SU(3) Relations and the CP Asymmetry in $B \to K_S K_S K_S$

Abstract

The CP asymmetry in the $B \to K_S K_S K_S$ decay is being measured by the two B factories. A large deviation of the CP asymmetry $S_{K_S K_S K_S}$ from $-S_{\psi K_S}$ and/or of $C_{K_S K_S K_S}$ from zero would imply new physics in $b \to s$ transitions. We try to put upper bounds on the Standard Model size of these deviations, using SU(3) flavor relations and experimental data on the branching ratios of various decay modes that proceed via $b \to d$ transitions. We point out several subtleties that distinguish the case of three body final states from two body ones. We present several simple relations that can become useful once all relevant modes are measured accurately enough.