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We present the first unquenched lattice-QCD calculation of the hadronic formfactors for the exclusive decay $\overline{B} \rightarrow D \ell\overline{\nu}$ at nonzero recoil. We carry out numerical simulations onfourteen ensembles of gauge-field configurations generated with 2+1 flavors ofasqtad-improved staggered sea quarks. The ensembles encompass a wide range oflattice spacings (approximately 0.045 to 0.12 fm) and ratios of light (up anddown) to strange sea-quark masses ranging from 0.05 to 0.4. For the $b$ and $c$valence quarks we use improved Wilson fermions with the Fermilabinterpretation, while for the light valence quarks we use asqtad-improvedstaggered fermions. We extrapolate our results to the physical point usingrooted staggered heavy-light meson chiral perturbation theory. We thenparameterize the form factors and extend them to the full kinematic range usingmodel-independent functions based on analyticity and unitarity. We present ourfinal results for $f_+(q^2)$ and $f_0(q^2)$, including statistical andsystematic errors, as coefficients of a series in the variable $z$ and thecovariance matrix between these coefficients. We then fit the latticeform-factor data jointly with the experimentally measured differential decayrate from BaBar to determine the CKM matrix element, $|V_{cb}|=(39.6 \pm1.7_{\rm QCD+exp} \pm 0.2_{\rm QED})\times 10^{-3}$. As a byproduct of thejoint fit we obtain the form factors with improved precision at large recoil.Finally, we use them to update our calculation of the ratio $R(D)$ in theStandard Model, which yields $R(D) = 0.299(11)$.

The $B \to D \ell \nu$ form factors at nonzero recoil and $|V_{cb}|$ from $2+1$-flavor lattice QCD