# The Doubly-heavy Tetraquarks ($qq^{\prime}\bar{Q}\bar{Q^{\prime}}$) in a
Constituent Quark Model with a Complete Set of Harmonic Oscillator Bases

Sungsik Noh<EMAIL_ADDRESS>Department of Physics and Institute of
Physics and Applied Physics, Yonsei University, Seoul 03722, Korea  Woosung
Park<EMAIL_ADDRESS>Department of Physics and Institute of Physics
and Applied Physics, Yonsei University, Seoul 03722, Korea  Su Houng Lee
<EMAIL_ADDRESS>Department of Physics and Institute of Physics and
Applied Physics, Yonsei University, Seoul 03722, Korea

###### Abstract

We have improved our previous variational method based constituent quark model
by introducing a complete set of 3-dimensional harmonic oscillator bases as
the spatial part of the total wave function. To assess the validity of our
approach, we compared the binding energy, thus calculated with the exact value
for the hydrogen model. After fitting to the masses of the ground state
hadrons, we apply our new method to analyze the doubly-heavy tetraquark states
$qq^{\prime}\bar{Q}\bar{Q^{\prime}}$ and compared the result for the binding
energies with that from other works. We also calculated the ground state
masses of $T_{sc}(ud\bar{s}\bar{c})$ and $T_{sb}(ud\bar{s}\bar{b})$ with
$(I,S)=(0,1),(0,2)$. We found that $T_{bb}(ud\bar{b}\bar{b})$ and
$us\bar{b}\bar{b}$, both with $(I,S)=(0,1)$, are stable against the two lowest
threshold meson states with binding energies $-145$ MeV and $-42$ MeV,
respectively. We further found that $T_{cb}(ud\bar{c}\bar{b})$ is near the
lowest threshold. The spatial sizes for the tetraquarks are also discussed.

##  I INTRODUCTION

Since the observation of X(3872)Choi:2003ue  and several exotic hadron
candidates that followed, the structure of these particles and other potential
flavor exotic configurations have become a central theme of study. Theoretical
approaches on these topics range from direct lattice calculationIkeda:2016zwx
, sum rule approachNielsen:2009uh , effective models and constituent quark
based modelsLiu:2019zoy . To better understand the data and to point to future
searches, it becomes necessary to improve any simple minded model calculations
to full details.

Quark model based on color and spin interaction was successful in describing
the mass differences between hadrons within a multipletDeRujula:1975qlm . A
constituent quark model with a more realistic potential was proposed in Ref.
Bhaduri  in the early 1980s, which gives a unified description of meson and
baryon spectra. The potential in BCN modelBhaduri  was applied to investigate
the baryon spectra in detailSilvestre:PRD1985 , and to various tetraquark
statesSilvestre:ZPC1993 . In Ref. Silvestre:ZPC1994 , the authors introduced
four different types of interquark potentials, which improves the simultaneous
fit to both the meson and the baryon spectra.

In addition to building up the realistic potentials, there have been many
works towards improving the accuracy of the spatial part of the total wave
function. Among them, in Refs. Silvestre:ZPC1986 ; Silvestre:ZPC1993 , the
harmonic oscillator bases were applied to construct the spatial part of the
wave function for the baryon structure. In particular, Ref. Silvestre:ZPC1993
discussed the validity of the harmonic oscillator bases within the light
baryons, composed of only $u$, $d$, and $s$ quarks. They also extended their
work to the tetraquarks in Refs. Silvestre:ZPC1986 ; Silvestre:ZPC1993 ;
Silvestre:ZPC1994 . In Refs. Silvestre:ZPC1986 ; Vijande:PRD2009 , the authors
built the spatial function using the hyper spherical coordinates for the
tetraquark systems. In Ref. Brink:1998 , using the spatial function that is
made up of multiple Gaussians which allow for internal angular momentum and
satisfy the permutation symmetries restricted by the Pauli principle, the
authors studied the properties of $T_{cc}$ and $T_{bb}$, and confirmed the
stability for the latter. Subsequently, in Ref. Janc:FewBody2004 , the authors
performed a similar calculations but with a more sophisticated bases of
multiple Gaussians.

In this work, we have introduced a complete set of 3-dimensional harmonic
oscillator bases with a rescaling factor that can be flexibly used for better
convergence compared to the harmonic oscillator bases in the
workSilvestre:ZPC1986 , and applied them to construct the spatial part of the
total wave function in a constituent quark model with hyperfine potential
given in Eq. (1). In doing so, our spatial bases more efficiently describes
the ground state wave functions not only for the meson and baryon structures,
but also for the tetraquark structure.

In the following section, we first introduce the constituent quark model and
compare the fittings in the present work to those in our previous
workWoosung:NPA2019 . In Section III, each part of the total wave function is
introduced, emphasizing the classificiation of the harmonic oscillator bases
with the notion of quanta of the harmonic oscillator bases. In Section IV, the
newly obtained numerical results are presented and compared with those in the
previous workWoosung:NPA2019 . Furthermore, we show the relative distances
between the quark pairs together with a pictorial description of the spatial
sizes for the tetraquarks. We also discuss the tetraquarks as a baryon
structure. Discussion and summary are given in Section V. In Appendix, to
assess the validity of our approach, we compare the result obtained with our
method for the hydrogen model with the exact solution. We also show the
details in constructing the spatial wave function with the complete set of
harmonic oscillator bases for the meson as well as the baryon structure. As a
special case, in Appendix C, we present the method for constructing the bases
for the proton.

##  II FORMALISM

We use a nonrelativistic Hamiltonian for the constituent quarks, which is the
same as in our previous workWoosung:NPA2019 .

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum^{4}_{i=4}\left(m_{i}+\frac{{\mathbf{p}}^{2}_{i}}{2m_{i}}\right)-\frac{3}{4}\sum^{4}_{i<j}\frac{\lambda^{c}_{i}}{2}\,\,\frac{\lambda^{c}_{j}}{2}\left(V^{C}_{ij}+V^{CS}_{ij}\right),\qquad$
(1)

where $m_{i}$ is the quark mass and $\lambda^{c}_{i}/2$ is the SU(3) color
operator for the $i$-th quark. The internal quark potentials $V^{C}_{ij}$ and
$V^{CS}_{ij}$ are taken to be the same forms as in our previous
workWoosung:NPA2019 :

$\displaystyle V^{C}_{ij}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{r_{ij}}+\frac{r_{ij}}{a^{2}_{0}}-D,$ (2)
$\displaystyle V^{CS}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}c^{2}\kappa^{\prime}}{m_{i}m_{j}c^{4}}\frac{e^{-\left(r_{ij}\right)^{2}/\left(r_{0ij}\right)^{2}}}{(r_{0ij})r_{ij}}\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}.$
(3)

Here

$\displaystyle r_{0ij}$ $\displaystyle=$ $\displaystyle
1/\left(\alpha+\beta\frac{m_{i}m_{j}}{m_{i}+m_{j}}\right),$ (4)
$\displaystyle\kappa^{\prime}$ $\displaystyle=$
$\displaystyle\kappa_{0}\left(1+\gamma\frac{m_{i}m_{j}}{m_{i}+m_{j}}\right),$
(5)

where $r_{ij}=|{\mathbf{r}}_{i}-{\mathbf{r}}_{j}|$ is the relative distance
between the $i$ and $j$ quarks, and $\bm{\sigma}_{i}$ is the spin operator.
The parameters appearing in Eqs. (2)-(5) are determined by fitting them to the
experimental ground state masses of the hadrons listed in Tables LABEL:mesons,
LABEL:baryons.

Table 1: The masses of mesons obtained (Column 3) with the fitting parameters set given in Eq. (6). Column 4 shows the variational parameter $a$ . Particle | Experimental | Mass | Variational | Error
---|---|---|---|---
Value (MeV) | (MeV) | Parameter (${\rm fm}^{-2}$) | (%)
$D$ | 1864.8 | 1853.8 |  $a$ = 7.5 | 0.59
$D^{*}$ | 2007.0 | 2006.2 |  $a$ = 5.7 | 0.04
$\eta_{c}$ | 2983.6 | 2986.0 |  $a$ = 25.2 | 0.08
$J/\Psi$ | 3096.9 | 3118.4 |  $a$ = 19.7 | 0.69
$D_{s}$ | 1968.3 | 1963.6 |  $a$ = 12.1 | 0.24
$D^{*}_{s}$ | 2112.1 | 2109.2 |  $a$ = 9.3 | 0.14
$K$ | 493.68 | 498.32 |  $a$ = 7.7 | 0.94
$K^{*}$ | 891.66 | 874.66 |  $a$ = 4.1 | 1.91
$B$ | 5279.3 | 5301.2 |  $a$ = 7.3 | 0.42
$B^{*}$ | 5325.2 | 5360.5 |  $a$ = 6.5 | 0.66
$\eta_{b}$ | 9398.0 | 9327.1 |  $a$ = 100.2 | 0.75
$\Upsilon$ | 9460.3 | 9456.6 |  $a$ = 81.9 | 0.04
$B_{s}$ | 5366.8 | 5375.3 |  $a$ = 13.0 | 0.16
$B_{s}^{*}$ | 5415.4 | 5439.3 |  $a$ = 11.5 | 0.44
$B_{c}$ | 6275.6 | 6268.4 |  $a$ = 38.7 | 0.11
$B_{c}^{*}$ | - | 6361.8 |  $a$ = 32.6 | -
Table 2: Same as Table LABEL:mesons but for baryons. In column 4, $a_{1}$ and $a_{2}$ are the variational parameters. Particle | Experimental | Mass |  Variational | Error
---|---|---|---|---
Value (MeV) | (MeV) |  Parameters (${\rm fm}^{-2}$) | (%)
$\Lambda$ | 1115.7 | 1111.7 |  $a_{1}$ = 4.2, $a_{2}$ = 3.5 | 0.36
$\Lambda_{c}$ | 2286.5 | 2269.4 |  $a_{1}$ = 4.4, $a_{2}$ = 4.4 | 0.75
$\Xi_{cc}$ | 3621.4 | 3621.1 |  $a_{1}$ = 11.3, $a_{2}$ = 4.5 | 0.01
$\Lambda_{b}$ | 5619.4 | 5634.6 |  $a_{1}$ = 4.5, $a_{2}$ = 5.0 | 0.27
$\Sigma_{c}$ | 2452.9 | 2438.5 |  $a_{1}$ = 2.8, $a_{2}$ = 5.5 | 0.59
$\Sigma_{c}^{*}$ | 2517.5 | 2523.2 |  $a_{1}$ = 2.5, $a_{2}$ = 4.7 | 0.23
$\Sigma_{b}$ | 5811.3 | 5841.6 |  $a_{1}$ = 2.8, $a_{2}$ = 5.8 | 0.52
$\Sigma_{b}^{*}$ | 5832.1 | 5875.1 |  $a_{1}$ = 2.8, $a_{2}$ = 4.2 | 0.74
$\Sigma$ | 1192.6 | 1192.4 |  $a_{1}$ = 2.9, $a_{2}$ = 4.6 | 0.02
$\Sigma^{*}$ | 1383.7 | 1395.8 |  $a_{1}$ = 2.3, $a_{2}$ = 3.2 | 0.87
$\Xi$ | 1314.9 | 1327.1 |  $a_{1}$ = 4.5, $a_{2}$ = 4.3 | 0.93
$\Xi^{*}$ | 1531.8 | 1540.4 |  $a_{1}$ = 4.1, $a_{2}$ = 2.8 | 0.56
$\Xi_{c}$ | 2467.8 | 2472.0 |  $a_{1}$ = 4.9, $a_{2}$ = 6.0 | 0.17
$\Xi_{c}^{*}$ | 2645.9 | 2649.7 |  $a_{1}$ = 3.3, $a_{2}$ = 6.1 | 0.15
$\Xi_{b}$ | 5787.8 | 5824.1 |  $a_{1}$ = 5.0, $a_{2}$ = 7.2 | 0.63
$\Xi_{b}^{*}$ | 5945.5 | 5989.1 |  $a_{1}$ = 3.4, $a_{2}$ = 7.6 | 0.73
$p$ | 938.27 | 938.05 |  $a_{1}$ = 2.9, $a_{2}$ = 2.9 | 0.02
$\Delta$ | 1232 | 1242.2 |  $a_{1}$ = 1.9, $a_{2}$ = 1.9 | 0.83

The fitting has been done mostly within 0.7% error as can be seen in Tables
LABEL:mesons, LABEL:baryons. In particular, we focused on trying to minimize
the error on the doubly-charmed baryon, $\Xi_{cc}$, and on the baryons
$\Lambda$, $p$ and also on the mesons $D$, $D^{*}$, $B$, $B^{*}$, $B_{s}$
which comprise the lowest thresholds for the doubly-heavy tetraquarks of
interest in this work.

In comparison to our previous workWoosung:NPA2019 , where only hadrons with at
least one $c$ or $b$ were fit, here we fit all hadrons including those with
only light quarks. The number of fitting hadrons is increased by almost a
factor of 2 compared to that in our previous workWoosung:NPA2019 . The better
fit is a consequence of using the complete set of harmonic oscillator bases
approach. Using the harmonic oscillator bases, the new fitting parameters are
as follows.

$\displaystyle\kappa=120.0\,\textrm{MeV fm},\quad
a_{0}=0.0334066\,\textrm{(MeV${}^{-1}$fm)${}^{1/2}$},$ $\displaystyle
D=917\,\textrm{MeV},$ $\displaystyle m_{u}=342\,\textrm{MeV},\qquad
m_{s}=642\,\textrm{MeV},$ $\displaystyle m_{c}=1922\,\textrm{MeV},\qquad
m_{b}=5337\,\textrm{MeV},$
$\displaystyle\alpha=1.0749\,\textrm{fm${}^{-1}$},\,\,\beta=0.0008014\,\textrm{(MeV
fm)${}^{-1}$},$
$\displaystyle\gamma=0.001380\,\textrm{MeV${}^{-1}$},\,\,\kappa_{0}=197.144\,\textrm{MeV}.$
(6)

##  III Wave Function

Here, we will study the tetraquark states with the total orbital angular
momentum equal to zero in the constituent quark model. However, it should be
noted that the $p$-wave or even higher orbital states between the quarks can
play a crucial role in lowering the total hadron energy and contribute to the
total wave function. This can be successfully done by introducing the complete
set of 3-dimensional harmonic oscillator bases. The total wave function of the
Hamiltonian consists of the spatial, color, spin, and the flavor parts of
bases. We adopt the harmonic oscillator bases as the spatial part of the wave
function. The other parts of the wave function are the same as in the
workWoosung:NPA2019 . We will thus discuss the spatial part in detail, while
the other parts will be mentioned briefly.

###  III.1 Jacobi Coordinates Sets

To set up the spatial function, we first set the Jacobi coordinates,
representing the relative positions of all the quarks in the tetraquark
configuration. The Jacobi coordinates sets in each configuration can be
written as follows.

