$\displaystyle\langle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}\rangle=\left(\begin{array}[]{cc}\frac{8}{3}&0\\\
0&8\\\ \end{array}\right).$ (737)

Here, the eigenvalue of the flavor $\mathbf{{15}^{\prime}}$ multiplet to the
$C^{F}_{4}$ is 16/3. We can also obtain the same values as those of Eq. (737)
through Eq. (353), because the Young-Yamanouchi bases in Eq. (734) obtained
from the coupling scheme belong to the multiplets of $SU(6)_{CS}$
representation, $[21^{2}]$ for particles 1-4. These states have an eigenvalue
of 26/3 in Table 1, as the eigenstates of the quadratic Casimir operator,
$C^{CS}_{4}$. These are given as follows:

$\displaystyle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\psi_{1}\rangle=(-4\times
26/3+2\times 4/3+$ $\displaystyle 4/3\times 1\times
2+32)|\psi_{1}\rangle=8/3|\psi_{1}\rangle,$
$\displaystyle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\psi_{2}\rangle=(-4\times
26/3+2\times 4/3+$ $\displaystyle 4/3\times 2\times
3+32)|\psi_{2}\rangle=8|\psi_{2}\rangle.$ (738)

On the one hand, from the point of view of the $SU(6)_{CS}$ representation of
pentaquark, both the $[32^{2}1^{2}]$ and $[1^{3}]$ in the $SU(6)_{CS}$
representation conjugate to the flavor $\mathbf{{15}^{\prime}}$ multiplet for
the four quarks contain the state which are in both the color singlet and
$S=3/2$ state, as shown in Table 4 and Table 5 and Eq. (349). Therefore,
another approach based on the $SU(6)_{CS}$ representation in Eq. (349) gives
the two orthonormal flavor $\otimes$ color $\otimes$ spin states for $S=3/2$
which satisfy fully antisymmetry in the same way as in the coupling scheme:

$\displaystyle|\Psi_{1}\rangle=$
$\displaystyle\frac{1}{\sqrt{3}}\bigg{(}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&3\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}}-$ (744)
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}}+$ (750)
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&3&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}}\bigg{)},$
(756)
$\displaystyle|\Psi_{2}\rangle=\frac{1}{\sqrt{3}}\bigg{(}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&3\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}}-$ (762)
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}}+\begin{tabular}[]{|c|c|c|}\hline\cr
1&3&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}}\bigg{)}.$ (773)

It should be noted that both $[32^{2}1^{2}]$ and $[1^{3}]$ multiplets in Eq.
(773) are eigenstates of $C^{CS}_{5}$ of the $SU(6)_{CS}$ representation.
Therefore, we can see from Table 5 that the following relations hold:

$\displaystyle C^{CS}_{5}|\Psi_{1}\rangle=49/4|\Psi_{1}\rangle,\quad\quad
C^{CS}_{5}|\Psi_{2}\rangle=21/4|\Psi_{2}\rangle.$ (774)

Due to the situation of a similar kind that occurs in the case of $S=1/2$, the
approach used in the previous subsection can also be applied to this case to
find a correspondence between Eq.(734) and Eq.(773). This can be achieved by
solving an algebraic problem involving a coupled equation with two unknown
variables. Specifically, we obtain the following relation:

$\displaystyle\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}}=\sqrt{\frac{5}{7}}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{1}}+\sqrt{\frac{2}{7}}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{2}},$ (784)
$\displaystyle\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}}=-\sqrt{\frac{2}{7}}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{1}}+\sqrt{\frac{5}{7}}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{1}}.$ (794)

The same procedure holds good for the rest of Young-Yamanouchi bases of
$[32^{2}1^{2}]$ ($[1^{3}]$). We can then find at once the relation between the
states in Eq. (734) and Eq. (773), by using Eq. (794), as the following:

$\displaystyle|\Psi_{1}\rangle=\sqrt{\frac{5}{7}}|\psi_{1}\rangle+\sqrt{\frac{2}{7}}|\psi_{2}\rangle,|\Psi_{2}\rangle=-\sqrt{\frac{2}{7}}|\psi_{1}\rangle+\sqrt{\frac{5}{7}}|\psi_{2}\rangle.$
(795)

We can now straightforwardly calculate the matrix element of Eq. (351) in
terms of $|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ in Eq. (773). Using Eq.
(795), we find the following 2 by 2 matrix elements:

.
$\displaystyle\langle\Psi_{1}|-{\sum}_{i<j}^{5}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi_{1}\rangle=4\times
49/4-8\times 26/3-$ $\displaystyle 2\times 0+4\times 4/3-4/3\times 3/2\times
5/2+8/3(5/7\times 2+$ $\displaystyle 2/7\times 2\times 3)+24=260/21,$
$\displaystyle\langle\Psi_{2}|-{\sum}_{i<j}^{5}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi_{2}\rangle=4\times
21/4-8\times 26/3-$ $\displaystyle 2\times 0+4\times 4/3-4/3\times 3/2\times
5/2+8/3(2/7\times 2+$ $\displaystyle 5/7\times 2\times 3)+24=-232/21,$
$\displaystyle\langle\Psi_{1}|-{\sum}_{i<j}^{5}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi_{2}\rangle=8/3\times(-\sqrt{10}/\sqrt{49}\times
2$ $\displaystyle+\sqrt{10}/\sqrt{49}\times 2\times 3)=32\sqrt{10}/21$ (796)

By diagonalizing the matrix, we find that the eigenvalue of Eq. (351) is
either -12 or 40/3.

We also examine the expectation value of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$ from
the symmetry property. Using Eq. (353) and Eq. (795), one can easily show that
the expectation value of Eq. (353) for particles 1-4 is given in terms of
$|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ in Eq. (773) as follows.

$\displaystyle\langle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}\rangle=\left(\begin{array}[]{cc}\frac{88}{21}&\frac{16\sqrt{10}}{21}\\\
\frac{16\sqrt{10}}{21}&\frac{136}{21}\\\ \end{array}\right).$ (799)

Then, the matrix in Eq. (799) is diagonalized to give the eigenvalues of 8 and
8/3, which are the same as those obtained from Eq. (737). The antisymmetry
property for particles 1-4 makes it possible to calculate the expectation
value of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$
($i<j$ =1, 2, 3, 4), immediately resulting from Eq. (799), as follows:

$\displaystyle\langle-\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}\rangle=\left(\begin{array}[]{cc}\frac{44}{63}&\frac{8\sqrt{10}}{63}\\\
\frac{8\sqrt{10}}{63}&\frac{68}{63}\\\ \end{array}\right).\quad(i<j=1,2,3,4).$
(802)

In a similar manner, we can calculate the expectation value of
$\lambda_{i}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{5}$ ($i$
=1, 2, 3, 4) in terms of $|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ in Eq.
(773), using Eq. (796) and Eq. (799), and the fully antisymmetric property for
the four quarks:

$\displaystyle\langle-\lambda_{i}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{5}\rangle=\left(\begin{array}[]{cc}\frac{43}{21}&\frac{4\sqrt{10}}{21}\\\
\frac{4\sqrt{10}}{21}&-\frac{92}{21}\\\ \end{array}\right).\quad(i=1,2,3,4).$
(805)

For the the flavor $\mathbf{3}$ multiplet, there is one fully antisymmetric
flavor $\otimes$ color $\otimes$ spin state coming from the coupling scheme in
Eq. (253), which is given as:

$\displaystyle|\psi\rangle=\frac{1}{\sqrt{3}}\bigg{(}$
$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|c|}\hline\cr
1&3&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{1}}$ (811)
$\displaystyle-\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{1}}$ (817)
$\displaystyle+\begin{tabular}[]{|c|c|}\hline\cr 1&4\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&3\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{3/2};S^{1}}\bigg{)},$ (823)

where the spin state for particles 1-4 is 1. Since the state in Eq. (823) is
an eigenstate of Eq. (352) for particles 1-4, we find from Eq. (352) the
following equation:

