# Masses of fully-heavy tetraquark states from a four-quark static potential
model

Z. Asadi111E-mail<EMAIL_ADDRESS>Department of Physics, Razi University,
Kermanshah 67149, Iran  G. R. Boroun222E-mail<EMAIL_ADDRESS>Department of
Physics, Razi University, Kermanshah 67149, Iran

###### Abstract

In this article, we study the ground-state mass of full-heavy tetraquarks
$cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ with solving the nonrelativistic
four-body systems. The flux-tube configurations, the tetraquark four-body
potential butterfly, and the dimeson potential flip-flop of SU(3) lattice
quantum chromodynamics have been applied to describe the tetraquark
interaction. Our numerical analysis indicates that the disconnected and
connected static potentials can predict the mass of tetraquark very close to
experimental data.

###### pacs:

21.65.-f, 26.60.-c, 64.70.-p

##  I INTRODUCTION

Gell-Mann and Zweig had schematized the multiquark states with four or more
quarks to interpret the observed spectrum of mesons and baryons products such
as $qq\bar{q}\bar{q}$ and $qqqq\bar{q}$ Gell ; Zweig . In a short time,
enormous works have been done to explain the structure of four-quark states
containing at least one light quark Brambilla ; Chen . The full-heavy
tetraquark states have not been investigated until that; the very large energy
states have been detected in the experiments of pair production of
$\Upsilon(1s)$ Khachatryan . The CMS Collaboration observed pair production of
$\Upsilon(1S)$ mesons in proton-proton collisions at $\sqrt{s}=8$ TeV with a
global significance of $3.6\sigma$ and a mass $18.4\pm 0.1(stat)\pm
0.2(syst)$GeV Khachatryan ; Durgut . After that, the LHCb Collaboration
studied the $\Upsilon(1S)_{\mu^{+}\mu^{-}}$ invariant-mass distribution to
seek a possible $bb\bar{b}\bar{b}$ exotic meson but they did not see any
significant excess in the range $17.5-20.0$ GeV Aaij . Another interesting
structure is the full charm tetraquark. The existence of these particles has
been confirmed by experimental searches of the LHCb observation of the
tetraquark containing only the charm quarks Aaij2 . The LHCb Collaboration
declared a narrow structure, matching the line shape of resonance and a broad
structure next to the $di-J/\Psi$ mass threshold with the data in the range of
6.2–7.2 GeV Aaij2 , which called X(6900) with the structure
$cc\bar{c}\bar{c}$. The existence and stability of such states have been
considered by applying several models with different interactions Berezhnoy ;
Maiani3 ; Wu ; Bai ; Hughes ; Buccella ; Karliner ; Maiani2 .

Creutz applied the lattice QCD simulations with the Wilson loop to describe
the interquark potential between a quark and an antiquark Creutz  after that a
large amount of effort has been devoted in lattice QCD to study the multiquark
force Creutz ; Bali ; Takahashi ; Alexandrou ; Okiharu . These potentials are
successful to calculate the energy and the mass of tetraquark systems
contained purely heavy quarks Csikor ; Chiu ; Sasaki . The experimental
discoveries of multiquark hadrons reveal new aspects of the interquark force
such as the quark confinement force, the color-magnetic interaction and the
diquark correlation Jaffe . According to these, the proper Hamiltonian for the
quark-model calculation of multiquarks has been suggested to investigate the
interquark force in the multiquark system directly based on QCD Stancu ;
Kanada-Enyo .

