Dalitz plot analysis of the decay $B^0({\overline{B}}^0)\to K^{\pm }{\pi }^{\mp }{\pi }^0$

We report a Dalitz-plot analysis of the charmless hadronic decays of neutral $B$ mesons to $K^{\pm }{\pi }^{\mp }{\pi }^0$. With a sample of $(231.8\pm 2.6)\times {10}^6\mathit{{\rm Y}}(4S)\to B\overline{B}$ decays collected by the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC, we measure the magnitudes and phases of the intermediate resonant and nonresonant amplitudes for $B^0$ and ${\overline{B}}^0$ decays and determine the corresponding $CP$-averaged branching fractions and charge asymmetries. The inclusive branching fraction and $CP$-violating charge asymmetry are measured to be $\mathcal{B}(B^0\to K^{+}{\pi }^{-}{\pi }^0)=({35.7}_{-1.5}^{+2.6}\pm 2.2)\times {10}^{-6}$ and ${\mathcal{A}}_{CP}=-{0.030}_{-0.051}^{+0.045}\pm 0.055$ where the first errors are statistical and the second systematic. We observe the decay $B^0\to K^{*0}(892){\pi }^0$ with the branching fraction $\mathcal{B}(B^0\to K^{*0}(892){\pi }^0)=({3.6}_{-0.8}^{+0.7}\pm 0.4)\times {10}^{-6}$. This measurement differs from zero by 5.6 standard deviations (including the systematic uncertainties). The selected sample also contains $B^0\to {\overline{D}}^0{\pi }^0$ decays where ${\overline{D}}^0\to K^{+}{\pi }^{-}$, and we measure $\mathcal{B}(B^0\to {\overline{D}}^0{\pi }^0)=(2.93\pm 0.17\pm 0.18)\times {10}^{-4}$.

Figure 1

The standard (a) and square (b) Dalitz plots of the selected data sample of 8014 events. The structures are more spread out in the square Dalitz plot. The ${\overline{D}}^0\to K^{+}{\pi }^{-}$ narrow band is preserved with the choice made for the $m^{\prime }$ variable.

Figure 2

Fraction of misreconstructed events ($f_{\mathrm{SCF}}$), in the square Dalitz plot. This figure includes $4.9\times {10}^6$ signal Monte Carlo events. The white area is the SCF-free region.

Figure 3

Ratio of the TM term ${\mathcal{L}}_{\mathrm{TM},i}$ [first relevant term in Eq. (22)] over the full likelihood function ${\mathcal{L}}_i$ [Eq. (20)] for all events. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and gray shaded areas represent the continuum and $B$ background, respectively. The light gray region shows the SCF contribution.

Figure 4

(a) $m_{\mathrm{ES}}$, (b) $\Delta {\mathrm{E}}^{\prime }$, and (c) NN distributions after restricting $r_{\mathrm{lik}}$ to enhance signal. The fractions of accepted signal and rejected continuum events are, respectively, 65% and 71% (a), 21% and 97% (b), and 49% and 83% (c). The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and gray shaded areas represent the continuum and $B$ background, respectively. The light gray region shows the SCF contribution.

Figure 5

The invariant mass spectra for all events: $m_{K^{+}{\pi }^{-}}$ (a), $m_{K^{+}{\pi }^0}$ (b), and $m_{{\pi }^{-}{\pi }^0}$ (c). The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and gray shaded areas represent the continuum and $B$ background, respectively. The light gray region shows the SCF contribution. The ${\overline{D}}^0$ mass peak is prominent in the top plot.

Figure 6

The signal-enriched spectra for masses between threshold and $1.75\mathrm{GeV}/c^2$: $m_{K^{+}{\pi }^{-}}$ (a), $m_{K^{+}{\pi }^0}$ (b), and $m_{{\pi }^{-}{\pi }^0}$ (c). The fractions of accepted signal and rejected continuum events are, respectively, 51% and 89%. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and gray shaded areas represent the continuum and $B$ background, respectively. The light gray region shows the SCF contribution.

Figure 7

Signal-enriched spectra of $m_{K^{+}{\pi }^{-}}$ in the low-mass resonance region and different ranges of the helicity angle, ${\theta }_{K^{+}{\pi }^{-}}$. (a) $0<{\theta }_{K^{+}{\pi }^{-}}<90\mathrm{degrees}$, (b) $90<{\theta }_{K^{+}{\pi }^{-}}<180\mathrm{degrees}$. The data sample is enriched in signal events as in Fig. 6. An interference between the vector and scalar $K^{*0}$ is apparent through a positive forward-backward asymmetry below the $K^{*}(892)$ and a negative one above. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and gray shaded areas represent the continuum and $B$ background, respectively. The light gray region shows the SCF contribution.

