Upper bounds on K → πvv̄ and K_L → π⁰e⁺e^- from ε′ / ε and K_L → μ⁺μ^-

We analyze rare kaon decays in models in which the dominant new effect is an enhanced s̄dZ vertex Z_ds. We point out that in spite of large theoretical uncertainties the CP-violating ratio ε^′ / ε provides at present the strongest constraint on Im Z_ds. Assuming 0 ≤ ε^′ / ε ≤ 2 × 10^-3 and Standard Model values for the CKM parameters we obtain the bounds BR(K_L → π⁰ vv̄) ≤ 2.4 × 10^-10 and BR(K_L → π⁰e⁺e^-) ≤ 3.6 × 10^-11 (which are substantially stronger than the bounds found recently by Colangelo and Isidori, using ε_K instead of ε′ / ε). We illustrate how these bounds can be improved with the help of the forthcoming data on ε′ / ε. Using the bound on Re Z_ds from K_L → μ⁺μ^- we find BR(K⁺ → π⁺vv̄) ≤ 2.3 × 10^-10. In this context we derive an analytic upper bound on BR(K⁺ → π⁺vv̄) as a function of BR(K^L → π⁰vv̄) and the short distance contribution to BR(K^L → μ⁺μ^-). We also discuss new physics scenarios in which in addition to an enhanced s̄dZ vertex also neutral meson mixing receives important new contributions. In this case larger values of the branching ratios in question cannot be excluded.