Branching fraction and form-factor shape measurements of exclusive charmless semileptonic $B$ decays, and determination of $|V_{ub}|$

We report the results of a study of the exclusive charmless semileptonic decays, $B^0\to {\pi }^{-}{\ell }^{+}\nu$, $B^{+}\to {\pi }^0{\ell }^{+}\nu$, $B^{+}\to \omega {\ell }^{+}\nu$, $B^{+}\to \eta {\ell }^{+}\nu$, and $B^{+}\to {\eta }^{\prime }{\ell }^{+}\nu$ ($\ell =e$ or $\mu$) undertaken with approximately $462\times {10}^6$ $B\overline{B}$ pairs collected at the ${\rm Y}(4S)$ resonance with the BABAR detector. The analysis uses events in which the signal $B$ decays are reconstructed with a loose neutrino reconstruction technique. We obtain partial branching fractions in several bins of $q^2$, the square of the momentum transferred to the lepton-neutrino pair, for $B^0\to {\pi }^{-}{\ell }^{+}\nu$, $B^{+}\to {\pi }^0{\ell }^{+}\nu$, $B^{+}\to \omega {\ell }^{+}\nu$, and $B^{+}\to \eta {\ell }^{+}\nu$. From these distributions, we extract the form-factor shapes $f_{+}(q^2)$ and the total branching fractions $\mathcal{B}(B^0\to {\pi }^{-}{\ell }^{+}\nu )=(1.45\pm {0.04}_{\mathrm{stat}}\pm {0.06}_{\mathrm{syst}})\times {10}^{-4}$ (combined ${\pi }^{-}$ and ${\pi }^0$ decay channels assuming isospin symmetry), $\mathcal{B}(B^{+}\to \omega {\ell }^{+}\nu )=(1.19\pm {0.16}_{\mathrm{stat}}\pm {0.09}_{\mathrm{syst}})\times {10}^{-4}$ and $\mathcal{B}(B^{+}\to \eta {\ell }^{+}\nu )=(0.38\pm {0.05}_{\mathrm{stat}}\pm {0.05}_{\mathrm{syst}})\times {10}^{-4}$. We also measure $\mathcal{B}(B^{+}\to {\eta }^{\prime }{\ell }^{+}\nu )=(0.24\pm {0.08}_{\mathrm{stat}}\pm {0.03}_{\mathrm{syst}})\times {10}^{-4}$. We obtain values for the magnitude of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{ub}|$ by direct comparison with three different QCD calculations in restricted $q^2$ ranges of $B\to \pi {\ell }^{+}\nu$ decays. From a simultaneous fit to the experimental data over the full $q^2$ range and the FNAL/MILC lattice QCD predictions, we obtain $|V_{ub}|=(3.25\pm 0.31)\times {10}^{-3}$, where the error is the combined experimental and theoretical uncertainty.

Figure 1

Left panels: Distributions of the selection values for the $q^2$-dependent selections on the variables used in the analysis of $B^0\to {\pi }^{-}{\ell }^{+}\nu$ decays. The vertical axis represents the selection value for a given $q^2$ value. We reject an event when its value is in the shaded region. Right panels: Corresponding distributions in the total fit region illustrating the effects of the $q^2$-dependent selections. The arrows indicate the rejected regions, as explained in the text. All the selections have been applied except for the one of interest. In each panel, the signal area is scaled to the area of the total background.

Figure 2

$\Delta E\mathrm{\text{−}}{m}_{ES}$ MC distributions, summed over all bins of $q^2$, for the six categories of events used in the signal extraction fit, after all the selections have been applied, in the case of the $B^0\to {\pi }^{-}{\ell }^{+}\nu$ decay channel. Also shown is the binning used for this decay mode.

Figure 3

Comparison of the on-resonance data and MC simulation, for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ decays, after all analysis cuts and MC simulation corrections have been applied. The $Y$ signal candidates related distributions are generated from events in the $\Delta E$ and $m_{ES}$ plane with the signal-enhanced region removed. The ratios of data/MC events are presented below each panel. The general level of agreement is better than 5%.

Figure 4

Projections of the data and fit results for the $B^0\to {\pi }^{-}{\ell }^{+}\nu$ decays, in the signal-enhanced region: (a,b) $m_{ES}$ with $-0.16<\Delta E<0.20\mathrm{GeV}$; and (c,d) $\Delta E$ with $m_{ES}>5.268\mathrm{GeV}$. The distributions (a,c) and (b,d) are projections for $q^2<16{\mathrm{GeV}}^2$ and for $q^2>16{\mathrm{GeV}}^2$, respectively.

Figure 5

Projections of the data and fit results for the $B^{+}\to {\pi }^0{\ell }^{+}\nu$ decays, in the signal-enhanced region: (a,b) $m_{ES}$ with $-0.16<\Delta E<0.20\mathrm{GeV}$; and (c,d) $\Delta E$ with $m_{ES}>5.268\mathrm{GeV}$. The distributions (a,c) and (b,d) are projections for $q^2<16{\mathrm{GeV}}^2$ and for $q^2>16{\mathrm{GeV}}^2$, respectively.

