We report the measurement of muon neutrino charged-current interactions on carbon without pions in the final state at the T2K beam energy using 5.734$times10^{20}$ protons on target. For the first time the measurement is reported as a flux-integrated, double-differential cross-section in muon kinematic variables ($costheta_mu$, $p_mu$), without correcting for events where a pion is produced and then absorbed by final state interactions. Two analyses are performed with different selections, background evaluations and cross-section extraction methods to demonstrate the robustness of the results against biases due to model-dependent assumptions. The measurements compare favorably with recent models which include nucleon-nucleon correlations but, given the present precision, the measurement does not solve the degeneracy between different models. The data also agree with Monte Carlo simulations which use effective parameters that are tuned to external data to describe the nuclear effects. The total cross-section in the full phase space is $sigma = (0.417 pm 0.047 text{(syst)} pm 0.005 text{(stat)})times 10^{-38} text{cm}^2$ $text{nucleon}^{-1}$ and the cross-section integrated in the region of phase space with largest efficiency and best signal-over-background ratio ($costheta_mu>0.6$ and $p_mu > 200$ MeV) is $sigma = (0.202 pm 0.0359 text{(syst)} pm 0.0026 text{(stat)}) times 10^{-38} text{cm}^2$ $text{nucleon}^{-1}$.
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