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

We analyze the rare kaon decays K_L→π^0 νν ̅,K^+→π^+ νν ̅ ,K_L→π^0 e^+ e^- and K_L→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^+→π^0 νν ̅)≲ 1.2 * 10⁻¹⁰, BR(,K^+→π^+ νν ̅) ≲ 1.7*10⁻¹⁰, BR〖(K_L→π^0 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〖(K_L→π^0 e^+ e^-)〗_dir; can also be expected.