Direct CP violation in $K\to \pi \pi$ decays and supersymmetry

The quantities $\epsilon_K^\prime$ and $\epsilon_K$ measure the amount of direct and indirect CP violation in $K\to \pi\pi$ decays, respectively. Using the recent lattice results from the RBC and UKQCD Collaborations and a new compact implementation of the $\Delta S=1$ renormalization group evolution we predict $\mbox{Re}\, \frac{\epsilon_{K}'}{\epsilon_{K}} = \left(1.06 \pm 5.07 \right) \times 10^{-4}$ in the Standard Model. This value is $2.8\,\sigma$ below the experimental value of $\mbox{Re}\, \frac{\epsilon_{K}'}{\epsilon_{K}} = \left(16.6 \pm 2.3 \right) \times 10^{-4}.$ In generic models of new physics the well-understood $\epsilon_K$ precludes large contributions to $\epsilon_K^\prime$, if the new contributions enter at loop level. However, one can resolve the tension in $\epsilon_{K}'/\epsilon_{K}$ within the Minimal Supersymmetric Standard Model. To this end two features of supersymmetry are crucial: First, one can have large isospin-breaking contributions (involving the strong instead of the weak interaction) which enhance $\epsilon_K^\prime$. Second, the Majorana nature of gluinos permits a suppression of the MSSM contribution to $\epsilon_K$, because two box diagrams interfere destructively.