We report a measurement of the flux-integrated $ u_{mu}$ charged-current cross sections on water, hydrocarbon, and iron in the T2K on-axis neutrino beam with a mean neutrino energy of 1.5 GeV. The measured cross sections on water, hydrocarbon, and iron are $sigma^{rm{H_{2}O}}_{rm{CC}}$ = (0.840$pm 0.010$(stat.)$^{+0.10}_{-0.08}$(syst.))$times$10$^{-38}$cm$^2$/nucleon, $sigma^{rm{CH}}_{rm{CC}}$ = (0.817$pm 0.007$(stat.)$^{+0.11}_{-0.08}$(syst.))$times$10$^{-38}$cm$^2$/nucleon, and $sigma^{rm{Fe}}_{rm{CC}}$ = (0.859$pm 0.003$(stat.) $^{+0.12}_{-0.10}$(syst.))$times$10$^{-38}$cm$^2$/nucleon respectively, for a restricted phase space of induced muons: $theta_{mu}<45^{circ}$ and $p_{mu}>$0.4 GeV/$c$ in the laboratory frame. The measured cross section ratios are ${sigma^{rm{H_{2}O}}_{rm{CC}}}/{sigma^{rm{CH}}_{rm{CC}}}$ = 1.028$pm 0.016$(stat.)$pm 0.053$(syst.), ${sigma^{rm{Fe}}_{rm{CC}}}/{sigma^{rm{H_{2}O}}_{rm{CC}}}$ = 1.023$pm 0.012$(stat.)$pm 0.058$(syst.), and ${sigma^{rm{Fe}}_{rm{CC}}}/{sigma^{rm{CH}}_{rm{CC}}}$ = 1.049$pm 0.010$(stat.)$pm 0.043$(syst.). These results, with an unprecedented precision for the measurements of neutrino cross sections on water in the studied energy region, show good agreement with the current neutrino interaction models used in the T2K oscillation analyses.
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