Development of the Range Counter for the COMET Phase-$\alpha$ Experiment

The COMET Phase-$\alpha$ experiment aims to evaluate the novel muon transport beamline for the muon-to-electron conversion search at J-PARC, Japan. A dedicated Range Counter (RC) was developed to measure the momentum spectrum of transported negative muons with momenta of 30--100 MeV/$c$. The RC consists of graphite momentum degraders, a muon absorber, and plastic scintillation counters ($\rm T_0$, $\rm T_1$, and $\rm T_2$) to detect decay-in-orbit (DIO) electrons from stopped muons. The number of muons stopped in the absorber is reconstructed from the decay time distribution. A copper absorber was selected due to the short lifetime of muonic atoms in copper, which enhances signal separation. The counters' performance was evaluated experimentally. The $\rm T_0$ Counter, made of a $200\times 200\times 0.5~{\rm mm^3}$ scintillator plate, achieved a muon-trigger efficiency exceeding 99.9%. The $\rm T_1$ and $\rm T_2$ Counters also demonstrated high electron-detection efficiencies of $>99$%. Based on these results, simulation studies estimate the acceptance for reconstructing the number of DIO electrons from the absorber to be approximately 47% with a corresponding signal purity of 60% against muon capture-induced backgrounds.