We report a measurement of the $
u_mu$ charged current quasi-elastic
cross-sections on carbon in the T2K on-axis neutrino beam. The measured charged
current quasi-elastic cross-sections on carbon at mean neutrino energies of
1.94 GeV and 0.93 GeV are $(11.95pm 0.19(stat.)_{-1.47}^{+1.82} (syst.)) imes
10^{-39}mathrm{cm}^2/mathrm{neutron}$ and $(10.64pm
0.37(stat.)_{-1.65}^{+2.03} (syst.)) imes
10^{-39}mathrm{cm}^2/mathrm{neutron}$, respectively. These results agree well
with the predictions of neutrino interaction models. In addition, we
investigated the effects of the nuclear model and the multi-nucleon
interaction.
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