Motivated by the recent indications of the possibility of sizable deviations of the mixing-induced $CP$ violation parameter, $S_f$, in the penguin-dominated $b\to sq\overline{q}$ transition decays such as $B^0\to (\varphi ,\omega ,{\rho }^0,{\eta }^{\prime },\eta ,{\pi }^0,f_0)K_S$ from $\sin 2\beta$ determined from $B\to J/\psi K_S$, we study final-state rescattering effects on their decay rates and $CP$ violation. Recent observations of large direct $CP$ asymmetry in modes such as $B^0\to K^{+}{\pi }^{-},{\rho }^{-}{\pi }^{+}$ suggest that final-state phases in two-body $B$ decays may not be small. It is therefore important to examine these long-distance effects on penguin-dominated decays. Such long-distance effects on $S_f$ are found to be generally small [i.e. $\mathcal{O}(1-2\%)$] or negligible except for the $\omega K_S$ and ${\rho }^0{K}_S$ modes where $S_f$ is lowered by around 15% for the former and increased by the same percentage for the latter. However, final-state rescattering can enhance the $\omega K_S$, $\varphi K_S$, ${\eta }^{\prime }{K}_S$, ${\rho }^0{K}_S$, and ${\pi }^0{K}_S$ rates significantly and flip the signs of direct $CP$ asymmetries of the last two modes. Direct $CP$ asymmetries in $\omega K_S$ and ${\rho }^0{K}_S$ channels are predicted to be ${\mathcal{A}}_{\omega K_S}\approx -0.13$ and ${\mathcal{A}}_{{\rho }^0{K}_S}\approx 0.47$, respectively. However, direct $CP$ asymmetry in all the other $b\to s$ penguin-dominated modes that we study is found to be rather small ($\lesssim$ a few percent), rendering these modes a viable place to search for the $CP$-odd phases beyond the standard model. Since $\Delta S_f$ ($\equiv -{\eta }_f{S}_f-S_{J/\psi K_S}$, with ${\eta }_f$ being the $CP$ eigenvalue of the final state $f$) and ${\mathcal{A}}_f$ are closely related, the theoretical uncertainties in the mixing-induced parameter $S_f$ and the direct $CP$ asymmetry parameter ${\mathcal{A}}_f$ are also coupled. Based on this work, it seems difficult to accommodate $|\Delta S_f|>0.10$ within the standard model for $B^0\to (\varphi ,\omega ,{\rho }^0,{\eta }^{\prime },\eta ,{\pi }^0)K_S$; in particular, ${\eta }^{\prime }{K}_S$ is especially clean in our picture. For $f_0{K}_S$, at present we cannot make reliable estimates. The sign of the central value of $\Delta S_f$ for all the modes we study is positive but quite small, compared to the theoretical uncertainties, so that at present conclusive statements on the sign are difficult to make.

• Received 1 March 2005

DOI:https://doi.org/10.1103/PhysRevD.72.014006