Difficulties in obtaining weak phases from b → d penguin decays

We re-examine a recent proposal for obtaining β using the measurements of the b → d penguin decays B_d⁰(t) → K⁰K⁰ and B_s⁰(t) → ΠK_S, along with a theoretical assumption. We show that there are in fact three assumptions one can make, so that the method can in principle be used to extract α, β or γ. We also show that the best assumption is the one which yields γ. However, the theoretical error on this assumption is still 25-30%, which leads to an error on → of at least 20-25%. Given our current understanding of hadronic physics, it does not seem possible to reduce this error.