Measurement of the inclusive electron spectrum from $B$ meson decays and determination of $|V_{ub}|$

Based on the full BABAR data sample of 466.5 million $B\overline{B}$ pairs, we present measurements of the electron spectrum from semileptonic $B$ meson decays. We fit the inclusive electron spectrum to distinguish Cabibbo-Kobayashi-Maskawa (CKM) suppressed $B\to X_{u}e\nu$ decays from the CKM-favored $B\to X_{c}e\nu$ decays, and from various other backgrounds, and determine the total semileptonic branching fraction $\mathcal{B}(B\to Xe\nu )=(10.34\pm {0.04}_{\mathrm{stat}}\pm 0.26_{\mathrm{syst}})\%$, averaged over $B^{\pm }$ and $B^0$ mesons. We determine the spectrum and branching fraction for charmless $B\to X_{u}e\nu$ decays and extract the CKM element $|V_{ub}|$, by relying on four different QCD calculations based on the heavy quark expansion. While experimentally, the electron momentum region above $2.1\mathrm{GeV}/c$ is favored, because the background is relatively low, the uncertainties for the theoretical predictions are largest in the region near the kinematic endpoint. Detailed studies to assess the impact of these four predictions on the measurements of the electron spectrum, the branching fraction, and the extraction of the CKM matrix element $|V_{ub}|$ are presented, with the lower limit on the electron momentum varied from $0.8\mathrm{GeV}/c$ to the kinematic endpoint. We determine $|V_{ub}|$ using each of these different calculations and find, $|V_{ub}|=(3.794\pm {0.107}_{\exp }{}_{-0.219\mathrm{SF}}^{+0.292}{}_{-0.068\text{theory}}^{+0.078})\times {10}^{-3}$ (De Fazio and Neubert), $(4.563\pm {0.126}_{\exp }{}_{-0.208\mathrm{SF}}^{+0.230}{}_{-0.163\text{theory}}^{+0.162})\times {10}^{-3}$ (Bosch, Lange, Neubert, and Paz), $(3.959\pm {0.104}_{\exp }{}_{-0.154\mathrm{SF}}^{+0.164}{}_{-0.079\text{theory}}^{+0.042})\times {10}^{-3}$ (Gambino, Giordano, Ossola, and Uraltsev), $(3.848\pm {0.108}_{\exp }{}_{-0.070\text{theory}}^{+0.084})\times {10}^{-3}$ (dressed gluon exponentiation), where the stated uncertainties refer to the experimental uncertainties of the partial branching fraction measurement, the shape function parameters, and the theoretical calculations.

Figure 1

MC-generated electron momentum spectra in the $\mathrm{\Upsilon }(4S)$ rest frame for charmless semileptonic $B$ decays. The full spectrum (solid line) is normalized to 1.0. The largest contribution is from decays involving higher-mass resonances and nonresonant states ($X_u^{\mathrm{n}r}$) (dash-three-dotted). The exclusive decays (scaled by a factor of 5) are $B\to \pi e\nu$ (dash-dotted), $B\to \rho e\nu$ (dashed), $B\to \omega e\nu$ (dotted), $B\to \eta e\nu$ (long-dashed), $B\to {\eta }^{\prime }e\nu$ (long-dash-dotted).

Figure 2

MC-generated electron momentum spectra for semileptonic decays to charm mesons, $B\to X_{c}e\nu$ with the total rate (solid line) normalized to 1.0. The individual components are: $B\to De\nu$ (dash-dotted), $B\to D^{*}e\nu$ (dashed), $B\to D^{**}e\nu +B\to D^{(*)}\pi e\nu$ (dotted). The highly suppressed $B\to X_{u}e\nu$ signal spectrum (long dashed) is shown for comparison.

