)$ and $X(4500)$, which
lead to the proposal that they might be the genuine resonances. In Ref Ebert ,
the masses of the excited heavy tetraquarks with hidden charm were calculated
within the relativistic diquark-antidiquark picture, and the results showed
that $X(3872)$, $Y(4260)$, $Y(4360)$, $Z(4248)$, $Z(4433)$ and $Y(4660)$ could
be tetraquark states with hidden charm.

In this work, to see whether these exotic resonances can be described by
$cs\bar{c}\bar{s}$ tetraquark systems with $J^{P}=0^{+},1^{+}$ and $2^{+}$, we
systematically study the properties of these exotic resonances by using the
quark delocalization color screening model (QDCSM) QDCSM_explain1 , which was
proposed particularly to study the similarities between nuclear and molecular
forces. According to the characteristics of QDCSM, it can give a good
description of the properties of the deuteron, nucleon-nucleon, and hyperon-
nucleon interactions QDCSM_explain2 ; QDCSM_explain3 . In the present
calculation, two configurations, the meson-meson ($q\bar{q}-q\bar{q}$) and the
diquark-antidiquark ($qq-\bar{q}\bar{q}$), are taken into account. Besides, to
be more convincing, the channel coupling effect of $cs\bar{c}\bar{s}$
tetraquark systems is also included.

This work is organized as follows. In section II, we present a review of the
quark delocalization color screening model and the wave functions of the total
system in the present work. The numerical results and a discussion for the
tetraquarks are given in Section III. Finally, the last section is devoted to
a brief summary.

##  II THE QUARK DELOCALIZATION COLOR SCREENING MODEL (QDCSM) AND WAVE
FUNCTIONS

###  II.1 The quark delocalization color screening model (QDCSM)

The quark delocalization color screening model (QDCSM) is an extension of the
native quark cluster model native  and was developed with aim of addressing
multiquark systems. The detail of QDCSM can be found in the Refs.
QDCSM_explain1 ; QDCSM1 ; QDCSM2 . Here, the general form of the four body
complex Hamiltonian is given by

$H=\sum_{i=1}^{4}\left(m_{i}+\frac{\boldsymbol{p}_{i}^{2}}{2m_{i}}\right)-T_{CM}+\sum_{j>i=1}^{4}V(r_{ij}),\\\
$ (1)

where the center-of-mass kinetic energy, $T_{CM}$ is subtracted without losing
generality since we mainly focus on the internal relative motions of the
multiquark system. The interplay is of two body potential which includes
color-confining, $V_{CON}$, one-gluon exchange, $V_{OGE}$, and Goldstone-boson
exchange, $V_{\chi}$, respectively,

$V(r_{ij})=V_{CON}(r_{ij})+V_{OGE}(r_{ij})+V_{\chi}(r_{ij})$ (2)

In this work, we focus on the low-lying positive parity $cs\bar{c}\bar{s}$
tetraquark states of $s-$wave, and the spin-orbit and tensor interactions are
not included. The potential $V_{OGE}(r_{ij})$ can be written as

$V_{OGE}(r_{ij})=\frac{1}{4}\alpha_{s}\boldsymbol{\lambda}^{c}_{i}\cdot\boldsymbol{\lambda}^{c}_{j}\left[\frac{1}{r_{ij}}-\frac{\pi}{2}\delta(\boldsymbol{r}_{ij})(\frac{1}{m^{2}_{i}}+\frac{1}{m^{2}_{j}}+\frac{4\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}}{3m_{i}m_{j}})\right]$
(3)

where $m_{i}$ and $\boldsymbol{\sigma}$ are the quark mass and the Pauli
matrices, respectively. The $\boldsymbol{\lambda^{c}}$ is SU(3) color matrix.
The QCD-inspired effective scale-dependent strong coupling constant,
$\alpha_{s}^{ij}$, offers a consistent description of mesons from light to
heavy quark sector.

Similary, the confining interaction $V_{CON}(r_{ij})$ can be expressed as

$V_{CON}(r_{ij})=-a_{c}\boldsymbol{\lambda^{c}_{i}\cdot\lambda^{c}_{j}}[f(r_{ij})+V_{0_{ij}}],$
(4)

and the $f(r_{ij})$ can be written as

$f(r_{ij})=\left\\{\begin{array}[]{ll}r_{ij}^{2}&\qquad\mbox{if }i,j\mbox{
occur in the same cluster}\\\
\frac{1-e^{-\mu_{ij}r_{ij}^{2}}}{\mu_{ij}}&\qquad\mbox{if }i,j\mbox{ occur in
different cluster}\\\ \end{array}\right.$ (5)

where the color screening parameter $\mu_{ij}$ is determined by fitting the
deuteron properties, $NN$ and $NY$ scattering phase shifts, with
$\mu_{qq}=0.45$, $\mu_{qs}=0.19$ and $\mu_{ss}=0.08$, satisfying the relation
$\mu_{qs}^{2}=\mu_{qq}\mu_{ss}$, where $q$ represents $u$ or $d$ quark. When
extending to the heavy-quark case, we found that the dependence of the
parameter $\mu_{cc}$ is not very significant in the calculation of the $P_{c}$
states Pc_huang1  by taking it from $0.0001$ to $0.01$. So here we take
$\mu_{cc}=0.01$. Then $\mu_{sc}$ and $\mu_{uc}$ are obtained by the relation
$\mu^{2}=\mu_{ss}\mu_{cc}$ and $\mu^{2}=\mu_{uu}\mu_{cc}$, respectively.

The Goldstone-boson exchange interactions between light quarks appear because
the dynamical breaking of chiral symmetry. For the $cs\bar{c}\bar{s}$ system,
the $\pi$ and $K$ exchange interactions do not appear because there is no up
or down quarks herein. Only the following $\eta$ exchange term works between
the $s\bar{s}$ pair.

$V_{\chi}(r_{ij})=v^{\eta}_{ij}\left[\left(\lambda_{i}^{8}\cdot\lambda_{j}^{8}\right)\cos\theta_{P}-(\lambda_{i}^{0}\cdot\lambda_{j}^{0})\sin\theta_{P}\right]$
(6)

with

$\displaystyle v^{\eta}_{ij}$ $\displaystyle=$
$\displaystyle{\frac{g_{ch}^{2}}{{4\pi}}}{\frac{m_{\chi}^{2}}{{\
12m_{i}m_{j}}}}{\frac{\Lambda_{\chi}^{2}}{{\Lambda_{\chi}^{2}-m_{\chi}^{2}}}}m_{\chi}$
(7)
$\displaystyle\left\\{(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j})\left[Y(m_{\chi}\,r_{ij})-{\frac{\Lambda_{\chi}^{3}}{m_{\chi}^{3}}}Y(\Lambda_{\chi}\,r_{ij})\right]\right\\}$

where $Y(x)=e^{-x}/x$ is the standard Yukawa function. The
$\boldsymbol{\lambda^{a}}$ is the SU(3) flavor Gell-Mann matrix. The mass of
the $\eta$ meson is taken from the experimental value PDG . Finally, the chair
coupling constant, $g_{ch}$, is determined from the $\pi NN$ coupling constant
through

$\frac{g_{ch}^{2}}{4\pi}=\left(\frac{3}{5}\right)^{2}\frac{g_{\pi
NN}^{2}}{4\pi}{\frac{m_{u,d}^{2}}{m_{N}^{2}}}$ (8)

which assumes that flavor SU(3) is an exact symmetry, only broken by the
different mass of the strange quark. The model parameters and the masses of
the ground mesons are listed in Tables 1 and 2, respectively.

