Study of $B\to \pi l\nu$ and $B\to \rho l\nu$ decays and determination of $|V_{ub}|$

We present an analysis of exclusive charmless semileptonic $B$-meson decays based on $377\times {10}^6$ $B\overline{B}$ pairs recorded with the BABAR detector at the $\mathit{{\rm Y}}(4S)$ resonance. We select four event samples corresponding to the decay modes $B^0\to {\pi }^{-}{\ell }^{+}\nu$, $B^{+}\to {\pi }^0{\ell }^{+}\nu$, $B^0\to {\rho }^{-}{\ell }^{+}\nu$, and $B^{+}\to {\rho }^0{\ell }^{+}\nu$ and find the measured branching fractions to be consistent with isospin symmetry. Assuming isospin symmetry, we combine the two $B\to \pi \ell \nu$ samples, and similarly the two $B\to \rho \ell \nu$ samples, and measure the branching fractions $\mathcal{B}(B^0\to {\pi }^{-}{\ell }^{+}\nu )=(1.41\pm 0.05\pm 0.07)\times {10}^{-4}$ and $\mathcal{B}(B^0\to {\rho }^{-}{\ell }^{+}\nu )=(1.75\pm 0.15\pm 0.27)\times {10}^{-4}$, where the errors are statistical and systematic. We compare the measured distribution in $q^2$, the momentum transfer squared, with predictions for the form factors from QCD calculations and determine the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$. Based on the measured partial branching fraction for $B\to \pi \ell \nu$ in the range $q^2<12{\mathrm{GeV}}^2$ and the most recent QCD light-cone sum-rule calculations, we obtain $|V_{ub}|=(3.78\pm {0.13}_{-0.40}^{+0.55})\times {10}^{-3}$, where the errors refer to the experimental and theoretical uncertainties. From a simultaneous fit to the data over the full $q^2$ range and the FNAL/MILC lattice QCD results, we obtain $|V_{ub}|=(2.95\pm 0.31)\times {10}^{-3}$ from $B\to \pi \ell \nu$, where the error is the combined experimental and theoretical uncertainty.

Figure 1

Simulated distributions of $q^2$ versus $E_{\ell }$ for (a) $B\to \pi \ell \nu$ and (b) $B\to \rho \ell \nu$ decays. $E_{\ell }$ is the lepton energy in the $B$-meson rest frame.

Figure 2

$q^2$ distributions for $B\to \pi \ell \nu$ (left) and $B\to \rho \ell \nu$ (right) decays, based on form-factor predictions from the ISGW2 model [14], LCSR calculations (LCSR 1 [15] and LCSR 2 [19] for $B\to \pi \ell \nu$ and LCSR [17] for $B\to \rho \ell \nu$), and the HPQCD [23] lattice calculation. The extrapolations of the QCD predictions to the full $q^2$ range are marked as dashed lines.

Figure 3

Predictions of the form factors $f_{+}$ and $f_0$ for $B\to \pi \ell \nu$ decays based on unquenched LQCD calculations by the FNAL/MILC [22] and HPQCD [23] Collaborations (data points with combined statistical and systematic errors) and LCSR calculations [15] (solid black lines). The dashed lines indicate the extrapolations of the LCSR predictions to $q^2>16{\mathrm{GeV}}^2$.

Figure 4

$q^2$ resolution for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ (left) and $B^0\to {\rho }^{-}{\ell }^{+}\nu$ (right) samples after the full event selection: two-dimensional distribution of $q_{\mathrm{true}}^2$ versus $q_{\mathrm{corr}}^2$ in the fit region and in the signal region. Top row: true signal decays; middle row: all signal decays (true and combinatorial); bottom row: distribution of $q_{\mathrm{corr}}^2-q_{\mathrm{true}}^2$ for true signal (black, solid histogram) and combinatorial signal (blue, dashed histogram) in the fit region. The fit of the sum of two Gaussian functions to the true signal distribution is shown as a solid red line; the contribution of the broader of the two functions is shown as a dotted red line.