A number of phenomenological models have been put forward to explain
multiquark stability. A type-II diquark-antidiquark model, proposed by L.
Maian et al. based on the new Ansatz on spin-spin couplings, in which the cq
(c the charm and q a light quark) interaction inside the diquark dominates
over all other possible pairings Maiani1 . In this model, diquarks are similar
to compact bosonic building blocks which supposed that the size of the entire
tetraquark is consistently larger than the size of these blocks therefore, the
spin-spin interactions between different diquarks are neglected. The color
configuration for the diquark-antidiquark pair is similar to one of the quark-
antiquark systems with a bound state of two pointlike color sources Maiani1 .
The dynamical diquark method was introduced to explain the nature of the
exotic XYZ states based on a color flux-tub configuration where the separated
diquark-antidiquark pair is connected by the shortest tube configurations
(energetically), Lebed2 ; Lebed5 ; Lebed3 ; Lebed4 . The flux tube has
different configurations, which we will discuss in the next section, but none
of these structures can be suitable to describe the dynamical diquark model
especially as the system remains in a state of rapid change until the moment
it decays Lebed5 . In this physical picture, each diquark is only in the
color-triplet combination Lebed . In 2020, Lebed et al. investigated the basic
spectroscopy of $c\bar{c}c\bar{c}$ tetraquark states in this model Lebed . The
full mass spectrum of all tetraquark states was predicted, using lattice-
calculated confining two-body potential computed with the Born-Oppenheimer
(BO) approximation, which includes spin and isospin-dependent operators with
the spin-orbit and tensor coupling operators, such a way that the spin-spin
couplings dominated between quarks (antiquarks), not between quark and
antiquark Lebed .

The flux-tube picture as an analytical model for the multiquark system
supported by lattice QCD has been considered for the structure and the
reaction of hadrons Okiharu ; Alexandrou ; Nambu ; Kogut ; Carlson ; Isgur ;
Suzuki . The evaluated tetraquark potentials in Refs. Okiharu ; Alexandrou
have been constructed the Wilson loops for investigating the interaction
between quarks in the four-quark system directly from QCD by using SU(3)
lattice QCD at the quenched level and studied the hypothetical flux-tube
pictures for the multiquark system. They [i.e, Refs. Okiharu ; Alexandrou ]
used three flux-tube configurations according to four quarks’ locations. All
quarks and antiquarks are connected with the single flux-tube in the connected
flux tube system while in the “dimeson” states there are two disconnected
flux-tubes Okiharu . The investigating of the properties of tetraquark using
these three types of four-quark potentials and the transition between the
connected and disconnected four quark states are physically important for
analyzing the tetraquark decay process into two mesons and the reaction
mechanism between two mesons. In this paper, we have calculated masses of
full-heavy tetraquarks $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ using three
types of connected and two-meson four-quark potentials.

This paper is organized as follows. Section. II gives a brief review on the
color flux-tube model. The calculation method is described in Sec. III. The
numerical results for both the spin-independent and spin-dependent potential
are presented in Sec. IV. Section V is devoted for summary and concluding
remarks.

##  II THE COLOR FLUX-TUBE MODEL

The short-distance one gluon exchange (OGE) Coulomb force as a quantity of
perturbative QCD and the long-distance confinement force as a typical
contribution of nonperturbative can be developed to an alternative color flux
tube based on the lattice QCD picture with the Wilson loops Takahashi ;
Alexandrou ; Okiharu . The tetraquark SU(3) Wilson loop is defined in a four
quarks gauge, at time $t=0$ which annihilated at a later time $t$. The
potential of the system has been extracted from the behavior of the Wilson
loop at the large time region Alexandrou . The multibody color flux-tube
dynamical mechanism has been used to describe multiquark states from this
phenomenological point of view Deng ; Deng2 .

The minimum energy of a system of two quarks and two antiquarks is recorded
when the two quarks and the two antiquarks are linked by the minimal value of
the total flux tube length. Therefore the flux tube is formed to achieve a
minimum of the total flux-tube length of the system for the low-lying state
Okiharu . The theoretical form of the tetraquark potential according to the
four quarks location has three candidates for the flux-tube configuration. One
for the four-quark state of two quarks and two antiquarks $(qq\bar{q}\bar{q})$
that fluxtube system connect all quarks and antiquarks with the single flux
tube as shown in the (a) panel of Fig. 1, so-called “butterfly” configuration.
Two others for the tetraquark states of two mesons $(q\bar{q}q\bar{q})$ with
two disconnected flux tubes as shown in the (b) panel of Fig. 1, the “flip-
flop” configurations. The theoretical form of the tetraquark potential for
each configuration is written as below.

