Order ${\alpha }_s^2{\beta }_0$ correction to the charged lepton spectrum in $b\to c\mathcal{l}{\overline{\nu }}_{\mathcal{l}}$ decays

We compute the ${\alpha }_s^2{\beta }_0$ part of the two-loop QCD corrections to the charged lepton spectrum in $b\to c\mathcal{l}{\overline{\nu }}_{\mathcal{l}}$ decays and find them to be about 50% of the first order corrections at all lepton energies, except those close to the end point. Including these corrections we extract the central values $\overline{\Lambda }=0.33\mathrm{}\hspace{0.5em}\mathrm{GeV}$ and ${\lambda }_1=-0.17{\mathrm{\hspace{0.5em}}\mathrm{GeV}}^2$ for the HQET matrix elements and use them to determine the $\overline{\mathrm{MS}}$ $b$ and $c$ quark masses, and $|V_{\mathrm{cb}}|$.