Measurement and interpretation of moments in inclusive semileptonic decays $\overline{B}\to X_c{\ell }^{-}\overline{\nu }$

We present results for the moments of observed spectra in inclusive semileptonic $B$-meson decays to charm hadrons $\overline{B}\to X_c{\ell }^{-}\overline{\nu }$. Moments of the hadronic-mass and the combined mass-and-energy spectra for different minimum electron or muon momenta between 0.8 and $1.9\mathrm{GeV}/c$ are obtained from a sample of $232\times {10}^6$ ${\rm Y}(4S)\to B\overline{B}$ events, collected with the BABAR detector at the PEP-II asymmetric-energy $B$-meson factory at SLAC. We also present a reevaluation of the moments of electron-energy spectra and partial decay fractions $\mathcal{B}(\overline{B}\to X_c{e}^{-}\overline{\nu })$ for minimum electron momenta between 0.6 and $1.5\mathrm{GeV}/c$ based on a sample of $51\times {10}^6$ ${\rm Y}(4S)\to B\overline{B}$ events. The measurements are used for the extraction of the total decay fraction, the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cb}|$, the quark masses $m_b$ and $m_c$, and four heavy-quark QCD parameters in the framework of a Heavy-Quark Expansion (HQE). We find $\mathcal{B}(\overline{B}\to X_c{\ell }^{-}\overline{\nu })=(10.64\pm 0.17\pm 0.06)\%$ and $|V_{cb}|=(42.05\pm 0.45\pm 0.70)\times {10}^{-3}$.

Figure 1

The $m_{\mathrm{ES}}$ spectrum of $B_{\mathrm{tag}}$ decays accompanied by a lepton with $p_{\ell }^{*}\ge 0.8\mathrm{GeV}/c$. The fit functions for the sum of signal and background (solid line) and the background (red dashed line) are overlaid. The crossed area shows the predicted background under the $B_{\mathrm{tag}}$ signal. The background control region in the $m_{\mathrm{ES}}$ sideband is indicated by the hatched area.

Figure 2

Hadronic-mass spectra after the kinematic fit for lepton momenta $p_{\ell }^{*}\ge 0.8\mathrm{GeV}/c$ (top) and $p_{\ell }^{*}\ge 1.9\mathrm{GeV}/c$ (bottom) together with distributions of combinatorial background and background from non-$B\overline{B}$ decays (red, hatched area) as well as residual background (blue, crossed area). The two background histograms are plotted on top of each other.

Figure 3

Examples of calibration functions for $⟨m_X^2⟩$ in bins of $N_{X_c}$, $E_{\mathrm{miss}}-c{p}_{\mathrm{miss}}$ and $p_{\ell }^{*}$. Shown are the extracted moments $⟨m_{X,\mathrm{reco}}^2⟩$ versus the true moments $⟨m_{X,\mathrm{true}}^2⟩$ for $0.8\le p_{\ell }^{*}<0.9\mathrm{GeV}/c$ (•), $1.4\le p_{\ell }^{*}<1.5\mathrm{GeV}/c$ ($\circ$), and $p_{\ell }^{*}\ge 1.9\mathrm{GeV}/c$ (▪). The results of fits of linear functions are overlaid as solid lines. A reference line with $⟨m_{X,\mathrm{reco}}^2⟩=⟨m_{X,\mathrm{true}}^2⟩$ is superimposed (dashed line). The calibration function for $p_{\ell }^{*}\ge 1.9\mathrm{GeV}/c$ is constructed independent of $N_{X_c}$ and $E_{\mathrm{miss}}-c{p}_{\mathrm{miss}}$. It is plotted in each of the bins for comparison only.

Figure 4

Calibrated (•) and uncorrected (□) moments $⟨m_X^2⟩$ (left) and $⟨m_X^4⟩$ (right) of individual hadronic modes for lepton momenta $p_{\ell }^{*}\ge 0.8\mathrm{GeV}/c$. A reference line with $⟨m_{X,\mathrm{calib}}⟩=⟨m_{X,\mathrm{true}}⟩$ is superimposed.

Figure 5

Radiation-corrected hadronic-mass moments $⟨m_X^k⟩$ with $k=1\dots 6$ for different selection criteria on the minimum lepton momentum $p_{\ell ,\min }^{*}$. The inner error bars correspond to the statistical uncertainties while the full error bars correspond to the total uncertainties. The moments, as well as their values for different $p_{\ell ,\min }^{*}$, are highly correlated.

