Higher-order perturbative corrections to b --> 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--Mack enzie 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.