$\displaystyle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\psi\rangle=-40/3|\psi\rangle,$
(824)

where the eigenvalue of the flavor $\mathbf{3}$ to $C_{4}^{F}$ is 4/3. We can
also obtain the same result of Eq. (824) through Eq. (353), because the Young-
Yamanouchi bases in color and spin part of Eq. (823) obtained from the
coupling scheme belong to the multiplets of $SU(6)_{CS}$ representation,
$[31]$ for particles 1-4. Then, by using Eq. (353), we obtain the following:

$\displaystyle-$
$\displaystyle{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\psi\rangle=(-4\times
38/3+2\times 4/3+$ $\displaystyle 4/3\times 1\times
2+32)|\psi\rangle=-40/3|\psi\rangle,$ (825)

On the one hand, from the point of view of the $SU(6)_{CS}$ representation of
pentaquark, the $[421^{3}]$ in the $SU(6)_{CS}$ representation conjugate to
the flavor $\mathbf{3}$ multiplet for the four quarks contains the state which
is in both the color singlet and $S=3/2$ state, as shown in Table 4 and Table
5 and Eq. (349). Also, it has the symmetry property for $q^{4}$ corresponding
to Young diagram $[21^{2}]$. Therefore, another approach based on the
$SU(6)_{CS}$ representation in Eq. (349) gives a flavor $\otimes$ color
$\otimes$ spin state for $S=3/2$ which satisfies fully antisymmetry in the
same way as in the coupling scheme:

$\displaystyle|\Psi\rangle=\frac{1}{\sqrt{3}}\bigg{(}$
$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&3&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{3/2}}-$ (831)
$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{3/2}}+$ (837)
$\displaystyle\begin{tabular}[]{|c|c|}\hline\cr 1&4\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&3\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{3/2}}\bigg{)}.$ (843)

Since the color-spin parts of Eq. (843) are the eigenstates with an eigenvalue
of 65/4 for $C^{CS}_{5}$, using Eq. (351), we can obtain the following
equation:

$\displaystyle-$
$\displaystyle{\sum}_{i<j}^{5}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi\rangle=(4\times
65/4-8\times 38/3-2\times 0+$ $\displaystyle 4\times 4/3-4/3\times 3/2\times
5/2+8/3\times 1\times 2+24)|\Psi\rangle$ $\displaystyle=-20/3|\Psi\rangle.$
(844)

For the the flavor $\mathbf{\bar{6}}$ multiplet, there is one fully
antisymmetric flavor $\otimes$ color $\otimes$ spin state coming from the
coupling scheme in Eq. (260). This state must be exactly equivalent to that
coming from Young diagram $[3^{2}1^{3}]$ of the $SU(6)_{CS}$ representation of
pentaquark, which is conjugate to the flavor $\mathbf{\bar{6}}$ multiplet, as
can be seen from Table 4, Table 5, and Eq. (349). It has to be so for the
reason that the multiplet of the $[3^{2}1^{3}]$ representation contains the
state in the color singlet and at the same time $S=3/2$ state. The fully
antisymmetric flavor $\otimes$ color $\otimes$ spin state for $S=3/2$ is given
as follows:

$\displaystyle|\Psi\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{(}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr 3&4\\\
\hline\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\
\hline\cr 3&4\\\
\hline\cr\end{tabular}_{CS^{3/2};S^{1}}-\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \hline\cr 2&4\\\
\hline\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\hline\cr 3&4\\\ \hline\cr\end{tabular}_{CS^{3/2};S^{1}}\bigg{)}$ (853)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{(}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr 3&4\\\
\hline\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \hline\cr 2&4\\\
\hline\cr\end{tabular},[3^{2}1^{3}]\big{\rangle}_{CS^{3/2}}$ (858)
$\displaystyle-\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr 2&4\\\
\hline\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr 3&4\\\
\hline\cr\end{tabular},[3^{2}1^{3}]\big{\rangle}_{CS^{3/2}}\bigg{)}.$ (863)

It should be noted that the color $\otimes$ spin states in Eq. (863) become
the eigenstates of $C^{CS}_{4}$ as well as $C^{CS}_{5}$. From either Eq. (353)
or Eq. (352), we find the eigenvalue equation for particles 1-4 as follows:

$\displaystyle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi\rangle=-16/3|\Psi\rangle.$
(864)

Moreover, we straightforwardly find the eigenvalue equation of Eq. (351) as
follows:

$\displaystyle-{\sum}_{i<j}^{5}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi\rangle=(4\times
57/4-8\times 32/3-2\times 0$ $\displaystyle+4\times 4/3-4/3\times 3/2\times
5/2+8/3\times 1\times 2+24)|\Psi\rangle$ $\displaystyle=4/3|\Psi\rangle.$
(865)

###  IV.4 the flavor $\otimes$ color $\otimes$ spin states in the case of
$S=5/2$

In the case of $S=5/2$, we have only to consider the flavor
$\mathbf{{15}^{\prime}}$ multiplet, because color $\otimes$ spin states coming
from the coupling scheme are given by Eq. (277) conjugate to the flavor
$\mathbf{{15}^{\prime}}$ states. With the color $\otimes$ spin states of the
coupling scheme, the fully antisymmetric flavor $\otimes$ color $\otimes$ spin
state for $S=5/2$ can be constructed, as follows:

$\displaystyle|\psi\rangle=$
$\displaystyle\frac{1}{\sqrt{3}}\bigg{(}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&3\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|}\hline\cr 1&4\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{5/2};S^{2}}-$ (871)
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{5/2};S^{2}}+\begin{tabular}[]{|c|c|c|}\hline\cr
1&3&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{CS^{5/2};S^{2}}\bigg{)}.$ (882)

This state in Eq. (882) must be exactly equivalent to that coming from Young
diagram $[32^{2}1^{2}]$ of the $SU(6)_{CS}$ representation of the pentaquark,
involving the state which are in the color singlet and $S=5/2$, as can be seen
from Table 4, Table 5, and Eq. (349). The fully antisymmetric flavor $\otimes$
color $\otimes$ spin state for $S=5/2$ coming from the Young diagram
$[32^{2}1^{2}]$ of the $SU(6)_{CS}$ representation of the pentaquark is given
as:

$\displaystyle|\Psi\rangle=\frac{1}{\sqrt{3}}\bigg{(}$
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{5/2}}-$ (888)
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{5/2}}+$ (894)
$\displaystyle\begin{tabular}[]{|c|c|c|}\hline\cr 1&3&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular}_{F}\otimes\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{5/2}}\bigg{)}.$
(900)

The color $\otimes$ spin states coming from the coupling scheme in Eq.(882)
become the eigenstates of $C^{CS}_{4}$. Also, those coming from the Young
diagram $[32^{2}1^{2}]$ of the $SU(6)_{CS}$ representation of pentaquark
become the eigenstates of $C^{CS}_{5}$. These indicate that these color
$\otimes$ spin states are the eigenstates of both Eq.(353) and Eq.(351). From
either Eq. (353) or Eq. (352), we find the eigenvalue equation for particles
1-4 as follows:

$\displaystyle-{\sum}_{i<j}^{4}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi\rangle=8|\Psi\rangle.$
(901)

Moreover, we straightforwardly find the eigenvalue equation of Eq. (351) as
follows:

$\displaystyle-$
$\displaystyle{\sum}_{i<j}^{5}\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}|\Psi\rangle=(4\times
49/4-8\times 26/3-2\times 0$ $\displaystyle+4\times 4/3-4/3\times 5/2\times
7/2+8/3\times 2\times 3+24)|\Psi\rangle$ $\displaystyle=40/3|\Psi\rangle.$
(902)