Table 1: Model parameters. The masses of mesons take their experimental values. $m_{\eta}=2.77$ fm-1. Quark masses |  $m_{u}$(MeV) | 313
---|---|---
|  $m_{s}$(MeV) | 536
|  $m_{c}$(MeV) | 1728
confinement | $b(fm)$ | 0.3
|  $a_{c}$(MeV $fm^{-2}$) | 101
|  $V_{0_{uu}}$(MeV) | -2.2543
|  $V_{0_{us}}$(MeV) | -1.7984
|  $V_{0_{uc}}$(MeV) | -1.3231
|  $V_{0_{ss}}$(MeV) | -1.3649
|  $V_{0_{sc}}$(MeV) | -0.6739
|  $V_{0_{cc}}$(MeV) | 0.7555
OGE | $\alpha_{s}^{uu}$ | 0.2567
| $\alpha_{s}^{us}$ | 0.2970
| $\alpha_{s}^{uc}$ | 0.3805
| $\alpha_{s}^{ss}$ | 0.1905
| $\alpha_{s}^{sc}$ | 0.6608
| $\alpha_{s}^{cc}$ | 1.6717
Table 2: The Masses (in MeV) of the ground mesons. Experimental values are taken from the Particle Data Group (PDG) PDG . |  $K$ |  $K^{*}$ |  $\pi$ |  $\rho$ |  $\eta_{s\bar{s}}$ |  $\phi$
---|---|---|---|---|---|---
Expt | 495 | 892 | 139 | 770 | 958 | 1020
Model | 495 | 892 | 139 | 770 | 958 | 1020
|  $D_{s}$ |  $D_{s}^{*}$ |  $\eta_{c\bar{c}}$ |  $J/\psi$ |  $D$ |  $D^{*}$
Expt | 1968 | 2112 | 2983 | 3096 | 1865 | 2007
Model | 1968 | 2112 | 2983 | 3096 | 1865 | 2007

In QDCSM, the quark delocalization is realized by specifying the single
particle orbital wave function of QDCSM as a linear combination of left and
right Gaussian, the single particle orbital wave functions used in the
ordinary quark cluster model,

$\displaystyle\psi_{r}(\boldsymbol{r},\boldsymbol{s}_{i},\epsilon)$
$\displaystyle=$
$\displaystyle(\phi_{R}(\boldsymbol{r},\boldsymbol{s}_{i})+\epsilon\phi_{L}(\boldsymbol{r},\boldsymbol{s}_{i}))/N(\epsilon),$
(9) $\displaystyle\psi_{l}(\boldsymbol{r},\boldsymbol{s}_{i},\epsilon)$
$\displaystyle=$
$\displaystyle(\phi_{L}(\boldsymbol{r},\boldsymbol{s}_{i})+\epsilon\phi_{R}(\boldsymbol{r},\boldsymbol{s}_{i}))/N(\epsilon),$
(10) $\displaystyle N(\epsilon)$ $\displaystyle=$
$\displaystyle\sqrt{1+\epsilon^{2}+2\epsilon e^{-s^{2}_{i}/{4b^{2}}}},$ (11)
$\displaystyle\phi_{R}(\boldsymbol{r},\boldsymbol{s}_{i})$ $\displaystyle=$
$\displaystyle(\frac{1}{\pi
b^{2}})^{\frac{3}{4}}e^{-\frac{1}{2b^{2}}(\boldsymbol{r}-\frac{2}{5}s_{i})^{2}},$
(12) $\displaystyle\phi_{L}(\boldsymbol{r},\boldsymbol{s}_{i})$
$\displaystyle=$ $\displaystyle(\frac{1}{\pi
b^{2}})^{\frac{3}{4}}e^{-\frac{1}{2b^{2}}(\boldsymbol{r}+\frac{3}{5}s_{i})^{2}},$
(13)

The $\boldsymbol{s}_{i}$, $i=1,2,...,n$, are the generating coordinates, which
are introduced to expand the relative motion wave function si_1 ; si_2 ; si_3
. The mixing parameter $\epsilon(s_{i})$ is not an adjusted one but determined
variationally by the dynamics of the multi-quark system itself. This
assumption allows the multi-quark system to choose its favorable configuration
in the interacting process. It has been used to explain the cross-over
transition between the hadron phase and the quark-gluon plasma phase si_4 .

###  II.2 The wave function

In this work, we focus on the double heavy $c\bar{c}s\bar{s}$ system by using
the resonance group method RGM . Figure 1 shows two kinds of configurations
for this system, which are the meson-meson structures shown in Fig. 1 (a) and
(b), and the diquark-antidiquark structure shown in Fig. 1(c). For the purpose
of solving a manageable 4-body problem, currently, the system calculation
considers only these two structures. But an economic way is used to combine
these two configurations to see the effect of the multi-channel coupling. Four
fundamental degrees of freedom, which are color, spin, flavor, and orbit are
generally accepted by the QCD theory at the quark level. The multiquark
system’s wave function is an internal product of the color, spin, flavor, and
orbit terms.

Figure 1: Two types of configurations in $c\bar{c}s\bar{s}$ tetraquarks.
Panel(a) and panel (b) is the meson-meson configuration, panel(c) is diquark-
antidiquark.

####  II.2.1 The color wave function

Plenty of color structures in multiquark systems will be available with
respect those of conventional hadrons such as $q\bar{q}$ mesons and $qqq$
baryons. In this section, the goal is to construct the colorless wave function
of a 4-quark system. For the meson-meson configurations, the color wave
functions of a $q\bar{q}$ cluster are listed.

$\displaystyle C^{1}_{[111]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}(r\bar{r}+g\bar{g}+b\bar{b})$ $\displaystyle
C^{2}_{[21]}$ $\displaystyle=$ $\displaystyle r\bar{b},C^{3}_{[21]}=-r\bar{g}$
$\displaystyle C^{4}_{[21]}$ $\displaystyle=$ $\displaystyle
g\bar{b},C^{5}_{[21]}=-b\bar{g}$ $\displaystyle C^{6}_{[21]}$ $\displaystyle=$
$\displaystyle g\bar{r},C^{7}_{[21]}=b\bar{r}$ $\displaystyle C^{8}_{[21]}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{1}{2}}(r\bar{r}-g\bar{g})$
$\displaystyle C^{9}_{[21]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{6}}(-r\bar{r}-g\bar{g}+2b\bar{b})$

where the subscript [111] and [21] stand for color-singlet($\textbf{1}_{c}$)
and color-octet($\textbf{8}_{c}$), respectively. So, the $SU(3)_{color}$ wave
functions of color-singlet (two color-singlet cluters,
$\textbf{1}_{c}\otimes\textbf{1}_{c}$) and hidden-color (two color-octet
clusters, $\textbf{8}_{c}\otimes\textbf{8}_{c}$) channels are given
respectively.