Figure 1: (a)The connected tetraquark system, which is the butterfly
configuration.(b)The disconnected tetraquark system, which corresponds to the
flip-flop configuration.

The OGE plus the double-Y-shaped flux-tube configurations have been suggested
for the theoretical form of the tetraquark $(qq\bar{q}\bar{q})$ potential. All
the quarks and antiquarks are in the same plane and link to each other with
the flux tubes that the angles between them must be $120^{0}$ Alexandrou . The
tetraquark potential for the butterfly is written as

$\displaystyle V_{2q2\bar{q}}^{Butterfly}$ $\displaystyle=$ $\displaystyle-
A_{4q}[\frac{1}{2}(\frac{1}{r_{13}}+\frac{1}{r_{14}}+\frac{1}{r_{23}}+\frac{1}{r_{24}})+(\frac{1}{r_{12}}+\frac{1}{r_{34}})]+\sigma_{4q}L_{min}$
(1)

with $A_{4q}$ the Coulomb coefficient, $\sigma_{4q}$ the string tension
extracted from the quark-antiquark potential and $r_{ij}=|r_{j}-r_{i}|$ with
$r_{i}$ which denotes the location of $i$th particle. $L_{min}$ is the minimal
value of the total flux-tube length in the (a) panel of Fig. 1. The minimal
length for the tetraquark system accrues when the flux tubes from each of the
quarks and the antiquarks meet at two Steiner points [Steiner point is the
junctions of two flux tubes $(s_{1},s_{2})$] Warren . $L_{min}$ can be
expressed as

$\displaystyle
L_{min}=r_{1s_{1}}+r_{2s_{1}}+r_{3s_{2}}+r_{4s_{2}}+r_{s_{1}s_{2}}.$ (2)

A disconnected flux-tube configuration the (b) panel of Fig. 1 is acceptable
when the nearest quark and antiquark pair is spatially close. Thus the system
can be regarded as a “dimeson state” rather than a connected four quark state.
The flip-flop configuration is summarized in the form

$\displaystyle V_{2(q\bar{q})}^{flip-flop}$ $\displaystyle=$ $\displaystyle-
A_{q\bar{q}}(\frac{1}{r_{13}}+\frac{1}{r_{24}})+\sigma_{q\bar{q}}(r_{13}+r_{24}).$
(3)

Where the potential for dimeson system the (b) panel of Fig. 1 is given by
exchanging indices $1\leftrightarrow 2$. The total tetraquark potential is
defined to represent the minimum energy of the connected four quarks; it is
taken by

$\displaystyle V_{4q}$ $\displaystyle=$ $\displaystyle
min(V_{2q2\bar{q}}^{Butterfly},V_{2(q\bar{q})}^{flip-flop}).$ (4)

The transition between the connected and disconnected four-quark states can be
demonstrated by Eq. (4) where these two potentials are equal.

The lattice QCD simulation parameters such as $\beta$, the lattice size, and
the corresponding lattice spacing $a$ with some related information are
estimated so as to reproduce the string tension $\sigma$. In Table. 1 the
lattice parameters, Coulomb coefficient, the string tension $\sigma$ and the
quark mass $m_{c}$ and $m_{b}$ have been presented. They have been obtained by
using the fitting analysis on the on axis data of the quark-antiquark
potential in lattice QCD Okiharu ; Takahashi2 . The extensive studies on the
interquark potentials in lattice QCD have indicated that
$\sigma_{4q}\simeq\sigma_{q\bar{q}}$ and the OGE results for the Coulomb
coefficient $A_{q\bar{q}}\simeq 2A_{4q}$ are fairly compatible with the
hypothetical flux-tube picture Okiharu .

