The Subleading Isgur-Wise Form Factor $\xi_3(v\cdot v')$ and its Implications for the Decays $\bar B\to D^{(*)}\ell\,\bar\nu$

We calculate, in the framework of QCD sum rules and to next-to-leading order in perturbation theory, the universal function $\xi_3(v\cdot v')$ which appears at order $1/m_Q$ in the heavy quark expansion of meson weak decay form factors. We find that radiative corrections of order $\alpha_s$ are very important. Over the kinematic range accessible in semileptonic decays, $\xi_3(v\cdot v')$ is proportional to the leading-order Isgur-Wise function $\xi(v\cdot v')$ to very good accuracy. Taking into account all sources of uncertainty, we estimate $\xi_3/\xi=(0.6\pm 0.2)$. This reduces the theoretical uncertainty in the extraction of $|\,V_{cb}|$ from $\bar B\to D\,\ell\, \bar\nu$ transitions. A measurement of the form factor ratio $A_2/A_1$ in $\bar B\to D^*\ell\,\bar\nu$ decays can be used to test our prediction.