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.