Figure 5

Distributions of $\Delta E$ and $m_{\mathrm{ES}}$ for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ (left) and $B^0\to {\rho }^{-}{\ell }^{+}\nu$ (right) candidates after the preselection, i.e., prior to the neural-network application. The stacked histograms show the predicted signal and background contributions prior to the fit. The expected signal distribution (with arbitrary normalization) is indicated as a magenta dashed histogram.

Figure 6

Background suppression for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ candidates. Distributions of the seven input variables to the neural network: after the preselection (left column), after the $q\overline{q}$ neural network (center column), and after the $B\to X_c\ell \nu$ neural network (right column). The data are compared to the sum of the MC-simulated background contributions; for a legend, see Fig. 5. The expected signal distribution is overlaid as a magenta, dashed histogram with arbitrary normalization.

Figure 7

The $q\overline{q}$ neural-network discriminators for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ candidates in the signal region, $-0.15<|\Delta E|<0.25\mathrm{GeV}$; $5.255<m_{\mathrm{ES}}<5.295\mathrm{GeV}$. The distributions are shown for four different $q^2$ bins, columns from left to right: $0<q^2<4{\mathrm{GeV}}^2$, $4<q^2<8{\mathrm{GeV}}^2$, $12<q^2<16{\mathrm{GeV}}^2$, $q^2>20{\mathrm{GeV}}^2$. Top row: Discriminator distributions for signal (magenta, dashed line) and $q\overline{q}$ background (blue, solid line), normalized to the same area. The arrows indicate the chosen cuts. Bottom row: Discriminator distributions for data compared with MC-simulated signal and background contributions. For a legend see Fig. 5.

Figure 8

The $B\to X_c\ell \nu$ neural-network discriminators for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ candidates in the signal region, $-0.15<|\Delta E|<0.25\mathrm{GeV}$; $5.255<m_{\mathrm{ES}}<5.295\mathrm{GeV}$. The distributions are shown for four different $q^2$ bins, columns from left to right: $0<q^2<4{\mathrm{GeV}}^2$, $4<q^2<8{\mathrm{GeV}}^2$, $12<q^2<16{\mathrm{GeV}}^2$, $q^2>20{\mathrm{GeV}}^2$. Top row: Discriminator distributions for signal (magenta, dashed line) and $B\to X_c\ell \nu$ background (blue, solid line), normalized to the same area. The arrows indicate the chosen cuts. Bottom row: Discriminator distributions for data compared with MC-simulated signal and background contributions. For a legend see Fig. 5.

Figure 9

The $B\to X_u\ell \nu$ neural-network discriminators for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ candidates in the signal region, $-0.15<|\Delta E|<0.25\mathrm{GeV}$; $5.255<m_{\mathrm{ES}}<5.295\mathrm{GeV}$. The distributions are shown for the three highest $q^2$ bins, columns from left to right: $12<q^2<16{\mathrm{GeV}}^2$, $16<q^2<20{\mathrm{GeV}}^2$, $q^2>20{\mathrm{GeV}}^2$. Top row: Discriminator distributions for signal (magenta, dashed line) and $B\to X_u\ell \nu$ background (blue, solid line), normalized to the same area. The arrows indicate the chosen cuts. Bottom row: Discriminator distributions for data compared with MC-simulated signal and background contributions. For a legend see Fig. 5.

Figure 10

The $q\overline{q}$ neural-network discriminators for $B^0\to {\rho }^{-}{\ell }^{+}\nu$ candidates in the signal region, $-0.15<|\Delta E|<0.25\mathrm{GeV}$; $5.255<m_{\mathrm{ES}}<5.295\mathrm{GeV}$. The distributions are shown for three different $q^2$ bins, columns from left to right: $0<q^2<8{\mathrm{GeV}}^2$, $8<q^2<16{\mathrm{GeV}}^2$, $q^2>16{\mathrm{GeV}}^2$. Top row: Discriminator distributions for signal (magenta, dashed line) and $q\overline{q}$ background (blue, solid line), normalized to the same area. The arrows indicate the chosen cuts. Bottom row: Discriminator distributions for data compared with MC-simulated signal and background contributions. For a legend see Fig. 5.

