Measurement of the $D^0\to {\pi }^{-}{e}^{+}{\nu }_e$ differential decay branching fraction as a function of $q^2$ and study of form factor parametrizations

Based on a sample of 500 million $e^{+}{e}^{-}\to c\overline{c}$ events recorded by the BABAR detector at c.m. energies of close to 10.6 GeV, we report on a study of the decay $D^0\to {\pi }^{-}{e}^{+}{\nu }_e$. We measure the ratio of branching fractions, $R_D=\mathcal{B}(D^0\to {\pi }^{-}{e}^{+}{\nu }_e)/\mathcal{B}(D^0\to K^{-}{\pi }^{+})=0.0713\pm 0.0017_{\text{stat}}\pm 0.0024_{\text{syst}}$, and use the present world average for $\mathcal{B}(D^0\to K^{-}{\pi }^{+})$ to obtain $\mathcal{B}(D^0\to {\pi }^{-}{e}^{+}{\nu }_e)=(2.770\pm 0.068_{\text{stat}}\pm 0.092_{\text{syst}}\pm 0.037_{\text{ext}})\times 10^{-3}$ where the third error accounts for the uncertainty on the branching fraction for the reference channel. The measured dependence of the differential branching fraction on $q^2$, the four-momentum transfer squared between the $D$ and the $\pi$ meson, is compared to various theoretical predictions for the hadronic form factor, $f_{+,D}^{\pi }(q^2)$, and the normalization $|V_{cd}|\times f_{+,D}^{\pi }(q^2=0)=0.1374\pm 0.0038_{\text{stat}}\pm 0.0022_{\text{syst}}\pm 0.0009_{\text{ext}}$. is extracted from a fit to data. Using the most recent LQCD prediction of $f_{+,D}^{\pi }(q^2=0)=0.666\pm 0.029$, we obtain $|V_{cd}|=0.206\pm 0.007_{\text{exp}}\pm 0.009_{\mathrm{LQCD}}$. Assuming, instead, $|V_{cd}|=|V_{us}|=0.2252\pm 0.0009$, we obtain $f_{+,D}^{\pi }(q^2=0)=0.610\pm 0.020_{\text{exp}}\pm 0.005_{\text{ext}}$. The $q^2$ dependence of $f_{+,D}^{\pi }(q^2)$ is compared to a variety of multipole parametrizations. This information is applied to $B^0\to {\pi }^{-}{e}^{+}{\nu }_e$ decays and, combined with an earlier $B^0\to {\pi }^{-}{e}^{+}{\nu }_e$ measurement by BABAR, is used to derive estimates of $|V_{ub}|$.

Figure 1

Distributions of the Fisher discriminants. Left: $F_{b\overline{b}}$ for signal and $B\overline{B}$ events. Right: $F_{c\overline{c}}$ for signal and other $c\overline{c}$ events. The vertical lines indicate the selection requirements: $F_{b\overline{b}}>1.2$ and $F_{c\overline{c}}>0.6$.

Figure 2

Mass difference $\mathrm{\Delta }(m)=m(D^0{\pi }_s^{+})-m(D^0)$ after all selection criteria and the additional requirement on the first (open circles) and second (full circles) kinematic fits probabilities. The distribution for MC-simulated signal and the different background distributions are superimposed for the final selections. These MC distributions are normalized to data based on the integrated luminosity and have been corrected to account for small differences between data and MC distributions.

Figure 3

The measured $q^2$ distribution (data points) for events selected in the $\mathrm{\Delta }(m)$ signal region is compared to the sum of the estimated backgrounds and the fitted signal components. Peaking and nonpeaking background contributions refer only to $c\overline{c}$ events.

Figure 4

Study of the uncorrected pion efficiency in data and in simulation versus the laboratory pion momentum. Top: measured efficiencies. Bottom: ratio of efficiencies in data and MC.

