Time-dependent analysis of $B^0\to K_S^0{\pi }^{-}{\pi }^{+}\gamma$ decays and studies of the $K^{+}{\pi }^{-}{\pi }^{+}$ system in $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ decays

We measure the time-dependent $CP$ asymmetry in the radiative-penguin decay $B^0\to K_S^0{\pi }^{-}{\pi }^{+}\gamma$, using a sample of $471\times {10}^6$ $\mathrm{\Upsilon }(4S)\to B\overline{B}$ events recorded with the BABAR detector at the PEP-II $e^{+}{e}^{-}$ storage ring at SLAC. Using events with $m_{K\pi \pi }<1.8\mathrm{GeV}/c^2$, we measure the branching fractions of $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ and $B^0\to K^0{\pi }^{-}{\pi }^{+}\gamma$, the branching fractions of the kaonic resonances decaying to $K^{+}{\pi }^{-}{\pi }^{+}$, as well as the overall branching fractions of the $B^{+}\to {\rho }^0{K}^{+}\gamma$, $B^{+}\to K^{*0}{\pi }^{+}\gamma$ and $S$-wave $B^{+}\to (K\pi {)}_0^{*0}{\pi }^{+}\gamma$ components. For events from the $\rho$ mass band, we measure the $CP$-violating parameters ${\mathcal{S}}_{K_S^0{\pi }^{+}{\pi }^{-}\gamma }=0.14\pm 0.25\pm 0.03$ and ${\mathcal{C}}_{K_S^0{\pi }^{+}{\pi }^{-}\gamma }=-0.39\pm 0.20_{-0.02}^{+0.03}$, where the first uncertainties are statistical and the second are systematic. We extract from this measurement the time-dependent $CP$ asymmetry related to the $CP$ eigenstate ${\rho }^0{K}_S^0$ and obtain ${\mathcal{S}}_{K_S^0\rho \gamma }=-0.18\pm 0.32_{-0.05}^{+0.06}$, which provides information on the photon polarization in the underlying $b\to s\gamma$ transition.

Figure 1

Combination of the efficiency maps for each kaonic resonance. The relative weights used for the combination are extracted from a fit to the $m_{K\pi \pi }$ spectrum (Sec. 4c). Large fluctuations at high $m_{K\pi }$ or $m_{\pi \pi }$ are due to the small number of events.

Figure 2

The $m_{K\pi }$ projections ${\mathcal{H}}_{{\mathrm{R}}_j}$ of the two-dimensional histograms ${\mathrm{H}}_{{\mathrm{R}}_j}(m_{K\pi },m_{\pi \pi })$ describing the $K^{*}(892{)}^0$ (upper left), the $\rho (770{)}^0$ (upper right) and the $(K\pi {)}_0^{*0}$ (bottom) contributions in the $m_{K\pi }$ fit model. The histograms, normalized to unit area, describe the expected $m_{K\pi }$ distributions of these components once reconstruction and resolution effects have been corrected.

Figure 3

Distributions of $m_{\mathrm{ES}}$ (top left), $\mathrm{\Delta }E$ (top right) and the Fisher discriminant (bottom) showing the fit results on the $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ data sample. The distributions have their signal/background ratio enhanced by means of the following requirements: $-0.10\le \mathrm{\Delta }E\le 0.075\mathrm{GeV}$ ($m_{\mathrm{ES}}$); $m_{\mathrm{ES}}>5.27\mathrm{GeV}/c^2$ ($\mathrm{\Delta }E$); $m_{\mathrm{ES}}>5.27\mathrm{GeV}/c^2$, $-0.10\le \mathrm{\Delta }E\le 0.075\mathrm{GeV}$ (Fisher). Points with error bars show data. The projection of the fit result is represented by stacked histograms, where the shaded areas represent the background contributions, as described in the legend. Some of the contributions are hardly visible due to their small fractions. Note that the same order is used for the various contributions in both the stacked histograms and the corresponding legend, in which the “Generic” and “Charmless” entries correspond to the generic $B$ background and the sum of $B^{+}\to a_1^{+}(\to {\rho }^0{\pi }^{+}){\pi }^0\gamma$ and $B^{+}\to K^{*0}(\to K\pi ){\pi }^{+}{\pi }^0\gamma$ event categories, respectively, as defined in Table 1.

