Beyond the Standard Model prospects for kaon physics at future experiments

Rare kaon decays offer a powerful tool for investigating new physics in $s\to d$ transitions. Currently, many of the interesting decay modes are either measured with rather large uncertainties compared to their theoretical predictions or have not yet been observed. The future HIKE programme at CERN will provide unprecedented sensitivity to rare kaon decays, allowing for strong constraints on new physics scenarios with lepton flavour universality violation. We present the overall picture that emerges from a study of the different decay modes with a global analysis considering projections based on the HIKE programme, both with and without KOTO-II future measurements. We also highlight the most relevant decays and identify that in addition to the "golden channel", $K^+\to\pi^+\nu\bar\nu$, the rare $K_L\to\pi^0 \ell\bar\ell$ decay, especially in the electron sector offers strong constraints on short-distance physics.