#  Equivalence among color singlet, color octet and diquark structure in a
chiral quark model

Yue Tan<EMAIL_ADDRESS>Department of Physics, Yancheng Institute of
Technology, Yancheng 224000, P. R. China  Xuejie Liu<EMAIL_ADDRESS>School
of Physics, Henan Normal University, Xinxiang 453007, P. R. of China  Xiaoyun
Chen<EMAIL_ADDRESS>College of Science, Jinling Institute of Technology,
Nanjing 211169, P. R.China  Yuheng Wu<EMAIL_ADDRESS>Department of
Physics, Nanjing Normal University, Nanjing 210023, P.R. China  Hongxia Huang
Hongxia<EMAIL_ADDRESS>(Corresponding author)  Department of Physics,
Nanjing Normal University, Nanjing 210023, P.R. China  Jialun Ping
<EMAIL_ADDRESS>(Corresponding author) Department of Physics, Nanjing
Normal University, Nanjing 210023, P.R. China

###### Abstract

Since the quark model was put forward, theoretical researchers have always
attached great importance to the study of hidden color channels (including
color octets and diquark structure). Because of the influence of color Van der
waals forces, the hidden color channel itself has strong attraction, which
provides a dynamic mechanism for the formation of resonance state or bound
state. In this paper, taking the $T_{cc}$ system as an example, under the
framework of multi-Gaussian expansion method, a set of relatively complete
color singlets (that is, the ground state of the color singlet plus its
corresponding higher-order component) is used to replace the contribution of
the color octet. Similarly, we endeavor to replace the diquark structure with
a relatively complete set of molecular states, encompassing both the ground
state and excited states. Our results demonstrate that the color octet
structure can be effectively replaced by a set of relatively complete color
singlet bases, while the diquark structure cannot be entirely substituted by
an equivalently comprehensive set of molecular state bases.

##  I Introduction

In the traditional quark model, mesons are composed of $q\bar{q}$ and baryons
are composed of $qqq$. Under this framework, the traditional quark model
successfully explained a large number of experimental data at that time and
predicted the $\Omega$ particle. However, the basic theory of strong
interaction, QCD, does not prohibit the existence of multiple quarks. As long
as the number of quarks satisfies the formula,
$q^{m}\bar{q}^{n},m-n=3k,m+n>3$. In fact, since 2003, many possible multi-
quark states, such as $X(3872)$ Choi:2003ue , $P_{c}$ states LHCb:2015yax ,
and so on, have been reported by various experimental groups. Theoretically,
there are varying perspectives: some contend that the predominant constituent
of exotic states is the color octet structure Tan:2022pzi , others propose
that diquarks may play a leading role in exotic states Wang:2019veq , and
there are also those who argue that experimentally observed exotic states near
the threshold are likely color singlet molecular states Ji:2022uie ;
Guo:2017jvc .

Recently, the LHCb collaboration reported that the mass of $T_{cc}$ is about 1
MeV lower than the threshold $DD^{*}$ LHCb:2021auc . However, within the
framework of the chiral quark model, if only molecular state structures
including color-singlet and color-octet states are taken into account, the
calculated results align well with experimental findings. On the other hand,
when diquark structures are considered, the lowest eigenenergy obtained is 100
MeV lower than the $DD^{*}$ threshold, significantly below the experimentally
observed results Tan:2020ldi ; Deng:2021gnb ; Chen:2021tnn ; Jin:2020yjn ;
Vijande:2003ki ; Luo:2017eub ; Yang:2019itm . If $T_{cc}$ with energy of 3875
MeV is confirmed by other collaboration, which may pose a challenge to the
chiral quark models. Hence, a profound understanding of the relationships
among color singlet, color octet, and diquark structures stands as a crucial
task within the current framework of the chiral quark model. The primary
objective of this paper is to investigate the relationship between color
singlet and color octet states, and diquarks, with the $T_{cc}$ system as a
case study.

Firstly, let’s demonstrate that in the molecular structure, the contribution
of the color octet state to the system can be substituted by a superposition
of the color singlet ground state and excited states. Suppose there are four
quarks in a four-quark system, two positive quarks are labeled as $q_{1}$ and
$q_{3}$, and two anti-quarks are labeled as $\bar{q}_{2}$ and $\bar{q}_{4}$.
Then, the three possible molecular states are as
follows:$[q_{1}\bar{q}_{2}]_{1}[q_{3}\bar{q}_{4}]_{1}$,
$[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8}$, and
$[q_{1}\bar{q}_{4}]_{1}[q_{3}\bar{q}_{2}]_{1}$. In the first case, ${q}_{1}$
and $\bar{q}_{2}$ form a color singlet, and ${q}_{3}$ and $\bar{q}_{4}$ form a
color singlet. These two color singlets form a colorless
$[q_{1}\bar{q}_{2}]_{1}[q_{3}\bar{q}_{4}]_{1}$. In the second case, ${q}_{1}$
and $\bar{q}_{2}$ form a color octet, ${q}_{3}$ and $\bar{q}_{4}$ form a color
octet, and these two color octets form a colorless
$[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8}$. In the third case, ${q}_{1}$
and $\bar{q}_{4}$ form a color singlet, and ${q}_{3}$ and $\bar{q}_{2}$ form a
color singlet. These two color singlets form a colorless
$[q_{1}\bar{q}_{4}]_{1}[q_{3}\bar{q}_{2}]_{1}$. In the quark model, any
physical wave function is composed of four parts, namely the spatial part, the
spin part, the flavor part, and the color part. In molecular structures, the
biggest difference among these three cases lies in the color wave functions.
Based on the expressions of color singlet and color octet wave functions, it
can be easily proven that $[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8}$ can
be composed of $[q_{1}\bar{q}_{2}]_{1}[q_{3}\bar{q}_{4}]_{1}$ and
$[q_{1}\bar{q}_{4}]_{1}[q_{3}\bar{q}_{2}]_{1}$, which can be written as
follows:

$\displaystyle[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8}=\frac{3}{\sqrt{8}}[q_{1}\bar{q}_{2}]_{1}[q_{3}\bar{q}_{4}]_{1}-\frac{1}{\sqrt{8}}[q_{1}\bar{q}_{4}]_{1}[q_{3}\bar{q}_{2}]_{1}$

Assuming that the relative motions between $q_{1}$-$\bar{q}_{2}$,
$q_{3}$-$\bar{q}_{4}$, and $q_{1}\bar{q}_{2}$-$q_{3}\bar{q}_{4}$ are in their
respective ground states in $[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8}$,
variations in quark order could lead to the three relative motions in
$[q_{1}\bar{q}_{4}]_{1}[q_{3}\bar{q}_{2}]_{1}$ being excited states. Thus, it
is plausible to assert that the color octet state in a tetraquark structure
can be substituted by a superposition of color singlet ground and excited
states, which can be depicted by FIG. 1,

Figure 1: A color octet structure can be replaced by superimposing color
singlets with the same quark order and another set of color singlets with a
different quark order.

Secondly, by constructing complete superposition states of color singlet and
color octet states, it can be proven that the diquark structure can be
replaced by the dimeson structure. Here, we also define two diquark
structures: $[q_{1}q_{3}]_{\bar{3}}[\bar{q}_{2}\bar{q}_{4}]_{3}$, and
$[q_{1}q_{3}]_{6}[\bar{q}_{2}\bar{q}_{4}]_{\bar{6}}$. In the first case,
${q}_{1}$ and ${q}_{3}$ form an anti-color triplet, and $\bar{q}_{2}$ and
$\bar{q}_{4}$ form a color triplet. Subsequently, the anti-color triplet and
the color triplet are coupled into a colorless
$[q_{1}q_{3}]_{\bar{3}}[\bar{q}_{2}\bar{q}_{4}]_{3}$. In the second case,
${q}_{1}$ and ${q}_{3}$ form the anti-color sextuple, and $\bar{q}_{2}$ and
$\bar{q}_{4}$ form the color sextuple. Subsequently, the anti-color sextuple
and the color sextuple are coupled into a colorless
$[q_{1}q_{3}]_{6}[\bar{q}_{2}\bar{q}_{4}]_{\bar{6}}$. Likewise, according to
the forms of the color wave functions for diquarks and dimesons, we can easily
obtain the transformation relationship between their color wave functions.

$\displaystyle[q_{1}q_{3}]_{\bar{3}}[\bar{q}_{2}\bar{q}_{4}]_{3}=\sqrt{\frac{1}{3}}[q_{1}\bar{q}_{2}]_{1}[q_{3}\bar{q}_{4}]_{1}-\sqrt{\frac{2}{3}}[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8},$
$\displaystyle[q_{1}q_{3}]_{6}[\bar{q}_{2}\bar{q}_{4}]_{\bar{6}}=\sqrt{\frac{2}{3}}[q_{1}\bar{q}_{2}]_{1}[q_{3}\bar{q}_{4}]_{1}+\sqrt{\frac{1}{3}}[q_{1}\bar{q}_{2}]_{8}[q_{3}\bar{q}_{4}]_{8}.$

As for the spatial part, we define the relative coordinates $\mathbf{r}_{ij}$
and $\mathbf{r}_{ij,kl}$ as follows:

$\displaystyle\begin{array}[]{cccccc}\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j},~{}\mathbf{r}_{ij,kl}=\frac{m_{i}\mathbf{r}_{i}+m_{j}\mathbf{r}_{j}}{m_{i}+m_{j}}-\frac{m_{k}\mathbf{r}_{k}+m_{l}\mathbf{r}_{l}}{m_{k}+m_{l}},\end{array}$

where {$\mathbf{r}_{12}$, $\mathbf{r}_{34}$, $\mathbf{r}_{12,34}$} and
{$\mathbf{r}_{13}$, $\mathbf{r}_{24}$, $\mathbf{r}_{13,24}$} serve as the
Jacobi coordinates for the dimeson configuration and the diquark
configuration, respectively. Then, expressing the orbit wave function
$\phi(\mathbf{r}_{12},\mathbf{r}_{34},\mathbf{r}_{12,34})$ in terms of the
explicit diquark configuration, denoted as
$\phi(\mathbf{r}_{12},\mathbf{r}_{34},\mathbf{r}_{12,34})=\phi^{\prime}(\mathbf{r}_{13},\mathbf{r}_{24},\mathbf{r}_{13,24})$.
If we adopt the Gaussian function as the orbit trial wave function, the
coordinate-related components can be expressed as

$\displaystyle
e^{-\nu_{i}\mathbf{r}^{2}_{12}-\nu_{j}\mathbf{r}^{2}_{34}-\nu_{k}\mathbf{r}^{2}_{12,34}}=e^{-a\mathbf{r}^{2}_{13}-b\mathbf{r}^{2}_{24}-c\mathbf{r}^{2}_{13,24}}$
(2) $\displaystyle\times
e^{-d\mathbf{r}_{13}\cdot\mathbf{r}_{24}-e\mathbf{r}_{13}\cdot\mathbf{r}_{13,24}-f\mathbf{r}_{24}\cdot\mathbf{r}_{13,24}},$

in which the coefficients $a$-$f$ depend on the variational parameters $\nu$s
and the elements of the transformation matrix from {$\mathbf{r}_{12}$,
$\mathbf{r}_{34}$, $\mathbf{r}_{12,34}$} to {$\mathbf{r}_{13}$,
$\mathbf{r}_{24}$, $\mathbf{r}_{13,24}$}. The radial wave function includes
all possible relative orbital angular momenta coupled to zero angular momentum
Vijande:2009ac ,

$\displaystyle
e^{-d\mathbf{r}_{13}\cdot\mathbf{r}_{24}-e\mathbf{r}_{13}\cdot\mathbf{r}_{13,24}-f\mathbf{r}_{24}\cdot\mathbf{r}_{13,24}}=\frac{1}{4\sqrt{\pi}}\sum_{l_{13}=0}^{\infty}\sum_{l_{24}=0}^{\infty}\sum_{l_{13,24}=0}^{\infty}\left[[Y_{l_{13}}(\hat{\mathbf{r}}_{13})Y_{l_{24}}(\hat{\mathbf{r}}_{24})]_{l_{13,24}}Y_{l_{13,24}}(\hat{\mathbf{r}}_{13,24})\right]_{0}$
$\displaystyle\times\sum_{l_{1},l_{2},l_{3}}\left(2l_{1}+1\right)\left(2l_{2}+1\right)\left(2l_{3}+1\right)\langle
l_{1}0l_{2}0|l_{13}\rangle\langle l_{1}0l_{3}0|l_{24}\rangle\langle
l_{2}0l_{3}0|l_{13,24}\rangle\left\\{\begin{array}[]{cccccc}l_{13}&l_{24}&l_{13,24}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr l_{3}&l_{2}&l_{1}\\\
\end{array}\right\\}$ (5)
$\displaystyle\times\left(\sqrt{\frac{\pi}{2dr_{13}r_{24}}}I_{l_{1}+\frac{1}{2}}(dr_{13}r_{24})\right)\left(\sqrt{\frac{\pi}{2er_{13}r_{13,24}}}I_{l_{2}+\frac{1}{2}}(er_{13}r_{13,24})\right)\left(\sqrt{\frac{\pi}{2fr_{24}r_{13,24}}}I_{l_{3}+\frac{1}{2}}(fr_{24}r_{13,24})\right),$

where $I_{a}(x)$ are the modified Bessel functions.