$\chi^{c}_{1}=C^{1}_{[111]}C^{1}_{[111]}$ (15)
$\begin{split}\chi^{c}_{2}=&\sqrt{\frac{1}{8}}(C^{2}_{[21]}C^{7}_{[21]}-C^{4}_{[21]}C^{5}_{[21]}-C^{3}_{[21]}C^{6}_{[21]}\\\
&+C^{8}_{[21]}C^{8}_{[21]}-C^{6}_{[21]}C^{3}_{[21]}+C^{9}_{[21]}C^{9}_{[21]}\\\
&-C^{5}_{[21]}C^{4}_{[21]}+C^{7}_{[21]}C^{2}_{[21]})\end{split}$ (16)

For the diquark-antidiquark structure, the color wave functions of the diquark
clusters is given,

$\displaystyle C^{1}_{[2]}$ $\displaystyle=$ $\displaystyle
rr,C^{2}_{[2]}=\sqrt{\frac{1}{2}}(rg+gr)$ $\displaystyle C^{3}_{[2]}$
$\displaystyle=$ $\displaystyle gg,C^{4}_{[2]}=\sqrt{\frac{1}{2}}(rb+br)$
$\displaystyle C^{5}_{[2]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(gb+bg),C^{6}_{[2]}=bb$ $\displaystyle
C^{7}_{[11]}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1}{2}}(rg-
gr),C^{8}_{[11]}=\sqrt{\frac{1}{2}}(rb-br)$ $\displaystyle C^{9}_{[11]}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{1}{2}}(gb-bg)$

And the color wave functions of the antidiquark clusters can be writen as:

$\displaystyle C^{1}_{[22]}$ $\displaystyle=$
$\displaystyle\bar{r}\bar{r},C^{2}_{[22]}=-\sqrt{\frac{1}{2}}(\bar{r}\bar{g}+\bar{g}\bar{r})$
$\displaystyle C^{3}_{[22]}$ $\displaystyle=$
$\displaystyle\bar{g}\bar{g},C^{4}_{[22]}=\sqrt{\frac{1}{2}}(\bar{r}\bar{b}+\bar{b}\bar{r})$
$\displaystyle C^{5}_{[22]}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}(\bar{g}\bar{b}+\bar{b}\bar{g}),C^{6}_{[22]}=\bar{b}\bar{b}$
$\displaystyle C^{7}_{[211]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\bar{r}\bar{g}-\bar{g}\bar{r}),C^{8}_{[211]}=-\sqrt{\frac{1}{2}}(\bar{r}\bar{b}-\bar{b}\bar{r})$
$\displaystyle C^{9}_{[211]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\bar{g}\bar{b}-\bar{b}\bar{g})$

The color wave functions of the diquark-antidiquark structure shown in Fig.
1(c) are $\chi^{c}_{3}$ (color sextet-antisextet clusters,
$\textbf{6}_{c}\otimes\bar{\textbf{6}}_{c}$) and $\chi^{c}_{4}$ (color-
triplet-antitriplet cluster, $\textbf{3}_{c}\otimes\bar{\textbf{3}}_{c}$).

$\begin{split}\chi^{c}_{3}=&\sqrt{\frac{1}{6}}(C^{1}_{[2]}C^{1}_{[22]}-C^{2}_{[2]}C^{[2]}_{[22]}+C^{3}_{[2]}C^{3}_{[22]}\\\
&+C^{4}_{[2]}C^{4}_{[22]}-C^{5}_{[2]}C^{5}_{[22]}+C^{6}_{2}C^{6}_{22})\end{split}$
(19)

$\begin{split}\chi^{c}_{4}=&\sqrt{\frac{1}{3}}(C^{7}_{[11]}C^{7}_{[211]}-C^{8}_{[11]}C^{8}_{[211]}+C^{9}_{[11]}C^{9}_{[211]})\end{split}$
(20)

####  II.2.2 The flavor wave function

For the flavor degree of freedom, since the quark content of the tetraquark
systems are two heavy quarks and two strange quarks, the isoscalar sector is
$I=0$. The flavor wave functions denoted as $F^{i}_{I}$, with the superscript
$I$ referring to isoscalar, can be written as

$\displaystyle F^{1}_{0}$ $\displaystyle=$ $\displaystyle c\bar{c}s\bar{s}$
$\displaystyle F^{2}_{0}$ $\displaystyle=$ $\displaystyle c\bar{s}s\bar{c}$
$\displaystyle F^{3}_{0}$ $\displaystyle=$ $\displaystyle cs\bar{c}\bar{s}$
(21)

####  II.2.3 The spin wave function

For the spin, the total spin $S$ of tetraquark states ranges from 0 to 2. All
of them are considered. The wave functions of two body clusters are

$\displaystyle\chi_{11}$ $\displaystyle=$ $\displaystyle\alpha\alpha,$
$\displaystyle\chi_{10}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\alpha\beta+\beta\alpha)$
$\displaystyle\chi_{1-1}$ $\displaystyle=$ $\displaystyle\beta\beta$
$\displaystyle\chi_{00}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\alpha\beta-\beta\alpha)$ (22)

Then, the total spin wave functions $S^{i}_{s}$ are obtained by considering
the coupling of two subcluster spin wave functions with SU(2) algebra, and the
total spin wave functions of four-quark states can be read as

$\displaystyle S^{1}_{0}$ $\displaystyle=$ $\displaystyle\chi_{00}\chi_{00}$
$\displaystyle S^{2}_{0}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}(\chi_{11}\chi_{1-1}-\chi_{10}\chi_{10}+\chi_{1-1}\chi_{11})$
$\displaystyle S^{3}_{1}$ $\displaystyle=$ $\displaystyle\chi_{00}\chi_{11}$
$\displaystyle S^{4}_{1}$ $\displaystyle=$ $\displaystyle\chi_{11}\chi_{00}$
$\displaystyle S^{5}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}(\chi_{11}\chi_{10}-\chi_{10}\chi_{11})$
$\displaystyle S^{6}_{2}$ $\displaystyle=$ $\displaystyle\chi_{11}\chi_{11}$
(23)

####  II.2.4 The orbital wave function

Among the different methods to solve the Schrödinger-like 4-body bound state
equation, we use the resonating group method (RGM) RGM , which is one of the
most extend tools to solve eigenvalue problems and scattering problems. The
total orbital wave functions can be constructed by coupling the orbital wave
function of two internal cluster and the relative motion wave function between
two clusters.

$\psi^{L}=\psi_{1}(R_{1})\psi_{2}(R_{2})\chi_{L}(R)$ (24)

where $R_{1}$ and $R_{2}$ are the internal coordinates for the cluster 1 and
cluster 2. $R=R_{1}-R_{2}$ is the relative coordinate between the two clusters
1 and 2. The $\psi_{1}$ and $\psi_{2}$ are the internal cluster orbital
functions of the clusters 1 and clusters 2, and $\chi_{L}(R)$ is the relative
motion wave function between two clusters, which is expanded by the gaussian
bases

$\displaystyle\begin{split}\chi_{L}(R)=&\sqrt{\frac{1}{4\pi}}(\frac{3}{2\pi
b^{2}})\sum^{n}_{i=1}C_{i}\\\
&\times\int{exp}[-\frac{3}{4b^{2}}(R-s_{i})^{2}]Y_{LM}(\hat{s_{i}})d\hat{s_{i}}\end{split}$
(25)

where n is the number of gaussian bases, which is determined by the stability
of the results.

Finally, to fulfill the Pauli principle, the complete wave function is written
as

$\psi=A[[\psi^{L}S^{j}_{s}]_{JM_{J}}F^{i}_{I}\chi^{c}_{k}]$ (26)

where A is the antisymmetry operator of double-heavy tetraquarks. In this
work, the operator A is defined as $A=1$ due to the absence of any homogeneous
quarks in the $c\bar{c}s\bar{c}$ system.

