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

Abstract

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 $\overline{\mathrm{MS}}$ 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.

Comments: 9 pages LaTeX, CERN-TH.7454/94