\[\begin{split}\begin{array}{rcl}
A^{1}_{- \frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} \\
A^{2}_{- \frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} \\
A^{1}_{- \frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} \\
A^{2}_{- \frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} \\
A^{1}_{\frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} \\
A^{2}_{\frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} \\
A^{1}_{\frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} \\
A^{2}_{\frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} \\
\end{array}\end{split}\]