##  V Hamiltonian

For the purpose of investigating the stability of a pentaquark system against
the strong decay, we present a nonrelativistic Hamiltonian in a constituent
quark model, which includes one gluon exchange potential. This model is an
effective tool for describing a multiquark configuration involving the short-
range interaction of the potentials. The Hamiltonian is given by

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum^{5}_{i=1}\left(m_{i}+\frac{{\mathbf{p}}^{2}_{i}}{2m_{i}}\right)-\frac{3}{4}\sum^{5}_{i<j}\frac{\lambda^{c}_{i}}{2}\,\,\frac{\lambda^{c}_{j}}{2}\left(V^{C}_{ij}+V^{CS}_{ij}\right),\qquad$
(903)

where $m_{i}$ is the quark mass and $\lambda^{c}_{i}/2$ is the SU(3) color
operator for the $i$-th quark. In Hamiltonian in Eq. (903), $V^{C}_{ij}$ and
$V^{CS}_{ij}$ are a confinement and a hyperfine potential, respectively.
$V^{C}_{ij}$ consist of a linear-rising and a coulomb in a potential form, and
$V^{CS}_{ij}$ is chosen to be entirely a delta function in the limit of two
extremely heavy quarks. These are given as the follows:

$\displaystyle V^{C}_{ij}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{r_{ij}}+\frac{r_{ij}}{a^{2}_{0}}-D,$ (904)
$\displaystyle V^{CS}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}c^{2}\kappa^{\prime}}{m_{i}m_{j}c^{4}}\frac{e^{-\left(r_{ij}\right)^{2}/\left(r_{0ij}\right)^{2}}}{(r_{0ij})r_{ij}}\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}.$
(905)

Here

$\displaystyle r_{0ij}$ $\displaystyle=$ $\displaystyle
1/\left(\alpha+\beta\frac{m_{i}m_{j}}{m_{i}+m_{j}}\right),$ (906)
$\displaystyle\kappa^{\prime}$ $\displaystyle=$
$\displaystyle\kappa_{0}\left(1+\gamma\frac{m_{i}m_{j}}{m_{i}+m_{j}}\right),$
(907)

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. (904)-(907) are determined by fitting them to
the experimental value of both mesons and baryons, demanding the requirement
that ${\chi}^{2}$ should be minimized for a number of set of fitting
parametersNoh:Prd2021 . These fitting parameters are given 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}$},\,\,\kappa_{0}=197.144\,\textrm{MeV}.$
(908)

We use a complete set of harmonic oscillator basis as a spatial function to
calculate the masses of mesons and baryons within our quark model. We employ
the variational method to obtain their exact ground states as accurately as
possible. In fact, the variational method for the calculation of the
Hamiltonian in Eq. (903) by the complete set of harmonic oscillator is a tool
of great significance, in a sense that it provides a uniform convergence of
the ground state mass in terms of ideal methodology. The direct consequence is
that in doing so, an exact solution on the part of variational method in a
given quark model enables us to understand the property of the structure of
multi-quark system, such as tetraquark, pentaquark, and so on.

Indeed, one can attain ultimately its convergence of the value, only if a
sufficient number of harmonic oscillator basis is considered as a spatial
function with which the variational method is concerned. To carry out this
with a higher level of precision than in the previous worksNoh:Prd2021 ;
Noh:2023fdy , it is necessary to consider additionally higher quanta of
harmonic oscillator bases. The result is that the ground state becomes very
close to a convergent state to the system. The detailed explanation about the
quanta will be added in the next section.

In our present work, we fit the masses of mesons and baryons using the
variational method and a complete set of harmonic oscillator basis up to the
6th quanta. The fitted values are presented in Tables LABEL:mesons and
LABEL:baryons, respectively, which show slight differences in the fitted
values compared to our previous workNoh:Prd2021 . The overall fitted value to
baryons has improved, resulting in a very slight reduction of the ${\chi}^{2}$
value in this case.

Table 6: The masses of mesons obtained (Column 3) with the fitting parameters set given in Eq. (908). Column 4 shows the variational parameter $a$. Particle | Experimental | Mass | Variational
---|---|---|---
Value (MeV) | (MeV) | Parameter (${\rm fm}^{-2}$)
$D$ | 1864.8 | 1853.83 |  $a$ = 7.5
$D^{*}$ | 2007.0 | 2006.22 |  $a$ = 5.7
$\eta_{c}$ | 2983.6 | 2985.97 |  $a$ = 25.2
$J/\Psi$ | 3096.9 | 3118.36 |  $a$ = 19.7
$D_{s}$ | 1968.3 | 1963.58 |  $a$ = 12.1
$D^{*}_{s}$ | 2112.1 | 2109.16 |  $a$ = 9.3
$K$ | 493.68 | 498.318 |  $a$ = 7.7
$K^{*}$ | 891.66 | 874.664 |  $a$ = 4.1
$B$ | 5279.3 | 5301.22 |  $a$ = 7.3
$B^{*}$ | 5325.2 | 5360.45 |  $a$ = 6.5
$\eta_{b}$ | 9398.0 | 9327.06 |  $a$ = 100.2
$\Upsilon$ | 9460.3 | 9456.56 |  $a$ = 81.9
$B_{s}$ | 5366.8 | 5375.25 |  $a$ = 13.0
$B_{s}^{*}$ | 5415.4 | 5439.34 |  $a$ = 11.5
$B_{c}$ | 6275.6 | 6268.41 |  $a$ = 38.7
$B_{c}^{*}$ | - | 6361.84 |  $a$ = 32.6
Table 7: Same as Table LABEL:mesons but for baryons. In column 4, $a_{1}$ and $a_{2}$ are the variational parameters. Particle | Experimental | Mass |  Variational
---|---|---|---
Value (MeV) | (MeV) |  Parameters (${\rm fm}^{-2}$)
$\Lambda$ | 1115.7 | 1111.21 |  $a_{1}$ = 4.3, $a_{2}$ = 3.9
$\Lambda_{c}$ | 2286.5 | 2269.04 |  $a_{1}$ = 4.4, $a_{2}$ = 5.0
$\Xi_{cc}$ | 3621.4 | 3620.81 |  $a_{1}$ = 11.8, $a_{2}$ = 4.6
$\Lambda_{b}$ | 5619.4 | 5634.20 |  $a_{1}$ = 4.5, $a_{2}$ = 5.6
$\Sigma_{c}$ | 2452.9 | 2438.13 |  $a_{1}$ = 3.0, $a_{2}$ = 5.7
$\Sigma_{c}^{*}$ | 2517.5 | 2523.1 |  $a_{1}$ = 2.7, $a_{2}$ = 4.9
$\Sigma_{b}$ | 5811.3 | 5841.38 |  $a_{1}$ = 2.9, $a_{2}$ = 6.1
$\Sigma_{b}^{*}$ | 5832.1 | 5874.89 |  $a_{1}$ = 2.8, $a_{2}$ = 5.7
$\Sigma$ | 1192.6 | 1191.78 |  $a_{1}$ = 3.0, $a_{2}$ = 5.0
$\Sigma^{*}$ | 1383.7 | 1395.70 |  $a_{1}$ = 2.4, $a_{2}$ = 3.4
$\Xi$ | 1314.9 | 1326.58 |  $a_{1}$ = 4.6, $a_{2}$ = 4.6
$\Xi^{*}$ | 1531.8 | 1540.28 |  $a_{1}$ = 4.3, $a_{2}$ = 2.9
$\Xi_{c}$ | 2467.8 | 2471.60 |  $a_{1}$ = 4.9, $a_{2}$ = 6.6
$\Xi_{c}^{*}$ | 2645.9 | 2649.58 |  $a_{1}$ = 3.5, $a_{2}$ = 6.3
$\Xi_{b}$ | 5787.8 | 5823.74 |  $a_{1}$ = 5.0, $a_{2}$ = 7.9
$\Xi_{b}^{*}$ | 5945.5 | 5988.91 |  $a_{1}$ = 3.6, $a_{2}$ = 7.8
$p$ | 938.27 | 936.67 |  $a_{1}$ = 2.8, $a_{2}$ = 2.8
$\Delta$ | 1232 | 1242.1 |  $a_{1}$ = 1.9, $a_{2}$ = 1.9

