Effects of final-state interactions on mixing-inducedCPviolation in penguin-dominatedBdecays

Abstract

Motivated by the recent indications of the possibility of sizable deviations of the mixing-induced $CP$ violation parameter, ${S}_{f}$, in the penguin-dominated $b\ensuremath{\rightarrow}sq\overline{q}$ transition decays such as ${B}^{0}\ensuremath{\rightarrow}(\ensuremath{\phi},\ensuremath{\omega},{\ensuremath{\rho}}^{0},{\ensuremath{\eta}}^{\ensuremath{'}},\ensuremath{\eta},{\ensuremath{\pi}}^{0},{f}_{0}){K}_{S}$ from $\mathrm{sin}2\ensuremath{\beta}$ determined from $B\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}_{S}$, we study final-state rescattering effects on their decay rates and $CP$ violation. Recent observations of large direct $CP$ asymmetry in modes such as ${B}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}},{\ensuremath{\rho}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ 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. $\mathcal{O}(1\ensuremath{-}2%)$] or negligible except for the $\ensuremath{\omega}{K}_{S}$ and ${\ensuremath{\rho}}^{0}{K}_{S}$ 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 $\ensuremath{\omega}{K}_{S}$, $\ensuremath{\phi}{K}_{S}$, ${\ensuremath{\eta}}^{\ensuremath{'}}{K}_{S}$, ${\ensuremath{\rho}}^{0}{K}_{S}$, and ${\ensuremath{\pi}}^{0}{K}_{S}$ rates significantly and flip the signs of direct $CP$ asymmetries of the last two modes. Direct $CP$ asymmetries in $\ensuremath{\omega}{K}_{S}$ and ${\ensuremath{\rho}}^{0}{K}_{S}$ channels are predicted to be ${\mathcal{A}}_{\ensuremath{\omega}{K}_{S}}\ensuremath{\approx}\ensuremath{-}0.13$ and ${\mathcal{A}}_{{\ensuremath{\rho}}^{0}{K}_{S}}\ensuremath{\approx}0.47$, respectively. However, direct $CP$ asymmetry in all the other $b\ensuremath{\rightarrow}s$ penguin-dominated modes that we study is found to be rather small ($\ensuremath{\lesssim}$ a few percent), rendering these modes a viable place to search for the $CP$-odd phases beyond the standard model. Since $\ensuremath{\Delta}{S}_{f}$ ($\ensuremath{\equiv}\ensuremath{-}{\ensuremath{\eta}}_{f}{S}_{f}\ensuremath{-}{S}_{J/\ensuremath{\psi}{K}_{S}}$, with ${\ensuremath{\eta}}_{f}$ 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 $|\ensuremath{\Delta}{S}_{f}|>0.10$ within the standard model for ${B}^{0}\ensuremath{\rightarrow}(\ensuremath{\phi},\ensuremath{\omega},{\ensuremath{\rho}}^{0},{\ensuremath{\eta}}^{\ensuremath{'}},\ensuremath{\eta},{\ensuremath{\pi}}^{0}){K}_{S}$; in particular, ${\ensuremath{\eta}}^{\ensuremath{'}}{K}_{S}$ 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 $\ensuremath{\Delta}{S}_{f}$ 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.