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SU(3) Relations and the CP Asymmetries in B Decays to $η' K_S$, $ϕK_S$ and $K^+ K^- K_S$

We consider CP asymmetries in neutral $B$ meson decays to $η' K_S$, $ϕK_S$, and $K^+ K^- K_S$. We use SU(3) relations to estimate or bound the contributions to these amplitudes proportional to $V_{ub}^*V_{us}$. Such contributions induce a deviation of the $S_f$ terms measured in these time dependent CP asymmetries from that measured for $ψK_S$. For the $K^+ K^- K_S$ mode, we estimate the deviation to be of order 0.1. For the $η' K_S$ mode, we obtain an upper bound on this deviation of order 0.3. For the $ϕK_S$ mode, we have to add a mild dynamical assumption to the SU(3) analysis due to insufficient available data, yielding an upper bound of order 0.25. These bounds may improve significantly with future data. While they are large at present compared to the usually assumed Standard Model contribution, they are obtained with minimal assumptions and hence provide more rigorous tests for new physics. If measurements yield $|S_f-S_{ψK}|$ that are much larger than our bounds, it would make a convincing case for new physics.