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Connections between epsilon'/epsilon and Rare Kaon Decays in Supersymmetry

We analyze the rare kaon decays $K_L \to π^0 ν\bar ν$, $K^+ \to π^+ ν\bar ν$, $K_L \to π^0 e^+ e^-$ and $K_L \to μ^+ μ^-$ in conjunction with the CP violating ratio $ε'/ε$ in a general class of supersymmetric models in which $Z$- and 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 ${\rm BR}(K_L \to π^0 ν\bar ν)\lsim 1.2\cdot 10^{-10}$, ${\rm BR}(K^+ \to π^+ ν\bar ν)\lsim 1.7\cdot 10^{-10}$, ${\rm BR}(K_L \to π^0 e^+ e^-)_{\rm dir}\lsim 2.0\cdot 10^{-11}$, 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 $BR(K_L \to π^0 e^+ e^-)_{\rm dir}$ can also be expected.