Footprints of leptoquarks: from $ R_{K^{(*)}} $ to $ K \to \pi \nu \bar\nu $

Rare $K \to \pi \nu \bar \nu$ decays, being dominated by short distance contributions within the Standard Model (SM), open a window for New Physics (NP) searches at low energies. The $ K \to \pi \nu \bar\nu $ branching ratios are expected to be measured with $ \sim 10\% $ accuracies by NA62/CERN and KOTO/JPARC. The theoretical uncertainties of branching ratios within the SM are well under control. In the $ B $ sector, it is tentative to explain the $ B $-meson anomalies $ R_{D^{(\ast)}} $ and/or $ R_{K^{(\ast)}} $ by effects of physics beyond the SM. Although NP seems to be present in the third fermion generation it might also manifest in the flavor changing neutral current transition $ s \to d $. Together with the anticipated good experimental sensitivities and accurate theoretical predictions for $ K\to \pi \nu \bar \nu $, this motivates studies of correlated effects of NP in rare $ K\to \pi \nu \bar \nu $ and $ B \to K^{(*)} \mu^+ \mu^- $ decays. Here we consider the loop induced effects in $ K \to \pi \nu \bar\nu $ in two leptoquark models designed to address lepton-flavor universality violation in the $ R_{K^{(\ast)}} $ anomalies.