Constraints on penguins from hadronic sum rules

We have derived an upper bound on the K-π transition amplitude of the penguin operator using hadronic sum rules. Combined with an anomalous-commutator contribution, this sets lower and upper limits on the physical K → 2π amplitude of the penguin operator. Applying this result to CP-violating effects, we find that the Gilman-Hagelin lower limit on ?′/? decreases by an order of magnitude.