First measurement of ${\varphi }_3$ with a model-independent Dalitz plot analysis of $B^{\mathbf{\pm }}\to D{K}^{\mathbf{\pm }}$, $D\to K_S^0{\pi }^{\mathbf{+}}{\pi }^{\mathbf{-}}$ decay

We present the first measurement of the angle ${\varphi }_3$ of the unitarity triangle using a model-independent Dalitz plot analysis of $B^{\pm }\to D{K}^{\pm }$, $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays. The method uses, as input, measurements of the strong phase of the $D\to K_S^0{\pi }^{+}{\pi }^{-}$ amplitude from the CLEO Collaboration. The result is based on the full data set of $772\times {10}^6$ $B\overline{B}$ pairs collected by the Belle experiment at the ${\rm Y}(4S)$ resonance. We obtain ${\varphi }_3=({77.3}_{-14.9}^{+15.1}\pm 4.1\pm 4.3)°$ and the suppressed amplitude ratio $r_B=0.145\pm 0.030\pm 0.010\pm 0.011$. Here the first error is statistical, the second is the experimental systematic uncertainty, and the third is the error due to the precision of the strong-phase parameters obtained by CLEO.

Figure 1

Optimal binning of the $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot. The color scale indicated corresponds to the absolute value of the bin index, $|i|$.

Figure 2

Comparison of phase terms $(c_i,s_i)$ for the optimal binning measured by CLEO, and calculated from the Belle $D\to K_S^0{\pi }^{+}{\pi }^{-}$ amplitude model.

Figure 3

Projections of the flavor-tagged $D^{*\pm }\to D{\pi }^{\pm }$, $D\to K_S^0{\pi }^{+}{\pi }^{-}$ data with $1.8\mathrm{GeV}/c<p_D<2.8\mathrm{GeV}/c$. (a) $M_D$ distribution for $144.5\mathrm{MeV}/c^2<\Delta M<146.5\mathrm{MeV}/c^2$. (b) $\Delta M$ distribution for $1854\mathrm{MeV}/c^2<M_D<1876\mathrm{MeV}/c^2$. Histograms show the fitted signal and background contributions, points with the error bars are the data. The full $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot is used.

Figure 4

Projections of the $B^{\pm }\to D{\pi }^{\pm }$ data. (a) $M_{\mathrm{bc}}$ distribution with $|\Delta E|<30\mathrm{MeV}$ and $\cos {\theta }_{\mathrm{thr}}<0.8$ requirements. (b) $\Delta E$ distribution with $M_{\mathrm{bc}}>5270\mathrm{MeV}/c^2$ and $\cos {\theta }_{\mathrm{thr}}<0.8$ requirements. (c) $\cos {\theta }_{\mathrm{thr}}$ and (d) $\mathcal{F}$distributions with $|\Delta E|<30\mathrm{MeV}$ and $M_{\mathrm{bc}}>5270\mathrm{MeV}/c^2$ requirements. Histograms show the fitted signal and background contributions, points with error bars are the data. The entire $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot is used.

Figure 5

Projections of the $B^{\pm }\to D{K}^{\pm }$ data. (a) $M_{\mathrm{bc}}$ distribution with $|\Delta E|<30\mathrm{MeV}$ and $\cos {\theta }_{\mathrm{thr}}<0.8$ requirements. (b) $\Delta E$ distribution with $M_{\mathrm{bc}}>5270\mathrm{MeV}/c^2$ and $\cos {\theta }_{\mathrm{thr}}<0.8$ requirements. (c) $\cos {\theta }_{\mathrm{thr}}$ and (d) $\mathcal{F}$ distributions with $|\Delta E|<30\mathrm{MeV}$ and $M_{\mathrm{bc}}>5270\mathrm{MeV}/c^2$ requirements. Histograms show the fitted signal and background contributions, points with error bars are the data. The entire $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot is used.

Figure 6

Results of the fit to the $B^{\pm }\to D{\pi }^{\pm }$ sample. (a) Signal yield in bins of the $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot: from $B^{-}\to D{\pi }^{-}$ (red), $B^{+}\to D{\pi }^{+}$ (blue) and flavor sample (histogram). (b) Difference of signal yields between the $B^{+}\to D{\pi }^{+}$ and $B^{-}\to D{\pi }^{-}$ decays. (c) Difference of signal yields between the $B^{-}\to D{\pi }^{-}$ and flavor samples (normalized to the total $B^{-}\to D{\pi }^{-}$ yield): yield from the separate fits (points with error bars), and as a result of the combined $(x,y)$ fit (horizontal bars). (d) Same as (c) for $B^{+}\to D{\pi }^{+}$ data.

Figure 7

Results of the fit of the $B^{\pm }\to D{K}^{\pm }$ sample. (a) Signal yield in bins of the $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot: from $B^{-}\to D{K}^{-}$ (red), $B^{+}\to D{K}^{+}$ (blue) and flavor sample (histogram). (b) Difference of signal yields between the $B^{+}\to D{K}^{+}$ and $B^{-}\to D{K}^{-}$ decays. (c) Difference of signal yields between the $B^{-}\to D{K}^{-}$ and flavor samples (normalized to the total $B^{-}\to D{K}^{-}$ yield): yield from the separate fits (points with error bars), and as a result of the combined $(x,y)$ fit (horizontal bars). (d) Same as (c) for $B^{+}\to D{K}^{+}$ data.

Figure 8

1, 2, and 3 standard deviation $(x,y)$ confidence levels for (a) $B^{\pm }\to D{\pi }^{\pm }$ and (b) $B^{\pm }\to D{K}^{\pm }$ decays (statistical only). Note different scale for $B^{\pm }\to D{\pi }^{\pm }$ and $B^{\pm }\to D{K}^{\pm }$ modes. The weak phase ${\varphi }_3$ appears as half the opening angle between $(x_{+},y_{+})$ and $(x_{-},y_{-})$ vectors.

Figure 9

Two-dimensional projections of confidence region onto $({\varphi }_3,{\delta }_B)$ and $({\varphi }_3,r_B)$ planes (1, 2, and 3 standard deviations, statistical only).

Figure 10

Projections of the combined fit of the $B^{+}\to D{K}^{+}$ sample on $\Delta E$ for each Dalitz plot bin, with the $M_{\mathrm{bc}}>5270\mathrm{MeV}/c^2$ and $\cos {\theta }_{\mathrm{thr}}<0.8$ requirements. The fill styles for the signal and background components are the same as in Fig. 5.

Figure 11

Projections of the combined fit of the $B^{-}\to D{K}^{-}$ sample on $\Delta E$ for each Dalitz plot bin, with the $M_{\mathrm{bc}}>5270\mathrm{MeV}/c^2$ and $\cos {\theta }_{\mathrm{thr}}<0.8$ requirements. The fill styles for the signal and background components are the same as in Fig. 5.