Probing dark-matter scenarios of Belle II excess in $B^{+}\to K^{+}{M}_X$ with $K^{+}\to {\pi }^{+}{M}_X$ decays

The hypothesis of top-philic dark matter as the origin of the Belle II excess events in $B\to K{M}_X$ decays has direct impact on $K\to \pi M_X$ decays: Both of these decays proceed via loop diagrams~with a top quark emitting some mediator field $R$ and the subsequent decay of $R$ into a pair of dark-matter fermions $\bar{\chi }\chi$. The Belle II results on $B\to K{M}_X$ allowed us to extract the parameters of the mediator $R$ with reasonable accuracy, whereas the sensitivity to the fermion mass $m_{\chi }$ proved to be rather low. In contrast to $B\to K{M}_X$ decays, the $K\to \pi M_X$ decays exhibit a strong sensitivity to $m_{\chi }$; therefore, if - upon reducing theoretical and experimental uncertainties - the tension between standard-model predictions for the $K\to \pi M_X$ decays and the most recent NA62 measurements becomes pronounced the latter decays may either enable the extraction of the value of $m_{\chi }$ or provide a lower limit on~$m_{\chi }$.