Figure 3

The shape function parameters $m_b$ and ${\mu }_{\pi }^2$ in the kinetic scheme (HFAG 2014): fit to $X_c$ data with constraint on the $c$-quark mass (solid line, solid triangle); fit to $X_c+X_s\gamma$ data (${\mu }_i=1.5\mathrm{GeV}$, $\mu ={\mu }_i$) (dotted line, solid square). Translation of fit to $X_c$ data with constraint on the $c$-quark mass (short dashed line, open triangle); translation of fit to $X_c+X_s\gamma$ data with ${\mu }_i=2.0\mathrm{GeV}$, $\mu ={\mu }_i$ (dash-dotted line, open square). The previous BABAR endpoint analysis [8] was based on a $X_s+X_c$ fit (long dashed line, open circle). The contours represent $\mathrm{\Delta }{\chi }^2=1$.

Figure 4

The number of events before the NN selection, as a function of (a) ${\mathcal{R}}_2$, (b) $l_2$, (c) $\cos {\theta }_{e-\mathrm{roe}}$: on-resonance data (triangles), the sum of simulated $B\overline{B}$ events and off-resonance data (solid histogram), MC simulated $B\overline{B}$ events (dashed histogram), and off-resonance data (dotted histogram). For comparison, the distributions for $B\overline{B}$ events with a signal $B\to X_{u}e\nu$ decay (dash-dotted histogram), are shown (scaled by a factor of 50). $\mathrm{Ratio}=[B\overline{B}(\mathrm{MC})+\mathrm{off}\text{−}\mathrm{resonance}]/\mathrm{on}\text{−}\mathrm{resonance}$.

Figure 5

The distribution of events as a function of the NN output: On-resonance data (triangles), the sum of MC simulated $B\overline{B}$ and off-resonance data (solid histogram), MC simulated $B\overline{B}$ events (dashed histogram), off-resonance data (dotted histogram). For comparison, the distributions for $B\overline{B}$ events with a signal $B\to X_{u}e\nu$ decay (dash-dotted histogram) are shown (scaled by a factor of 50). $\mathrm{Ratio}=[B\overline{B}(\mathrm{MC})+\mathrm{off}\text{−}\mathrm{resonance}]/\mathrm{on}\text{−}\mathrm{resonance}$.

Figure 6

Selection efficiency for $B\overline{B}$ events with a $B\to X_{u}e\nu$ decay as a function of the MC-generated electron momentum. The error bars represent the statistical uncertainties only.

Figure 7

The combined fit to off-resonance data in the momentum interval of 0.8 to $3.5\mathrm{GeV}/c$ and to on-resonance data in the momentum interval of 2.8 to $3.5\mathrm{GeV}/c$, with the constraint on $r_L/r_L^{(0)}$; (a) comparison of off-resonance data (solid squares), on-resonance data (open circles) and fitted function; (b) $(\text{Data Fit})/\sigma$: off-resonance data (solid histogram), on-resonance data (dotted histogram)

Figure 8

Electron momentum spectra in the $\mathrm{\Upsilon }(4S)$ rest frame: (a) on-resonance data (solid squares), scaled off-resonance data (solid triangles), the solid line shows the results of the fit to the continuum component using both on-resonance and off-resonance data. (b) On-resonance data with non-$B\overline{B}$ background subtracted (open squares), $B\overline{B}$ MC without $B\to X_{u}e\nu$ decays (open triangles).

Figure 9

The simulated contributions to $B\overline{B}$ events as a function of the momentum for electron candidates (a) all events (solid histogram), primary electrons (dashed histogram), secondary electrons (dotted histogram), misidentified hadrons (dash-dotted histogram). (b) Primary electrons: $B\to De\nu (\text{solid}\text{histogram})$, $B\to D^{*}e\nu$ (dashed histogram), $B\to D^{(*)}\pi e\nu$ (dotted histogram), $B\to (D_0^{*}+D_1^{*})e\nu$ (long-dashed histogram), $B\to (D_1+D_2^{*})e\nu$ (long-dash-dotted histogram), signal $B\to X_{u}e\nu$ decays (dash-dotted histogram). (c) Secondary electrons from $D^{\pm }$ (solid histogram), $D^0({\overline{D}}^0)$ (dashed histogram), $D_s$ (dotted histogram), $J/\psi$ (dash-dotted histogram), $\tau$ (long-dashed histogram), $\gamma$ conversion (long-dash-dotted histogram), other $e^{\pm }$ (dash-three-dotted histogram).

