The Lippman-Schwinger equation for two-nucleon scattering

The general structure of the \( T \)-matrix in partial waves is $$ T_{ll'}^{\alpha}(kk'K\omega)=V_{ll'}^{\alpha}(kk') $$ $$ \begin{equation} +{\displaystyle \frac{2}{\pi}\sum_{l''m_{l''}M_S}\int_{0}^{\infty} d \mathbf{q} (\langle l''m_{l''}Sm_S|{\cal J}M\rangle)^2 \frac{Y_{l''m_{l''}}^*(\hat{\mathbf{q}}) Y_{l''m_{l''}}(\hat{\mathbf{q}}) V_{ll''}^{\alpha}(kq) T_{l''l'}^{\alpha}(qk'K\omega)} {\omega -H_0}}, \tag{15} \end{equation} $$