Similar to the spatial part, the spin part also differs only between the
dimeson structure and the diquark structure. Due to the constraints of quantum
numbers, the total spin of the ground state $T_{cc}$ system is always
$\mathbf{1}$. Thus, its spin of subgroups could have three possible
combinations: {$\mathbf{1}\oplus\mathbf{1}$, $\mathbf{1}\oplus\mathbf{0}$,
$\mathbf{0}\oplus\mathbf{1}$}. We denote the spin structures of dimesons and
diquarks as {$\mathbf{1}_{12}\oplus\mathbf{1}_{34}$,
$\mathbf{1}_{12}\oplus\mathbf{0}_{34}$,
$\mathbf{0}_{12}\oplus\mathbf{1}_{34}$} and
{$\mathbf{1}_{13}\oplus\mathbf{1}_{24}$,
$\mathbf{1}_{13}\oplus\mathbf{0}_{24}$,
$\mathbf{0}_{13}\oplus\mathbf{1}_{24}$}, respectively. The transformation
relations between them are expressed as follows.

$\displaystyle\left(\begin{array}[]{cccccc}\mathbf{1}_{12}\oplus\mathbf{1}_{34}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\mathbf{1}_{12}\oplus\mathbf{0}_{34}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\mathbf{0}_{12}\oplus\mathbf{1}_{34}\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr-\frac{1}{\sqrt{2}}&\frac{1}{2}&-\frac{1}{2}\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr-\frac{1}{\sqrt{2}}&-\frac{1}{2}&\frac{1}{2}\\\
\end{array}\right)\left(\begin{array}[]{cccccc}\mathbf{1}_{13}\oplus\mathbf{1}_{24}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\mathbf{1}_{13}\oplus\mathbf{0}_{24}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\mathbf{0}_{13}\oplus\mathbf{1}_{24}\\\
\end{array}\right).$ (15)

Considering the $T_{cc}$ system, the flavor wave functions of dimesons and the
flavor wave functions of diquark structures can also be mutually transformed.
Therefore, from a theoretical perspective, we have demonstrated that if the
wave functions are complete, then dimeson structures and diquark structures
can be mutually transformed, which can be depicted by FIG. 2,

Figure 2: The diquark structure can be obtained by superimposing color
singlets and color octets with identical quark content.

In this paper, we will take the $T_{cc}$ system as an example, with the help
of the multi-Gaussian expansion method. Two cases are studied systematically,
namely, the equivalence between color octet and higher-order component wave,
and the equivalence between diquark structure and molecular structure. The
structure of this paper is organized as follows: In Section I, we provide an
introduction to the study. In Section II, we elaborate on the details of the
Chiral Quark Model (ChQM) and Gaussian Expansion Method (GEM) employed in our
investigation. Subsequently, in Section III, we describe a method for
identifying and calculating the decay width of the genuine resonance state. In
Section IV, we present the numerical results. Finally, in Section IV, we
summarize our findings and conclude this work.

##  II Chiral quark model, wave function of $T_{cc}$ system

###  II.1 Chiral quark model

The chiral quark model has been applied successfully in describing the hadron
spectra and hadron-hadron interactions. The details of the model can be found
in Refs. Vijande:2004he ; Hu:2021nvs; Tan:2020ldi . In this writing, we
introduce a chiral quark model with scalar nonet exchange. The Hamiltonian of
the chiral quark model is given as follows,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(m_{i}+\frac{{\vec{p}_{i}}^{2}}{2m_{i}})-T_{CM}+\sum_{i<j=1}^{n}[V(\textbf{r}_{ij})],$
(16)

where $m_{i}$ is the constituent masse of $i$-th quark (antiquark),
$\frac{\vec{p}_{i}^{2}}{2m_{i}}$ is the kinetic energy of $i$-th quark
(antiquark), $\textbf{r}_{ij}$ refers to the relative coordinates between
$i$-th and $j$-th quark(antiquark). $T_{CM}$ is centroid kinetic energy, which
can be written as

$\displaystyle
T_{CM}=\frac{(\vec{p}_{1}+\vec{p}_{2}+\vec{p}_{3}+\vec{p}_{4})^{2}}{2(m_{1}+m_{2}+m_{3}+m_{4})}$
(17)

The interaction potential $V(\textbf{r}_{ij})$, shown in Eq. 18, stems from
the derivation of field theory and embodies three essential aspects of QCD:
the confinement of quarks, the phenomenon of asymptotic freedom, and the
spontaneous breaking of chiral symmetry.

$\displaystyle
V(\textbf{r}_{ij})=V_{CON}(\textbf{r}_{ij})+V_{OGE}(\textbf{r}_{ij})+V_{OBE}(\textbf{r}_{ij})$
(18)

The form of the quark confinement potential $V_{CON}(r_{ij})$ has not been
rigorously proven. However, it is widely accepted that the exchange of
multiple gluons results in a potential with a linearly increasing behavior.
This potential is proportional to the distance between quarks, and this notion
has been informally demonstrated in Ref. Bali:2000gf . The $V_{CON}(r_{ij})$
term includes central force $V_{con}^{C}(r_{ij})$ and spin-orbit force
$V_{con}^{SO}(r_{ij})$.

$\displaystyle V_{con}^{C}(r_{ij})$
$\displaystyle=(-a_{c}r_{ij}^{2}-\Delta)\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}$
(19a) $\displaystyle V_{con}^{SO}(r_{ij})$
$\displaystyle=-\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}\frac{a_{c}}{4m_{i}^{2}m_{j}^{2}}[((m_{i}^{2}+m_{j}^{2})(1-2a_{s})$
$\displaystyle+4m_{i}m_{j}(1-a_{s}))(\vec{S}_{+}\cdot\vec{L})+$
$\displaystyle((m_{j}^{2}-m_{i}^{2})(1-2a_{s}))(\vec{S}_{-}\cdot\vec{L})]$
(19b)

Due to the absence of a rigorous proof for the form of the confinement
potential, there are currently three prevalent models: linear confinement
potential Yang:2020atz , quadratic confinement potential, and confinement
potential with screening effects Vijande:2004he . In this paper, we adopt the
quadratic confinement potential. In Eqs. 19, the parameters $a_{c}$, $a_{s}$,
and $\Delta$ are defined and listed in Table 1, and
$\vec{S}_{\pm}=\vec{S}_{i}\pm\vec{S}_{j}$. $\boldsymbol{\lambda}^{c}$ are
$SU(3)$ color Gell-Mann matrices.

Table 1: Quark model parameters ($m_{\pi}=0.7$ $fm^{-1}$, $m_{\sigma}=3.42$ $fm^{-1}$, $m_{\eta}=2.77$ $fm^{-1}$). Quark masses |  $m_{u}=m_{d}$(MeV) | 313
---|---|---
|  $m_{c}$(MeV) | 1728
Goldstone bosons | $\Lambda_{\pi}(fm^{-1})$ | 4.2
| $\Lambda_{\eta}(fm^{-1})$ | 5.2
| $\Lambda_{a0}(fm^{-1})$ | 4.2
| $\Lambda_{f0}(fm^{-1})$ | 5.8
| $g_{ch}^{2}/(4\pi)$ | 0.54
| $\theta_{p}(^{\circ})$ | -15
Confinement |  $a_{c}$(MeV$\cdot fm^{-2}$) | 101
|  $\Delta$(MeV) | -78.3
OGE | $\alpha_{qq}$ | 0.5723
| $\alpha_{qc}$ | 0.4938
| $\alpha_{cc}$ | 0.3753
|  $\hat{r}_{0}$(MeV) | 28.17
|  $\hat{r}_{g}$(MeV) | 34.5
| $a_{s}$ | 0.777

Beyond the scale of chiral symmetry breaking, one expects that the dynamics
are influenced by QCD perturbative effects. These effects mimic gluon
fluctuations around the instanton vacuum and are incorporated through the
$V_{OGE}(\textbf{r}_{ij})$ which contains central force $V_{oge}^{C}(r_{ij})$,
spin-orbit force $V_{oge}^{SO}(r_{ij})$ and tensor force
$V_{oge}^{T}(r_{ij})$. The central potential can be written as

$\displaystyle
V_{oge}^{C}(r_{ij})=\frac{\alpha_{s}}{4}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}\left[\frac{1}{r_{ij}}-\frac{2\pi}{3}\frac{\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}}{m_{i}m_{j}}\delta(r_{ij})\right],$
(20)

where $\boldsymbol{\sigma}$ are the $SU(2)$ Pauli matrices,
$r_{0}(\mu_{ij})=\frac{r_{0}}{\mu_{ij}}$ and $\alpha_{s}$ is an effective
scale-dependent running coupling,

$\alpha_{s}(\mu_{ij})=\frac{\alpha_{0}}{\ln\left[(\mu_{ij}^{2}+\mu_{0}^{2})/\Lambda_{0}^{2}\right]}.$
(21)

The $\delta(r_{ij})$ function, arising as a consequence of the non-
relativistic reduction of the one-gluon exchange diagram between point-like
particles, has to be regularized in order to perform exact calculations. It
reads

$\displaystyle\delta{(\boldsymbol{r}_{ij})}=\frac{e^{-r_{ij}/r_{0}(\mu_{ij})}}{4\pi
r_{ij}r_{0}^{2}(\mu_{ij})}.$

The spin-orbit force $V_{oge}^{SO}(r_{ij})$ and tensor force
$V_{oge}^{T}(r_{ij})$ can be written as

$\displaystyle V_{oge}^{SO}(r_{ij})$
$\displaystyle=-\frac{1}{16}\frac{\alpha_{s}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}}{4m_{i}^{2}m_{j}^{2}}[\frac{1}{r_{ij}^{3}}-\frac{e^{-r_{ij}/r_{g}(\mu)}}{r_{ij}^{3}}(1+\frac{r_{ij}}{r_{g}(\mu)})]$
$\displaystyle[(m_{i}^{2}+m_{j}^{2}+4m_{i}m_{j})(\vec{S}_{+}\cdot\vec{L})$
$\displaystyle+(m_{j}^{2}-m_{i}^{2})(\vec{S}_{-}\cdot\vec{L})],$ (22a)
$\displaystyle V_{oge}^{T}(r_{ij})$
$\displaystyle=-\frac{1}{16}\frac{\alpha_{s}\boldsymbol{\lambda}_{i}^{c}\cdot\boldsymbol{\lambda}_{j}^{c}}{4m_{i}^{2}m_{j}^{2}}[\frac{1}{r_{ij}^{3}}-\frac{e^{-r_{ij}/r_{g}(\mu)}}{r_{ij}}(\frac{1}{r_{ij}^{2}}$
$\displaystyle+\frac{1}{3r^{2}_{g}(\mu)}+\frac{1}{r_{ij}r_{g}(\mu)})]S_{ij},$
(22b)

where
$S_{ij}=3(\vec{\sigma}_{i}\cdot\hat{r}_{ij})(\vec{\sigma}_{j}\cdot\hat{r}_{ij})-(\vec{\sigma}_{i}\cdot\vec{\sigma}_{j})$,
$r_{g}(\mu_{ij})=\frac{r_{g}}{\mu_{ij}}$.

In the chiral quark model, quarks, referred to as constituent quarks, acquire
their masses from the interaction with nearly massless current quarks and the
gluon medium. To ensure the chiral invariance of QCD, there should be the
exchange of Goldstone bosons between constituent quarks, giving rise to new
Lagrangian interaction terms. Therefore, this interaction can be expressed as

$\displaystyle
V_{OBE}(\textbf{r}_{ij})=V_{\chi}(\textbf{r}_{ij})+V_{S}(\textbf{r}_{ij}).$
(23)

There are two different contributions including pseudoscalar meson exchange
$V_{\chi=\pi,\eta,K}(\textbf{r}_{ij})$ and scalar meson exchange
$V_{S=a_{0},f_{0},\kappa,\sigma}(\textbf{r}_{ij})$ to the Goldstone boson
exchange. The pseudoscalar meson exchange potential
$V_{\chi}(\textbf{r}_{ij})$ consists of two components: the central potential
$v_{\chi=\pi,K,\eta}^{C}$ and the tensor force $v_{{\chi}=\pi,K,\eta}^{T}$.
Simultaneously, the scalar meson exchange potential $V_{S}(\textbf{r}_{ij})$
encompasses the central potential $v_{s=\sigma,a_{0},f_{0},\kappa}^{C}$, along
with the spin-orbit coupling potential $v_{s=\sigma,a_{0},f_{0},\kappa}^{SO}$.

