Measurement of $CP$-violating asymmetries in $B^0\to (\rho \pi {)}^0$ decays using a time-dependent Dalitz plot analysis

We present results for a time-dependent Dalitz plot measurement of $CP$-violating asymmetries in the mode $B^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$. The data set is derived from the complete sample of $471\times {10}^6$ $B\overline{B}$ meson pairs collected with the BABAR detector at the PEP-II asymmetric-energy $e^{+}{e}^{-}$ collider at the SLAC National Accelerator Laboratory operating on the ${\rm Y}(4S)$ resonance. We extract parameters describing the time-dependent $B^0\to \rho \pi$ decay probabilities and $CP$ asymmetries, including $\mathcal{C}=0.016\pm 0.059\pm 0.036$, $\Delta \mathcal{C}=0.234\pm 0.061\pm 0.048$, $\mathcal{S}=0.053\pm 0.081\pm 0.034$, and $\Delta \mathcal{S}=0.054\pm 0.082\pm 0.039$, where the uncertainties are statistical and systematic, respectively. We perform a two-dimensional likelihood scan of the direct $CP$-violation asymmetry parameters for $B^0\to {\rho }^{\pm }{\pi }^{\mp }$ decays, finding the change in ${\chi }^2$ between the minimum and the origin (corresponding to no direct $CP$ violation) to be $\Delta {\chi }^2=6.42$. We present information on the $CP$-violating parameter $\alpha$ in a likelihood scan that incorporates $B^{\pm }\to \rho \pi$ measurements. To aid in the interpretation of our results, statistical robustness studies are performed to assess the reliability with which the true values of the physics parameters can be extracted. Significantly, these studies indicate that $\alpha$ cannot be reliably extracted with our current sample size, though the other physics parameters are robustly extracted.

Figure 1

Nominal (left) and square (right) $B^0\to \rho \pi$ Dalitz plots obtained from Monte Carlo generated events without detector simulation [2]. The amplitudes in Eq. (1) are generated with values $A^{+}=A^{-}=A^0=1$ so that they interfere destructively for equal $\pi \pi$ masses. The hatched areas indicate the main overlap regions between the different $\rho$ bands. The dashed lines in the square Dalitz plot correspond to $m({\rho }^{+,-,0})=1.5\mathrm{GeV}/c^2$. The middle plot depicts the Jacobian determinant of the transformation and shows the distribution in the square Dalitz plot for uniformly distributed events in the nominal Dalitz plot.

Figure 2

$B^0\to {\rho }^{\pm }{\pi }^{\mp }$ continuum background rejection versus signal efficiency for final neural network implementation, with continuum background represented by off-resonance data, and signal represented by signal MC. Points on the curve correspond to different requirements on the neural network output. Our selection criterion retains 75% of signal events.

Figure 3

NN output distributions for signal $B^0\to {\rho }^{\pm }{\pi }^{\mp }$ (solid blue) and continuum background (hatched red) with an arbitrary vertical scale. Continuum background is represented by off-resonance data while signal is represented by signal MC. Our selection criterion requires $\mathrm{NN}>0.58$.

Figure 4

Distribution of $\Delta t$ in simulated $B^0\to {\rho }^{\pm }{\pi }^{\mp }$ signal events (top) and in off-resonance data (bottom).

Figure 5

Square Dalitz plot map of TM (top) and SCF (bottom) selection efficiencies. The color scale on the TM plot has a maximum of 0.60 while that on the SCF plot has a maximum of 0.075.

Figure 6

Overlay plots of fit variable distributions in on-resonance data (points with error bars) and in a parametrized MC sample generated from the final fit results (red line) with 10 times the number of events in data. The MC histograms are scaled to have the same integral as the data histograms onto which they are overlaid. A restrictive selection criterion is applied to the NN output (${\mathrm{NN}}_{\mathrm{new}}>0.8$) to enhance the signal component.

Figure 7

Combined statistical and systematic two-dimensional likelihood scan of ${\mathcal{A}}_{\rho \pi }^{+-}$ versus ${\mathcal{A}}_{\rho \pi }^{-+}$ with 68.3%, 95.5%, and 99.7% confidence-level contours ($\Delta {\chi }^2=\{2.30,6.18,11.83\}$). The yellow dot inside the contours indicates the central value.

Figure 8

Isospin-constrained (solid red) and unconstrained (dashed black) scans of minimum ${\chi }^2$ values as a function of $\alpha$. The scans are based on the fit to the full on-resonance data set and include contributions from both statistical and systematic uncertainties. Note that the origin is suppressed.

Figure 9

Isospin-constrained (solid red) and unconstrained (dashed black) scans of $\Sigma$ values as a function of $\alpha$. The scans are based on the fit to the full on-resonance data set and include contributions from both statistical and systematic uncertainties. The upper and lower horizontal dashed lines correspond to $\Sigma =0.05$ and 0.32, respectively.

Figure 10

Plot of the sum of 25 normalized Gaussians with means and widths determined by the peak positions and mean errors extracted from the $25\alpha$ scans of parametrized MC generated with $\alpha =89°$. Also plotted are the individual Gaussians that contribute to the total PDF.

Figure 11

Plot of the total $\Sigma$ distribution obtained by summing all 25 $\Sigma$ $\alpha$ scans of parametrized MC generated with $\alpha =89°$ after normalizing their areas to 1. The total distribution is scaled so that it peaks at 1. Also plotted are the individual scaled $\Sigma$ scans that contribute to the total distribution.