Lattice calculation of the zero-recoil form factor of $\overline{B}\to D^{*}l\overline{\nu }:$ Toward a model independent determination of $|V_{\mathrm{cb}}|$

We develop a new method in lattice QCD to calculate the form factor ${\mathcal{F}}_{\overrightarrow{B}{D}^{*}}\mathrm{}\left(1\right)\mathrm{}$ at zero recoil. This is the main theoretical ingredient needed to determine $|V_{\mathrm{cb}}|$ from the exclusive decay $\overline{B}\to D^{*}l\overline{\nu }.$ We introduce three ratios, in which most of statistical and systematic error cancels, making a precise calculation possible. We fit the heavy-quark mass dependence directly, and extract the ${1/m}_Q^2$ and three of the four ${1/m}_Q^3$ corrections in the heavy-quark expansion. In this paper we show how the method works in the quenched approximation, obtaining ${\mathcal{F}}_{\overrightarrow{B}{D}^{*}}{\mathrm{}\left(1\right)=0.913\mathrm{}}_{-0.017}^{+0.024\mathrm{}}\pm {0.016}_{-0.014-0.016-0.014}^{+0.003+0.000+0.006\mathrm{}}$ where the uncertainties come, respectively, from statistics and fitting, matching lattice gauge theory to QCD, lattice spacing dependence, light quark mass effects, and the quenched approximation. We also discuss how to reduce these uncertainties and, thus, to obtain a model-independent determination of $|V_{\mathrm{cb}}|.$