Figure 8

Signal-enriched spectra of $m_{K^{+}{\pi }^0}$ in the low-mass resonance region and different ranges of the helicity angle, ${\theta }_{K^{+}{\pi }^0}$. (a) $0<{\theta }_{K^{+}{\pi }^0}<90\mathrm{degrees}$, (b) $90<{\theta }_{K^{+}{\pi }^0}<180\mathrm{degrees}$. The data sample is enriched in signal events as in Fig. 6. An interference between the vector and scalar $K^{*}$ is apparent through a negative forward-backward asymmetry below the $K^{*}(892)$ and a positive one above. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and gray shaded areas represent the continuum and $B$ background, respectively. The light gray region shows the SCF contribution.

Figure 9

The NLL close to its minimum as a function of the $K^{*0}(892){\pi }^0$ branching fraction. The shapes of the curves are nonparabolic and shallow because the plotted $\Delta {\chi }^2$ is the smallest of those from all solutions. The dashed line scan corresponds to the statistical uncertainty. The solid line is the scan after smearing by the systematic uncertainties described in Sec. 7.

Figure 10

The phase difference between the $K^{*0}$ and the charged $K^{*}$ for the $P$-wave $K^{*}(892)$. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ (b) decays as well as their differences (c). The dashed line gives the statistical uncertainty, and the solid line, the total uncertainty. The data do not indicate preferred angles, except for the $[-131,-75]$ degree range which is excluded for ${\overline{B}}^0$ at the 2 standard deviation level. The four fit solutions find their NLL minimum for distinct phases. The vertical scale stops at $2\Delta (\mathrm{NLL})=4$ slightly above 3.84 which is the 95% confidence level. A horizontal dashed line at $2\Delta (\mathrm{NLL})=1$ shows the 1 standard deviation level.

Figure 11

The phase difference between the $(K\pi {)}_0^{*0}$ and the $(K\pi {)}_0^{*\pm }$ $S$-waves. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ decays (b) as well as their differences (c). The lines are drawn as in Fig. 10. The data do not indicate preferred angles. The four fit solutions find their NLL minimum for distinct phases.

Figure 12

The phase difference between the charged $K\pi$ $P$- and $S$-waves. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ (b) decays as well as their differences (c). The lines are drawn as in Fig. 10. The data provide significant constraints on these angles. The four fit solutions find their NLL minimum at approximately the same phase differences.

Figure 13

The phase difference between the neutral $K\pi$ $P$- and $S$-waves. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ (b) decays as well as their differences (c). The lines are drawn as in Fig. 10. The data provide significant constraints on these angles. The four fit solutions find their NLL minimum at approximately the same phase differences.

Figure 14

The phase difference between the $\rho K$ and the charged $K\pi$ $P$-wave. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ (b) decays as well as their differences (c). The lines are drawn as in Fig. 10. The vertical scale cuts off $\Delta {\chi }^2=4$; however, it has been checked that all phase differences are consistent with the data at the 3 standard deviation level.

Figure 15

The phase difference between the nonresonant $K^{+}{\pi }^{-}{\pi }^0$ and the charged $K\pi$ $S$-wave. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ (b) decays as well as their differences (c). The lines are drawn as in Fig. 10. Ranges with widths of 140 (a) and 190 (b) degrees are excluded at the 95% confidence level for the $B^0$ and ${\overline{B}}^0$ decays. No significant difference between the $B^0$ and ${\overline{B}}^0$ interference patterns is seen.

Figure 16

The phase difference between the nonresonant $K^{+}{\pi }^{-}{\pi }^0$ and the neutral $K\pi$ $S$-wave. The three diagrams are the NLL scans for the $B^0$ (a) and ${\overline{B}}^0$ (b) decays as well as their differences (c). The lines are drawn as in Fig. 10. Ranges with widths of 90 (a) and 110 (b) degrees are excluded at the 95% confidence level for the $B^0$ and ${\overline{B}}^0$ decays. No significant difference between the $B^0$ and ${\overline{B}}^0$ interference patterns is seen.

Figure 17

The NLL functions for the isobar fractions. The NLL functions of each solution are shown with dashed lines. The fitted values for the $B\to (K\pi {)}_0^{*0}\pi$ in the four solutions are quite distinct. The envelope curves (solid lines) are used to quote the physical results.

Figure 18

The NLL functions for the $CP$ asymmetries. The NLL functions of each solution are shown with dashed lines. The fitted values for the $B\to (K\pi {)}_0^{*0}\pi$ in the four solutions are quite distinct. The envelope curves (solid lines) are used to quote the physical results.

Figure 19

NLL scans for the global $CP$ asymmetry (a) and the total isobar fraction (b). The scans of each solution are shown with dashed lines. The envelope curves (solid lines) are the scans that are used to quote the physical results.