Amplitude analysis of $B^0\to K^{+}{\pi }^{-}{\pi }^0$ and evidence of direct $CP$ violation in $B\to K*\pi$ decays

We analyze the decay $B^0\to K^{+}{\pi }^{-}{\pi }^0$ with a sample of $4.54\times {10}^8$ $B\overline{B}$ events collected by the BABAR detector at the PEP-II asymmetric-energy $B$ factory at SLAC, and extract the complex amplitudes of seven interfering resonances over the Dalitz plot. These results are combined with amplitudes measured in $B^0\to K_S^0{\pi }^{+}{\pi }^{-}$ decays to construct isospin amplitudes from $B^0\to K^{*}\pi$ and $B^0\to \rho K$ decays. We measure the phase of the isospin amplitude ${\Phi }_{3/2}$, useful in constraining the Cabibbo-Kobayashi-Maskawa unitarity triangle angle $\gamma$ and evaluate a $CP$ rate asymmetry sum rule sensitive to the presence of new physics operators. We measure direct $CP$ violation in $B^0\to K^{*+}{\pi }^{-}$ decays at the level of $3\sigma$ when measurements from both $B^0\to K^{+}{\pi }^{-}{\pi }^0$ and $B^0\to K_S^0{\pi }^{+}{\pi }^{-}$ decays are combined.

Figure 1

The helicity angle $({\theta }_H)$ and momenta of particles $(\overrightarrow{q},\overrightarrow{p})$ in the rest frame of a resonance $k$.

Figure 2

The DP of the selected data sample of 23 683 events. The selection criteria are described in Sec. 4. The $B^0\to {\overline{D}}^0{\pi }^0$ decay is visible as a band near $m_{K^{+}{\pi }^{-}}^2=3.5(\mathrm{GeV}/c^2{)}^2$. The remaining resonances populate the borders of the DP.

Figure 3

The $m_{{\pi }^{-}{\pi }^0}$ invariant mass distributions in the entire kinematic range (top) and below $1.8\mathrm{GeV}/c^2$ (bottom) for all selected events. The $\rho (770{)}^{-}$ is visible as a broad peak near $0.77\mathrm{GeV}/c^2$. The data are shown as points with error bars. The solid histograms show the projection of the fit result for charmless signal and $D$ events (white), $B$ background (medium), and continuum (dark).

Figure 4

The $m_{K^{+}{\pi }^0}$ invariant mass distributions in the entire kinematic range (top) and below $1.8\mathrm{GeV}/c^2$ (bottom) for all selected events. The $K^{*}(892{)}^{+}$ is visible as a narrow peak near $0.89\mathrm{GeV}/c^2$ while the broad distribution near $1.40\mathrm{GeV}/c^2$ is the $(K\pi {)}_0^{*+}$. The data are shown as points with error bars. The solid histograms show the projection of the fit result for charmless signal and $D$ events (white), $B$ background (medium), and continuum (dark).

Figure 5

The $m_{K^{+}{\pi }^{-}}$ invariant mass distributions in the entire kinematic range (top) and below $1.8\mathrm{GeV}/c^2$ (bottom) for all selected events. The $K^{*}(892{)}^0$ is visible as a narrow peak near $0.89\mathrm{GeV}/c^2$ while the broad distribution near $1.40\mathrm{GeV}/c^2$ is the $(K\pi {)}_0^{*0}$. The narrow peak near $1.8\mathrm{GeV}/c^2$ is the $D^0$ meson. The data are shown as points with error bars. The solid histograms show the projection of the fit result for charmless signal and $D$ events (white), $B$ background (medium), and continuum (dark).

Figure 6

Distributions of $\Delta {\mathrm{E}}^{\prime }$ (left), $m_{\mathrm{ES}}$ (center), and $NN$ output (right) for all selected events. The data are shown as points with error bars. The solid histograms show the projection of the fit result for charmless signal and $D$ events (white), $B$ background (medium) and continuum (dark).

