Measurements of the $B\to D^{*}$ form factors using the decay ${\overline{B}}^0\to D^{*+}{e}^{-}{\overline{\nu }}_e$

We measure the dependence of ${\overline{B}}^0\to D^{*+}{e}^{-}{\overline{\nu }}_e$ on the decay angles and momentum transfer. The data sample consists of $\sim 86\times {10}^6$ $B\overline{B}$-pairs accumulated on the $\mathit{{\rm Y}}(4\mathrm{S})$ resonance by the BABAR detector at the asymmetric $e^{+}{e}^{-}$ collider PEP-II. We specify the three form factors by two ratios $R_1$ and $R_2$, and by a single parameter ${\rho }^2$ characterizing the polynomial representing $h_{A_1}$, the function which describes the momentum-transfer dependence of the form factor $A_1$. We determine $R_1$, $R_2$, and ${\rho }^2$ using an unbinned maximum likelihood fit to the full decay distribution. The results are $R_1=1.396\pm 0.060\pm 0.035\pm 0.027$, $R_2=0.885\pm 0.040\pm 0.022\pm 0.013$, and ${\rho }^2=1.145\pm 0.059\pm 0.030\pm 0.035$. The stated uncertainties are the statistical from the data, statistical from the size of the Monte Carlo sample and the systematic uncertainty, respectively. In addition, based on this measurement, we give an updated value for the CKM matrix element $|V_{cb}|$.

Figure 1

Quark-level diagram showing the weak interaction vertices in the decay $\overline{B}0\to D^{*+}\ell -\overline{\nu }$.

Figure 2

Kinematics of a $B\to D^{*}\ell \nu$ decay, mediated by a weak virtual intermediate vector boson $W$. This diagram defines the three kinematic variable angles ${\theta }_{\ell }$, ${\theta }_V$ and $\chi$.

Figure 3

Reconstructed particle directions and $\cos {\theta }_{BY}$ in the CM frame. The momentum vectors ${\overrightarrow{\mathbf{P}}}_{D^{*}}$ and ${\overrightarrow{\mathbf{P}}}_{\ell }$ are measured. Their sum is ${\overrightarrow{\mathbf{P}}}_Y$. The magnitude of the $B$ momentum, ${\wp }_B\equiv |{\overrightarrow{\mathbf{P}}}_B|$, and the angle ${\theta }_{BY}$ the $B$ makes with ${\overrightarrow{\mathbf{P}}}_Y$ are known, but the azimuthal orientation ${\varphi }_{BY}$ of ${\overrightarrow{\mathbf{P}}}_B$ around ${\overrightarrow{\mathbf{P}}}_Y$ is not known. The points at ${\varphi }_{BY}=0$, $\pi$ are in the $D^{*}-\ell$ plane.

Figure 4

Monte Carlo assessment of the experimental resolution for the variables $w$, $\cos {\theta }_{\ell }$, $\cos {\theta }_V$, and $\chi$. For each variable the difference between reconstructed and generated values is shown. The resolutions are small compared to the kinematic ranges of the variables as shown in Fig. 5.

Figure 5

One-dimensional projections from Monte Carlo simulation of kinematic variables $w$, $\cos {\theta }_{\ell }$, $\cos {\theta }_V$, and $\chi$ for selected events. The blank region represents signal, and the shaded regions are estimated backgrounds; the histograms are a sum of both the signal and the estimated background beneath them.

Figure 6

Events vs the cosine of the slow pion angle relative to the line-of-flight of the $D^{*}$, shown for bins of CM momentum of the $D^{*}$. The momentum ranges for each plot, in $\mathrm{GeV}/c$, are indicated in the upper left-hand corners. Data (points) and MC simulation (histogram) are shown for slow pions from the decay of inclusively produced $D^{*}$ mesons. The vertical scale is arbitrary, but corresponds to the number of background-subtracted reconstructed $D^{*}$ candidates in a given bin. Only the differences between data and MC points are relevant for our purposes. The errors are comparable to the dots representing the data. The simulation has been weighted by restricted functions to fit to the data as described in the text.