Table 1:  The quark mass for each $\beta$, the lattice size, and the lattice spacing a with the Coulomb coefficient and string tension.  $\beta$ Okiharu  | Lattice size Okiharu  | a(fm) Okiharu  |  $\sigma_{q\bar{q}}(GeV)^{2}$ Okiharu  |  $A_{q\bar{q}}$ Okiharu  | $m_{c}(GeV)$ | $m_{b}(GeV)$ | S.N
---|---|---|---|---|---|---|---
$6.0$ | $16^{3}\times 32$ | 0.10 | 0.09 | 0.27 | 1.70 | 4.61 | I
$5.8$ | $16^{3}\times 32$ | 0.14 | 0.18 | 0.27 | 1.88 | 4.79 | II

For the four-quark system, the Hamiltonian can be expressed as follows:

$\displaystyle
H=\sum_{i=1}^{4}{(m_{i}+\frac{{p_{i}}^{2}}{2m_{i}})}-T_{cm}+V_{4q},$ (5)

$m_{i}$ and $P_{i}$ are , respectively the mass and momentum of the ith quark
(antiquark). $T_{cm}$ is the center-of-mass kinetic energy of the states. The
solution of this four-body problem is very difficult. We partially reduce the
difficulty of the calculation with the some simple assumptions. The quarks are
assumed to be heavy quarks and nonrelativistic. All of them, quarks and
antiquarks, have the same mass.

The model parameters in Table. 1 have been adopted to calculate the mass of
experimentally observed full charm tetraquark X(6900) and predict the mass of
the full heavy bottom tetraquark.

##  III Numerical Calculation Method

The wave function of tetraquark can be constructed of the spatial degrees of
freedom and the internal degrees of freedom of spin, color, and flavor.
Tetraquarks have been comprised of two couples of identical fermions; thus,
their wave functions must be antisymmetric for the exchange of the two quarks
and the two antiquarks, and like all physical states, they are the color
singlet. The trial wave function of the tetraquark can be defined as

$\displaystyle\Psi^{q_{1}q_{2}\bar{q}_{3}\bar{q}_{4}}_{JM_{J}}=\mathcal{A}\\{[[[\phi_{nlm}(\rho_{12})\chi_{s}(12)]_{\Lambda
M_{\Lambda}}^{q_{1}q_{2}}[\phi_{NLM}(\rho_{34})\chi_{s^{\prime}}(34)]_{IM_{I}}^{\bar{q}_{3}\bar{q}_{4}}]^{q_{1}q_{2}\bar{q}_{3}\bar{q}_{4}}_{\lambda
M_{\lambda}}$ (6)
$\displaystyle\times\phi_{\nu\lambda\mu}(r_{12-34})]_{JM_{J}}^{q_{1}q_{2}\bar{q}_{3}\bar{q}_{4}}\times[\eta_{c}(12)\eta_{c}(34)]_{CW_{c}}\times[\xi_{f}(12)\xi_{f}(34)]_{FW_{f}}\\}.$

The factor $\mathcal{A}$ is the antisymmetrization operator and $\phi$,
$\chi$, $\eta$, and $\xi$ respectively, express space, spin, color, and flavor
states. All the indexes are all possible flavor–spin–color–spatial
intermediate quantum numbers.

To use the planar geometry, it has been supposed all the quarks are in one
plate. Thus, reducing the number of variables of the four body potentials is
useful to simplify the calculation method. The new relative spatial
coordinates can be defined as

$\displaystyle\rho_{12}$ $\displaystyle=$ $\displaystyle r_{1}-r_{2}$ (7)
$\displaystyle\rho_{34}$ $\displaystyle=$ $\displaystyle r_{3}-r_{4}$ (8)
$\displaystyle r_{12,34}$ $\displaystyle=$
$\displaystyle\frac{m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}}-\frac{m_{3}r_{3}+m_{4}r_{4}}{m_{3}+m_{4}}$
(9) $\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{m_{1}r_{1}+m_{2}r_{2}+m_{3}r_{3}+m_{4}r_{4}}{m_{1}+m_{2}+m_{3}+m_{4}}$
(10)

In our model, the interdistance of two quarks with the interdistance of two
antiquarks are same, $\rho_{12}=\rho_{34}=\rho$. With this simplification,
only two dimensions remain in Eq. (5) r the inner distance between quark and
antiquark and $\rho$ the distance between two quarks and two antiquarks. The
excitations orbital angular momenta associated with the Jacobi coordinates are
$l_{\rho}$ and $l_{r}$. The parity of the tetraquark system can be expressed
in terms of the relative orbital angular momentum as
$P=(-1)^{l_{\rho}+l_{r}}$. To calculate the ground state of energies, the
system has been considered in positive parity with $l_{\rho}=0$ and $l_{r}=0$.

