Measurements of $q^2$ moments of inclusive $B\to X_c{\ell }^{+}{\nu }_{\ell }$ decays with hadronic tagging

We present the measurement of the first to fourth order moments of the four-momentum transfer squared, $q^2$, of inclusive $B\to X_c{\ell }^{+}{\nu }_{\ell }$ decays using the full Belle dataset of $711{\mathrm{fb}}^{-1}$ of integrated luminosity at the $\mathrm{\Upsilon }(4S)$ resonance where $\ell =e$, $\mu$. The determination of these moments and their systematic uncertainties open new pathways to determine the absolute value of the Cabibbo-Kobayashi-Maskawa matrix element $V_{cb}$ using a reduced set of matrix elements of the heavy quark expansion. In order to identify and reconstruct the $X_c$ system, we reconstruct one of the two $B$-mesons using machine learning techniques in fully hadronic decay modes. The moments are measured with progressively increasing threshold selections on $q^2$ starting with a lower value of $3.0{\mathrm{GeV}}^2$ in steps of $0.5{\mathrm{GeV}}^2$ up to a value of $10.0{\mathrm{GeV}}^2$. The measured moments are further unfolded, correcting for reconstruction and selection effects as well as QED final state radiation. We report the moments separately for electron and muon final states and observe no lepton flavor universality violating effects.

Figure 1

The inclusive $B\to X_c{\ell }^{+}{\nu }_{\ell }$ semileptonic processes for a $B^{+}$ (left) or a $B^0$ (right) meson decay.

Figure 2

The resolution of the reconstructed $M_X$ and $q^2$ values for the $B\to X_c{\ell }^{+}{\nu }_{\ell }$ signal is shown as a residual with respect to the generated values.

Figure 3

The reconstructed $M_X$ and $q^2$ distributions are shown. The error band of the simulated samples incorporates the full set of systematic uncertainties discussed in Sec. 5.

Figure 4

The reconstructed $q^2$ distributions with an example $q^2$ threshold selection of $3.0{\mathrm{GeV}}^2$ (top) for electron (left) and muon (right) candidates and the determined binned signal probabilities (bottom) are shown. The background contributions are scaled to their estimated values using the fit described in the text. The binned signal probabilities are obtained by a fit of a polynomial of a given order $n$ (red curve).

Figure 5

The generator-level, $⟨q_{\mathrm{true}}^{2m}⟩$, and reconstructed, $⟨q_{\mathrm{reco}}^{2m}⟩$, $B\to X_c{\ell }^{+}{\nu }_{\ell }$ moments are shown for the first to fourth $q^2$ moment for electrons. The different markers represent different threshold selections on the generator-level and reconstructed $q^2$ and the relation between the moments can be described with a linear parametrization of the form $a_m+⟨q_{\mathrm{true}}^{2m}⟩·b_m=⟨q_{\mathrm{reco}}^{2m}⟩$. The corresponding plots for muons can be found in Appendix pp2.

Figure 6

The residual bias and acceptance correction factors, denoted as $C_{\mathrm{cal}}$ (circles) and $C_{\mathrm{acc}}$ (diamonds), respectively, are shown for the first to fourth $q^2$ moment for electrons. The corresponding plots for muons can be found in Appendix pp2.

Figure 7

Statistical correlations between the first and the second, third, and fourth moment for the electron final state.

Figure 8

The expectation of lepton flavor universality of the moments is tested for the first to fourth $q^2$ moments: in the ratio of electron to muon moments many of the associated systematic uncertainties cancel and all reported moments are compatible with the expectation of lepton flavor universality (bottom top). Note that the individual electron and muon moments are highly correlated. Furthermore, the measured and generated-level moments for all the threshold selections on $q^2$ are compared as a ratio (bottom middle) and difference (bottom lower) for both electrons and muons.

Figure 9

Comparisons between the different $B\to D_{\mathrm{gap}}^{**}\ell {\overline{\nu }}_{\ell }$ decay models for the reconstructed lepton energy in the signal $B$ rest frame $E_{\ell }^B$ (upper left), the $M_X$ (upper right), and the $q^2$ (bottom) distributions.

Figure 10

The determined background in the $q^2$ spectrum, after obtaining the expected background yields from the fit to the $M_X$ spectrum, for different $q^2$ threshold selections for electrons are shown.

Figure 11

The determined signal probabilities for different $q^2$ threshold selections for electrons are shown.

Figure 12

The determined background in the $q^2$ spectrum, after obtaining the expected background yields from the fit to the $M_X$ spectrum, for different $q^2$ threshold selections for muons are shown.

Figure 13

The determined signal probabilities for different $q^2$ threshold selections for muons are shown.

Figure 14

The calibration curves, bias, and acceptance calibration factors for the first to the fourth moment of $q^2$ for muons.

Figure 15

Selection efficiencies of different $B\to X_c{\ell }^{+}{\nu }_{\ell }$ components for electron (left) and muon (right) final states with basic transverse momentum and angular acceptance requirements in bins of $q^2$.

Figure 16

Color map of the full experimental correlation coefficients determined for the moments for the electron final state.

Figure 17

Color map of the systematic correlation coefficients determined for the moments for the muon final state.

Figure 18

The measured second (left), third (right), and fourth (bottom) central $q^2$ moments for both the electron and muon final states.