Discriminating Tauphilic Leptoquark Explanations of the $B$ Anomalies via $K\to \pi \nu \bar{\nu }$ and $B\to K\nu \bar{\nu }$

Leptoquark models are prime candidates for new physics (NP) explanations of the long-standing anomalies in semi-leptonic $B$ decays; $b\to c\tau \bar{\nu }$ (encoded in $R(D^{(\ast )})$) and $b\to s\mathrm{\ell }\bar{\mathrm{\ell }}(\mathrm{\ell }=e,\mu )$ transitions. Furthermore, Belle II and NA62 reported weaker-than-expected limits on $B^{+}\to K^{+}\nu \bar{\nu }$ and $K^{+}\to {\pi }^{+}\nu \bar{\nu }$, respectively. While the $R(D^{(\ast )})$ and $b\to s\mathrm{\ell }\bar{\mathrm{\ell }}$ measurements can be explained with NP contributions at the $O(10\mathrm{\%})$ level, the neutrino channels suggest that the NP effect could be comparable in size to the Standard Model one. In this context, we consider the two types of leptoquark models with minimal sets of the couplings that can best describe the semi-leptonic $B$ anomalies and lead at the same time to effects in the neutrino modes, the singlet-triplet scalar leptoquark model ($S_1+S_3$) and the singlet vector leptoquark model ($U_1$). More specifically, the neutrino channels pose non-trivial constraints on the parameter space, and we find that large effects (i.e., accounting for the current central value) in $B\to K^{(\ast )}\nu \bar{\nu }$ are only possible in the $S_1+S_3$ setup, while both models can account for the central value of $K^{+}\to {\pi }^{+}\nu \bar{\nu }$.