  * •

Coordinates Set 1

$\displaystyle\mathbf{x}_{1}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{1}-{\mathbf{r}}_{2}),\qquad\mathbf{x}_{2}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{3}-{\mathbf{r}}_{4})\,,$
$\displaystyle\mathbf{x}_{3}=\frac{1}{\mu}\left(\frac{m_{1}{\mathbf{r}}_{1}+m_{2}{\mathbf{r}}_{2}}{m_{1}+m_{2}}-\frac{m_{3}{\mathbf{r}}_{3}+m_{4}{\mathbf{r}}_{4}}{m_{3}+m_{4}}\right)\,,$
(7)

  * •

Coordinates Set 2

$\displaystyle\mathbf{y}_{1}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{1}-{\mathbf{r}}_{3}),\qquad\mathbf{y}_{2}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{4}-{\mathbf{r}}_{2})\,,$
$\displaystyle\mathbf{y}_{3}=\frac{1}{\mu}\left(\frac{m_{1}{\mathbf{r}}_{1}+m_{2}{\mathbf{r}}_{3}}{m_{1}+m_{2}}-\frac{m_{3}{\mathbf{r}}_{2}+m_{4}{\mathbf{r}}_{4}}{m_{3}+m_{4}}\right)\,,$
(8)

  * •

Coordinates Set 3

$\displaystyle\mathbf{z}_{1}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{1}-{\mathbf{r}}_{4}),\qquad\mathbf{z}_{2}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{2}-{\mathbf{r}}_{3})\,,$
$\displaystyle\mathbf{z}_{3}=\frac{1}{\mu}\left(\frac{m_{1}{\mathbf{r}}_{1}+m_{2}{\mathbf{r}}_{4}}{m_{1}+m_{2}}-\frac{m_{3}{\mathbf{r}}_{2}+m_{4}{\mathbf{r}}_{3}}{m_{3}+m_{4}}\right)\,,$
(9)

where

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\left[\frac{m_{1}^{2}+m_{2}^{2}}{(m_{1}+m_{2})^{2}}+\frac{m_{3}^{2}+m_{4}^{2}}{(m_{3}+m_{4})^{2}}\right]^{1/2}\,,$

and

$\displaystyle m_{u}=m_{d};$ $\displaystyle
m_{1}=m_{2}=m_{u},\,\,m_{3}=m_{4}=m_{c}$ $\displaystyle\quad{\rm
for}\,\,ud\bar{c}\bar{c},$ $\displaystyle
m_{1}=m_{2}=m_{u},\,\,m_{3}=m_{4}=m_{b}$ $\displaystyle\quad{\rm
for}\,\,ud\bar{b}\bar{b},$ $\displaystyle
m_{1}=m_{2}=m_{u},\,\,m_{3}=m_{c},\,\,m_{4}=m_{b}$ $\displaystyle\quad{\rm
for}\,\,ud\bar{c}\bar{b},$ $\displaystyle
m_{1}=m_{2}=m_{u},\,\,m_{3}=m_{s},\,\,m_{4}=m_{c}$ $\displaystyle\quad{\rm
for}\,\,ud\bar{s}\bar{c},$ $\displaystyle
m_{1}=m_{2}=m_{u},\,\,m_{3}=m_{s},\,\,m_{4}=m_{b}$ $\displaystyle\quad{\rm
for}\,\,ud\bar{s}\bar{b},$ $\displaystyle
m_{1}=m_{u},\,\,m_{2}=m_{s},\,\,m_{3}=m_{4}=m_{b}$ $\displaystyle\quad{\rm
for}\,\,us\bar{b}\bar{b}.$

For symmetry reason, we take the coordinates set 1 as our reference, and use
the transformations between the above sets of Jacobi coordinates to calculate
relevant matrix elements involving two quarks.

###  III.2 Color, Spin Bases and Flavor

The most stable state for the doubly-heavy tetraquarks can be found in the
spin 1 channelWoosung:NPA2019 . The colosr-spin (CS) space for the spin 1
tetraquark system is spanned by the six CS bases due to the fact that the
tetraquark configuration in the total spin 1 state can be described by two
color bases, and three spin bases. In the configuration of the Jacobi
coordinates set 1, the six CS bases are as followsWoosung:NPA2019 .

$\displaystyle\psi^{CS}_{1}=\left(q_{1}q_{2}\right)^{\mathbf{6}}_{1}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{\bar{6}}}_{1},\,\,\psi^{CS}_{2}=\left(q_{1}q_{2}\right)^{\mathbf{\bar{3}}}_{1}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{3}}_{1},$
$\displaystyle\psi^{CS}_{3}=\left(q_{1}q_{2}\right)^{\mathbf{6}}_{1}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{\bar{6}}}_{0},\,\,\psi^{CS}_{4}=\left(q_{1}q_{2}\right)^{\mathbf{\bar{3}}}_{1}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{3}}_{0},$
$\displaystyle\psi^{CS}_{5}=\left(q_{1}q_{2}\right)^{\mathbf{6}}_{0}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{\bar{6}}}_{1},\,\,\psi^{CS}_{6}=\left(q_{1}q_{2}\right)^{\mathbf{\bar{3}}}_{0}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{3}}_{1}.$
(10)

where the superscript indicates the color state, and the subscript indicates
the spin state for the subparticle systems in the tetraquark structure.

For the falvor part, we will consider the isospin 0 for
$T_{cc}(ud\bar{c}\bar{c})$, $T_{bb}(ud\bar{b}\bar{b})$,
$T_{cb}(ud\bar{c}\bar{b})$, $T_{sc}(ud\bar{s}\bar{c})$,
$T_{sb}(ud\bar{s}\bar{b})$, and the isospin 1/2 for $us\bar{b}\bar{b}$. The
basis of the Hamiltonian is determined to satisfy the symmetry constraint due
to the Pauli principle, and constructed by combining the CS basis with the
spatial part. The permutation symmetries for the CS bases and for the flavor
part are summarized in Table LABEL:SymmetryCSF.

Table 3: The permutation symmetries of the color, spin, and flavor parts for the tetraquarks studied in this work. Here, $+1(-1)$ indicates that it is symmetric (antisymmetric) under the corresponding permutations in the configuration of the Jacobi coordinates set 1. The blank in the table indicates that there is no symmetry constraint under the corresponding permutation. $T_{QQ^{\prime}}$ stands for $ud\bar{Q}\bar{Q^{\prime}}$. Permutation |  | CS Bases
---|---|---
| $\psi^{CS}_{1}$ | $\psi^{CS}_{2}$ | $\psi^{CS}_{3}$ | $\psi^{CS}_{4}$ | $\psi^{CS}_{5}$ | $\psi^{CS}_{6}$
(12) |  | +1 | $-1$ | +1 | $-1$ | $-1$ | +1
(34) |  | +1 | $-1$ | $-1$ | +1 | +1 | $-1$
Permutation |  | Flavor
| $T_{QQ}$ | $T_{QQ^{\prime}}$ | $us\bar{b}\bar{b}$
$(12)$ |  | $-1$ | $-1$ |
$(34)$ |  | $+1$ |  | +1

###  III.3 Harmonic Oscillator Bases as The Spatial Function

Generalizing the method used in Appendix A, and B for the meson and baryon
structures, we construct the complete set of harmonic oscillator bases for the
tetraquarks. Since there are three internal orbital angular momenta $l_{1}$,
$l_{2}$, and $l_{3}$, there are three ways to combine them depending on the
order of addition. Choosing to combine $l_{1}$ and $l_{2}$ first, the spatial
function can be constructed as follows.

$\displaystyle\psi(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3})^{Spatial}_{[n_{1},n_{2},n_{3},l_{1},l_{2},l_{3}]}$
$\displaystyle=\sum_{m_{1},m_{2},m_{3}}C(l_{1},m_{1},l_{2},m_{2};L_{1,2}=l_{3},m_{1,2}=-m_{3})$
$\displaystyle\hskip
2.84544pt\times\,C(L_{1,2}=l_{3},m_{1,2}=-m_{3},l_{3},m_{3};l=0,m=0)$
$\displaystyle\hskip
2.84544pt\times\,R_{n_{1},l_{1}}(x_{1})R_{n_{2},l_{2}}(x_{2})R_{n_{3},l_{3}}(x_{3})$
$\displaystyle\hskip
2.84544pt\times\,Y^{m_{1}}_{l_{1}}(\theta_{1},\phi_{1})Y^{m_{2}}_{l_{2}}(\theta_{2},\phi_{2})Y^{m_{3}}_{l_{3}}(\theta_{3},\phi_{3})\,.$
(11)

where the coefficient $C(l_{1},m_{1},l_{2},m_{2};L_{1,2},m_{1,2})$ is the
Clebsch-Gordan(CG) coefficient for the decomposition of the subtotal angular
state $|L_{1,2},m_{1,2}\rangle$ in terms of
$|l_{1},m_{1}\rangle|l_{2},m_{2}\rangle$, and the subtotal angular state of
the subparticle corresponding to the quark pair $(1,2)$ is restricted to
$|L_{1,2}=l_{3},m_{1,2}=-m_{3}\rangle$ to satisfy the total $l=0$ state.
$R_{n_{i},l_{i}}(x_{i})$ has the same form as in Eq. (V.1.2), and
$Y^{m_{i}}_{l_{i}}(\theta_{i},\phi_{i})$ is the spherical harmonic function
for the angular part of the $i$-th Jacobi coordinate $\mathbf{x}_{i}$. For the
other two types of combinations, Eq. (11) is modified by adding $l_{1},l_{3}$
or $l_{2},l_{3}$ first.

Since there are a large number of harmonic oscillator bases in the tetraquark
system, there has to be a criterion when adding the harmonic oscillator bases
as a spatial function into the calculations. The magnitude of the eigenvalues
of the Hamiltonian is mostly affected by the diagonal elements rather than the
off-diagonal ones. Especially, except for the rest masses, the largest part in
the diagonal element is the kinetic energy. In addition, non-zero kinetic
energy contribution is mostly coming from the diagonal elements. Therefore, in
classifying the harmonic oscillator bases, we define the quanta of the
harmonic oscillator bases by the expectation value of the kientic energy of
the diagonal component for the extreme case where all the constituent quark
masses are identical. Introducing the center of mass frame, the kinetic energy
denoted by $T_{c}$ becomes as follows.

$\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\sum^{4}_{i=1}\frac{{\mathbf{p}}^{2}_{i}}{2m_{i}}-\frac{{\mathbf{p}}^{2}_{rC}}{2M}=\frac{{\mathbf{p}}^{2}_{1}}{2m^{\prime}_{1}}+\frac{{\mathbf{p}}^{2}_{2}}{2m^{\prime}_{2}}+\frac{{\mathbf{p}}^{2}_{3}}{2m^{\prime}_{3}}\,,$
(12)

where

$\displaystyle
m^{\prime}_{1}=m_{u},\,\,m^{\prime}_{2}=m_{c},\,\,m^{\prime}_{3}=\frac{2m_{u}m_{c}}{m_{u}+m_{c}}\quad\qquad{\rm
for}\,\,ud\bar{c}\bar{c},$ $\displaystyle
m^{\prime}_{1}=m_{u},\,\,m^{\prime}_{2}=m_{b},\,\,m^{\prime}_{3}=\frac{2m_{u}m_{b}}{m_{u}+m_{b}}\quad\qquad{\rm
for}\,\,ud\bar{b}\bar{b},$ $\displaystyle
m^{\prime}_{1}=m_{u},\,\,m^{\prime}_{2}=\frac{2m_{c}m_{b}}{m_{c}+m_{b}},$
$\displaystyle\hskip
7.96674ptm^{\prime}_{3}=\frac{(3m_{c}^{2}+2m_{c}m_{b}+3m_{b}^{2})m_{u}}{(m_{c}+m_{b})(2m_{u}+m_{c}+m_{b})}\,\,\quad\qquad{\rm
for}\,\,ud\bar{c}\bar{b},$ $\displaystyle
m^{\prime}_{1}=m_{u},\,\,m^{\prime}_{2}=\frac{2m_{s}m_{c}}{m_{s}+m_{c}},$
$\displaystyle\hskip
7.96674ptm^{\prime}_{3}=\frac{(3m_{s}^{2}+2m_{s}m_{c}+3m_{c}^{2})m_{u}}{(m_{s}+m_{c})(2m_{u}+m_{s}+m_{c})}\,\,\quad\qquad{\rm
for}\,\,ud\bar{s}\bar{c},$ $\displaystyle
m^{\prime}_{1}=m_{u},\,\,m^{\prime}_{2}=\frac{2m_{s}m_{b}}{m_{s}+m_{b}},$
$\displaystyle\hskip
7.96674ptm^{\prime}_{3}=\frac{(3m_{s}^{2}+2m_{s}m_{b}+3m_{b}^{2})m_{u}}{(m_{s}+m_{b})(2m_{u}+m_{s}+m_{b})}\,\,\quad\qquad{\rm
for}\,\,ud\bar{s}\bar{b},$ $\displaystyle
m^{\prime}_{1}=\frac{2m_{u}m_{s}}{m_{u}+m_{s}},\,\,m^{\prime}_{2}=m_{b},$
$\displaystyle\hskip
7.96674ptm^{\prime}_{3}=\frac{(3m_{u}^{2}+2m_{u}m_{s}+3m_{s}^{2})m_{b}}{(m_{u}+m_{s})(m_{u}+m_{s}+2m_{b})}\,\,\quad\qquad{\rm
for}\,\,us\bar{b}\bar{b}.$ (13)

For the extreme case where all the masses of the constituent quarks are
identical ($m_{1}=m_{2}=m_{3}=m_{4}\equiv m$), and thus also the variational
parameters ($a_{1}=a_{2}=a_{3}\equiv a$), the diagonal component of the
kinetic energy reduces to

$\displaystyle\langle T_{c}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}c^{2}a}{m}\bigg{[}\left(2n_{1}+l_{1}+\frac{3}{2}\right)$
(14)
$\displaystyle\qquad\qquad\qquad+\left(2n_{2}+l_{2}+\frac{3}{2}\right)+\left(2n_{3}+l_{3}+\frac{3}{2}\right)\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\hbar^{2}c^{2}a}{m}\bigg{[}2n_{1}+l_{1}+2n_{2}+l_{2}+2n_{3}+l_{3}+\frac{9}{2}\bigg{]}\,.$

Therefore, one notes for this special cases that the kinetic energy is the
same for all the possible combinations of the quantum numbers
($n_{1},n_{2},n_{3},l_{1},l_{2},l_{3}$) if the sum
$(Q=2n_{1}+2n_{2}+2n_{3}+l_{1}+l_{2}+l_{3})$ is unchanged. One can now
organize the harmonic oscillator bases according to the quanta $Q$ appearing
in Eq. (14). Then, for mesons, each of the spatial functions is categorized in
the different quanta. For the tetraquarks, the spatial functions included in
our calculations are classified as follows. In terms of
$\psi^{spatial}_{[n_{1},n_{2},n_{3},l_{1},l_{2},l_{3}]}$ without specifying
the arguments $(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3})$,

  * •

1st Quanta ($Q=0$)
$\psi^{Spatial}_{[0,0,0,0,0,0]}$.

  * •

2nd Quanta ($Q=2$)
$\psi^{Spatial}_{[1,0,0,0,0,0]}$, $\psi^{Spatial}_{[0,1,0,0,0,0]}$,
$\psi^{Spatial}_{[0,0,1,0,0,0]}$,
$\psi^{Spatial}_{[0,0,0,1,1,0]}$, $\psi^{Spatial}_{[0,0,0,1,0,1]}$,
$\psi^{Spatial}_{[0,0,0,0,1,1]}$.

