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Interpreting the time-dependent CP asymmetry in $B^0 \to π^0 K_S$

Flavor SU(3) is used for studying the time-dependent CP asymmetry in $B^0 \to π^0 K_S$ by relating this process to $B^0\to π^0π^0$ and $B^0 \to K^+K^-$. We calculate correlated bounds on $S_{πK} - \sin 2β$ and $C_{πK}$, with maximal magnitudes of 0.2 and 0.3, where $S_{πK}$ and $C_{πK}$ are coefficients of $\sinΔmt$ and $\cosΔmt$ in the asymmetry. Stronger upper limits on $B^0 \to K^+K^-$ are expected to reduce these bounds and to imply nonzero lower limits on these observables. The asymmetry is studied as a function of a strong phase and the weak phase $γ$.