Leptoquarks meet ε ^′/ε and rare Kaon processes

We analyse for the first time the CP violating ratio ε^′/ε in K → ππ decays in leptoquark (LQ) models. Assuming a mass gap to the electroweak (EW) scale, the main mechanism for LQs to contribute to ε^′/ε 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 ε^′/ε with rare Kaon processes (K_L→ π⁰, K⁺→ π⁺, K_L→ π⁰, K_S→ , ΔM_Kand ε_K) and find that even imposing only a moderate enhancement of (ε^′/ε)_NP= 5 × 10⁻⁴to 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 ε^′/ε anomaly and put strong-to-serious constraints on parameter spaces of the remaining models. Future results on K⁺→ π⁺from the NA62 collaboration, K_L→ π⁰from the KOTO experiment and K_S→ from LHCb will even stronger exhibit the difficulty of LQ models in explaining the measured ε^′/ε, in case the ε^′/ε anomaly will be confirmed by improved lattice QCD calculations. Hopefully also improved measurements of K_L→ π⁰decays will one day help in this context.