  * •

3rd Quanta ($Q=4$)
$\psi^{Spatial}_{[2,0,0,0,0,0]}$, $\psi^{Spatial}_{[0,2,0,0,0,0]}$,
$\psi^{Spatial}_{[0,0,2,0,0,0]}$,
$\psi^{Spatial}_{[1,0,0,1,1,0]}$, $\psi^{Spatial}_{[0,1,0,1,1,0]}$,
$\psi^{Spatial}_{[0,0,1,1,1,0]}$,
$\psi^{Spatial}_{[1,0,0,1,0,1]}$, $\psi^{Spatial}_{[0,1,0,1,0,1]}$,
$\psi^{Spatial}_{[0,0,1,1,0,1]}$,
$\psi^{Spatial}_{[1,0,0,0,1,1]}$, $\psi^{Spatial}_{[0,1,0,0,1,1]}$,
$\psi^{Spatial}_{[0,0,1,0,1,1]}$,
$\psi^{Spatial}_{[0,0,0,2,2,0]}$, $\psi^{Spatial}_{[0,0,0,2,0,2]}$,
$\psi^{Spatial}_{[0,0,0,0,2,2]}$,
$\psi^{Spatial}_{[0,0,0,1,1,2]}$, $\psi^{Spatial}_{[0,0,0,1,2,1]}$,
$\psi^{Spatial}_{[0,0,0,2,1,1]}$,
$\psi^{Spatial}_{[1,1,0,0,0,0]}$, $\psi^{Spatial}_{[1,0,1,0,0,0]}$,
$\psi^{Spatial}_{[0,1,1,0,0,0]}$.

  * •

4th Quanta ($Q=6$)
$\psi^{Spatial}_{[3,0,0,0,0,0]}$, $\psi^{Spatial}_{[0,3,0,0,0,0]}$,
$\psi^{Spatial}_{[0,0,3,0,0,0]}$,
$\psi^{Spatial}_{[2,0,0,1,1,0]}$, $\psi^{Spatial}_{[0,2,0,1,1,0]}$,
$\psi^{Spatial}_{[0,0,2,1,1,0]}$,
$\psi^{Spatial}_{[2,0,0,1,0,1]}$, $\psi^{Spatial}_{[0,2,0,1,0,1]}$,
$\psi^{Spatial}_{[0,0,2,1,0,1]}$,
$\psi^{Spatial}_{[2,0,0,0,1,1]}$, $\psi^{Spatial}_{[0,2,0,0,1,1]}$,
$\psi^{Spatial}_{[0,0,2,0,1,1]}$,
$\psi^{Spatial}_{[1,1,0,1,1,0]}$, $\psi^{Spatial}_{[1,0,1,1,1,0]}$,
$\psi^{Spatial}_{[0,1,1,1,1,0]}$,
$\psi^{Spatial}_{[1,1,0,1,0,1]}$, $\psi^{Spatial}_{[1,0,1,1,0,1]}$,
$\psi^{Spatial}_{[0,1,1,1,0,1]}$,
$\psi^{Spatial}_{[1,1,0,0,1,1]}$, $\psi^{Spatial}_{[1,0,1,0,1,1]}$,
$\psi^{Spatial}_{[0,1,1,0,1,1]}$,
$\psi^{Spatial}_{[1,0,0,2,2,0]}$, $\psi^{Spatial}_{[0,1,0,2,2,0]}$,
$\psi^{Spatial}_{[0,0,1,2,2,0]}$,
$\psi^{Spatial}_{[1,0,0,2,0,2]}$, $\psi^{Spatial}_{[0,1,0,2,0,2]}$,
$\psi^{Spatial}_{[0,0,1,2,0,2]}$,
$\psi^{Spatial}_{[1,0,0,0,2,2]}$, $\psi^{Spatial}_{[0,1,0,0,2,2]}$,
$\psi^{Spatial}_{[0,0,1,0,2,2]}$,
$\psi^{Spatial}_{[0,0,0,3,3,0]}$, $\psi^{Spatial}_{[0,0,0,3,0,3]}$,
$\psi^{Spatial}_{[0,0,0,0,3,3]}$,
$\psi^{Spatial}_{[1,0,0,1,1,2]}$, $\psi^{Spatial}_{[0,1,0,1,1,2]}$,
$\psi^{Spatial}_{[0,0,1,1,1,2]}$,
$\psi^{Spatial}_{[1,0,0,1,2,1]}$, $\psi^{Spatial}_{[0,1,0,1,2,1]}$,
$\psi^{Spatial}_{[0,0,1,1,2,1]}$,
$\psi^{Spatial}_{[1,0,0,2,1,1]}$, $\psi^{Spatial}_{[0,1,0,2,1,1]}$,
$\psi^{Spatial}_{[0,0,1,2,1,1]}$,
$\psi^{Spatial}_{[1,2,0,0,0,0]}$, $\psi^{Spatial}_{[2,1,0,0,0,0]}$,
$\psi^{Spatial}_{[1,0,2,0,0,0]}$, $\psi^{Spatial}_{[2,0,1,0,0,0]}$,
$\psi^{Spatial}_{[0,1,2,0,0,0]}$, $\psi^{Spatial}_{[0,2,1,0,0,0]}$,
$\psi^{Spatial}_{[1,1,1,0,0,0]}$.

  * •

5th Quanta ($Q=8$)
$\psi^{Spatial}_{[4,0,0,0,0,0]}$, $\psi^{Spatial}_{[0,4,0,0,0,0]}$,
$\psi^{Spatial}_{[0,0,4,0,0,0]}$,
$\psi^{Spatial}_{[3,0,0,1,1,0]}$, $\psi^{Spatial}_{[0,3,0,1,1,0]}$,
$\psi^{Spatial}_{[0,0,3,1,1,0]}$,
$\psi^{Spatial}_{[3,0,0,1,0,1]}$, $\psi^{Spatial}_{[0,3,0,1,0,1]}$,
$\psi^{Spatial}_{[0,0,3,1,0,1]}$,
$\psi^{Spatial}_{[3,0,0,0,1,1]}$, $\psi^{Spatial}_{[0,3,0,0,1,1]}$,
$\psi^{Spatial}_{[0,0,3,0,1,1]}$,
$\psi^{Spatial}_{[1,2,0,1,1,0]}$, $\psi^{Spatial}_{[2,1,0,1,1,0]}$,
$\psi^{Spatial}_{[1,0,2,1,1,0]}$, $\psi^{Spatial}_{[2,0,1,1,1,0]}$,
$\psi^{Spatial}_{[0,1,2,1,1,0]}$, $\psi^{Spatial}_{[0,2,1,1,1,0]}$,
$\psi^{Spatial}_{[1,2,0,1,0,1]}$, $\psi^{Spatial}_{[2,1,0,1,0,1]}$,
$\psi^{Spatial}_{[1,0,2,1,0,1]}$, $\psi^{Spatial}_{[2,0,1,1,0,1]}$,
$\psi^{Spatial}_{[0,1,2,1,0,1]}$, $\psi^{Spatial}_{[0,2,1,1,0,1]}$,
$\psi^{Spatial}_{[1,2,0,0,1,1]}$, $\psi^{Spatial}_{[2,1,0,0,1,1]}$,
$\psi^{Spatial}_{[1,0,2,0,1,1]}$, $\psi^{Spatial}_{[2,0,1,0,1,1]}$,
$\psi^{Spatial}_{[0,1,2,0,1,1]}$, $\psi^{Spatial}_{[0,2,1,0,1,1]}$,
$\psi^{Spatial}_{[1,1,1,1,1,0]}$, $\psi^{Spatial}_{[1,1,1,1,0,1]}$,
$\psi^{Spatial}_{[1,1,1,0,1,1]}$,
$\psi^{Spatial}_{[2,0,0,2,2,0]}$, $\psi^{Spatial}_{[0,2,0,2,2,0]}$,
$\psi^{Spatial}_{[0,0,2,2,2,0]}$,
$\psi^{Spatial}_{[2,0,0,2,0,2]}$, $\psi^{Spatial}_{[0,2,0,2,0,2]}$,
$\psi^{Spatial}_{[0,0,2,2,0,2]}$,
$\psi^{Spatial}_{[2,0,0,0,2,2]}$, $\psi^{Spatial}_{[0,2,0,0,2,2]}$,
$\psi^{Spatial}_{[0,0,2,0,2,2]}$,
$\psi^{Spatial}_{[1,1,0,2,2,0]}$, $\psi^{Spatial}_{[1,0,1,2,2,0]}$,
$\psi^{Spatial}_{[1,0,1,2,2,0]}$,
$\psi^{Spatial}_{[1,1,0,2,0,2]}$, $\psi^{Spatial}_{[1,0,1,2,0,2]}$,
$\psi^{Spatial}_{[1,0,1,2,0,2]}$,
$\psi^{Spatial}_{[1,1,0,0,2,2]}$, $\psi^{Spatial}_{[1,0,1,0,2,2]}$,
$\psi^{Spatial}_{[0,1,1,0,2,2]}$,
$\psi^{Spatial}_{[1,0,0,3,3,0]}$, $\psi^{Spatial}_{[0,1,0,3,3,0]}$,
$\psi^{Spatial}_{[0,0,1,3,3,0]}$,
$\psi^{Spatial}_{[1,0,0,3,0,3]}$, $\psi^{Spatial}_{[0,1,0,3,0,3]}$,
$\psi^{Spatial}_{[0,0,1,3,0,3]}$,
$\psi^{Spatial}_{[1,0,0,0,3,3]}$, $\psi^{Spatial}_{[0,1,0,0,3,3]}$,
$\psi^{Spatial}_{[0,0,1,0,3,3]}$,
$\psi^{Spatial}_{[2,0,0,1,1,2]}$, $\psi^{Spatial}_{[0,2,0,1,1,2]}$,
$\psi^{Spatial}_{[0,0,2,1,1,2]}$,
$\psi^{Spatial}_{[2,0,0,1,2,1]}$, $\psi^{Spatial}_{[0,2,0,1,2,1]}$,
$\psi^{Spatial}_{[0,0,2,1,2,1]}$,
$\psi^{Spatial}_{[2,0,0,2,1,1]}$, $\psi^{Spatial}_{[0,2,0,2,1,1]}$,
$\psi^{Spatial}_{[0,0,2,2,1,1]}$,
$\psi^{Spatial}_{[1,1,0,1,1,2]}$, $\psi^{Spatial}_{[1,0,1,1,1,2]}$,
$\psi^{Spatial}_{[0,1,1,1,1,2]}$,
$\psi^{Spatial}_{[1,1,0,1,2,1]}$, $\psi^{Spatial}_{[1,0,1,1,2,1]}$,
$\psi^{Spatial}_{[0,1,1,1,2,1]}$,
$\psi^{Spatial}_{[1,1,0,2,1,1]}$, $\psi^{Spatial}_{[1,0,1,2,1,1]}$,
$\psi^{Spatial}_{[0,1,1,2,1,1]}$,
$\psi^{Spatial}_{[2,2,0,0,0,0]}$, $\psi^{Spatial}_{[2,0,2,0,0,0]}$,
$\psi^{Spatial}_{[0,2,2,0,0,0]}$,
$\psi^{Spatial}_{[1,3,0,0,0,0]}$, $\psi^{Spatial}_{[3,1,0,0,0,0]}$,
$\psi^{Spatial}_{[1,0,3,0,0,0]}$, $\psi^{Spatial}_{[3,0,1,0,0,0]}$,
$\psi^{Spatial}_{[0,1,3,0,0,0]}$, $\psi^{Spatial}_{[0,3,1,0,0,0]}$,
$\psi^{Spatial}_{[1,1,2,0,0,0]}$, $\psi^{Spatial}_{[1,2,1,0,0,0]}$,
$\psi^{Spatial}_{[2,1,1,0,0,0]}$.

As can be seen in the above classification, we have considered bases with
internal angular momenta up to the $l_{i}=3$ states because the contribution
from the higher angular momentum basis is not so significant. There are other
types of harmonic oscillator bases constituting the complete set, such as
$\psi^{Spatial}_{[0,0,0,1,1,1]}$. However, there is no transitional matrix
element of the Hamiltonian between this type of bases and the other bases
presented above. Thus, they do not contribute to the ground state masses of
the tetraquarks. In doing so, we have observed the convergence behavior of the
ground state mass, and have included the harmonic oscillator bases up to the
5th quanta in the calculations.

On the other hand, in order to satisfy the Pauli principle, it is necessary to
consider the permutation symmetries of the spatial functions as well as the
other parts of the wave function. In the Jacobi coordinates set 1 of our
reference, it is obvious that $T_{QQ}$ configuration has symmetries under the
permutations (12) and (34). $T_{QQ^{\prime}}$ has symmetry under the
permutation (12), while $us\bar{b}\bar{b}$ has symmetry under the permutation
(34) as summarized in Table LABEL:SymmetryCSF. The symmetry property of the
spatial functions can be summarized for each tetraquark configuration in Table
LABEL:SymmetrySpatial.

Table 4: The permutation symmetry properties of the spatial functions and the corresponding CS bases for each tetraquark configuration. In the table, $(12)$ and $(34)$ indicate the permutations. $T_{QQ^{\prime}}$ stands for $ud\bar{Q}\bar{Q^{\prime}}$. Here, $+1(-1)$ indicates that the spatial function is symmetric (antisymmetric) under the corresponding permutations. The blank in the table implies that there is no symmetry constraint under the corresponding permutation. Type |  | $T_{QQ}$ |  |  | $T_{QQ^{\prime}}$ |  |  | $us\bar{b}\bar{b}$
---|---|---|---|---|---|---|---|---
| (12) | (34) | CS bases |  |  | (12) | (34) | CS bases |  |  | (12) | (34) | CS bases
$l_{1}=l_{2}=$ odd, $l_{3}=$ even |  | $-1$ | $-1$ |  $\psi^{CS}_{4}$, $\psi^{CS}_{5}$ |  |  | $-1$ |  |  $\psi^{CS}_{2}$, $\psi^{CS}_{4}$, $\psi^{CS}_{5}$ |  |  |  | $-1$ |  $\psi^{CS}_{1}$, $\psi^{CS}_{4}$, $\psi^{CS}_{5}$
$l_{1}=l_{3}=$ odd, $l_{2}=$ even |  | $-1$ | +1 | $\psi^{CS}_{2}$ |  |  | $-1$ |  |  $\psi^{CS}_{2}$, $\psi^{CS}_{4}$, $\psi^{CS}_{5}$ |  |  |  | +1 |  $\psi^{CS}_{2}$, $\psi^{CS}_{3}$, $\psi^{CS}_{6}$
$l_{2}=l_{3}=$ odd, $l_{1}=$ even |  | +1 | $-1$ | $\psi^{CS}_{1}$ |  |  | +1 |  |  $\psi^{CS}_{1}$, $\psi^{CS}_{3}$, $\psi^{CS}_{6}$ |  |  |  | $-1$ |  $\psi^{CS}_{1}$, $\psi^{CS}_{4}$, $\psi^{CS}_{5}$
$l_{1}=l_{2}=l_{3}=$ even |  | +1 | +1 |  $\psi^{CS}_{3}$, $\psi^{CS}_{6}$ |  |  | +1 |  |  $\psi^{CS}_{1}$, $\psi^{CS}_{3}$, $\psi^{CS}_{6}$ |  |  |  | +1 |  $\psi^{CS}_{2}$, $\psi^{CS}_{3}$, $\psi^{CS}_{6}$
Table 5: The masses and binding energies $B_{T}$ of the tetraquark states obtained with the fitting parameters in Eq. (6). The binding energy $B_{T}$ is defined by the difference between the tetraquark mass and the sum of the masses of the lowest threshold mesons, $B_{T}\equiv M_{Tetraquark}-M_{meson1}-M_{meson2}$. The values in the parentheses are the results in our previous workWoosung:NPA2019 . Type |  | ($I$, $S$) |  | Mass(MeV) |  | Variational Parameters(fm-2) |  |  $B_{T}$(MeV)
---|---|---|---|---|---|---|---|---
$ud\bar{b}\bar{b}$ |  | (0,1) |  | 10517 (10518) |  | $a_{1}=3.9(2.8),\,a_{2}=25.0(20.9),\,a_{3}=3.8(2.8)$ |  |  $-145$ ($-121$)
$ud\bar{c}\bar{c}$ |  | (0,1) |  | 3873 (3965) |  | $a_{1}=2.6(2.8),\,a_{2}=4.6(7.6),\,a_{3}=4.6(2.7)$ |  | +13 (+99)
$ud\bar{c}\bar{b}$ |  | (0,1) |  | 7212 (7262) |  | $a_{1}=3.1(3.1),\,a_{2}=8.0(10.3),\,a_{3}=5.0(2.7)$ |  |  $-3$ (+49)
$us\bar{b}\bar{b}$ |  | (1/2,1) |  | 10694 (10684) |  | $a_{1}=4.0(3.5),\,a_{2}=21.4(20.6),\,a_{3}=6.0(3.5)$ |  |  $-42$ ($-7$)
$ud\bar{s}\bar{c}$ |  | (0,1) |  | 2596 |  | $a_{1}=2.4,\,a_{2}=3.9,\,a_{3}=5.3$ |  | +91
$ud\bar{s}\bar{c}$ |  | (0,2) |  | 2938 |  | $a_{1}=1.6,\,a_{2}=2.1,\,a_{3}=4.0$ |  | +57
$ud\bar{s}\bar{b}$ |  | (0,1) |  | 5949 |  | $a_{1}=2.6,\,a_{2}=5.1,\,a_{3}=6.3$ |  | +90
$ud\bar{s}\bar{b}$ |  | (0,2) |  | 6298 |  | $a_{1}=1.8,\,a_{2}=2.6,\,a_{3}=4.9$ |  | +63