Figure 7

Distributions of the log-likelihood for all events (top) and for events excluding the $D^0$ region $1.8\mathrm{GeV}/c^2<m_{K^{+}{\pi }^{-}}<1.9\mathrm{GeV}/c^2$ (bottom). The data are shown as points with error bars. The solid histograms show the projection of the fit result for charmless signal and $D$ events (white), $B$ background (medium), and continuum (dark).

Figure 8

Likelihood scans illustrating the measurements of ${\Phi }_{K^{*0}{\pi }^0}-{\Phi }_{K^{*+}{\pi }^{-}}=-22\pm 24(\mathrm{stat})\pm 18(\mathrm{syst})°$ (top) and ${\overline{\Phi }}_{K^{*0}{\pi }^0}-{\overline{\Phi }}_{K^{*-}{\pi }^{+}}=-4\pm 24(\mathrm{stat})\pm 19(\mathrm{syst})°$ (bottom). The solid (dashed) line shows the $2\Delta \log (L)$ for the total (statistical) uncertainty.

Figure 9

Isospin triangles drawn to scale for $B\to K^{*}\pi$ decays. The isobar amplitudes are summarized in Table 8 as solution I. Note that the isospin triangle for $\overline{B}$ decays is relatively flat and ${\overline{\mathcal{A}}}_{3/2}(K^{*}\pi )$ is consistent with 0.

Figure 10

Likelihood scans illustrating the measurements of ${\Phi }_{{\rho }^{-}{K}^{+}}-{\Phi }_{K^{*+}{\pi }^{-}}=-33\pm 22(\mathrm{stat})\pm 20(\mathrm{syst})°$ (top) and ${\overline{\Phi }}_{{\rho }^{+}{K}^{-}}-{\overline{\Phi }}_{K^{*-}{\pi }^{+}}=-39\pm 25(\mathrm{stat})\pm 20(\mathrm{syst})°$ (bottom). The solid (dashed) line shows the $2\Delta \log (L)$ for the total (statistical) uncertainty.

Figure 11

Likelihood scans illustrating the measurement of ${\Phi }_{3/2}=-{10}_{-20}^{+10}(\mathrm{stat}{)}_{-22}^{+7}(\mathrm{syst})°$. The solid (dashed) line shows the $2\Delta \log (L)$ for the total (statistical) uncertainty.

Figure 12

Isospin triangles drawn to scale for $B\to \rho K$ decays. The isobar amplitudes are summarized in Table 8 as solution I and in Ref. [5].

Figure 13

Likelihood scan for ${\Sigma }_{3/2}$. The solid (dashed) line shows the $2\Delta \log (L)$ for the total (statistical) uncertainty. We measure ${\Sigma }_{3/2}={0.82}_{-0.92}^{+0.18}(\mathrm{stat}{)}_{-1.35}^{+0.11}(\mathrm{syst})$.

Figure 14

Likelihood scans for $A_{CP}(K^{*}(892{)}^{+}{\pi }^{-})$ using only the $B^0\to K^{+}{\pi }^{-}{\pi }^0$ analysis (top) and the combined measurement with the $B^0\to K_S^0{\pi }^{+}{\pi }^{-}$ analysis (bottom). The solid (dashed) line shows the $2\Delta \log (L)$ for the total (statistical) uncertainty. We measure $A_{CP}(K^{*+}{\pi }^{-})=-0.29\pm 0.11(\mathrm{stat})\pm 0.02(\mathrm{syst})$ (Table 6) in $B^0\to K^{+}{\pi }^{-}{\pi }^0$ and $A_{CP}(K^{*+}{\pi }^{-})=-0.24\pm 0.07(\mathrm{stat})\pm 0.02(\mathrm{syst})$ when the measurement in $B^0\to K_S^0{\pi }^{+}{\pi }^{-}$ is combined. The vertical line highlights $A_{CP}=0$ and the horizontal line corresponds to $2\Delta \log \mathcal{L}=9$, i.e. $3\sigma$.