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Improving Bounds on Penguin Pollution in B -> pi pi

In the presence of penguin contributions, the indirect CP asymmetry in B0(t) -> pi+ pi- measures sin(2α+ 2θ), where 2θparametrizes the size of the penguin ``pollution.'' We derive a new upper bound on |2θ|, requiring the measurement of of BR(B+ -> pi+ pi0) and an upper bound on B00 \equiv {1\over 2} [BR(B0 -> pi0 pi0) + BR(B0bar -> pi0 pi0)]. The new bound is stronger than those previously discussed in the literature. We also present a lower bound on B00. Current data may suggest that it is not very small, in which case θcan be determined using a complete isospin analysis.