Figure 10

Inclusive $B\to Xe\nu$ decays: (a) The differential branching fraction as a function of the electron momentum [in the $\mathrm{\Upsilon }(4S)$ rest frame] after background subtraction and corrections for bremsstrahlung and final state radiation. (b) The relative statistical uncertainties on the background subtraction combined with the uncertainties of the fit parameters.

Figure 11

The differential branching fraction for charmless semileptonic $B$ decays (data points) as a function of the electron momentum [in the $\mathrm{\Upsilon }(4S)$ rest frame] after background subtraction and corrections for bremsstrahlung and final state radiation, compared to the Monte Carlo simulation (histogram). The uncertainties indicate the statistical uncertainties on the background subtraction, including the uncertainties of the fit parameters. The shaded area indicates the momentum interval for which the on-resonance data are combined into a single bin.

Figure 12

The differential branching fraction for charmless semileptonic $B$ decays (data points) as a function of the electron momentum (in the $B$ rest frame) after background subtraction and corrections for bremsstrahlung and final state radiation, compared to the Monte Carlo simulation (histogram). The uncertainties indicate the statistical uncertainties on the background subtraction, including the uncertainties of the fit parameters. The shaded area indicates the momentum interval for which the on-resonance data are combined into a single bin.

Figure 13

The comparison of the theoretical differential branching fraction for charmless semileptonic $B$ decays with normalization based on the fit as a function of the electron momentum [in the $\mathrm{\Upsilon }(4S)$ rest frame] for DN (solid), BLNP (dashed), GGOU (dotted) and DGE (dash-dotted). The shaded area indicates the momentum interval for which the on-resonance data are combined into a single bin.

Figure 14

Total branching fraction for $B\to X_{u}e\nu$ decays as a function $p_{\min }$, the lower limit of the electron momentum range used in the extraction of the signal and the total uncertainty which include experimental, SF parametrization and theoretical uncertainties, separately for DN, ${\mathrm{BLNP}}_1$, ${\mathrm{GGOU}}_1$, and DGE predictions of the decay rate used in the fit.

Figure 15

Relative uncertainties on the signal BF for DN, ${\mathrm{BLNP}}_1$, ${\mathrm{GGOU}}_1$, and DGE calculations as a function of $p_{\min }$, the lower limit of the electron momentum range used in the signal extraction, open and solid circles: total uncertainties (positive and negative uncertainties), solid line: experimental, long-dashed and long-dash-dotted lines: theoretical, dashed and dotted lines: SF uncertainty.

Figure 16

$|V_{ub}|$ as a function of $p_{\min }$, the lower limit of the momentum range used in the extraction of the signal, for DN, ${\mathrm{BLNP}}_1$, ${\mathrm{GGOU}}_1$, and DGE predictions of the decay rate.

Figure 17

Relative uncertainties on $|V_{ub}|$ for four different predictions as a function of the lower limit $p_{\min }$ of the momentum range used to extract the signal, for DN, ${\mathrm{BLNP}}_1$, ${\mathrm{GGOU}}_1$, and DGE predictions of the signal rate: total uncertainties (open and solid dots for positive and negative uncertainties), experimental (solid histogram), SF parametrization (dashed and dotted histograms for positive and negative uncertainties), and theoretical (long-dashed and long-dotted histograms for positive and negative uncertainties).

Figure 18

Results for (a) $\mathcal{B}(B\to X_{u}e\nu )$, and (b) $|V_{ub}|$ based on the momentum range $p_e=0.8–2.7\mathrm{GeV}/\mathrm{c}$ for different QCD calculations and SF parametrizations (Table 6). For BLNP and GGOU: solid squares—$X_c\ell \nu$ moment fits with $m_c$ constraint (${\mathrm{BLNP}}_1$ and ${\mathrm{GGOU}}_1$), solid triangles—$X_c\ell \nu$ and $X_s\gamma$ moment fits (${\mathrm{BLNP}}_3$ and ${\mathrm{GGOU}}_2$), open squares or triangles—the same constraints for translation of SF parameters from “kinetic” to “shape-function” scheme with ${\mu }_i=1.5\mathrm{GeV}$ (${\mathrm{BLNP}}_2$ and ${\mathrm{BLNP}}_4$).