$K\to \pi \nu \overline{\nu }$ in the MSSM in Light of the ${ϵ}_K^{\prime }/{ϵ}_K$ 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 $\sigma$. 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 $K_L\to {\pi }^0\nu \bar{\nu }$ and $K^{+}\to {\pi }^{+}\nu \bar{\nu }$ are similarly sensitive to very high mass scales and the first one also measures CP violation. In this article, we analyze the correlations between ${ϵ}_K^{\prime }/{ϵ}_K$ and $B(K_L\to {\pi }^0\nu \bar{\nu })$ and $B(K^{+}\to {\pi }^{+}\nu \bar{\nu })$ within the MSSM aiming at an explanation of ${ϵ}_K^{\prime }/{ϵ}_K$ via gluino-squark box diagrams. The dominant MSSM contribution to the $K\to \pi \nu \bar{\nu }$ 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 $B(K_L\to {\pi }^0\nu \bar{\nu })/B^{SM}(K_L\to {\pi }^0\nu \bar{\nu })\lesssim 2(1.2)$ and $B(K^{+}\to {\pi }^{+}\nu \bar{\nu })/B^{SM}(K^{+}\to {\pi }^{+}\nu \bar{\nu })\lesssim 1.4(1.1)$, 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 $B(K_L\to {\pi }^0\nu \bar{\nu })$ and the hierarchy between the masses $m_{\bar{U}}$, $m_{\bar{D}}$ of the right-handed up-squark and down-squark: sgn$[B(K_L\to {\pi }^0\nu \bar{\nu })-B^{SM}(K_L\to {\pi }^0\nu \bar{\nu })]=$sgn$(m_{\bar{U}}-m_{\bar{D}})$.

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