Perturbative corrections to zero recoil inclusive $B$ decay sum rules

Comparing the result of inserting a complete set of physical states in a time ordered product of $b$ decay currents with the operator product expansion gives a class of zero recoil sum rules. They sum over physical states with excitation energies less than $\Delta$, where $\Delta$ is much greater than the QCD scale and much less than the heavy charm and bottom quark masses. These sum rules have been used to derive an upper bound on the zero recoil limit of the $B\to D^*$ form-factor, and on the matrix element of the kinetic energy operator between $B$ meson states. Perturbative corrections to the sum rules of order $\alpha_s(\Delta) \Delta^2/m_{c,b}^2$ have previously been computed. We calculate the corrections of order $\alpha_s(\Delta)$ and $\alpha_s^2(\Delta) \beta_0$ keeping all orders in $\Delta/m_{c,b}$, and show that these perturbative QCD corrections suppressed by powers of $\Delta/m_{c,b}$ significantly weaken the upper bound on the zero recoil $B\to D^*$ form-factor, and also on the kinetic energy operator's matrix element.