Supersymmetric Explanation of CP Violation in $K\to \pi\pi$ Decays

Recent progress in the determination of hadronic matrix elements has revealed a tension between the measured value of $\epsilon_K^{\prime}/\epsilon_K$, which quantifies direct $CP$ violation in $K \to \pi\pi$ decays, and the Standard-Model prediction. The well-understood indirect $CP$ violation encoded in the quantity $\epsilon_K$ typically precludes large new-physics contributions to $\epsilon_K^{\prime}/\epsilon_K$ and challenges such an explanation of the discrepancy. We show that it is possible to cure the $\epsilon_K^{\prime}/\epsilon_K$ anomaly in the Minimal Supersymmetric Standard Model with squark masses above 3 TeV without overshooting $\epsilon_K$. This solution exploits two features of supersymmetry: the possibility of large isospin-breaking contributions (enhancing $\epsilon_K^{\prime}$) and the Majorana nature of gluinos (permitting a suppression of $\epsilon_K$). Our solution involves no fine-tuning of $CP$ phases or other parameters.