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}V_{\chi}(r_{ij})&=v_{\pi}({{\bf
r}_{ij}})\sum_{a=1}^{3}\boldsymbol{\lambda}_{i}^{a}\boldsymbol{\lambda}_{j}^{a}+v_{K}({{\bf
r}_{ij}})\sum_{a=4}^{7}\boldsymbol{\lambda}_{i}^{a}\boldsymbol{\lambda}_{j}^{a}\\\
&+v_{\eta}({{\bf
r}_{ij}})[\cos\theta_{P}(\boldsymbol{\lambda}_{i}^{8}\boldsymbol{\lambda}_{j}^{8})-\sin\theta_{P}(\boldsymbol{\lambda}_{i}^{0}\boldsymbol{\lambda}_{j}^{0})],\\\
v_{\chi=\pi,K,\eta}^{C}&=\frac{g^{2}_{ch}}{4\pi}\frac{m_{\chi}^{2}}{\Lambda_{\chi}^{2}-m_{\chi}^{2}}\frac{\Lambda^{2}_{\chi}}{\Lambda^{2}_{\chi}-m^{2}_{\chi}}m_{\chi}[Y(m_{{\chi}}r_{ij})-\frac{\Lambda_{\chi}^{3}}{m_{\chi}^{3}}\\\
&Y(\Lambda_{{\chi}}r_{ij})](\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}),\\\
v_{{\chi}=\pi,K,\eta}^{T}&=\frac{g^{2}_{ch}}{4\pi}\frac{m_{{\chi}}^{2}}{\Lambda_{\chi}^{2}-m_{\chi}^{2}}\frac{\Lambda^{2}_{\chi}}{\Lambda^{2}_{\chi}-m^{2}_{\chi}}m_{\chi}[H(m_{{\chi}}r_{ij})-\frac{\Lambda_{\chi}^{3}}{m_{\chi}^{3}}\\\
&H(\Lambda_{{\chi}}r_{ij})]S_{ij},\\\ \end{array}\right.\end{split}$ (24)

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}V_{S}(r_{ij})&=v_{\sigma}({{\bf
r}_{ij}})\boldsymbol{\lambda}_{i}^{0}\boldsymbol{\lambda}_{j}^{0}+v_{a_{0}}({{\bf
r}_{ij}})\sum_{a=1}^{3}\boldsymbol{\lambda}_{i}^{a}\boldsymbol{\lambda}_{j}^{a}\\\
&+v_{\kappa}({{\bf
r}_{ij}})\sum_{a=4}^{7}\boldsymbol{\lambda}_{i}^{a}\boldsymbol{\lambda}_{j}^{a}+v_{f_{0}}({{\bf
r}_{ij}})\boldsymbol{\lambda}_{i}^{8}\boldsymbol{\lambda}_{j}^{8},\\\
v_{s=\sigma,a_{0},f_{0},\kappa}^{C}&=-\frac{g^{2}_{ch}}{4\pi}\frac{\Lambda^{2}_{s}}{\Lambda^{2}_{s}-m^{2}_{s}}m_{s}\left[Y(m_{s}r_{ij})-\frac{\Lambda_{s}}{m_{s}}Y(\Lambda_{s}r_{ij})\right],\\\
v_{s=\sigma,a_{0},f_{0},\kappa}^{SO}&=-\frac{g^{2}_{ch}}{4\pi}\frac{\Lambda^{2}_{s}}{\Lambda^{2}_{s}-m^{2}_{s}}\frac{m_{s}^{3}}{2m_{i}m_{j}}[G(m_{s}r_{ij})\\\
&-\frac{\Lambda_{s}^{3}}{m_{s}^{3}}G(\Lambda_{s}r_{ij})]\vec{L}\cdot\vec{S},\end{array}\right.\end{split}$
(25)

where $\boldsymbol{\lambda}^{a}$ are $SU(3)$ flavor Gell-Mann matrices,
$m_{\chi(s)}$ is the masses of Goldstone bosons, $\Lambda_{\chi(s)}$ is the
cut-offs, $g^{2}_{ch}/4\pi$ is the Goldstone-quark coupling constant. Finally,
$Y(x)$ is the standard Yukawa function defined by $Y(x)=e^{-x}/x$,
$G(x)=(1+1/x)Y(x)/x$ and $H(x)=(1+3/x+3/x^{2})Y(x)/x$.

All the parameters are determined by fitting the meson spectrum, taking into
account only a quark-antiquark component. They are shown in Table 1. The
calculated masses of the mesons involved in the present work are shown in
Table 2.

Table 2:  Numerical results for the meson spectrum (in MeV) for different models. (unit: MeV). Meson | Our work | GMGodfrey:2015dva  | ChQMVijande:2004he  | RMNi:2021pce  | EXP.(PDG)
---|---|---|---|---|---
$D$ | 1892.34 | 1877 | 1883 | 1865 | 1864.84$\pm$0.05
$D^{*}$ | 1980.37 | 2041 | 2010 | 2008 | 2006.85$\pm$0.05
$D_{0}^{*}$ | 2332.70 | 2399 | - | 2313 | 2343$\pm$10
$D_{1}^{*}$ | 2436.03 | 2456 | 2492 | 2424 | $2420.1\pm 0.6$
$D_{2}^{*}$ | 2472.99 | 2502 | 2502 | 2475 | $2461.1^{+0.7}_{-0.8}$
$D_{1}^{\prime}$ | 2448.54 | 2467 | - | 2453 | $2412\pm 9$

###  II.2 The wave function of $T_{cc}$ system

The wave function of $T_{cc}$ four quark system includes two important
structures, namely dimeson structure and diquark structure. The wave function
of each structure includes four parts, namely orbital, spin, flavor and color.
The wave function of each part is constructed in two steps, first write down
the two-body wave functions, Then coupling two sub-clusters wave functions to
form the four-body one.

In this paper, we employ the $J$-$J$ coupling scheme to derive our wave
functions. In the initial stage, we couple the orbital wave functions
$\phi_{nlm}$ and spin wave functions $\chi_{S,mS}$ of the two subgroups to
obtain the wave function $\psi_{J,mJ}(\textbf{r})$, which can be written as

$\displaystyle\psi_{J_{1},mJ_{1}}(\boldsymbol{\lambda})=\phi_{n_{1}L_{1}m_{1}}(\boldsymbol{\lambda})\otimes\chi_{S_{1},mS_{1}},$
(26)
$\displaystyle\psi_{J_{2},mJ_{2}}(\boldsymbol{\rho})=\phi_{n_{2}L_{2}m_{2}}(\boldsymbol{\rho})\otimes\chi_{S_{2},mS_{2}}.$
(27)

Subsequently, the wave functions of these two subgroups,
$\psi_{J_{1},mJ_{1}}(\textbf{r})$ and $\psi_{J_{2},mJ_{2}}(\textbf{r})$, are
further coupled to form $\psi_{J_{12},mJ_{12}}(\textbf{r})$. This obtained
wave function is then combined with the relative motion wave function
$\phi_{n_{3}l_{3}m_{3}}(\hat{R})$ between the two subgroups to obtain the
orbit-spin wave function $\psi^{SO}_{i}(\textbf{r})$, where $J$ denotes the
total angular momentum.

$\displaystyle\psi_{J_{12},mJ_{12}}=\psi_{J_{1},mJ_{1}}(\boldsymbol{\lambda})\otimes\psi_{J_{2},mJ_{2}}(\boldsymbol{\rho}),$
(28)
$\displaystyle\psi^{SO}_{i}(\textbf{r})=\psi_{J_{12},mJ_{12}}\otimes\phi_{n_{3}L_{3}m_{3}}(\textbf{R}),$
(29) $\displaystyle
i\equiv\\{L_{1},S_{1},J_{1},L_{2},S_{2},J_{2},J_{12},L_{3}\\}.$ (30)

Table 3:  Different combinations of J-J coupling. $J_{12}=0,L_{3}=1$ | $J_{1}=1$ |  | $J_{2}=1$ | i
---|---|---|---|---
|  | $L_{1}=0$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=0$ | 1
|  | $L_{1}=0$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 2
|  | $J_{1}=2$ |  | $J_{2}=2$ | i
|  | $L_{1}=2$ | $S_{1}=0$ |  | $L_{2}=1$ | $S_{2}=0$ | 3
|  | $L_{1}=2$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 4
$J_{12}=1,L_{3}=0$ | $J_{1}=0$ |  | $J_{2}=1$ | i
|  | $L_{1}=0$ | $S_{1}=0$ |  | $L_{2}=0$ | $S_{2}=1$ | 5
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=0$ | 6
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 7
|  | $J_{1}=1$ |  | $J_{2}=1$ | i
|  | $L_{1}=0$ | $S_{1}=1$ |  | $L_{2}=0$ | $S_{2}=1$ | 8
|  | $L_{1}=0$ | $S_{1}=1$ |  | $L_{2}=2$ | $S_{2}=1$ | 9
|  | $L_{1}=1$ | $S_{1}=0$ |  | $L_{2}=1$ | $S_{2}=0$ | 10
|  | $L_{1}=1$ | $S_{1}=0$ |  | $L_{2}=1$ | $S_{2}=1$ | 11
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 12
|  | $L_{1}=2$ | $S_{1}=1$ |  | $L_{2}=2$ | $S_{2}=1$ | 13
$J_{12}=1,L_{3}=2$ | $J_{1}=0$ |  | $J_{2}=1$ | i
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=0$ | 14
|  | $L_{1}=0$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 15
|  | $L_{1}=0$ | $S_{1}=0$ |  | $L_{2}=0$ | $S_{2}=1$ | 16
|  | $L_{1}=1$ | $S_{1}=0$ |  | $L_{2}=1$ | $S_{2}=0$ | 17
|  | $J_{1}=1$ |  | $J_{2}=1$ | i
|  | $L_{1}=1$ | $S_{1}=0$ |  | $L_{2}=1$ | $S_{2}=0$ | 18
|  | $L_{1}=1$ | $S_{1}=0$ |  | $L_{2}=1$ | $S_{2}=1$ | 19
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 20
|  | $L_{1}=0$ | $S_{1}=1$ |  | $L_{2}=0$ | $S_{2}=1$ | 21
|  | $J_{1}=2$ |  | $J_{2}=1$ | i
|  | $L_{1}=2$ | $S_{1}=0$ |  | $L_{2}=0$ | $S_{2}=1$ | 22
|  | $L_{1}=2$ | $S_{1}=1$ |  | $L_{2}=0$ | $S_{2}=1$ | 23
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=0$ | 24
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 25
|  | $J_{1}=2$ |  | $J_{2}=2$ | i
|  | $L_{1}=1$ | $S_{1}=1$ |  | $L_{2}=1$ | $S_{2}=1$ | 27

In this coupling process, there are a total of 8 variables, denoted as
$L_{1},S_{1},J_{1},L_{2},S_{2},J_{2},J_{12},L_{3}$. For simplicity, we use “i”
to represent their combinations. Considering the symmetry of $T_{cc}$, such
combinations are considered in 27 cases, all listed in Table 3. Because of no
difference between spin of quark and antiquark, the meson-meson structure has
the same spin wave function as the diquark-antidiquark structure. The spin
wave functions of the sub-cluster are shown below,

$\displaystyle\chi^{11}_{\sigma}=\alpha\alpha,~{}~{}\chi^{10}_{\sigma}=\frac{1}{\sqrt{2}}(\alpha\beta+\beta\alpha),~{}~{}\chi^{1-1}_{\sigma}=\beta\beta,$
$\displaystyle\chi^{00}_{\sigma}=\frac{1}{\sqrt{2}}(\alpha\beta-\beta\alpha).$

In the context of quark spin, $\alpha$ and $\beta$ represent the third
component of quark spin, taking values of $\frac{1}{2}$ and $-\frac{1}{2}$,
respectively, in two distinct cases.

