and only an exchange diagram contributes in this sector.

The examples just discussed demonstrate the following. In the diagonal $\Lambda N\rightarrow\Lambda N$ potential the two low-energy constants, $H_1$ and $H_2$, those associated with the three-body contact term using decuplet dominance, appear in the combination $H_1 + 3H_2$. This leaves only one parameter in this channel, as it has been the case in previous calculations of $\Lambda$ single-particle potentials in nuclear and neutron matter performed in refs.[@Kohno2018; @Haidenbauer2017]. In contrast, the explicit inclusion of $\Lambda NN\leftrightarrow\Sigma NN$ coupled channels introduces $H_1$ and $H_2$ in various different combinations so that these parameters have to be constrained independently.

Brueckner-Hartree-Fock approach to hyperon single-particle potentials
=====================================================================

The two- and three-body potentials described in the previous section are used[^1] as input for computing a hyperon-nucleon $G$-matrix within Brueckner theory at first order in the hole-line expansion (the Brueckner-Hartree-Fock approximation [@Day1967]). We focus on the single-particle potentials, i.e. the self-energies of hyperons in nuclear matter. The relevant formalism is briefly summarized below. For more details we refer to refs.[@Schulze1998; @Rijken1999; @Vidana2000; @Kohno2000].

The Brueckner reaction matrix or G-matrix is determined by solving the coupled-channel Bethe-Goldstone equation (in symbolic form) $$G(\omega) = V + V \frac Q{e(\omega)+\mathrm 