# Exotic fully-heavy $Q\bar{Q}Q\bar{Q}$ tetraquark states in
$\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}$ color configuration

Qi-Nan Wang  Zi-Yan Yang  Wei Chen<EMAIL_ADDRESS>School of
Physics, Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

We have systematically calculated the mass spectra for S-wave and P-wave
fully-charm $c\bar{c}c\bar{c}$ and fully-bottom $b\bar{b}b\bar{b}$ tetraquark
states in the $\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}$ color
configuration, by using the moment QCD sum rule method. The masses for the
fully-charm $c\bar{c}c\bar{c}$ tetraquark states are predicted about $6.3-6.5$
GeV for S-wave channels and $7.0-7.2$ GeV for P-wave channels. These results
suggest the possibility that there are some
$\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$ components in LHCb’s
di-$J/\psi$ structures. For the fully-bottom $b\bar{b}b\bar{b}$ system, their
masses are calculated around 18.2 GeV for S-wave tetraquark states while
18.4-18.8 GeV for P-wave ones, which are below the $\eta_{b}\eta_{b}$ and
$\Upsilon(1S)\Upsilon(1S)$ two-meson decay thresholds.

Di-$J/\psi$, Exotic state, QCD sum rules

###### pacs:

12.39.Mk, 12.38.Lg, 14.40.Ev, 14.40.Rt

##  I Introduction

The existence of multiquark states was first suggested by Gell-Mann and Zweig
at the birth of quark model Gell-Mann:1964ewy ; ZweigSU(3) . Since 2003,
plenty of charmoniumlike exotic states and $P_{c}$ states have been observed
Belle:2003nnu ; BESIII:2013mhi ; BESIII:2013ris ; BaBar:2005hhc ;
BaBar:2006ait ; Belle:2011aa ; Belle:2007umv ; LHCb:2015yax ; LHCb:2019kea ;
LHCb:2020jpq  , many of which are unexpected and can not be fitted into the
conventional quark model. To understand the nature of these new resonances,
many exotic hadron configurations have been proposed such as hadron molecules,
compact multiquarks, hybrid mesons and so on 2016-Chen-p1-121 ;
2017-Ali-p123-198 ; 2017-Lebed-p143-194 ; 2018-Guo-p15004-15004 ;
2019-Liu-p237-320 ; 2020-Brambilla-p1-154 . Among these theoretical models,
the loosely bound hadron molecule and compact multiquark are two especially
appealing configurations. For the charmoniumlike XYZ and $P_{c}$ states, it is
complicated and difficult to distinguish these two different hadron
configurations experimentally and theoretically since the existence of light
quarks.

In 2017, an exotic structure around $18.4$ GeV was reported by CMS
Collaboration in the $\Upsilon(1S)\mu^{+}\mu^{-}$ channel CMS:2016liw , which
was once regarded as a fully-bottom $bb\bar{b}\bar{b}$ tetraquark state. In
2019, the ANDY Collaboration at RHIC reported an evidence of a significance
peak at around $18.12$ GeV ANDY:2019bfn . Although these structures were not
confirmed by some other experiments LHCb:2018uwm ; CMS:2020qwa , their
observations still attracted a lot of research interests in fully heavy
tetraquark states Chen:2016jxd ; Anwar:2017toa ; Esposito:2018cwh ;
Hughes:2017xie ; Karliner:2016zzc ; Wu:2016vtq ; Richard:2017vry ; Bai:2016int
; Chen:2019dvd ; Debastiani:2017msn . Very recently, the LHCb Collaboration
declared a narrow resonance $X(6900)$ in the di-$J\psi$ mass spectrum with the
significance more than $5\sigma$ LHCb:2020bwg . Moreover, a broad structure
ranging from 6.2 to 6.8 GeV and a hint for another structure around 7.2 GeV
were also reported at the same time LHCb:2020bwg . These exotic structures
observed in LHCb immediately attracted great attention to study the fully-
charm $cc\bar{c}\bar{c}$ tetraquarks for their mass spectra
Albuquerque:2020hio ; Giron:2020wpx ; Gordillo:2020sgc ; Guo:2020pvt ;
Jin:2020jfc ; Karliner:2020dta ; Ke:2021iyh ; Li:2021ygk ; Liang:2021fzr ;
Liu:2019zuc ; liu:2020eha ; Pal:2021gkr ; Sonnenschein:2020nwn ; Wan:2020fsk ;
Wang:2018poa ; Wang:2019rdo ; Wang:2020ols ; Wang:2021kfv ; Weng:2020jao ;
Yang:2020rih ; Yang:2020wkh ; Zhang:2020xtb ; Zhao:2020zjh ; Zhao:2020nwy ;
Zhu:2020xni ; Cao:2020gul ; Mutuk:2021hmi ; Yang:2021hrb , their production
mechanisms Huang:2021vtb ; Feng:2020riv ; Wang:2020gmd ; Feng:2020qee ;
Maciula:2020wri ; Goncalves:2021ytq ; Ma:2020kwb ; Wang:2020tpt ; Zhao:2020nwy
; Zhu:2020sn ; Gong:2020bmg  and their decay properties Guo:2020pvt ;
Lu:2020cns ; Li:2019uch ; Chen:2020xwe ; Becchi:2020uvq ; Sonnenschein:2020nwn
. Since the absence of light quarks, a fully-heavy tetraquark system is more
likely to form a compact tetraquark state via the gluon-exchange color
interaction, but rather than a loosely hadron molecule combined by the light
meson exchanged interaction Maiani:2020pur ; Chao:2020dml .

