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An angular analysis of the decay ${\overline{B} \rightarrow D^\ast \ell^-\overline{\nu}_\ell}$, $\ell\in\{e,\mu\}$, is reported using the full $e^+e^-$collision data set collected by the \babar experiment at the $\Upsilon(4S)$resonance. One $B$ meson from the ${\Upsilon(4S)\to B\overline{B}}$ decay isfully reconstructed in a hadronic decay mode, which constrains the kinematicsand provides a determination of the neutrino momentum vector. The kinematics ofthe semileptonic decay is described by the di-lepton mass squared, $q^2$, andthree angles. The first unbinned fit to the full four-dimensional decay rate inthe Standard Model is performed in the so-called Boyd-Grinstein-Lebed approach,which employs a generic $q^2$ parameterization of the underlying form factorsbased on crossing symmetry, analyticity and QCD dispersion relations for theamplitudes. A fit using the more model-dependent Caprini-Lellouch-Neubert (CLN)approach is performed as well. Our form factor shapes show deviations fromprevious fits based on the CLN parameterization. The latest form factors alsoprovide an updated prediction for the branching fraction ratio$\mathcal{R}(D^\ast)\equiv \mathcal{B}(\overline{B}\to D^\ast \tau^-\bar{\nu}_\tau)/\mathcal{B}({\overline{B} \rightarrow D^\ast \ell^-\overline{\nu}_\ell})=0.253 \pm 0.005$. Finally, using the well measuredbranching fraction for the ${\overline{B} \rightarrow D^\ast \ell^-\overline{\nu}_\ell}$ decay, a value of $|V_{cb}|=(38.36\pm 0.90)\times10^{-3}$is obtained that is consistent with the current world average for exclusive${\overline{B}\to D^{(\ast)}\ell^- \overline{\nu}_\ell}$ decays and remains intension with the determination from inclusive semileptonic $B$ decays to finalstates with charm.

Extraction of form Factors from a Four-Dimensional Angular Analysis of $\overline{B} \rightarrow D^\ast \ell^- \overline{\nu}_\ell$