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The missing mass spectroscopy of $\Xi^{-}$ hypernuclei with the$(K^{-},K^{+})$ reaction is planned to be performed at the J-PARC K1.8 beamline by using a new magnetic spectrometer, Strangeness $-2$ Spectrometer(S-2S). A $\v{C}$cerenkov detector with a radiation medium of pure water(refractive index of 1.33) is designed to be used for on-line proton rejectionfor a momentum range of 1.2 to 1.6 GeV/$c$ in S-2S. Prototype water$\v{C}$erenkov detectors were developed and tested with positron beams andcosmic rays to estimate their proton-rejection capability. We achieved anaverage number of photoelectrons of greater than 200 with the latest prototypefor cosmic rays, which was stable during an expected beam time of one month.The performance of the prototype in the cosmic-ray test was well reproducedwith a Monte Carlo simulation in which some input parameters were adjusted.Based on the Monte Carlo simulation, we expect to achieve $>90\%$proton-rejection efficiency while maintaining $>95\%$ $K^{+}$ survival ratio inthe whole S-2S acceptance. The performance satisfies the requirements toconduct the spectroscopic study of $\Xi^{-}$ hypernuclei at J-PARC.

Development of Water $\v{C}$erenkov Detector for On-line Proton Rejection in $\Xi^{-}$ Hypernuclear Spectroscopy via the $(K^{-},K^{+})$ Reaction