Nevertheless, the authors of Ref. Dong:2021lkh  discussed the interaction
between two $J/\psi$ mesons via the exchange of soft gluons, which hadronise
into two light mesons at large distance. By studying the correlated $\pi\pi$
and $K\bar{K}$ exchanges, they found that it is possible for two $J/\psi$
mesons to form a bound state. In Ref. Albuquerque:2020hio , the authors
studied the di-charmonia states in
$\mathbf{1}_{[c\bar{c}]}\otimes\mathbf{1}_{[c\bar{c}]}$ configuration with
$J^{PC}=0^{++}$ and predicted their masses around 6.0-6.7 GeV in the method of
QCD sum rules. The existence of di-charmonia bound states were also studied in
Ref. Yang:2021hrb , in which the authors investigated the
$\eta_{c}\eta_{c},J/\psi J/\psi$ bound states in both
$\mathbf{1}_{[c\bar{c}]}\otimes\mathbf{1}_{[c\bar{c}]}$ and
$\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$ color structures by
using a non-relativistic quark model. The $0^{++}$ di-charmonia states in the
$\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$ color structure were
also investigated in Ref. Yang:2020wkh  by the Laplace QCD sum rule method.

In our previous works in Refs. Chen:2016jxd ; Yang:2021zrc ; Wang:2021taf , we
have studied the fully-heavy tetraquark states in diquark-antidiquark
configuration with both
$\mathbf{6}_{[QQ]}\otimes\mathbf{\bar{6}}_{[\bar{Q}\bar{Q}]}$ and
$\mathbf{\bar{3}}_{[QQ]}\otimes\mathbf{3}_{[\bar{Q}\bar{Q}]}$ color
structures. In this work, we shall further investigate the possibility of
fully-heavy tetraquark states in meson-meson configuration with
$\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}$ color structure by
using the method of QCD moment sum rules Reinders:1984sr ; Shifman:1978bx .

This paper is organized as follows. In Sec. II, we construct the interpolating
currents for $\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}$ meson-
meson tetraquark states. In Sec. III, we evaluate the correlation functions
for these interpolating currents. We extract the masses for these tetraquark
states by performing the moment QCD sum rule analyses in Sec. IV. The last
section is a brief summary.

##  II Interpolating currents

The color structure of a meson-meson operator $[Q\bar{Q}][Q\bar{Q}]$ can be
written via the SU(3) symmetry

$\displaystyle(\mathbf{3}\otimes\overline{\mathbf{3}})_{[Q\bar{Q}]}\otimes(\mathbf{3}\otimes\overline{\mathbf{3}})_{[Q\bar{Q}]}$
$\displaystyle=(\mathbf{1}\oplus\mathbf{8})_{[Q\bar{Q}]}\otimes(\mathbf{1}\oplus\mathbf{8})_{[Q\bar{Q}]}$
(1)
$\displaystyle=(\mathbf{1}\otimes\mathbf{1})\oplus(\mathbf{1}\otimes\mathbf{8})\oplus(\mathbf{8}\otimes\mathbf{1})\oplus(\mathbf{8}\otimes\mathbf{8})$
$\displaystyle=\mathbf{1}\oplus\mathbf{8}\oplus\mathbf{8}\oplus(\mathbf{1}\oplus\mathbf{8}\oplus\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus\mathbf{27})\,,$

in which the color singlet structures come from the
$\left(\mathbf{1}_{[Q\bar{Q}]}\otimes\mathbf{1}_{[Q\bar{Q}]}\right)$ and
$\left(\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}\right)$ terms.
Following Ref. Chen:2015ata , we can construct the S-wave and P-wave
$[Q\bar{Q}][Q\bar{Q}]$ interpolating currents as below:

  * •

The S-wave interpolating currents are

$\displaystyle J^{PC}=0^{++}:\leavevmode\nobreak$ $\displaystyle
J_{1}=(\bar{Q}_{a}\gamma_{5}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\gamma_{5}\lambda_{de}^{n}Q_{e})\,,$
(2) $\displaystyle
J_{2}=(\bar{Q}_{a}\gamma_{\mu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\gamma_{\mu}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle J^{PC}=1^{+-}:\leavevmode\nobreak$ $\displaystyle
J_{1\mu}=(\bar{Q}_{a}\gamma_{\mu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\gamma_{5}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle J^{PC}=2^{++},\,0^{++}:\leavevmode\nobreak$ $\displaystyle
J_{\mu\nu}=(\bar{Q}_{a}\gamma_{\mu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\gamma_{\nu}\lambda_{de}^{n}Q_{e})\,,$

in which we only obtain $[Q\bar{Q}][Q\bar{Q}]$ currents with
$J^{PC}=0^{++},1^{+-}$ and $2^{++}$ in the S-wave channel. The tensor current
$J_{\mu\nu}$ can couple to both the $J^{PC}=0^{++}$ and $2^{++}$ quantum
numbers, but not the $J^{PC}=1^{++}$ channel since the Lorentz symmetry
restriction.