##  VI Wave Function

In this section, we discuss about the wave function of the pentaquark,
primarily involving the spatial function. In our present paper, we deal with a
pentaquark system consisting of $u(1)d(2)c(3)c(4)\bar{s}(5)$, which is in $S$
wave for the total angular momentum, $L$, $S=1/2$ for the total spin, and
$I=0$ for the isospin. Generally, in setting up the Hamiltonian by means of a
spatial function in the form of a Gaussian, we adopt Jacobian coordinates in
the center of mass frame. For the pentaquark system, there are several sets of
four Jacobi coordinates available, depending on the method of connecting the
position vectors of the constituent quarks. In our study, we choose a specific
set of Jacobi coordinates that provides an effective description for our
purpose to investigate the stability of the pentaquark configuration. These
are given as follows:

$\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}),\
$
$\displaystyle\mathbf{x}_{3}=\sqrt{\frac{2}{3}}\left({\mathbf{r}}_{5}-\frac{1}{2}{\mathbf{r}}_{1}-\frac{1}{2}{\mathbf{r}}_{2}\right),\,$
$\displaystyle\mathbf{x}_{4}=\mu\left(\frac{m_{u}{\mathbf{r}}_{1}+m_{u}{\mathbf{r}}_{2}+m_{s}{\mathbf{r}}_{5}}{m_{u}+m_{u}+m_{s}}-\frac{m_{c}{\mathbf{r}}_{3}+m_{c}{\mathbf{r}}_{4}}{m_{c}+m_{c}}\right),$
$\displaystyle\mu=\frac{\sqrt{2}m_{c}(m_{s}+2m_{u})}{m_{c}\sqrt{3{m_{s}}^{2}+4m_{s}m_{u}+8{m_{u}}^{2}}}.$
(909)

In order to obtain a more precise value for a solution of the Hamiltonian in
Eq. (903), it is essential to understand a three dimensional harmonic
oscillator system by using the variational method. One of the most important
features which are found in a complicated system like our own is that one
should manifestly use the wave function of three dimensional harmonic
oscillator system for describing the exact solution, and for investigating the
property of its structure.

The wave function of a three-dimensional harmonic oscillator system is well
known as consisting of the associated Laguerre polynomials for a radial part
and the spherical harmonics for the angular part:

$\displaystyle\psi(r,\theta,\phi)=R(r)_{n,l}Y^{m}_{l}(\theta,\phi)$
$\displaystyle=\sqrt{\frac{2\,\Gamma(n+1)}{\Gamma\left(n+l+\frac{3}{2}\right)}}\,r^{l}\exp\left[-\frac{r^{2}}{2}\right]L^{l+\frac{1}{2}}_{n}\left(r^{2}\right)Y^{m}_{l}(\theta,\phi).$
(910)

However, in applying this to the variational method, it is necessary to
rescale the argument of the exponential in the radial part to become the wave
function appropriate for our scheme. This is achieved by replacing the radial
distance $r$ with $\sqrt{2a}\,x$, where $x$ is the magnitude of the position
vector similar to those in Eq. (909), and $a$ is the variational parameter.
Unlike the wave function in Eq. (910), the modified wave function in a three
dimensional harmonic oscillator system is useful to deal with a multiquark
system such as $T_{cc}$ through the variational method. In addition,
introducing several variational parameters corresponding to the Jacobian
coordinates is the best optimum tool for making such a system converge faster.
Finally, for the main purpose of carrying out the variational method, we
search for the variational parameter which gives a minimal value of the
diagonalized Hamiltonian represented by the complete set of harmonic
oscillator bases.

In particular, concerning the convergence, it is convenient to use the concept
of quanta, which appears in the expectation value of kinetic term with respect
to the complete set of harmonic oscillator bases. In the extreme case of which
the constituent quarks and all the variational parameters are identical, the
expectation value of kinetic term increases directly as the value of quanta
increases. The diagonal term, denoted by $\langle T_{c}\rangle$, is given by:

$\displaystyle\langle T_{c}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}c^{2}a}{m}\bigg{[}\left(2n_{1}+l_{1}+\frac{3}{2}\right)+\left(2n_{2}+l_{2}+\frac{3}{2}\right)+$
(911)
$\displaystyle\left(2n_{3}+l_{3}+\frac{3}{2}\right)+\left(2n_{4}+l_{4}+\frac{3}{2}\right)\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\hbar^{2}c^{2}a}{m}\bigg{[}2n_{1}+l_{1}+2n_{2}+l_{2}+2n_{3}+l_{3}+$
$\displaystyle 2n_{4}+l_{4}+6\bigg{]}.$

Here, $m_{1}=m_{2}=m_{3}=m_{4}=m_{5}\equiv m$, and
$a_{1}=a_{2}=a_{3}=a_{4}\equiv a$. For a given set of quantum numbers,
$(n_{1},n_{2},n_{3},n_{4},l_{1},l_{2},l_{3},l_{4})$ that specifies the spatial
functions of the pentaquark, the quanta defined in our previous
workNoh:Prd2021  can be expressed as:

$\displaystyle Q\equiv 2n_{1}+2n_{2}+2n_{3}+2n_{4}+l_{1}+l_{2}+l_{3}+l_{4}$
(912)

It shows that the diagonal term in Eq. (911) remains unchanged for the same
quanta, even though there are enumerable sets of the quantum numbers that
comprise the same quanta. Moreover, in this concept of quanta, it is
remarkable that the value of the Hamiltonian tends to decrease markedly with
each step of increasing quanta. Thus, instead of counting up all the spatial
functions, using quanta as a criterion to gauge the extent of convergence is
more convenient.

In our investigation of $T_{cc}$, composed of two light quarks and two heavy
antiquarks, we observed a noticeable decrease in the mass of the system with
each quanta, leading to its convergence of the ground state. This finding
highlights the usefulness of the quanta as a standard for analyzing the
convergence process. Therefore, in our present work, we employ the concept of
quanta to facilitate the analysis of the entire convergence process.

Now, for the first time, to investigate the existence of the pentaquark and
enhance the precision of calculating the Hamiltonian in a variational method,
we use the complete set of harmonic oscillator basis, which can be generally
expanded in terms of three Jacobian coordinates in Eq.(909). In particular, in
the absence of any internal orbital angular momentum, the complete set of
harmonic oscillator for $S$ wave can be generated by Laguerre polynomials in
the form of the square of a single Jacobian coordinate. In this case, the
simplest spatial function of the pentaquark is given by

$\displaystyle\frac{8{a_{1}}^{3/4}{a_{2}}^{3/4}{a_{3}}^{3/4}{a_{4}}^{3/4}}{{\pi}^{3}}\times$
$\displaystyle{\rm
exp}\big{[}-(a_{1}{\mathbf{x}_{1}}^{2}+a_{2}{\mathbf{x}_{2}}^{2}+a_{3}{\mathbf{x}_{3}}^{2}+a_{4}{\mathbf{x}_{4}}^{2})\big{]},$
(913)

where $a_{i}$($i$=1, 2, 3, 4) is the variational parameter. This spatial
function in Eq. (913), where each quantum number $n_{i}=0$ in the Laguerre
polynomials, gives relatively the largest contribution to the value of the
mass of the pentaquark. This only represents the lowest quanta.

The next highest contribution to the mass of the pentaquark comes from a
spatial function obtained by increasing the quantum number, $n_{i}$, of the
Laguerre polynomials in any one of the four by one. In this case, the value of
quanta $Q$ in Eq. (912) is equal to 2.