The spatial wave functions are Gaussians with range parameters chosen to lie
in a geometrical progression:

$\displaystyle\phi_{n_{1}L_{1}m_{1}}(\boldsymbol{\lambda})$
$\displaystyle=N_{n_{1}L_{1}}r^{L_{1}}e^{-(\lambda/\lambda_{n1})^{2}}Y_{L_{1}m_{1}}(\hat{\lambda}),$
(31a) $\displaystyle\phi_{n_{2}L_{2}m_{2}}(\boldsymbol{\rho})$
$\displaystyle=N_{n_{2}L_{2}}r^{L_{2}}e^{-(\rho/\rho_{n2})^{2}}Y_{L_{2}m_{2}}(\hat{\rho}),$
(31b) $\displaystyle\phi_{n_{3}L_{3}m_{3}}(\boldsymbol{R})$
$\displaystyle=N_{n_{3}L_{3}}r^{L_{3}}e^{-(R/R_{n3})^{2}}Y_{L_{3}m_{3}}(\hat{R}),$
(31c)

where $N_{n_{1}L_{1}},N_{n_{2}L_{2}}$ and $N_{n_{3}L_{3}}$ are normalization
constants, which can be expressed by a general formula

$\displaystyle
N_{nL}=\left[\frac{2^{L+2}(2\nu_{n})^{L+\frac{3}{2}}}{\sqrt{\pi}(2L+1)!!}\right]^{\frac{1}{2}}.$
(32)

In this Eq. 32, $\nu_{n}$ represents the expansion width determined by the
range parameter $r_{n}$ of the wave function, which can be expressed as
$\nu_{n}=\frac{1}{r_{n}^{2}}$. For the motion between different quarks, the
expression for the range parameter is as follows:

$\displaystyle\lambda_{n_{1}}$
$\displaystyle=\lambda_{min}a_{1}^{n_{1}-1},a_{1}={\frac{\lambda_{max}}{\lambda_{min}}}^{\frac{1}{n_{1}-1}},$
(33a) $\displaystyle\rho_{n_{2}}$
$\displaystyle=\rho_{min}a_{2}^{n_{2}-1},a_{2}={\frac{\rho_{max}}{\rho_{min}}}^{\frac{1}{n_{2}-1}},$
(33b) $\displaystyle R_{n_{3}}$
$\displaystyle=R_{min}a_{3}^{n_{3}-1},a_{3}={\frac{R_{max}}{R_{min}}}^{\frac{1}{n_{3}-1}},$
(33c)

where, $n_{1}$, $n_{2}$, and $n_{3}$ represent the Gaussian expansion orders
for the three relative motions among the four quarks. $\lambda_{min}$ and
$\lambda_{max}$ denote the minimum and maximum values for the Gaussian
expansion of the first subgroup’s relative motion. Similarly, $\rho_{min}$ and
$\rho_{max}$ represent the minimum and maximum values for the Gaussian
expansion of the second subgroup’s relative motion. $R_{min}$ and $R_{max}$
indicate the minimum and maximum values for the Gaussian expansion between
these two groups. The range parameters here are all obtained through fitting
the meson spectrum.

(a)

(b)

(c)

(d)

(e)

Figure 3:  The spatial structures of the dimeson are represented (a), (b), (c)
and (d), while (e) represents the spatial structure of the diquark wave
function.

.

We have two flavor wave functions of the $T_{cc}$ system,

$\displaystyle\xi_{1}=(c\bar{q})(c\bar{q}),\xi_{2}=(cc)(\bar{q}\bar{q}).$

$\xi_{1}$ is for meson-meson structure, and $\xi_{2}$ is for diquark-
antidiquark structure. The colorless tetraquark system has four color wave
functions, two for meson-meson structure, $1\otimes 1$ ($\varphi_{1}$),
$8\otimes 8$ ($\varphi_{2}$), and two for diquark-antidiquark structure,
$\bar{3}\otimes 3$ ($\varphi_{3}$) and $6\otimes\bar{6}$ ($\varphi_{4}$).

$\displaystyle\varphi_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{9}}(\bar{r}r\bar{r}r+\bar{r}r\bar{g}g+\bar{r}r\bar{b}b+\bar{g}g\bar{r}r+\bar{g}g\bar{g}g$
$\displaystyle+\bar{g}g\bar{b}b+\bar{b}b\bar{r}r+\bar{b}b\bar{g}g+\bar{b}b\bar{b}b),$
$\displaystyle\varphi_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{72}}(3\bar{b}r\bar{r}b+3\bar{g}r\bar{r}g+3\bar{b}g\bar{g}b+3\bar{g}b\bar{b}g+3\bar{r}g\bar{g}r$
$\displaystyle+3\bar{r}b\bar{b}r+2\bar{r}r\bar{r}r+2\bar{g}g\bar{g}g+2\bar{b}b\bar{b}b-\bar{r}r\bar{g}g$
$\displaystyle-\bar{g}g\bar{r}r-\bar{b}b\bar{g}g-\bar{b}b\bar{r}r-\bar{g}g\bar{b}b-\bar{r}r\bar{b}b),$
$\displaystyle\varphi_{3}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{12}}(rg\bar{r}\bar{g}-rg\bar{g}\bar{r}+gr\bar{g}\bar{r}-gr\bar{r}\bar{g}+rb\bar{r}\bar{b}$
$\displaystyle-
rb\bar{b}\bar{r}+br\bar{b}\bar{r}-br\bar{r}\bar{b}+gb\bar{g}\bar{b}-gb\bar{b}\bar{g}$
$\displaystyle+bg\bar{b}\bar{g}-bg\bar{g}\bar{b}),$ $\displaystyle\varphi_{4}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{1}{24}}(2rr\bar{r}\bar{r}+2gg\bar{g}\bar{g}+2bb\bar{b}\bar{b}+rg\bar{r}\bar{g}+rg\bar{g}\bar{r}$
(34)
$\displaystyle+gr\bar{g}\bar{r}+gr\bar{r}\bar{g}+rb\bar{r}\bar{b}+rb\bar{b}\bar{r}+br\bar{b}\bar{r}$
$\displaystyle+br\bar{r}\bar{b}+gb\bar{g}\bar{b}+gb\bar{b}\bar{g}+bg\bar{b}\bar{g}+bg\bar{g}\bar{b}).$

Index $|i,j,k\rangle$ color singlet: $J_{12}=0,L_{3}=1$ Index $|i,j,k\rangle$
color singlet:$J_{12}=1,L_{3}=2$ 1 $|1,1,1\rangle$
$D^{*}D_{1}^{{}^{\prime}}$-$P$ wave 12 $|14,1,1\rangle$
$D_{0}^{*}D_{1}^{{}^{\prime}}$-$D$ wave 2 $|2,1,1\rangle$
$D^{*}D_{1}^{*}(P)$-$P$ wave 13 $|15,1,1\rangle$ $D_{0}^{*}D_{1}^{*}(P)$-$D$
wave 3 $|3,1,1\rangle$ $D_{2}^{{}^{\prime}}D_{2}^{*}(P)$-$P$ wave 14
$|16,1,1\rangle$ $DD^{*}$-$D$ wave 4 $|4,1,1\rangle$
$D_{2}^{*}(D)D_{2}^{*}(P)$-$P$ wave 15 $|17,1,1\rangle$
$D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}$-$D$ wave Index $|i,j,k\rangle$ color
singlet: $J_{12}=1,L_{3}=0$ 16 $|18,1,1\rangle$
$D_{1}^{{}^{\prime}}D_{1}^{*}(P)$-$D$ wave $|5,1,1\rangle$ $DD^{*}$-$S$ wave
17 $|19,1,1\rangle$ $D_{1}^{*}(P)D_{1}^{*}(P)$-$D$ wave 5 $|6,1,1\rangle$
$D_{0}^{*}D_{1}^{{}^{\prime}}(P)$-$S$ wave 18 $|20,1,1\rangle$
$D^{*}D^{*}$-$D$ wave 6 $|7,1,1\rangle$ $D_{0}^{*}D_{1}^{*}(P)$-$S$ wave 19
$|21,1,1\rangle$ $D_{2}^{{}^{\prime}}D^{*}$-$D$ wave $|8,1,1\rangle$
$D^{*}D^{*}$-$S$ wave 20 $|22,1,1\rangle$ $D_{2}^{*}(D)D^{*}$-$D$ wave 7
$|9,1,1\rangle$ $D^{*}D_{1}^{*}(D)$-$S$ wave 21 $|23,1,1\rangle$
$D_{2}^{*}(P)D_{1}^{{}^{\prime}}$-$D$ wave 8 $|10,1,1\rangle$
$D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}$-$S$ wave 22 $|24,1,1\rangle$
$D_{2}^{*}(P)D_{1}^{*}(P)$-$D$ wave 9 $|11,1,1\rangle$
$D_{1}^{{}^{\prime}}D_{1}^{*}(P)$-$S$ wave 23 $|25,1,1\rangle$
$D_{2}^{*}(P)D_{2}^{*}(P)$-$D$ wave 10 $|12,1,1\rangle$
$D_{1}^{*}(P)D_{1}^{*}(P)$-$S$ wave 11 $|13,1,1\rangle$
$D_{1}^{*}(D)D_{1}^{*}(D)$-$S$ wave Index $|i,j,k\rangle$ color octet:
$J_{12}=0,L_{3}=1$ Index $|i,j,k\rangle$ color octet:$J_{12}=1,L_{3}=2$ 24
$|1,1,2\rangle$ $[D^{*}D_{1}^{{}^{\prime}}]_{8}$-$P$ wave 37 $|14,1,2\rangle$
$[D_{0}^{*}D_{1}^{{}^{\prime}}]_{8}$-$D$ wave 25 $|2,1,2\rangle$
$[D^{*}D_{1}^{*}(P)]_{8}$-$P$ wave 38 $|15,1,2\rangle$
$[D_{0}^{*}D_{1}^{*}(P)]_{8}$-$D$ wave 26 $|3,1,2\rangle$
$[D_{2}^{{}^{\prime}}D_{2}^{*}(P)]_{8}$-$P$ wave 39 $|16,1,2\rangle$
$[DD^{*}]_{8}$-$D$ wave 27 $|4,1,2\rangle$
$[D_{2}^{*}(D)D_{2}^{*}(P)]_{8}$-$P$ wave 40 $|17,1,2\rangle$
$[D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}]_{8}$-$D$ wave Index $|i,j,k\rangle$
color octet: $J_{12}=1,L_{3}=0$ 41 $|18,1,2\rangle$
$[D_{1}^{{}^{\prime}}D_{1}^{*}(P)]_{8}$-$D$ wave 28 $|5,1,2\rangle$
$[DD^{*}]_{8}$-$S$ wave 42 $|19,1,2\rangle$
$[D_{1}^{*}(P)D_{1}^{*}(P)]_{8}$-$D$ wave 29 $|6,1,2\rangle$
$[D_{0}^{*}D_{1}^{{}^{\prime}}(P)]_{8}$-$S$ wave 43 $|20,1,2\rangle$
$[D^{*}D^{*}]_{8}$-$D$ wave 30 $|7,1,2\rangle$
$[D_{0}^{*}D_{1}^{*}(P)]_{8}$-$S$ wave 44 $|21,1,2\rangle$
$[D_{2}^{{}^{\prime}}D^{*}]_{8}$-$D$ wave 31 $|8,1,2\rangle$
$[D^{*}D^{*}]_{8}$-$S$ wave 45 $|22,1,2\rangle$ $[D_{2}^{*}(D)D^{*}]_{8}$-$D$
wave 32 $|9,1,2\rangle$ $[D^{*}D_{1}^{*}(D)]_{8}$-$S$ wave 46 $|23,1,2\rangle$
$[D_{2}^{*}(P)D_{1}^{{}^{\prime}}]_{8}$-$D$ wave 33 $|10,1,2\rangle$
$[D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}]_{8}$-$S$ wave 47 $|24,1,2\rangle$
$[D_{2}^{*}(P)D_{1}^{*}(P)]_{8}$-$D$ wave 34 $|11,1,2\rangle$
$[D_{1}^{{}^{\prime}}D_{1}^{*}(P)]_{8}$-$S$ wave 48 $|25,1,2\rangle$
$[D_{2}^{*}(P)D_{2}^{*}(P)]_{8}$-$D$ wave 35 $|12,1,2\rangle$
$[D_{1}^{*}(P)D_{1}^{*}(P)]_{8}$-$S$ wave 36 $|13,1,2\rangle$
$[D_{1}^{*}(D)D_{1}^{*}(D)]_{8}$-$S$ wave Index $|i,j,k\rangle$ diquark:
$J_{12}=0,L_{3}=1$ 49 $|5,2,4\rangle$
$[cc]_{6}^{0}[\bar{q}\bar{q}]_{\bar{6}}^{1}$-$S$ wave 50 $|8,2,3\rangle$
$[cc]_{\bar{3}}^{1}[\bar{q}\bar{q}]_{3}^{1}$-$S$ wave

Table 4: The masses of $D_{s}$ mesons and the percentages of $c\bar{s}$
component in the states. ‘S’, ‘P’ and ‘D’ in the parentheses denote $S$-,
$P$\- and $D$-wave states.

The symbols $r(\bar{r})$, $g(\bar{g})$, and $b(\bar{b})$ correspond to the
colors of quarks (antiquarks), representing red, green, and blue,
respectively.

