Connections between ε^'/ε and Rare Kaon Decays in Supersymmetry

We analyze the rare kaon decays KL→π⁰ νν ̅, K^+→π^+ νν ̅, KL→π⁰ e^+ e^- and KL→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 BR (K^+→π⁰ νν ̅) ≲ 1. 2 * 10^-10, BR (, K^+→π^+ νν ̅) ≲ 1. 7*10^-10, BR〖 (KL→π⁰ e^+ e^-) 〗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→π⁰ e^+ e^-) 〗dir; can also be expected.