Figure 4

Distribution of $m_{K\pi \pi }$ for correctly reconstructed $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ signal events (${}_s\mathcal{P}lot$), extracted from the maximum likelihood fit to $m_{\mathrm{ES}}$, $\mathrm{\Delta }E$, and $\mathcal{F}$. Points with error bars give the sum of ${}_s\text{Weights}$ [19]. The blue solid curve is the result of the fit performed directly to this $m_{K\pi \pi }$ distribution to extract the contributions from kaonic resonances decaying to $K^{+}{\pi }^{-}{\pi }^{+}$. Below each bin are shown the residuals, normalized in error units. The parallel dotted and full lines mark the one- and two-standard-deviation levels, respectively.

Figure 5

One-dimensional scans of $-2\mathrm{\Delta }\ln \mathcal{L}$ as a function of magnitudes (top and middle) and phases (bottom). The horizontal dashed lines mark the one- and two-standard-deviation levels.

Figure 6

Distribution of $m_{K\pi }$ for correctly reconstructed $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ signal events (${}_s\mathcal{P}lot$), extracted from the maximum-likelihood fit to $m_{\mathrm{ES}}$, $\mathrm{\Delta }E$, and $\mathcal{F}$. Points with error bars give the sum of ${}_s\text{Weights}$. The blue solid curve corresponds to the total PDF fit projection. The small-dashed red, medium-dashed green and dotted magenta curves correspond to the $K^{*}(892{)}^0$, $\rho (770{)}^0$ and $(K\pi {)}_0^{*0}$ contributions, respectively. The dashed-dotted gray curve corresponds to the interference between the two $P$-wave components, i.e. the $K^{*}(892{)}^0$ and the $\rho (770{)}^0$, and the dashed-triple-dotted light blue curve corresponds to the interference between the $(K\pi {)}_0^{*0}$ and the $\rho (770{)}^0$. Below the $m_{K\pi }$ spectrum, we also show the residuals normalized in units of standard deviations, where the parallel dotted and full lines mark the one- and two-standard-deviation levels, respectively.

Figure 7

One-dimensional scans of $-2\mathrm{\Delta }\ln \mathcal{L}$ as a function of the magnitudes (left) and phases (right) of the $\rho (770{)}^0$ and $(K\pi {)}_0^{*0}$ components. The horizontal dashed lines mark the one- and two-standard-deviation levels.

Figure 8

Distributions of $m_{\mathrm{ES}}$ (top left), $\mathrm{\Delta }E$ (top right), the Fisher discriminant (bottom left), and $\mathrm{\Delta }t$ (bottom right), showing the results of the fit to the $B^0\to K_S^0{\pi }^{-}{\pi }^{+}\gamma$ data sample. The distributions have their signal/background ratio enhanced by means of the following requirements: $-0.15\le \mathrm{\Delta }E\le 0.10\mathrm{GeV}$ ($m_{\mathrm{ES}}$); $m_{\mathrm{ES}}>5.27\mathrm{GeV}/c^2$ ($\mathrm{\Delta }E$); $m_{\mathrm{ES}}>5.27\mathrm{GeV}/c^2$, $-0.15\le \mathrm{\Delta }E\le 0.10\mathrm{GeV}$ (Fisher and $\mathrm{\Delta }t$). Points with error bars show the data. The projection of the fit result is represented by stacked histograms, where the shaded areas represent the background contributions, as described in the legend. Some of the contributions are hardly visible due to their small fractions. Note that the same order is used for the various contributions in both the stacked histograms and the corresponding legend.