Leptoquarks meet ${\epsilon }^{\prime }/\epsilon$ and rare Kaon processes

Abstract

We analyse for the first time the CP violating ratio ${\epsilon }^{\prime }/\epsilon$ 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 ${\epsilon }^{\prime }/\epsilon$ 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 $Q_{8,8^{\prime }}$ already at the LQ scale. We then investigate correlations of ${\epsilon }^{\prime }/\epsilon$ 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 ${\epsilon }_K$) and find that even imposing only a moderate enhancement of $({\epsilon }^{\prime }/\epsilon {)}_{\mathrm{NP}}=5\times 10^{-4}$ 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 ${\epsilon }^{\prime }/\epsilon$ 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 ${\epsilon }^{\prime }/\epsilon$, in case the ${\epsilon }^{\prime }/\epsilon$ 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.

53 pages, 9 figures; v2: minor extensions, matches published version; v3: corrected eq.(B.10) and figure 2 (right plot U_1,L), conclusions not affected; v4: corrected eqs.(C.8, C.12, C.16), numerics and conclusions not affected

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