Higher order QCD corrections tob→cc¯s

We calculate the $O\left({\alpha }_s^2{\beta }_0\right)$ corrections to the decay rate $b\to c\overline{c}s$. For reasonable values of $m_c{/m}_b$ this term is of the same order as both the one-loop and $O\left({\alpha }_s^2{\ln }^2{m}_W{/m}_b\right)$ corrections to the decay rate. For $m_c{/m}_b=0.3\mathrm{}$ the $O\left({\alpha }_s^2{\beta }_0\right)$ corrections enhance the rate by $\sim 18\%$. We also discuss the $O\left({\alpha }_s^2{\beta }_0\right)$ corrections to $b\to c\tau {\overline{\nu }}_{\tau },$ the $B$ semileptonic branching fraction, and the charm multiplicity.