Leptoquarks meet $\varepsilon'/\varepsilon$ and rare Kaon processes

We analyse for the first time the CP violating ratio $\varepsilon'/\varepsilon$ in $K\to \pi\pi$ decays in leptoquark (LQ) models. Assuming a mass gap to the electroweak (EW) scale, the main mechanism for LQs to contribute to $\varepsilon'/\varepsilon$ is EW gauge-mixing of semi-leptonic into non-leptonic operators, which we treat in the Standard Model effective theory (SMEFT). We perform also the one-loop decoupling for scalar LQs, finding that in all models with both left-handed and right-handed LQ couplings box-diagrams generate numerically strongly enhanced EW-penguin operators $Q_{8,8'}$ already at the LQ scale. We then investigate correlations of $\varepsilon'/\varepsilon$ with rare Kaon processes ($K_L\to\pi^0\nu\bar\nu$, $K^+\to\pi^+\nu\bar\nu$, $K_L\to\pi^0\ell\bar\ell$, $K_S\to\mu\bar\mu$, $\Delta M_K$ and $\varepsilon_K$) and find that even imposing only a moderate enhancement of $(\varepsilon'/\varepsilon)_{\rm NP} = 5 \times 10^{-4}$ to explain the current anomaly hinted by the Dual QCD approach and RBC-UKQCD lattice QCD calculations leads to conflicts with experimental upper bounds on rare Kaon processes. They exclude all LQ models with only a single coupling as an explanation of the $\varepsilon'/\varepsilon$ anomaly and put strong-to-serious constraints on parameter spaces of the remaining models. Future results on $K^+\to\pi^+\nu\bar\nu$ from the NA62 collaboration, $K_L\to\pi^0\nu\bar\nu$ from the KOTO experiment and $K_S\to\mu\bar\mu$ from LHCb will even stronger exhibit the difficulty of LQ models in explaining the measured $\varepsilon'/\varepsilon$, in case the $\varepsilon'/\varepsilon$ anomaly will be confirmed by improved lattice QCD calculations. Hopefully also improved measurements of $K_L\to\pi^0\ell\bar\ell$ decays will one day help in this context.