Based on the planer geometry of tetraquark in Fig. 2, the butterfly and flip-
flop potentials can simplify as follows:

$\displaystyle V_{2q2\bar{q}}^{Butterfly}$ $\displaystyle=$ $\displaystyle-
A_{4q}[\frac{1}{r}+\frac{1}{\sqrt{r^{2}+\rho^{2}}}+\frac{2}{\rho}]+\sigma_{4q}(r+\sqrt{3}\rho)$
(11)

$\displaystyle V_{2(q\bar{q})}^{flip-flop}$ $\displaystyle=$ $\displaystyle-
A_{q\bar{q}}\frac{2}{r}+\sigma_{q\bar{q}}(2r).$ (12)

Figure 2: A planar configuration of the tetraquark system.

The total tetraquark potential in this two-dimensional system has been defined
in Eq. (4). Finally, the Schrodinger equation corresponding to two relative
motions $\rho$ and $r$ has been defined as:

$\displaystyle[\sum_{i=1}^{4}m_{i}-\frac{\hbar}{2m}(\nabla^{2}_{\rho}+\nabla^{2}_{r})-T_{cm}+V_{2(q\bar{q})}^{flip-
flop}(r,\rho)]\Psi(r,\rho)=E\Psi(r,\rho)$ (13)

This equation is somewhat complicated to get a reliable result; thus, a high-
precision numerical method is required. The Gaussian expansion method (GEM)
Hiyama  has been used to study this four-quark system. The GEM was proposed to
a variety of few-body systems and has been powerful to solve three and four-
body problems. In this method, the spatial wave function has been expanded in
terms of a set of Gaussian basis functions,

$\displaystyle\Psi_{lm}^{G}(R)=\sum_{n=1}^{n_{max}}c_{nl}N_{nl}R^{l}e^{-\nu_{n}R^{2}}Y_{lm}(R)$
(14) $\displaystyle
N_{n}l=(\frac{2^{l+2(2\nu_{n})^{(l+\frac{3}{2})}}}{\sqrt{\pi}(2l+1)!!})^{\frac{1}{2}}\quad\textrm{}\quad$
(15) $\displaystyle\nu_{n}=\frac{1}{R^{2}_{n}}\quad
a=(\frac{R_{n_{max}}}{R_{1}})^{\frac{1}{n_{max}-1}}\quad.$ (16)

There are three parameters $\\{n_{max},R_{1},R_{n_{max}}\\}$ that we have
calculated the numerical result with $n_{max}=10$, $R_{1}=0.1fm$ and
$R_{n_{max}}=3.0$. The spin wave functions base on total spin of the
tetraquark $S_{tot}=0,1,2$ in the ($qq)(\bar{q}\bar{q}$) configuration, are in
the six below states.

$\displaystyle\chi_{s=0}(12,34)=\left\\{\begin{array}[]{ll}|\chi_{s=0}(12)\chi_{s=0}(34)>\\\
|\chi_{s=1}(12)\chi_{s=1}(34)>\par\end{array}\right.$ (19)
$\displaystyle\chi_{s=1}(12,34)=\left\\{\begin{array}[]{ll}|\chi_{s=0}(12)\chi_{s=1}(34)>\\\
|\chi_{s=1}(12)\chi_{s=0}(34)>\end{array}\right.$ (22)
$\displaystyle\chi_{s=2}(12,34)=|\chi_{s=1}(12)\chi_{s=1}(34)>$ (23)

The color representation for the tetraquark is only one single and also the
configuration of diquark and diantiquark are antisymmetric
$[q_{1}q_{2}]_{\bar{3}_{c}}$ and $[\bar{q}_{1}\bar{q}_{2}]_{3_{c}}$ and
symmetric $[q_{1}q_{2}]_{6_{c}}$ and $[\bar{q}_{1}\bar{q}_{2}]_{\bar{6}_{c}}$
Deng . Therefore, the allowed color tetraquark $[qq][\bar{q}\bar{q}]$ state is
in the two representations $[3]_{c}\bigotimes[\bar{3}]_{c}$ and
$[6]_{c}\bigotimes[\bar{6}]_{c}$.

