Measurement of $CP$ asymmetries in $B^0\to \varphi K^0$ and $B^0\to K^{+}{K}^{-}{K}_s^0$ decays

We measure the time-dependent $CP$ asymmetry parameters in $B^0\to K^{+}{K}^{-}{K}^0$ based on a data sample of approximately $227\times {10}^6$ $B$-meson pairs recorded at the ${\rm Y}(4S)$ resonance with the BABAR detector at the PEP-II $B$-meson Factory at SLAC. We reconstruct two-body $B^0$ decays to $\varphi (1020)K_S^0$ and $\varphi (1020)K_L^0$, and the three-body decay $K^{+}{K}^{-}{K}_S^0$ with $\varphi (1020)K_S^0$ excluded. For the $B^0\to \varphi K^0$ decays, we measure $\sin 2{\beta }_{\mathrm{eff}}(\varphi K^0)=+0.50\pm 0.25(\mathrm{stat}{)}_{-0.04}^{+0.07}(\mathrm{syst})$. The $B^0\to K^{+}{K}^{-}{K}_S^0$ decays are dominated by $K^{+}{K}^{-}$ $S$ wave, as determined from an angular analysis; we measure $\sin 2{\beta }_{\mathrm{eff}}(K^{+}{K}^{-}{K}_S^0)=+0.55\pm 0.22(\mathrm{stat})\pm 0.04(\mathrm{syst})\pm 0.11(CP)$, where the last error is due to the uncertainty in the fraction of $CP$-even contributions to the decay amplitude. We find no evidence for direct $CP$ violation.