Measurement of $B\to D^{(\ast )}\tau \nu$ using full reconstruction tags

Charged particle tracks are reconstructed from hits in the SVD and CDC. They are required to satisfy track quality cuts based on their impact parameters relative to the measured profile of the interaction point (IP) of the two beams. Charged kaons are identified by combining information on ionization loss ($dE/dx$) in the CDC, Cherenkov light yields in the ACC and time-of-flight measured by the TOF system. For the nominal requirement, the kaon identification efficiency is approximately $88\%$ and the probability of misidentifying a pion as a kaon is about $8\%$. Hadron tracks that are not identified as kaons are treated as pions. Tracks satisfying the lepton identification criteria, as described below, are removed from consideration. Electron identification is based on a combination of $dE/dx$ in CDC, the response of the ACC, shower shape in the ECL, position matching between ECL clusters and the track, and the ratio of the energy deposited in the ECL to the momentum measured by the tracking system. Muon candidates are selected using range of tracks measured in KLM and the deviation of hits from the extrapolated track trajectories. The lepton identification efficiencies are estimated to be about 90% for both electrons and muons in the momentum region above 1.2 GeV/$c$. The hadron misidentification rate is measured using reconstructed $K_S^0\to {\pi }^{+}{\pi }^{-}$ decays and found to be less than 0.2% for electrons and 1.5% for muons in the same momentum region. $K_S^0$ mesons are reconstructed using pairs of charged tracks that have an invariant mass within $\pm 30$ MeV/$c^2$ of the known $K_S^0$ mass and a well reconstructed vertex that is displaced from the IP. Candidate $\gamma$’s are required to have a minimum energy deposit $E_{\gamma }\ge 50$ MeV. Candidate ${\pi }^0$ mesons are reconstructed using $\gamma \gamma$ pairs with invariant masses between 117 and 150 MeV/$c^2$. For slow ${\pi }^0$’s used in $D^{\ast }$ reconstruction, the minimum $\gamma$ energy requirement is lowered to 30 MeV.

$B_{tag}$ candidates are reconstructed in the following decay modes: $B^{+}\to \overline{D}{}^{(\ast )0}{\pi }^{+}$, $\overline{D}{}^{(\ast )0}{\rho }^{+}$, $\overline{D}{}^{(\ast )0}{a}_1^{+}$, $\overline{D}{}^{(\ast )0}{D}_s^{(\ast )+}$, and $B^0\to D^{(\ast )-}{\pi }^{+}$, $D^{(\ast )-}{\rho }^{+}$, $D^{(\ast )-}{a}_1^{+}$, $D^{(\ast )-}{D}_s^{(\ast )+}$. Candidate ${\rho }^{+}$ and ${\rho }^0$ mesons are reconstructed in the ${\pi }^{+}{\pi }^0$ and ${\pi }^{+}{\pi }^{-}$ decay modes, by requiring their invariant masses to be within $\pm 225$ MeV/$c^2$ of the nominal $\rho$ mass. We then select $a_1^{+}$ candidates by combining a ${\rho }^0$ candidate and a pion with invariant mass between 0.7 and 1.6 GeV/$c^2$ and require that the charged tracks form a good vertex. $D$ meson candidates are reconstructed in the following decay modes: $\overline{D}{}^0\to K^{+}{\pi }^{-}$, $K^{+}{\pi }^{-}{\pi }^0$, $K^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, $K_S^0{\pi }^0$, $K_S^0{\pi }^{-}{\pi }^{+}$, $K_S^0{\pi }^{-}{\pi }^{+}{\pi }^0$, $K^{-}{K}^{+}$, and $D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}$, $K^{+}{\pi }^{-}{\pi }^{-}{\pi }^0$, $K_S^0{\pi }^{-}$, $K_S^0{\pi }^{-}{\pi }^0$, $K_S^0{\pi }^{+}{\pi }^{-}{\pi }^{-}$, $K^{+}{K}^{-}{\pi }^{-}$, $D_s^{+}\to K_S^0{K}^{+}$, $K^{+}{K}^{-}{\pi }^{+}$. The $D$ candidates are required to have an invariant mass $M_D$ within $4-5\sigma$ ($\sigma$ is the mass resolution) of the nominal $D$ mass value depending on the mode. $D^{\ast }$ mesons are reconstructed as $D^{\ast +}\to D^0{\pi }^{+}$, $D^{+}{\pi }^0$, $D^{\ast 0}\to D^0{\pi }^0$, $D^0\gamma$, and $D_s^{\ast +}\to D_s^{+}\gamma$. $D^{\ast }$ candidates from modes that include a pion are required to have a mass difference $\Delta M=M_{D\pi }-M_D$ within $\pm 5$ MeV/$c^2$ of its nominal value. For decays with a photon, we require that the mass difference $\Delta M=M_{D\gamma }-M_D$ be within $\pm 20$ MeV/$c^2$ of the nominal value.