##  IV Numerical results and analysis

###  IV.1 Spin-independent potential

We consider solving the two-dimensional Schrodinger equation, which is now
equivalent to a matrix eigenvalue equation of dimension $n_{r}n_{\rho}\times
n_{r}n_{\rho}$, using the Rayleigh–Ritz variational method,

$\displaystyle<\Psi_{lm,n}^{G}|H-M|\Psi_{lm,n^{\prime}}^{G}>=0.$ (24)

Here, M is the mass of tetraquark, which has been calculated by applying the
parameters of Table. 1. The visualization of the ground state energy for the
flip-flop potential with ignoring the OGE part is presented in Fig. 3. Adding
the OGE part changes the visualization which is depicted in Fig. 4. As one can
see, the matrix elements of the ground state without the OGE part are positive
with a big peak at the first element, which is caused by the arrangement of
Gaussian size parameters. After adding the OGE part to the calculation the
elements of the matrix started to be negative, which means the contribution of
the OGE part leads to the achievement of the bound state energy points, which
are visible with small pits in Fig. 4. These points help to produce the
eigenvalues that reported the best value of the tetraquark mass with different
static potentials.

Figure 3: The visualization of ground state energy for the flip-flop potential
with ignoring the OGE part

Figure 4: The visualization of ground state energy for the flip-flop potential

The $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ masses without a spin-dependent
correction with two sets of parameters Table. 1 for different static
potentials have been presented in Table. 2. In this table, the calculated
masses compared with the experimental mass of the X(6900) for
$cc\bar{c}\bar{c}$ and the observed mass for pair production of $\Upsilon(1S)$
mesons for $bb\bar{b}\bar{b}$. As one can see, the flip-flop potential result
in S.N $I$ for $cc\bar{c}\bar{c}$ is $6.8878$ GeV and has a considerable
agreement with the experimental mass of the X(6900). The obtained result for
the butterfly is not significant as the flip-flop one but this is a
considerable result with just 0.04 GeV different from experimental data. Our
results show that the flip-flop and butterfly potentials have almost the same
results for the heavier tetraquark. The masses of $bb\bar{b}\bar{b}$, which
have been measured with these two potentials in S.N $I$ presented in Table. 2,
are in good agreement with the experimental mass of pair $\Upsilon(1S)$
mesons.

Table 2:  The ground state masses of the fully heavy tetraquarks, unit in GeV .  System | M($flip-flop$) | M($butterfly$) | M($flip-flop+butterfly$) | S.N | $M_{exp}$
---|---|---|---|---|---
$cc\bar{c}\bar{c}$ | 6.8878 | 6.9456 | 6.8218 | I | $6.905\pm 11$ Aaij2
7.559 | 7.701 | 7.694 | II
$bb\bar{b}\bar{b}$ | 18.4397 | 18.4441 | 18.4404 | I | $18.4\pm 0.1\pm.2$ Durgut
19.186 | 19.302 | 19.295 | II

The radial probability density distributions of the initial state wave
functions for different static potentials using S.N $I$ have been depicted in
Fig. 5. The two-dimensional matrix eigenvectors of the tetraquark
$cc\bar{c}\bar{c}$ have been used to generate the wave functions. We can see
that curves are localized in the range 2 to 3 $fm$ with an exponential tail at
large distances. The maximum probability for the butterfly is located at
$r=2.14$ $fm$. The location of maxima for the flip-flop shift towards higher
$r$ values at $2.42$ $fm$. The butterfly potential is localized sooner and
stronger than the flip-flop potential, its squared wave function is around
$22.5\%$ localized larger than the flip-flop one.

