Corrections to the Bjorken and Voloshin sum rules

We calculate near zero recoil the order $\alpha_s$ corrections to the Bjorken and Voloshin sum rules that bound the $B\to D^{(*)}\ell\bar\nu$ form factors. These bounds are derived by relating the result of inserting a complete set of physical states in a time ordered product of weak currents to the operator product expansion. The sum rules sum over physical states with excitation energies less than a scale $\Delta$. We find that the corrections to the Bjorken bound are moderate, while the Voloshin bound receives sizable corrections enhanced by $\Delta/\Lambda_{QCD}$. With some assumptions, we find that the slope parameter for the form factor $h_{A_1}$ in $B\to D^*\ell\bar\nu$ decay satisfies $0.4\lesssim\rho_{A_1}^2 \lesssim 1.3$.