Annihilation signatures of neutron dark decay models in neutron oscillation and proton decay searches

We point out that two models that reconcile the neutron lifetime anomaly via dark decays of the neutron, also predict dark matter-neutron ($\bar{\chi}-n$) annihilation that may be observable in neutron-antineutron oscillation and proton decay searches at Super-Kamiokande, Hyper-Kamiokande and DUNE. We study signatures of $\bar{\chi}n\to \gamma\pi^0$ (or multi-$\pi^0$) and $\bar{\chi}n\to \phi\gamma\pi^0$ (or $\phi+$multi-$\pi^0$), where $\phi$ is an almost massless boson in one of the two models.