Figure 11

The $B\to X_c\ell \nu$ neural-network discriminators for $B^0\to {\rho }^{-}{\ell }^{+}\nu$ candidates in the signal region, $-0.15<|\Delta E|<0.25\mathrm{GeV}$; $5.255<m_{\mathrm{ES}}<5.295\mathrm{GeV}$. The distributions are shown for three different $q^2$ bins, columns from left to right: $0<q^2<8{\mathrm{GeV}}^2$, $8<q^2<16{\mathrm{GeV}}^2$, $q^2>16{\mathrm{GeV}}^2$. Top row: Discriminator distributions for signal (magenta, dashed line) and $B\to X_c\ell \nu$ background (blue, solid line), normalized to the same area. The arrows indicate the chosen cuts. Bottom row: Discriminator distributions for data compared with MC-simulated signal and background contributions. For a legend see Fig. 5.

Figure 12

Comparison of off-resonance background to $B^0\to {\pi }^{-}{\ell }^{+}\nu$ samples for data and MC distributions. Top row: $m_{\mathrm{ES}}$, center row: $\Delta E$, and bottom row: $q^2$, separately for the electron (left column) and muon (right column) samples. The shaded histograms indicate the true leptons; the hatched histograms indicate the fake leptons. The distributions are obtained from the full event selection, except for the $q\overline{q}$ neural-network discrimination. Linear corrections have been applied to the simulation.

Figure 13

Comparison of off-resonance background to $B^0\to {\rho }^{-}{\ell }^{+}\nu$ samples for data and MC distributions. Top row: $m_{\mathrm{ES}}$, center row: $\Delta E$, and bottom row: $q^2$, separately for the electron (left column) and muon (right column) samples. The shaded histograms indicate the true leptons; the hatched histograms indicate the fake leptons. The distributions are obtained from the full event selection, except for the $q\overline{q}$ neural-network discrimination. Linear corrections have been applied to the simulation.

Figure 14

Comparison of data with MC distributions for the charm-enhanced selection for the $B^0\to {\pi }^{-}{\ell }^{+}\nu$ sample. Top row: $m_{\mathrm{ES}}$ and $\Delta E$ for the signal bands, center row: $m_{\mathrm{ES}}$ and $\Delta E$ for the side bands, and bottom row: $q_{\mathrm{corr}}^2$ for the whole fit region. The bin-by-bin ratio of data over the sum of all MC contributions is given in the plots below each histogram.

Figure 15

Comparison of data with MC distributions for the charm-enhanced selection for the $B^0\to {\rho }^{-}{\ell }^{+}\nu$ sample. Top row: $m_{\mathrm{ES}}$ and $\Delta E$ for the signal bands, center row: $m_{\mathrm{ES}}$ and $\Delta E$ for the side bands, and bottom row: $q_{\mathrm{corr}}^2$ for the whole fit region. The bin-by-bin ratios of data over the sum of all MC contributions is given in the plots below each histogram.

Figure 16

Comparison of data and MC-simulated distributions for the $B^0\to D^{*-}{\ell }^{+}\nu$, ${\overline{D}}^0\to K^{+}{\pi }^{-}$ sample, after selection criteria have been applied on all variables except the one presented. (a) $\cos {\theta }_{BY}$, (b) $p_{\mathrm{miss}}$, (c) ${\theta }_{\mathrm{miss}}$, (d) $\cos {\theta }_{W\ell }$, (e) $m_{\mathrm{miss}}^2/2E_{\mathrm{miss}}$, and (f) the total charge per event $Q_{\mathrm{tot}}$. The background to the sample is indicated as a shaded (yellow) histogram. The combinatorial signal contribution is indicated as a dashed histogram.

Figure 17

Comparison of data and MC-simulated distributions for the $B^0\to D^{*-}{\ell }^{+}\nu$, ${\overline{D}}^0\to K^{+}{\pi }^{-}$ sample, after all selection cuts have been applied, (a) $\Delta E$ for events in the $m_{\mathrm{ES}}$ signal band, (b) $m_{\mathrm{ES}}$ for events in the $\Delta E$ signal band, (c) $\Delta E$ for events in the $m_{\mathrm{ES}}$ side band, and (d) $m_{\mathrm{ES}}$ for events in the $\Delta E$ side bands. The background to the sample is indicated as a shaded (yellow) histogram. The combinatorial signal contribution is indicated as a dashed histogram.

