Left-left squark mixing in $K^+\rightarrow\pi^+\nu\bar{\nu}$ and minimal SUSY

We analyze the role of the left-left squark mixing in the rare $K^+ \rightarrow \pi^+\nu\bar{\nu}$ decay within the minimal supersymmetry with a large $\tan\beta$. A Higgs boson mass of 125 GeV has been taken into account leading to correlation between stop masses and trilinear soft supersymmetry breaking coupling $A_{\tilde{t}}$. We find that measurable effects, similar to that of the well known $LR$ squark mixing terms, are possible for large $A_{\tilde{t}}$ combined with the off-diagonal $LL$-insertions. Precise measurements of the decay rate are expected from the ongoing NA62 experiment at CERN. We emphasize that the effect we present can put certain limits on the left-left flavor changing structure of the squark mass matrix.