Figure 5

Comparison of the measured event yields (black data points with statistical errors), as a function of $\cos {\theta }_e$, with the corrected sum of the expected signal and background distributions after all corrections. Left: observed events in data and in simulation. Right: the ratio (Data-MC)/MC.

Figure 6

Variation of the ratio of numbers of $D^0\to K^{-}{\pi }^{+}$ events measured in MC and data, as a fraction of events selected by restrictions on $F_{c\overline{c}-2}$. The vertical line shows the fraction of selected $D^0\to {\pi }^{-}{e}^{+}{\nu }_e$ events after the requirement on $F_{c\overline{c}}$. All events satisfy $F_{b\overline{b}}>1.2$. The horizontal lines indicate the value adopted for this ratio (full line) and the corresponding uncertainty (dashed lines).

Figure 7

Unfolded $q^2$ distribution for $D^0\to e^{+}{\pi }^{-}{\nu }_e$ decays.

Figure 8

Measured values of $|V_{cd}|\times f_{+,D}^{\pi }(q^2)$ are compared with the results of a fit using a $z$-expansion parametrization of the hadronic form factor (full blue line). The dashed (green) lines show the comparison with a fit using the effective three-pole ansatz in which the mass of an effective third pole is fitted. The superconvergence condition and constraints on the two first residues are imposed (see Sec. 6c4).

Figure 9

Comparison of this measurement with an average by HFAG of all other results listed in Table 7, both obtained from fits to the $z$ expansion. For the curves and their error bands the BABAR results in Fig. 8 have been subtracted. The continuous (blue) lines illustrate total uncertainties in the BABAR measurement, for which the central values are, by construction, equal to zero. Dashed (black) lines are the results of HFAG.

Figure 10

Contributions of high-mass poles to $|V_{cd}|\times f_{+,D}^{\pi }(q^2)$. For all data and projections the $D^{*+}$ contributions are subtracted. Data points (red) represent the measurements (Fig. 8), and the full (blue) curve, with thin lines on both sides, represents the fit result and uncertainties in the $z$-expansion parametrization. The dash-dotted lines indicate the additional uncertainties from the pole estimate. The dashed black lines mark the expected contribution from the $D_1^{*\prime }$ pole [see Eq. (a3)] and corresponding uncertainties.

Figure 11

Comparison of the $d{\mathcal{B}}^B/dw$ differential decay rate for $B^0\to {\pi }^{-}{e}^{+}{\nu }_e$ decays measured by BABAR with an extrapolation of the $D^0\to {\pi }^{-}{e}^{+}{\nu }_e$ form factor measurement. The solid red line is the result of the fit to the fixed three-pole ansatz with $m_{\text{pole}3}=3.1\mathrm{GeV}/c^2$, the short-dash red line marks the extrapolation beyond the physical region for the $D^0\to {\pi }^{-}{e}^{+}{\nu }_e$ decay. The two thin red lines indicate the impact of the 12% uncertainty on the form factor ratio $R_{BD}$. The long-dash magenta line marks the fit result, to $D^0\to {\pi }^{-}{e}^{+}{\nu }_e$ data, for the effective three-pole ansatz. In these comparisons, the value ${|V_{ub}|}^{\text{excl}}$ is used.

Figure 12

Comparison of the measured differential branching fraction for $B^0\to {\pi }^{-}{e}^{+}{\nu }_e$ [46], integrated over $2{\mathrm{GeV}}^2$ $q^2$ intervals (apart from the last bin which extends from 22 to $26.2{\mathrm{GeV}}^2$), with expectations from the effective three-pole ansatz. The two lines indicate theoretical uncertainties on these predictions. The contributions from the $B^{*}$ pole and from the sum of the $B^{*}$ and $B_1^{*\prime }$ poles are indicated.

Figure 13

Variation of the ratio $f_{+,B}^{\pi }(w_B)/f_{+,D}^{\pi }(w_D)$ (thick line) computed from the evaluation of the two form factors obtained in LQCD [45, 58]. Thin lines give the uncertainties in this evaluation.