i\epsilon} G(\omega)\,.$$ The energy denominator $e(\omega)$ depends on the starting energy $\omega$. The Pauli blocking operator $Q$ excludes particles in intermediate states from scattering into the filled Fermi sea. The potential $V$ is a matrix (labeled by the outgoing and ingoing two-baryon channels) including two-body and three-body contibutions, with the latter given in their density-dependent effective two-body form.

After angle-averaging (see appendix A for details) the Bethe-Goldstone equation decomposes into partial waves with total angular momentum $J$: $$\begin{aligned}
\label{eq:BGE}
&&G^{J}_{\alpha\beta}(p',p; P,\omega) = V^{J}_{\alpha\beta}(p',p)  +\\
&&\sum_{\nu}\int_0^\infty\frac{\mathrm d k\, k^2}{(2\pi)^3} \, V^{J}_{\alpha\nu}(p',k) 
\frac{\bar Q_{\nu}(P,k)}{\bar e_{\nu}(P,k;\omega)+\mathrm i \epsilon}G^{J}_{\nu\beta}(k,p;P,\omega) \,.\nonumber \end{aligned}$$ Here, $P$ is the total momentum of the two baryons. The indices $\alpha, \beta, \nu$ represent complete sets of channels, including partial wave quantum numbers and pairs of interacting baryons, $(B_1,B_2)$. In our case of interest, $B_1$ in the initial state is a $\Lambda$ hyperon, while $B_2$ is a nucleon within its Fermi sea. The channel coupling turns $B_1$ into a $\Sigma$ which subsequently interacts with the nuclear medium and turns back into a $\Lambda$ in the final state.

The $G$-matrix elements of eq.(\[eq:BGE\]) are calculated at the on-shell starting energy $$\begin{aligned}
\omega &=& E_{B_1}(p_1) +  E_{B_2}(p_2)~,\nonumber\\
\mathrm{with}~~~~~~E_{B_i}(p) &=& M_i + {p^2\over 2M_i} 