Studies of the decays $D^0\to K_S^0{K}^{-}{\pi }^{+}$ and $D^0\to K_S^0{K}^{+}{\pi }^{-}$

The first measurements of the coherence factor $R_{K_S^{0}K\pi }$ and the average strong-phase difference ${\delta }_D^{K_S^{0}K\pi }$ in $D^0\to K_S^0{K}^{\mp }{\pi }^{\pm }$ decays are reported. These parameters can be used to improve the determination of the unitary triangle angle $\gamma$ in $B^{-}\to \tilde{D}{K}^{-}$ decays, where $\tilde{D}$ is either a $D^0$ or a ${\overline{D}}^0$ meson decaying to the same final state, and also in studies of charm mixing. The measurements of the coherence factor and strong-phase difference are made using quantum-correlated, fully reconstructed $D^0{\overline{D}}^0$ pairs produced in $e^{+}{e}^{-}$ collisions at the $\psi (3770)$ resonance. The measured values are $R_{K_S^{0}K\pi }=0.73\pm 0.08$ and ${\delta }_D^{K_S^{0}K\pi }=(8.3\pm 15.2)°$ for an unrestricted kinematic region and $R_{K^{*}K}=1.00\pm 0.16$ and ${\delta }_D^{K^{*}K}=(26.5\pm 15.8)°$ for a region where the combined $K_S^0{\pi }^{\pm }$ invariant mass is within $100\mathrm{MeV}/c^2$ of the $K^{*}(892{)}^{\pm }$ mass. These results indicate a significant level of coherence in the decay. In addition, isobar models are presented for the two decays, which show the dominance of the $K^{*}(892{)}^{\pm }$ resonance. The branching ratio $\mathcal{B}(D^0\to K_S^0{K}^{+}{\pi }^{-})/\mathcal{B}(D^0\to K_S^0{K}^{-}{\pi }^{+})$ is determined to be $0.592\pm 0.044(\mathrm{stat})\pm 0.018(\mathrm{syst})$, which is more precise than previous measurements.

Figure 1

The reconstructed mass distributions (points) of (a) $D^0\to K_S^0{K}^{-}{\pi }^{+}$ and (b) $D^0\to K_S^0{K}^{+}{\pi }^{-}$ in CLEO III data. The signal window selection in $\Delta m_D$ has been applied. The total fit projection (dark line) and background (light line) component are also shown. The signal window in the reconstructed mass variable is indicated by the vertical lines.

Figure 2

CLEO-c data distribution of fully-reconstructed double tags on the [$m_{bc}(D_{\mathrm{sig}})$, $m_{bc}(D_{\mathrm{tag}})$] plane (a) where the signal region is labeled S and the other indicated boxes are the sideband regions. The projections of the $m_{bc}(D_{\mathrm{sig}})$ are shown separately for candidates that are tagged by (b) $CP$ eigenstates, (c) the flavor states, and (d) $K_S^0{\pi }^{+}{\pi }^{-}$.

Figure 3

The distribution for the missing mass squared for the $K_L^0{\pi }^{+}{\pi }^{-}$ tagged candidates. The points show the data and the solid line is the Monte Carlo background prediction, scaled to the data luminosity. The region between the two straight solid lines is the signal region, marked S and the regions between the two dotted lines are the sidebands, marked L and H, used to determine the combinatorial background.

Figure 4

Dalitz plot distributions for (a)[(c)] $D^0\to K_S^0{K}^{-}{\pi }^{+}$ and (b)[(d)] $D^0\to K_S^0{K}^{+}{\pi }^{-}$ for flavor-tagged CLEO III [CLEO-c] data.

Figure 5

Fit projections for the $D^0\to K_S^0{K}^{-}{\pi }^{+}$ decay mode using favored model 1, the best model containing the $K^{*}(892{)}^0$ resonance. The data are shown as points and the total fit projection is overlaid with the solid line. The projection with the dotted line is the contribution of background. Plots (a)–(c)[(d)–(f)] show the CLEO III [CLEO-c] data for the three pairs of invariant mass distributions.

Figure 6

Fit projections for the $D^0\to K_S^0{K}^{-}{\pi }^{+}$ decay mode using favored model 2, the best model. The data are shown as points and the total fit projection is overlaid with the solid line. The projection with the dotted line is the contribution of background. Plots (a)-(c)[(d)-(f)] show the CLEO III [CLEO-c] data for the three pairs of invariant mass distributions.

Figure 7

Fit projections for the $D^0\to K_S^0{K}^{+}{\pi }^{-}$ decay mode using suppressed model 1, the best model containing the $K^{*}(892{)}^0$ resonance. The data are shown as points and the total fit projection is overlaid with the solid line. The projection with the dotted line is the contribution of background. Plots (a)–(c)[(d)–(f)] show the CLEO III [CLEO-c] data for the three pairs of invariant mass distributions.

Figure 8

Fit projections for the $D^0\to K_S^0{K}^{+}{\pi }^{-}$ decay mode using suppressed model 2, the best model. The data are shown as points and the total fit projection is overlaid with the solid line. The projection with the dotted line is the contribution of background. Plots (a)–(c)[(d)–(f)] show the CLEO III [CLEO-c] data for the three pairs of invariant mass distributions.

Figure 9

The best-fit point for $R_{K_S^{0}K\pi }$ and ${\delta }_D^{K_S^{0}K\pi }$ measured over the whole Dalitz plot, and the regions enclosing 1, 2, and 3 standard deviations from that point.

Figure 10

The best-fit point for $R_{K^{*}K}$ and ${\delta }_D^{K^{*}K}$ which are measured over the restricted region, and the regions enclosing 1, 2, and 3 standard deviations from that point.