  * •

The P-wave interpolating currents are

$\displaystyle J^{PC}=0^{-+}:\leavevmode\nobreak$
$\displaystyle\eta_{1}=(\bar{Q}_{a}\gamma_{5}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\lambda_{de}^{n}Q_{e})\,,$
(3)
$\displaystyle\eta_{2}=(\bar{Q}_{a}\sigma_{\mu\nu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\sigma_{\mu\nu}\gamma_{5}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle J^{PC}=0^{--}:\leavevmode\nobreak$
$\displaystyle\eta_{3}=(\bar{Q}_{a}\gamma_{\mu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\gamma_{\mu}\gamma_{5}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle J^{PC}=1^{--}:\leavevmode\nobreak$
$\displaystyle\eta_{1\mu}=(\bar{Q}_{a}\gamma_{\mu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle\eta_{2\mu}=(\bar{Q}_{a}\gamma_{\alpha}\gamma_{5}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\sigma_{\alpha\mu}\gamma_{5}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle J^{PC}=1^{-+}:\leavevmode\nobreak$
$\displaystyle\eta_{3\mu}=(\bar{Q}_{a}\gamma_{5}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\gamma_{\mu}\gamma_{5}\lambda_{de}^{n}Q_{e})\,,$
$\displaystyle\eta_{4\mu}=(\bar{Q}_{a}\gamma_{\mu}\lambda_{ab}^{n}Q_{b})(\bar{Q}_{d}\sigma_{\mu\nu}\lambda_{de}^{n}Q_{e})\,,$

where only one P-wave $[Q\bar{Q}]$ operator is contained in these
interpolating currents. One should note that these tetraquark interpolating
currents with $\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}$ color
structure can be written as combinations of the diquark-antidiquark operators
through the Fierz transformation and the color rearrangement. Their decay
properties should be the same with the $[QQ][\bar{Q}\bar{Q}]$ tetraquark
states as discussed in Refs. Chen:2016jxd ; Chen:2020xwe . Thus we shall not
discuss the decay behaviors for these $[Q\bar{Q}][Q\bar{Q}]$ tetraquarks in
this work.

##  III QCD sum rules

In this section, we investigate the two-point correlation functions of the
interpolating currents constructed above. For the scalar and pseudo-scalar
currents, the correlation function are

$\displaystyle\Pi\left(p^{2}\right)$ $\displaystyle=i\int d^{4}xe^{ip\cdot
x}\left\langle
0\left|T\left[J(x)J^{\dagger}(0)\right]\right|0\right\rangle\,,$ (4)

while for the vector and axial-vector currents

$\displaystyle\Pi_{\mu\nu}\left(p^{2}\right)=i\int d^{4}xe^{ip\cdot
x}\left\langle
0\left|T\left[J_{\mu}(x)J_{\nu}^{\dagger}(0)\right]\right|0\right\rangle\,.$
(5)

The correlation function $\Pi_{\mu\nu}(p^{2})$ in Eq. (5) can be divided into
two part

$\Pi_{\mu\nu}\left(p^{2}\right)=\left(\frac{p_{\mu}p_{\nu}}{p^{2}}-g_{\mu\nu}\right)\Pi_{1}\left(p^{2}\right)+\frac{p_{\mu}p_{\nu}}{p^{2}}\Pi_{0}\left(p^{2}\right)\,,$
(6)

where $\Pi_{0}\left(p^{2}\right)$ and $\Pi_{1}\left(p^{2}\right)$ represent
the spin-0 and spin-1 invariant functions, respectively. For the tensor
currents $J_{\mu\nu}(x)$ in Eq. (2),

$\displaystyle\Pi_{\mu\nu,\rho\sigma}\left(p^{2}\right)=i\int d^{4}xe^{ip\cdot
x}\left\langle
0\left|T\left[J_{\mu\nu}(x)J_{\rho\sigma}^{\dagger}(0)\right]\right|0\right\rangle\,.$
(7)

The correlation function $\Pi_{\mu\nu,\rho\sigma}(p^{2})$ in Eq. (7) can be
expressed as

$\Pi_{\mu\nu,\rho\sigma}\left(p^{2}\right)=\left(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2}{3}\eta_{\mu\nu}\eta_{\rho\sigma}\right)\Pi_{2}\left(p^{2}\right)+\cdots\,,$
(8)

where

$\ \eta_{\mu\nu}=\frac{p_{\mu}p_{\nu}}{p^{2}}-g_{\mu\nu}\,.$ (9)

The invariant function $\Pi_{2}\left(p^{2}\right)$ relates to the spin-2
intermediate state, and the $``\cdots"$ represents the contribution from the
spin-0 state.

