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 $\bar c,\gamma^\mu\gamma_5,b$bycalculatingthetermsoforder$n_f,\alpha_s^2$ 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 $\overline{\mathrm{M}\mathrm{S}}$ 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 $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ are briefly discussed.