Figure 6

Projections of the data and fit results for the combined $B^0\to {\pi }^{-}{\ell }^{+}\nu$ and $B^{+}\to {\pi }^0{\ell }^{+}\nu$ decays, in the signal-enhanced region: (a,b) $m_{ES}$ with $-0.16<\Delta E<0.20\mathrm{GeV}$; and (c,d) $\Delta E$ with $m_{ES}>5.268\mathrm{GeV}$. The distributions (a,c) and (b,d) are projections for $q^2<16{\mathrm{GeV}}^2$ and for $q^2>16{\mathrm{GeV}}^2$, respectively.

Figure 7

Projections of the data and fit results for the $B^{+}\to \omega {\ell }^{+}\nu$ and $B^{+}\to {\eta }^{(\prime )}{\ell }^{+}\nu$ decays, in the signal-enhanced region: (a,b,c) $m_{ES}$ with $-0.16<\Delta E<0.20\mathrm{GeV}$; and (d,e,f) $\Delta E$ with $m_{ES}>5.268\mathrm{GeV}$. The distributions (a,d), (b,e), and (c,f) are projections for the $B^{+}\to \omega {\ell }^{+}\nu$, combined $B^{+}\to \eta {\ell }^{+}\nu$, and $B^{+}\to {\eta }^{\prime }{\ell }^{+}\nu$ decays, respectively.

Figure 8

Partial $\Delta \mathcal{B}(q^2)$ spectra in 12 bins of $q^2$ for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ and 11 bins of $q^2$ for$B^{+}\to {\pi }^0{\ell }^{+}\nu$ decays. The data points are placed in the middle of each bin whose width is defined in Table 24. The smaller error bars are statistical only while the larger ones also include systematic uncertainties. The solid blue curve shows the result of the fit to the $B^0\to {\pi }^{-}{\ell }^{+}\nu$ data of the BGL [17] parametrization while the dashed red curve shows the result of the fit to the $B^{+}\to {\pi }^0{\ell }^{+}\nu$ data of the same parametrization.

Figure 9

Partial $\Delta \mathcal{B}(q^2)$ spectrum in 12 bins of $q^2$ for $B\to \pi {\ell }^{+}\nu$ decays. The data points are placed in the middle of each bin whose width is defined in Table 23. The smaller error bars are statistical only while the larger ones also include systematic uncertainties. The solid black curve shows the result of the fit to the data of the BGL [17] parametrization. The data are also compared to unquenched LQCD calculations (HPQCD [5], FNAL [6]) and a LCSR calculation [3].

Figure 10

Partial $\Delta \mathcal{B}(q^2)$ spectrum in 5 bins of $q^2$ for $B^{+}\to \omega {\ell }^{+}\nu$ decays. The data points are placed in the middle of each bin whose width is defined in Table 26. The smaller error bars are statistical only while the larger ones also include systematic uncertainties. The data are also compared to a LCSR calculation [18].

Figure 11

Partial $\Delta \mathcal{B}(q^2)$ spectrum in 5 bins of $q^2$ for $B^{+}\to \eta {\ell }^{+}\nu$ decays. The data points are placed in the middle of each bin whose width is defined in Table 27. The smaller error bars are statistical only while the larger ones also include systematic uncertainties. The data are also compared to a LCSR calculation [4].

Figure 12

Simultaneous fit of the BGL parametrization [17] to our experimental data (black solid points) and to four of the points of the FNAL/MILC predictions [6] (magenta full triangles) for the $B\to \pi {\ell }^{+}\nu$ decays. The shaded band shows the uncertainty of the fitted function. The remaining points of the FNAL/MILC predictions (magenta empty triangles) are not used in the fit.

Figure 13

$\Delta E$ yield fit projections in the signal-enhanced region, with $m_{ES}>5.2675\mathrm{GeV}$, obtained in 12 $q^2$ bins from the fit to the experimental data for combined $B^0\to {\pi }^{-}{\ell }^{+}\nu$ and $B^{+}\to {\pi }^0{\ell }^{+}\nu$ decays. The fit was done using the full $\Delta E\mathrm{\text{−}}{m}_{ES}$ fit region.

Figure 14

$m_{ES}$ yield fit projections in the signal-enhanced region, with $-0.16<\Delta E<0.20\mathrm{GeV}$, obtained in 12 $q^2$ bins from the fit to the experimental data for combined $B^0\to {\pi }^{-}{\ell }^{+}\nu$ and $B^{+}\to {\pi }^0{\ell }^{+}\nu$ decays. The fit was done using the full $\Delta E\mathrm{\text{−}}{m}_{ES}$ fit region.