##  III RESULTS AND DISCUSSIONS

The low-lying $S-$wave states of $c\bar{c}s\bar{s}$ tetraquark are
systematically investigated herein. The parity for $c\bar{c}s\bar{s}$
tetraquark is positive under our assumption that the total orbital angular
momenta $L$ is 0. Accordingly, the total angular momenta, $J$, can take values
of 0, 1, 2. The value of isospin can only be 0 for the $c\bar{c}s\bar{s}$
tetraquark system. Two structures of $c\bar{c}s\bar{s}$ tetraquark, meson-
meson and diquark-antidiquark structures, are investigated. In each structure,
all possible states are considered, which are listed in Table 3. The
$F^{i}_{I};S^{j}_{s};\chi^{c}_{k}$ shows the necessary basis combination in
flavor $(F^{i}_{I})$, spin $(S^{j}_{s})$ and color $(\chi^{c}_{k})$ degrees of
freedom. For meson-meson structure, only the color singlet-singlet $(1\times
1)$ is taken into account because of the effect of hidden color channel
coupling is considered in QDCSM QDCSM1 ; QDCSM2 .

Table 3:  All possible channels for all quantum numbers $IJ^{P}=00^{+}$ | $IJ^{P}=01^{+}$ | $IJ^{P}=02^{+}$
---|---|---
index | $F^{i}_{I};S^{j}_{s};\chi^{c}_{k}$ | channels | index | $F^{i}_{I};S^{j}_{s};\chi^{c}_{k}$ | channels | index | $F^{i}_{I};S^{j}_{s};\chi^{c}_{k}$ | channels
| [i;j;k] |  |  | [i;j;k] |  |  | [i;j;k] |
1 | [1,1,1] | $\eta_{c}\eta_{s}$ | 1 | [1,3,1] | $\eta_{c}\phi$ | 1 | [1,6,1] | $J/\psi\phi$
2 | [2,1,1] | $D_{s}\bar{D}_{s}$ | 2 | [2,3,1] | $D_{s}{D}_{s}^{*}$ | 2 | [2,6,1] | $D_{s}^{*}{D}_{s}^{*}$
3 | [1,2,1] | $J/\psi\phi$ | 3 | [1,4,1] | $J/\psi\eta_{s}$ | 3 | [3,6,3] | $(cs)(\bar{c}\bar{s})$
4 | [2,2,1] | $D_{s}^{*}\bar{D}_{s}^{*}$ | 4 | [2,4,1] | $D_{s}^{*}{D}_{s}$ | 4 | [3,6,4] | $(cs)(\bar{c}\bar{s})$
5 | [3,1,3] | $(cs)(\bar{c}\bar{s})$ | 5 | [1,5,1] | $J/\psi\phi$ |
6 | [3,1,4] | $(cs)(\bar{c}\bar{s})$ | 6 | [2,5,1] | $D_{s}^{*}{D}_{s}^{*}$ |
7 | [3,2,3] | $(cs)(\bar{c}\bar{s})$ | 7 | [3,3,3] | $(cs)(\bar{c}\bar{s})$ |
8 | [3,2,4] | $(cs)(\bar{c}\bar{s})$ | 8 | [3,3,4] | $(cs)(\bar{c}\bar{s})$ |
| 9 | [3,4,3] | $(cs)(\bar{c}\bar{s})$ |
| 10 | [3,4,4] | $(cs)(\bar{c}\bar{s})$ |
| 11 | [3,5,3] | $(cs)(\bar{c}\bar{s})$ |
| 12 | [3,5,4] | $(cs)(\bar{c}\bar{s})$ |

The energy of $c\bar{c}s\bar{s}$ tetraquark system with $IJ^{P}=00^{+}$,
$01^{+}$, and $02^{+}$ for both the meson-meson and diquark-antidiquark
structures, as well as the channel coupling of these two structures are listed
in Table 4, 5, 6, respectively. In those tables, the first column represents
the index of every possible channel; the second column lists the corresponding
physical channels; the third column indicates the theoretical threshold of
every channel; the fourth column ($E_{sc}$) is the energy of every single
channel; the fifth column ($E_{cc}$) shows the energy by channel coupling of
one certain configuration; the last column ($E_{mix}$) is the lowest energy of
the system by coupling all channels of both two configurations.

Table 4:  The lowest-lying eigenenergies of $c\bar{c}s\bar{s}$ tetraquarks with $IJ^{P}=00^{+}$ in the QDCSM.  Index | Channel | Threshold | $E_{sc}$ | $E_{cc}$ | $E_{mix}$
---|---|---|---|---|---
1 | $\eta_{s\bar{s}}\eta_{c\bar{c}}$ | 3942 | 3944 | 3938 | 3930
2 | $D_{s}\bar{D_{s}}$ | 3936 | 3938 |  |
3 | $J/\psi\phi$ | 4117 | 4119 |  |
4 | $D_{s}^{*}\bar{D}_{s}^{*}$ | 4224 | 4226 |  |
5 | $(cs)(\bar{c}\bar{s})$ |  | 4324 | 4219 |
6 | $(cs)(\bar{c}\bar{s})$ |  | 4442 |  |
7 | $(cs)(\bar{c}\bar{s})$ |  | 4405 |  |
8 | $(cs)(\bar{c}\bar{s})$ |  | 4305 |  |
Table 5:  The lowest-lying eigenenergies of $c\bar{c}s\bar{s}$ tetraquarks with $IJ^{P}=01^{+}$ in the QDCSM.  Index | Channel | Threshold | $E_{sc}$ | $E_{cc}$ | $E_{mix}$
---|---|---|---|---|---
1 | $\eta_{c\bar{c}}\phi$ | 4004 | 4006 | 4006 | 4006
2 | $D_{s}\bar{D_{s}^{*}}$ | 4080 | 4082 |  |
3 | $J/\psi\eta_{s\bar{s}}$ | 4055 | 4057 |  |
4 | $D_{s}^{*}\bar{D_{s}}$ | 4080 | 4082 |  |
5 | $J/\psi\phi$ | 4117 | 4119 |  |
6 | $D_{s}^{*}\bar{D}_{s}^{*}$ | 4224 | 4226 |  |
7 | $(cs)(\bar{c}\bar{s})$ |  | 4375 | 4327 |
8 | $(cs)(\bar{c}\bar{s})$ |  | 4419 |  |
9 | $(cs)(\bar{c}\bar{s})$ |  | 4375 |  |
10 | $(cs)(\bar{c}\bar{s})$ |  | 4419 |  |
11 | $(cs)(\bar{c}\bar{s})$ |  | 4413 |  |
12 | $(cs)(\bar{c}\bar{s})$ |  | 4352 |  |
Table 6:  The lowest-lying eigenenergies of $c\bar{c}s\bar{s}$ tetraquarks with $IJ^{P}=02^{+}$ in the QDCSM.  Index | Channel | Threshold | $E_{sc}$ | $E_{cc}$ | $E_{mix}$
---|---|---|---|---|---
1 | $J/\psi\phi$ | 4117 | 4122 | 4121 | 4119
2 | $D_{s}^{*}\bar{D}_{s}^{*}$ | 4224 | 4229 |  |
3 | $(cs)(\bar{c}\bar{s})$ |  | 4429 | 4420 |
4 | $(cs)(\bar{c}\bar{s})$ |  | 4437 |  |