Figure 5:  The radial probability density distributions of the initial state
wave functions for different static potentials using S.N $I$

###  IV.2 Spin-dependent corrections

The general form of the total spin-dependent interaction in QCD between the
particles i and j for the four-body system can be expressed as follows Rujula
:

$\displaystyle
H^{SS}=\sum_{i<j=1}^{4}-\frac{2\pi\alpha_{s}\lambda_{i}.\lambda_{i}\sigma_{i}.\sigma_{j}\delta(r_{ij})}{12m_{i}m_{j}}.$
(25)

Here $\alpha_{s}$ is the quark-gluon coupling constant and $\lambda$ and
$\sigma$ represent the Gell-Mann matrices and Pauli matrices, respectively.
The values of $<\lambda_{i}.\lambda_{i}>$ and $<\sigma_{i}.\sigma_{j}>$ have
been measured according to the symmetry properties of the tetraquark wave
function Deng3 . A tetraquark state with total quantum numbers $l=0$, and
$S=0$, $1$ and $2$ with positive parity for two representations
$[\bar{3}]_{c}\bigotimes[3]_{c}$ and $[6]_{c}\bigotimes[\bar{6}]_{c}$ has been
considered. The spin-dependent correction with calculating the elements matrix
$<\Psi|H^{ss}|\Psi>$ for three wave functions with total spin $S=0,$ $1$ and
$2$ has been added to the spin-independent matrix. The masses of states
$cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ for two color configurations with
the spin-dependent correction are presented in Table. 3. It is worth
highlighting that from Table. 3 the color configuration has a significant role
in the effect of spin correction in the mass of tetraquark. But in our
calculation for heaver tetraquark, this effect has almost vanished. As one can
see in Table. 3 the mass of $cc\bar{c}\bar{c}$ for different spin quantum
numbers have considerable change while no change has been measured for the
mass of $bb\bar{b}\bar{b}$. There are no exact experimental observations for
the mass of X(6900) with different spin quantum numbers; nevertheless, the
obtained masses with $J^{P}=2^{+}$ for flip-flop and $J^{P}=1^{+}$ for
butterfly are approximately the same with the mass of X(6900).

Table 3:  The mass spectra of the ground state $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ for different static potentials, unit in GeV.  System | Color | $J^{P}$ | $M(flip-flop)$ | $M(butterfly)$ | $M(flip-flop+butterfly)$
---|---|---|---|---|---
$cc\bar{c}\bar{c}$ | $[\bar{3}]_{c}\bigotimes[3]_{c}$ | $0^{+}$ | 6.8500 | 6.8742 | 6.8217
$1^{+}$ | 6.8704 | 6.9133 | 6.8218
$2^{+}$ | 6.9127 | 6.9899 | 6.8218
$[6]_{c}\bigotimes[\bar{6}]_{c}$ | $0^{+}$ | 6.9285 | 7.0174 | 6.8218
$1^{+}$ | 6.9660 | 6.8992 | 6.8217
$2^{+}$ | 6.8783 | 6.9281 | 6.8218
$bb\bar{b}\bar{b}$ | $[\bar{3}]_{c}\bigotimes[3]_{c}$ | $0^{+}$ | 18.43971 | 18.44416 | 18.44045
$1^{+}$ | 18.43971 | 18.44417 | 18.44045
$2^{+}$ | 18.43972 | 18.44417 | 18.44045
$[6]_{c}\bigotimes[\bar{6}]_{c}$ | $0^{+}$ | 18.43972 | 18.44417 | 18.44045
$1^{+}$ | 18.43971 | 18.44417 | 18.44045
$2^{+}$ | 18.43971 | 18.44416 | 18.44045

In fact, there are many theoretical calculations for measuring the mass of
fully-heavy tetraquark states which predict similar or different mass
spectrums for $J^{P}=0^{+}$, $1^{+}$ and $2^{+}$ corresponding to their method
Wang2 ; yang1 . The full mass spectrum of all tetraquark states was predicted
in Ref. Lebed  a state or states within the 2S multiplet were suggested as the
best interpretation for identifying $X(6900)$ Lebed . The masses of the ground
state of tetraquark $cc\bar{c}\bar{c}$ for different $J^{P}$ quantum numbers
have been compared with the other nonrelativistic quark models in Table. 4.
Our results are compatible with the other approaches; however, they are larger
than the average of the other works. It has proved that four-body interaction
models are completely suitable to apply for studying the tetraquark properties
as well as the other models.

