(H_1\!+\!3H_2)^2{ \rho_p\over 3}\bigg]\bigg\}\,. \nonumber\\ &&\end{aligned}$$ One-pion exchange part: $$\begin{aligned}
&&V^{(1)}_\text{{eff}}(\Lambda n\!\rightarrow\!\Sigma^- p)={\sqrt{6}g_A C \over 27f^2\Delta}\, {\vec\sigma_1\cdot\vec q\,\vec\sigma_2\cdot\vec q\over m_\pi^2+\vec q^2}\nonumber\\ && \qquad\qquad \times \Big[(2H_1-3H_2)\rho_n-(2H_1+9H_2)\rho_p\Big]\,.\end{aligned}$$ Two-pion exchange part: $$\begin{aligned}
V^{(2)}_\text{{eff}}(\Lambda n\!\rightarrow\!\Sigma^- p)=-{16g_A D C^2\over 9\sqrt{6}f^4\Delta}{\vec\sigma_1\!\cdot\!\vec q\,\vec\sigma_2\!\cdot\!\vec q\over( m_\pi^2+\vec q^2)^2}(\rho_p+\rho_n) \vec q^2 , \nonumber\\ &&\end{aligned}$$ where the SU(3) axial vector coupling constant $D$ stems from the $\Lambda\Sigma\pi$ vertex. For the $\Lambda p\rightarrow \Sigma^+ n$ channel, corresponding expressions hold with $\rho_n\leftrightarrow\rho_p$ interchanged.

Finally, the ladder summations in the coupled hyperon-nucleon channels require also as input the $\Sigma N\leftrightarrow\Sigma N$ effective two-body potentials in different charge combinations. As representative examples we list here expressions for the in-medium $\Sigma^-n\rightarrow\Sigma^- n$ potential:\
Contact term: $$\begin{aligned}
&&V^{(0)}_\text{{eff}}(\Sigma^- n) = 
 {1\over 6\Delta}\,\Big\{\big[(H_1+H_2)^2\rho_p + 4H_1^2\rho_n\big]
\nonumber\\ && \qquad \qquad\qquad\qquad -{1\over 3}\,\vec\sigma_1\cdot\vec\sigma_2\,(H_1+H_2)^2\,\rho_p\Big\}\, .\end{aligned}$$ One-pion exchange part: $$\begin{aligned}
&&V^{(1)}_\text{{eff}}(\Sigma^- n)=
-{4g_A C\over 9f^2\Delta}\,{\vec\sigma_1\!\cdot\!\vec q\,\vec\sigma_2\!\cdot\!\vec q\over m_\pi^2+\vec q^2}\Big[H_1\rho_n \!+\!(H_1\!+\!H_2)\rho_p\Big]\,,\nonumber\\ &&\end{aligned}$$ Two-pion exchange part: $$V^{(2)}_\text{{eff}}(\Sigma^- n)=-{8g_A F C^2\over 9f^4\Delta}\,{\vec\sigma_1\cdot\vec q\,\vec\sigma_2\cdot\vec q\over( m_\pi^2+\vec q^2)^2}\,(\rho_n+\rho_p)\,\vec q^2\,,$$ where now the SU(3) axial vector coupling constant $F$ enters. The $\Sigma^+ p\rightarrow\Sigma^+ p$ effective potential is obtained by interchanging $\rho_n\leftrightarrow\rho_p$. The $\Sigma^0 n\rightarrow\Sigma^0 n$ and $\Sigma^0 p\rightarrow\Sigma^0 p$ effective potentials have similar expressions, except that for their two-pion exchange parts the direct term as shown in fig.\[fig:3BFeff\] vanishes 