$K\to πν\overlineν$ in the MSSM in Light of the $ε^{\prime}_K/ε_K$ Anomaly

Abstract

The Standard-Model (SM) prediction for the CP-violating quantity $ε_K^{\prime}/ε_K$ deviates from its measured value by 2.8 $σ$. It has been shown that this tension can be resolved within the Minimal Supersymmetric Standard Model (MSSM) through gluino-squark box diagrams, even if squarks and gluinos are much heavier than 1 TeV. The rare decays $K_L \to π^0ν\barν$ and $K^+ \to π^+ν\barν$ are similarly sensitive to very high mass scales and the first one also measures CP violation. In this article, we analyze the correlations between $ε^{\prime}_K/ε_K$ and $B(K_L \to π^0ν\barν)$ and $B(K^+ \to π^+ν\barν)$ within the MSSM aiming at an explanation of $ε_K^{\prime}/ε_K$ via gluino-squark box diagrams. The dominant MSSM contribution to the $K \to πν\barν$ branching fractions stems from box diagrams with squarks, sleptons, charginos, and neutralinos, and the pattern of the correlations is different from the widely studied $Z$-penguin scenarios. This is interesting in light of future precision measurements by KOTO and NA62 at J-PARC and CERN, respectively. We find $B(K_L \to π^0ν\barν)/B^{SM} (K_L \to π^0ν\barν)\lesssim 2\,(1.2)$ and $B(K^+ \to π^+ν\barν)/B^{SM}(K^+ \to π^+ν\barν) \lesssim 1.4\,(1.1)$, if all squark masses are above 1.5 TeV, gaugino masses obey GUT relations, and if one allows for a fine-tuning at the $1\%\,(10\%)$ level for the gluino mass. Larger values are possible for a tuned CP violating phase. Furthermore, the sign of the MSSM contribution to $ε_K^{\prime}$ imposes a strict correlation between $B(K_L \to π^0ν\barν)$ and the hierarchy between the masses $m_{\bar{U}}$, $m_{\bar{D}}$ of the right-handed up-squark and down-squark: sgn$[B(K_L \to π^0ν\barν)-B^{SM} (K_L \to π^0ν\barν)] = $sgn$(m_{\bar{U}}-m_{\bar{D}}) $.

9 pages, 5 figures; references added, version published in PRD