Note that there is antisymmetric part in some of the spatial bases due to the
symmetry property of the harmonic oscillator bases as seen in Table.
LABEL:SymmetrySpatial. Thus, contrary to the previous workWoosung:NPA2019
where only the fully symmetric spatial basis $\psi^{Spatial}_{[0,0,0,0,0,0]}$
was used in the calculations, all the CS bases are used in the present work.

##  IV Numerical Analysis

We use the total wave function discussed in the previous section, and perform
a variational method to determine the ground state masses for the tetraquarks.
The results are shown in Table LABEL:result. We found that $ud\bar{b}\bar{b}$
and $us\bar{b}\bar{b}$ are stable against the lowest strong decay threshold.
In particular, one can see that the effect from the harmonic oscillator bases
appears to lower the binding energies. As a result, $T_{cb}(ud\bar{c}\bar{b})$
is found to be below the lowest threshold. One of the major contributions to
this is coming from the excited orbital states.

In this section, we first discuss a similar tendency of the ground state wave
functions between the tetraquark and the meson structures. Then we will
discuss the effects of the excited orbital states. Also, we investigate the
spatial distibutions and the sizes for the tetraquarks. Finally, we compare
our model with other works.

###  IV.1 Ground State of Tetraquarks

We first analyze the ground state of the tetraquarks by expanding the wave
function in terms of the complete set of harmonic oscillator bases. Using the
variational method, we find that the expansion coefficients of the bases
rapidly converge to zero.

As discussed in Section III.3, each of the harmonic oscillator bases for
mesons is classified by different quanta. Then, as can be seen in Figure 1 for
the $D$ meson, the expansion coefficient monotonically decreases as desired
when the quanta of the bases increases. This tendency can be seen also for the
tetraquark states in Figures 2-5, respectively. In addition to the convergence
of the expansion coefficients, one can observe the convergence of the ground
state masses as the number of the harmonic oscillator bases and their quanta
increase as seen in Figure 10 for the $D$ meson. We then determined the ground
state masses for the mesons, baryons, and the tetraquarks when the convergent
values change by just few MeV for

Table 6: The changes in masses of the tetraquarks when the indicated spatial bases are included in the calculations with the corresponding dominant CS bases. Spatial Bases |  |  |  $M_{T_{bb}}$ (MeV) |  |  |  $M_{T_{cc}}$ (MeV) |  |  |  $M_{T_{cb}}$ (MeV) |  |  |  $M_{us\bar{b}\bar{b}}$ (MeV)
---|---|---|---|---|---|---|---|---|---|---|---|---
$\psi^{Spatial}_{[0,0,0,0,0,0]}$ |  |  | 10578 |  |  | 4002 |  |  | 7316 |  |  | 10764
$\psi^{Spatial}_{[0,0,0,0,0,0]}$, $\psi^{Spatial}_{[0,0,0,1,1,0]}$ |  |  | 10567 |  |  | 3968 |  |  | 7294 |  |  | 10748
$\psi^{Spatial}_{[0,0,0,0,0,0]}$, $\psi^{Spatial}_{[0,0,0,1,1,0]}$ |  |  | 10566 |  |  | 3967 |  |  | 7290 |  |  | 10746
$\psi^{Spatial}_{[0,0,0,1,0,1]}$, $\psi^{Spatial}_{[0,0,0,0,1,1]}$ |  |  |  |  |  |  |  |
Total Change in Mass |  |  | -12 |  |  | -35 |  |  | -26 |  |  | -18
Figure 1: The expansion coefficients of the ground state wave function for $D$
meson. The total number of the Hamiltonian bases (the total wave functions) is
31, each of which is categorized according to a given quanta in the meson
structure. Figure 2: Same as Figure 1 but for $T_{cc}$. The figure is
separated into three parts to clearly see the coefficients with appropriate
scales. The dashed-lines separate them into each quanta of the bases. The
total number of Hamiltonian bases is 266 for $T_{cc}$, which are composed of
the five quanta of harmonic oscillator bases as listed in Section III.3.
Figure 3: Same as Figure 2 but for $T_{bb}$. The Hamiltonian bases for
$T_{bb}$ are the same with those for $T_{cc}$. Figure 4: Same as Figure 2 but
for $T_{cb}$. The number of Hamiltonian bases for $T_{cb}$ is 501, but the
spatial bases are the same with those for $T_{cc}$. Figure 5: Same as Figure 2
but for $us\bar{b}\bar{b}$. The number of Hamiltonian bases for
$us\bar{b}\bar{b}$ is 501, but the spatial bases are the same with those for
$T_{cc}$.

the entire bases contained in the last quanta, which turns out to be the 5th
quanta. On the other hand, the convergence behavior of the expansion
coefficients for the tetraquarks is not monotonic. This is so because while
the nonzero orbital bases are ordered later, their contributions to the
tetraquark configurations are important due to the attraction coming from the
dipole and quadrupole moments. Still the average value of the coefficients in
each quanta monotonically decreases.

Discussing the tetraquark state in some more detail, one finds that from the
second quanta on, many of the largest coefficients correspond to the harmonic
oscillator bases with $l_{1}=l_{2}\neq 0$, which implies that the
contributions from this type of bases are important to obtain the exact ground
state masses for the tetraquarks. This can be quantitatively seen by the mass
changes when this type of bases are additively included in the calculations.
For simplicity, considering only the dominant CS bases, we first evaluate the
mass with only the spatial basis $\psi^{Spatial}_{[0,0,0,0,0,0]}$, and then
compare this with the mass obtained by adding the contribution from the basis
$\psi^{Spatial}_{[0,0,0,1,1,0]}$. The results for such a calculation for some
of the tetraquarks are summarized in Table LABEL:AngularContribution. The
contribution from the basis $\psi^{Spatial}_{[0,0,0,1,1,0]}$ appears to lower
the tetraquark masses by 11 MeV in $T_{bb}$ and 34 MeV in $T_{cc}$,
respectively. The effect is larger in $T_{cc}$ because the magnitudes of the
corresponding expansion coefficients in $T_{cc}$ are obviously larger than the
others in the second quanta, while it is not so in $T_{bb}$.

Also, considering only the dominant CS bases, the coefficient in the first
quanta corresponding to the spatial basis $\psi^{Spatial}_{[0,0,0,0,0,0]}$ is
larger in $T_{bb}$ than that in $T_{cc}$. The first coefficient in $T_{bb}$ is
0.94, which implies that there are less contributions from the other bases in
the tetraquarks of heavier antiquarks than in the tetraquarks of lighter
antiquarks. However, compared to the total changes in masses from the value
only with the first basis in Table LABEL:AngularContribution to the exact
ground state masses in Table LABEL:result, the contribution from the basis
$\psi^{Spatial}_{[0,0,0,1,1,0]}$ is still important in $T_{bb}$ as well.

Let us now discuss the coefficients corresponding to the rest types of excited
orbital bases. For $T_{cc}$ in Figure 2 and $T_{cb}$ in Figure 4, the
coefficients in $T_{cb}$ are not so small relative to those in $T_{cc}$.
Obviously in the second quanta, it is comparable to the others in $T_{cb}$,
while they are not so in $T_{cc}$. As a result, as can be seen in Table
LABEL:AngularContribution, the mass is lowered by 1 MeV in $T_{cc}$ but it is
4 MeV in $T_{cb}$ when adding the bases $\psi^{Spatial}_{[0,0,0,1,0,1]}$ and
$\psi^{Spatial}_{[0,0,0,0,1,1]}$ to the calculations. This is due to the
symmetry breaking in the flavor part of antiquarks, relative to $T_{cc}$
structure. A similar behavior appears also in the comparison between $T_{bb}$
and $us\bar{b}\bar{b}$ depicted in Figures 3, 5. In this case, the flavor
symmetry is bronken in the quark part. Comparing the changes in masses between
$T_{bb}$ and $us\bar{b}\bar{b}$, the contributions from the bases
$\psi^{Spatial}_{[0,0,0,1,0,1]}$ and $\psi^{Spatial}_{[0,0,0,0,1,1]}$ are
little larger in $us\bar{b}\bar{b}$. Therefore, all the types of the excited
orbital states are necessary to obtain the exact ground state masses for the
tetraquark states.

###  IV.2 Spatial Size of Tetraquarks

It is also useful to investigate the spatial size of the tetraquarks, and the
spatial distribution of the constituent quarks in the tetraquark structure.
The relative distances between the quarks are listed in Table
LABEL:RelDistances.

Table 7: The relative distances between the quarks in the tetraquarks in fm unit. The distances are obtained with the ground state of the tetraquakrs. In the table, $(i,j)$ denotes the distance between $i$ and $j$ quarks, and $(1,2)$-$(3,4)$ that between the center of masses of the pairs $(1,2)$ and $(3,4)$. Quark Pair |  |  |  | $T_{bb}$ |  | $T_{cc}$ |  | $T_{cb}$ |  | $us\bar{b}\bar{b}$
---|---|---|---|---|---|---|---|---|---|---
$(1,2)$ |  |  |  | 0.676 |  | 0.830 |  | 0.753 |  | 0.644
$(1,3)$ |  |  |  | 0.592 |  | 0.672 |  | 0.631 |  | 0.584
$(1,4)$ |  |  |  | 0.592 |  | 0.672 |  | 0.612 |  | 0.584
$(2,3)$ |  |  |  | 0.592 |  | 0.672 |  | 0.631 |  | 0.490
$(2,4)$ |  |  |  | 0.592 |  | 0.672 |  | 0.612 |  | 0.490
$(3,4)$ |  |  |  | 0.268 |  | 0.610 |  | 0.464 |  | 0.287
$(1,2)$-$(3,4)$ |  |  |  | 0.463 |  | 0.433 |  | 0.441 |  | 0.397
Table 8: The spatial sizes of the lowest decay threshold mesons for each tetraquark state in fm unit. In the last columns for each section, “Total” indicates the sum of the sizes of two threshold mesons. Tetraquarks |  | $T_{bb}$ |  |  | $T_{cc}$
---|---|---|---|---|---
Lowest Threshold |  | $B$ | $B^{*}$ |  | Total |  |  | $D$ | $D^{*}$ |  | Total
Size |  | 0.525 | 0.551 |  | 1.076 |  |  | 0.519 | 0.586 |  | 1.105
Tetraquarks |  | $T_{cb}$ |  |  | $us\bar{b}\bar{b}$
Lowest Threshold |  | $D$ | $B^{*}$ |  | Total |  |  | $B_{s}$ | $B^{*}$ |  | Total
Size |  | 0.519 | 0.551 |  | 1.070 |  |  | 0.397 | 0.551 |  | 0.948

The relative distance between the heavier quarks are in general closer than
that of the lighter quarksWoosung:NPA2019 . This tendency is maintained in
each tetraquark state as seen in Table LABEL:RelDistances. Looking into the
relative distances, it is found that the relative distances except for $(1,2)$
and $(3,4)$ pairs are all the same in $T_{QQ}$ structure. For
$T_{QQ^{\prime}}$ structure, the relative distances for the pairs $(1,3)$ and
$(2,3)$ are the same, and also for the pairs $(1,4)$ and $(2,4)$. Likewise,
the relative distances for the pairs $(1,3)$ and $(1,4)$ are the same as are
those for the pairs $(2,3)$ and $(2,4)$ in $us\bar{b}\bar{b}$. This is due to
the flavor symmetry in each tetraquark structure, and can be simply evaluated
through the permutation symmetry for the ground state wave function in each
tetraquark state. Since the total wave function satisfies the Pauli principle
in each tetraquark, if we denote the ground state wave function by
$|\Psi^{Tetraquark}_{G}\rangle\equiv|\psi^{Spatial}\rangle\times|\psi^{CS}\rangle$,
the permutation symmetries for each ground state are as follows.

