  Three-body forces acting on a $\Lambda$ hyperon in a nuclear medium are
investigated, with special focus on the so-called hyperon puzzle in neutron
stars. The hyperon-nucleon two-body interaction deduced from SU(3) chiral
effective field theory is employed at next-to-leading order. Hyperon-nucleon
three-body forces are approximated using saturation by decuplet baryons and are
transcribed to density-dependent effective two-body interactions. These
together are taken as input in a Brueckner-Bethe-Goldstone equation with
explicit treatment of the $\Lambda N\leftrightarrow\Sigma N$ and $\Lambda
NN\leftrightarrow\Sigma NN$ coupled channels. Single-particle potentials of a
$\Lambda$ hyperon in symmetric nuclear matter and neutron matter are
calculated. With parameters of the $\Lambda NN$ three-body force constrained by
hypernuclear phenomenology, extrapolations to high baryon density are
performed. By comparison of the $\Lambda$ and neutron chemical potentials at
densities characteristic of the core of neutron stars it is found that the
combined repulsive effects of two- and three-body correlations makes the
appearance of $\Lambda$ hyperons in neutron stars energetically unfavourable,
thus offering a possible solution to a longstanding question.
