Determination of the form factors for the decay $B^0\to D^{*-}{l}^{+}{\nu }_l$ and of the CKM matrix element $|V_{cb}|$

We present a combined measurement of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ and of the parameters ${\rho }^2$, $R_1(1)$, and $R_2(1)$, which fully characterize the form factors for the $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ decay in the framework of heavy-quark effective field theory. The results, based on a selected sample of about 52 800 $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ decays, recorded by the BABAR detector, are ${\rho }^2=1.157\pm 0.094\pm 0.027$, $R_1(1)=1.327\pm 0.131\pm 0.043$, $R_2(1)=0.859\pm 0.077\pm 0.021$, and $\mathcal{F}(1)|V_{cb}|=(34.7\pm 0.4\pm 1.0)\times {10}^{-3}$. The first error is the statistical and the second is the systematic uncertainty. Combining these measurements with the previous BABAR measurement of the form factors, which employs a different fit technique on a partial sample of the data, we improve the statistical precision of the result, ${\rho }^2=1.191\pm 0.048\pm 0.028$, $R_1(1)=1.429\pm 0.061\pm 0.044$, $R_2(1)=0.827\pm 0.038\pm 0.022$, and $\mathcal{F}(1)|V_{cb}|=(34.4\pm 0.3\pm 1.1)\times {10}^{-3}$. Using lattice calculations for the axial form factor $\mathcal{F}(1)$, we extract $|V_{cb}|=(37.4\pm 0.3\pm 1.2{\pm }_{1.4}^{1.2})\times {10}^{-3}$, where the third error is due to the uncertainty in $\mathcal{F}(1)$. We also present a measurement of the exclusive branching fraction, $\mathcal{B}=(4.69\pm 0.04\pm 0.34)\%$.

Figure 1

Quark-level Feynman diagram for the decay $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$.

Figure 2

Definition of the angles ${\theta }_{\ell }$, ${\theta }_V$, and $\chi$ for the $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ decay, mediated by a vector boson $W$; ${\pi }_s$ refers to the low-momentum pion from the decay $D^{*-}\to {\overline{D}}^0{\pi }_s^{-}$.

Figure 3

$\Delta m$ distributions for the decays $D^{*-}\to {\overline{D}}^0{\pi }_s^{-}$ with ${\overline{D}}^0\to K^{+}{\pi }^{-}$ for events in which the ${\pi }_s^{-}$ track is reconstructed in the SVT only, (a) for on-resonance, (b) for off-resonance data in which an electron or muon was identified, and (c) for on-resonance data in which no charged lepton was found in the event. The data (points with statistical errors), integrated over the full $w$ range, are compared to the result of the simultaneous fit (solid line) to all three distributions. The shaded area shows the fitted combinatorial background; the remainder (white area) represents the sum of the signal and peaking backgrounds.

Figure 4

The $\cos {\theta }_{B,D^{*}\ell }$ distributions (points with statistical errors) for selected events in the six subsamples (integrated over the full range in $w$): (a,b) $D\to K\pi$, (c,d) $D\to K\pi \pi \pi$, and (e,f) $D\to K\pi {\pi }^0$, compared to the result of the fits to signal and background contributions.

Figure 5

Comparison of the measured distributions (data points), (a) $w$, (b) $\cos {\theta }_{\ell }$, (c) $\cos {\theta }_V$, and (d) $\chi$, with the result of the fit, shown as the sum of the fitted signal yield and the estimated background distributions. The statistical uncertainties of the data are too small to be visible.

Figure 6

The measured $w$ dependence of $\mathcal{F}(w)|V_{cb}|$ (data points) compared to the theoretical function with the fitted parameters (solid line). The experimental uncertainties are too small to be visible.

Figure 7

Correlations between fitted variables and their uncertainties, both for the present analysis with the statistical uncertainties (dotted line, central value marked with an open circle) and for the combined result with the total experimental uncertainties, i.e. the statistical and systematic uncertainties combined (solid line, central value marked with a solid dot): (a) ${\rho }^2-R_1(1)$, (b) ${\rho }^2-R_2(1)$, (c) ${\rho }^2-\mathcal{F}(1)|V_{cb}|\times {10}^3$, (d) $R_1(1)-R_2(1)$, (e) ${\mathcal{R}}_1(1)-\mathcal{F}(1)|V_{cb}|\times {10}^3$, and (f) $R_2(1)-\mathcal{F}(1)|V_{cb}|\times {10}^3$ projections. The contours correspond to $\Delta {\chi }^2=1$, i.e. 39% CL.