The $IJ^{P}=00^{+}$ system: Four possible meson-meson channels,
$\eta_{s\bar{s}}\eta_{c\bar{c}},D_{S}\bar{D_{s}},J/\psi\phi,D_{s}^{*}\bar{D}_{s}^{*}$,
and two diquark-antiquark channels, $((cs)(\bar{c}\bar{s}))(3\times\bar{3})$
and $((cs)(\bar{c}\bar{s}))(6\times\bar{6})$, are studied in QDCSM. All
results with the $IJ^{P}=00^{+}$ are given in Table 4. We can see that the
energy of every single channel for the meson-meson structure is higher than
the corresponding theoretical threshold, which indicates that there is no any
bound state. For the diquark-antidiquark configuration, all the masses are
higher than the lowest energy of $D_{s}\bar{D_{s}}$ in our model calculation,
and the minimum energy is 4305 MeV. Then, we perform the channel-coupling
calculation on both the meson-meson and diquark-antidiquark structure,
respectively. The energy of the meson-meson structure is 3938 MeV, almost the
same as the lowest single channel $D_{s}\bar{D_{s}}$, which indicates that the
effect of the channel coupling is quite weak and no bound state is found for
the meson-meson structure. For the diquark-antidiquark structure, although the
coupling is rather stronger than the meson-meson structure, the energy is
still higher than the theoretical threshold of the lowest channel
$D_{s}\bar{D_{s}}$.

However, the lowest energy of $3930$ MeV is obtained by coupling all channels
of two structures, which is 6 MeV lower than the threshold of the lowest
channel $D_{s}\bar{D_{s}}$, which means that there is a bound state for the
$IJ^{P}=00^{+}$ $c\bar{c}s\bar{s}$ tetraquark system with mass of $3930$ MeV.
In addition, to explore the structure of this bound state, we calculate the
proportion of each channel, and find that the percentage of the
$D_{s}\bar{D}_{s}$ state is about $85\%$, while the percentages of the other
seven channels are much smaller. This means that the largest contribution to
forming this bound state comes from the $D_{s}\bar{D}_{s}$ channel, so this
bound state tends to be a molecular state. Moreover, this value is in
proximity to the $\chi_{c0}(3930)$ observed by the LHCb collaboration. So we
can explain the $\chi_{c0}(3930)$ as a molecular state $D_{s}\bar{D}_{s}$ in
present quark model calculation. This result is consistent with the work of
Ref. lattice_X3930 , in which the lattice QCD calculation with $m_{\pi}\simeq
280$ MeV indicated the existence of the scalar $\bar{D}_{s}D_{s}$ bound state,
which might correspond to the $\chi_{c0}(3930)$ observed by the LHCb
collaboration LHCb_X3930.1 ; LHCb_X3930.2 . Also, in Ref BS_X3930 , two pole
positions of $\bar{D}_{s}D_{s}$ system were obtained by solving the Bethe-
Salpeter equation, which explained the properties of new exotic resonance
$\chi_{c0}(3930)$.

The $IJ^{P}=01^{+}$ system: From Table 3, there are six meson-meson channels
and six diquark-antidiquark channels. Table 5 lists the calculated masses of
these channels and also their coupling results. The energy range of every
single-channel of the meson-meson structure is about $4.0-4.2$ GeV, and the
mass of the diquark-antidiquark channel is around 4.4 GeV. All these single
channels are unbound. By coupling the channels with the same configuration,
the lowest masses are located at 4006 MeV for the meson-meson structure and
4327 MeV for the diquark-antidiquark structure, both of which are still above
the threshold of the lowest channel $\eta_{c\bar{c}}\phi$, indicating that no
any bound state exists in the meson-meson structure or the diquark-antidiquark
structure. Meanwhile, the lowest energy is still 4006 MeV by the full channel
coupling calculation, which means that the effect of all channel coupling is
very minor here, and there is no any bound state in the $IJ^{P}=01^{+}$
system.

The $IJ^{P}=02^{+}$ system: Table 6 shows that there are two channels
($J/\psi\phi$ and $D_{s}^{*}\bar{D}_{s}^{*}$) of the meson-meson structure and
two channels of the diquark-antidiquark configurations for the $IJ^{P}=02^{+}$
system. The situation is similar to the $IJ^{P}=01^{+}$ case. The energy of
each channel is above the threshold of the corresponding channel. Meanwhile,
the channel coupling cannot help too much, the lowest energy is still higher
than the threshold of the lowest channel $J/\psi\phi$. Therefore, there is no
any bound state in the $IJ^{P}=02^{+}$ system at present calculation.

Although there is no any bound state for the $IJ^{P}=01^{+}$ and
$IJ^{P}=02^{+}$ system, some resonance states are still possible in the
$c\bar{c}s\bar{s}$ tetraquark system. The colorful subclusters diquark and
antidiquark cannot fall apart directly due to the color confinement, so it is
possible for them to be resonance states. To find out if there is any
resonance state, a stabilization method (also named a real scaling method),
which has been successfully applied in other multiquark systems real_method2 ;
real_method3 , is used in this work. To realize the real scaling method in our
calculation, the distance between two clusters is defined as $S$. With the
increase of $S$, each state will fall off towards its threshold, except the
resonance state, the energy of which will be stable because it will not be
affected by the boundary at a large distance. So we calculate the energy
eigenvalues of the $c\bar{c}s\bar{s}$ systems by taking the value of $S$ from
4.1 fm to 9.0 fm to see if there is any stable state. The results of the
$c\bar{c}s\bar{s}$ tetraquark systems with $IJ^{P}=0{0^{+}},0{1^{+}}$ and
$0{2^{+}}$ are shown in Fig 2, Fig 3 and Fig 4, respectively.

Figure 2: The stabilization plots of the energies of the $c\bar{c}s\bar{s}$
with $IJ^{P}=00^{+}$ in QDCSM.  Figure 3: The stabilization plots of the
energies of the $c\bar{c}s\bar{s}$ with $IJ^{P}=01^{+}$ in QDCSM.  Figure 4:
The stabilization plots of the energies of the $c\bar{c}s\bar{s}$ with
$IJ^{P}=02^{+}$ in QDCSM.

For $IJ^{P}=0{0^{+}}$ system in Fig 2, it is obvious that the lowest
horizontal line locates at the energy of $3930$ MeV, which represents the
bound state of this system. Then, three horizontal lines, which stand for the
thresholds of $\eta_{s\bar{s}}\eta_{c\bar{c}}$, $J/\psi\phi$ and
$D^{*}_{s}\bar{D}^{*}_{s}$, are marked in Fig 2. Besides, another four
horizontal lines appear in Fig 2, corresponding to four resonance states with
the energy around 4035 MeV, 4385 MeV, 4524 MeV, and 4632 MeV, respectively. By
comparing with the experimental results, we find that the energy of 4385 MeV
is close to the $X(4350)$, and the quantum number $IJ^{P}=0{0^{+}}$ is
consistent with the reported data by the Belle Collaboration Bell-CCSS . So we
explain the $X(4350)$ as a compact tetraquark resonance state with
$IJ^{P}=0{0^{+}}$ in present calculation. Our result is also agrees with the
results of the Born-Oppenheimer approach, in which a mass of $4370$ MeV was
obtained X(4350)CCSS . Besides, in Ref CHQM-CCSS , $X(4350)$ was also a good
candidate of the compact tetraquark state with $IJ^{P}=0{0^{+}}$ in the chiral
quark model. Similarly, the resonance energy of 4524 MeV is close to the
X(4500), and the quantum number $IJ^{P}=0{0^{+}}$ is also consistent with the
reported result of the LHCb Collaboration LHCb-CCSS-2017.2 . So $X(4500)$ is
possible to be a compact tetraquark resonance state with $IJ^{P}=0{0^{+}}$ in
present calculation. In addition to the $X(4350)$ and $X(4500)$, another
resonance state with energy around 4632 MeV is obtained. Although the mass is
very close to the $X(4630)$, the quantum number $IJ^{P}=0{0^{+}}$ is different
from the reported one $IJ^{P}=0{1^{-}}$ LHCb-CCSS-2021 . However, the mass is
also close to the state $X(4700)$, and the quantum number is also fit to the
experimental data of the LHCb Collaboration LHCb-CCSS-2017.2 . So we prefer to
assign the resonance state with energy 4632 MeV to be the exotic state
$X(4700)$.

