Footprints of leptoquarks: from $$ R_{K^{()}} $$ RK(∗) to $$ K \rightarrow \pi \nu \bar{\nu }$$ K→πνν

Abstract Rare $$K \rightarrow \pi \nu \bar{\nu }$$ K→πνν¯ decays, being dominated by short distance contributions within the standard model (SM), open a window for new physics (NP) searches at low energies. The $$ K \rightarrow \pi \nu \bar{\nu }$$ K→πνν¯ branching ratios are expected to be measured with $$ \sim 10\% $$ ∼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 $$ R_{D^{(*)}} $$ RD(∗) and/or $$ R_{K^{(*)}} $$ RK(∗) 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 $$ s \rightarrow d $$ s→d . Together with the anticipated good experimental sensitivities and accurate theoretical predictions for $$ K\rightarrow \pi \nu \bar{\nu }$$ K→πνν¯ , this motivates studies of correlated effects of NP in rare $$ K\rightarrow \pi \nu \bar{\nu }$$ K→πνν¯ and $$ B \rightarrow K^{(*)} \mu ^+ \mu ^- $$ B→K(∗)μ+μ- decays. Here we consider the loop induced effects in $$ K \rightarrow \pi \nu \bar{\nu }$$ K→πνν¯ in two leptoquark models designed to address lepton-flavor universality violation in the $$ R_{K^{(*)}} $$ RK(∗) anomalies.