At the hadron level, the invariant functions can be expressed through the
dispersion relation

$\Pi\left(p^{2}\right)=\frac{\left(p^{2}\right)^{N}}{\pi}\int_{16m_{Q}^{2}}^{\infty}\frac{\operatorname{Im}\Pi(s)}{s^{N}\left(s-p^{2}-i\epsilon\right)}ds+\sum_{n=0}^{N-1}b_{n}\left(p^{2}\right)^{n}\,,$
(10)

where $b_{n}$ is the subtraction constant. In QCD sum rules, the imaginary
part of the correlation functions are usually simplified as the following
“pole plus continuum” spectral function

$\rho(s)=\frac{1}{\pi}\text{Im}\Pi(s)=f_{H}^{2}\delta(s-m_{H}^{2})+\text{QCD
continuum and higher states}\,,$ (11)

in which the $\delta$ function represents the lowest-lying state. The
parameters $f_{H}$ and $m_{H}$ are the coupling constant and mass of the
lowest-lying hadronic resonance $H$ respectively

$\displaystyle\langle 0|J|H\rangle$ $\displaystyle=f_{H}\,,$ (12)
$\displaystyle\left\langle 0\left|J_{\mu}\right|H\right\rangle$
$\displaystyle=f_{H}\epsilon_{\mu}\,,$ $\displaystyle\left\langle
0\left|J_{\mu\nu}\right|H\right\rangle$
$\displaystyle=f_{H}\epsilon_{\mu\nu}\,,$

with the polarization vector $\epsilon_{\mu}$ and polarization tensor
$\epsilon_{\mu\nu}$.

To extract the lowest-lying resonance, we first define the moments by taking
derivatives of the correlation function $\Pi(q^{2})$ in Euclidean region
$Q^{2}=-q^{2}>0$

$M_{n}\left(Q_{0}^{2}\right)=\left.\frac{1}{n!}\left(-\frac{d}{dQ^{2}}\right)^{n}\Pi\left(Q^{2}\right)\right|_{Q^{2}=Q_{0}^{2}}=\int_{16m_{Q}^{2}}^{\infty}\frac{\rho(s)}{\left(s+Q_{0}^{2}\right)^{n+1}}ds\,.$
(13)

We then rewrite the moments by applying the above equation to the Eq.(10) and
obtain

$M_{n}\left(Q_{0}^{2}\right)=\frac{f_{H}^{2}}{\left(m_{H}^{2}+Q_{0}^{2}\right)^{n+1}}\left[1+\delta_{n}\left(Q_{0}^{2}\right)\right]\,,$
(14)

where $\delta_{n}(Q_{0}^{2})$ represents higher excited states and continuum
contribution, and it is a function of $n$ and $Q_{0}^{2}$. It should be noted
that for a specific value of $Q_{0}^{2}$, $\delta_{n}(Q_{0}^{2})$ will tend to
zero as $n$ going to infinity. Considering the ratio of the moments to remove
the unknown coupling constant $f_{H}$

$r\left(n,Q_{0}^{2}\right)\equiv\frac{M_{n}\left(Q_{0}^{2}\right)}{M_{n+1}\left(Q_{0}^{2}\right)}=\left(m_{H}^{2}+Q_{0}^{2}\right)\frac{1+\delta_{n}\left(Q_{0}^{2}\right)}{1+\delta_{n+1}\left(Q_{0}^{2}\right)}\,,$
(15)

where the relation $\delta_{n}(Q_{0}^{2})\approx\delta_{n+1}(Q_{0}^{2})$ will
be satisfied when $n$ is large enough, and we can extract the hadron mass as

$m_{H}=\sqrt{r\left(n,Q_{0}^{2}\right)-Q_{0}^{2}}\,.$ (16)

At the quark-gluon level, we can evaluate the invariant functions $\Pi(p^{2})$
via the operator product expansion (OPE) method. The Wilson coefficients can
be calculated by adopting the following heavy quark propagator in momentum
space

$iS_{Q}^{ab}(p)=\frac{i\delta^{ab}}{\hat{p}-m_{Q}}+\frac{i}{4}g_{s}\frac{\lambda_{ab}^{n}}{2}G_{\mu\nu}^{n}\frac{\sigma^{\mu\nu}\left(\hat{p}+m_{Q}\right)+\left(\hat{p}+m_{Q}\right)\sigma^{\mu\nu}}{12}+\frac{i\delta^{ab}}{12}\left\langle
g_{s}^{2}GG\right\rangle
m_{Q}\frac{p^{2}+m_{Q}\hat{p}}{(p^{2}-m_{Q}^{2})^{4}}\,,$ (17)

where $Q$ represents the charm quark or bottom quark field. The superscripts
$a,b$ are the color indices and $\hat{p}=p^{\mu}\gamma_{\mu}$. In this work,
we will only evaluate the perturbative term and gluon condensate term in the
correlation function, the contributions from higher non-perturbative terms
such as tri-gluon condensate are small enough to be neglected.