On the other hand, given two non-zero internal orbital angular momentum
associated with any two Jacobian coordinates, it is possible to construct the
spatial function of the pentaquark for $S$ wave from a general theory of
angular momentum addition. Such a spatial function plays an important role in
inducing the value of the Hamiltonian to converge uniformly. We also consider
the possible combination of $L=0$ of the pentaquark among any non-zero four
internal orbital angular momenta, which in some case have complicated patterns
to establish.

Nonetheless, it is easy to build the spatial function of the pentaquark for
the $S$ wave, as these wave functions are invariant forms under a rotation
group whose types are scalar, such as $\mathbf{x}_{1}\cdot\mathbf{x}_{2}$,
$\mathbf{x}_{1}\cdot\mathbf{x}_{3}$, $\mathbf{x}_{2}\cdot\mathbf{x}_{3}$,
$\mathbf{x}_{1}\cdot(\mathbf{x}_{2}\times\mathbf{x}_{3})$, and etc., in a
three dimensional vector space. For instance, in the case of
$\mathbf{x}_{1}\cdot\mathbf{x}_{2}$, where both $l_{1}$ and $l_{2}$ are equal
to 1, the value of quanta $Q$ in Eq. (912) is equal to 2. Apart from such
scalars as the inner product between two Jacobian coordinates, there remains
another kind of scalar,
$\mathbf{x}_{1}\cdot(\mathbf{x}_{2}\times\mathbf{x}_{3})$, where $l_{1}$,
$l_{2}$, and $l_{3}$ are all equal to 1, respectively. In this case, the value
of quanta $Q$ in Eq. (912) is equal to 3. However, since such scalars do
contribute little to the value of the off-diagonal Hamiltonian matrix element,
the kind of spatial function which contains those types in the angular part is
completely neglected. Thus, the quanta only takes on even values.

In an analogy with the case of color $\otimes$ spin states represented by the
$SU(6)_{CS}$ representation, the spatial function of the pentaquark for the
$S$ wave has a certain symmetry between two light quarks and two heavy quarks,
respectively. From the Jacobian coordinates in Eq. (909), we find the
following symmetry properties for the Jacobian coordinates: $\mathbf{x}_{1}$
is antisymmetric for the two light quarks and $\mathbf{x}_{2}$ is
antisymmetric for the two heavy quarks. $\mathbf{x}_{3}$ is symmetric for the
two light quarks and $\mathbf{x}_{4}$ is symmetric for the two light quarks
and for the two heavy quarks, respectively. From the standpoint of the quantum
number, all the possible spatial functions characterized by only a quantum
number, $n_{i}$ ($i$=1, 2, 3, 4), are symmetric between two light quarks and
two heavy quarks, respectively, because the power of the Jacobian coordinate
corresponding to the quantum number $n_{i}$ in the Laguerre polynomials is
always even. However, for the spatial functions of the pentaquark involved in
the quantum number, $l_{i}$ ($i$=1, 2, 3, 4), the symmetry properties depend
entirely upon whether $l_{i}$ is even or odd. This is because the increase in
$l_{i}$ is attended by the same increase in the number of the power of the
corresponding Jacobian coordinate. In general, we can easily deduce this rule
from the condition where the spatial function should be a scalar form for a
given relative orbital angular momentum quantum number.

We are now in a position to organize the full wave function for the pentaquark
consisting of $u(1)d(2)c(3)c(4)\bar{s}(5)$, which is used to calculate the
Hamiltonian in Eq. (903) using the variational method. Since the Pauli
principle restricts the full wave function to be antisymmetric between the two
light quarks and the two heavy quarks, respectively, we have to choose the
full wave function that satisfies these symmetries. Such full wave functions
can be constructed straightforwardly by combining the symmetry properties of
both the color $\otimes$ spin states and the spatial functions.

In particular, for the symmetry between the third and fourth quarks, there are
some considerations regarding the color $\otimes$ spin states obtained from
either the coupling scheme or the $SU(6)_{CS}$ representation. One can
determine definitely the symmetry property between the first and second
quarks, but not that between the third and fourth quarks: for examples, as can
be seen in Eq. (175), we cannot determine the symmetry between the third and
fourth quarks for the second and third Young-Yamanouchi bases in Eq. (175)
since they are not neighboring. However, the symmetry between the third and
fourth quarks can be directly determined for the first Young-Yamanouchi basis
since they are neighboring.

Fortunately, it is well-known that the method for constructing either symmetry
or antisymmetry between two non-neighboring particles is to combine two
related Young-Yamanouchi bases. For example, the symmetries between the third
and fourth quarks depend entirely on how to combine the first and second
states in Eq. (VII.1.41).

##  VII Numerical Analysis

In this section, we analyze the pentaquark system consisting of
$u(1)d(2)c(3)c(4)\bar{s}(5)$ with a total spin of $S=1/2$ and isospin $I=0$ in
the $S$ wave. We investigate the convergence of the expectation value of the
Hamiltonian with each quanta and also discuss the numerical results regarding
the stability of the system against strong decay into a baryon and a meson.

Calculating the pentaquark system is technically more challenging than a
tetraquark due to the vast number of numerical inputs required. Our approach
method makes it possible to achieve a precise assessment of the calculation
despite these technical difficulties. However, this technical method may still
be a decisive factor in obtaining the exact solution for the pentaquark
system.

In addition, according to our workNoh:2023fdy , it should be noted that the
study for the tetraquarks shows that the structure depends critically upon the
functional form of the hyperfine potential. In particular, we found that a
Yukawa type, rather than a Gaussian type, is necessary to describe the compact
bound state in $T_{cc}$. Therefore, the former potential should be adopted to
investigate the stability as well as the structure even in the pentaquark
system.

In the pentaquark system, the precise calculation of the Hamiltonian requires
a vast number of the spatial functions of the pentaquark within the 6th
quanta. Specifically, for the first six quanta, the number of spatial
functions in the pentaquark system is estimated to be more than 1800. However,
due to the enormous time-consuming model calculation, we limit it to only 1356
in the actual calculation. This limitation on the number of spatial functions
may lead us to overlook the possibility of at least 3 MeV drop in the mass of
the pentaquark.

On the other hand, in carrying out the expectation value of the Hamiltonian
using the variational method, we have to search for exact four variational
parameters in such a way that minimize the eigenvalue of the diagonalized
Hamiltonian represented by the full wave functions. However, due to the
complexity of the system, it is challenging to achieve this perfectly for
higher quanta. For this reason, we find the exact four variational parameters
only up to the 4th quanta.

In our practical evaluation, we first find the exact four variational
parameters which give the diagonalized Hamiltonian its minimum eigenvalue with
the 4th quanta of the spatial bases. Subsequently, we calculate the eigenvalue
with the 5th quanta and 6th quanta simply by putting the fixed values of the
four variational parameters obtained from the evaluation with the 4th quanta.
However, up to the 5th quanta, we consider all possible spatial functions,
whereas for the 6th quanta, we only select a limited number of spatial
functions that are expected to relatively contribute to the mass value. This
selection of spatial functions allows for reducing the computational burden.

As can be seen in Table 8, we find that there is a tendency for the value of
the mass to decrease noticeably in accordance with each quanta. The result in
Table 8 shows that the bound pentaquark configuration is less likely to be
found within our quark model, because the mass of the pentaquark has +18.5 MeV
above its lowest threshold.

However, as pointed out above, it is highly probable that the value of the
mass could be varied within a range of 3-6 MeV, if our technical limitation in
the evaluation process is entirely eliminated. Nonetheless, the possibility of
the existence of the pentaquark system is not still high due to a repulsion in
the energy of at least 12 MeV.

Besides, in the light of the result in Table 8, it is reasonable to expect
that the system will converge beyond the 6th quanta, perhaps in the 7th quanta
or beyond. In this case, the mass of the pentaquark system would be very close
to its threshold value. However, it is important to note that achieving
convergence in higher quanta may pose additional technical challenges, and it
remains to be seen whether the pentaquark system can be experimentally
observed.

