Connections between ε′ / ε and rare kaon decays in supersymmetry

We analyze the rare kaon decays K_L → π⁰ν̄ν, K⁺ → π⁺ν̄ν, K_L → π⁰e⁺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_L → π⁰ν̄ν) ≲ 1.2 × 10^-10, BR(K⁺ → π⁺ ν̄ν) ≲ 1.7 × 10^-10, BR(K_L → π⁰e⁺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 → π⁰e⁺e^-)_dir can also be expected.