##  IV Numerical analysis

We now perform the QCD moment sum rule analyses by adopting the following
values of heavy quark masses and gluon condensate Nielsen:2009uh ;
Narison:2018nbv ; ParticleDataGroup:2020ssz

$\begin{array}[]{l}{m_{c}\left(m_{c}\right)=(1.27_{-0.02}^{+0.02})\mathrm{GeV}}\,,\vspace{1ex}\\\
{m_{b}\left(m_{b}\right)=(4.18_{-0.02}^{+0.03})\mathrm{GeV}}\,,\vspace{1ex}\\\
{\left\langle g_{s}^{2}GG\right\rangle=(0.88\pm
0.25)\mathrm{GeV}^{4}}\,.\end{array}$ (18)

As mentioned in Sec. III, there remain two important parameters $n$ and
$Q_{0}^{2}$ in the extracted hadron mass in Eq. (16). In the original
literatures on moment sum rules, the authors set $Q_{0}^{2}=0$, which may lead
to a bad OPE convergence. To avoid such bad behavior, we follow the Refs.
Chen:2016jxd ; Yang:2021zrc ; Wang:2021taf  to choose $Q_{0}^{2}>0$ and
introduce $\xi=Q_{0}^{2}/(4m_{c})^{2}$ to perform sum rule analysis. The
parameters $n$ and $\xi$ are related to each other through the following
respects: (1) a large enough $n$ will reduce the higher excited states and
continuum region contributions, but it will also decrease the convergence of
OPE series. (2) a large $\xi$ (or $Q_{0}^{2}$) can compensate the OPE
convergence (see Fig. 1), but may cause a bad convergence of
$\delta_{n}(Q_{0}^{2})$ and make it difficult to obtain the parameters of the
lowest lying resonance. One needs to find suitable working regions for these
two parameters to establish stable sum rules.

We take the interpolating current $J_{1}(x)$ with $J^{PC}=0^{++}$ as an
example to show the numerical analysis details. The correlation function of
this current is evaluated as the following

$\displaystyle\Pi^{pert}(Q^{2})$
$\displaystyle=\frac{1}{192\pi^{6}}\int_{0}^{1}dx\int_{0}^{1}dy\int_{0}^{1}dz\Bigg{\\{}\left(\frac{-13x(1-x)y(1-y)^{3}(1-z)}{z^{3}}\right)F(m_{c},Q^{2})^{4}+\left(\frac{40m_{c}^{2}y(1-y)}{z^{2}}\right.$
$\displaystyle\left.-\frac{52Q^{2}x(1-x)y(1-y)^{3}(1-z)^{2}}{z^{2}}\right)F(m_{c},Q^{2})^{3}+\left(\frac{60m_{c}^{2}Q^{2}y(1-y)(1-z)}{z}-\frac{26m_{c}^{4}(1-y)}{z^{2}}\right.$
$\displaystyle\left.-\frac{26Q^{4}x(1-x)y(1-y)^{3}(1-z)^{3}}{z}\right)F(m_{c},Q^{2})^{2}\Bigg{\\}}\mbox{Log}\left[F(m_{c},Q^{2})\right]\,,$
$\displaystyle\Pi^{GG}(Q^{2})$ $\displaystyle=\frac{\langle
g_{s}^{2}GG\rangle}{288\pi^{6}}\int_{0}^{1}dx\int_{0}^{1}dy\int_{0}^{1}dz\Bigg{\\{}\left(\frac{30m_{c}^{2}x(1-x)(1-y)^{3}(1-z)^{3}}{z^{3}}-\frac{13m_{c}^{2}x(1-x)y(1-y)^{3}(1-z)^{4}}{z^{3}}\right)$
$\displaystyle\times
F(m_{c},Q^{2})+\left(\frac{10m_{c}^{4}x(1-x)y(1-y)^{3}(1-z)^{4}}{z^{3}}-\frac{13m_{c}^{4}(1-y)(1-z)^{2}}{z^{2}}-\frac{10\text{mc}^{4}(1-y)y(1-z)^{3}}{z^{2}}\right.$
$\displaystyle+\left.\frac{15m_{c}^{2}Q^{2}(1-x)x(1-y)^{3}(1-z)^{4}}{z^{2}}-\frac{26m_{c}^{2}Q^{2}(1-x)x(1-y)^{3}(1-z)^{5}}{z^{2}}\right)\Bigg{\\}}\mbox{Log}\left[F(m_{c},Q^{2})\right]$
$\displaystyle+\frac{\langle
g_{s}^{2}GG\rangle}{864\pi^{6}}\int_{0}^{1}dx\int_{0}^{1}dy\int_{0}^{1}dz\frac{-1}{F(m_{c},Q^{2})}\Bigg{\\{}\frac{13m_{c}^{6}(1-y)(1-z)^{3}}{z^{2}}-\frac{15m_{c}^{4}Q^{2}y(1-y)(1-z)^{4}}{z^{2}}$
$\displaystyle-\frac{15m_{c}^{2}Q^{2}x(1-x)(1-y)^{3}(1-z)^{5}}{z^{2}}+\frac{13m_{c}^{2}Q^{4}x(1-x)y(1-y)^{3}(1-z)^{6}}{z}\Bigg{\\}}$
$\displaystyle+\frac{\langle
g_{s}^{2}GG\rangle}{1152\pi^{6}}\int_{0}^{1}dx\int_{0}^{1}dy\int_{0}^{1}dz\Bigg{\\{}\left(\frac{18(1-x)x(1-y)^{3}(1-z)^{2}}{z^{2}}\right)F(m_{c},Q^{2})^{2}+\left(\frac{-12m_{c}^{2}(1-y)(1-z)}{z}\right.$
$\displaystyle\left.-\frac{14m_{c}^{2}x(1-x)(1-y)^{3}(1-z)^{2}}{yz^{2}}-\frac{24m_{c}^{2}y(1-z)}{xz}+\frac{36Q^{2}x(1-x)(1-y)^{3}(1-z)^{3}}{z}\right)F(m_{c},Q^{2})$
$\displaystyle+\left(\frac{12m_{c}^{4}(1-y)(1-z)}{yz}-\frac{12m_{c}^{2}Q^{2}y(1-z)^{2}}{x}-6m_{c}^{2}Q^{2}(1-y)(1-z)^{2}\right.$
$\displaystyle\left.-\frac{7m_{c}^{2}Q^{2}(1-x)x(1-y)^{3}(1-z)^{3}}{yz}+6Q^{4}(1-x)x(1-y)^{3}(1-z)^{4}\right)\Bigg{\\}}\mbox{Log}\left[F(m_{c},Q^{2})\right]\,,$
(19)