Table 4:  The ground state masses of $cc\bar{c}\bar{c}$ in various models, unit in GeV.  $J^{P}$ | M($flip-flop$) | M($butterfly$) | MChen2  | MWu  | MDeng  | MLloyd  | MAder
---|---|---|---|---|---|---|---
$0^{+}$ | 6.8500 | 6.8742 | $6.82\pm 0.18$ | 7.016 | 6.491 | 6.477 | 6.437
$1^{+}$ | 6.8704 | 6.9133 | $6.51\pm 0.15$ | 6.899 | 6.580 | 6.528 | 6.437
$2^{+}$ | 6.9127 | 6.9899 | $6.51\pm 0.15$ | 6.956 | 6.607 | 6.573 | 6.437

In Table. 5 we have compared the $J^{P}=0^{+}$ , $1^{+}$, and $2^{+}$ ground
state masses of tetraquark $bb\bar{b}\bar{b}$ with the works that reported
results on $bb\bar{b}\bar{b}$ using different theoretical techniques. It would
be noted that in many works authors used calculation methods Wu ; Deng ; Deng2
; Deng3  and physical pictures Gordillo ; Lebed4 ; Maiani2 ; Maiani3  which
are so close to our method but the potentials we have applied to describe the
connection between four quarks are different. We believe that the main reason
for the slight difference between our results and other works is the
difference in potentials.

Table 5:  The ground state masses of $bb\bar{b}\bar{b}$ in various models, unit in GeV.  $J^{P}$ | M($flip-flop$) | M($butterfly$) | MChen2  | MWang  | MWu  | MLiu  | MGordillo
---|---|---|---|---|---|---|---
$0^{+}$ | 18.43972 | 18.44417 | $18.45\pm 0.15$ | 18.840 | 20.155 | 19.322 | 19.199
$1^{+}$ | 18.43972 | 18.44417 | $18.33\pm 0.17$ | 18.840 | 20.212 | 19.329 | 19.276
$2^{+}$ | 18.43972 | 18.44417 | $18.32\pm 0.17$ | 18.850 | 20.243 | 19.341 | 19.289

##  V CONCLUSION

Masses of fully-heavy tetraquark for different flux-tube configurations, the
connected butterfly, and the disconnected flip-flop configuration have been
computed with solving the nonrelativistic four-body systems. The measured mass
of $cc\bar{c}\bar{c}$ with the flip-flop potential is $6.8878$ GeV which is so
close to the experimental mass of X(6900) while for butterfly configuration
prediction is about $6.9456$ Gev that is about $0.046$ GeV above the
experimental data. After adding spin-dependent correction in color
configuration $[3]_{c}\bigotimes[\bar{3}]_{c}$ with total spin 2 the flip-flop
potential has reported the best result for the mass of $cc\bar{c}\bar{c}$, it
is $6.9127$ GeV. The mass of $bb\bar{b}\bar{b}$ measured with the flip-flop is
about $18.440$ GeV with S.N I which is in good agreement with the experimental
mass of pair $\Upsilon(1S)$ mesons. There was a perceived no significant
change for the masses of the $bb\bar{b}\bar{b}$ tetraquark with the different
spin quantum numbers after adding the spin-dependent correction. Altogether
both potentials provide good results for the mass of tetraquarks. However, the
results have shown that disconnected configuration is a little more suitable
structure for the heavy tetraquark. The comparison between our results and the
works which used the two-body potential proves that the four-body potentials
are also reliable for studying the properties of tetraquark.

###### Acknowledgements.

The authors are thankful to the Razi University for financial support of this
project.

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