$\displaystyle(12)\left|\Psi^{T_{QQ}}_{G}\right\rangle=(34)\left|\Psi^{T_{QQ}}_{G}\right\rangle=-\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle\quad\,{\rm for}\,\,\,T_{QQ}\,,$
$\displaystyle(12)\left|\Psi^{T_{QQ^{\prime}}}_{G}\right\rangle=-\left|\Psi^{T_{QQ^{\prime}}}_{G}\right\rangle$
$\displaystyle\,\quad{\rm for}\,\,\,T_{QQ^{\prime}}\,,$
$\displaystyle(34)\left|\Psi^{us\bar{b}\bar{b}}_{G}\right\rangle=-\left|\Psi^{us\bar{b}\bar{b}}_{G}\right\rangle$
$\displaystyle\quad\,{\rm for}\,\,\,us\bar{b}\bar{b}\,.$

Then the relative distances in each tetraquark structure can be obtained as
follows.
For $T_{QQ}$,

$\displaystyle\left\langle\Psi^{T_{QQ}}_{G}\right|(12)^{-1}(12)|\mathbf{r}_{1}-\mathbf{r}_{3}|(12)^{-1}(12)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ}}_{G}\right||\mathbf{r}_{2}-\mathbf{r}_{3}|\left|\Psi^{T_{QQ}}_{G}\right\rangle\,,$
$\displaystyle\left\langle\Psi^{T_{QQ}}_{G}\right|(34)^{-1}(34)|\mathbf{r}_{2}-\mathbf{r}_{3}|(34)^{-1}(34)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ}}_{G}\right||\mathbf{r}_{2}-\mathbf{r}_{4}|\left|\Psi^{T_{QQ}}_{G}\right\rangle\,,$
$\displaystyle\left\langle\Psi^{T_{QQ}}_{G}\right|(12)^{-1}(12)|\mathbf{r}_{2}-\mathbf{r}_{4}|(12)^{-1}(12)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ}}_{G}\right||\mathbf{r}_{1}-\mathbf{r}_{4}|\left|\Psi^{T_{QQ}}_{G}\right\rangle\,.$
(15)

For $T_{QQ^{\prime}}$,

$\displaystyle\left\langle\Psi^{T_{QQ^{\prime}}}_{G}\right|(12)^{-1}(12)|\mathbf{r}_{1}-\mathbf{r}_{3}|(12)^{-1}(12)\left|\Psi^{T_{QQ^{\prime}}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ^{\prime}}}_{G}\right||\mathbf{r}_{2}-\mathbf{r}_{3}|\left|\Psi^{T_{QQ^{\prime}}}_{G}\right\rangle\,,$
$\displaystyle\left\langle\Psi^{T_{QQ^{\prime}}}_{G}\right|(12)^{-1}(12)|\mathbf{r}_{1}-\mathbf{r}_{4}|(12)^{-1}(12)\left|\Psi^{T_{QQ^{\prime}}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ^{\prime}}}_{G}\right||\mathbf{r}_{2}-\mathbf{r}_{4}|\left|\Psi^{T_{QQ^{\prime}}}_{G}\right\rangle\,.$
(16)

For $us\bar{b}\bar{b}$,

$\displaystyle\left\langle\Psi^{us\bar{b}\bar{b}}_{G}\right|(34)^{-1}(34)|\mathbf{r}_{1}-\mathbf{r}_{3}|(34)^{-1}(34)\left|\Psi^{us\bar{b}\bar{b}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{us\bar{b}\bar{b}}_{G}\right||\mathbf{r}_{1}-\mathbf{r}_{4}|\left|\Psi^{us\bar{b}\bar{b}}_{G}\right\rangle\,,$
$\displaystyle\left\langle\Psi^{us\bar{b}\bar{b}}_{G}\right|(34)^{-1}(34)|\mathbf{r}_{2}-\mathbf{r}_{3}|(34)^{-1}(34)\left|\Psi^{us\bar{b}\bar{b}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{us\bar{b}\bar{b}}_{G}\right||\mathbf{r}_{2}-\mathbf{r}_{4}|\left|\Psi^{us\bar{b}\bar{b}}_{G}\right\rangle\,.$

Therefore, the relative distances reflect the symmetry property.

Figure 6: The spatial distributions of the quarks for $T_{bb}$ in fm unit.
$\mathbf{R_{c}}$ is the geometric center of the four quarks (the center of the
sphere). The quark positions are the same in both figures. In the bottom
figure, the diameter of the sphere is 0.725 fm. Figure 7: Same as Figure 6 but
for $T_{cc}$. The diameter of the sphere is 0.866 fm. Figure 8: Same as Figure
6 but for $T_{cb}$. The diameter of the sphere is 0.789 fm. Figure 9: Same as
Figure 6 but for $us\bar{b}\bar{b}$. The diameter of the sphere is 0.677 fm.

On the other hand, without loss of generality, one can place one of the quarks
at the origin of the Cartesian system, and also the second quark can be
located on one of the axes. Then the degrees of freedom for the tetraquark
system is in general 7. By the symmetry property, the degrees of freedom
reduces to 3 for $T_{QQ}$ or 4 for $T_{QQ^{\prime}}$ and $us\bar{b}\bar{b}$.
Now we can prove for $T_{QQ}$ structure that the three independent vectors,
$\mathbf{R}_{(1,2)}\equiv(\mathbf{r}_{1}-\mathbf{r}_{2})$,
$\mathbf{R}_{(3,4)}\equiv(\mathbf{r}_{3}-\mathbf{r}_{4})$, and
$\mathbf{R}^{\prime}\equiv\frac{1}{2}(\mathbf{r}_{1}+\mathbf{r}_{2}-\mathbf{r}_{3}-\mathbf{r}_{4})$,
which correspond to the Jacobi coordinates $\mathbf{x}_{1}$, $\mathbf{x}_{2}$,
and $\mathbf{x}_{3}$ are orthogonal to each other.

$\displaystyle\left\langle\Psi^{T_{QQ}}_{G}\right|\left(\mathbf{R}_{(1,2)}\cdot\mathbf{R}_{(3,4)}\right)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ}}_{G}\right|(12)^{-1}(12)\left(\mathbf{R}_{(1,2)}\cdot\mathbf{R}_{(3,4)}\right)(12)^{-1}(12)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=-\left\langle\Psi^{T_{QQ}}_{G}\right|\left(\mathbf{R}_{(1,2)}\cdot\mathbf{R}_{(3,4)}\right)\left|\Psi^{T_{QQ}}_{G}\right\rangle=0\,,$
(18)
$\displaystyle\left\langle\Psi^{T_{QQ}}_{G}\right|\left(\mathbf{R}_{(1,2)}\cdot\mathbf{R}^{\prime}\right)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ}}_{G}\right|(12)^{-1}(12)\left(\mathbf{R}_{(1,2)}\cdot\mathbf{R}^{\prime}\right)(12)^{-1}(12)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=-\left\langle\Psi^{T_{QQ}}_{G}\right|\left(\mathbf{R}_{(1,2)}\cdot\mathbf{R}^{\prime}\right)\left|\Psi^{T_{QQ}}_{G}\right\rangle=0\,,$
(19)
$\displaystyle\left\langle\Psi^{T_{QQ}}_{G}\right|\left(\mathbf{R}_{(3,4)}\cdot\mathbf{R}^{\prime}\right)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=\left\langle\Psi^{T_{QQ}}_{G}\right|(34)^{-1}(34)\left(\mathbf{R}_{(3,4)}\cdot\mathbf{R}^{\prime}\right)(34)^{-1}(34)\left|\Psi^{T_{QQ}}_{G}\right\rangle$
$\displaystyle=-\left\langle\Psi^{T_{QQ}}_{G}\right|\left(\mathbf{R}_{(3,4)}\cdot\mathbf{R}^{\prime}\right)\left|\Psi^{T_{QQ}}_{G}\right\rangle=0\,.$
(20)

These relations show that we can describe the positions of the four quarks in
$T_{cc}$ or $T_{bb}$ with the three independent vectors as shown in Figures 6,
7.

For $T_{cb}$, $\mathbf{R}_{(1,2)}$ and $\mathbf{R}_{(3,4)}$ can be taken as
two orthogonal vectors, but for the third vector
$\left\langle\Psi^{T_{cb}}_{G}\right|\left(\mathbf{R}_{(3,4)}~{}\cdot~{}\mathbf{R}^{\prime}\right)\left|\Psi^{T_{cb}}_{G}\right\rangle\neq
0$ although
$\left\langle\Psi^{T_{cb}}_{G}\right|\left(\mathbf{R}_{(1,2)}~{}\cdot~{}\mathbf{R}^{\prime}\right)\left|\Psi^{T_{cb}}_{G}\right\rangle=0$.
However,
$\mathbf{R}^{\prime}=\left(\mathbf{r}_{1}+\mathbf{r}_{2}\right)/2-\left(m_{3}\mathbf{r}_{3}+m_{4}\mathbf{r}_{4}\right)/\left(m_{3}+m_{4}\right)$
and linearly independent from $\mathbf{R}_{(3,4)}$ and thus span the linearly
independent third direction as shown in Figure 8. One can also show that the
non-vanishing component along the $\mathbf{R}_{(3,4)}$ direction for $T_{cb}$
and $\mathbf{R}_{(1,2)}$ direction for $us\bar{b}\bar{b}$ in the wave
functions are described through the excited orbital states that produce the
asymmetry in the configuration of the Jacobi coordinates set 1 as can be seen
in Figures 8 and 9, respectively. Specifically, since the spatial bases with
$(l_{1},l_{2},l_{3})=(0,0,0)$ are of even powers with respect to the Jacobi
coordinates $\mathbf{x}_{1}$, $\mathbf{x}_{2}$, and $\mathbf{x}_{3}$, one can
find that the spatial part integration of
$\left\langle\Psi^{T_{cb}}_{G}\right|\left(\mathbf{R}_{(3,4)}~{}\cdot~{}\mathbf{R}^{\prime}\right)\left|\Psi^{T_{cb}}_{G}\right\rangle$
becomes zero if considering only the spatial bases with
$(l_{1},l_{2},l_{3})=(0,0,0)$. This implies that the calculations only with
the spatial basis $\psi^{Spatial}_{[0,0,0,0,0,0]}$ lead to the $T_{QQ}$ like
structure even for both $T_{cb}$ and $us\bar{b}\bar{b}$, which is far away
from the real structures in nature.

In addition, using the relations of the relative distances in Table
LABEL:RelDistances, it is possible to specify the spatial positions of the
quarks in the tetraquark structure as the points on the surface of a sphere.
The results are depicted in Figures 6-9 with the center specified as
$\mathbf{R_{c}}$ in the same scales.

###  IV.3 Tetraquarks in a Baryon Structure

The structure ($QQ\bar{q}\bar{q}$) is expected to be similar to that of an
antibaryon $\bar{Q}_{1}\bar{q}\bar{q}$ with the heavy antiquark $\bar{Q}_{1}$
replaced by the diquark $(QQ)$Silvestre:ZPC1986 . However, in $qqQ_{1}$
structure, $Q_{1}$ is in fact composed of two heavy antiquarks so that it can
be of either color [$\mathbf{3}$] or [$\bar{\mathbf{6}}$]. For an antiquark
pair of color [$\bar{\mathbf{6}}$], it cannot be regarded as a point particle
because, at short distance, the Coulomb potential gives strong repulsion due
to the fact that the color matrix element
$\langle\bm{\lambda}^{c}_{i}\bm{\lambda}^{c}_{j}\rangle$ is $4/3$ for the
color [$\bar{\mathbf{6}}$] while it is $-8/3$ for the color [$\mathbf{3}$].
Thus, in terms of color structure, this means that the
$\left(q_{1}q_{2}\right)^{\mathbf{\bar{3}}}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{3}}$
channel, which is the only color configuration in $qqQ_{1}$, dominates the
$\left(q_{1}q_{2}\right)^{\mathbf{6}}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{\bar{6}}}$
channel in $T_{QQ^{\prime}}$ tetraquark structure. This assumption can be
tested by computing the antidiquark $(\bar{Q}\bar{Q^{\prime}})$ mass in the
color $[\mathbf{3}]$ state, and then putting it into the evaluation of the
baryon-like mass ($q$-$q^{\prime}$-$Q_{1}$) where $Q_{1}$ replaces the
antidiquark $(\bar{Q}\bar{Q^{\prime}})$. An isolated diquark is in principle
an ill defined concept as its non-zero color makes it a gauge dependent
quantity so that one can add any other gauge dependent gluon field
configuration to change its mass. What that means is that inside a tetraquark
or baryon we are free to include any fractional amount of the interaction
between the diquark and other color source to define the diquark mass. For
example, the division of the interaction terms between light quark and heavy
quark pairs in a tetraquark given in Table LABEL:TetraSeparation is in
principle arbitrary. Furthermore, the different spin structure between a heavy
quark and diquark, which can be of either spin 1 or 0, could induce different
spin interaction for the two different configurations depending on the quantum
numbers. Thus, we define the diquark mass in a constituent quark model to be
the sum of the masses, the interactions between the two antiquarks, and their
relative kinetic term all in an isolated color $[\mathbf{3}]$ and spin 0 or 1
configuration. We can then calculate the hypothetical baryon-like mass
assuming a ($q$-$q^{\prime}$-$Q_{1}$) structure and call it the tetraquark in
a baryon structure, where the subscript 1 denotes the spin of the antidiquark.
As shown in Table LABEL:Baryonlike, the tetraquark mass is almost reproduced
in the case of $T_{bb}$ and the difference becomes larger in the other
tetraquark states.

Table 9: Masses of tetraquarks ($qq^{\prime}\bar{Q}\bar{Q^{\prime}}$) calculated in a baryon-like structure ($q$-$q^{\prime}$-$Q_{1}$), where the isolated antidiquark ($\bar{Q}\bar{Q^{\prime}}$) mass is used for the $Q_{1}$ mass. For comparison, the values in Table LABEL:result are presented in Column 3. All the masses are in MeV unit. Configuration |  |  |  | Baryon |  |  | Tetraquark
---|---|---|---|---|---|---|---
|  |  | Structure |  |  | Structure
$T_{cc}$ |  |  |  | 3920 |  |  | 3873
$T_{cb}$ |  |  |  | 7238 |  |  | 7212
$us\bar{b}\bar{b}$ |  |  |  | 10702 |  |  | 10694
$T_{bb}$ |  |  |  | 10517 |  |  | 10517

The difference is related to how much the color
$\left(q_{1}q_{2}\right)^{\mathbf{\bar{3}}}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{3}}$
channel dominates over the
$\left(q_{1}q_{2}\right)^{\mathbf{6}}\otimes\left(\bar{q}_{3}\bar{q}_{4}\right)^{\mathbf{\bar{6}}}$
channel. As shown in Figures 2-5, the contribution of the dominant CS basis in
the first quanta is much larger than the others in each tetraquark state,
tendency of which is more apparent in $T_{bb}$ than the other tetraquarks.
Moreover, looking at the sizes of the antidiquarks in Table LABEL:AntiDiquark,
the corresponding relative distances in the tetraquark structure in Table
LABEL:RelDistances are larger. However, it becomes larger only by 0.006 fm in
$T_{bb}$ while it is 0.149 fm in $T_{cc}$. This implies that treating the
antiquark pair such as ($\bar{b}\bar{b}$) in $T_{bb}$ as an isolated diquark
seems a better approximation when the mass becomes heavier.

Table 10: The masses and sizes of isolated antidiquarks in the color $[\mathbf{3}]$ and the spin $1$ state. Comparing the sizes to those in Table LABEL:RelDistances, the effect of surrounding charge becomes smaller for heavier quarks. Antidiquark |  |  | Mass(MeV) |  | Variational |  | Size(fm)
---|---|---|---|---|---|---|---
|  |  | Parameter (fm-2) |
$\bar{c}\bar{c}$ |  |  | 3609 |  | $a=9.5$ |  | 0.461
$\bar{b}\bar{b}$ |  |  | 10234 |  | $a=30.5$ |  | 0.262
$\bar{c}\bar{b}$ |  |  | 6947 |  | $a=14.4$ |  | 0.378
Table 11: The relative distances between the quarks in various configuration. The size of an isolated $ud$ diquark in the color $[\bar{\mathbf{3}}]$ and spin $0$ state is given in Column 2. The distances in the baryon-like ($q$-$q^{\prime}$-$Q_{1}$) structure are given in Columns 3-5. The distances in the baryons $\Lambda_{c}$ and $\Lambda_{b}$ are given in Columns 6-7. Quark Pair |  | Diquark |  | ($q$-$q^{\prime}$-$Q_{1}$) Structure |  | Baryon
---|---|---|---|---|---|---
|  | $T_{cc}$ |  | $T_{cb}$ |  | $T_{bb}$ |  | $\Lambda_{c}$ |  | $\Lambda_{b}$
(1,2) |  | 0.780 |  | 0.664 |  | 0.661 |  | 0.660 |  | 0.668 |  | 0.662
(1,3) |  | - |  | 0.590 |  | 0.578 |  | 0.574 |  | 0.611 |  | 0.582
(2,3) |  | - |  | 0.590 |  | 0.578 |  | 0.574 |  | 0.611 |  | 0.582

Now we look at the size of $(ud)$ pair in various configurations in Table
LABEL:udDiquark. The size of $(ud)$ pair in the diquark configuration becomes
smaller in the baryons $\Lambda_{c}$ and $\Lambda_{b}$ due to the interaction
with the heavy quark. Comparing the size of ($ud$) pair in $\Lambda_{b}$ and
$\Lambda_{c}$, one finds that the size of ($ud$) pair becomes smaller when the
heavy quark is closer to the ($ud$) pair. This tendency can be seen also in
the baron-like ($q$-$q^{\prime}$-$Q_{1}$) structure. However, in the
tetraquark configuration as can be seen in Table LABEL:RelDistances, the size
of ($ud$) pair becomes smaller in $T_{bb}$ than that in $T_{cc}$ even though
the relative distance (1,2)-(3,4) is smaller in $T_{cc}$, which shows an
opposite tendency to that in the baryon structure.