where
$F(m_{c},Q^{2})=m_{c}^{2}\left(1-z+\frac{z}{y}+\frac{z}{x(1-y)}+\frac{z}{(1-x)(1-y)}\right)+Q^{2}z(1-z)$.
We shall not evaluate the dimension-6 tri-gluon condensate $\langle
G^{3}\rangle$ $(\sim$$g_{s}^{3})$ and dimension-8 $\langle G^{4}\rangle$
$(\sim$$g_{s}^{4})$ condensate in the OPE series. The $\langle G^{3}\rangle$
term gives negligible contribution to the correlation functions even at
$\xi=0$ for the charmonium system Nikolaev:1981ff  and four-charm tetraquark
system Zhang:2020xtb . For the dimension-8 $\langle G^{4}\rangle$ condensate,
it was proven in the charmonium moment sum rules that this term was much
larger suppressed comparing to the dimension-4 gluon condensate $\langle
G^{2}\rangle$ at $\xi\neq 0$, and thus can also be neglected for the mass sum
rule analysis Nikolaev:1982ra ; Reinders:1984sr .

Figure 1: $|\Pi^{GG}/\Pi^{Pert}|$ with respect to $n$ for different values of
$\xi$ from $J_{1}(x)$ with $J^{PC}=0^{++}$.

To obtain convergent OPE series, we require that the contribution of the gluon
condensate be smaller than the perturbative term, and obtain the upper bound
$n_{max}=47,61,77,91$ for $\xi=2,3,4,5$ respectively. We show the ratio
$|\Pi^{GG}/\Pi^{Pert}|$ in Fig. 1 to display the convergence of the OPE series
with respect to $n$ and $\xi$, which indicates that the OPE convergence
becomes better with increasing of $\xi$ and decreasing of $n$.

In Fig. 2, we show the variation of the extracted mass with $n$ for different
value of $\xi$, and get stable mass prediction plateaus
$(n,\xi)=(32,2),(42,3),(52,4),(62,5)$. To choose the values of $\xi$, one
should consider both the existence of the mass plateaus and the stability of
the hadron mass for growing $\xi$. Both of these two criteria can be satisfied
for $\xi\geq 2$, as shown in Fig. 2. Accordingly, the mass of such a
$c\bar{c}c\bar{c}$ tetraquark state is finally predicted to be

$m_{{c\bar{c}c\bar{c}}}=6.54_{-0.18}^{+0.19}\leavevmode\nobreak\
\text{GeV}\,,$ (20)

in which the errors come from the uncertainties of $\xi$ and $n$, charm quark
mass and the gluon condensate.

Figure 2: Hadron mass for the fully-charm $c\bar{c}c\bar{c}$ tetraquark state
with $J^{PC}=0^{++}$ from $J_{1}(x)$ with respect to $n$ for different value
of $\xi$.

The same numerical analyses can be done for the other interpolating currents
in Eqs. (2-3). Then we obtain the masses for all $c\bar{c}c\bar{c}$ tetraquark
states in $\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$
configuration in Table LABEL:ccccResultTab. In this mass spectra, we predict
three S-wave $c\bar{c}c\bar{c}$ tetraquarks with $J^{PC}=0^{++},1^{+-}$ and
$2^{++}$ and four P-wave $c\bar{c}c\bar{c}$ tetraquarks with
$J^{PC}=0^{-+},0^{--},1^{--}$ and $1^{-+}$. The masses are predicted to be
around $6.3-6.5$ GeV for the S-wave states while $7.0-7.2$ GeV for the P-wave
states. Comparing to the mass spectra obtained in Ref. Chen:2016jxd , the
masses for S-wave fully-charm tetraquark states are consistent with each other
in both the $[c\bar{c}][c\bar{c}]$ and $[cc][\bar{c}\bar{c}]$ configurations.
However, the P-wave $[c\bar{c}][c\bar{c}]$ tetraquarks are predicted to be
200-300 MeV higher than those in the diquark-antidiquark configuration
Chen:2016jxd . In Table LABEL:ccccResultTab, we also list the masses for some
S-wave $c\bar{c}c\bar{c}$ tetraquark states in
$\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$ configuration obtained
by the Laplace QCD sum rule Yang:2020wkh  and a non-relativistic quark model
Yang:2021hrb . Our results for these $c\bar{c}c\bar{c}$ tetraquark states are
in good agreement with those in Refs. Yang:2020wkh ; Yang:2021hrb .

