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Rare kaon decays with missing energy, $K\to\pi$+$E_{\rm miss}$, have receivedconsiderable attention because their rates can be calculated quite preciselywithin the standard model (SM), where the missing energy is carried away by anundetected neutrino-antineutrino pair. Beyond the SM, clean theoreticalpredictions can also be made regarding these processes. One such prediction isthe so-called Grossman-Nir (GN) bound, which states that the branchingfractions of the $K_L$ and $K^+$ modes must satisfy the relation$\mathcal{B}(K_L\to\pi^0$+$E_{\rmmiss})\lesssim4.3\,\mathcal{B}(K^+\to\pi^+$+$E_{\rm miss})$ and applies withinand beyond the SM, as long as the hadronic transitions change isospin by$\Delta I=1/2$. In this paper we extend the study of these modes to includenew-physics scenarios where the missing energy is due to unobservedlepton-number-violating neutrino pairs, invisible light new scalars, or pairsof such scalars. The new interactions are assumed to arise above theelectroweak scale and described by an effective field theory. We explore thepossibility of violating the GN bound through $\Delta I=3/2$ contributions tothe $K\to\pi$ transitions within these scenarios and find that large violationsare only possible in the case where the missing energy is due to an invisiblelight new scalar.

Evading the Grossman-Nir bound with $\Delta I=3/2$ new physics