Figure 18

Comparison of the data and MC simulation of $q^2$ distributions for the $B^0\to D^{*-}{\ell }^{+}\nu$, ${\overline{D}}^0\to K^{+}{\pi }^{-}$ sample after all selection criteria have been applied, (a) the raw $q^2$, and (b) the corrected $q^2$. The background to the sample is indicated as a shaded (yellow) histogram. The combinatorial signal contribution is indicated as a dashed histogram.

Figure 19

Distribution of $\Delta E$ vs $m_{\mathrm{ES}}$ for true $B^0\to {\pi }^{-}{\ell }^{+}\nu$ signal events (left) and definition of bins in the $\Delta E-m_{\mathrm{ES}}$ plane used in the fit for all samples (right).

Figure 20

$m_{\mathrm{ES}}$ and $\Delta E$ distributions in each $q^2$ bin for $B^0\to {\pi }^{-}{\ell }^{+}\nu$ after the fit. The distributions are shown in the $\Delta E$ and $m_{\mathrm{ES}}$ signal bands, respectively. Legend: see Fig. 5.

Figure 21

$m_{\mathrm{ES}}$ and $\Delta E$ distributions in each $q^2$ bin for $B^{+}\to {\pi }^0{\ell }^{+}\nu$ after the fit. The distributions are shown in the $\Delta E$ and $m_{\mathrm{ES}}$ signal bands, respectively. Legend: see Fig. 5.

Figure 22

$m_{\mathrm{ES}}$ and $\Delta E$ distributions in each $q^2$ bin for $B^0\to {\rho }^{-}{\ell }^{+}\nu$ after the fit. The distributions are shown in the $\Delta E$ and $m_{\mathrm{ES}}$ signal bands, respectively. Legend: see Fig. 5.

Figure 23

$m_{\mathrm{ES}}$ and $\Delta E$ distributions in each $q^2$ bin for $B^{+}\to {\rho }^0{\ell }^{+}\nu$ after the fit. The distributions are shown in the $\Delta E$ and $m_{\mathrm{ES}}$ signal bands, respectively. Legend: see Fig. 5.

Figure 24

Measured $\Delta \mathcal{B}/\Delta q^2$ distribution for $B\to \pi \ell \nu$. The vertical error bars correspond to the combined statistical and systematic uncertainties. The positions of the data points have been adjusted to correspond to the mean $q^2$ value in each bin, based on the quadratic BGL ansatz. Left: fits of four different form-factor parametrizations to the $\Delta \mathcal{B}/\Delta q^2$ data spectrum. The fit result for the BZ and BCL parametrizations are barely visible, since they overlap almost completely with the BGL result. The shaded band illustrates the uncertainty of the quadratic BGL fit to data. Right: shape comparisons of the data to various $B\to \pi \ell \nu$ form-factor predictions (LCSR 1 [15], LCSR 2 [19], HPQCD [23], ISGW2 [14]), which have been normalized to the measured total branching fraction. The extrapolations of the QCD predictions to the full $q^2$ range are marked as dashed lines.

Figure 25

Measured $\Delta \mathcal{B}/\Delta q^2$ distribution for $B\to \rho \ell \nu$. The inner and outer error bars correspond to the statistical uncertainty and the combined statistical and systematic uncertainty, respectively. The data are compared with the $B\to \rho \ell \nu$ form-factor predictions from LCSR [17] and from the ISGW2 quark model [14].

Figure 26

Simultaneous fits of the BGL parametrization to data (solid points with vertical error bars representing the total experimental uncertainties) and to four of the 12 points of the FNAL/MILC lattice prediction (magenta, closed triangles). Left: linear ($2+1$-parameter) BGL fit, right: quadratic ($3+1$-parameter) BGL fit. The LQCD results are rescaled to the data according to the $|V_{ub}|$ value obtained in the fit. The shaded band illustrates the uncertainty of the fitted function. For comparison, the HPQCD (blue, open squares) lattice results are also shown. They are used in an alternate fit.