Here, it is necessary to return to the problem of how a Yukawa type hyperfine
potential affects the organization of $T_{cc}$ more attractively than a
Gaussian type hyperfine potential at short range. In the previous
workNoh:Prd2021  that used a Gaussian type hyperfine potential, similar to our
present work, there was a significant discrepancy in the value of the binding
energy compared to experimental results, as the mass of $T_{cc}$ was found to
have about 13 MeV above its threshold. However, there is a remarkable finding
from the use of the Yukawa type hyperfine potential. The $T_{cc}$ in our
studyNoh:2023fdy  forms a compact and bound configuration, with a mass that is
almost identical to the experimentally observed value. Therefore, it is
crucial to consider the use of a Yukawa type potential to obtain more accurate
results.

Since a similar situation will arise in the pentaquark system, what is at
least as certain is that the choice of the Yukawa type potential is most
preferable to that of the Gaussian type potential to achieve our purpose.
Therefore, it is probable that the significant characteristics of the Yukawa
type potential in our quark model lead to a different result from what has
been obtained in this work.

Table 8: Mass of the pentaquark $udcc\bar{s}$ in each step of quanta. The lowest threshold of the pentaquark configuration for $S$=1/2 and $I$=0 is $\Xi_{cc}K$ with a mass of 4119.1 MeV. The masses of $\Xi_{cc}$ and $K$ are obtained from the quark model calculation with the variational method. In each step, the value of Q is obtained from $Q=2n_{1}+2n_{2}+2n_{3}+2n_{4}+l_{1}+l_{2}+l_{3}+l_{4}$. Q |  | Mass(MeV) |  | Variational Parameters(fm-2) |  | The number of |  | The number of |  | Binding
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | Spatial Wave function |  | Total Wave function |  | energy
$Q_{1}=0$ |  | 4406.4 |  | $a_{1}=2.5,\,a_{2}=7.5,\,a_{3}=4.0,\,a_{4}=3.2$ |  | 1 |  | 4 |  |
$Q_{2}=2$ |  | 4266.9 |  | $a_{1}=2.4,\,a_{2}=7.3,\,a_{3}=4.0,\,a_{4}=3.0$ |  | 11 |  | 42 |  |
$Q_{3}=4$ |  | 4196.3 |  | $a_{1}=2.5,\,a_{2}=7.8,\,a_{3}=4.4,\,a_{4}=2.8$ |  | 66 |  | 250 |  |
$Q_{4}=6$ |  | 4164.9 |  | $a_{1}=2.5,\,a_{2}=8.0,\,a_{3}=4.6,\,a_{4}=2.7$ |  | 282 |  | 1064 |  |
$Q_{5}=8$ |  | 4145.9 |  | $a_{1}=2.5,\,a_{2}=8.0,\,a_{3}=4.6,\,a_{4}=2.7$ |  | 916 |  | 3452 |  |
$Q_{6}=10$ |  | 4137.6 |  | $a_{1}=2.5,\,a_{2}=8.0,\,a_{3}=4.6,\,a_{4}=2.7$ |  | 1356 |  | 5179 |  | +18.5 MeV

## Acknowledgments

This work was supported by the Korea National Research Foundation under the
grant number 2021R1A2C1009486(NRF).

## Appendix

In this appendix, we present the color $\otimes$ spin states obtained from a
correspondence between the coupling scheme and the $SU(6)_{CS}$ representation
of the pentaquark with respect to the total spin $S$. These states are
represented by a specific Young-Yamanouchi basis for particles 1-4
corresponding to the Young diagram of the $SU(6)_{CS}$ irreducible
representation of the pentaquark. In particular, the color part indicates the
color singlet of the pentaquark. Subsequently, we will examine the method of
calculating the expectation values of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$.

###  VII.1 Color $\otimes$ spin states with $S=1/2$ in terms of the
irreducible $SU(6)_{CS}$ representation of the pentaquark

There are a total of 15 color $\otimes$ spin states that are in a color
singlet and have $S=1/2$, as shown in Eq.(55) and Eq.(67). These states
correspond to 15 orthonormal states obtained from the systematic analysis of
Eq. (349), each of which is associated with a specific Young-Yamanouchi basis
for particles 1-4 and has a certain symmetry. These states arise from the
$SU(6)_{CS}$ representation of the pentaquark, and we will examine the method
of calculating the expectation values of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$
using these states:

$\displaystyle\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&3&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&3\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},$ (VII.1.11)
$\displaystyle\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&3&4\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},$ (VII.1.26)
$\displaystyle\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr 3&4\\\
\hline\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \hline\cr 2&4\\\
\hline\cr\end{tabular},[21]\big{\rangle}_{CS^{1/2}},\quad\big{|}\begin{tabular}[]{|c|}\hline\cr
1\\\ \hline\cr 2\\\ \hline\cr 3\\\ \hline\cr 4\\\
\hline\cr\end{tabular},[2^{4}1]\big{\rangle}_{CS^{1/2}}.$ (VII.1.41)

###  VII.2 Color $\otimes$ spin states with $S=3/2$ in terms of the
irreducible $SU(6)_{CS}$ representation of the pentaquark

There are a total of 12 color $\otimes$ spin states that are in a color
singlet and have $S=3/2$, as shown in Eq. (55) and Eq. (94). These states
correspond to 12 orthonormal states obtained from the systematic analysis of
Eq. (349), each of which is associated with a specific Young-Yamanouchi basis
for particles 1-4 and has a certain symmetry. These states arise from the
$SU(6)_{CS}$ representation of the pentaquark, and we will examine the method
of calculating the expectation values of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$
using these states:

$\displaystyle\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr 1&2&3\\\
\cline{1-3}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&2&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|c|}\hline\cr
1&3&4\\\ \cline{1-3}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[421^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}},$ (VII.2.13)
$\displaystyle\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&4\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[1^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr 3&4\\\
\hline\cr\end{tabular},[3^{2}1^{3}]\big{\rangle}_{CS^{3/2}},$ (VII.2.28)
$\displaystyle\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&3\\\ \hline\cr 2&4\\\
\hline\cr\end{tabular},[3^{2}1^{3}]\big{\rangle}_{CS^{3/2}},\quad\big{|}\begin{tabular}[]{|c|}\hline\cr
1\\\ \hline\cr 2\\\ \hline\cr 3\\\ \hline\cr 4\\\
\hline\cr\end{tabular},[1^{3}]\big{\rangle}.$ (VII.2.35)

###  VII.3 Color $\otimes$ spin states with $S=5/2$ in terms of the
irreducible $SU(6)_{CS}$ representation of the pentaquark

According to Eq.(55) and Eq.(110), there are 3 color $\otimes$ spin states
that are in a color singlet and have $S=5/2$. From the systematic analysis of
Eq. (349), we find that these states correspond to the multiplet
$[32^{2}1^{2}]$ representation of the pentaquark. Consequently, these 3
orthonormal states can be obtained only from this representation:

$\displaystyle\big{|}\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\cline{1-2}\cr\vrule\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\vrule\lx@intercol\hfil 4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle},\quad\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&3\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle},\quad\quad\big{|}\begin{tabular}[]{|c|c|}\hline\cr
1&4\\\ \cline{1-2}\cr\vrule\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol \\\ \cline{1-1}\cr\vrule\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol \\\
\cline{1-1}\cr\end{tabular},[32^{2}1^{2}]\big{\rangle}.$ (VII.3.10)

###  VII.4 The expectation value of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$

In this section, we examine the expectation values of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$ in
terms of color $\otimes$ spin states in Eq. (VII.1.41), Eq. (VII.2.35) and Eq.
(VII.3.10). Though this problem can be completely understood through a
systematic analysis in Section IV, the process is not easy, but very
complicated. However, there is a simpler way to approach this calculation by
taking advantage of the symmetry properties between the first and second
quarks, since all the states in Eq. (VII.1.41), Eq. (VII.2.35) and Eq.
(VII.3.10) are made up of either symmetric or antisymmetric combinations of
color and spin states. Then, it is easy to show that the expectation values of
$\lambda_{1}^{c}\lambda_{2}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{2}$ can
be calculated directly through the following:

