Measurement of the $ \bar{B}^0 \to D*^+ l^- \bar{nu}_l$ Decay Rate and $|V_{cb}|$

We present a measurement of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ based on a sample of about 53,700 \mbox{$\Bzb \rightarrow D^{*+} \ell^- \bar{\nu}_{\ell}$} decays observed by the \babar\ detector. We obtain the branching fraction averaged over $\ell = e,\mu$, ${\cal B}(\Bzb \rightarrow D^{*+} \ell^- \bar{\nu}_{\ell}) = (4.90 \pm 0.07\mathrm{(stat.)}^{+0.36}_{-0.35}\mathrm{(syst.)})\%$. We measure the differential decay rate as a function of $w$, the relativistic boost $\gamma$ of the $D^{*+}$ in the ${\Bzb}$ rest frame. By extrapolating $d\Gamma/dw$ to the kinematic limit $w \rightarrow 1$, we extract the product of $|V_{cb}|$ and the axial form factor ${\cal A}_1(w=1)$. We combine this measurement with a lattice QCD calculation of ${\cal A}_1(w=1)$ to determine $ |V_{cb}|$ = (38.7 \pm 0.3 \mathrm{(stat.)} \pm 1.7 \mathrm{(syst.)} ^{+1.5} _{-1.3} \mathrm{(theory)})\times 10^{-3} . $