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Two-loop QCD corrections to semileptonic b decays at maximal recoil

We present a complete $O(α_s^2)$ correction to the differential width of the inclusive semileptonic decay $b\to clν_l$ at the kinematical point of vanishing invariant mass of the leptons, $q^2=0$. Together with the recently computed $O(α_s^2)$ correction at the upper boundary of the lepton invariant mass spectrum, this new information permits an estimate of the $O(α_s^2)$ effect in the total inclusive semileptonic decay width $b\to clν_l$. We argue that the non-BLM part of the $O(α_s^2)$ correction gives at most 1% correction to the inclusive semileptonic decay width $b\to clν_l$. This significantly improves the credibility of extracting $|V_{cb}|$ from the inclusive semileptonic decays of the b-hadrons.