Measurement of the moments of the hadronic invariant mass distribution in semileptonic $B$ decays

Using 180 $\mathrm{p}{\mathrm{b}}^{-1}$ of data collected with the CDF II detector at the Tevatron, we measure the first two moments of the hadronic invariant mass-squared distribution in charmed semileptonic $B$ decays. From these we determine the nonperturbative Heavy Quark Effective Theory parameters $\Lambda$ and ${\lambda }_1$ used to relate the $B$ meson semileptonic branching ratio to the CKM matrix element $|V_{cb}|$. For a minimum lepton momentum of 0.7 GeV/$c$ in the $B$ rest frame we measure the first two moments of the $D^{**}\to D^{(*)}\pi$ component to be $⟨m_{D^{**}}^2⟩=(5.83\pm {0.16}_{\mathrm{stat}}\pm {0.08}_{\mathrm{syst}}){\mathrm{GeV}}^2/c^4$ and $⟨(m_{D^{**}}^2-⟨m_{D^{**}}^2⟩{)}^2⟩=(1.30\pm {0.69}_{\mathrm{stat}}\pm {0.22}_{\mathrm{syst}})\mathrm{Ge}{\mathrm{V}}^4/c^8$. Combining these with the discrete mass terms from the $D$ and $D^{*}$ mesons, we find the total moments to be $⟨M_{X_c}^2⟩-{\overline{m}}_D^2=(0.467\pm {0.038}_{\mathrm{stat}}\pm {0.068}_{\mathrm{syst}})\mathrm{Ge}{\mathrm{V}}^2/c^4$ and $⟨(M_{X_c}^2-⟨M_{X_c}^2⟩{)}^2⟩=(1.05\pm {0.26}_{\mathrm{stat}}\pm {0.13}_{\mathrm{syst}})\mathrm{Ge}{\mathrm{V}}^4/c^8$, where ${\overline{m}}_D$ is the spin-averaged $D$ mass. The systematic error is dominated by the uncertainties in the world-average branching ratios used to combine the $D$, $D^{*}$, and $D^{**}$ contributions. The analysis makes no assumptions about the shape or resonant structure of the $D^{**}\to D^{(*)}\pi$ invariant mass distribution.

Figure 1

Left: Mass-difference $\Delta m^{*}=M(D^0{\pi }_{*}^{+})-M(D^0)$ in the $D^0\to K^{-}{\pi }^{+}$ and $D^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ channels (narrow peak) and $\Delta m^{*}=M(K^{-}{\pi }^{+}{\pi }_{*}^{+})-M(K^{-}{\pi }^{+})$ in the $D^0\to K^{-}{\pi }^{+}{\pi }^0$ channel (broad peak). Right: Mass distribution for the $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}$ channel. The signal areas are shaded; side-band regions are also indicated.

Figure 2

Side-band-subtracted invariant mass distribution for the $D^{*+}{\pi }_{**}$ channels (left) and for the $D^{+}{\pi }_{**}$ channel (right). The mass regions shown are limited at 3.5 GeV$/c^2$ for illustration only. No explicit mass cut is applied in the analysis.

Figure 3

Fully corrected invariant mass distribution $m(D^{(*)+}{\pi }^{-})$. The number of events in each bin has been background subtracted and corrected for mass-dependent and $D^{*}/D^{+}$ relative efficiency corrections. The plotted errors take into account all corrections and subtractions.