Two-loop corrections to the Isgur-Wise function in QCD sum rules

We complete the QCD sum rule analysis of the Isgur-Wise form factor $\xi \left(v·v^{\prime }\right)$ at next-to-leading order in renormalization-group improved perturbation theory. To this end, the exact result for the two-loop corrections to the perturbative contribution is derived using the heavy quark effective theory. Several techniques for the evaluation of two-loop integrals involving two different types of heavy quark propagators are discussed in detail, among them the methods of integration by parts and differential equations. The order-${\alpha }_s$ corrections to the Isgur-Wise function turn out to be small and well under control. At large recoil, they tend to decrease the form factor by 5-10%.