Two-loop QCD Corrections to $b \to c$ Transitions at Zero Recoil: Analytical Results

We present analytical results for the $O(\alpha_s^2)$ contributions to the functions $\eta _A$ and $\eta _V$ which parameterize QCD corrections to semileptonic $b \to c$ transitions at zero recoil. Previously obtained approximate results are confirmed. The methods of computing the relevant two-loop diagrams with two mass scales are discussed in some detail.