Charm Mass Dependence of the $\text{CO}({\alpha }_s^2{n}_f)$ Correction to Inclusive $B\to X_{c}e{\bar{\nu }}_e$ Decay

We compute the ${\alpha }_s^2{n}_f$ perturbative QCD contribution to semileptonic B decay, including the finite mass of the charm quark. This result provides an estimate of the size of the two-loop correction, which is found to be about 50% of the one-loop correction. We use these results to set the scale for the one-loop correction using the scheme of Brodsky, Lepage and Mackenzie and find a BLM scale of ${\mu }_{\mathrm{B}\mathrm{L}\mathrm{M}}=0.13m_b$, when the inclusive semileptonic rate is expressed in terms of the $b$ and $c$ quark pole masses and the \msbar strong coupling. The two loop correction lies roughly midway between that obtained at $m_c=0$ and that obtained in the Shifman-Voloshin limit $m_b, m_c\gg m_b-m_c\gg\Lambda_{\rm QCD}$ while the corresponding BLM scale is somewhat closer to that obtained in the former case.