We enumerate the conditions necessary for $CP$ violation to be manifest in $n-\overline{n}$ oscillations and build a simple model that can give rise to such effects. We discuss a possible connection between neutron oscillations and dark matter, provided the mass of the latter lies between $m_p-m_e$ and $m_p+m_e$. We apply our results to a possible baryogenesis scenario involving $CP$ violation in the oscillations of the ${\mathrm{\Xi }}^0$.

• Received 24 December 2015

DOI:https://doi.org/10.1103/PhysRevD.94.076002