Considering the tetraquarks $T_{cc}$ and $T_{bb}$ as a baryon-like
($q$-$q$-$Q_{1}$) structure, the size of $(ud)$ pair in the baryon-like
$T_{cc}$($T_{bb}$) is close to that in the baryon
$\Lambda_{c}$($\Lambda_{b}$). However, as can be seen in Table
LABEL:RelDistances, the size of $(ud)$ pair in $T_{cc}$ is in fact 0.830 fm,
which is not so close to that in the baryon-like $T_{cc}$, and is even larger
than the size of $(ud)$ pair in the diquark configuration, while the size of
$(ud)$ pair in $T_{bb}$ is close to that in the baryon-like $T_{bb}$. The
analysis so far points out that $T_{bb}$ indeed has a similar structure as a
baryon, while $T_{cc}$ does not.

As discussed in the workWoosung:NPA2019 , the total mass obtained with the
Hamiltonian in Eq. (1) can be divided into ‘light quark’, ‘heavy quark’, ‘CS’
parts as given in Table LABEL:TetraSeparation. The constant $-D$ term in the
Hamiltonian is divided into each quark by multiplying a factor of 1/2. The
relative kinetic energy involving $\mathbf{p}_{3}$, which corresponds to the
relative coordinate $\mathbf{x}_{3}$ connecting the quark and the antiquark
pairs can be divided according to the relative contributions depending on the
mass of either the quark pair or the antiquark pair.

Considering only the spatial basis $\psi^{Spatial}_{[0,0,0,0,0,0]}$, the
relative distance between the light quarks in $T_{cc}$ is the same with that
in $T_{bb}$Woosung:NPA2019 . Comparing the results between the full
calculation and the simplified calculation denoted by 1-Basis in Table
LABEL:TetraSeparation, the hyperfine interaction $\sum V^{CS}(i,j)$ in
$T_{cc}$ becomes much stronger than that in $T_{bb}$. This can be understood
from the change in $V^{CS}(1,2)$, which implies that the attraction coming
from $V^{CS}(1,2)$ partially spreads into the attraction coming from $\sum
V^{CS}(i,j)$ in $T_{cc}$. As a result, for $T_{bb}$ as shown in Table
LABEL:TetraSeparation, the attraction from the hyperfine part between
$\bar{b}$ and $u$ (or $d$) quarks is just -5.7 MeV, while that in $T_{cc}$ is
-69.4 MeV. This is one of the major reasons that the distance between the
light quark pair and the heavy antiquark pair is closer in $T_{cc}$ than that
in $T_{bb}$ as seen in Table LABEL:RelDistances. This closer distance in
$T_{cc}$ causes the slightly larger size of ($ud$) pair in $T_{cc}$ than in
$T_{bb}$.

Table 12: Contributions to the $T_{bb}(ud\bar{b}\bar{b})$ and $T_{cc}(ud\bar{c}\bar{c})$ masses from the present work. $(i,j)$ denotes the $i$ and $j$ quarks, where $i,j=1,2$ label the light quarks, and 3, 4 are for the heavy antiquarks in each configuration. $\sum V^{C}(i,j)$ and $\sum V^{CS}(i,j)$ covers pairs ($i,j$) except for (1,2) and (3,4) pairs. $D$ is separately added and not included in $V^{C}(i,j)$. $m_{Q}$ is the heavy quark mass, and $m_{i}^{\prime}$ are defined in Eq. (13) for each configuration. $\mathbf{p}_{i}$ is the relative momentum corresponding to the $i$-th Jacobi coordinate $\mathbf{x}_{i}$. 1-Basis is the result with only one spatial basis $\psi^{Spatial}_{[0,0,0,0,0,0]}$ and the corresponding dominant CS basis. Overall |  | Contribution |  | $T_{bb}$ |  | $T_{cc}$
---|---|---|---|---|---|---
|  | Full Calculation | 1-Basis |  | Full Calculation | 1-Basis
Heavy Quark |  | 2$m_{Q}$ |  | 10674.0 | 10674.0 |  | 3844.0 | 3844.0
|  | $\frac{\mathbf{p}^{2}_{2}}{2m^{\prime}_{2}}$ |  | 206.8 | 220.0 |  | 142.5 | 221.8
|  | $\frac{m_{q}}{m_{Q}+m_{q}}\frac{\mathbf{p}^{2}_{3}}{2m^{\prime}_{3}}$ |  | 16.4 | 15.3 |  | 53.8 | 38.0
|  | $V^{C}(3,4)$ |  | $-188.8$ | $-190.8$ |  | 19.3 | 4.2
|  | $\frac{1}{2}\sum V^{C}(i,j)$ |  | 115.8 | 137.6 |  | 159.1 | 168.5
|  | $-D$ |  | $-917.0$ | $-917.0$ |  | $-917.0$ | $-917.0$
Subtotal |  |  |  | 9907.2 | 9939.1 |  | 3301.8 | 3359.5
Light Quark |  | 2$m_{q}$ |  | 684.0 | 684.0 |  | 684.0 | 684.0
|  | $\frac{\mathbf{p}^{2}_{1}}{2m^{\prime}_{1}}$ |  | 494.1 | 495.3 |  | 424.1 | 478.2
|  | $\frac{m_{Q}}{m_{Q}+m_{q}}\frac{\mathbf{p}^{2}_{3}}{2m^{\prime}_{3}}$ |  | 255.8 | 239.1 |  | 302.2 | 213.5
|  | $V^{C}(1,2)$ |  | 171.3 | 181.6 |  | 91.3 | 188.8
|  | $\frac{1}{2}\sum V^{C}(i,j)$ |  | 115.8 | 137.6 |  | 159.1 | 168.5
|  | $-D$ |  | $-917.0$ | $-917.0$ |  | $-917.0$ | $-917.0$
Subtotal |  |  |  | 804.0 | 820.6 |  | 743.7 | 816.0
CS Interaction |  | $V^{CS}(3,4)$ |  | 7.0 | 6.8 |  | 5.3 | 9.3
|  | $V^{CS}(1,2)$ |  | $-195.3$ | $-188.1$ |  | $-108.6$ | $-182.6$
|  | $\sum V^{CS}(i,j)$ |  | $-5.7$ | 0.0 |  | $-69.4$ | 0.0
Subtotal |  |  |  | $-194.0$ | $-181.3$ |  | $-172.7$ | $-173.3$
Total |  |  |  | 10517.2 | 10578.4 |  | 3872.8 | 4002.2

###  IV.4 Comparison with Other Models

We now compare our present results with other works in Ref. Karliner:PRL2017
and Ref. Silvestre:ZPC1993 . In a simplified constituent quark
modelKarliner:PRL2017 , the masses for $T_{cc}$ and $T_{bb}$ are given as

$\displaystyle M_{T_{cc}}$ $\displaystyle=$ $\displaystyle
2m^{b}_{c}+B(cc)+2m^{b}_{q}+\frac{a_{cc}}{(m^{b}_{c})^{2}}-\frac{3a}{(m^{b}_{q})^{2}}$
$\displaystyle=$ $\displaystyle 3421.0-129.0+726.0+14.2-150.0=3882.2\,,$
$\displaystyle M_{T_{bb}}$ $\displaystyle=$ $\displaystyle
2m^{b}_{b}+B(bb)+2m^{b}_{q}+\frac{a_{bb}}{(m^{b}_{b})^{2}}-\frac{3a}{(m^{b}_{q})^{2}}$
$\displaystyle=$ $\displaystyle 10087.0-281.4+726.0+7.8-150.0=10389.4\,,$

where $m^{b}_{c,b,q}$ are the constituent quark masses of $c$, $b$, and light
quark $q$ inside a baryon, and $B(cc)$ ($B(bb)$) is the binding between the
charm(bottom) quarks, which can be understood as coming from the extra
attraction between two $c(b)$ quarks due to that shorter interquark distance
as compared to two light quarks: this attraction can be estimated by studying
the quark attractions inside $\Lambda_{c}$ and $\Xi_{cc}$Woosung:NPA2019 .
$a$’s are multiplicative constants for the CS interaction. Here,
$a_{QQ}/(m^{b}_{Q})^{2}$ is the CS interaction between the two heavy quarks
$Q$ corresponding to $V^{CS}(3,4)$ in our model, while $-3a/(m^{b}_{q})^{2}$
is that between the light quarks $q$ denoted by $V^{CS}(1,2)$ in our model.
Then, treating $B(cc)$ ($B(bb)$) as part of the two charm(bottom) quark
system, the energy in the simplified model in Eq. (LABEL:SimEq) can be divided
into the charm(bottom) quark, light quark, and CS interaction parts. The
subtotal values in Table LABEL:TetraSeparation can be regarded as the
constituent quark masses in the simplified quark model in Eq. (LABEL:SimEq).
The additional attraction for the heavy quark pairs with respect to the light
quark pair in the simple modelKarliner:PRL2017  can be seen from $V^{C}(3,4)$
being much more attractive than $V^{C}(1,2)$. While the importance of the
large additional attraction for heavier quarks denoted by $B(QQ)$ remains a
valid point, detailed mass and size differences change other parameters such
as the effective quark masses in Eq. (LABEL:SimEq) so that a simple
parametrization formula given in Eq. (LABEL:SimEq) for the tetraquark masses
might become problematic. Furthermore, while Eq. (LABEL:SimEq) does not allow
for the color spin interaction between light and heavy quarks, our full
calculations show the presence of such terms as given in Table XII, which
becomes $-5.7$ and $-69.4$ MeV for $T_{bb}$ and $T_{cc}$, respectively. Also,
the discrepancy becomes larger for $T_{bb}$ suggesting that nonlinear quark
mass dependence becomes also important. Therefore, care should be taken when
simple parametrizations that work for normal hadrons are generalized to more
complicated configurations.

There is another workSilvestre:ZPC1993  using the complete set of harmonic
oscillator bases, which is similar to ours but different in part as we
introduced the rescaled form of the harmonic oscillator bases. The hamiltonian
in their model is as follows.

$\displaystyle
H=\sum_{i}\left(m_{i}+\frac{\mathbf{p}_{i}^{2}}{2m_{i}}\right)+\sum_{i<j}V_{ij}(r_{ij})\,,$
(22)

with

$\displaystyle V_{ij}(r_{ij})$ $\displaystyle=$
$\displaystyle-\frac{3}{16}\lambda_{i}\lambda_{j}\bigg{[}-\frac{\kappa}{r_{ij}}+\frac{r_{ij}}{a_{0}^{2}}-D$
(23) $\displaystyle\hskip
56.9055pt+\frac{\hbar^{2}c^{2}\kappa}{m_{i}m_{j}c^{4}}\frac{e^{-r_{ij}/r_{0}}}{r_{0}^{2}r_{ij}}\mathbf{\sigma}_{i}\mathbf{\sigma}_{j}\bigg{]}\,,$

where the hyperfine interaction is of $\exp[-r_{ij}]$, instead of
$\exp[-r^{2}_{ij}]$ in our model. Also, they fixed the parameters $\kappa$ and
$(r_{0})_{ij}$ by fitting them to the experimental values, while the hyperfine
part in our model has additional mass dependence appearing as in Eqs. (4)-(5).
The fitting parameters in their model are as follows.

$\displaystyle\kappa=102.67\,\textrm{MeV fm},\quad
a_{0}=0.0326\,\textrm{(MeV${}^{-1}$fm)${}^{1/2}$},$ $\displaystyle
D=913.5\,\textrm{MeV},\quad r_{0}=0.4545\,\textrm{fm},$ $\displaystyle
m_{u}=337\,\textrm{MeV},\qquad m_{s}=600\,\textrm{MeV},$ $\displaystyle
m_{c}=1870\,\textrm{MeV},\qquad m_{b}=5259\,\textrm{MeV}.$ (24)

Comparing with our results in Table LABEL:ComparisonSilvestre, the two
independent models give almost the same binding energies for each tetraquark
state. However, there is a large difference in the mass of $T_{cc}$, which is
due to the larger obtained masses for the $D$ and $D^{*}$ mesons in their
model, which were found to be 1891 MeV and 2021 MeV, respectively. Compared to
the total mass of the $D$ and $D^{*}$ mesons in our model, the difference is
52 MeV, which approximately accounts for the difference in the mass of
$T_{cc}$ between the two models. Also, in their model, the lowest threshold
for $us\bar{b}\bar{b}$ is $B$ $B_{s}^{*}$, while it is $B^{*}$ $B_{s}$ in our
model as in the experiment. In the experiment, the total mass of the $B$ and
$B_{s}^{*}$ mesons is slightly larger than that of the $B^{*}$ and $B_{s}$
mesons by approximately 2.7 MeV, while the corresponding difference in our
model calculation is $4.7$ MeV.

Table 13: The masses and the binding energies $B_{T}$ of the tetraquark states obtained in the present work and in Ref. Silvestre:ZPC1993 . The masses and $B_{T}$ are in MeV unit. Type |  | ($I$, $S$) |  | Present Work |  |  | Ref. Silvestre:ZPC1993
---|---|---|---|---|---|---|---
|  | Mass |  | $B_{T}$ |  |  | Mass |  | $B_{T}$
$ud\bar{b}\bar{b}$ |  | (0,1) |  | 10517 |  | $-145$ |  |  | 10525 |  | $-131$
$ud\bar{c}\bar{c}$ |  | (0,1) |  | 3873 |  | +13 |  |  | 3931 |  | +19
$ud\bar{c}\bar{b}$ |  | (0,1) |  | 7212 |  | $-3$ |  |  | 7244 |  | +1
$us\bar{b}\bar{b}$ |  | (1/2,1) |  | 10694 |  | $-42$ |  |  | 10680 |  | $-40$

##  V Summary and Discussion

We have improved our constituent quark model by introducing the complete set
of 3-dimensional harmonic oscillator bases. We also assessed the validity by
comparing the ground state wave function for the meson structure with the
exact solution of the hydrogen model. The effect turns out to lower the
binding energies for the tetraquark systems. In particular, the harmonic
oscillator bases for the excited orbital states play a crucial role in
obtaining the exact ground state for the tetraquark systems. We have also
successfully fitted the parameters in the Hamiltonian in Eq. (1) to most of
the observed mesons and baryons allowed in our model. The results are
summarized in Table LABEL:result. Also, from the relative distances between
the quarks given in Table LABEL:RelDistances, we have described the spatial
distributions of the quarks in the tetraquark structure in Figures 6-9. By
comparing with our earlier work shown in Table LABEL:result, and the
discussions in Sections IV.2, IV.3, we have found that a simple Gaussian
spatial function fails to provide precise information on the stability and the
structure for the tetraquarks so that the detailed treatment as presented in
this work should be performed. Also, by comparing with other work, fitting
procedure is important to evaluate the exact values for the masses. One can
conclude that while simplified constituent quark model based on universal
constants that does not depend on specific configuration and/or simple models
based on universal diquarks are intuitively important to identify possible
attractive configurations, a detailed full constituent quark model calculation
is needed to assess the stability and existence of a compact exotic multiquark
configuration.

