Two-loop QCD Corrections to SemileptonicbDecays at Maximal Recoil

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