The selection of $B_{tag}$ candidates is based on the beam-constrained mass $M_{bc}\equiv \sqrt{E_{beam}^2-p_B^2}$ and the energy difference $\Delta E\equiv E_B-E_{beam}$. Here, $E_B$ and $p_B$ are the reconstructed energy and momentum of the $B_{tag}$ candidate in the $e^{+}{e}^{-}$ c.m. system, and $E_{beam}$ is the beam energy in the c.m. frame. The background from jet-like continuum events ($e^{+}{e}^{-}\to q\overline{q},q=u,d,s,c$) is suppressed on the basis of event topology: we require the normalized second Fox-Wolfram moment ($R_2$) R2 () to be smaller than 0.5, and $|\cos {\theta }_{th}|<0.8$, where ${\theta }_{th}$ is the angle between the thrust axis of the $B$ candidate and that of the remaining tracks in the event. The latter requirement is not applied to $B^{+}\to \overline{D}{}^0{\pi }^{+}$, $\overline{D}{}^{\ast 0}(\to \overline{D}{}^0{\pi }^0){\pi }^{+}$ and $B^0\to D^{\ast -}(\to \overline{D}{}^0{\pi }^{-}){\pi }^{+}$ decays, where the continuum background is small. For the $B_{tag}$ candidate, we require $5.27\text{GeV}/c^2<M_{bc}<5.29\text{GeV}/c^2$ and $-80\text{MeV}<\Delta E<60\text{MeV}$. If an event has multiple $B_{tag}$ candidates, we choose the one having the smallest ${\chi }^2$ based on deviations from the nominal values of $\Delta E$, the $D$ candidate mass, and the $D^{\ast }-D$ mass difference if applicable. The number of $B^{+}$ and $B^0$ candidates in the selected region are $1.75\times {10}^6$ and $1.18\times {10}^6$, respectively. By fitting the distribution to the sum of an empirical parameterization of the background shape Albrecht () and a signal shape Bloom:1983pc (), we estimate that in the selected region there are $(10.11\pm 0.03)\times {10}^5$ (with purity=0.58) $B^{+}$ and $(6.05\pm 0.03)\times {10}^5$ (with purity=0.51) $B^0$ events, respectively.

In the events where a $B_{tag}$ is reconstructed, we search for decays of $B_{sig}$ into a $D^{(\ast )}$, $\tau$ and a neutrino. In the present analysis, the $\tau$ lepton is identified in the leptonic decay modes, ${\mu }^{-}\overline{\nu }\nu$ and $e^{-}\overline{\nu }\nu$. We require that the charge/flavor of the $\tau$ daughter particles and the $D$ meson are consistent with the $B_{sig}$ flavor, opposite to the $B_{tag}$ flavor. The loss of signal due to $B^0-\overline{B^0}$ mixing is estimated by the MC simulation.

The procedures to reconstruct charged particles ($e^{\pm },{\mu }^{\pm },{\pi }^{\pm },K^{\pm }$) and neutral particles (${\pi }^0,K_S^0$) for the signal side are the same as those used for the tag side. For $\gamma$ candidates, we require a minimum energy threshold of 50 MeV for the barrel, and 100 (150) MeV for the forward (backward) end-cap ECL. A higher threshold is used for the endcap ECL, where the effect of beam background is more severe. We also require that the lepton momentum in the laboratory frame exceeds 0.6 GeV/$c$ to ensure good lepton identification efficiency.

The decay modes used for $D$ reconstruction are slightly different from those used for the tagging side: $\overline{D}{}^0\to K^{+}{\pi }^{-}$, $K^{+}{\pi }^{-}{\pi }^0$, $K^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, $K^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}{\pi }^0$, $K_S^0{\pi }^0$, $K_S^0{\pi }^{-}{\pi }^{+}$, $K_S^0{\pi }^{-}{\pi }^{+}{\pi }^0$, and $D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}$, $K^{+}{\pi }^{-}{\pi }^{-}{\pi }^0$, $K_S^0{\pi }^{-}$. The $D$ candidates are required to have an invariant mass $M_D$ within $5\sigma$ of the nominal $D$ mass value. $D^{\ast }$ mesons are reconstructed using the same decay modes as on the tagging side: $D^{\ast +}\to D^0{\pi }^{+}$, $D^{+}{\pi }^0$, and $D^{\ast 0}\to D^0{\pi }^0$, $D^0\gamma$. $D^{\ast }$ candidates are required to have a mass difference $\Delta M=M_{D\pi \left(\gamma \right)}-M_D$ within $5\sigma$ of the nominal value.

