Complete $O\left({\alpha }_s^2\right)$ corrections to zero-recoil sum rules for $\overrightarrow{B}{D}^{*}$ transitions

We present the complete $O\left({\alpha }_s^2\right)$ corrections to the Wilson coefficient of the unit operator in the zero-recoil sum rule for the $\overrightarrow{B}{D}^{*}$ transition. We include both perturbative and power-suppressed nonperturbative effects in a manner consistent with the operator product expansion. The impact of these corrections on $|V_{\mathrm{cb}}|$ extracted from semileptonic $\overrightarrow{B}{D}^{*}$ decays near zero recoil is discussed. The mixing of the heavy quark kinetic operator with the unit operator at the two loop level is obtained. $O\left({\alpha }_s\right)$ corrections to a number of power-suppressed operators are calculated.