over the simpler gap choice (with potentials set to zero for momenta above the Fermi sea), because it allows for a reliable determination of the single-particle potentials including their imaginary parts. Some further technical details are given in Appendix B.

Results {#sec:res}
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Single-particle potentials of a $\Lambda$ hyperon in nuclear and neutron matter {#subsec:spp}
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The next step is now to calculate the single-particle potential, $U_\Lambda(p_1=0,\rho)$, for a $\Lambda$ at rest in symmetric nuclear matter as a function of baryon density $\rho=\rho_p+\rho_n$. Using as input the NLO hyperon-nucleon potentials together with the three-body interactions as specified in Section \[sec:int\], it is still necessary to constrain the two free parameters of the three-body contact terms. We recall that the calculation explicitly includes the $\Lambda NN \leftrightarrow\Sigma NN$ coupled channels, so the respective constants $H_1$ and $H_2$ enter independently.

![image](USNM)

![image](3BFH1H2)

A key constraint comes from hypernuclear phenomenology. Bulk properties and the shell structure of $\Lambda$ hypernuclei are well described by a $\Lambda$-nuclear single-particle potential parametrized in Wood-Saxon form, with an accurately determined potential depth at nuclear central densities [@Gal2016]: $$U_\Lambda(\rho \simeq \rho_0) = - 30~ \mathrm{MeV}~,
\label{eq:USNM}$$ corresponding to about half of the attractive strength of the nucleon single-particle potential 