Connections between epsilon'/epsilon and Rare Kaon Decays in Supersymmetry

Abstract

We analyze the rare kaon decays $K_L\to {\pi }^0\nu \bar{\nu }$, $K^{+}\to {\pi }^{+}\nu \bar{\nu }$, $K_L\to {\pi }^0{e}^{+}{e}^{-}$ and $K_L\to {\mu }^{+}{\mu }^{-}$ in conjunction with the CP violating ratio ${ϵ}^{\prime }/ϵ$ 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 ${ϵ}^{\prime }/ϵ$ we find $\mathrm{BR}(K_L\to {\pi }^0\nu \bar{\nu })\text{lsim}1.2\cdot 10^{-10}$, $\mathrm{BR}(K^{+}\to {\pi }^{+}\nu \bar{\nu })\text{lsim}1.7\cdot 10^{-10}$, $\mathrm{BR}(K_L\to {\pi }^0{e}^{+}{e}^{-}{)}_{\mathrm{dir}}\text{lsim}2.0\cdot 10^{-11}$, assuming the usual determination of the CKM parameters and neglecting the possibility of cancellations among different supersymmetric effects in ${ϵ}^{\prime }/ϵ$. 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 ${ϵ}^{\prime }/ϵ$, in order to shed light on the realization of various supersymmetric scenarios. We reemphasize that the most natural enhancement of ${ϵ}^{\prime }/ϵ$, within supersymmetric models, comes from chromomagnetic penguins and show that in this case sizable enhancements of $BR(K_L\to {\pi }^0{e}^{+}{e}^{-}{)}_{\mathrm{dir}}$ can also be expected.

Comments: 40 pages, 5 figures