$\displaystyle\lambda_{1}^{c}\lambda_{2}^{c}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr\end{tabular}_{C}=4/3\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\hline\cr\end{tabular}_{C},\quad\quad\lambda_{1}^{c}\lambda_{2}^{c}\begin{tabular}[]{|c|}\hline\cr
1\\\ \hline\cr 2\\\
\hline\cr\end{tabular}_{C}=-8/3\begin{tabular}[]{|c|}\hline\cr 1\\\ \hline\cr
2\\\
\hline\cr\end{tabular}_{C},\quad\quad{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{2}\begin{tabular}[]{|c|c|}\hline\cr
1&2\\\ \hline\cr\end{tabular}_{S}=\begin{tabular}[]{|c|c|}\hline\cr 1&2\\\
\hline\cr\end{tabular}_{S},\quad\quad\quad{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{2}\begin{tabular}[]{|c|}\hline\cr
1\\\ \hline\cr 2\\\
\hline\cr\end{tabular}_{S}=-3\begin{tabular}[]{|c|}\hline\cr 1\\\ \hline\cr
2\\\ \hline\cr\end{tabular}_{S}.$ (VII.4.13)

In Eq. (VII.4.13), the subscript $C$ indicates the color state, and $S$
indicates the spin state. In the case of $S=1/2$, we can calculate the
expectation values of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$ in
terms of color $\otimes$ spin states in Eq. (VII.1.41), resulting in a 15 by
15 matrix form denoted by
$\langle\lambda_{1}^{c}\lambda_{2}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{2}\rangle$.

In order to calculate the other expectation values, such as
$\langle\lambda_{1}^{c}\lambda_{3}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{3}\rangle$,
it is necessary to consider a transposition operator, $(ij)$ to take an
important role in this situation. Here, for the purpose, we use the following
formula:

$\displaystyle(ij)\lambda_{1}^{c}\lambda_{i}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{i}(ij)=\lambda_{1}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{j},$
(VII.4.14)

where $(ij)$ is the transposition operator of the permutation group, $S_{4}$,
which acts on the Young-Yamanouchi states corresponding to a given Young
diagram of $q^{4}$. As presented in Ref. Park:2015nha , by the use of the
generator of $SU(3)_{C}$ and $SU(2)_{S}$, the $(ij)$ operator ($i<j$ = 1, 2,
3, 4) can be replaced by $(1/3I+1/2\lambda_{i}^{c}\lambda_{j}^{c})$ $\otimes$
$(1/2I+1/2{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j})$, acting on the color and
spine space, respectively. Furthermore, when the transposition operator is
represented by the Young-Yamanouchi bases of the color $\otimes$ spin states
of the pentaquark in Eq. (VII.1.41), it becomes a block diagonal matrix
because the Young-Yamanouchi bases of a specific $SU(6)_{CS}$ representation
form an invariant subspace under the transposition operator.

Now, we can calculate the expectation value of
$\lambda_{1}^{c}\lambda_{3}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{3}$ by
means of
$\langle\lambda_{1}^{c}\lambda_{3}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{3}\rangle=(23)\langle\lambda_{1}^{c}\lambda_{2}^{c}{\vec{\sigma}}_{1}\cdot{\vec{\sigma}}_{2}\rangle(23)$.
Here, we present the transposition operator, $(23)$, which is represented in a
block diagonal matrix with respect to the color $\otimes$ spin states of the
pentaquark in Eq. (VII.1.41):

$\displaystyle(23)=\left(\begin{array}[]{ccccccccccccccc}1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&-\frac{1}{2}&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&\frac{\sqrt{3}}{2}&\frac{1}{2}&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&-\frac{1}{2}&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{\sqrt{3}}{2}&\frac{1}{2}&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&-\frac{1}{2}&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\frac{\sqrt{3}}{2}&\frac{1}{2}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&-1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&-\frac{1}{2}&\frac{\sqrt{3}}{2}&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&\frac{\sqrt{3}}{2}&\frac{1}{2}&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&-1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&-\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&\frac{\sqrt{3}}{2}&\frac{1}{2}&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&-1\\\ \end{array}\right).$ (VII.4.30)

In a similar manner, we calculate the remaining expectation values of
$\lambda_{i}^{c}\lambda_{j}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{j}$
($i<j$ = 1, 2, 3, 4) by using other transposition operators.

Additionally, we can obtain the expectation values of
$\lambda_{i}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{5}$ ($i$
= 1, 2, 3, 4) between a quark and an antiquark by deriving the following
formula:

$\displaystyle\lambda_{4}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{4}\cdot{\vec{\sigma}}_{5}=-$
$\displaystyle
4C^{CS}_{(4+5)}+2C^{C}_{(4+5)}+\frac{4}{3}(\vec{S}\cdot\vec{S})_{(4+5)}+16I.$
(VII.4.31)

In order to calculate the expectation value of
$\lambda_{4}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{4}\cdot{\vec{\sigma}}_{5}$, it
is necessary to decompose $\mathbf{6}_{CS}\otimes\mathbf{\bar{6}}_{CS}$ into
the direct sum of the composition of $SU(3)_{C}$ and $SU(2)_{S}$:
$\mathbf{6}_{CS}\otimes\mathbf{\bar{6}}_{CS}=\mathbf{35}_{CS}\oplus\mathbf{1}_{CS}$.
The dimensions of the SU(6)CS representation corresponding to the Young
diagrams $[21^{4}]$ and $[1^{6}]$ are 35 and 1, respectively. Table IX shows
the composition of $SU(3)_{C}$ and $SU(2)_{S}$ concerning the $SU(6)_{CS}$
representation, and its eigenvalue of the quadratic Casimir operator, $C^{CS}$
for $q\bar{q}$.

Table 9: The composition of $SU(3)_{C}$ and $SU(2)_{S}$ concerning the $SU(6)_{CS}$ representation for $q\bar{q}$. $SU(6)_{CS}$ |  |  |
---|---|---|---
Young digram |  $SU(3)_{C}$ $\otimes$ $SU(2)_{S}$ | Dimesion | Eigenvalue
$[21^{4}]$ |  $(\mathbf{8},\mathbf{3})$, $(\mathbf{8},\mathbf{1})$, $(\mathbf{1},\mathbf{3})$ | 35 | 6
$[1^{6}]$ | $(\mathbf{1},\mathbf{1})$ | 1 | 0

On the other hand, it should be noted that, in Eq. (55), the color states
between the 4th and 5th quarks are octet in the $|C_{1}\rangle$ and
$|C_{2}\rangle$, and singlet in the $|C_{3}\rangle$. For the spin part, since
most of the spin states between the 4th and 5th quarks in Eq. (67) and Eq.
(94) cannot be directly determined, these Young-Yamanouchi spin states should
be transformed into those associated with the decay mode. With the
availability of these eigenvalues of such Casimir operators, we can calculate
the expectation values of
$\lambda_{4}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{4}\cdot{\vec{\sigma}}_{5}$, and
the $\lambda_{i}^{c}\lambda_{5}^{c}{\vec{\sigma}}_{i}\cdot{\vec{\sigma}}_{5}$
($i$ = 1, 2, 3) through the relevant permutations.