For signal selection, we use the following variables that characterize the signal decay: the missing mass squared in the event ($M_{miss}^2$), the momentum (in the c.m. system) of the $\tau$ daughter leptons ($P_{\ell }^{\ast }$), and the extra energy in the ECL ($E_{extra}^{ECL}$). The missing mass squared is calculated as $M_{miss}^2=(E_{B_{tag}}-E_D-E_{\tau \to X}{)}^2-(-{\overrightarrow{P}}_{B_{tag}}-{\overrightarrow{P}}_{D^{(\ast )}}-{\overrightarrow{P}}_{\tau \to X}{)}^2$, using the energy and momenta of the $B_{tag}$, the $D^{(\ast )}$ candidate and the lepton from the $\tau$ decay. The signal decay is characterized by large $M_{miss}^2$ due to the presence of more than two neutrinos in the final state. The lepton momenta ($P_{\ell }^{\ast }$) distribute lower than those from primary $B$ decays. The extra energy in the ECL ($E_{extra}^{ECL}$) is the sum of the energies of photons that are not associated with either the $B_{tag}$ or the $B_{sig}$ reconstruction. ECL clusters with energies greater than 50 MeV in the barrel, and 100 (150) MeV in the forward (backward) end-cap ECL are used to calculate $E_{extra}^{ECL}$. For signal events, $E_{extra}^{ECL}$ must be either zero or a small value arising from beam background hits, therefore, signal events peak at low $E_{extra}^{ECL}$. On the other hand, background events are distributed toward higher $E_{extra}^{ECL}$ due to the contribution from additional neutral clusters. We also require that the event has no extra charged tracks and no ${\pi }^0$ candidate other than those from the signal decay and those used in the $B_{tag}$ reconstruction. Table 1 summarizes the cuts to define the signal region. The cuts are optimized by maximizing the figure of merit (F.O.M.), defined as $F.O.M.=N_S/\sqrt{N_S+N_B}$, where $N_S\left(N_B\right)$ are the number of signal (total background) events in the signal region, assuming the SM branching fractions for the $D\tau \nu$ and the $D^{\ast }\tau \nu$ modes.

Table 2 lists the signal detection efficiencies, which are estimated from signal MC simulation, with the selection criteria shown in Table 1. Taking account of the cross talks between $B\to D\tau \nu$ and $B\to D^{\ast }\tau \nu$ modes, the signal detection efficiency (${ϵ}_{ij}$) is defined as,

where $N_{ij}$ represents the yield of the generated $j$-th mode reconstructed in the $i$-th mode. $B{}_j$ is the branching fraction of the $j$-th mode including the sub-decay ($\tau$ and $D^{(\ast )}$) branching fractions. $N_{tag}$ is the number of $B$ events fully reconstructed on the tagging side. Table 2 also shows, in parentheses, the efficiencies without cuts on $M_{miss}^2$ and $E_{extra}^{ECL}$. These are the two variables used to extract the signal yields.

According to the MC simulation, the expected number of signal (background) events in the signal region is 19(48) for $B^{+}\to \overline{D}{}^0{\tau }^{+}\nu$, 7(13) for $B^0\to D^{-}{\tau }^{+}\nu$, 18(25) for $B^{+}\to \overline{D}{}^{\ast 0}{\tau }^{+}\nu$, and 7(6) for $B^0\to D^{\ast -}{\tau }^{+}\nu$. The major background sources are semileptonic $B$ decays, $B\to D\ell \nu$, $D^{\ast }\ell \nu$ and $D^{\ast \ast }\ell \nu$ (60-70% depending on the decay mode). The remaining background comes from hadronic $B$ decays including a $D$ meson in the final state. Background from $q\overline{q}$ processes are found to be small (less than one event). As shown in Table 2, the cross talk between $B\to D\tau \nu$ and $B\to D^{\ast }\tau \nu$ arises, when a pion or a photon is missed in the reconstruction of $D^{\ast }$, or when a random photon is combined with a $D$ to form a fake $D^{\ast }$. The cross-feed to other $B^0$ or $B^{+}$ tag samples is negligibly small.