 Solution

For $N_p$ protons, exposure $T$ and efficiency $ϵ$, the expected count in this channel is $\mu =ϵN_p{\Gamma }_{\mu {\pi }^0}T$. A partial decay rate at the quoted upper bound gives one event per year whenMore generally, $M_p\simeq 11\mathrm{kilotonnes}\mu /[ϵ(T/\mathrm{yr})]$ at the limiting rate. This is an estimate of the required proton inventory, not necessarily the whole detector mass. For water, each molecule contains ten protons and has mass approximately $18$ atomic mass units; the corresponding ideal inventory is about $19.7\mathrm{kilotonnes}$ of water. Finite efficiency and a useful sample larger than one event increase the required exposure.

An upper limit is not a measured nonzero rate. The true rate could be much smaller, including zero, so no finite target guarantees a measurable signal. The estimate describes sensitivity to a rate near the existing limit.