For $c\bar{c}s\bar{s}$ system with $IJ^{P}=0{1^{+}}$ in Fig. 3, the first six
horizontal lines located at the corresponding physical threshold of six
channels, which are $\eta_{c\bar{c}}\phi$, $D_{s}\bar{D_{s}^{*}}$,
$J/\psi\eta_{s\bar{s}}$, $D_{s}^{*}\bar{D_{s}}$, $J/\psi\phi$ and
$D_{s}^{*}\bar{D}_{s}^{*}$. Obviously, a resonant state is obtained at the
energy around 4327 MeV. Although the mass is very close to the $X(4350)$, the
quantum number is not quite applicable. However, the LHCb Collaboration
claimed the existence of the $X(4274)$ and the measured quantum number was
$J^{P}=1^{+}$ LHCb-CCSS-2017.1 . Therefore, we tend to use the resonance state
around 4327 MeV to explain the $X(4274)$ state in this work. This is similar
to the result of Ref 2011_CCSS , in which a resonance state with energy near
4.3 GeV is considered as the $X(4274)$.

For the last system $IJ^{P}=0{2^{+}}$ in Fig. 4, the first two horizontal
lines represent obviously the thresholds of two channels: $J/\psi\phi$ and
$D_{s}^{*}\bar{D}_{s}^{*}$. Another two horizontal lines stand for two
resonance states, the energy of which is about 4419 MeV and 4526 MeV,
respectively. One may note that the energy of 4526 MeV is also very close to
the mass of the $X(4500)$, but the quantum number is not consistent with the
experimental data. So these two resonance state maybe some new exotic states.

##  IV Summary

The $cs\bar{c}\bar{s}$ tetraquark systems with $IJ^{P}=00^{+}$, $01^{+}$ and
$02^{+}$ have been systemically investigated by using the RGM in the framework
of QDCSM. Our goal is to search for any bound state or resonance state to
explain the exotic states, which have been recently observed in the invariant
mass distribution of $J/\psi\phi$ and another exotic state $\chi_{c0}(3930)$
observed by the LHCb collaboration. In this work, two structures: the meson-
meson and diquark-antidiquark structures are taken into account. Both single-
channel and channel-coupling calculations are performed. Besides, to search
for any resonance state, a stabilization method is applied to the coupling
calculation of all channels of both two configurations.

The numerical results show that we obtain a bound molecular state
$\bar{D}_{s}D_{s}$ with the quantum number $IJ^{P}=00^{+}$ and the energy 3930
MeV, which can be used to explain the observed $\chi_{c0}(3930)$. Moreover,
several resonant states are obtained in this work, which are four
$IJ^{P}=00^{+}$ states with the resonance masses around 4035 MeV, 4385 MeV,
4524 MeV, and 4632 MeV, respectively; one $IJ^{P}=01^{+}$ state with the
resonance mass around 4327 MeV; and two $IJ^{P}=02^{+}$ states with the
resonance masses around 4419 MeV and 4526 MeV, respectively. All of them are
obtained by coupling all channels of both the meson-meson and diquark-
antidiquark structures, so they are compact tetraquarks in present quark model
calculations. By comparing with the experimental data, we are inclined to
explain the exotic states $X(4350)$, $X(4500)$ and $X(4700)$ as the compact
tetraquark state with $IJ^{P}=00^{+}$. The $X(4274)$ is possible to be a
candidate of the compact tetraquark state with $IJ^{P}=01^{+}$.

All these resonance states are worth searching by experiments. We suggest more
experimental tests to check the existence of all these possible resonance
states. In addition, to confirm the existence of these $cs\bar{c}\bar{s}$
tetraquark, the study of the scattering process of the corresponding open
channels is needed in future work.

###### Acknowledgements.

This work is supported partly by the National Natural Science Foundation of
China under Contract No. 11675080, No. 11775118, No. 11535005, and No.
11775050.

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# Congruity of Crab pulsar’s gamma-ray spectrum with the spectral distribution
of the radiation by the current sheet in its magnetosphere

Houshang Ardavan
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3
0HA, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The spectrum derived here for the most tightly-focused component of the
radiation generated by the superluminally moving current sheet in the
magnetrosphere of a non-aligned neutron star has a distribution function that
fits the entire gamma-ray spectrum of the Crab pulsar on its own. This is the
first time that the undivided breadth of this spectrum, from $10^{2}$ to
$10^{6}$ MeV, is not only described by a single distribution function but is
also explained by means of a single emission mechanism.

###### keywords:

pulsars: individual: J0534+2200 – gamma-rays: stars – stars: neutron –
methods: data analysis – radiation mechanisms: non-thermal – radiation:
dynamics.

††pubyear: 2022††pagerange: Congruity of Crab pulsar’s gamma-ray spectrum with
the spectral distribution of the radiation by the current sheet in its
magnetosphere–Congruity of Crab pulsar’s gamma-ray spectrum with the spectral
distribution of the radiation by the current sheet in its magnetosphere

##  1 Introduction

Numerical computations based on the force-free and particle-in-cell formalisms
have now firmly established that the magnetosphere of a non-aligned neutron
star entails a current sheet outside its light cylinder whose rotating
distribution pattern moves with linear speeds exceeding the speed of light in
vacuum (see the review article Philippov & Kramer, 2022, and the references
therein). A study of the characteristics of the radiation that is generated by
this superluminally moving current sheet has in turn provided an all-
encompassing explanation for the salient features of the radiation received
from pulsars: its brightness temperature, polarization, spectrum and profile
with microstructure and with a phase lag between the radio and gamma-ray peaks
(Ardavan, 2021, 2022a).

The radiation field generated by a constituent volume element of the current
sheet in the magnetosphere of a neutron star embraces a synergy between the
superluminal version of the field of synchrotron radiation and the vacuum
version of the field of Čerenkov radiation. Once superposed to yield the
emission from the entire volume of the source, the contributions from the
volume elements of the current sheet that approach the observation point with
the speed of light and zero acceleration at the retarded time interfere
constructively and form caustics in certain latitudinal directions relative to
the spin axis of the neutron star. The waves that embody these caustics are
more focused the further they are from their source: as their distance from
their source increases, two nearby stationary points of their phases draw
closer and eventually coalesce at infinity (Ardavan, 2021, Section 4). By
virtue of their extremely narrow peaks in the time domain, such focused pulses
have broad spectra that encompass X-ray and gamma-ray frequencies (Ardavan,
2021, Table 1 and Section 5.3). In directions where the stationary points of
their phases are isolated, the emitted pulses are not focused as strongly and
so are detectable at lower frequencies (Ardavan, 2021, Section 5.4).

