NN\!\rightarrow\!\Lambda NN$, $\Lambda NN\leftrightarrow\Sigma NN$, and $\Sigma NN \rightarrow\Sigma NN$ coupled channels are translated into effective density-dependent hyperon-nucleon potentials, which additionally enter the coupled Bethe-Goldstone equations. This is done as in refs.[@Petschauer2016; @Petschauer2017; @Haidenbauer2017; @Holt2010] by integrating one of the two nucleons over the filled Fermi sea, as illustrated schematically in fig.\[fig:3BFeff\]:

![Density-dependent in-medium potentials generated from 3-body forces. Upper left: contact term; upper right: one-meson exchange term; lower part: two-meson exchange term. Loops with arrows indicate integration over the filled nucleon Fermi sea. []{data-label="fig:3BFeff"}](FBBeff)

$$\label{eq:red}
V_{12}^{\text{eff}}(\rho_p,\rho_n) = \sum_{N=p,n}\int\limits_{|\vec k|\leq k_F^{(N)}}\frac{\mathrm d^3k}{(2\pi)^3}\,{\mathrm tr}_3V_{123} \,,$$

where ${\mathrm tr}_3$ denotes the spin-trace over the third particle and the summation over nucleons (protons and neutrons) in the Fermi sea is performed. The resulting effective two-body potentials depend explicitly on the proton and neutron densities, $$\rho_p = {(k_F^{(p)})^3\over 3\pi^2}~,~~~\rho_n = {(k_F^{(n)})^3\over 3\pi^2}~.$$ As a first example, we present expressions for the $\Lambda N\to \Lambda N$ effective two-body potential in (asymmetric) nuclear matter with density $\rho = \rho_p + \rho_n$. Only the $\Lambda n $ effective potential in the nuclear medium needs to be given since the $\Lambda p$ potential follows by simply interchanging $\rho_n\leftrightarrow\rho_p$. The three parts have the following structure [@Petschauer2017]:\
Contact term: 