Based on the above discussion, the $T_{cc}$ tetraquark system is composed of
three parts: the spin-orbit coupled wave function $\psi^{SO}_{i}(\textbf{r})$,
the flavor wave function $\xi_{j}$, and the color wave function $\varphi_{k}$.
We have listed all possible physical channels in Table 7. The total wave
function $\Psi_{i,j,k}(\textbf{r})$ can be written as below

$\displaystyle\Psi_{i,j,k}(\textbf{r})$
$\displaystyle=\mathcal{A}\psi^{SO}_{i}(\textbf{r})\xi_{j}\varphi_{k}$ (35)

The total wave function $\Psi_{i,j,k}(\textbf{r})$ can be expanded as follows:

$\displaystyle\Psi_{i,j,k}(\textbf{r})=C_{J_{12},mJ_{12};L_{3},mL_{3}}^{J,mJ}C_{J_{1},mJ_{1};J_{2},mJ_{2}}^{J_{12},mJ_{12}}C_{L_{1},mL_{1};S_{1},mS_{1}}^{J_{1},mJ_{1}}$
$\displaystyle
C_{L_{2},mL_{2};S_{2},mS_{2}}^{J_{2},mJ_{2}}\mathcal{A}\phi_{n_{1}L_{1}m_{1}}(\boldsymbol{\lambda})\phi_{n_{2}L_{2}m_{2}}(\boldsymbol{\rho})\phi_{n_{3}L_{3}m_{3}}(\boldsymbol{R})$
$\displaystyle\chi_{S_{1},mS_{1}}\chi_{S_{2},mS_{2}}\xi_{j}\varphi_{k}$ (36)

In Eq. II.2, $C$ represents the Clebsch-Gordan coefficient. $\mathcal{A}$ is
the antisymmetrization operator, and for the dimeson structure in the $T_{cc}$
system, ${\cal A}=1-P_{13}-P_{24}+P_{13}P_{24}$, where $P_{ij}$ denotes the
antisymmetric operator, and the four exchange terms correspond to the first
four quark coupling orders in Fig. 3(e). However, for the diquark structure in
the $T_{cc}$ system, due to symmetry considerations, ${\cal A}=1$,
corresponding to the fifth quark coupling order in Fig. 3(e). The relevant
coordinates in each set are defined as

$\displaystyle\boldsymbol{\lambda}_{a}$
$\displaystyle=\textbf{r}_{1}-\textbf{r}_{2},\boldsymbol{\rho}_{a}=\textbf{r}_{3}-\textbf{r}_{4},$
$\displaystyle\textbf{R}_{a}$
$\displaystyle=\frac{m_{1}\textbf{r}_{1}+m_{2}\textbf{r}_{2}}{m_{1}+m_{2}}-\frac{m_{3}\textbf{r}_{3}+m_{4}\textbf{r}_{4}}{m_{3}+m_{4}},$
(37) $\displaystyle\boldsymbol{\lambda}_{b}$
$\displaystyle=\textbf{r}_{1}-\textbf{r}_{4},\boldsymbol{\rho}_{b}=\textbf{r}_{2}-\textbf{r}_{3},$
$\displaystyle\textbf{R}_{b}$
$\displaystyle=\frac{m_{1}\textbf{r}_{1}+m_{4}\textbf{r}_{4}}{m_{1}+m_{4}}-\frac{m_{2}\textbf{r}_{2}+m_{3}\textbf{r}_{3}}{m_{2}+m_{3}},$
(38) $\displaystyle\boldsymbol{\lambda}_{c}$
$\displaystyle=\textbf{r}_{3}-\textbf{r}_{2},\boldsymbol{\rho}_{c}=\textbf{r}_{1}-\textbf{r}_{4},$
$\displaystyle\textbf{R}_{c}$
$\displaystyle=\frac{m_{3}\textbf{r}_{3}+m_{2}\textbf{r}_{2}}{m_{3}+m_{2}}-\frac{m_{1}\textbf{r}_{1}+m_{4}\textbf{r}_{4}}{m_{1}+m_{4}},$
(39) $\displaystyle\boldsymbol{\lambda}_{d}$
$\displaystyle=\textbf{r}_{3}-\textbf{r}_{4},\boldsymbol{\rho}_{d}=\textbf{r}_{1}-\textbf{r}_{2},$
$\displaystyle\textbf{R}_{d}$
$\displaystyle=\frac{m_{3}\textbf{r}_{4}+m_{4}\textbf{r}_{4}}{m_{3}+m_{4}}-\frac{m_{1}\textbf{r}_{1}+m_{2}\textbf{r}_{2}}{m_{1}+m_{2}},$
(40) $\displaystyle\boldsymbol{\lambda}_{e}$
$\displaystyle=\textbf{r}_{1}-\textbf{r}_{3},\boldsymbol{\rho}_{e}=\textbf{r}_{2}-\textbf{r}_{4},$
$\displaystyle\textbf{R}_{e}$
$\displaystyle=\frac{m_{1}\textbf{r}_{1}+m_{3}\textbf{r}_{3}}{m_{1}+m_{3}}-\frac{m_{2}\textbf{r}_{2}+m_{4}\textbf{r}_{4}}{m_{2}+m_{4}}.$
(41)

In all cases, the center-of-mass coordinate is

$\textbf{R}=\frac{m_{1}\textbf{r}_{1}+m_{2}\textbf{r}_{2}+m_{3}\textbf{r}_{3}+m_{4}\textbf{r}_{4}}{m_{1}+m_{2}+m_{3}+m_{4}},$
(42)

where $m_{1}$, $m_{2}$, $m_{3}$, and $m_{4}$ are the masses of particles 1, 2,
3, and 4, respectively.

At last, we solve the following Schrödinger equation to obtain eigen-energies
of the system, with the help of the Rayleigh-Ritz variational principle.

$H\Psi_{i,j,k}(\textbf{r})=E\Psi_{i,j,k}(\textbf{r}),$ (43)

the eigenvalues ($E_{n}$) and eigenvectors ($C_{n}$) of the system are
obtained by solving the diagonalization problem,

$\displaystyle[(H)-E_{n}(N)][C_{n}]=0,$ (44)

with

$\displaystyle(H)=\langle\Psi_{i,j,k}(\textbf{r})|H|\Psi_{i,j,k}(\textbf{r})\rangle,$
$\displaystyle(N)=\langle\Psi_{i,j,k}(\textbf{r})|1|\Psi_{i,j,k}(\textbf{r})\rangle.$

In the quark model, obtaining matrix elements that involve tensor forces and
spin-orbit coupling is a complex process, especially when dealing with wave
functions associated with orbital excited states. Taking the structure in Fig.
3(e)(a) as an example, where the orbital quantum number between $c_{1}$ and
$\bar{q}_{2}$ is 1, after Jacobi coordinate transformation to the structure in
Fig. 3(e)(b), where $c_{1}$ forms a $c_{1}$-$\bar{q}_{4}$ subgroup with
$\bar{q}_{4}$, the orbital quantum number between $c_{1}$ and $\bar{q}_{4}$
cannot be directly determined. In this paper, we employ the spherical
harmonics expansion method (Eq. (139) in Ref. Hiyama:2003cu ). If there is the
following relationship between Jacobi coordinates:
$\bf{r_{a}}=\alpha\bf{r_{b}}+\beta\bf{R_{b}}$, then the transformation
relationship between their spherical harmonics is given by

$\displaystyle
r_{a}^{L}Y_{L,mL}(\textbf{r}_{a})=\sum\limits_{\lambda=0}^{L}\left[\frac{4\pi(2L+1)!}{(2k+1)!(2(L-k)!+1)!}\right]^{1/2}(\alpha
r_{b})^{k}$ $\displaystyle(\beta
R_{b})^{L-k}\sum\limits_{mk=-k}^{k}C_{k,mk;L-k,mL-
mk}^{L,mL}Y_{k,mk}(\textbf{r}_{b})$ $\displaystyle Y_{L-k,mL-
mk}(\textbf{R}_{b}).$ (46)

In Eq. II.2, $k$ represents the possible orbital angular momentum of
$\textbf{r}_{b}$, $L-k$ denotes the orbital angular momentum of
$\textbf{R}_{b}$, and their sum corresponds to the orbital angular momentum of
$\textbf{r}_{a}$. The subscripts $mk$ and $mL-mk$ refer to the respective
third components. In the $T_{cc}$ system, we need to handle matrix elements
like
$\langle\Psi_{i,j,k}(\textbf{r}_{a})|V(r_{c})|\Psi_{i,j,k}(\textbf{r}_{b})\rangle$.
This involves Jacobi coordinate transformations, i.e., transforming
$\Psi_{i,j,k}(\textbf{r}_{a})$ and $\Psi_{i,j,k}(\textbf{r}_{b})$ into the
form of $\Psi_{i,j,k}(\textbf{r}_{c})$, and then considering spatial
integrations. We first assume that there is the following Jacobi coordinate
transformation relationship between the Jacobi coordinates of $r_{a}$ and
$r_{c}$,

$\displaystyle\lambda_{a}=\alpha_{1}\lambda_{c1}+\beta_{1}\rho_{c2}+\gamma_{1}R_{c3},$
(47)
$\displaystyle\rho_{a}=\alpha_{2}\lambda_{c1}+\beta_{2}\rho_{c2}+\gamma_{2}R_{c3},$
(48) $\displaystyle
R_{a}=\alpha_{3}\lambda_{c1}+\beta_{3}\rho_{c2}+\gamma_{3}R_{c3},$ (49)

where, $\alpha$, $\beta$, and $\gamma$ are transformation coefficients for the
coordinates $\textbf{r}_{a}$ to transform into $\textbf{r}_{c}$. Then,
$\lambda_{a}^{L_{a1}}Y_{L_{a1},mL_{a1}}(\boldsymbol{\lambda}_{a})$,$\rho_{a}^{L_{a2}}Y_{L_{a2},mL_{a2}}(\boldsymbol{\rho}_{a})$,$R_{a}^{L_{a3}}Y_{L_{a3},mL_{a3}}(\boldsymbol{R}_{a})$
can be expanded as:

$\displaystyle\lambda_{a}^{L_{a1}}Y_{L_{a1},mL_{a1}}(\boldsymbol{\lambda}_{a1})$
$\displaystyle=\sum\limits_{k_{1}=0}^{L_{a1}}\sum\limits_{t_{1}=0}^{k_{1}}\left[\frac{4\pi(2L_{a1}+1)!}{(2k_{1}+1)!(2(L_{a1}-k_{1})!+1)!}\right]^{1/2}\left[\frac{4\pi(2k_{1}+1)!}{(2t_{1}+1)!(2(k_{1}-t_{1})!+1)!}\right]^{1/2}(\alpha_{1}\lambda_{c1})^{t_{1}}(\beta_{1}\rho_{c2})^{k_{1}-t_{1}}$
$\displaystyle(\gamma_{1}R_{c3})^{L_{a1}-k_{1}}\sum\limits_{mk_{1}=-k_{1}}^{k_{1}}\sum\limits_{mt_{1}=-t_{1}}^{t_{1}}C_{k_{1},mk_{1};L_{a1}-k_{1},mL_{a1}-mk_{1}}^{L_{a1},mL_{a1}}C_{t_{1},mt_{1};k_{1}-t_{1},mk_{1}-mt_{1}}^{k_{1},mt_{1}}Y_{t_{1},mt_{1}}(\boldsymbol{\lambda}_{c1})$
$\displaystyle
Y_{k_{1}-t_{1},mk_{1}-mt_{1}}(\boldsymbol{\rho}_{c2})Y_{L_{a1}-k_{1},mL_{a1}-mk_{1}}(\textbf{R}_{c3}).$
$\displaystyle\rho_{a}^{L_{a1}}Y_{L_{a2},mL_{a2}}(\boldsymbol{\rho}_{a2})$
$\displaystyle=\sum\limits_{k_{2}=0}^{L_{a2}}\sum\limits_{t_{2}=0}^{k_{2}}\left[\frac{4\pi(2L_{a2}+1)!}{(2k_{2}+1)!(2(L_{a2}-k_{2})!+1)!}\right]^{1/2}\left[\frac{4\pi(2k_{2}+1)!}{(2t_{2}+1)!(2(k_{2}-t_{2})!+1)!}\right]^{1/2}(\alpha_{2}\lambda_{c1})^{t_{2}}(\beta_{2}\rho_{c2})^{k_{2}-t_{2}}$
$\displaystyle(\gamma_{2}R_{c3})^{L_{a2}-k_{2}}\sum\limits_{mk_{2}=-k_{2}}^{k_{2}}\sum\limits_{mt_{2}=-t_{2}}^{t_{2}}C_{k_{2},mk_{2};L_{a2}-k_{2},mL_{a2}-mk_{2}}^{L_{a2},mL_{a2}}C_{t_{2},mt_{2};k_{2}-t_{2},mk_{2}-mt_{2}}^{k_{2},mt_{2}}Y_{t_{2},mt_{2}}(\boldsymbol{\lambda}_{c1})$
$\displaystyle
Y_{k_{2}-t_{2},mk_{2}-mt_{2}}(\boldsymbol{\rho}_{c2})Y_{L_{a2}-k_{2},mL_{a2}-mk_{2}}(\textbf{R}_{c3}).$
$\displaystyle R_{a}^{L_{a3}}Y_{L_{a3},mL_{a3}}(\boldsymbol{R}_{a3})$
$\displaystyle=\sum\limits_{k_{3}=0}^{L_{a3}}\sum\limits_{t_{3}=0}^{k_{3}}\left[\frac{4\pi(2L_{a3}+1)!}{(2k_{3}+1)!(2(L_{a3}-k_{3})!+1)!}\right]^{1/2}\left[\frac{4\pi(2k_{3}+1)!}{(2t_{3}+1)!(2(k_{3}-t_{3})!+1)!}\right]^{1/2}(\alpha_{3}\lambda_{c1})^{t_{3}}(\beta_{3}\rho_{c2})^{k_{3}-t_{3}}$
$\displaystyle(\gamma_{3}R_{c3})^{L_{a3}-k_{3}}\sum\limits_{mk_{3}=-k_{3}}^{k_{3}}\sum\limits_{mt_{3}=-t_{3}}^{t_{3}}C_{k_{3},mk_{3};L_{a3}-k_{3},mL_{a3}-mk_{3}}^{L_{a3},mL_{a3}}C_{t_{3},mt_{3};k_{3}-t_{3},mk_{3}-mt_{3}}^{k_{3},mt_{3}}Y_{t_{3},mt_{3}}(\boldsymbol{\lambda}_{c1})$
$\displaystyle
Y_{k_{3}-t_{3},mk_{3}-mt_{3}}(\boldsymbol{\rho}_{c2})Y_{L_{a3}-k_{3},mL_{a3}-mk_{3}}(\textbf{R}_{c3}),$
(50)

