# $K\to \pi \nu\overline{\nu}$ in the MSSM in Light of the $\epsilon^{\prime}_K/\epsilon_K$ Anomaly

## TL;DR

This paper investigates how the MSSM can explain the $ ext{ε'}_K/ ext{ε}_K$ anomaly through gluino-squark box diagrams and explores the resulting correlations with rare kaon decay branching ratios, emphasizing implications for future experiments.

## Contribution

It provides a detailed analysis of MSSM contributions to kaon decay processes and their correlation with the $ ext{ε'}_K/ ext{ε}_K$ anomaly, highlighting distinctive patterns from $Z$-penguin scenarios.

## Key findings

- MSSM can enhance $K o
uar{
u}$ decay rates up to 2 times the SM prediction.
- Correlations depend on squark mass hierarchy and CP-violating phases.
- Sign of $ ext{ε'}_K$ contribution correlates with the mass difference between right-handed up- and down-squarks.

## Abstract

The Standard-Model (SM) prediction for the CP-violating quantity $\epsilon_K^{\prime}/\epsilon_K$ 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 $\epsilon^{\prime}_K/\epsilon_K$ 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 $\epsilon_K^{\prime}/\epsilon_K$ 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 $\epsilon_K^{\prime}$ 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}}) $.