Measurement of $CP$ violating asymmetries in $B^0\to K^{+}{K}^{-}{K}_S^0$ decays with a time-dependent Dalitz approach

We report a measurement of $CP$ violating asymmetries in $B^0({\overline{B}}^0)\to K^{+}{K}^{-}{K}_S^0$ decays with a time-dependent Dalitz approach. This analysis is based on a data sample of $657\times {10}^6$ $B\overline{B}$ pairs accumulated at the ${\rm Y}(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. As the result of an unbinned maximum likelihood fit to the selected candidates, the mixing-induced and direct $CP$ violation parameters ${\varphi }_1^{\mathrm{eff}}$ and ${\mathcal{A}}_{CP}$ are obtained for $B^0\to \varphi (1020)K_S^0$, $B^0\to f_0(980)K_S^0$, and other $B^0\to K^{+}{K}^{-}{K}_S^0$ decays. We find four solutions that describe the data. There are ${\varphi }_1^{\mathrm{eff}}(B^0\to \varphi (1020)K_S^0)=(32.2\pm 9.0\pm 2.6\pm 1.4)°;{\varphi }_1^{\mathrm{eff}}(B^0\to \varphi (1020)K_S^0)=(26.2\pm 8.8\pm 2.7\pm 1.2)°;{\varphi }_1^{\mathrm{eff}}(B^0\to \varphi (1020)K_S^0)=(27.3\pm 8.6\pm 2.8\pm 1.3)°;\mathrm{\text{and}}{\varphi }_1^{\mathrm{eff}}(B^0\to \varphi (1020)K_S^0)=(24.3\pm 8.0\pm 2.9\pm 5.2)°.$ The values for the $CP$ violating phase in $B^0\to \varphi (1020)K_S^0$ are similar, but other properties of the Dalitz plot are quite different for the four solutions. These four solutions have consistent ${\varphi }_1^{\mathrm{eff}}$ values for all three $B$-meson decay channels and none of them deviates significantly from the values measured in $B\to (c\overline{c})K^0$ decays with the currently available statistics. In addition, we find no significant direct $CP$ violation.

Figure 1

Signal enhanced total projections of (a) $\Delta E$ with $5.272\mathrm{GeV}/c^2<M_{\mathrm{bc}}<5.288\mathrm{GeV}/c^2$, (b) $M_{\mathrm{bc}}$ with $|\Delta E|<0.045\mathrm{GeV}$, and (c)  $l$ in the $(\Delta E,M_{\mathrm{bc}})$ signal region for the $B^0\to K^{+}{K}^{-}{K}_S^0$ candidate events. The solid curves show the fit projections, the hatched areas show the continuum background component, and the dotted curves show the total background contribution. The points with error bars are the data.

Figure 2

The Dalitz distribution based on our signal model of GEANT-based signal MC [23](a) with the normal Dalitz parameterization $s_{+}$ and $s_{-}$, and (b) with the square Dalitz parameterization $m^{\prime }$ and ${\theta }^{\prime }$. The dashed boxes indicate the regions where most of the signal components and the background are located.

Figure 3

The mass projections onto (a) $M(K_S^0{K}^{+})$, (b) $M(K_S^0{K}^{-})$, and (c) $M(K^{+}{K}^{-})$ distributions (the inset shows the projection near the $\varphi (1020)$ resonance) for the $B^0\to K^{+}{K}^{-}{K}_S^0$ candidate events in the signal region using Solution 1. In (a–c), the solid curves show the fit projections while the hatched areas and the dashed curves show the $q\overline{q}$ and total background components, respectively. The points with error bars are the data.

Figure 4

(a) $\Delta t$ distribution and (b) raw asymmetry for the $B^0\to K^{+}{K}^{-}{K}_S^0$ candidates in the $\varphi (1020)K_S^0$ region $|M_{K^{+}{K}^{-}}-M_{\varphi (1020)}|<0.01\mathrm{GeV}/c^2$, with good tags $r>0.5$, using Solution 1. In (a), the solid and dashed curves show the fitted results with $B^0$ and ${\overline{B}}^0$ tags, respectively. The dotted curve shows the background component with $B^0$ and ${\overline{B}}^0$ tags. In (b), the solid curve shows the fit projection and the dashed curve shows the standard model expectation from the time-dependent $CP$ asymmetry measurement in $b\to c\overline{c}s$ decays.

Figure 5

Likelihood scans of ${\varphi }_1^{\mathrm{eff}}(f_0(980)K_S^0)$ for (a) Solution 1, (b) Solution 2, (c) Solution 3, and (d) Solution 4. The solid (dashed) curve contains the total (statistical) error and the dotted box indicates the parameter range corresponding to $\pm 1\sigma$.

Figure 6

Likelihood scans of ${\varphi }_1^{\mathrm{eff}}(\varphi (1020)K_S^0)$ for (a) Solution 1, (b) Solution 2, (c) Solution 3, and (d) Solution 4. The solid (dashed) curve contains the total (statistical) error and the dotted box indicates the parameter range corresponding to $\pm 1\sigma$.