Short-distance expansion of heavy-quark currents

Leading and next-to-leading QCD corrections to current-induced transitions of heavy quarks are analyzed in renormalization-group-improved perturbation theory, employing an effective field theory approach. The equivalence of different ways in which the transition from QCD to the effective theory can be performed is emphasized. We develop a consistent scheme that allows us to take into account the full dependence on the heavy-quark masses, while leading and subleading logarithms are summed to all orders in perturbation theory. Our results are accurate up to corrections of order ${\alpha }_s^2{\left(z \mathrm{In}z\right)}^n$ with $z\le 1$ being the ratio of the heavy-quark masses, and $n=0,1,2\mathrm{}$. The application to hadronic matrix elements of currents between heavy baryon or meson states is discussed in detail. In the heavy-quark-mass limit, the associated form factors can be written as products of a universal renormalized function, which is independent of the heavy-quark masses and normalized at zero recoil, with nonuniversal short-distance coefficients. As an example, we calculate the QCD correction to the $\overline{B}\to D^{*}\ell {\overline{\nu }}_{\ell }$ decay rate at zero recoil and obtain, from a comparison to experimental data, an updated value for $V_{\mathrm{cb}}$.