One notes that the two interpolating currents in the same channel
($J^{PC}=0^{++},0^{-+},1^{-+},1^{--}$) lead to almost the same hadron masses.
To specify if these two currents couple to the same physical state or not, we
calculate their cross correlation functions of two different currents with the
same quantum number, e.g., the $J_{1}(x)$ and $J_{2}(x)$ with $J^{PC}=0^{++}$

$\Pi_{12}\left(p^{2}\right)=i\int d^{4}xe^{ip\cdot x}\left\langle
0\left|T\left[J_{1}(x)J_{2}^{\dagger}(0)\right]\right|0\right\rangle\,.$ (21)

Our calculations show that all these cross correlation functions are large
enough and comparable to the diagonal correlators, implying that they couple
to the same physical states. Since the two interpolating currents in the same
channel give almost the same hadron masses, we don’t reanalyze the mass sum
rules by using their mixing current, avoiding more errors from the uncertain
mixing angle.

Table 1: The mass spectra for the fully-charm $c\bar{c}c\bar{c}$ tetraquark states in $\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$ color configuration. Current |  $J^{PC}$ |  Mass(GeV) |  Ref. Yang:2020wkh (GeV) |  Ref. Yang:2021hrb  (MeV)
---|---|---|---|---
$J_{1}$ |  $0^{++}$ |  $6.54_{-0.18}^{+0.19}$ |  $6.44_{-0.11}^{+0.11}$ |  $6403$
$J_{2}$ |  $0^{++}$ |  $6.36_{-0.16}^{+0.16}$ |  $6.52_{-0.11}^{+0.11}$ |  $6346$
$J_{1\mu}$ |  $1^{+-}$ |  $6.47_{-0.17}^{+0.18}$ |  - |  $6325$
$J_{\mu\nu}$ |  $2^{++}$ |  $6.52_{-0.17}^{+0.17}$ |  - |  $6388$
$\eta_{1}$ |  $0^{-+}$ |  $7.00_{-0.20}^{+0.23}$ |  - |  -
$\eta_{2}$ |  $0^{-+}$ |  $7.02_{-0.20}^{+0.24}$ |  - |  -
$\eta_{3}$ |  $0^{--}$ |  $7.00_{-0.20}^{+0.23}$ |  - |  -
$\eta_{1\mu}$ |  $1^{--}$ |  $6.99_{-0.20}^{+0.23}$ |  - |  -
$\eta_{2\mu}$ |  $1^{--}$ |  $7.17_{-0.22}^{+0.28}$ |  - |  -
$\eta_{3\mu}$ |  $1^{-+}$ |  $6.98_{-0.19}^{+0.21}$ |  - |  -
$\eta_{4\mu}$ |  $1^{-+}$ |  $7.07_{-0.19}^{+0.21}$ |  - |  -

We can also study the fully-bottom tetraquark states in
$\mathbf{8}_{[b\bar{b}]}\otimes\mathbf{8}_{[b\bar{b}]}$ configuration. For the
fully-bottom system, we define $\xi=Q_{0}^{2}/(m_{b})^{2}$ and find that the
two criteria of mass plateaus and $\xi$ stability can be achieved for
$\xi=0.2-0.8$. By requiring that the contribution of the gluon condensate be
smaller than the perturbative term, we obtain the upper bound on the parameter
$n_{max}=119,121,123,125$ for $\xi=0.2,0.4,0.6,0.8$ respectively. We show the
variation of the extracted mass with $n$ for different value of $\xi$ in Fig.
(3), and get stable mass prediction plateaus
$(n,\xi)=(75,0.2),(77,0.4),(77,0.6),(79,0.8)$. The mass of such a
$b\bar{b}b\bar{b}$ tetraquark state is finally predicted as

$m_{b\bar{b}b\bar{b}}=18.15_{-0.10}^{+0.14}\leavevmode\nobreak\ \text{GeV}\,.$
(22)

Figure 3: Hadron mass for fully-bottom $b\bar{b}b\bar{b}$ tetraquark state
with $J^{PC}=0^{++}$ from $J_{1}(x)$ with respect to $n$ for different value
of $\xi$.