## Acknowledgments

This work was supported by Samsung Science and Technology Foundation under
Project Number SSTF-BA1901-04, and by the Korea National Research Foundation
under the grant number 2017R1D1A1B03028419(NRF).

## Appendix

###  V.1 Harmonic Oscillator Bases in Mesons

To construct the spatial function for the meson structure, we solve the
Schr$\ddot{\rm o}$dinger equation for the 3-dimensional symmetric harmonic
oscillator. In the spherical coordinates system, the wave function can be
separated into the radial part and the angular part such as
$\psi(r,\theta,\phi)=R(r)Y^{m}_{l}(\theta,\phi)$. The solution of the angular
part is known to be the spherical harmonics. For the radial part of the
equation, the solution is obtained in terms of the associated Laguerre
polynomials. The orthonormalized radial part wave function is known as
follows.

$\displaystyle
R_{n,l}(r)=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma\left(n+l+\frac{3}{2}\right)}}\,r^{l}\exp\left[-\frac{r^{2}}{2}\right]L^{l+\frac{1}{2}}_{n}\left(r^{2}\right).\qquad\quad$
(V.1.1)

where $L^{l+\frac{1}{2}}_{n}(r^{2})$ is the associated Laguerre polynomials.
For our purpose of introducing the harmonic oscillator bases to our model, we
have modifed Eq. (V.1.1) by rescaling the radial distance $r$ to
$\sqrt{2a}\,x$, where $x$ is the magnitude of the Jacobi coordinate
$\mathbf{x}$, connecting the quark and the antiquark in the meson structure,
and $a$ is the variational parameter corresponding to the coordinate
$\mathbf{x}$. Then the spatial part of the total wave function is constructed
by combining the spherical harmonics as follows.

$\displaystyle\psi^{Spatial}_{[n,l,m]}(\mathbf{x})=\psi(x,\theta,\phi)^{Spatial}_{[n,l,m]}=R_{n,l}(x)Y^{m}_{l}(\theta,\phi)\,.$
(V.1.2)

where $R_{n,l}(x)$ is the rescaled radial part wave function. Here, the
quantum numbers $n$, $l$ indicate the principal quantum number, and the
orbital angular momentum, respectively. For mesons in the $l=0$ state, each of
the harmonic oscillator bases should be of $l=0$, and they compose the spatial
part of the wave function in the meson structure.

On the other hand, the permutation symmetry of the spatial bases depends on
the power of the Jacobi coordinate $|\mathbf{x}|$, which is contained only in
the radial part of the spatial function. From Eq. (V.1.1), it is recognized
that the permutation symmetry is determined by the angular momentum quantum
number $l$. In our case, for all the bases in the calculations, the angular
momentum is $l=0$. Therefore, for the mesons in the ground state, all the
spatial bases are symmetric under the permutation (12).

To assess the validity of the harmonic oscillator bases approach, we compared
the meson structure to the hydrogen atom in hadron picture. To do this, we
considered only the kinetic energy and the Coulomb potential in the
Hamiltonian of Eq. (1). Then the Hamiltonian reduces to the following form
with the kinetic energy in the center of mass frame.

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{p}}^{2}_{\mathbf{x}}}{2m^{\prime}}-\frac{3}{4}\frac{\lambda^{c}_{1}}{2}\,\,\frac{\lambda^{c}_{2}}{2}\left(-\frac{\kappa}{r_{12}}\right).\qquad$
(V.1.3)

where $m^{\prime}=(2m_{1}m_{2})/(m_{1}+m_{2})$, and
${\mathbf{p}}_{\mathbf{x}}$ is the relative momentum between the quark and the
antiquark. We take $m_{1}=m_{c},m_{2}=m_{q}$, and the values in Eq. (6) for
the parameters $m_{c}$, $m_{q}$, and $\kappa$. For our purpose of the
comparison to our model, we also modify the exact solution of the hydrogen
atom as that of the relative coordinate $\mathbf{x}$. Then, in hadron picture,
the radial part of the ground state wave function for the hydrogen,
$R^{Hydrogen}_{[0,0]}(x)$, becomes as follows.

$\displaystyle
R^{Hydrogen}_{[0,0]}(x)=2\left(\frac{\sqrt{2}\kappa}{\hbar^{2}}\mu\right)^{3/2}\exp[-a|\mathbf{x}|]\,.$
(V.1.4)

where $\mu$ is the reduced mass of the system in MeV unit. In comparing, we
choose the $D$ meson as a target, then the reduced mass $\mu$ in Eq. (V.1.4)
becomes $\frac{m_{u}m_{c}}{m_{u}+m_{c}}$. The results are shown in Figure 10.
It is obvious from Figure 10 that the more harmonic oscillator bases are
included, the more precisely it describes the actual ground state of the meson
structure.

Figure 10: Comparisons between the exact solution and the guasian basis for
the $D$ meson with coulomb potential. In the figures, Hydrogen indicates the
exact solutions . The upper figure represents the radial parts of the ground
state wave functions as the number of the harmonic oscillator bases increases.
The bottom figure represents the convergence behavior of the binding energy as
the number of bases increases. The binding energy in the hydrogen model is
-53.69 MeV, and the binding energy with the 31 harmonic oscillator bases is
-53.64 MeV, which is very close to the value in the hydrogen model.

We have included the harmonic oscillator bases up to $n=4$, which is enough to
obtain the convergent values for the ground state masses of the tetraquarks of
interest in this work.

###  V.2 Harmonic Oscillator Bases in Baryons

In baryons, there arise additional degrees of freedom due to the second Jacobi
coordinate $\mathbf{x}_{2}$. Furthermore, there are contributions from
combinations of nonzero internal relative orbital angular momenta, satisfying
zero total orbital angular momentum of the ground state baryon structure.
Using a similar method as in the meson structure, we construct the complete
set of orthonormalized harmonic oscillator bases.

$\displaystyle\psi(\mathbf{x}_{1},\mathbf{x}_{2})^{Spatial}_{[n_{1},n_{2},l_{1},1_{2}]}$
$\displaystyle=\sum_{m_{1},m_{2}}C(l_{1},m_{1},l_{2},m_{2};l=0,m=0)$
$\displaystyle\hskip
22.76228pt\times\,R_{n_{1},l_{1}}(x_{1})R_{n_{2},1_{2}}(x_{2})Y^{m_{1}}_{l_{1}}(\theta_{1},\phi_{1})Y^{m_{2}}_{l_{2}}(\theta_{2},\phi_{2})\,.$

where $R_{n_{i},l_{i}}(x_{i})$ has the same form as in Eq. (V.1.2), and
$Y^{m_{i}}_{l_{i}}(\theta_{i},\phi_{i})$ is the spherical harmonic function
for the angular part of the $i$-th Jacobi coordinate $\mathbf{x}_{i}$.
$C(l_{1},m_{1},l_{2},m_{2};l=0,m=0)$ is the Clebsch-Gordan(CG) coefficient for
the decomposition of $|l,m\rangle$ in terms of
$|l_{1},m_{2}\rangle|l_{2},m_{2}\rangle$, but the total angular momentum is
fixed at $l=0$, and thus $m=0$.

The following are the Jacobi coordinates for the baryon structure: we choose
the coordinates set 1 as our reference.

  * •

Coordinates Set 1

$\displaystyle\mathbf{x}_{1}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{1}-{\mathbf{r}}_{2}),\qquad\mathbf{x}_{2}=\frac{1}{\mu}\left(\frac{m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{2}}{m_{1}+m_{2}}-\mathbf{r}_{3}\right)$
$\displaystyle\,,$

  * •

Coordinates Set 2

$\displaystyle\mathbf{y}_{1}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{1}-{\mathbf{r}}_{3}),\qquad\mathbf{y}_{2}=\frac{1}{\mu}\left(\mathbf{r}_{2}-\frac{m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{3}}{m_{1}+m_{2}}\right)$
$\displaystyle\,,$

  * •

Coordinates Set 3

$\displaystyle\mathbf{z}_{1}=\frac{1}{\sqrt{2}}({\mathbf{r}}_{2}-{\mathbf{r}}_{3}),\qquad\mathbf{z}_{2}=\frac{1}{\mu}\left(\frac{m_{1}\mathbf{r}_{2}+m_{2}\mathbf{r}_{3}}{m_{1}+m_{2}}-\mathbf{r}_{1}\right)$
$\displaystyle\,,$

where

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\left[\frac{2\left(m_{1}^{2}+m_{2}^{2}+m_{1}m_{2}\right)}{(m_{1}+m_{2})^{2}}\right]^{1/2}\,.$

As in the mesons, the permutation symmetry of the spatial bases is determined
by the angular momentum quantum numbers $l_{1}$ and $l_{2}$. As can be seen in
the coordinates set 1, $\mathbf{x}_{1}$ is antisymmetric under the permutation
(12), while $\mathbf{x}_{2}$ is symmetric. Therefore, the bases with even
number of $l_{1}$ is symmetric, while the bases with odd number of $l_{1}$ is
antisymmetric under the permutation (12).

In the case of the proton where all the constituent quarks are identical, the
total wave function should be fully antisymmetric. Thus, we need to specify
the symmetries for the permutations (13) and (23) as well. However, it is not
clear in the coordinates set 1. Therefore, it is instructive to investigate
the method of constructing the spatial bases for the proton.

###  V.3 Harmonic Oscillator Bases in Proton

The proton is composed of three light quarks so that the symmetry property for
the proton becomes complicated, compared to the other types of baryons. To
satisfy the symmetry constraints on the proton structure, we can construct the
spatial bases by using linear combinations of the harmonic oscillator bases.
First, for the permutation group S3, there are four possible Young tableaux as
follows.

$\displaystyle\begin{tabular}[]{|c|c|c|}\cline{1-3}\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}\,,\quad\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}\,,\quad\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}\,,\quad\begin{tabular}[]{|c|}\cline{1-1}\cr\,1\\\
\cline{1-1}\cr\,2\\\ \cline{1-1}\cr\,3\\\ \cline{1-1}\cr\end{tabular}\,.$
(V.3.9)

There are correspondence relations between the Young tableaux and the harmonic
oscillator bases such that, as the simplest examples,

$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\hline\cr\end{tabular}=\psi^{Spatial}_{[0,0,0,0]}(\mathbf{x}_{1},\mathbf{x}_{2})\,,$
(V.3.11) $\displaystyle\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}=\psi^{Spatial}_{[0,0,1,1]}(\mathbf{x}_{1},\mathbf{x}_{2})\,.$
(V.3.14)

Using the follwing relation, we also obtain the basis corresponding to the
Young tableau  1 2 3 .

1 |  2
---|---
3 |
$\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}}\left[(23)\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}-\frac{1}{2}\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\ \cline{1-1}\cr\end{tabular}\,\right]$ $\displaystyle=$
$\displaystyle\frac{2\alpha}{\sqrt{3}}\left[-\mathbf{y}_{1}\cdot\mathbf{y}_{2}-\frac{1}{2}\mathbf{x}_{1}\cdot\mathbf{x}_{2}\right]\exp\big{[}-a_{1}x_{1}^{2}-a_{1}x_{2}^{2}\big{]}$
$\displaystyle=$
$\displaystyle\frac{2\alpha}{\sqrt{3}}\left[-\frac{\sqrt{3}}{4}x_{1}^{2}+\frac{\sqrt{3}}{4}x_{2}^{2}\right]\exp\big{[}-a_{1}x_{1}^{2}-a_{1}x_{2}^{2}\big{]}$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}\psi^{Spatial}_{[1,0,0,0]}(\mathbf{x}_{1},\mathbf{x}_{2})+\sqrt{\frac{1}{2}}\psi^{Spatial}_{[0,1,0,0]}(\mathbf{x}_{1},\mathbf{x}_{2})\,,$

where
$\alpha=-\sqrt{\frac{16}{3}}\left(\frac{2}{\pi}\right)^{\frac{3}{2}}a_{1}^{\frac{5}{2}}$
is the normalization factor, and the two variational parameters are taken to
be the same $a_{1}=a_{2}$, which is due to the symmetry in the proton. For the
Young tableau  1 2 3 , using the form of $\psi^{Spatial}_{[1,1,1,1]}$,

1
---
2
3
$\displaystyle=$
$\displaystyle\alpha\bigg{\\{}\psi^{Spatial}_{[1,1,1,1]}(\mathbf{x}_{1},\mathbf{x}_{2})$
$\displaystyle\hskip
28.45274pt+\psi^{Spatial}_{[1,1,1,1]}(\mathbf{y}_{1},\mathbf{y}_{2})+\psi^{Spatial}_{[1,1,1,1]}(\mathbf{z}_{1},\mathbf{z}_{2})\bigg{\\}}\,,$

where $\alpha$ is the normalization constant, and is manifestly antisymmetric
under all the permutations (12), (13), and (23) as desired. After the
transformations into the reference coordinates set
$\\{\mathbf{x}_{1},\mathbf{x}_{2}\\}$, and rearranging, it becomes:

1
---
2
3
$\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{4}\psi^{Spatial}_{[2,0,1,1]}(\mathbf{x}_{1},\mathbf{x}_{2})-\frac{\sqrt{3}}{4}\psi^{Spatial}_{[0,2,1,1]}(\mathbf{x}_{1},\mathbf{x}_{2})$
$\displaystyle+\sqrt{\frac{21}{40}}\psi^{Spatial}_{[1,1,1,1]}(\mathbf{x}_{1},\mathbf{x}_{2})-\sqrt{\frac{1}{10}}\psi^{Spatial}_{[0,0,3,3]}(\mathbf{x}_{1},\mathbf{x}_{2})\,.$

In such a way above, it is possible to relate the harmonic oscillator bases to
each of the Young tableaux for the permutation group S3.

