Measurement of the hadronic form factor in $D^0\to K^{-}{e}^{+}{\nu }_e$ decays

The shape of the hadronic form factor $f_{+}(q^2)$ in the decay $D^0\to K^{-}{e}^{+}{\nu }_e$ has been measured in a model independent analysis and compared with theoretical calculations. We use $75{\mathrm{fb}}^{-1}$ of data recorded by the BABAR detector at the PEP-II electron-positron collider. The corresponding decay branching fraction, relative to the decay $D^0\to K^{-}{\pi }^{+}$, has also been measured to be $R_D=\mathcal{B}(D^0\to K^{-}{e}^{+}{\nu }_e(\gamma ))/\mathcal{B}(D^0\to K^{-}{\pi }^{+}(\gamma ))=0.927\pm 0.007\pm 0.012$. From these results, and using the present world average value for $\mathcal{B}(D^0\to K^{-}{\pi }^{+})$, the normalization of the form factor at $q^2=0$ is determined to be $f_{+}(0)=0.727\pm 0.007\pm 0.005\pm 0.007$ where the uncertainties are statistical, systematic, and from external inputs, respectively.

Figure 1

Comparison of the $\delta (m)$ distributions from data and simulated events. MC events have been normalized to the sample luminosity according to the different cross sections. An excess of background events of the order of 5% is observed for large values of $\delta (m)$. The background component expected from light quarks is not visible with this scale. The arrow indicates the additional selection applied for the $q^2$ distribution measurement.

Figure 2

MC simulations of distributions of the variables used in the Fisher discriminant analysis to reduce the $B\overline{B}$ event background: (a) the normalized second Fox-Wolfram moment ($R_2$), (b) the event particle multiplicity, (c) the slow-pion momentum distribution, in the c.m. frame, (d) the Fisher variable for $B\overline{B}$ and for charm signal events.

Figure 3

MC simulations of some of the variables used in the Fisher discriminant analysis to reduce the $c\overline{c}$-event background: (a) the $D^0$ momentum after the kinematic fit, (b) the mass of the spectator system (peaks, at low mass values correspond to events with a single charged pion or photon reconstructed in the spectator system), (c) the cosine of the angle between the spectator system momentum and the thrust direction, (d) the cosine of the angle of the positron direction, relative to the kaon direction, in the $e{\nu }_e$ c.m. frame.

Figure 4

Distribution of the values of the Fisher variable in the signal region ($\delta (m)<0.16\mathrm{GeV}/c^2$ in (a), and for masses above the signal region ($\delta (m)>0.16\mathrm{GeV}/c^2$ in (b).

Figure 5

The measured $q_r^2$ distribution (data points) compared to the sum of the estimated background and of the fitted signal components.

Figure 6

The efficiency as a function of $q^2$, measured with simulated signal events, after all selection criteria applied.

Figure 7

Comparison between the normalized unfolded $q^2$ distribution obtained from this analysis, and those corresponding to two fitted models. Lower plot gives the difference between measured and fitted distributions. The error bars represent statistical errors only.

Figure 8

Contours at 70% and 90% C.L. resulting from the fit of the parameterization of the form-factor $q^2$ dependence with two parameters as given in Eq. (10). The value ${\delta }_{\mathrm{pole}}=0$ corresponds to the modified pole ansatz.

Figure 9

Distribution of the difference between the true and the reconstructed value of $q^2$. $D^0\to K^{-}{\pi }^{+}{\pi }^0$ data events correspond to dark squares and open circles are used for simulated ones. Ratio (Data/MC) of the two distributions are displayed. The distributions in a) compare data and simulated events before applying corrections measured with $D^0\to K^{-}{\pi }^{+}$ events, whereas these corrections have been applied for plots in (b).

Figure 10

Comparison of the measured variation of $f_{+}(q^2)/f_{+}(0)$ obtained in the present analysis and in the FOCUS experiment [5]. The band corresponds to lattice QCD [19] with the estimated uncertainty.

Figure 11

Measured values for $P\times \Phi \times f_{+}(z)$, normalized to unity for $z=z_{\max }=0.051$, are plotted versus $-z$. The straight lines represent the result for the modified pole ansatz, the fit in the center and the statistical and total uncertainty.

Figure 12

Events selected for the reference channel $D^{*+}\to D^0{\pi }^{+}$, $D^0\to K^{-}{\pi }^{+}$. (a) $K\pi$ mass distribution for events selected in the range $\delta (m)\in [0.143,0.148]\mathrm{GeV}/c^2$. (b) $\delta (m)$ distribution for events selected in the range $m(K\pi )\in [1.83,1.89]\mathrm{GeV}/c^2$. (c) same distribution as in (b), displayed on a larger mass range with the nonpeaking background indicated (shaded area). (d) $\delta (m)$ distribution after nonpeaking background subtraction, data (points with statistical errors) and simulated events (shaded histogram).

Figure 13

$\delta (m)$ distribution for $D^0\to K^{-}{\pi }^{+}{\pi }^0$ events analyzed as if they were semileptonic decays. Distributions have been normalized to unity; note that the bin size is not uniform. The bottom plot shows the ratio of the two distributions above.