Quantum numbers of the pentaquark states ${{\rm{P}}}_{{\rm{c}}}^{+}$ via symmetry analysis

Chong-yao Chen,Muyang Chen,Yu-Xin Liu

Table 5. IRREPs of the rotation group, j is the angular moment, m1 and m2 denote the rows and columns of the matrix, α, β, γ is the Eulerian angular.

${D}_{{m}_{1},{m}_{2}}^{j}(\alpha ,\beta ,\gamma )$ $=\,{\sum }_{k}\tfrac{{\left(-1\right)}^{k}\sqrt{(j+{m}_{1})!(j-{m}_{1})!(j+{m}_{2})!(j-{m}_{2})!}\exp (-{\rm{i}}\alpha {m}_{1}-{\rm{i}}\gamma {m}_{2}){\sin }^{2k-{m}_{1}+{m}_{2}}\left(\tfrac{\beta }{2}\right){\cos }^{2j-2k+{m}_{1}-{m}_{2}}\left(\tfrac{\beta }{2}\right)}{k!(j-k+{m}_{1})!(j-k-{m}_{2})!(k-{m}_{1}+{m}_{2})!}$ L = 1 ${R}_{\pi }^{\hat{k}}=D(0,0,\pi )=\left(\begin{array}{ccc}-1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\end{array}\right)$, ${R}_{\pi }^{\hat{j}}=D(0,\pi ,0)=\left(\begin{array}{ccc}0 & 0 & 1\\ 0 & -1 & 0\\ 1 & 0 & 0\end{array}\right)$, ${R}_{\pi /2}^{\hat{k}}=D(0,0,\pi /2)=\left(\begin{array}{ccc}-{\rm{i}} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & {\rm{i}}\end{array}\right)$, ${R}_{2\pi /3}^{\hat{n}}=D(0,-\theta ,0)\cdot D(2\pi /3,\theta ,0)\,=\,\left(\begin{array}{ccc}-{\rm{i}}/2 & 1/2+{\rm{i}}/2 & -1/2\\ 1/2+{\rm{i}}/2 & 0 & -1/2+{\rm{i}}/2\\ -1/2 & -1/2+{\rm{i}}/2 & {\rm{i}}/2\end{array}\right)$, $\quad \theta =\mathrm{Arccos}(\tfrac{1}{\sqrt{3}})$.