Here, the contribution toward the spectrum of this radiation from the vicinity
of its tightly-focused caustics is derived in Section 2 and it is shown that
the derived spectral distribution function fits the observed gamma-ray
spectrum of the Crab pulsar (Abdo et al., 2010; Aleksić et al., 2011; Ansoldi
et al., 2016) over its entire breadth from $10^{2}$ to $10^{6}$ MeV (Section
3). The contrast between the single radiation mechanism underlying the present
spectral distribution function and the disparate radiation mechanisms and
spectral distribution functions currently invoked in the literature for
fitting the data in different sections of the Crab pulsar’s gamma-ray spectrum
(see the review articles Zanin, 2017; Amato & Olmi, 2021, and the references
therein) is briefly commented on in Section 4.

##  2 Spectral distribution function of tightly-focused caustics

The frequency spectrum of the radiation that is generated by the
superluminally moving current sheet in the magnetosphere of a non-aligned
neutron star was presented, in its general form, in Ardavan (2021, Section
5.3). Here we derive the spectral distribution function of the most tightly-
focused component of this radiation from the general expression given in
equation (177) of that paper.

In a case where the magnitudes of the vectors denoted by $\boldsymbol{\cal
P}_{l}$ and $\boldsymbol{\cal Q}_{l}$ in eqaution (177) of Ardavan (2021) are
appreciably larger than those of their counterparts, ${\bar{\boldsymbol{\cal
P}}}_{l}$ and ${\bar{\boldsymbol{\cal Q}}}_{l}$, and the dominant contribution
towards the flux density $S_{\nu}$ of the radiation is made by only one of the
two terms corresponding to $l=1$ and $l=2$, eqaution (177) of Ardavan (2021)
can be written as

$S_{\nu}=\kappa_{0}\,k^{-2/3}\left|\boldsymbol{\cal P}{\rm
Ai}(-k^{2/3}\sigma^{2})-{\rm i}k^{-1/3}\boldsymbol{\cal Q}{\rm
Ai}^{\prime}(-k^{2/3}\sigma^{2})\right|^{2},$ (1)

where Ai and ${\rm Ai}^{\prime}$ are the Airy function and the derivative of
the Airy function with respect to its argument, respectively,
$k=2\pi\nu/\omega$ is the frequency $\nu$ of the radiation in units of the
rotation frequency $\omega/2\pi$ of the central neutron star, and the positive
constants $\kappa_{0}$ and $\sigma$ and the complex vectors $\boldsymbol{\cal
P}$ and $\boldsymbol{\cal Q}$ are determined by the characteristics of the
current sheet. The above spectrum is emblematic of any radiation that entails
caustics (see Stamnes, 1986).

Evaluation of the right-hand side of equation (1) results in

$\displaystyle S_{\nu}$ $\displaystyle=$
$\displaystyle\kappa_{1}\,k^{-2/3}\left[{\rm
Ai}^{2}(-k^{2/3}\sigma^{2})+a^{2}k^{-2/3}{{\rm
Ai}^{\prime}}^{2}(-k^{2/3}\sigma^{2})\right.$ (2)
$\displaystyle\left.+2a\cos\beta\,k^{-1/3}{\rm Ai}(-k^{2/3}\sigma^{2}){\rm
Ai}^{\prime}(-k^{2/3}\sigma^{2})\right],$

where

$\kappa_{1}=|\boldsymbol{\cal P}|^{2}\kappa_{0},\qquad
a=\frac{|{\boldsymbol{\cal Q}}|}{|{\boldsymbol{\cal
P}}|},\qquad\cos\beta=\frac{\Im(\boldsymbol{\cal Q}\cdot\boldsymbol{\cal
P}^{*})}{|\boldsymbol{\cal Q}||\boldsymbol{\cal P}|},$ (3)

and $\Im{}$ and $*$ denote an imaginary part and the complex conjugate,
respectively. The positive parameter $\sigma$ has a vanishingly small value at
the peak of a high-frequency pulse (Ardavan, 2021, Section 4.5). For the
purposes of the present analysis, we may therefore replace $\boldsymbol{\cal
P}$ and $\boldsymbol{\cal Q}$ by their limiting values for $k\gg 1$ and
$\sigma\ll 1$ and treat the quantities $\kappa_{1}$, $a$ and $\beta$ that
appear in equation (2) as constant parameters.

To take account of the fact that the parameter $\sigma$ assumes a non-zero
range of values (bordering on $\sigma=0$) across the width of a high-frequency
pulse (Ardavan, 2022a, Section 3), we must integrate $S_{\nu}$ with respect to
$\sigma$ over a finite interval $0\leq\sigma\leq\sigma_{0}$ with
$\sigma_{0}\ll 1$. Performing the integration of the Airy functions in
equation (2) with respect to $\sigma$ by means of Mathematica, we thus obtain

$\displaystyle{\cal S}_{\nu}$ $\displaystyle=$
$\displaystyle\int_{0}^{\sigma_{0}}S_{\nu}\,{\rm d}\sigma$ (4)
$\displaystyle=$
$\displaystyle\kappa\left[f_{1}(\chi)+\frac{b^{2}}{4\sqrt{3}}f_{2}(\chi)-\frac{b\cos\beta}{2\sqrt{3}}f_{3}(\chi)\right],$

where

$\kappa=\left(\frac{{\sigma_{0}}}{\sqrt{3\pi}}\right)^{3}\kappa_{1},\qquad
b=\sigma_{0}a,\qquad\chi=\frac{2}{3}{\sigma_{0}}^{3}k,$ (5) $\displaystyle
f_{1}(\chi)$ $\displaystyle=$ $\displaystyle
3\Gamma\left(\frac{7}{6}\right)\chi^{-2/3}{}_{2}F_{3}\left(\begin{matrix}1/6&1/6&{}\\\
1/3&2/3&7/6\end{matrix};-\chi^{2}\right)$ (6)
$\displaystyle+\pi^{1/2}{}_{2}F_{3}\left(\begin{matrix}1/2&1/2&{}\\\
2/3&4/3&3/2\end{matrix}\,;-\chi^{2}\right)$
$\displaystyle+\frac{9}{20}\Gamma\left(\frac{5}{6}\right)\chi^{2/3}{}_{2}F_{3}\left(\begin{matrix}5/6&5/6&{}\\\
4/3&5/3&11/6\end{matrix}\,;-\chi^{2}\right),$
$f_{2}(\chi)=\chi^{-4/3}\,{}_{24}G^{31}\left(-\chi^{2/3},\frac{1}{3}\left|\,\begin{matrix}5/6&7/6&{}&{}\\\
0&2/3&4/3&-1/6\end{matrix}\right),\right.$ (7)
$f_{3}(\chi)=\chi^{-1}\,{}_{24}G^{31}\left(-\chi^{2/3},\frac{1}{3}\left|\,\begin{matrix}5/6&1/2&{}&{}\\\
0&1/3&2/3&-1/6\end{matrix}\right),\right.$ (8)

and ${}_{2}F_{3}$ and ${}_{24}G^{31}$ are respectively the generalised
hypergeometric function (see Olver et al., 2010) and the generalised Meijer
G-Function (see https://mathworld.wolfram.com/MeijerG-Function.html). The
variable $\chi$ that appears in the above expressions is related to the
frequency $\nu$ of the radiation via $\chi=4\pi{\sigma_{0}}^{3}\nu/(3\omega)$.