To fully construct the total wave function for the proton, it is necessary to
construct the remaining parts of the wave function. For the color basis, it is
always fully antisymmetric for the baryons. Thus, we focus on the spin and the
isospin parts, and $\psi^{Spatial}\times\psi^{Isospin}\times\psi^{Spin}$ part
should be fully symmetric. Since both the spin and the isospin bases are
constructed by SU(2), we obtain the isospin-spin basis from the inner product
of the flavor SU(2) and the spin SU(2). For the proton,

$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr&\\\ \hline\cr\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!I^{1/2}}\times\,\,\,\begin{tabular}[]{|c|c|}\hline\cr&\\\
\hline\cr\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!S^{1/2}}$ (V.3.36)
$\displaystyle=\begin{tabular}[]{|c|c|c|}\hline\cr&&\\\
\cline{1-3}\cr\end{tabular}_{\,\,I^{1/2}S^{1/2}}+\,\,\,\begin{tabular}[]{|c|c|}\hline\cr&\\\
\hline\cr\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I^{1/2}S^{1/2}}+\,\,\,\begin{tabular}[]{|c|}\hline\cr\\\
\hline\cr\\\ \hline\cr\\\ \cline{1-1}\cr\end{tabular}_{\,\,I^{1/2}S^{1/2}}\,,$
(V.3.43)

where the subscript $I^{1/2}$ ($S^{1/2}$) indicates the Young diagram
corresponding to the isospin (spin) 1/2 state, and the Young diagrams on the
right hand side in Eq. (V.3.43) stand for SU(4), which correspond to the
irreducible representations of the isospin-spin space for the proton. The
irreducible representations on the right hand side in Eq. (V.3.43) are spanned
by the corresponding isospin-spin bases as follows. In terms of Young
tableaux,

  * •

$\begin{tabular}[]{|c|c|}\hline\cr&\\\ \hline\cr\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I^{1/2}S^{1/2}}$

$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\ \hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!(IS)_{1}}$ (V.3.46) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{1}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{1}}-\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{2}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{2}}\right)\,,$
(V.3.58) $\displaystyle\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!(IS)_{2}}$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{1}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{2}}+\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{2}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{1}}\right)\,.$
(V.3.67)

  * •

$\begin{tabular}[]{|c|c|c|}\hline\cr&&\\\
\cline{1-3}\cr\end{tabular}_{\,\,I^{1/2}S^{1/2}}$

$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,(IS)_{3}}$ (V.3.70) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{1}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{1}}+\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{2}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{2}}\right)\,.$
(V.3.79)

  * •

$\begin{tabular}[]{|c|}\hline\cr\\\ \hline\cr\\\ \hline\cr\\\
\cline{1-1}\cr\end{tabular}_{\,\,I^{1/2}S^{1/2}}$

$\displaystyle\hskip 19.91684pt\begin{tabular}[]{|c|}\hline\cr\,1\\\
\hline\cr\,2\\\ \hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\,\,(IS)_{4}}=\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{1}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{2}}-\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{2}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!S_{1}}\right)\,.$
(V.3.92)

where $\small\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\ \hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{1}(S_{1})}$ and
$\small\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\ \hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I_{2}(S_{2})}$ are the isospin (spin)
bases spanning the irreducible representation
$\small\begin{tabular}[]{|c|c|}\hline\cr&\\\ \hline\cr\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!I^{1/2}(S^{1/2})}$ for the proton. On
the other hand, in constructing the SU(4) irreducible representation of the
type  , there are two more different methods of performing the inner product
between any two of sptial, isospin, spin bases. However, these three methods
of construction are equivalent.

We are now ready to construct the total wave function for the proton. Before
proceeding, to avoid confusion, it is convenient to label the spatial bases as
in the isospin and spin bases as follows.

$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\ \hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!R_{1}}\,,\quad\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!R_{2}}\,,\quad\begin{tabular}[]{|c|c|c|}\cline{1-3}\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,R_{3}}\,,\quad\begin{tabular}[]{|c|}\cline{1-1}\cr\,1\\\
\cline{1-1}\cr\,2\\\ \cline{1-1}\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\,\,R_{4}}\,.$ (V.3.102)

There are three types of constructing the fully symmetric
$\psi^{Spatial}\times\psi^{Isospin}\times\psi^{Spin}$.

$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,(RIS)_{1}}$ (V.3.104) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!(IS)_{1}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,2\\\
\hline\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!R_{1}}+\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\
\cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!(IS)_{2}}\times\,\begin{tabular}[]{|c|c|}\hline\cr\,1&\,3\\\
\hline\cr\,2\\\ \cline{1-1}\cr\end{tabular}_{\\!\\!\\!\\!R_{2}}\right)\,,$
(V.3.125) $\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,(RIS)_{2}}=\,\,\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,(IS)_{3}}\times\,\,\,\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,R_{3}}\,,$
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr\,1&\,2&\,3\\\
\cline{1-3}\cr\end{tabular}_{\,\,(RIS)_{3}}=\,\,\begin{tabular}[]{|c|}\hline\cr\,1\\\
\cline{1-1}\cr\,2\\\ \cline{1-1}\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\,\,(IS)_{4}}\\!\\!\\!\\!\times\,\,\,\begin{tabular}[]{|c|}\hline\cr\,1\\\
\cline{1-1}\cr\,2\\\ \cline{1-1}\cr\,3\\\
\cline{1-1}\cr\end{tabular}_{\,\,R_{4}}\,.$

###  V.4 Color, Spin Bases of Tetraquarks in a Baryon Structure

As discussed in Section IV.3, we regard the antidiquark as a point particle
$Q_{1}$ which replaces ($\bar{Q}\bar{Q^{\prime}}$). Labeling the quarks in the
order of $q(1)$-$q^{\prime}(2)$-$Q_{1}(3)$, for the total $S=1$ system, the
spin bases can be constructed with the explicit spin numbers written in
subscripts as follows.

$\displaystyle\left|1\right\rangle_{1}$ $\displaystyle\equiv$
$\displaystyle\left|\left[q(1)q^{\prime}(2)\right]_{1}\left[Q_{1}(3)\right]_{0}\right\rangle\,,$
$\displaystyle\left|2\right\rangle_{1}$ $\displaystyle\equiv$
$\displaystyle\left|\left[q(1)q^{\prime}(2)\right]_{1}\left[Q_{1}(3)\right]_{1}\right\rangle\,,$
$\displaystyle\left|3\right\rangle_{1}$ $\displaystyle\equiv$
$\displaystyle\left|\left[q(1)q^{\prime}(2)\right]_{0}\left[Q_{1}(3)\right]_{1}\right\rangle\,.$
(V.4.1)

Then the spin matrices with this bases set are obtained as follows.

$\displaystyle\bm{\sigma}_{1}^{BL}\cdot\bm{\sigma}_{2}^{BL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&-3\\\
\end{array}\right)\,,$ (V.4.5)
$\displaystyle\bm{\sigma}_{1}^{BL}\cdot\bm{\sigma}_{3}^{BL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\ 0&-2&-2\sqrt{2}\\\
0&-2\sqrt{2}&0\\\ \end{array}\right)\,,$ (V.4.9)
$\displaystyle\bm{\sigma}_{2}^{BL}\cdot\bm{\sigma}_{3}^{BL}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\ 0&-2&2\sqrt{2}\\\
0&2\sqrt{2}&0\\\ \end{array}\right)\,,$ (V.4.13)

where the superscript $BL$ indicates that they are the matrices in the baryon-
like structure. On the other hand, the spin matrices in the tetraquark
structure are as follows.

$\displaystyle\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&-3\\\
\end{array}\right)\,,$ (V.4.17)
$\displaystyle\bm{\sigma}_{1}\cdot\bm{\sigma}_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\sqrt{2}&1\\\
\sqrt{2}&-1&-\sqrt{2}\\\ 1&-\sqrt{2}&0\\\ \end{array}\right)\,,$ (V.4.21)
$\displaystyle\bm{\sigma}_{1}\cdot\bm{\sigma}_{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&-\sqrt{2}&-1\\\
-\sqrt{2}&-1&-\sqrt{2}\\\ -1&-\sqrt{2}&0\\\ \end{array}\right)\,,$ (V.4.25)
$\displaystyle\bm{\sigma}_{2}\cdot\bm{\sigma}_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\sqrt{2}&-1\\\
\sqrt{2}&-1&\sqrt{2}\\\ -1&\sqrt{2}&0\\\ \end{array}\right)\,,$ (V.4.29)
$\displaystyle\bm{\sigma}_{2}\cdot\bm{\sigma}_{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&-\sqrt{2}&1\\\
-\sqrt{2}&-1&\sqrt{2}\\\ 1&\sqrt{2}&0\\\ \end{array}\right)\,,$ (V.4.33)
$\displaystyle\bm{\sigma}_{3}\cdot\bm{\sigma}_{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}-3&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{array}\right)\,,$ (V.4.37)

with the spin bases

$\displaystyle\left|1\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\left[q(1)q^{\prime}(2)\right]_{1}\left[\bar{Q}(3)\bar{Q}^{\prime}(4)\right]_{0}\right\rangle\,,$
$\displaystyle\left|2\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\left[q(1)q^{\prime}(2)\right]_{1}\left[\bar{Q}(3)\bar{Q}^{\prime}(4)\right]_{1}\right\rangle\,,$
$\displaystyle\left|3\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\left[q(1)q^{\prime}(2)\right]_{0}\left[\bar{Q}(3)\bar{Q}^{\prime}(4)\right]_{1}\right\rangle.$
(V.4.38)

Comparing the spin matrices in Eq. (V.4.13) and Eq. (V.4.37), one can find the
following relations,

$\displaystyle\bm{\sigma}_{1}^{BL}\cdot\bm{\sigma}_{3}^{BL}=\bm{\sigma}_{1}\cdot\bm{\sigma}_{3}+\bm{\sigma}_{1}\cdot\bm{\sigma}_{4}\,,$
$\displaystyle\bm{\sigma}_{2}^{BL}\cdot\bm{\sigma}_{3}^{BL}=\bm{\sigma}_{2}\cdot\bm{\sigma}_{3}+\bm{\sigma}_{2}\cdot\bm{\sigma}_{4}\,.$
(V.4.39)

The color bases in this baryon-like structure can be constructed with the
explicit color states as follows.

$\displaystyle\psi^{1}_{1}$ $\displaystyle\equiv$
$\displaystyle\left[q(1)q^{\prime}(2)\right]^{\mathbf{6}}\left[Q_{1}(3)\right]^{\bar{\mathbf{6}}}\,,$
$\displaystyle\psi^{1}_{2}$ $\displaystyle\equiv$
$\displaystyle\left[q(1)q^{\prime}(2)\right]^{\bar{\mathbf{3}}}\left[Q_{1}(3)\right]^{\mathbf{3}}\,.$
(V.4.40)

Then the color matrices with this bases set are obtained as follows.

$\displaystyle{\bm{\lambda}^{c}_{1}}^{BL}{\bm{\lambda}^{c}_{2}}^{BL}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}\frac{4}{3}&0\\\
0&-\frac{8}{3}\\\ \end{array}\right)\,,$ (V.4.43)
$\displaystyle{\bm{\lambda}^{c}_{1}}^{BL}{\bm{\lambda}^{c}_{3}}^{BL}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-\frac{20}{3}&0\\\
0&-\frac{8}{3}\\\ \end{array}\right)\,,$ (V.4.46)
$\displaystyle{\bm{\lambda}^{c}_{2}}^{BL}{\bm{\lambda}^{c}_{3}}^{BL}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-\frac{20}{3}&0\\\
0&-\frac{8}{3}\\\ \end{array}\right)\,,$ (V.4.49)

The color matrices in the tetraquark structure are as follows.

$\displaystyle\bm{\lambda}^{c}_{1}\bm{\lambda}^{c}_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\frac{4}{3}&0\\\ 0&-\frac{8}{3}\\\
\end{array}\right)\,,$ (V.4.52)
$\displaystyle\bm{\lambda}^{c}_{1}\bm{\lambda}^{c}_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-\frac{10}{3}&-2\sqrt{2}\\\
-2\sqrt{2}&-\frac{4}{3}\\\ \end{array}\right)\,,$ (V.4.55)
$\displaystyle\bm{\lambda}^{c}_{1}\bm{\lambda}^{c}_{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-\frac{10}{3}&2\sqrt{2}\\\
2\sqrt{2}&-\frac{4}{3}\\\ \end{array}\right)\,,$ (V.4.58)
$\displaystyle\bm{\lambda}^{c}_{2}\bm{\lambda}^{c}_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-\frac{10}{3}&2\sqrt{2}\\\
2\sqrt{2}&-\frac{4}{3}\\\ \end{array}\right)\,,$ (V.4.61)
$\displaystyle\bm{\lambda}^{c}_{2}\bm{\lambda}^{c}_{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-\frac{10}{3}&-2\sqrt{2}\\\
-2\sqrt{2}&-\frac{4}{3}\\\ \end{array}\right)\,,$ (V.4.64)
$\displaystyle\bm{\lambda}^{c}_{3}\bm{\lambda}^{c}_{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\frac{4}{3}&0\\\ 0&-\frac{8}{3}\\\
\end{array}\right)\,,$ (V.4.67)

with the color bases

$\displaystyle\psi^{1}_{1}$ $\displaystyle\equiv$
$\displaystyle\left[q(1)q^{\prime}(2)\right]^{\mathbf{6}}\left[\bar{Q}(3)\bar{Q^{\prime}}(4)\right]^{\bar{\mathbf{6}}}\,,$
$\displaystyle\psi^{1}_{2}$ $\displaystyle\equiv$
$\displaystyle\left[q(1)q^{\prime}(2)\right]^{\bar{\mathbf{3}}}\left[\bar{Q}(3)\bar{Q^{\prime}}(4)\right]^{\mathbf{3}}\,.$
(V.4.68)

Likewise in the spin matrices, comparing Eq. (V.4.49) and Eq. (V.4.67), one
can also find similar relations as follows.

$\displaystyle{\bm{\lambda}^{c}_{1}}^{BL}{\bm{\lambda}^{c}_{3}}^{BL}=\bm{\lambda}^{c}_{1}\bm{\lambda}^{c}_{3}+\bm{\lambda}^{c}_{1}\bm{\lambda}^{c}_{4}\,,$
$\displaystyle{\bm{\lambda}^{c}_{2}}^{BL}{\bm{\lambda}^{c}_{3}}^{BL}=\bm{\lambda}^{c}_{2}\bm{\lambda}^{c}_{3}+\bm{\lambda}^{c}_{2}\bm{\lambda}^{c}_{4}\,.$
(V.4.69)

## References

  * (1) S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett. 91, 262001 (2003).
  * (2) Y. Ikeda et al. [HAL QCD], Phys. Rev. Lett. 117, no.24, 242001 (2016) doi:10.1103/PhysRevLett.117.242001 [arXiv:1602.03465 [hep-lat]].
  * (3) M. Nielsen, F. S. Navarra and S. H. Lee, Phys. Rept. 497, 41-83 (2010) doi:10.1016/j.physrep.2010.07.005 [arXiv:0911.1958 [hep-ph]].
  * (4) Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237-320 (2019) doi:10.1016/j.ppnp.2019.04.003 [arXiv:1903.11976 [hep-ph]].
  * (5) A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12, 147-162 (1975) doi:10.1103/PhysRevD.12.147
  * (6) R. K. Bhaduri, L. E. Cohler, and Y. Nogami, Nuovo Cim. A 65, 376-390 (1981).
  * (7) B. Silvestre-Brac and C. Gignoux, Phys. Rev. D 32, 743 (1985).
  * (8) B. Silvestre-Brac and C. Semay, Z. Phys. C 57, 273-282 (1993).
  * (9) C. Semay and B. Silvestre-Brac, Z. Phys. C 61, 271-275 (1994).
  * (10) C. Semay and B. Silvestre-Brac, Z. Phys. C 30, 457 (1986).
  * (11) J. Vijande, A. Valcarce, and N. Barnea, Phys. Rev. D 79, 074010 (2009).
  * (12) D.M. Brink and Fl. Stancu, Phys. Rev. D 57, 6778 (1998).
  * (13) D. Janc and M. Rosina, Few Body Syst. 35, 175 (2004).
  * (14) Woosung Park, Sungsik Noh, and Su Houng Lee, Nucl. Phys. A 983, 1-19 (2019).
  * (15) Marek Karliner, and Jonathan L. Rosner, Phys. Rev. Lett. 119, 20, 202001 (2017)