Applying the similar moment sum rule analyses, we obtain the mass spectra for
these $b\bar{b}b\bar{b}$ tetraquark states and list them in Table
LABEL:bbbbResultTab. Accordingly, the S-wave $b\bar{b}b\bar{b}$ tetraquark
states are obtained to be around 18.2 GeV while the P-wave states are about
18.4-18.8 GeV. Such results are several hundreds MeV below the masses of
diquark-antidiquark $bb\bar{b}\bar{b}$ tetraquarks predicted in Ref.
Chen:2016jxd . As shown in Table LABEL:bbbbResultTab, our results for the
S-wave $\mathbf{8}_{[b\bar{b}]}\otimes\mathbf{8}_{[b\bar{b}]}$ tetraquarks are
much smaller than those predicted in the non-relativistic quark model
Yang:2021hrb , but in roughly agreement with the results in Laplace QCD sum
rules Yang:2020wkh .

Table 2: The mass spectra for the fully-bottom $b\bar{b}b\bar{b}$ tetraquark states in $\mathbf{8}_{[b\bar{b}]}\otimes\mathbf{8}_{[b\bar{b}]}$ color configuration. Current |  $J^{PC}$ |  Mass(GeV) |  Ref. Yang:2020wkh (GeV) |  Ref. Yang:2021hrb  (MeV)
---|---|---|---|---
$J_{1}$ |  $0^{++}$ |  $18.15_{-0.10}^{+0.14}$ |  $18.38_{-0.11}^{+0.11}$ |  $19243$
$J_{2}$ |  $0^{++}$ |  $18.13_{-0.09}^{+0.13}$ |  $18.44_{-0.10}^{+0.10}$ |  $19237$
$J_{1\mu}$ |  $1^{+-}$ |  $18.14_{-0.09}^{+0.14}$ |  - |  $19126$
$J_{\mu\nu}$ |  $2^{++}$ |  $18.15_{-0.09}^{+0.14}$ |  - |  $19197$
$\eta_{1}$ |  $0^{-+}$ |  $18.45_{-0.11}^{+0.15}$ |  - |  -
$\eta_{2}$ |  $0^{-+}$ |  $18.54_{-0.12}^{+0.16}$ |  - |  -
$\eta_{3}$ |  $0^{--}$ |  $18.47_{-0.11}^{+0.15}$ |  - |  -
$\eta_{1\mu}$ |  $1^{--}$ |  $18.46_{-0.11}^{+0.15}$ |  - |  -
$\eta_{2\mu}$ |  $1^{--}$ |  $18.46_{-0.11}^{+0.15}$ |  - |  -
$\eta_{3\mu}$ |  $1^{-+}$ |  $18.56_{-0.11}^{+0.16}$ |  - |  -
$\eta_{4\mu}$ |  $1^{-+}$ |  $18.79_{-0.13}^{+0.18}$ |  - |  -

##  V Conclusion and Discussion

We have studied the fully-heavy $Q\bar{Q}Q\bar{Q}$ tetraquark states in the
$\mathbf{8}_{[Q\bar{Q}]}\otimes\mathbf{8}_{[Q\bar{Q}]}$ color structure by
using the moment QCD sum rule method. We construct the S-wave and P-wave
interpolating tetraquark currents with various quantum numbers and calculate
their two-point correlation functions containing perturbative term and gluon
condensate term. Choosing suitable parameter working regions, we have
established stable moment sum rules for all interpolating currents and
extracted the mass spectra for the fully-charm and fully-bottom tetraquark
states.

For the fully-charm $c\bar{c}c\bar{c}$ system, our results suggest that the
S-wave tetraquark states with $J^{PC}=0^{++},1^{+-},2^{++}$ lie around
$6.3-6.5$ GeV while the P-wave tetraquark states with
$J^{PC}=0^{-+},0^{--},1^{--},1^{-+}$ are about $7.0-7.2$GeV. Especially, the
masses for the $c\bar{c}c\bar{c}$ tetraquarks with $J^{PC}=0^{++}$ and
$2^{++}$ are consistent with the broad structure observed by LHCb LHCb:2020bwg
. The P-wave fully-charm tetraquarks with $J^{PC}=0^{-+}$ and $1^{-+}$ are
predicted to be roughly in agreement with the mass of $X(6900)$ within errors.
Such results suggest the possibility that there are some
$\mathbf{8}_{[c\bar{c}]}\otimes\mathbf{8}_{[c\bar{c}]}$ components in LHCb’s
di-$J/\psi$ structures. More investigations are needed in both theoretical and
experimental aspects to study the nature of these structures.

For the fully-bottom $b\bar{b}b\bar{b}$ system, the numerical results show
that the S-wave tetraquark states are about $18.2$ GeV while the P-wave states
are around $18.4-18.8$ GeV. All these fully-bottom $b\bar{b}b\bar{b}$
tetraquarks are predicted to below the $\eta_{b}\eta_{b}$ and
$\Upsilon(1S)\Upsilon(1S)$ two-meson decay thresholds, indicating that these
tetraquark states will be stable against the strong interaction. Such results
are consistent with our previous prediction for the diquark-antidiquark
$bb\bar{b}\bar{b}$ tetraquarks in Ref. Chen:2016jxd . More efforts are
expected to search for such fully-bottom tetraquark states in the future
experiments, such as LHCb, CMS and so on.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
under Grant No. 12175318 and the National Key R$\&$D Program of China under
Contracts No. 2020YFA0406400.

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