= Solution
A large, optically clean water <Cherenkov detector> provides many <protons>, a transparent radiator and a relatively inexpensive homogeneous target. A surrounding array of inward-facing <photomultipliers> records hit times and charge. Reconstructing the vertex and cone intersections identifies track directions and visible <energies>; an outer active veto rejects entering particles. Water has refractive index near $1.33$, and the charged decay products are above its relevant thresholds.
For a free stationary <proton>, <relativistic two-body decay> gives
$$
p^*=\frac{\sqrt{[m_p^2-(m_\mu+m_{\pi^0})^2][m_p^2-(m_\mu-m_{\pi^0})^2]}}{2m_p}\simeq453\ \mathrm{MeV}/c.
$$
The <muon> and neutral <pion> are back-to-back. The <pion> decays essentially at the vertex into two <gamma rays>: $c\tau_{\pi^0}\simeq2.4\times10^{-8}\ \mathrm m$ before its modest Lorentz factor is included. Each <gamma ray> converts and initiates an <electromagnetic shower>, producing a diffuse ring. The <muon> gives a relatively sharp ring and, if it stops, a delayed Michel <Positron> on the $2.2\ \mathrm{microsecond}$ scale. The ideal topology therefore has one muon-like and two electron-like rings, although overlaps or a weak photon may merge or hide a ring.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2016-water-proton-decay.png]
{title=Water Cherenkov proton-decay topology with prompt rings and a delayed muon-decay tag}
Require the two-<photon> invariant <mass> to match $m_{\pi^0}$, the total reconstructed <mass> to match $m_p$, and small total <momentum>. Bound <oxygen> <protons> have Fermi motion and their hadronic products can rescatter, so those cuts need realistic nuclear corrections; the water target also contains free <hydrogen> <protons>. Install the detector deep underground to suppress cosmic-ray <muons>, with shielding and an active veto. Important irreducible backgrounds are atmospheric-<neutrino> interactions producing a <muon> and neutral <pion>, potentially with an undetected recoil nucleon.
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