where the total wave function $\Psi_{i,j,k}(\textbf{r}_{a})$ for the three
relative motions is expanded into nine relative motion wave functions. Within
these nine functions, there are actually three different orbital quantum
numbers denoted as $Y_{L_{1},m_{1}}(\boldsymbol{\lambda}_{c1})$,
$Y_{L_{2},m_{2}}(\boldsymbol{\rho}_{c2})$, and
$Y_{L_{3},m_{3}}(\boldsymbol{R}_{c3})$. Next, we need to combine the spherical
harmonics with different orbital quantum numbers using the following formula:

$\displaystyle
Y_{L_{1},m_{1}}(\textbf{r})Y_{L_{2},m_{2}}(\textbf{r})=\sum\limits_{L=|L_{1}-L_{2}|}^{L_{1}+L_{2}}C_{L_{1},m_{1};L_{2},m_{2}}^{L,m_{1}+m_{2}}C_{L_{1},0;L_{2},0}^{L,0}$
$\displaystyle\left[\frac{(2L_{1}+1)(2L_{2}+1)}{4\pi(2L+1)}\right]^{1/2}Y_{L,m_{1}+m_{2}}(\textbf{r})$
(51)

Therefore, the total wave function $\Psi_{i,j,k}(\textbf{r}_{a})$ undergoes a
transformation based on Jacobi coordinate change (Eqs. LABEL:Jacobi).
Initially, the spatial wave function for $r_{a}$ transforms from three terms
to nine terms, introducing six additional CG coefficients according to Eq.
II.2, when transforming from the initial $r_{a}$ to any coordinate $r_{c}$.
The spherical harmonics with the same orbital quantum numbers in the
transformed wave function are then combined using Eq. II.2. The final total
wave function is expressed as follows:

$\displaystyle\Psi_{i,j,k}(\textbf{r})=$ $\displaystyle
C_{J_{12},mJ_{12};L_{3},mL_{3}}^{J,mJ}C_{J_{1},mJ_{1};J_{2},mJ_{2}}^{J_{12},mJ_{12}}C_{L_{1},mL_{1};S_{1},mS_{1}}^{J_{1},mJ_{1}}C_{L_{2},mL_{2};S_{2},mS_{2}}^{J_{2},mJ_{2}}\chi_{S_{1},mS_{1}}\chi_{S_{2},mS_{2}}\xi_{j}\varphi_{k}\sum\limits_{k_{1}=0}^{L_{a1}}\sum\limits_{t_{1}=0}^{k_{1}}\sum\limits_{k_{2}=0}^{L_{a2}}\sum\limits_{t_{2}=0}^{k_{2}}\sum\limits_{k_{3}=0}^{L_{a3}}\sum\limits_{t_{3}=0}^{k_{3}}$
$\displaystyle\left[\frac{4\pi(2L_{a1}+1)!}{(2k_{1}+1)!(2(L_{a1}-k_{1})!+1)!}\right]^{1/2}\left[\frac{4\pi(2k_{1}+1)!}{(2t_{1}+1)!(2(k_{1}-t_{1})!+1)!}\right]^{1/2}\left[\frac{4\pi(2L_{a2}+1)!}{(2k_{2}+1)!(2(L_{a2}-k_{2})!+1)!}\right]^{1/2}$
$\displaystyle\left[\frac{4\pi(2k_{2}+1)!}{(2t_{2}+1)!(2(k_{2}-t_{2})!+1)!}\right]^{1/2}\left[\frac{4\pi(2L_{a3}+1)!}{(2k_{3}+1)!(2(L_{a3}-k_{3})!+1)!}\right]^{1/2}\left[\frac{4\pi(2k_{3}+1)!}{(2t_{3}+1)!(2(k_{3}-t_{3})!+1)!}\right]^{1/2}$
$\displaystyle(\alpha_{1}\lambda_{c1})^{t_{1}}(\beta_{1}\rho_{c2})^{k_{1}-t_{1}}(\gamma_{1}R_{c3})^{L_{a1}-k_{1}}(\alpha_{2}\lambda_{c1})^{t_{2}}(\beta_{2}\rho_{c2})^{k_{2}-t_{2}}(\gamma_{2}R_{c3})^{L_{a2}-k_{2}}(\alpha_{3}\lambda_{c1})^{t_{3}}(\beta_{3}\rho_{c2})^{k_{3}-t_{3}}(\gamma_{3}R_{c3})^{L_{a3}-k_{3}}$
$\displaystyle\sum\limits_{mk_{1}=-k_{1}}^{k_{1}}\sum\limits_{mt_{1}=-t_{1}}^{t_{1}}\sum\limits_{mk_{2}=-k_{2}}^{k_{2}}\sum\limits_{mt_{2}=-t_{2}}^{t_{2}}\sum\limits_{mk_{3}=-k_{3}}^{k_{3}}\sum\limits_{mt_{3}=-t_{3}}^{t_{3}}C_{k_{1},mk_{1};L_{a1}-k_{1},mL_{a1}-mk_{1}}^{L_{a1},mL_{a1}}C_{t_{1},mt_{1};k_{1}-t_{1},mk_{1}-mt_{1}}^{k_{1},mt_{1}}$
$\displaystyle
C_{k_{2},mk_{2};L_{a2}-k_{2},mL_{a2}-mk_{2}}^{L_{a2},mL_{a2}}C_{t_{2},mt_{2};k_{2}-t_{2},mk_{2}-mt_{2}}^{k_{2},mt_{2}}C_{k_{3},mk_{3};L_{a3}-k_{3},mL_{a3}-mk_{3}}^{L_{a3},mL_{a3}}C_{t_{3},mt_{3};k_{3}-t_{3},mk_{3}-mt_{3}}^{k_{3},mt_{3}}$
$\displaystyle\sum\limits_{t_{12}=|t_{1}-t_{2}|}^{t_{1}+t_{2}}\sum\limits_{t_{123}=|t_{12}-t_{3}|}^{t_{12}+t_{3}}\sum\limits_{s_{12}=|(k_{1}-t_{1})-(k_{2}-t_{2})|}^{|(k_{1}-t_{1})+(k_{2}-t_{2})|}\sum\limits_{s_{123}=|s_{12}-(k_{3}-t_{3})|}^{s_{12}+(k_{3}-t_{3})}\sum\limits_{q_{12}=|(L_{a1}-k_{1})-(L_{a2}-k_{2})|}^{|(L_{a1}-k_{1})+(L_{a2}-k_{2})|}\sum\limits_{q_{123}=|q_{12}-(L_{a3}-k_{3})|}^{q_{12}+(L_{a3}-k_{3})}$
$\displaystyle\left[\frac{(2t_{1}+1)(2t_{2}+1)}{4\pi(2t_{12}+1)}\right]^{1/2}\left[\frac{(2t_{12}+1)(2t_{3}+1)}{4\pi(2t_{123}+1)}\right]^{1/2}\left[\frac{(2(k_{1}-t_{1})+1)(2(k_{2}-t_{2})+1)}{4\pi(2s_{12}+1)}\right]^{1/2}$
$\displaystyle\left[\frac{(2s_{12}+1)(2(k_{3}-t_{3})+1)}{4\pi(2s_{123}+1)}\right]^{1/2}\left[\frac{(2(L_{a1}-k_{1})+1)(2(L_{a2}-k_{2})+1)}{4\pi(2q_{12}+1)}\right]^{1/2}\left[\frac{(2q_{12}+1)(2(L_{a3}-k_{3})+1)}{4\pi(2q_{123}+1)}\right]^{1/2}$
$\displaystyle
C_{t_{1},mt_{1};t_{2},mt_{2}}^{t_{12},mt_{12}}C_{t_{1},0;t_{2},0}^{t_{12},0}C_{t_{12},mt_{12};t_{3},mt_{3}}^{t_{123},mt_{123}}C_{t_{12},0;t_{3},0}^{t_{123},0}C_{(L_{a1}-k_{1}),(mL_{a1}-mk_{1});(L_{a2}-k_{2}),(mL_{a2}-mk_{2})}^{q_{12},mq_{12}}C_{(L_{a1}-k_{1}),0;(L_{a2}-k_{2}),0}^{q_{12},0}$
$\displaystyle
C_{q_{12},mq_{12};(L_{a3}-k_{3}),(mL_{a3}-mk_{3})}^{q_{123},mq_{123}}C_{q_{12},0;(L_{a3}-k_{3}),0}^{q_{123},0}C_{(k_{1}-t_{1}),(mk_{1}-mt_{1});(k_{2}-t_{2}),(mk_{2}-mt_{2})}^{s_{12},ms_{12}}C_{(k_{1}-t_{1}),0;(k_{2}-t_{2}),0}^{s_{12},0}$
$\displaystyle
C_{s_{12},ms_{12};(k_{3}-t_{3}),(mk_{3}-mt_{3})}^{s_{123},ms_{123}}C_{s_{12},0;(k_{3}-t_{3}),0}^{s_{123},0}Y_{t_{123},mt_{123}}(\boldsymbol{\lambda}_{c1})Y_{s_{123},ms_{123}}(\boldsymbol{\rho}_{c2})Y_{q_{123},mq_{123}}(\textbf{R}_{c3}),$
(52)