Figure 6

Spectra of $n_X^2$ after the kinematic fit together with distributions of combinatorial background and background from non-$B\overline{B}$ decays (red, hatched area) as well as residual background (blue, crossed area) for different minimum lepton momenta (a) $p_{\ell ,\min }^{*}=0.8\mathrm{GeV}/c$ and (b) $p_{\ell ,\min }^{*}=1.9\mathrm{GeV}/c$. The two background histograms are plotted on top of each other.

Figure 7

Example of the calibration verification procedure for different minimum lepton momenta (a) $p_{\ell ,\min }^{*}=0.8\mathrm{GeV}/c$ and (b) $p_{\ell ,\min }^{*}=1.7\mathrm{GeV}/c$. Moments $⟨n_X^2⟩$ of exclusive modes on simulated events before (□) and after (•) calibration are plotted against the true moments for each mode. The dotted line shows the result of a fit to the calibrated moments, the resulting parameters are given.

Figure 8

Radiation-corrected moments (a) $⟨n_X^2⟩$, (b) $⟨n_X^4⟩$, and (c) $⟨n_X^6⟩$ for different cuts on the lepton momentum $p_{\ell }^{*}$. The inner error bars correspond to the statistical uncertainties while the full error bars correspond to the total uncertainties. The moments are highly correlated.

Figure 9

Previously measured momentum spectrum (points) [9] and estimated backgrounds (histograms) for electron candidates in (a) the unlike-sign sample, and (b) the like-sign sample. The background spectra are updated w.r.t. the previous publication with more recent branching-fraction measurements [21, 41].

Figure 10

Electron-momentum spectrum from $B\to Xe\nu (\gamma )$ decays in the ${\rm Y}(4S)$ frame after correction for efficiencies and bremsstrahlung in the detector, with combined statistical and systematic errors.

Figure 11

Measured branching fraction (a) and moments $M_1$ (b), $M_2$ (c), and $M_3$ (d) of the inclusive electron-energy spectrum of $B\to X_{c}e\nu (\gamma )$ decays as a function of the cutoff energy $E_0$ in the $B$-meson rest frame.

Figure 12

The measured hadronic-mass moments $⟨m_X^k⟩$, combined mass-and-energy moments $⟨n_X^k⟩$, electron-energy moments $⟨E_{\ell }^k⟩$, partial branching fractions B, and photon-energy moments $⟨E_{\gamma }^n⟩$, as a function of the minimum lepton momentum $p_{\ell ,\min }^{*}$ and minimum photon energy $E_{\gamma ,\min }$ compared with the result of the simultaneous fit (solid line) to moments of the hadronic mass spectrum, electron-energy moments, and photon-energy moments. The solid data points mark the measurements included in the fit. Moments of semileptonic decays $\overline{B}\to X_c{\ell }^{-}\overline{\nu }$ are marked by (•). Photon-energy moments of Ref. 14 are marked by (▪), of Ref. 15 by (•), and of Ref. 16 by (★). Open data points are not used in the fit. The vertical bars indicate the experimental errors. The dashed lines correspond to the total fit uncertainty as obtained by converting the fit errors of each individual HQE parameter into an error for the individual moment.

Figure 13

The measured hadronic-mass moments $⟨m_X^k⟩$, combined mass-and-energy moments $⟨n_X^k⟩$, electron-energy moments $⟨E_{\ell }^k⟩$, partial branching fractions B, and photon-energy moments $⟨E_{\gamma }^n⟩$, as a function of the minimum lepton momentum $p_{\ell ,\min }^{*}$ and minimum photon energy $E_{\gamma ,\min }$ compared with the result of the simultaneous fit (solid line) to moments of the combined mass-and-energy spectrum, electron-energy moments, and photon-energy moments. The solid data points mark the measurements included in the fit. Moments of semileptonic decays $\overline{B}\to X_c{\ell }^{-}\overline{\nu }$ are marked by (•). Photon-energy moments of Ref. 14 are marked by (▪), of Ref. 15 by (•), and of Ref. 16 by (★). Open data points are not used in the fit. The vertical bars indicate the experimental errors. The dashed lines correspond to the total fit uncertainty as obtained by converting the fit errors of each individual HQE parameter into an error for the individual moment.

Figure 14

$\Delta {\chi }^2=1$ contours for different fits in the $(m_b,|V_{cb}|)$ and $(m_b,{\mu }_{\pi }^2)$ planes. We compare the results of the two fits including the full sets of measured moments, one including hadronic-mass moments (black line) and one including moments of the $n_X^2$ distribution instead (blue dashed line), with a fit including only hadronic-mass and lepton-energy moments (red dotted line) and a fit including only hadronic-mass moments and partial branching-fraction measurements (magenta dashed-dotted line). We do not include the additional uncertainty of 1.4% due to the expansion of ${\Gamma }_{SL}$ in the plotted values of $|V_{cb}|$.