Higher-Order Perturbative Corrections to $b\to c$ Transitions at Zero Recoil

We estimate the two-loop perturbative corrections to zero-recoil matrix elements of the flavour-changing currents $\bar c\,\gamma^\mu b$ and $\bar c\,\gamma^\mu\gamma_5\,b$ by calculating the terms of order $n_f\,\alpha_s^2$ and substituting the dependence on the number of flavours by the first coefficient of the $\beta$-function. Both for vector and axial vector currents, we find moderate two-loop corrections below 1\% in magnitude. Using the Brodsky--Lepage--Mackenzie prescription to set the scale in the order-$\alpha_s$ corrections in the $\overline{\rm MS}$ scheme, we obtain $\mu_V\simeq 0.92\sqrt{m_b m_c}$ and $\mu_A\simeq 0.51\sqrt{m_b m_c}$ in the two cases. These scales are sufficiently large for perturbation theory to be well-behaved. The implications of our results to the extraction of $|\,V_{cb}|$ are briefly discussed.