Measurement of $bar{ u}_{mu}$ and $ u_{mu}$ charged current inclusive cross sections and their ratio with the T2K off-axis near detector

We report a measurement of cross section $sigma( u_{mu}+{rm nucleus}rightarrowmu^{-}+X)$ and the first measurements of the cross section $sigma(bar{ u}_{mu}+{rm nucleus}rightarrowmu^{+}+X)$ and their ratio $R(frac{sigma(bar u)}{sigma( u)})$ at (anti-)neutrino energies below 1.5 GeV. We determine the single momentum bin cross section measurements, averaged over the T2K $bar{ u}/ u$-flux, for the detector target material (mainly Carbon, Oxygen, Hydrogen and Copper) with phase space restricted laboratory frame kinematics of $theta_{mu}<32^circ$ and $p_{mu}>$500 MeV/c. The results are $sigma(bar{ u})=left( 0.900pm0.029{rm (stat.)}pm0.088{rm (syst.)}right)times10^{-39}$ and $sigma( u)=left( 2.41 pm0.022{rm{(stat.)}}pm0.231{rm (syst.)} right)times10^{-39}$ in units of cm$^{2}$/nucleon and $Rleft(frac{sigma(bar{ u})}{sigma( u)}right)= 0.373pm0.012{rm (stat.)}pm0.015{rm (syst.)}$.