We present analytic expressions for three-flavor neutrino oscillations in presence of invisible neutrino decay and matter effects. Using the well-known Cayley-Hamilton formalism, the leading-order terms are derived for oscillation probabilities in all major channels assuming the neutrino mass eigenstate v3 to decay. Our work extends and complements previous studies utilizing the Cayley-Hamilton theorem, providing the series expansions for ve ve, ve v mu, ve va, v mu v mu, and v mu va. The accuracy of the analytical formulas is investigated by comparing the results with numerically calculated probabilities. We also comment on the implications on unitarity violation.

Presently under construction in Lund, Sweden, the European Spallation Source (ESS) will be the world's brightest neutron source. As such, it has the potential for a particle physics program with a unique reach and which is complementary to that available at other facilities. This paper describes proposed particle physics activities for the ESS. These encompass the exploitation of both the neutrons and neutrinos produced at the ESS for high precision (sensitivity) measurements (searches).

Proton decay is a hypothetical form of particle decay in which protons are assumed to decay into lighter particles. This form of decay has yet to be detected. In this contribution to the proceedings of Neutrino 2022, we review the current status of proton decay, covering both experimental results and theoretical models, including their predictions.