Figure 7

The fit (solid line) to the $\Delta m$ distribution for $D^{*}e\nu$ candidates as described in the text. The distribution is shown logarithmically to emphasize details of the functional fit, but the actual level of background is more clearly seen in the linear data plot shown in Fig. 16.

Figure 8

The $\cos {\theta }_{BY}$ data distribution and the result of the fit with the estimated signal and background contributions, as described in the text. Shading indicates the various components in the fit.

Figure 9

Ratio of data to Monte Carlo events for combinatorial- background for the four kinematic variables $w$, $\cos {\theta }_{\ell }$, $\cos {\theta }_V$, and $\chi$, obtained from 5-bin fits to data and Monte Carlo in each kinematic variable. The lines are result of linear fits (see text).

Figure 10

Ratio of data to Monte Carlo events for $D^{**}$ background for the four kinematic variables $w$, $\cos {\theta }_{\ell }$, $\cos {\theta }_V$, and $\chi$, obtained from 5-bin fits to data and Monte Carlo in each kinematic variable. The lines are result of linear fits (see text).

Figure 11

Plots showing empty regions (due primarily to cuts) in the phase space of the kinematic variables $w$, $\cos {\theta }_{\ell }$, and $\cos {\theta }_V$ using the Monte Carlo sample: (a) shows $\cos {\theta }_{\ell }$ vs $w$, where the high $w$, low $\cos {\theta }_{\ell }$ region is removed in the goodness-of-fit calculation; (b) shows $\cos {\theta }_V$ vs $w$, the high $\cos {\theta }_V$, low $w$ region is removed in the goodness-of-fit calculation.

Figure 12

Background-subtracted data (points) overlaid on Monte Carlo (histograms) for all four kinematic variable distributions and for the electron momentum ($p_{\ell }^{*}$) and transverse momentum ($p_t$) of the slow pion for our best fit. In the bottom panel of each figure is shown the pull (difference over error) plot. The line in the pull plot is not a fit, but is the line at zero shown for comparison (similarly for all following pull plots).

Figure 13

Background-subtracted data (points) overlaid on Monte Carlo (histograms) and pull plots for $\cos {\theta }_{\ell }$ for six $\chi$ cuts.

Figure 14

Background-subtracted data (points) overlaid on Monte Carlo (histograms) and pull plots for $\cos {\theta }_V$ for six $\chi$ bins. These plots illustrate our sensitivity to interference effects.

Figure 15

${\chi }^2$ of fit vs fractions of combinatorial ($f_{\mathrm{comb}}$), $D^{**}(f_{D^{**}})$ and other peaking backgrounds ($f_{\mathrm{otherPk}}$). From this we see that our method of ascertaining the goodness-of-fit is independently indicating that the values we obtained for these fractions from the $\cos {\theta }_{BY}$ and $\Delta m$ fits were correct.

Figure 16

The $\Delta m$ data distribution for $D^{*}e\nu$ candidates on a linear scale.

Figure 17

The $\Delta m$ distribution for combinatorial background selected from the Monte Carlo with fit to the unextended threshold function (a) and to the extended function.

Figure 18

The $\Delta m$ distribution for (a) pure $D^{*}$ signal selected from the Monte Carlo with fit to five Gaussians and (b) the $\Delta m$ distributions on the full Monte Carlo sample, fit to the five Gaussian signal shape plus extended threshold function.

Figure 19

The $\Delta m$ distribution for data in kinematic variable bins. The rows are fits for $w$, $\cos {\theta }_{\ell }$, $\cos {\theta }_V$ and $\chi$ in each of the five bins spanning the full range of the variable.

Figure 20

The $\cos {\theta }_{BY}$ distributions fit to background plus signal in five bins in each of the kinematic variables. Each row shows the plots for the five bins corresponding to the variables $w$, $\cos {\theta }_{\ell }$, $\cos {\theta }_V$ and $\chi$. The shading is the same as that in Fig. 10.