Semileptonicb→uDecays: Lepton Invariant Mass Spectrum

We compute $O\left({\alpha }_s^2\right)$ QCD corrections to the lepton invariant mass spectrum in the decay $b\to \mathrm{ul}{\nu }_l$, relevant for the determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{\mathrm{ub}}|$. Our method can also be used to evaluate moments of the lepton energy distribution with an $O\left({\alpha }_s^2\right)$ accuracy. The Abelian part of our result gives the neutrino invariant mass spectrum in the muon decay and, upon integration, the $O\left({\alpha }^2\right)$ correction to the muon lifetime.