Upper Bounds on $K\to \pi \nu \bar{\nu }$ and $K_L\to {\pi }^0{e}^{+}{e}^{-}$ from ${ϵ}^{\prime }/ϵ$ and $K_L\to {\mu }^{+}{\mu }^{-}$

Abstract

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

Comments: LaTeX, 21 pages, 1 figure