Connections between ε ′ / ε and Rare Kaon Decays in Supersymmetry

We analyze the rare kaon decays KL → π0νν̄, K+ → π+νν̄, KL → π0e+e− and KL → μ+μ− in conjunction with the CP violating ratio ε/ε in a general class of supersymmetric models in which Zand magnetic-penguin contributions can be substantially larger than in the Standard Model. We point out that radiative effects relate the double left-right mass insertion to the single left-left one, and that the phenomenological constraints on the latter reflect into a stringent bound on the supersymmetric contribution to the Z penguin. Using this bound, and those coming from recent data on ε/ε, we find BR(KL → π0νν̄) < ∼ 1.2·10−10, BR(K+ → π+νν̄) < ∼ 1.7·10−10, BR(KL → πee)dir < ∼ 2.0·10−11, assuming the usual determination of the CKM parameters and neglecting the possibility of cancellations among different supersymmetric effects in ε/ε. Larger values are possible, in principle, but rather unlikely. We stress the importance of a measurement of these three branching ratios, together with improved data and improved theory of ε/ε, in order to shed light on the realization of various supersymmetric scenarios. We reemphasize that the most natural enhancement of ε/ε, within supersymmetric models, comes from chromomagnetic penguins and show that in this case sizable enhancements of BR(KL → πee)dir can also be expected.