The scale and shape of the spectral distribution function ${\cal
S}_{\nu}(\nu)$ given in equation (4) depend on the four parameters $\kappa$,
$b$, $\beta$ and $\sigma_{0}$: parameters whose values are dictated by the
characteristics of the magnetospheric current sheet. The parameters $\kappa$
and $\sigma_{0}$ respectively determine the scale of the flux density and the
range of frequencies over which the flux density has a detectable value, while
the parameters $b$ and $\beta$ determine the shape of this distribution.

##  3 Fitting the derived distribution function to the data on the gamma-ray
spectrum of the Crab pulsar

The phase-averaged flux density of the gamma-rays received from the Crab
pulsar is plotted (in unitts of erg cm-2 s-1 MeV-1) versus photon energy (in
units of MeV) in figure 1. The data points below $25$ GeV in this figure are
the ones detected by the FERMI Large Area Telescope (FERMI-LAT) which were
reported in Abdo et al. (2010). The three data points between $25$ and $100$
GeV are those measured by the Major Atmospheric Gamma Imaging Cherenkov
(MAGIC) telescope and were reported in Aleksić et al. (2011).

Having been derived by a Fourier decomposition of the pulse profile with
respect to time, equation (1) and hence the expression for ${\cal S}_{\nu}$ in
equation (4) describe a spectral distribution that is averaged over the whole
pulse period (see Ardavan, 2021). On the other hand, the latest data by MAGIC
on photon energies exceeding $100$ GeV are reported by Ansoldi et al. (2016)
for the main peak and the interpulse of the Crab pulsar separately. To
estimate the phase-averaged fluxes at the five photon energies for which the
fluxes associated with both the main peak and the interpulse are given in
Table 3 of Ansoldi et al. (2016), we have here summed the contributions from
these two components of the pulse profile. Given that the reported upper
limits on the contribution of the main pulse toward the fluxes at $781$ and
$1211$ GeV are comparable to the observational errors in the contribution of
the interpulse to the fluxes at these energies (see Table 3 of Ansoldi et al.,
2016), we have equated the phase averaged fluxes at $781$ and $1211$ GeV to
the fluxes of the interpulse and have incorporated the upper limits on the
contributions of the main pulse toward the phase averaged fluxes at these two
photon energies in the error bars of the last two data points.

The data point with the highest photon energy ($81$ GeV) in the data set
reported by Aleksić et al. (2011) and the data point with the lowest photon
energy ($87$ GeV) in the data set reported by Ansoldi et al. (2016) correspond
to the same photon energy to within the limits of their horizontal error bars.
Hence, any discrepancy between the reported fluxes of these two data points
must be due to the difference partly in the phase intervals over which these
two group of authors have made their measurements and partly in the energy-
dependent systematic uncertainties associated with their experiments (see
Aleksić et al., 2012). Accordingly, we have removed the discrepancy between
the phase-averaged fluxes of the lowest-energy data point extracted from
Ansoldi et al. (2016) and the highest-energy data point given in Aleksić et
al. (2011) by multiplying the former by the factor $9.77$. Once the fluxes of
each of the remaining six data points extracted from Ansoldi et al. (2016) are
multiplied by the same factor, these points are collectively raised to the
level shown in figure 1.

The curve in figure 1 depicts the spectral distribution function ${\cal
S}_{\nu}$ that is described by equation (4) for the following values of its
free parameters:

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle 7.94\times
10^{-11}\quad{\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1},$
$\displaystyle\chi$ $\displaystyle=$ $\displaystyle 5.62\times
10^{-4}\times(h\nu)_{\rm MeV},$ $\displaystyle\sigma_{0}$ $\displaystyle=$
$\displaystyle 4.72\times 10^{-8},\quad b=0.4\quad{\rm and}\quad\beta=0.3,$
(9)

in which $h$ stands for the Planck constant and $2\pi/\omega$ has been set
equal to the period of the Crab pulsar. As the photon energy $h\nu$ increases
beyond $1.2$ TeV, the slope of this curve continues to decrease at the
relatively slow rate in which it decreases past $50$ GeV. Thus the upper
limits given by Ansoldi et al. (2016) for the flux density of gamma-rays whose
energies exceed $1.2$ TeV all lie above the continuation of the curve shown in
figure 1.

Figure 1: Gamma-ray spectrum of the Crab pulsar. The FERMI-LAT data points
below $25$ GeV (coloured black) are those reported in Abdo et al. (2010). The
MAGIC data points between $25$ and $100$ GeV (coloured blue) are those
reported in Aleksić et al. (2011). The MAGIC data points beyond $100$ Gev
(coloured green) are extracted (by the procedure described in Section 3) from
those reported by Ansoldi et al. (2016). The curve (coloured red) is a plot of
the spectral distribution function described by equation (4) for the
parameters given in equation (9).

##  4 Concluding remarks

In the current literature on the gamma-ray spectrum of the Crab pulsar (see
the review articles Zanin, 2017; Amato & Olmi, 2021, and the references
therein), this spectrum is parametrised partly (between $0.1$ and $100$ GeV)
by a power-law function with an exponential cut off and partly (between $100$
GeV and $1$ TeV) by a pure power-law with another exponent. The part of the
emission between $0.1$ and $100$ GeV is commonly attributed to synchrotron,
curvature or synchro-curvature processes. It is generally acknowledged,
however, that the very high energy component of the emission is unlikely to be
generated by such processes. The emission at higher than a few hundred GeV is
thought to be produced either by inverse Compton scattering of lower energy
photons or as a consequence of magnetic reconnection in the magnetospheric
current sheet.

In contrast, we have here fitted the entire gamma-ray spectrum of the Crab
pulsar by a single function representing the spectral distribution function of
a single emission mechanism. That the spectral distribution function of the
radiation generated by the superluminally moving current sheet in the
magnetosphere of the Crab pulsar should provide such an all-encompassing fit
to the observational data on its emission is not surprising given the firm
theoretical grounds on which both the existence of the current sheet in
question (see Philippov & Kramer, 2022) and the characterisation of the
radiation it generates (see Ardavan, 2021) are based. Clearly, the relevance
of the spectral distribution function given in equation (4) is not limited to
the Crab pulsar: this function describes the spectral distribution of the high
frequency emission from any non-aligned neutron star.

As already mentioned, the caustics whose spectra are described by equation (4)
are more focused the further they are from their source (Ardavan, 2021,
Section 4). As a result, flux densities of the pulses that are generated by
the emission mechanism discussed here diminish with the distance $D$ from the
star as $D^{-3/2}$ (rather than $D^{-2}$) in the latitudinal directions along
which the high frequency pulses are detectable (Ardavan, 2021, Section 5.5).
This slower rate of decay of the flux density applies to the high frequency
radiation from all non-aligned neutron stars, including the gamma-ray and/or
X-ray emission from any pulsar or magnetar, thereby reducing the conventional
estimates of their luminosities (see Ardavan, 2022b, c).

## Data availability

The data used in this paper can be found in Abdo et al. (2010), Aleksić et al.
(2011) and Ansoldi et al. (2016).

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