Study of $\overline{B}\to X_u\ell \overline{\nu }$ decays in $B\overline{B}$ events tagged by a fully reconstructed $B$-meson decay and determination of $|V_{ub}|$

We report measurements of partial branching fractions for inclusive charmless semileptonic $B$ decays $\overline{B}\to X_u\ell \overline{\nu }$ and the determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix element $|V_{ub}|$. The analysis is based on a sample of $467\times {10}^6$ ${\rm Y}(4S)\to B\overline{B}$ decays recorded with the BABAR detector at the PEP-II $e^{+}{e}^{-}$ storage rings. We select events in which the decay of one of the $B$ mesons is fully reconstructed and an electron or a muon signals the semileptonic decay of the other $B$ meson. We measure partial branching fractions $\Delta \mathcal{B}$ in several restricted regions of phase space and determine the CKM element $|V_{ub}|$ based on different QCD predictions. For decays with a charged lepton momentum $p_{\ell }^{*}>1.0\mathrm{GeV}$ in the $B$ meson rest frame, we obtain $\Delta \mathcal{B}=(1.80\pm {0.13}_{\mathrm{stat}}\pm {0.15}_{\mathrm{sys}}\pm {0.02}_{\mathrm{theo}})\times {10}^{-3}$ from a fit to the two-dimensional $M_X-q^2$ distribution. Here, $M_X$ refers to the invariant mass of the final state hadron $X$ and $q^2$ is the invariant mass squared of the charged lepton and neutrino. From this measurement we extract $|V_{ub}|=(4.33\pm {0.24}_{\exp }\pm {0.15}_{\mathrm{theo}})\times {10}^{-3}$ as the arithmetic average of four results obtained from four different QCD predictions of the partial rate. We separately determine partial branching fractions for ${\overline{B}}^0$ and $B^{-}$ decays and derive a limit on the isospin breaking in $\overline{B}\to X_u\ell \overline{\nu }$ decays.

Figure 1

MC distribution of the kinematic variables for which we apply restrictions sequentially as listed in Table 2, for $\overline{B}\to X_u\ell \overline{\nu }$ (solid line), $\overline{B}\to X_c\ell \overline{\nu }$ (dashed line), and “other” component (dotted line). All distributions are normalized to unity, and selection criteria have been applied cumulatively, except those affecting directly the variable shown. The arrows indicate the selection requirement for a specific variable, as described in Sec. 3c.

Figure 2

Resolution for MC simulated for signal $\overline{B}\to X_u\ell \overline{\nu }$ events passing all event selection criteria, (left) $M_{X\mathrm{reco}}-M_{X\mathrm{true}}$, (center) $q^2{}_{\mathrm{reco}}-q^2{}_{\mathrm{true}}$, and (right) $P_{+,\mathrm{reco}}-P_{+,\mathrm{true}}$. The curve shows a fit result for the sum of two Gaussian functions.

Figure 3

The $m_{\mathrm{ES}}$ distribution for the inclusive semileptonic sample, for fully reconstructed hadronic decays of $B^{-}$ (left) and ${\overline{B}}^0$ mesons (right). The solid line shows the result of the maximum-likelihood fit to signal and combinatorial backgrounds; the dashed line indicates the shape of the background described by an ARGUS function.

Figure 4

Measured distributions (data points) of (a) $M_X$, (b) $P_{+}$, (c) $q^2$ with $M_X<1.7\mathrm{GeV}$, and (d) $p_{\ell }^{*}$. Upper row: comparison with the result of the ${\chi }^2$ fit with varying bin size for the sum of two scaled MC contributions (histograms), $\overline{B}\to X_u\ell \overline{\nu }$ decays generated inside (white) or outside (light shading) the selected kinematic region, and the background (dark shading). Lower row: corresponding spectra with equal bin size after background subtraction based on the fit. The data are not corrected for efficiency.

Figure 5

Projections of measured distributions (data points) of (a) $q^2$ and (b) $M_X$ with varying bin size, for the fit to the $M_X-q^2$ distribution without constraints other than $p_{\ell }^{*}>1\mathrm{GeV}$. Upper row: comparison with the result of the ${\chi }^2$ fit to the two-dimensional $M_X-q^2$ distribution for the sum of two scaled MC contributions (histograms), $\overline{B}\to X_u\ell \overline{\nu }$ decays (white), and the background (dark shading). Lower row: corresponding spectra with equal bin size after background subtraction based on the fit. The data are not corrected for efficiency.

Figure 6

Comparison of the measured $q^2$ distributions (data points) for $M_X<1.7\mathrm{GeV}$ for charmless semileptonic decays of (a) charged and (b) neutral $B$ mesons to the results of the fit (histogram), after $\overline{B}\to X_c\ell \overline{\nu }$ and “other” background subtraction.

Figure 7

Fit results to the $M_X-q^2$ distribution for the signal-depleted sample. The $q^2$ distribution is reported separately for the four $M_X$ bins: (a) $M_X\le 1.5\mathrm{GeV}$, (b) $1.5<M_X\le 2.0\mathrm{GeV}$, (c) $2.0<M_X\le 2.5\mathrm{GeV}$, and (d) $2.5<M_X\le 3.0\mathrm{GeV}$. The three MC contributions shown here are $\overline{B}\to X_u\ell \overline{\nu }$ decays vetoed by the selection (no shading), $\overline{B}\to D\ell \overline{\nu }$, $\overline{B}\to D^{*}\ell \overline{\nu }$, and “other” background (light shading), and the $\overline{B}\to D^{**}\ell \overline{\nu }$ component as defined in the text (dark shading).

Figure 8

Comparison of data (points with statistical uncertainties) and MC (histograms) simulated distributions of (a),(d) the missing mass squared, (b),(e) the missing momentum, and (c),(f) the missing energy for $\overline{B}\to X_u\ell \overline{\nu }$ enhanced (top row) and depleted (bottom row) event samples.

Figure 9

Comparison of data (points with statistical uncertainties) and MC (histograms) simulated distributions of (a),(d) the charged track multiplicity, (b),(e) the photon multiplicity, and (c),(f) the total charge per event for $\overline{B}\to X_u\ell \overline{\nu }$ enhanced (top row) and depleted (bottom row) event samples.