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

The quantities ${ϵ}_K^{\mathrm{\prime }}$ and ${ϵ}_K$ 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 $\mathrm{\Delta }S=1$ renormalization group evolution we predict $ \mbox{Re}, \frac{\epsilon_{K}‘}{\epsilon_{K}} = \left(1.06 \pm 5.07 \right) \times 10^{-4}$intheStandardModel\mathrm{.}Thisvalueis$2.8,\sigma$belowtheexperimentalvalueof$ \mbox{Re}, \frac{\epsilon_{K}’}{\epsilon_{K}} = \left(16.6 \pm 2.3 \right) \times 10^{-4}.$Ingenericmodelsofnewphysicsthewell-understood$\epsilon_K$precludeslargecontributionsto$\epsilon_K^\prime$, if the new contributions enter at loop level. However, one can resolve the tension in ${ϵ}_K^{\mathrm{\prime }}\mathrm{/}{ϵ}_K$ 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 ${ϵ}_K^{\mathrm{\prime }}$. Second, the Majorana nature of gluinos permits a suppression of the MSSM contribution to ${ϵ}_K$, because two box diagrams interfere destructively.