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Standard Model and New Physics contributions to $K_L$ and $K_S$ into four leptons

We study the $K_L$ and $K_S$ decays into four leptons ($e\bar{e}e\bar{e}, \, μ\barμμ\barμ,\, e\bar{e}μ\barμ$) where we use a form factor motivated by vector meson dominance, and show the dependence of the branching ratios and spectra from the slopes. A precise determination of short distance contribution to $K_L\to μμ$ is affected by our ignorance on the sign of the amplitude $\mathcal{A}(K_L\to γγ)$ but we show a possibility to measure the sign of this amplitude by studying $K_L$ and $K_S$ decays in four leptons. We also investigate the effect of New Physics contributions for these decays.