## References

  * (1) Sungsik Noh and Woosung Park, Observation of $T_{cc}$ and a quark model, [arXiv:2303.03285 [hep-ph]].
  * (2) S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett. 91, 262001 (2003).
  * (3) R. Aaij et al. [LHCb], Nature Phys. 18, no.7, 751-754 (2022) doi:10.1038/s41567-022-01614-y
  * (4) J. l. Ballot and J. M. Richard, Phys. Lett. B 123, 449-451 (1983) doi:10.1016/0370-2693(83)90991-7
  * (5) S. Zouzou, B. Silvestre-Brac, C. Gignoux and J. M. Richard, Z. Phys. C 30, 457 (1986) doi:10.1007/BF01557611
  * (6) J. Carlson, L. Heller, and J.A. Tjon Stability of Dimesons, Phys. Rev. D 37, 744 (1988).
  * (7) B. Silvestre-Brac and C. Semay, Systematics of $L=0$ $q^{2}\bar{q}^{2}$ systems, Z. Phys. C 57, 273-282 (1993).
  * (8) C. Semay and B. Silvestre-Brac Diquonia and potential models, Z. Phys. C 61, 271-275 (1994).
  * (9) S. Pepin, Fl. Stancu, M. Genovese, and J.-M. Richard, Tetraquarks with colour-blind forces in chiral quark models, Phys. Lett. B 393, 119-123 (1997).
  * (10) Boris A. Gelman and Shmuel Nussinov, Does a narrow tetraquark cc anti-u anti-d state exist?, Phys. Lett. B 551, 296-304 (2003).
  * (11) J. Vijande, F. Fernandez, A. Valcarce, and B. Silvestre-Brac, Tetraquarks in a chiral constituent-quark model, Eur. Phys. J. A 19, 383-389 (2004).
  * (12) D. Janc and M. Rosina, The $T_{cc}=DD^{*}$ Molecular State, Few Body Syst. 35, 175-196 (2004).
  * (13) D. Ebert, R. N. Faustov, V. O. Galkin, and W. Lucha, Masses of tetraquarks with two heavy quarks in the relativistic quark model, Phys. Rev. D 76, 114015 (2007).
  * (14) Youchang Yang, Chengrong Deng, Jailun Ping, and T. Goldman, $S$-wave $QQ\bar{q}\bar{q}$ state in the constituent quark model, Phys. Rev. D 80, 114023 (2009).
  * (15) Yoichi Ikeda, Bruno Charron, Sinya Aoki, Takumi Doi, Tetsuo Hatsuda, Takashi Inoue, Noriyoshi Ishii, Keiko Murano, Hidekatsu Nemura, and Kenji Sasaki, Charmed tetraquarks $T_{cc}$ and $T_{cs}$ from dynamical lattice QCD simulations, Phys. Lett. B 729, 85-90 (2014).
  * (16) W. Park and S. H. Lee, Nucl. Phys. A 925 (2014), 161-184 doi:10.1016/j.nuclphysa.2014.02.008 [arXiv:1311.5330 [nucl-th]].
  * (17) Marek Karliner and Jonathan L. Rosner, Discovery of the Doubly Charmed $\Xi_{cc}$ Baryon Implies a Stable $bb\bar{u}\bar{d}$ Tetraquark, Phys. Rev. Lett. 119, 202001 (2017).
  * (18) Estia J. Eichten and Chris Quiqq, Heavy-Quark Symmetry Implies Stable Heavy Tetraquark Mesons $Q_{i}Q_{j}\bar{q}_{k}\bar{q}_{l}$, Phys. Rev. Lett. 119, 202002 (2017).
  * (19) Si-Qiang Luo, Kan Chen, Xiang Liu, Yan-Rui Liu, and Shi-Lin Zhu, Exotic tetraquark states with the $qq\bar{Q}\bar{Q}$ configuration, Eur. Phys. J. C 77, 709 (2017).
  * (20) Hadron Spectrum Collaboration, Gavin K. C. Cheung, Christopher E. Thomas, Jozef J. Dudek, and Robert G. Edwards, Tetraquark operators in lattice QCD and exotic flavour states in the charm sector, JHEP 11, 033 (2017).
  * (21) Zhi-Gang Wang, Analysis of the axialvector doubly heavy tetraquark states with QCD sum rules, Acta Phys. Polon. B 49, 1781 (2018).
  * (22) Woosung Park, Sungsik Noh, and Su Houng Lee, Masses of the doubly heavy tetraquarks in a constituent quark model, Nucl. Phys. A 983, 1-19 (2019).
  * (23) Parikshit Junnarkar, Nilmani Mathur, and M. Padmanath, Study of doubly heavy tetraquarks in lattice QCD, Phys. Rev. D 99, 034507 (2019).
  * (24) Anthony Francis, Renwick J. Hudspith, Randy Lewis, and Kim Maltman, Evidence for charm-bottom tetraquarks and the mass dependence of heavy-light tetraquark states from lattice QCD, Phys. Rev. D 99, 054505 (2019).
  * (25) Ming-Zhu Liu, Tian-Wei Wu, Manuel Pavon Valderrama, Ju-Jun Xie, and Li-Sheng Geng, Heavy-quark spin and flavor symmetry partners of the $X(3872)$ revisited: What can we learn from the one boson exchange model?, Phys. Rev. D 99, 094018 (2019).
  * (26) Chengrong Deng, Hong Chen, and Jialun Ping, Systematical investigation on the stability of doubly heavy tetraquark states, Eur. Phys. J. A 56, 9 (2020).
  * (27) Gang Yang, Jialun Ping, and Jorge Segovia, Doubly-heavy tetraquarks, Phys. Rev. D 101, 014001 (2020).
  * (28) Qi-Fang L$\ddot{\rm u}$, Dian-Yong Chen, and Yu-Bing Dong, Masses of doubly heavy tetraquarks $T_{QQ^{\prime}}$ in a relativized quark model, Phys. Rev. D 102, 034012 (2020).
  * (29) Eric Braaten, Li-Ping He, and Abhishek Mohapatra, Masses of doubly heavy tetraquarks with error bars, Phys. Rev. D 103, 016001 (2021).
  * (30) Jian-Bo Cheng, Shi-Yuan Li, Yan-Rui Liu, Zong-Guo Si, and Tao Yao Double-heavy tetraquark states with heavy diquark-antiquark symmetry, Chin. Phys. C 45, 043102 (2021).
  * (31) Sungsik Noh, Woosung Park, and Su Houng Lee, Doubly heavy tetraquarks, $qq^{\prime}\bar{Q}\bar{Q^{\prime}}$, in a nonrelativistic quark model with a complete set of harmonic oscillator bases, Phys. Rev. D 103, 114009 (2021)
  * (32) Rudolf N. Faustov, Vladimir O. Galkin, and Elena M. Savchenko, Heavy Tetraquarks in the Reltativistic Quark Model, Universe 7, 94 (2021).
  * (33) W. Park, S. Cho and S. H. Lee, Phys. Rev. D 99, no.9, 094023 (2019) doi:10.1103/PhysRevD.99.094023 [arXiv:1811.10911 [hep-ph]].
  * (34) W. Park, A. Park, S. Cho and S. H. Lee, Phys. Rev. D 95, no.5, 054027 (2017) doi:10.1103/PhysRevD.95.054027 [arXiv:1702.00381 [hep-ph]].
  * (35) F. Stancu and S. Pepin, Few Body Syst. 26, 113 (1999).
  * (36) R. L. Jaffe, Phys. Rev. D 15, 281 (1977) doi:10.1103/PhysRevD.15.281
  * (37) A. T. M. Aerts, P. J. G. Mulders and J. J. de Swart, Phys. Rev. D 17, 260 (1978) doi:10.1103/PhysRevD.17.260
  * (38) W. Park, A. Park and S. H. Lee, Phys. Rev. D 92, no.1, 014037 (2015) doi:10.1103/PhysRevD.92.014037 [arXiv:1506.01123 [nucl-th]].
  * (39) J. Q. Chen, J. L. Ping and F. Wang, “Group representation theory for physicists,” River Edge, USA: World Scientific (2002)
  * (40) J. Vijande, E. Weissman, A. Valcarce, and N. Barnea, Are there compact heavy four-quark bound states?, Phys. Rev. D 76, 094027 (2007).