$$V^{(0)}_{\text{eff}}(\Lambda n) = {(H_1+3H_2)^2\over 18 \Delta}\,(\rho_n+2\rho_p)~,$$ with the decuplet-octet baryon mass difference $\Delta$, for which we take an average value of $\Delta = 270$ MeV.\
One-pion exchange part: $$V^{(1)}_{\text{eff}}(\Lambda n) =
{g_A C\over 9f^2 \Delta}(H_1+3H_2)\left(\rho_n+2\rho_p - {m_\pi^2\over 2\pi^2}\Gamma_0\right),$$ where $\Gamma_0(p;k_F^{(N)})$ is a function of the $\Lambda N$ center-of-mass momentum $p$, and the proton or neutron Fermi momentum $ k_F^{(N)}$, given explicitly in ref.[@Petschauer2017].\
Two-pion exchange part: $$V^{(2)}_{\text{eff}}(\Lambda n) = {g_A^2 C^2\over 6f^4 \Delta}\left[\rho_n+2\rho_p+ {\cal F}(p,q;k_F^{(n)},k_F^{(p)})\right]\,,$$ where the function ${\cal F}$ depends additionally on the momentum transfer $q$ in the reduced two-body system. It is a lengthy expression involving the spin-orbit operator $\mathrm i \vec \sigma_2\cdot(\vec q\times \vec p\,)$, that is explicitly given in eq.(46) of ref.[@Petschauer2017].

Next, we present some selected expressions for the effective $\Lambda N \rightarrow \Sigma N$ interaction. As representative examples, consider the direct terms, shown in fig.\[fig:3BFeff\], of the $\Lambda n\rightarrow \Sigma^- p$ effective transition potential. Exchange terms are also included in the actual calculations but are not displayed here for simplicity. In the case of symmetric nuclear matter, the corresponding lengthy formulas can be found in subsection III.D of ref.[@Petschauer2017].\
Contact term: $$\begin{aligned}
 && V^{(0)}_\text{{eff}}(\Lambda n\!\rightarrow\!\Sigma^- p) = {1\over 6\sqrt{6}\Delta}\bigg\{(\rho_p\!+\!\rho_n) \big[4H_2^2\!-\!(H_1\!+\!H_2)^2\big]\nonumber\\ &&+ \vec\sigma_1\!\cdot\!\vec\sigma_2\bigg[\bigg(H_1^2\!+\!H_2^2\!+\!{10\over 3}H_1 H_2\bigg)\rho_n+ 