where it becomes evident that the original 9 spherical harmonics (Eq. II.2)
are reduced to 3 after undergoing coupling. The detailed procedure involves
initially coupling $Y_{t_{1},m_{t_{1}}}(\boldsymbol{\lambda}_{c_{1}})$ and
$Y_{t_{2},m_{t_{2}}}(\boldsymbol{\lambda}_{c_{1}})$ into
$Y_{t_{12},m_{t_{12}}}(\boldsymbol{\lambda}_{c_{1}})$, accompanied by the
relevant CG coefficients $C_{t_{1},mt_{1};t_{2},mt_{2}}^{t_{12},mt_{12}}$,
$C_{t_{1},0;t_{2},0}^{t_{12},0}$, and normalization factor
$\left[\frac{(2t_{1}+1)(2t_{2}+1)}{4\pi(2t_{12}+1)}\right]^{1/2}$. Here,
$t_{12}$ and $m_{t_{12}}$ are novel variables derived from coupling $t_{1}$
and $t_{2}$. Subsequently, the newly obtained
$Y_{t_{12},m_{t_{12}}}(\boldsymbol{\lambda}_{c_{1}})$ couples with the
original $Y_{t_{3},m_{t_{3}}}(\boldsymbol{\lambda}_{c_{1}})$ to yield
$Y_{t_{123},m_{t_{123}}}(\boldsymbol{\lambda}_{c_{1}})$, accompanied by the
relevant CG coefficients $C_{t_{12},mt_{12};t_{3},mt_{3}}^{t_{123},mt_{123}}$,
$C_{t_{12},0;t_{3},0}^{t_{123},0}$, and normalization factor
$\left[\frac{(2t_{12}+1)(2t_{3}+1)}{4\pi(2t_{123}+1)}\right]^{1/2}$, where
$t_{123}$ and $m_{t_{123}}$ derive from coupling $t_{12}$ and $t_{3}$.
Likewise, we couple the three spherical harmonics
$Y_{k_{1}-t_{1},mk_{1}-mt_{1}}(\boldsymbol{\rho}_{c_{2}})$,
$Y_{k_{2}-t_{2},mk_{2}-mt_{2}}(\boldsymbol{\rho}_{c_{2}})$, and
$Y_{k_{3}-t_{3},mk_{3}-mt_{3}}(\boldsymbol{\rho}_{c_{2}})$ associated with the
same motion $\rho_{c_{2}}$ but with different quantum numbers, resulting in a
new spherical harmonic $Y_{s_{123},ms_{123}}(\boldsymbol{\rho}_{c_{2}})$. The
process involves introducing new variables $s_{12},ms_{12}$,
$s_{123},ms_{123}$ obtained from the coupling of variables
$k_{1}-t_{1},mk_{1}-mt_{1}$, $k_{2}-t_{2},mk_{2}-mt_{2}$, and
$k_{3}-t_{3},mk_{3}-mt_{3}$. Similarly, the three spherical harmonics
$Y_{L_{a1}-k_{1},mL_{a1}-mk_{1}}(\textbf{R}_{c_{3}})$,
$Y_{L_{a2}-k_{2},mL_{a2}-mk_{2}}(\textbf{R}_{c_{3}})$, and
$Y_{L_{a3}-k_{3},mL_{a3}-mk_{3}}(\textbf{R}_{c_{3}})$ associated with the same
motion $R_{c_{3}}$ but different quantum numbers are coupled to produce the
new spherical harmonic $Y_{q_{123},mq_{123}}(\boldsymbol{R}_{c_{3}})$. Here,
${q_{12},mq_{12}}$ and ${q_{123},mq_{123}}$ represent the orbital quantum
numbers resulting from the coupling of these three spherical harmonics.

##  III Results and discussions

In this part, we will discuss the equivalence of the two cases:① The color
octet state can be replaced by a superposition of color singlet ground and
excited states. ② In the case of a complete basis, diquarks and dimesons are
equivalent.

Table 5: Results of the coupling of two ground-state channels in $T_{cc}$ system. (unit: MeV) $|[LS]_{i}F_{j}C_{k}\rangle$ | Channel | $E$ | $E^{Theo}_{th}$ | $E_{B}$ | $E^{Exp}_{th}$
---|---|---|---|---|---
$|5,1,1\rangle$ |  $DD^{*}$-$S$ wave | 3874.12 | 3872.71 | 0 | 3871.69
$|8,1,1\rangle$ |  $D^{*}D^{*}$-$S$ wave | 3961.50 | 3960.74 | 0 | 4013.70
$E_{0}=E(DD^{*}+D^{*}D^{*}$) | 3871.59 | 3872.71 | 1.12 |
Table 6: Comparison between the results of the accumulation approach and the complete-channel coupling calculations. (unit: MeV) Index | Channel | $E$
---|---|---
5 |  $DD^{*}$-$S$ wave | 3874.12
8 |  $D^{*}D^{*}$-$S$ wave | 3961.50
1 |  $D^{*}D_{1}^{\prime}$-$P$ wave | 4432.14
2 |  $D^{*}D_{1}^{*}(P)$-$P$ wave | 4419.64
3 |  $D_{2}^{\prime}D_{2}^{*}(P)$-$P$ wave | 5360.48
4 |  $D_{2}^{*}(D)D_{2}^{*}(P)$-$P$ wave | 5379.45
Complete-coupled-channels: | 3871.19
Index | Channel | $\Delta E_{i}$
1 |  $DD^{*}+D^{*}D^{*}+D^{*}D_{1}^{\prime}$-$P$ wave | -0.21
2 |  $DD^{*}+D^{*}D^{*}+D^{*}D_{1}^{*}(P)$-$P$ wave | -0.12
3 |  $DD^{*}+D^{*}D^{*}+D_{2}^{\prime}D_{2}^{*}(P)$-$P$ wave | -0.04
4 |  $DD^{*}+D^{*}D^{*}+D_{2}^{*}(D)D_{2}^{*}(P)$-$P$ wave | -0.03
$E_{0}$+$\sum\limits_{i=1}^{4}\Delta E_{i}$: | 3871.17

In the chiral quark model, $T_{cc}$ system is special, because $DD^{*}$ and
$D^{*}D^{*}$ are coupled to form a bound state, which is denoted as $E_{0}$,
shown in Table 5. However, the expansion of the wave function of the highly
excited state is too complicated for us (see Eq. II.2), and the number of
channels of the wave function of the highly excited state is very large. In
principle, the bound states of a system should be calculated directly by
calculating the complete-channel coupling of all channels. In this article, we
take an accumulation approach. Since the two channels $DD^{*}$ and
$D^{*}D^{*}$ can already form a bound state, we coupled all highly excited
physical channels with $DD^{*}+D^{*}D^{*}$ ($E_{0}$) respectively to find the
contribution of each channel to the bounding energy of the system, which can
be expressed by $\Delta{E_{i}}=E(DD^{*}+D^{*}D^{*}+ith~{}channel)-E_{0}$.
Finally, The total energy of the system can be expressed as the coupling
energy of $E_{0}$ plus all high excited state’s contribution ($\Delta{E_{i}}$)
to the system of binding energy. Its calculation formula of the accumulation
approach can be expressed as:

$\displaystyle E=E_{0}+\sum\limits_{i=1}^{n}\Delta{E_{i}}.$ (53)

To assess the accuracy of the accumulation approach, we performed calculations
for the six channels in the $T_{cc}$ system, which involved both direct
calculation with complete channel couplings and calculation using the
accumulation approach (see in Table 6). The six channels include two states
with $S$-wave ($DD^{*}$ and $D^{*}D^{*}$) and four states with $P$-wave
($D^{*}D_{1}^{\prime}$, $D^{*}D_{1}^{*}(P)$, $D_{2}^{\prime}D_{2}^{*}(P)$, and
$D_{2}^{*}(D)D_{2}^{*}(P)$). The results of the complete-channel coupling
calculations indicate that the lowest eigenenergy for these six channels in
the $T_{cc}$ system is $3871.19$ MeV. According to the accumulation approach,
the coupling energy for the channels $DD^{*}$ and $D^{*}D^{*}$ is $3871.59$
MeV ($E_{0}$=$3871.59$ MeV in Table 5). The coupling of the first channel
$D^{*}D_{1}^{\prime}$ with ($DD^{*}$ and $D^{*}D^{*}$) reduces the energy of
$E_{0}$ by 0.21 MeV, denoted as $\Delta E_{1}$. Similarly, the coupling of the
second channel $D^{*}D_{1}^{*}(P)$ further lowers the energy of $E_{0}$ by
0.12 MeV, denoted as $\Delta E_{2}$. The contributions to the coupling energy
from $D^{*}D_{1}^{*}(P)$ and $D_{2}^{\prime}D_{2}(P)$ are denoted as $\Delta
E_{3}$ and $\Delta E_{4}$, respectively. Finally, utilizing the calculation
formula in the accumulation approach,
$E=E_{0}+\sum\limits_{i=1}^{4}\Delta{E_{i}}$, the calculated final energy is
$3871.17$ MeV. This implies that the accuracy of the accumulation approach is
applicable in the calculation of bound states. Therefore, in this paper, we
employ the accumulation approach for the calculation of the remaining
channels.

###  III.1 Equivalence between color octet and orbital excited state

For the $T_{cc}$ system, there are two ground-state molecular states:
$DD^{*}$and $D^{*}D^{*}$. They have two corresponding color octet,
$[D]_{8}[D^{*}]_{8}$ and $[D^{*}]_{8}[D^{*}]_{8}$. Here, we consider the
equivalence between higher-order partial wave and color octet, so the color
octet here only considers the two ground states, and the excited color octet
is not taken into account. As shown in Table 7, the contributions of
$[D]_{8}[D^{*}]_{8}$ and $[D^{*}]_{8}[D^{*}]_{8}$ to the binding energy of the
system are $\Delta E_{28}=-0.37$ MeV and $\Delta E_{31}=-0.56$ MeV,
respectively. According to the calculation Eq. 53 of the accumulation
approach, the total energy is $E_{1}=E_{0}+\Delta E_{28}+\Delta
E_{29}=3870.66$ MeV.

Index(i) Channel $\Delta E_{i}$ Index(i) Channel $\Delta E_{i}$ 1
$DD^{*}+D^{*}D^{*}+D^{*}D_{1}^{\prime}$-$P$ wave -0.21 24
$DD^{*}+D^{*}D^{*}+[D^{*}D_{1}^{{}^{\prime}}]_{8}$-$P$ wave -0.17 2
$DD^{*}+D^{*}D^{*}+D^{*}D_{1}^{*}(P)$-$P$ wave -0.12 25
$DD^{*}+D^{*}D^{*}+[D^{*}D_{1}^{*}(P)]_{8}$-$P$ wave -0.34 3
$DD^{*}+D^{*}D^{*}+D_{2}^{\prime}D_{2}^{*}(P)$-$P$ wave -0.04 26
$DD^{*}+D^{*}D^{*}+[D_{2}^{{}^{\prime}}D_{2}^{*}(P)]_{8}$-$P$ wave -0.02 4
$DD^{*}+D^{*}D^{*}+D_{2}^{*}(D)D_{2}^{*}(P)$-$P$ wave -0.03 27
$DD^{*}+D^{*}D^{*}+[D_{2}^{*}(D)D_{2}^{*}(P)]_{8}$-$P$ wave -0.10 5
$DD^{*}+D^{*}D^{*}+D_{0}^{*}D_{1}^{\prime}(P)$-$S$ wave -0.31 28
$DD^{*}+D^{*}D^{*}+[DD^{*}]_{8}$-$S$ wave -0.37 6
$DD^{*}+D^{*}D^{*}+D_{0}^{*}D_{1}^{*}(P)$-$S$ wave -1.03 29
$DD^{*}+D^{*}D^{*}+[D_{0}^{*}D_{1}^{{}^{\prime}}(P)]_{8}$-$S$ wave -0.57 7
$DD^{*}+D^{*}D^{*}+D^{*}D_{1}^{*}(D)$-$S$ wave -0.01 30
$DD^{*}+D^{*}D^{*}+[D_{0}^{*}D_{1}^{*}(P)]_{8}$-$S$ wave -0.67 8
$DD^{*}+D^{*}D^{*}+D_{1}^{\prime}D_{1}^{\prime}$-$S$ wave -0.01 31
$DD^{*}+D^{*}D^{*}+[D^{*}D^{*}]_{8}$-$S$ wave -0.56 9
$DD^{*}+D^{*}D^{*}+D_{1}^{\prime}D_{1}^{*}(P)$-$S$ wave -0.25 32
$DD^{*}+D^{*}D^{*}+[D^{*}D_{1}^{*}(D)]_{8}$-$S$ wave -0.01 10
$DD^{*}+D^{*}D^{*}+D_{1}^{*}(P)D_{1}^{*}(P)$-$S$ wave -0.11 33
$DD^{*}+D^{*}D^{*}+[D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}]_{8}$-$S$ wave
-0.00 11 $DD^{*}+D^{*}D^{*}+D_{1}^{*}(D)D_{1}^{*}(D)$-$S$ wave -0.01 34
$DD^{*}+D^{*}D^{*}+[D_{1}^{{}^{\prime}}D_{1}^{*}(P)]_{8}$-$S$ wave -1.05 12
$DD^{*}+D^{*}D^{*}+D_{0}^{*}D_{1}^{\prime}$-$D$ wave -0.09 35
$DD^{*}+D^{*}D^{*}+[D_{1}^{*}(P)D_{1}^{*}(P)]_{8}$-$S$ wave -0.45 13
$DD^{*}+D^{*}D^{*}+D_{0}^{*}D_{1}^{*}(P)$-$D$ wave -0.05 36
$DD^{*}+D^{*}D^{*}+[D_{1}^{*}(D)D_{1}^{*}(D)]_{8}$-$S$ wave -0.01 14
$DD^{*}+D^{*}D^{*}+DD^{*}$-$D$ wave -0.54 37
$DD^{*}+D^{*}D^{*}+[D_{0}^{*}D_{1}^{{}^{\prime}}]_{8}$-$D$ wave -0.04 15
$DD^{*}+D^{*}D^{*}+D_{1}^{\prime}D_{1}^{\prime}$-$D$ wave -0.01 38
$DD^{*}+D^{*}D^{*}+[D_{0}^{*}D_{1}^{*}(P)]_{8}$-$D$ wave -0.02 16
$DD^{*}+D^{*}D^{*}+D_{1}^{\prime}D_{1}^{*}(P)$-$D$ wave -0.01 39
$DD^{*}+D^{*}D^{*}+[DD^{*}]_{8}$-$D$ wave -0.02 17
$DD^{*}+D^{*}D^{*}+D^{*}D^{*}$-$D$ wave -0.02 40
$DD^{*}+D^{*}D^{*}+[D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}]_{8}$-$D$ wave
-0.00 18 $DD^{*}+D^{*}D^{*}+D_{2}^{\prime}D^{*}$-$D$ wave -0.29 41
$DD^{*}+D^{*}D^{*}+[D_{1}^{{}^{\prime}}D_{1}^{*}(P)]_{8}$-$D$ wave -0.06 19
$DD^{*}+D^{*}D^{*}+D_{2}^{*}D^{*}$-$D$ wave -0.04 42
$DD^{*}+D^{*}D^{*}+[D_{1}^{*}(P)D_{1}^{*}(P)]_{8}$-$D$ wave -0.05 20
$DD^{*}+D^{*}D^{*}+D_{2}^{*}(D)D^{*}$-$D$ wave -0.05 43
$DD^{*}+D^{*}D^{*}+[D^{*}D^{*}]_{8}$-$D$ wave -0.01 21
$DD^{*}+D^{*}D^{*}+D_{2}^{*}(P)D_{1}^{\prime}$-$D$ wave -0.02 44
$DD^{*}+D^{*}D^{*}+[D_{2}^{{}^{\prime}}D^{*}]_{8}$-$D$ wave -0.06 22
$DD^{*}+D^{*}D^{*}+D_{2}^{*}(P)D_{1}^{*}(P)$-$D$ wave -0.01 45
$DD^{*}+D^{*}D^{*}+[D_{2}^{*}(D)D^{*}]_{8}$-$D$ wave -0.17 23
$DD^{*}+D^{*}D^{*}+D_{2}^{*}(P)D_{2}^{*}(P)$-$D$ wave -0.03 46
$DD^{*}+D^{*}D^{*}+[D_{2}^{*}(P)D_{1}^{{}^{\prime}}]_{8}$-$D$ wave -0.00 47
$DD^{*}+D^{*}D^{*}+[D_{2}^{*}(P)D_{1}^{*}(P)]_{8}$-$D$ wave -0.03 48
$DD^{*}+D^{*}D^{*}+[D_{2}^{*}(P)D_{2}^{*}(P)]_{8}$-$D$ wave -0.01
$E_{1}$=$E_{0}$+$\Delta E_{28}$+$\Delta E_{29}$=3870.66
$E_{2}$=$E_{0}$+$\sum\limits_{i=1}^{23}\Delta E_{i}$=3868.31
$E_{3}$=$E_{0}$+$\sum\limits_{i=1}^{48}\Delta E_{i}$=3863.75:

Table 7: Contributions to the binding energy from all possible channels in the
$T_{cc}$ system.

For the higher order component of $T_{cc}$ system, we only consider three
cases: the quantum numbers of $J_{12}$ of the two clusters are 0, 1, and 2,
and the orbital quantum numbers of the intercluster motion $L_{3}$ are 1, 0,
and 2, respectively. Other $J_{12}\times L_{3}$ combinations are not
considered in this article due to computational complexity. For the first case
$J_{12}=0$, $L_{3}=1$, we can see that in the table, there are only 4 physical
channels with intergroup orbits are $P$-wave due to the limitation of
symmetry. They are $D^{*}D_{1}^{\prime}$, $D^{*}D_{1}^{*}$,
$D_{2}^{\prime}D_{2}^{*}$ and $D_{2}^{*}(D)D_{2}^{*}(P)$. The two physical
channels, $D^{*}D_{1}^{\prime}$ and $D^{*}D_{1}^{*}$, contribute relatively
large to the bound energy of the $T_{cc}$ system, about -0.1 MeV. While the
other two physical channels, $D_{2}^{\prime}D_{2}^{*}$ and
$D_{2}^{*}(D)D_{2}^{*}(P)$, because they are highly excited states, their
contribution to the bound energy of the $T_{cc}$ system is much smaller, about
-0.03 MeV. For the second case of $J_{12}=1$, $L_{3}=0$, there are only seven
physical channels with $S$-wave between the clusters due to the symmetry. Also
due to the $S$-wave, their contribution to the binding energy of the $T_{cc}$
system is obviously larger. The largest contribution comes from the
$D_{0}^{*}D_{1}^{*}(P)$, which reaches -1.03 MeV. The contribution of the two
physical channels containing $D$-wave, $D^{*}D_{1}^{*}(D)$ and
$D_{1}^{*}(D)D_{1}^{*}(D)$, is almost zero. The contribution of other channels
is within -0.3 MeV. In the third case, $J_{12}=2$, $L_{3}=2$, there are 12
different physical channels with $D$-wave between the clusters. Because they
are $D$-wave between the clusters, most of the physical channels have
relatively small contributions to the bound state of the $T_{cc}$ system, with
absolute values less than 0.1 MeV. But there are two physical channels with
large contributions, which are $D$-wave excitation of $DD^{*}$ and
$D^{*}D^{*}$ . Because of the large $S$-$D$ coupling effect, they respectively
contribute -0.54 MeV and -0.29 MeV to the bound state of the $T_{cc}$ system.
The 12 physical channels with $D$-wave between the clusters contribute a total
of -1.15 MeV to the bound state of the $T_{cc}$ system. Combined with the
contributions of all the 7 physical channels with S wave between the clusters,
4 physical channels with P wave between the clusters, and 12 physical channels
with $D$-wave between the clusters, the lowest energy of the $T_{cc}$ system
can be suppressed to 3868.31 MeV ($E_{2}$=$E_{0}$+$\sum_{i=1}^{23}\Delta
E_{i}$=3868.31 MeV).

For the contribution of the color-octet state, we only consider two channels,
namely, $[DD^{*}]_{8}$ and $[D^{*}D^{*}]_{8}$. The contribution of the two
color-octet channels is relatively large, reaching -0.37 MeV and -0.56 MeV
respectively. Combined with the original color-singlet energy of
$DD^{*}+D^{*}D^{*}$, the energy of the system is $T_{cc}$ is 3870.66 MeV. We
can find that the contribution of the high-order excited states and the color-
octet states are almost the same. It is worth noting that the contribution of
the two color-octet channels, $[DD^{*}]_{8}$ and $[D^{*}D^{*}]_{8}$, and the
contribution of $D$-wave of $DD^{*}$ and $D^{*}D^{*}$ is almost the same,
which is about -0.9 MeV. Therefore, we believe that the contribution of the
color-octet channel can be replaced by the high-order excited states in the
molecular state.

###  III.2 Equivalence between diquark structure and molecular structure

The $T_{cc}$ system has two significant diquark channels, namely
$\bar{3}\times 3$ and $6\times\bar{6}$. In the quark models, the energy
associated with $\bar{3}\times 3$ typically lies below the threshold $DD^{*}$.
In this study, it is observed that $\bar{3}\times 3$ possesses an intrinsic
energy of 3709 MeV, representing a bound state with a binding energy of 163
MeV in Table 8. After channel coupling, the coupling energy decreases to 3705
MeV and its total binding energy increases to 167 MeV.

Table 8: Calculation results for the diquark structure. (unit: MeV) $|[LS]_{i}F_{j}C_{k}\rangle$ | Channel | $E$ | $E^{Theo}_{th}$ | $E_{B}$ | $E^{Exp}_{th}$
---|---|---|---|---|---
$|5,2,4\rangle$ |  $[cc]_{6}^{0}[\bar{q}\bar{q}]_{\bar{6}}^{1}$-$S$ wave | 4134.6 | 3872.71 | 0 | 3871.69
$|8,2,3\rangle$ |  $[cc]_{\bar{3}}^{1}[\bar{q}\bar{q}]_{3}^{1}$-$S$ wave | 3709.3 | 3872.71 | 163.4 | 3871.69
$E([cc]_{6}^{0}[\bar{q}\bar{q}]_{\bar{6}}^{1}+[cc]_{\bar{3}}^{1}[\bar{q}\bar{q}]_{3}^{1}$) | 3705.4 | 3872.71 | 167.3 |

In the molecular structure of $T_{cc}$, there are 25 color singlet states and
25 corresponding color octet states. They can be categorized based on the
difference in inter-subgroups orbital functions: $S$-wave, $P$-wave, and
$D$-wave. As discussed previously, the channel coupling result for all color
singlet states is 3868.31 MeV with a binding energy of 3.8 MeV. Therefore, we
only need to consider the contributions of all $S$-, $P$-, and $D$-wave color
octet states. The contribution from four color octet states with $P$ wave
orbital function is $\Delta E_{24}=-0.17$ MeV, $\Delta E_{25}=-0.34$ MeV,
$\Delta E_{26}=-0.02$ MeV, $\Delta E_{27}=-0.10$ MeV respectively.Indices from
27 to 36 in Table 7 correspond to the contributions of $S$-wave color octet
states. Due to the $S$-wave nature of inter-cluster motion, the contributions
of these nine channels are relatively bigger. With the exceptions of
$[D^{*}D_{1}^{*}(D)]_{8}$, $[D_{1}^{{}^{\prime}}D_{1}^{{}^{\prime}}]_{8}$, and
$[D_{1}^{*}(D)D_{1}^{*}(D)]_{8}$, whose contributions to the binding energy
are approximately zero, the remaining six channels contribute to the system’s
binding energy in the range of -0.4 to -1.0 MeV. The contribution from color
octets with $D$ wave orbital function has little impact on the binding energy,
as their absolute value for $\Delta E_{i}$ is less than 0.2 MeV. Finally, the
lowest energy of the $T_{cc}$ system is finally determined to be
$E_{3}$=3863.75 MeV, taking into account all excited color singlets and color
octet molecular channels.

According to the aforementioned discussion, it is challenging to conclude that
the diquark structure is equivalent to the molecular state structure. There
are two potential reasons for this assertion. Firstly, due to the intricacy of
calculations, it becomes arduous to consider additional molecular state
excitation channels in these computations. Although highly excited states’
physics contributes minimally to bound states of the system, a significant
contribution always exists when there is a large number of highly excited
states. However, we do not believe that relying on highly excited states’
physics can sufficiently reduce the minimum energy of the $T_{cc}$ system
close to that of diquarks. Secondly, while equivalence between diquark and
molecular state structures is derived from a wave function perspective,
determining direct parameters for diquarks in Hamiltonian remains unfeasible.
This discrepancy arises because parameters in Hamiltonian for molecular states
can be directly determined based on experimental discoveries related to
mesons; however, such determination cannot be applied directly towards
establishing parameters for diquarks in Hamiltonian as they only pertain to
molecular state parameters. All these factors may lead us towards concluding
that both diquark and molecular structures are indeed equivalent.

##  IV Summary

In the framework of the chiral quark model, we have investigated the
equivalence between different quark structures in our system with help of
Gaussian expansion method. This includes exploring whether the influence of
color octet states on the ground state can be replaced by high-excited states
of color singlet, and the equivalence between diquark and dimeson structures.
Due to the complexity in handling the wave functions of highly excited states,
the accumulation approach is employed in this paper to calculate the binding
energies of various structures.

The calculation results show that the energies coupled for the two $S$-wave
states, $DD^{*}$ and $D^{*}D^{*}$, and their corresponding color-octet states,
$[DD^{*}]_{8}$ and $[D^{*}D^{*}]_{8}$, amount to 3870.66 MeV. When considering
all possible color-singlet excited states, the energy is calculated to be
3868.31 MeV. The results from these two cases are approximately equal,
implying that the impact of color-octet states on the ground state is
comparable to that of color-singlet excited states.

The lowest energy of the diquark structure in the $T_{cc}$ system is 3705.4
MeV, with a binding energy of 167 MeV. We considered all possible excited
states in the dimeson structure, including color singlet and color octet
states. The calculational results show a binding energy of 3863.7 MeV.
Although the binding energy is further reduced, it is still significantly
higher than the energy of the diquark structure. This result could stem from
the limited consideration of highly excited states. Additionally, it is
possible that errors may arise in the calculation of the diquark structure
because all parameters in the Hamiltonian are directly determined by the meson
spectrum.

In summary, we believe that in the dimeson structure, the contribution of
color octet states can be replaced by highly excited states of color singlet.
Despite considering a sufficient number of bases, the diquark structure cannot
be entirely replaced by the dimeson structure.

###### Acknowledgements.

This work is supported partly by the National Natural Science Foundation of
China under Grant Nos. 12205249, 12205125, 11675080, 11775118 and 11535005 by
the Natural Science Foundation of Jiangsu Province (BK20221166), and the
Funding for School-Level Research Projects of Yancheng Institute of Technology
(No. xjr2022039).

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